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March 2014 Volume 105 No. 1www.saiee.org.za

Africa Research JournalResearch Journal of the South African Institute of Electrical Engineers

Incorporating the SAIEE Transactions

ISSN 1991-1696

Vol.105(1) March 2014SOUTH AFRICAN INSTITUTE OF ELECTRICAL ENGINEERS2

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ISSN 1991-1696

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SPECIALIST EDITORSCommunications and Signal Processing:Prof. L.P. Linde, Dept. of Electrical, Electronic & Computer Engineering, University of Pretoria, SA Prof. S. Maharaj, Dept. of Electrical, Electronic & Computer Engineering, University of Pretoria, SADr O. Holland, Centre for Telecommunications Research, London, UKProf. F. Takawira, School of Electrical and Information Engineering, University of the Witwatersrand, Johannesburg, SAProf. A.J. Han Vinck, University of Duisburg-Essen, GermanyDr E. Golovins, DCLF Laboratory, National Metrology Institute of South Africa (NMISA), Pretoria, SAComputer, Information Systems and Software Engineering:Dr M. Weststrate, Newco Holdings, Pretoria, SAProf. A. van der Merwe, Department of Infomatics, University of Pretoria, SA Prof. E. Barnard, Faculty of Economic Sciences and Information Technology, North-West University, SAProf. B. Dwolatzky, Joburg Centre for Software Engineering, University of the Witwatersrand, Johannesburg, SAControl and Automation:Dr B. Yuksel, Advanced Technology R&D Centre, Mitsubishi Electric Corporation, Japan Prof. T. van Niekerk, Dept. of Mechatronics,Nelson Mandela Metropolitan University, Port Elizabeth, SAElectromagnetics and Antennas:Prof. J.H. Cloete, Dept. of Electrical and Electronic Engineering, Stellenbosch University, SA Prof. T.J.O. Afullo, School of Electrical, Electronic and Computer Engineering, University of KwaZulu-Natal, Durban, SA Prof. R. Geschke, Dept. of Electrical and Electronic Engineering, University of Cape Town, SADr B. Jokanović, Institute of Physics, Belgrade, SerbiaElectron Devices and Circuits:Dr M. Božanić, Azoteq (Pty) Ltd, Pretoria, SAProf. M. du Plessis, Dept. of Electrical, Electronic & Computer Engineering, University of Pretoria, SADr D. Foty, Gilgamesh Associates, LLC, Vermont, USAEnergy and Power Systems:Prof. M. Delimar, Faculty of Electrical Engineering and Computing, University of Zagreb, Croatia Engineering and Technology Management:Prof. J-H. Pretorius, Faculty of Engineering and the Built Environment, University of Johannesburg, SAProf. L. Pretorius, Dept. of Engineering and Technology Management, University of Pretoria, SA

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VOL 105 No 1March 2014

SAIEE Africa Research Journal


March 2014 Volume 105 No. 1www.saiee.org.za

Africa Research JournalResearch Journal of the South African Institute of Electrical Engineers

Incorporating the SAIEE Transactions

ISSN 1991-1696

Low-Complexity Detection and Transmit Antenna Selection forSpatial ModulationN. Pillay and H. Xu ....................................................................... 4

Reed-Solomon Code Symbol AvoidanceT. Shongwe and A.J. Han Vinck .................................................... 13

Determination of Specific Rain Attenuation using DifferentTotal Cross Section Models for Southern AfricaS.J. Malinga, P.A. Owolawi and T.J.O. Afullo ............................. 20

The Influence of Disdrometer Channels on Specific AttenuationDue to Rain Over Microwave Links in Southern AfricaO. Adetan and T. Afullo ............................................................... 31

SAIEE AFRICA RESEARCH JOURNAL EDITORIAL STAFF ...................... IFC


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LOW-COMPLEXITY DETECTION AND TRANSMIT ANTENNA SELECTION FOR SPATIAL MODULATION N. Pillay* and H. Xu** * School of Engineering, Dept. of Electrical, Electronic & Computer Engineering, King George V Avenue, Durban 4041, University of KwaZulu-Natal, South Africa, E-mail: [email protected] ** School of Engineering, Dept. of Electrical, Electronic & Computer Engineering, King George V Avenue, Durban 4041, University of KwaZulu-Natal, South Africa, E-mail: [email protected] Abstract: In this paper, we propose a novel low-complexity detection technique for conventional Spatial Modulation (SM). The proposed scheme is compared to SM with optimal detection (SM-OD), SM with signal vector based detection (SM-SVD) and another reduced complexity detection technique, presented in the literature. The numerical results show that the proposed scheme can match the error performance of SM-OD very closely, even at low bit-error rate (BER). The computational complexity at the receiver, is shown to be independent of the symbol constellation size, and hence, offers a much lower complexity compared to existing schemes. To further improve the error performance, we then propose two closed-loop transmit antenna selection (TAS) schemes for conventional SM. We assume the receiver has knowledge of the channel and a perfect low-bandwidth feedback path to the transmitter exists. From evaluation of the BER versus normalized signal-to-noise ratio (SNR) and the complexity analysis, the proposed schemes exhibit a significant improvement over SM-OD and other improved SM schemes in terms of error performance and/or complexity. Keywords: Spatial modulation, transmit antenna selection, multiple-input multiple-output, adaptive spatial modulation, reduced complexity detection, signal vector based detection, optimal detection.

I. INTRODUCTION The use of multiple transmit and receive antennas in mobile communication systems will become more prominent in 4G systems and beyond. These multiple-input multiple-output (MIMO) systems allow unprecedented improvements (in terms of throughput and error performance) compared to their single-input single-output (SISO) counterparts due to diversity, multiplexing or antenna gain (smart antennas) or a combination of these. The primary inhibiting factors for the implementation of these MIMO techniques have been the system complexity and cost. SM is an attractive MIMO technique that has been presented fairly recently in the literature, and holds the promise of reduced system complexity and cost [1]. The key idea behind SM is to select only one antenna from the available transmit antennas for transmission of the symbol in the transmission interval. Thus, only one transmit antenna is active at any particular transmission instant. The SM symbol has two parts. The first part of the symbol represents the chosen antenna index, while the second part of the symbol is based on the amplitude and/or phase modulation (APM), e.g. quadrature amplitude modulation (QAM) and phase shift keying (PSK). It is important to note that the first part of the symbol is not transmitted explicitly, but rather by the particular antenna index. This implies that the active antenna index must be detected at the receiver to estimate the complete symbol.

The advantages of SM compared to other MIMO schemes, e.g. vertical Bell laboratories layered space-time architecture (V-BLAST), and Alamouti schemes [1], [2] are, a) avoids inter-channel interference (ICI) at the receiver [1], b) reduced receiver complexity [1], [3], c) inter-antenna synchronization (IAS) at the transmitter is not required, due to the single active antenna per signaling interval [1], d) high spectral efficiency with no ICI or IAS [4], e) SM requires a single radio frequency (RF) chain [1], and f) SM has been shown to outperform many existing MIMO schemes including V-BLAST [1], [2]. All of these also contribute to g) lower cost. These important advantages make SM an important technique for future high performance wireless communications. The detection scheme employed for SM has much importance, and has been investigated intensively. In [1], Mesleh et al. proposed the use of iterative maximal ratio combining (i-MRC). The i-MRC technique iteratively computes the product of the channel gain and received signal vectors to estimate the active transmit antenna index. Then, by using the active transmit antenna index estimate, the transmit symbol is estimated. However, this method was shown to be sub-optimal [2]. In [2], Jeganathan et al. derived the optimal detection rule by means of the maximum-likelihood (ML) principle. The proposed detector exhibited a gain of approximately 4 dB (constrained channels [2]) at a BER of 10 [2], compared to the sub-optimal technique [1], and was further supported by analytical results. However, the complexity analysis in [2] showed that both the optimal


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and sub-optimal detection techniques exhibit a similar level of complexity, which is still relatively high. Meanwhile, [3] showed that for high-order modulation schemes (𝑀𝑀𝑀𝑀 ≥ 16), the optimal detector [2] complexity can increase by approximately 125% in comparison to the sub-optimal scheme [1], [2]. Based on this observation, [3] presented a sub-optimal multiple stage detection technique. In [3], in the first stage the original SM detector for conventional channels [2] is used to determine the most probable estimates of the active transmit antenna index. A final estimate of the active transmit antenna index and transmit symbol is then computed by means of the ML principle [2]. This method showed a decrease in the receiver computational complexity, compared to the schemes in [1] and [2]. In addition, the error performance closely matched that of the optimal detector. To further reduce complexity, Xu [4] presented a simplified ML based detection scheme for multilevel-quadrature amplitude modulation (M-QAM) SM; this scheme was based on searching partitioned signal sets for the active transmit antenna index and transmit symbol, thus reducing the computational complexity. Xu [4] then applied the method to the multiple stage detection proposed in [3], which again exhibited a further reduction in complexity, compared to existing work. In [5], [6], signal vector based detection (SVD) was proposed. SVD determines the active transmit antenna index at the receiver by searching for the smallest angle between the channel gain vectors and the receive vector. The transmit symbol is then estimated using the ML principle [2], [5], [7]. Wang et al. [5] then motivated that the SVD detector BER matched that of the optimal detector very closely at a much reduced complexity. However, in [6], it was shown that for moderate to high SNRs the error performance does not match that of SM-OD. To improve on the error performance of SVD, Zheng presented SVD-List detection [7]. However, the complexity was higher [7] compared to SVD [5], due to the selection and search of a list of candidate antennas based on the 𝐿𝐿𝐿𝐿 smallest angles between the channel gain vectors and receive vector. In [8], a reduced complexity scheme (we will refer to as SM-RC) for achieving optimal error performance for SM was presented. In [8], the transmit antenna index and symbol are detected separately; however, by taking the correlation into account (since the antenna index and transmit symbol fade together) the scheme retains the error performance of SM-OD. However, the scheme works with demodulating bits rather than symbols, and the computational complexity is a function of the symbol constellation size (as with all the existing detection schemes for SM), meaning that for large constellation dimension the complexity will still be relatively high.

Sphere decoding for SM has also been presented in the literature and exhibits good BER performance [9], [10]; however, it has been shown that the lower bound of the computational complexity, at the receiver, is once again a function of the symbol constellation size. Motivated by this, in this paper, we first propose a low-complexity detection technique for conventional SM (SM-LCD), which has a computational complexity independent of the symbol constellation size. The motivation for our proposed scheme is based on a single antenna transmission system and is as follows: For a single antenna transmission system, it is relatively straightforward to compute the equalized symbol, and consequently, the estimate of the transmit symbol [11]. However, for SM, although only one transmit antenna is active per transmission instant (the active antenna is selected by mapping the first log 𝑁𝑁𝑁𝑁 symbol bits to the antenna index), at the receiver, knowledge of the active antenna is unknown. Thus, in the proposed detection scheme, the estimate of the transmit symbol for each transmit antenna in the array is determined using the corresponding equalized symbol estimate. Finally, the estimated transmit symbol for each of the transmit antennas are compared to the received signal vector and the closest estimate (based on the ML principle) is chosen as the final estimate of the antenna index/transmit symbol. The optimal detector performs the search over 𝑀𝑀𝑀𝑀 symbol constellation points for each antenna [2]. On the contrary, in SM-LCD there is only a single constellation point for each transmit antenna, hence reducing the computational complexity at the receiver, especially for high-order symbol constellations. In Section III, the proposed SM-LCD scheme is explained in more detail. As noted earlier, [2] demonstrated how optimal error performance can be achieved for SM using the ML principle. However, the error performance of SM is still dependent on the selected transmit antenna and the resulting propagation paths between the transmitter and receiver. Several works have considered antenna selection for MIMO to improve performance. In [12], Molisch et al. motivated that optimum selection of the antenna elements requires an exhaustive search of all possible combinations for either the best SNR (spatial diversity) or capacity (spatial multiplexing). This requires computation of determinants for each channel realization, thus making this impractical [12]. On this note, Molisch et al. [12] motivated for antenna selection at only one end of the MIMO system. Several works have investigated the application of single-ended transmit or receive antenna selection for MIMO systems. In [13], [14], the authors investigate TAS to reduce energy consumption in MIMO systems. Single TAS based on selecting the antenna with the highest equivalent receive SNR was proposed in [13]. Multiple TAS was investigated in [14], where the antennas corresponding to the highest channel gains were selected


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to form the active subset of the antenna array. In this paper, we refer to this method as maximal gain TAS (MG-TAS). A correlation-based method (CBM) was proposed in [15] for antenna selection at the receiver for MIMO systems, and showed improved performance. However, none of these schemes have been applied to SM. Furthermore, recently several works have shown that the BER performance of SM can be improved through closed-loop design. In [16], an adaptive SM scheme was proposed to achieve an improved system performance for fixed data rates. The key idea behind the scheme presented in [16] was to select the optimum modulation candidate (the candidate that maximizes the received minimum distance) for transmission. The candidate was chosen by searching among a set of all possible candidates. The transmitter then employs the chosen modulation orders in the next channel use. On the same note, in [17], the authors presented two more adaptive closed-loop SM schemes, which additionally use transmit-mode switching to further improve on the performance. The first scheme, optimal-hybrid SM (OH-SM) proposed in [17], uses both adaptive modulation and transmit-mode switching. The second scheme, concatenated SM (C-SM) was then proposed in [17], and used a multistage adaptation technique to balance the trade-off between the computational complexity and error performance. However, the existing closed-loop schemes have serious drawbacks. The first drawback of the schemes proposed in [16] and [17] was the very high complexity, e.g. for 4 bits/s/Hz the complexity of OH-SM was approximately 40,000 real multiplications, while the complexity of C-SM was approximately 20,000 real multiplications [17]. Secondly, in SM the antenna indices carry information; however, in the proposed closed-loop schemes [16], [17], the antenna indices also carry information about the transmission mode and there is no method presented to take care of the errors when the information at the receiver is inconsistent with that at the transmitter and the transmission mode is unknown. In addition, in SM, based on SVD [5], the BER of SM not only depends on the amplitude of the channel gain vectors but also on the angles between the 𝑗𝑗𝑗𝑗 channel gain vectors. Thus, based on these factors as well as the drawbacks of the existing closed-loop designs for SM [16], [17], we propose a more practical closed-loop design for SM, which uses antenna selection at the transmitter. Specifically, we propose two TAS schemes for SM which exhibits much lower complexity and/or an improved error performance compared to the existing schemes. The first scheme selects transmit antennas based on the largest channel gain amplitudes, then the largest angles between channel gain vectors. The second scheme selects transmit antennas based on the channel gain vectors which have the largest amplitudes followed by the smallest

correlations. The new schemes are also compared to MG-TAS, which we have applied to SM (MG-TAS-SM). The structure of the remainder of this paper is as follows: In Section II, the system model for conventional SM is presented and supported with brief background theory. In Section III, we present the proposed SM-LCD scheme and the complexity analysis, where we show SM-LCD has a computational complexity, at the receiver (in terms of the number of complex multiplications and additions), independent of the symbol constellation size. The TAS algorithms for SM are then presented in Section IV and the complexity analysis is included. Section V presents the numerical analysis for the proposed schemes and draws comparisons with existing schemes. Conclusions are finally drawn in Section VI.

II. SYSTEM MODEL & BACKGROUND THEORY Consider a MIMO communication system with transmit antenna array of length 𝑁𝑁𝑁𝑁 and receive antenna array of length 𝑁𝑁𝑁𝑁 . The received signal vector 𝒚𝒚𝒚𝒚 can be denoted as [1], [2],

𝒚𝒚𝒚𝒚 = 𝜌𝜌𝜌𝜌𝑯𝑯𝑯𝑯𝑯𝑯𝑯𝑯 + 𝜼𝜼𝜼𝜼 (1) where 𝑿𝑿𝑿𝑿 is an 𝑁𝑁𝑁𝑁 ×1 vector, with one non-zero entry corresponding to the transmit symbol (we consider M-QAM with Gray mapping). 𝑯𝑯𝑯𝑯 is an 𝑁𝑁𝑁𝑁 ×𝑁𝑁𝑁𝑁 complex channel gain matrix with independent and identically distributed (i.i.d) entries with distribution 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶(0,1). 𝜌𝜌𝜌𝜌 represents the average SNR per receive antenna and 𝜼𝜼𝜼𝜼 represents an 𝑁𝑁𝑁𝑁 ×1 complex additive white Gaussian noise (AWGN) vector with i.i.d entries with distribution 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶(0,1). In [1], [2], the sub-optimal detection rule for conventional channels [2] was defined as,

𝚥𝚥𝚥𝚥 = argmax|𝒉𝒉𝒉𝒉 𝒚𝒚𝒚𝒚|

𝒉𝒉𝒉𝒉 (2)

𝑞𝑞𝑞𝑞 = 𝒟𝒟𝒟𝒟𝒉𝒉𝒉𝒉 𝒚𝒚𝒚𝒚

𝒉𝒉𝒉𝒉 (3)

where 𝚥𝚥𝚥𝚥, 𝑞𝑞𝑞𝑞 represent the estimates of the active transmit antenna index 𝑗𝑗𝑗𝑗 and symbol index 𝑞𝑞𝑞𝑞, respectively. Note that 𝑗𝑗𝑗𝑗 ∈ 1:𝑁𝑁𝑁𝑁 and 𝑞𝑞𝑞𝑞 ∈ 1:𝑀𝑀𝑀𝑀 . 𝒉𝒉𝒉𝒉 represents a column vector of 𝑯𝑯𝑯𝑯 and is defined as 𝒉𝒉𝒉𝒉 = [ℎ ℎ … ℎ ] . † represents the transpose conjugate operator, argmax(∙) represents the argument of

the maximum with respect to 𝑗𝑗𝑗𝑗, ∙ represents the Frobenius norm, ∙ represents the magnitude operation, 𝒟𝒟𝒟𝒟(∙) represents the constellation demodulator or slicing function, which extracts the in-phase and quadrature-phase components from the symbol argument, and ∙ represents the matrix or vector transposition.


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Jeganathan et al. [2] presented the optimal joint detection rule based on the ML principle. The detection rule was defined as [2],

[𝚥𝚥𝚥𝚥, 𝑞𝑞𝑞𝑞] = argmin ,

𝜌𝜌𝜌𝜌 𝒍𝒍𝒍𝒍 − 2𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 𝒚𝒚𝒚𝒚 𝒍𝒍𝒍𝒍 (4)

where 𝒍𝒍𝒍𝒍 = 𝒉𝒉𝒉𝒉 𝑥𝑥𝑥𝑥 and 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 ∙ represents the real part of the complex argument. Note that (4) is a joint detection rule and 𝑀𝑀𝑀𝑀 symbol constellation points are searched for each transmit antenna 𝑗𝑗𝑗𝑗, thus the method has a high computational complexity at the receiver, and yields optimal error performance [2], [3]. To reduce the high complexity due to joint detection of antenna and symbol indices, Xu et al. [8] proposed a detection technique where the active transmit antenna index and symbol are detected separately using decorrelating vectors [8]. To detect the active antenna index, the maximum metric [8] for the known constellation is searched over 𝑁𝑁𝑁𝑁 candidates. Finally, the log 𝑀𝑀𝑀𝑀 symbol bits are demodulated based on a decision rule involving the elements of the decorrelating vectors [8]. This method was shown to reduce complexity at the receiver and retain the error performance of SM-OD [8]. In Section III, we propose a new low-complexity detection technique for SM. We refer to the scheme as SM-LCD. In addition, the complexity analysis of the proposed SM-LCD scheme is included. The complexity analysis shows that the computational complexity, at the receiver, for the proposed detector is independent of the symbol constellation size. This significantly reduces the complexity, compared to the existing schemes (from the literature survey we found that the computation complexity for all existing detection schemes for SM is a function of the symbol constellation size). Motivated by the work performed for antenna selection in MIMO systems [12-15], the potential improvement in error performance by employing closed-loop design for SM [16-17] and the drawbacks of the existing closed-loop designs for SM, in Section IV we propose TAS for SM. In the open literature, MG-TAS has been proposed for single or multiple TAS for MIMO to decrease the average energy consumption [13], [14]. In this paper, we will denote the configuration of MG-TAS as 𝑁𝑁𝑁𝑁 ,𝑁𝑁𝑁𝑁 ,𝑁𝑁𝑁𝑁 , where 𝑁𝑁𝑁𝑁 is the total number of

antennas in the transmit array and 𝑁𝑁𝑁𝑁 > 𝑁𝑁𝑁𝑁 . The MG-TAS scheme [14] proceeds as follows: Step 1: Compute the channel power gains,

𝑃𝑃𝑃𝑃 = 𝒉𝒉𝒉𝒉 1 ≤ 𝑥𝑥𝑥𝑥 ≤ 𝑁𝑁𝑁𝑁 (5) Step 2: Arrange the channel power gains in descending order,

𝑃𝑃𝑃𝑃 ≥ 𝑃𝑃𝑃𝑃 ≥ ⋯ ≥ 𝑃𝑃𝑃𝑃 (6)

Step 3: Select the antennas for transmission corresponding to the first 𝑁𝑁𝑁𝑁 elements. This algorithm has not been applied to SM or to improve error performance for MIMO systems. In Section V (Numerical results), we apply the algorithm to SM (MG-TAS-SM) and utilize it for comparison with the proposed schemes. The detection of SM symbols requires detection of the active transmit antenna index and the APM symbol at the receiver. Optimal detection of SM was presented in [2]; however, the complexity was relatively high [3]. In [5], Wang et al. proposed SVD, a low-complexity technique for detecting the active transmit antenna index at the receiver, for SM. Once the active transmit antenna index was estimated, the ML principle was applied to estimate the APM symbol. Estimation of the active transmit antenna index at the receiver was based on computing the angle between the 𝑗𝑗𝑗𝑗 channel gain vector and the received signal vector [5]. The antenna 𝑗𝑗𝑗𝑗 corresponding to the smallest angle was then selected as the active antenna. This choice was based on the motivation that the received signal vector lies within a sphere of the channel gain vector due to the summation of the scalar product of channel gain vector and QAM symbol and the noise/interference [5]. Motivated by this reasoning, for the first of the proposed TAS algorithms we propose to use in part, a similar technique [5], not for detection but rather to determine the angles between all pairs of channel gain vectors at the transmitter (we assume full channel knowledge at the receiver and a perfect low-bandwidth feedback path to the transmitter). By eliminating the transmit antennas corresponding to the smallest channel gain amplitudes followed by the smallest angles between the channel gain vectors the error performance can be improved by using the reduced transmit antenna array for SM. The second algorithm is a CBM-based method for SM for multiple TAS and involves selecting antennas based on amplitude of the channel gain vectors and correlation between pairs of channel gain vectors.

III. LOW-COMPLEXITY DETECTION FOR SPATIAL MODULATION

A. DETECTION OF SM USING A LOW-COMPLEXITY APPROACH Assume a SISO system, and suppose the received signal is defined by 𝑦𝑦𝑦𝑦 = ℎ𝑥𝑥𝑥𝑥 + 𝑛𝑛𝑛𝑛, where ℎ is the channel gain, 𝑥𝑥𝑥𝑥 is the transmit symbol and 𝑛𝑛𝑛𝑛 is the AWGN component, then the equalized transmit symbol is given by 𝑥𝑥𝑥𝑥 =

∗

where ∙ ∗ represents the conjugate operator. The final estimate of the transmit symbol is given by 𝑥𝑥𝑥𝑥 = 𝒟𝒟𝒟𝒟(𝑥𝑥𝑥𝑥 ). In the SM system, the selected transmit antenna used to transmit the symbol is unknown. Thus, the estimate of the transmit symbol for each possible transmit antenna is needed. A comparison of each of the estimated transmit symbols with the receive vector is then performed to determine the most probable final estimate of the antenna index/transmit symbol.


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Following this logic, the algorithm for the proposed low-complexity detection of SM proceeds as follows: Step 1: Compute the equalized symbol for each antenna,

𝑧𝑧𝑧𝑧 =𝒉𝒉𝒉𝒉 𝒚𝒚𝒚𝒚

𝒉𝒉𝒉𝒉 𝑗𝑗𝑗𝑗 ∈ 1:𝑁𝑁𝑁𝑁 (7)

Step 2: Demodulate the estimated symbol for each transmit antenna using,

𝑞𝑞𝑞𝑞 = 𝒟𝒟𝒟𝒟 𝑧𝑧𝑧𝑧 (8) where the constellation demodulator function extracts the in-phase and quadrature-phase components from the equalized symbol. Step 3: Perform detection using the ML principle [2] to estimate the antenna index and transmit symbol,

[𝚥𝚥𝚥𝚥, 𝑞𝑞𝑞𝑞 ] = argmin 𝜌𝜌𝜌𝜌

𝒈𝒈𝒈𝒈 − 2𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 𝒚𝒚𝒚𝒚 𝒈𝒈𝒈𝒈 (9)

where 𝒈𝒈𝒈𝒈 = 𝒉𝒉𝒉𝒉 𝑞𝑞𝑞𝑞 . It is evident from Step 3, that the search is performed only over a single symbol constellation point for each transmit antenna, compared to the 𝑀𝑀𝑀𝑀 constellation points searched for each transmit antenna in SM-OD [2], thus reducing complexity. In the next sub-section, the complexity analysis for SM-LCD is presented.

B. COMPLEXITY ANALYSIS FOR SM-LCD In this sub-section, we represent the computational complexity using the same approach as [1] and [4], i.e. in terms of complex multiplications and additions. Table 1 presents the complexity at the receiver for each of the existing schemes that are used for comparison with SM-LCD. Table 1: Computational complexity for SM detection schemes Scheme Computational complexity

SM-OD [4] 𝛿𝛿𝛿𝛿 = 3𝑁𝑁𝑁𝑁 +𝑀𝑀𝑀𝑀 − 1 𝑁𝑁𝑁𝑁 +𝑀𝑀𝑀𝑀 (10)

SM-SVD 𝛿𝛿𝛿𝛿 = 3𝑁𝑁𝑁𝑁 𝑁𝑁𝑁𝑁 + 𝑁𝑁𝑁𝑁 + 2𝑀𝑀𝑀𝑀 (11)

SM-RC [8] 𝛿𝛿𝛿𝛿 ∈ 𝑂𝑂𝑂𝑂 𝑀𝑀𝑀𝑀𝑁𝑁𝑁𝑁 + 4 (12)

Next, we formulate the complexity of the proposed scheme SM-LCD. Step 1 involves the computation of the equalized symbol (7). Assuming a single transmit antenna. The numerator of (7) 𝒉𝒉𝒉𝒉 𝒚𝒚𝒚𝒚 is the multiplication of a 1×𝑁𝑁𝑁𝑁 row vector and the 𝑁𝑁𝑁𝑁 ×1 received signal vector. This equates to 𝑁𝑁𝑁𝑁 complex multiplications and

𝑁𝑁𝑁𝑁 − 1 complex additions. The denominator 𝒉𝒉𝒉𝒉 requires 𝑁𝑁𝑁𝑁 complex multiplications and zero complex additions for a single transmit antenna. Thus, for 𝑁𝑁𝑁𝑁 transmit antennas the complexity of Step 1 is given by,

𝛿𝛿𝛿𝛿 _ _ = 3𝑁𝑁𝑁𝑁 𝑁𝑁𝑁𝑁 − 𝑁𝑁𝑁𝑁 (13) Step 2 demodulates the estimated symbol using the constellation demodulator function [2], and similarly to [2], [4] we can ignore the complexity of this stage, since the mapping is one-to-one and does not involve any complex multiplications or additions. Therefore, 𝛿𝛿𝛿𝛿 _ _ = 0. Step 3 involves the final estimation of the antenna index and transmit symbol and is based on the detection rule (9), similar to [2]. However, as discussed earlier, in this scenario instead of searching over 𝑀𝑀𝑀𝑀 symbol constellation points for each transmit antenna, the search is performed only over a single constellation point for each transmit antenna. In addition, both 𝒉𝒉𝒉𝒉 and 𝒚𝒚𝒚𝒚 𝒉𝒉𝒉𝒉 have been computed in Step 1 and could be stored, thus not further adding to the complexity. Therefore, it can be verified that the complexity of Step 3 is defined by,

𝛿𝛿𝛿𝛿 _ _ = 2𝑁𝑁𝑁𝑁 (14) Hence the total complexity for SM-LCD can be defined as,

𝛿𝛿𝛿𝛿 = 3𝑁𝑁𝑁𝑁 𝑁𝑁𝑁𝑁 + 𝑁𝑁𝑁𝑁 (15) From examination of the expressions in Table 1 and (15), it is immediately clear that the computational complexity of the proposed SM-LCD is not a function of 𝑀𝑀𝑀𝑀 compared to SM-OD [2], SM-SVD [5] and SM-RC (lower bound) [8]. Thus, we can conclude that the complexity will be significantly reduced, especially as the symbol constellation order increases. The numerical results for the SM-LCD scheme are presented in Section V and compared with existing schemes. In the next section, the proposed antenna selection schemes for SM are presented.

IV. TRANSMIT ANTENNA SELECTION FOR SPATIAL MODULATION

A. TAS-SM In this sub-section, we present the proposed algorithms for multiple TAS for SM. We refer to the first scheme as TAS-SM and denote its configuration as (𝑁𝑁𝑁𝑁 ,𝑁𝑁𝑁𝑁 ,𝑁𝑁𝑁𝑁 ,𝑁𝑁𝑁𝑁 ), where 𝑁𝑁𝑁𝑁 is the number of antennas in a subset of the array. Note that 𝑁𝑁𝑁𝑁 >𝑁𝑁𝑁𝑁 > 𝑁𝑁𝑁𝑁 . The algorithm begins by selecting 𝑁𝑁𝑁𝑁 out of 𝑁𝑁𝑁𝑁 antennas corresponding to the maximum amplitudes of the respective channel gain vectors [14], and follows by discarding 𝑁𝑁𝑁𝑁 − 𝑁𝑁𝑁𝑁 antennas corresponding to the smallest angles for all possible antenna pairs. The algorithm proceeds as follows:


6

Step 1: Arrange the channel gain vectors for 𝑁𝑁𝑁𝑁 antennas into the matrix

𝑯𝑯𝑯𝑯 = 𝒉𝒉𝒉𝒉 𝒉𝒉𝒉𝒉 … 𝒉𝒉𝒉𝒉 (16) Step 2: Compute the sum of the magnitudes of each element in the column vectors to yield,

𝑯𝑯𝑯𝑯 = ℎ ℎ … ℎ (17) where ℎ = ℎ . Step 3: Sort the 𝑁𝑁𝑁𝑁 elements in matrix 𝑯𝑯𝑯𝑯 in descending order to form the vector,

𝑯𝑯𝑯𝑯 = ℎ ℎ … ℎ (18) where ℎ ≥ ℎ ≥ ⋯ ≥ ℎ . Step 4: Select the first 𝑁𝑁𝑁𝑁 elements of 𝑯𝑯𝑯𝑯 to form the vector

𝑯𝑯𝑯𝑯 = ℎ ℎ … ℎ (19) Step 5: Let 𝑯𝑯𝑯𝑯 = 𝒉𝒉𝒉𝒉 𝒉𝒉𝒉𝒉 … 𝒉𝒉𝒉𝒉 represent the matrix of 𝑁𝑁𝑁𝑁 channel gain vectors from 𝑯𝑯𝑯𝑯 (Step 1) chosen and ordered according to the elements of matrix 𝑯𝑯𝑯𝑯 (Step 4). Generate 𝑁𝑁𝑁𝑁 combinations in 2 ways for the vectors of 𝑯𝑯𝑯𝑯 using the binomial coefficient 𝑁𝑁𝑁𝑁

2 . We assume the

pairs are of the form 𝒉𝒉𝒉𝒉 ,𝒉𝒉𝒉𝒉 . Step 6: Compute the angle 𝜗𝜗𝜗𝜗 between the vectors 𝒉𝒉𝒉𝒉 and 𝒉𝒉𝒉𝒉

𝜗𝜗𝜗𝜗 = cos𝒉𝒉𝒉𝒉 𝒉𝒉𝒉𝒉

𝒉𝒉𝒉𝒉 𝒉𝒉𝒉𝒉 (20)

Step 7: Arrange the 𝑛𝑛𝑛𝑛 elements of 𝝑𝝑𝝑𝝑 in ascending order. Step 8: The vectors of the pair 𝒈𝒈𝒈𝒈 ,𝒈𝒈𝒈𝒈 corresponding to the first element of 𝝑𝝑𝝑𝝑 are compared in the following manner, IF 𝒈𝒈𝒈𝒈 < 𝒈𝒈𝒈𝒈 THEN Discard antenna corresponding to 𝒈𝒈𝒈𝒈 ELSE Discard antenna corresponding to 𝒈𝒈𝒈𝒈 ENDIF

Step 9: At this point there are 𝑁𝑁𝑁𝑁 − 1 channel gains. The remainder of the algorithm proceeds as follows, IF 𝑁𝑁𝑁𝑁 = 𝑁𝑁𝑁𝑁 − 1 THEN Stop ELSE Repeat algorithm from Step 5

until 𝑁𝑁𝑁𝑁 = 𝑁𝑁𝑁𝑁 − 1 ENDIF where 𝑁𝑁𝑁𝑁 is the new antenna subset size and replaces 𝑁𝑁𝑁𝑁 . The final 𝑁𝑁𝑁𝑁 antennas are then utilized as the antennas for transmission with SM.

B. CBM-SM The second proposed algorithm will be referred to as CBM-SM. Once again, the configuration is denoted in the form 𝑁𝑁𝑁𝑁 ,𝑁𝑁𝑁𝑁 ,𝑁𝑁𝑁𝑁 ,𝑁𝑁𝑁𝑁 . All steps are identical to the preceding scheme with the exception of Step 6 and 7. Step 6 and 7 for CBM-SM are as follows: Step 6: For all combinations of the channel gain vectors find the correlation,

𝜌𝜌𝜌𝜌 = 𝒉𝒉𝒉𝒉 ,𝒉𝒉𝒉𝒉 (21) where 𝒂𝒂𝒂𝒂,𝒃𝒃𝒃𝒃 represents the inner product between vectors a and b. Step 7: Arrange the 𝑛𝑛𝑛𝑛 elements of 𝝆𝝆𝝆𝝆 in descending order. Step 8 then proceeds by using 𝝆𝝆𝝆𝝆 instead of 𝝑𝝑𝝑𝝑.

C. COMPLEXITY ANALYSIS FOR TAS-SM AND CBM-SM It can be verified that the complexity of the TAS-SM selection algorithm in terms of real multiplications (we use real multiplications here to match the complexity analysis in [16], [17]) is given by,

𝛿𝛿𝛿𝛿 = 2𝑁𝑁𝑁𝑁 𝑁𝑁𝑁𝑁 + 3𝑁𝑁𝑁𝑁 −

2

(22)

Similarly, for the CBM-SM scheme it can be verified that the complexity is defined by,

𝛿𝛿𝛿𝛿 = 2𝑁𝑁𝑁𝑁 𝑁𝑁𝑁𝑁 + 2𝑁𝑁𝑁𝑁 −

2

(23)

From (22), (23) it is immediately evident that the proposed antenna selection schemes for SM have much lower complexity compared to the schemes proposed in [16] and [17].

V. NUMERICAL ANALYSIS

A. LOW-COMPLEXITY DETECTION OF SM In this sub-section, the proposed SM-LCD method is compared to SM-OD [2] and SM-SVD [5]. Since SM-RC [8] closely matches the error performance of SM-OD, we do not include the simulation results for comparison. The lower bound for SM was also evaluated using Eq. (9) from [3], and has been included for comparison.


7

In the model, we have considered an uncoded SM system. QAM modulation has been used and two constellation sizes have been considered, viz. 𝑀𝑀𝑀𝑀 = 16, 64. The channel considered is a quasi-static Rayleigh flat fading channel [2].

Figure 1: BER performance comparison of SM-OD including theoretical result, SM-LCD and SM-SVD for 16-QAM. Figure 1 depicts the results for 16-QAM, where the 4×2 and the 4×4 antenna configurations have been considered. It is evident from the results, that the proposed SM-LCD technique matches the SM-OD almost perfectly for the 4×4 configuration, and for the 4×2 system there is less than 0.5 dB difference at a BER of 10 . These results have also been compared to SM-SVD. It is clear that the SM-SVD scheme only matches the SM-OD scheme at low SNRs (as is consistent with the discussion in [6]). Figure 2 presents similar results for 64-QAM. Once again, it is evident that the proposed SM-LCD simulation curves match the SM-OD simulation results very closely.

Figure 2: BER performance comparison of SM-OD including theoretical result, SM-LCD and SM-SVD for 64-QAM.

To further demonstrate the significance of the proposed SM-LCD scheme, the numerical complexity comparison of SM-LCD with SM-OD and SM-SVD is presented in Table 2 for several configurations. Table 2: Comparison of computational complexity for SM-OD, SM-SVD and the proposed SM-LCD scheme

Configuration Computational Complexity SM-OD

(10) SM-SVD

(11)

SM-LCD (Proposed)

(15) 𝑁𝑁𝑁𝑁 = 4, 𝑁𝑁𝑁𝑁 = 2,

𝑀𝑀𝑀𝑀 = 16 100 58 28

𝑁𝑁𝑁𝑁 = 4, 𝑁𝑁𝑁𝑁 = 4, 𝑀𝑀𝑀𝑀 = 16 124 84 52

𝑁𝑁𝑁𝑁 = 4, 𝑁𝑁𝑁𝑁 = 2, 𝑀𝑀𝑀𝑀 = 64 340 154 28

𝑁𝑁𝑁𝑁 = 4, 𝑁𝑁𝑁𝑁 = 4, 𝑀𝑀𝑀𝑀 = 64 364 180 52

For 𝑁𝑁𝑁𝑁 = 4, 𝑁𝑁𝑁𝑁 = 2 and 𝑀𝑀𝑀𝑀 = 16, SM-LCD yields a percentage drop of 72% compared to SM-OD and 52% compared to SM-SVD. When the number of receive antennas are increased to 𝑁𝑁𝑁𝑁 = 4 for the same configuration this percentage drop becomes 58% and 38%, respectively. For 𝑁𝑁𝑁𝑁 = 4, 𝑁𝑁𝑁𝑁 = 2 and 𝑀𝑀𝑀𝑀 = 64, SM-LCD yields a percentage drop of 92% and 82% compared to SM-OD and SM-SVD. Finally, when 𝑁𝑁𝑁𝑁 is increased to 4 antennas for the same configuration, the percentage drops are 86% and 71%, respectively. Thus, it is clear that due to the computational complexity of the proposed SM-LCD scheme being independent of the constellation size, the imposed complexity is significantly reduced. Thus, we can conclude that the SM-LCD scheme can match the error performance of SM-OD very closely at a much lower computational complexity.

B. TRANSMIT ANTENNA SELECTION FOR SM In this sub-section, the proposed multiple TAS-SM and CBM-SM schemes are evaluated. The analytical and simulation results for SM-OD [2], [3] are included for comparison. In addition, we have applied MG-TAS to SM. The curves have been included. In our model, we have considered a SM system with no coding. Optimal detection has been used at the receiver [2]. In each of the comparisons we have kept the primary number of antennas and the data rate the same. M-QAM modulation with Gray coded constellation is used. A quasi-static Rayleigh flat fading channel has been assumed. Figure 3 presents the results for data rates of a) 3 bits/transmission and b) 4 bits/transmission, for two receive antennas. The TAS-SM scheme has been compared to SM-OD [2] and the APM result for AWGN, similarly to [16], [17].

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Figure 3: BER performance comparison for a) 3 bits/transmission and b) 4 bits/transmission for optimal SM, TAS-SM and APM maximum ratio receive combining in AWGN. It is evident that for both data rates the TAS-SM scheme exhibits a significant improvement in the error rate. Example, at a BER of 10 an improvement of approximately 7.5 dB is realized in both instances and is of great significance. Consider Figure 3 in [17]; comparing the scheme OH-SM (high complexity) to APM AWGN at a BER of 10 the OH-SM is approximately 1.6 dB worse for 4 bits/transmission and C-SM (complexity lower than OH-SM but still relatively high) is approximately 2 dB worse. In comparison to the same APM AWGN curves for 4bits/transmission the proposed scheme TAS-SM is approximately 0.5 dB worse. From this observation, we can conclude that the proposed TAS-SM offers an improved error performance compared to the closest competing schemes from [16] and [17].

Figure 4: BER performance comparison for 4 bits/transmission and 8 bits/transmission for optimal SM, TAS-SM 8,6,4,4 , CBM-SM 8,6,4,4 and MG-TAS-SM 8,4,4 .

Figure 4 compares TAS-SM to optimal SM, CBM-SM and MG-TAS-SM for 4 and 8 bits/transmission and 4 receive antennas. For 4 bits/transmission, at a BER of 10 CBM-SM and MG-TAS-SM have a gain of approximately 0.5 dB compared to optimal SM, however TAS-SM has a gain of approximately 3.5 dB. For 8 bits/transmission and BER 10 CBM-SM exhibits approximately 1 dB improvement over optimal SM, slightly lower than MG-TAS-SM. On the other hand, as expected, TAS-SM exhibits an improvement of approximately 2.8 dB. Thus, the proposed antenna selection scheme for SM can yield a significant improvement in the error performance.

VI. CONCLUSION In this paper, we first proposed a low-complexity detection scheme for conventional SM. From comparisons with the existing schemes it is evident that the proposed SM-LCD scheme matches the optimal detector error performance very closely down to low BER. The receiver computational complexity analysis also shows that SM-LCD exhibits very low complexity, especially for large symbol constellation size, compared to the competing schemes, due to the complexity being independent of the symbol constellation size. Secondly, two TAS schemes to improve the error performance of optimal SM were proposed. From comparisons with the existing schemes, CBM-SM has shown a small improvement in BER at a much lower complexity and TAS-SM has shown a significant improvement in BER at a much lower complexity.

REFERENCES [1] R. Mesleh, H. Haas, C.W. Ahn, S. Yun, “Spatial

Modulation – A New Low Complexity Spectral Efficiency Enhancing Technique,” in Proc. of the Conf. on Communications and Networking in China, Oct. 2006.

[2] J. Jeganathan, A. Ghrayeb, L. Szczecinski, “Spatial Modulation: Optimal Detection and Performance Analysis,” IEEE Communications Letters, vol. 12, no. 8, pp. 545-547, Aug. 2008.

[3] N.R. Naidoo, H.J. Xu, T.A. Quazi, “Spatial modulation: optimal detector asymptotic performance and multiple-stage detection,” IET Communications, vol. 5, no. 10, pp. 1368-1376, 2011.

[4] H. Xu, “Simplified ML Based Detection Schemes for M-QAM Spatial Modulation,” IET Communications, vol. 6, no. 11, pp. 1356-1363, 2012.

[5] J. Wang, S. Xia, J. Song, “Signal Vector Based Detection Scheme for Spatial Modulation,” IEEE Communications Letters, vol. 16, no. 1, pp. 19-21, Jan. 2012.

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9

[6] N. Pillay, H. Xu, “Comments on “Signal Vector Based Detection Scheme for Spatial Modulation”,” IEEE Communications Letters, vol. 17, no. 1, pp. 2-3, Jan. 2013.

[7] J. Zheng, “Signal Vector Based List Detection for Spatial Modulation,” IEEE Wireless Communications Letters, vol. 1, no. 4, pp. 265-267, Aug. 2012.

[8] C. Xu, S. Sugiura, S.X. Ng, L. Hanzo, “Spatial Modulation and Space-Time Shift Keying: Optimal Performance at a Reduced Detection Complexity,” IEEE Transactions on Communications, 2012 (in press).

[9] A. Younis, M.D. Renzo, R. Mesleh, H. Haas, “Sphere decoding for Spatial Modulation,” in Proc. of the IEEE Conf. on Communications, pp. 1-6, Jun. 2011.

[10] A. Younis, R. Mesleh, H. Haas, P.M. Grant, “Reduced Complexity Sphere Decoder for Spatial Modulation Detection Receivers,” in Proc. of the IEEE Global Telecommunications Conf. (GLOBECOM), pp. 1-5, Dec. 2010.

[11] J.G. Proakis, “Digital Communications,” 4th Edition, McGraw-Hill, 2001.

[12] A.F. Molisch, M.Z. Win, “MIMO systems with antenna selection,” IEEE Microwave Magazine, vol. 5, no. 1, pp. 46-56, Mar. 2004.

[13] S.R. Meraji, “Performance Analysis of Transmit Antenna Selection in Nakagami-m Fading Channels,” Wireless Personal Communications, vol. 43, no. 2, pp. 327-333, 2007.

[14] S. Goel, J. Abawajy, “Transmit Antenna Selection for Hybrid Relay Systems,” Technical Report, TR C11/8, August 2011.

[15] Y-S. Choi, A.F. Molisch, M.Z. Win, J.H. Winters, “Fast algorithms for antenna selection in MIMO systems,” in Proc. of the Vehicular Technology Conf., vol. 3, pp. 1733-1737, 2003.

[16] P. Yang, Y. Xiao, Y. Yu, S. Li, “Adaptive Spatial Modulation for Wireless MIMO Transmission Systems,” IEEE Communications Letters, vol. 15, no. 6, pp. 602-604, Jun. 2011.

[17] P. Yang, Y. Xiao, L. Li, Q. Tang, Y. Yu, S. Li, “Link Adaptation for Spatial Modulation with Limited Feedback,” IEEE Transactions on Vehicular Technology, vol. 61, no. 8, pp. 3808-3813, Oct. 2012.


REED-SOLOMON CODE SYMBOL AVOIDANCE

T. Shongwe∗ and A. J. Han Vinck†

∗ Department of Electrical and Electronic Engineering Science, University of Johannesburg, P.O. Box524, Auckland Park, 2006, Johannesburg, South Africa E-mail: [email protected]† University of Duisburg-Essen, Institute for Experimental Mathematics, Ellernstr. 29, 45326 Essen,Germany E-mail: [email protected]

Abstract: A Reed-Solomon code construction that avoids or excludes particular symbols in a linearReed-Solomon code is presented. The resulting code, from our symbol avoidance construction, hasthe same or better error-correcting capabilities compared to the original Reed-Solomon code, butwith reduced efficiency in terms of rate. The codebook of the new code is a subset of the originalReed-Solomon code and the code may no longer be linear. We also present computer search results forthe bound on the number of symbols that can be avoided, and we make an attempt to find an expressionfor the bound. Such a code, by symbol avoidance, can be well suited to a number of applications,some of which include markers for synchronization, frequency hopping signatures, and pulse positionmodulation.

Key words: Reed-Solomon codes, Subcodes.

1. INTRODUCTION

In the work presented by Solomon [1] on alphabet codesand fields of computation, he showed the following. Givenan alphabet (Q) and a field of computation (F) which isprime or power of a prime, where the size of Q is less thanthat of F , it is possible to form a code over Q from anothercode over F with the field of computation of the encodingprocedure being F . The field F is defined as the field ofencoding because that is where the encoding process takesplace and the resulting code being over Q. It was alsoshown in [1] that the price to pay for this modificationin the code is reduction in efficiency in the code over Q,when comparing it to the corresponding code over F . Afitting example of a code over the field F was taken as an(n, k, d) Reed-Solomon (RS) code, which has well definedencoding and decoding operations, and is known to havegood error-correcting capabilities. In the (n, k, d) RS code,n is the length, k is the dimension and d is the minimumHamming distance of the code. The reader who wantsto get acquainted with the principles of Reed-Solomonencoding is referred to [2] and [3]. Solomon [1] used theexample of the RS code over F (F is normally a Galoisfield–GF) to show that from the RS code, a code overalphabet Q can be derived which is “almost” a RS code.

The motivation for the work by Solomon [1] was the factthat not all alphabets we encounter in systems are of thesame size as the field F in which most codes are definedover. For example, the decimal number system with thealphabet {0,1, . . . ,9}, can have GF(11) as the field ofcomputation for the encoding, and the English alphabetwith 26 letters, can have GF(33) or GF(29) as the fieldof computation for the encoding. This idea can thereforebe seen as reducing the size of the field F to form codesover a smaller alphabet Q, or avoiding using some of thesymbols of the field F in the alphabet Q. Solomon [4] didanother work, related to his previous work in [1], where

he introduced non-linear, non-binary cyclic group codes.These codes were associated with GF(2m) such that theywere of length 2m −1, but had (m− j)-bit symbols insteadof the usual m-bit symbols, for m and j positive integers.Further work, in which Solomon was one of the authors,was done on subspace subcodes of Reed-Solomon codesin [5]. The work in [5] presented RS codes that weresubsets of parent RS codes, and the main contributionwas coming up with a formula for the dimension for suchcodes. The work in [5] was limited to RS codes overGF(2m). Recently, Urivskiy [6] published work on subsetsubcodes of linear codes. The subcodes mentioned in [6]were over a set which was a subset of GF(q), where q ispower of a prime. This idea was similar to the idea in [1].However, the focus of the work in [6] was not specificallyon RS codes, it was on linear block codes and findingbounds on the cardinality of the subcodes.

In this paper we take the idea in [1] further; we presentan encoding procedure by which new codes can be formedover an alphabet that is smaller than the size of the fieldof computation. We call this procedure symbol avoidancebecause it allows for the avoidance of particular symbols inthe new code. Our symbol avoidance procedure is a moregeneralised case of the construction in [1]. We will alsoattempt to give an expression for a bound on the number ofsymbols that can be avoided for each code and show whyit may not be possible to get a good estimate of the bound.The codes resulting from our symbol avoidance procedureof a RS code may have the same distance properties as theoriginal RS code or better and may be non-linear.

The applications to symbol avoidance include thosementioned in [1] and the following. Frame synchro-nization: this involves using the avoided symbols forsynchronization to mark the start/end of a frame, or as pilotsymbols. Frequency hopping signatures and pulse positionmodulation: in multiple access communications where


each user is given a unique signature to communicate,it may be desired to avoid particular symbols fromoccurring in the signatures in order to avoid or minimiseconflicts/collision among users.

2. SYMBOL AVOIDANCE

Let us define a linear RS code as (n, k, d)W over GF(q),where n is the length, k is the dimension, d is the minimumHamming distance and q is the size of the field which ispower of a prime. From the linear RS code (n, k, d)W weproduce a new code (n, k′, d′)W ′, of length n, dimensionk′ and minimum Hamming distance d′, over an alphabet ofsize q′. We call the operation by which W ′ is producedfrom W , symbol avoidance. This operation is given insimplified form as

(n, k, d)W →

SymbolAvoidanceOperation

→ (n, k′, d′)W ′,

where d′ ≥ d, q′ < q, k′ < k, and (n, k′, d′)W ′ maybe non-linear. q′ = q − |A|, where A is a set ofelements/symbols to be avoided in (n, k′, d′)W ′. Next, weexplain the symbol avoidance operation in detail.

The conventional systematic generator matrix of the RScode (n, k d)W , G = [Ik|Pn−k] with the symbols taken froma Galois field GF(q), is decomposed into two parts. Thefirst part of G, which is composed of k′ rows of G, will bedenoted Gk′ . The second part of G, which is composed ofr rows of G such that k = k′+ r, will be denoted Gr.

G =

[Ik′ 0k′

r | Pk′n−k

0rk′ Ir | Pr

n−k

],

where Gk′ = [Ik′ 0k′r |Pk′

n−k], Gr = [0rk′ Ir|Pr

n−k] and

Pk′n−k =

Pk′11 Pk′

21 . . . Pk′1(n−k)

......

...Pk′

k′1 Pk′k′2 . . . Pk′

k′(n−k)

, (1)

Prn−k =

Pr11 Pr

21 . . . Pr1(n−k)

......

...Pr

r1 Prr2 . . . Pr

r(n−k)

. (2)

Gk′ is used to encode a k′-tuple (M = m1m2 . . .mk′ , wheremi ∈ GF(q)), and this results into a codeword C = MGk′ .Gr encodes an r-tuple (V = v1v2 . . .vr, where vi ∈ GF(q)),resulting into what we shall call a control vector R =V Gr.The difference between C and R will be in their usage,otherwise they are both results of RS encoding. Thecontrol vectors (collection of the vectors R) are used tocontrol the presence/absence of a particular symbol(s) in

each codeword C, as will be demonstrated shortly. For aq-ary linear block code with a systematic generator matrix,undesired symbols in the codeword, due to the identity part,can be avoided by simply not including those symbols inthe message to be encoded. However, for the parity partof the generator, a different method to avoid undesiredsymbols needs to be applied. It is therefore the maintask of this paper to show that we can avoid undesiredsymbols in a Reed-Solomon code while maintaining orimproving its minimum Hamming distance even thoughthe new code may be non-linear. Using control vectors,we control which symbols to avoid in the parity part of theRS code. C is the RS codeword and R is a control vector tobe used on C, in case C has an undesired symbol.

We now focus on the parity parts of C and R. We denoteby PC

1 ,PC2 , . . .P

Cn−k and PR

1 ,PR2 , . . .P

Rn−k the parity symbols

for C and R, respectively. Using (1), each parity symbolof a codeword, PC

i , becomes PCi = m1Pk′

1i + . . .+mk′Pk′k′i,

for 1 ≤ i ≤ n − k. Using (2), each parity symbol of acontrol vector, PR

j , becomes PRj = v1Pr

1 j + . . .+ vrPrr j, for

1 ≤ j ≤ n− k. Let us define a set A ={

a1,a2, . . . ,a|A|}

,which is a set of symbols taken from GF(q), where |.| hereindicates the cardinality of a set. The symbols in set A arethe symbols we want to avoid in all codewords that resultedfrom the encoding by Gk′ . If any PC

i ∈ A then we ought toeliminate/avoid that PC

i using a corresponding PRj , where

j = i. This is done by adding the corresponding parity partsof C and R such that the resulting parity symbols in the newcodeword, Pi (= PC

i +PRi ), do not have any of the symbols

in A. This procedure can be expressed mathematically asPi �= ax, or PC

i +PRi �= ax, for 1 ≤ x ≤ |A| and 1 ≤ i ≤ n−k.

Depending on their suitability in a sentence, the wordseliminate and avoid will be used interchangeably sincethey convey the same message.

The next example illustrates the symbol avoidanceprocedure.

Example 1 For this example, we take an (n = 7, k = 3)RS code over GF(23) with the generator G in (3). The fieldGF(23) is generated by a primitive polynomial, p(I) = I3+I +1.

G =

1 0 0 | 6 1 6 70 1 0 | 4 1 5 50 0 1 | 3 1 2 3

. (3)

Then by choosing r = 1, we have k′ = 2,

Gk′ =

[1 0 0 | 6 1 6 70 1 0 | 4 1 5 5

](4)

and

Gr =[0 0 1 | 3 1 2 3

]. (5)

Let 7 be the symbol to avoid in codewords encoded by Gk′

in (4), hence A = {7}.


The Gr in (5) gives the following control vectors we canuse when we encounter a codeword with symbol 7,

0 0 0 | 0 0 0 00 0 1 | 3 1 2 30 0 2 | 6 2 4 60 0 3 | 5 3 6 50 0 4 | 7 4 3 70 0 5 | 4 5 1 40 0 6 | 1 6 7 10 0 7 | 2 7 5 2

. (6)

The last control vector, [0 0 7 | 2 7 5 2], is included forcompleteness, otherwise it cannot be used to eliminatesymbol 7 since it has 7 in its information part. To illustratethe elimination of symbol 7, we pick one codeword withthe symbol 7 (in the parity part) from the list of codewordsproduced by Gk′ , and that codeword is C = [0 3 0 | 7 3 4 4].From C, we have PC

1 = 7, PC2 = 3, PC

3 = 4 and PC4 = 4.

Taking the control vector to use as R = [0 0 1 | 3 1 2 3]from (6), we have PR

1 = 3, PR2 = 1, PR

3 = 2 and PR4 =

3. Adding the corresponding parity symbols, we haveP1 = PC

1 +PR1 = 4, P2 = PC

2 +PR2 = 2, P3 = PC

3 +PR3 = 6

and P4 = PC4 + PR

4 = 7. The new codeword has P1 = 4,P2 = 2, P3 = 6 and P4 = 7, which still contains symbol 7,hence R = [0 0 1 | 3 1 2 3] is not a suitable control vector.We repeat this procedure with every control vector in (6)until we find one that is suitable, and in this case it isR = [0 0 3 | 5 3 6 5], which gives P1 = 2, P2 = 0, P3 = 2and P4 = 1. It can also be verified that R = [0 0 5 | 4 5 1 4],[0 0 6 | 1 6 7 1] and [0 0 2 | 6 2 4 6] are all suitable controlvector.

Table 1: Different representations of the elements ofGF(23) generated by the primitive polynomial

p(I) = I3 + I +1, where α is a primitive element ofGF(23).

Elements Binary Polynomial Decimal representationof GF(23) representation representation of the binary

0 000 01 001 1 1α 010 α 2α2 100 α2 4α3 011 α+1 3α4 110 α2 +α 6α5 111 α2 +α+1 7α6 101 α2 +1 5

Remark: It is common practice to represent the elements ofGF(q) by the powers of some primitive element, say α. Forease of presentation, we represented the elements of GF(q)in decimal format in this example. Throughout this paper,we use the decimal representation to represent the elementsof GF(q). The operations are still done the conventionalway using the primitive element. As an example, therelationship between the powers of α and their decimalequivalents is shown in Table 1, where the elements ofGF(23) are generated by a primitive polynomial p(I) =I3 + I +1. �

Note that the key operation of the symbol avoidanceprocedure, we just demonstrated with in Example 1, isabout adding two codewords of the original RS code Wto form a new codeword without the symbols in A. Thisnew codeword then belongs to W ′ together with all othercodewords without the symbols in A. Since W is a linearcode, then it always holds that W ′ ⊆ W . According to thedefinition of linear codes, W ′ can be non-linear, but itsminimum Hamming distance can be the same as that ofW or even greater.

2.1 Table Format Representation

Having shown in Example 1 that some control vectorsare not suitable for symbol avoidance for a particularcodeword, we now want to look at the number of suchcontrol vectors. To do this we take a look at ther-tuples that generate these unsuitable control vectors. Thelist of all possible combinations of v1v2 . . .vr (r-tuples)generating the unsuitable control vectors given ax and PC

i ,is tabulated. Remember that the situation PC

i +PRi = ax or

PRi = ax −PC

i should be avoided, which is a representationof a list of PR

i that should be avoided. Again, rememberingthat PR

j = v1Pr1 j+ . . .+vrPr

r j, then PRi = v1Pr

1i+ . . .+vrPrri =

ax − PCi . Since ax and PC

i are given, the task is to finda list of all possible r-tuples (v1v2 . . .vr) for which PR

i =ax −PC

i is satisfied. This will give us all the r-tuples to beavoided for each codeword, given set A. Each r-tuple, to beavoided, multiplied by Gr corresponds to a control vectorthat is not suitable for symbol avoidance.

At this point we want to stress the following. All thepossible r-tuples when multiplied by Gr, give a list ofavailable control vectors. Among the available controlvectors, there are those that are suitable and those that areunsuitable for symbol avoidance. This means that somer-tuples generate control vectors not suitable for symbolavoidance (r-tuples satisfying PR

i = ax − PCi ), and hence

have to be avoided. The remainder of the r-tuples generatesuitable control vectors (r-tuples satisfying PR

i �= ax −PCi ).

We will pay more attention to the r-tuples satisfying PRi =

ax−PCi (r-tuples to be avoided), as they indicate to us when

symbol avoidance is not possible. It will be more clear laterwhy these r-tuples satisfying PR

i = ax −PCi are important.

Using Example 1, a list of all the possible combinations ofv1v2 . . .vr satisfying each ax −PC

i is found and tabulated asshown in Table 2. The field of computation is GF(23) asbefore, and since the field is an extension of a binary fieldwhere addition and subtraction mean the same thing, thenax −PC

i is the same as ax +PCi . Since r = 1 and |A|= 1,

v1Pr1i = a1 +PC

i . (7)

Then for C = [0 3 0 | 7 3 4 4] and a1 = 7, for each of the(n−k= 4) parity symbols, the equations to solve accordingto (7) are listed as follows.


3v1 = 7+7v1 = 0. (8)

v1 = 7+3v1 = 4. (9)

2v1 = 7+4v1 = 4. (10)

3v1 = 7+4v1 = 1. (11)

The solution of each equation gives the r-tuple (onesymbol in this case) to be avoided for each correspondingparity symbol in the codeword, given set A. In thisparticular example, we have only v1 = 0 or 4 or 1 to beavoided, which means only three control vectors cannot beused. We will refer to the r-tuple(s) to be avoided for eachcodeword, given set A, simply as r-tuple to be avoided.The results of equations (8)–(11) are summarised in Table2.

Table 2: Table listing all r-tuples to be avoided.PC

1 = 7 PC2 = 3 PC

3 = 4 PC4 = 4

a1 = 7 a1 = 7 a1 = 7 a1 = 7v1 v1 v1 v1

0 4 4 1

We shall use this table format (Table 2) as it makes analysisof results, for larger numbers of r-tuples to be avoided,easier. The next example will demonstrate this. Using thistable format will help in an attempt to derive an intuitivebound on |A|, as will be shown later on.

Example 2 Using the same RS code in Example 1, butnow having r = 2, A = {5,6,7}, |A| = 3 and codeword,C = [0 0 4 | 7 4 3 7]. With r = 2, we have k′ = 1,

Gk′ =[0 0 1 | 3 1 2 3

]

and

Gr =

[1 0 0 | 6 1 6 70 1 0 | 4 1 5 5

].

Note: Here we intentionally set the last row of G in (3) to

Gk′ and the remaining rows of G to Gr, to show that it doesnot matter which rows of G are assigned to Gk′ and Gr.

For each PCi and each ax, the equation for all possible

combinations of v1 and v2 to be avoided is given byv1Pr

1i + v2Pr2i = ax +PC

i . The results of all r-tuples to beavoided are given in Table 3.

Table 3: Table listing all r-tuples to be avoided, forn− k = 4, r = 2 and A = {5,6,7}.

6 01 13 24 37 40 52 65 7

PC3 = 3

v1 v2v1 v2v1 v2 v1 v2

|A|(n−k) = 3×4 = 12

v1 v2 v1 v2v1 v2v1 v2v1 v2v1 v2v1 v2 v1 v2

a1 =6

PC1 = 7 PC

4 = 7PC2 = 4

3 04 16 21 32 45 57 60 7

1 00 13 22 35 44 57 66 7

2 03 10 21 36 47 54 65 7

3 02 11 20 37 46 55 64 7

1 05 12 26 37 43 54 60 7

4 00 17 23 32 46 51 65 7

7 03 14 20 31 45 52 66 7

3 01 17 25 30 42 54 66 7

4 06 10 22 37 45 53 61 7

0 02 14 26 33 41 57 65 7

0 07 15 22 31 46 54 63 7

a1 =5

a1 =6

a1 =7

a1 =5

a1 =6

a1 =7

a1 =5

a1 =6

a1 =7

a1 =5

a1 =7

qr−1=

8

It can be seen in Table 3 that the total number of ther-tuples to be avoided is the product of the following:

• the number of symbols in set A: |A|,• the number of parity symbols in the RS code: n− k,

• and the number of unique r-tuples with symbols fromGF(q): qr−1.

This results in the expression, |A|(n− k)qr−1. �

2.2 Bound

The previous subsection addressed the matter of countingthe r-tuples to be avoided, given a symbol(s) to be avoided(set A) and a codeword of the RS code. An interestingquestion to be answered is, what is the limit to the numberof symbols that can be avoided given a RS code whengiven r? An answer or an attempt to provide an answerto this question involves estimating an upper bound on|A|. We have shown that the number of r-tuples (ornumber of control vectors) to be avoided given ax ∈ A,is |A|(n− k)qr−1. We also know that the control vectorsgenerated by the combinations of v1v2 . . .vr when given qis qr. However, with the condition that |A| symbols areto be avoided, only (q − |A|) symbols will be availableto generate control vectors from sequences of length r.The number of available control vectors is then, (q−|A|)r,and the number of control vectors that are not suitable forsymbol avoidance (generated by the r-tuples to be avoided)is |A|(n− k)qr−1. It stands to reason that for successfulavoidance of symbols, the control vectors not suitable forsymbol avoidance should be less than the total number ofavailable control vectors. This leads to


|A|(n− k)qr−1 < (q−|A|)r, (12)

as a sufficient condition for symbol avoidance.

We know that given a linear RS code (n, k d)W and r, wecan apply the symbol avoidance operation on W such thatthe resulting code is (n, k′ d′)W ′, where k′ = k − r andd′ ≥ d. We now attempt to estimate an upper bound on|A| (the number of symbols that can be avoided) using thecondition in (12) as a guideline.

A close observation of Equation (12) shows that term qr−1

on the LHS counts all the r-tuples to be avoided eventhose having symbols from set A (see Table 4), while theRHS has already eliminated all control vectors producedby symbols in set A. This requires an adjustment ofqr−1, which takes into account the fact that the r-tuplesto be avoided which have symbols from set A should notbe counted. An adjustment in (12) results in the LHSbecoming |A|(n− k)(q−|A|)r−1, where we subtracted |A|from q to yield (q−|A|)r−1.

For brevity, let X = |A|(n− k) and Y = (q− |A|) in (12).It should be noted that the value of X is limited by qr−1

because there are exactly qr−1 unique combinations ofv1v2 . . .vr, after which there are repetitions.

Even with the adjustment in (12), of qr−1 to (q−|A|)r−1 ,there are still two problematic cases not taken into account.

1. Not all the r-tuples to be avoided, which have symbolsfrom set A, are always included in the adjustment ofqr−1 to (q−|A|)r−1. There still remains an unknownnumber of r-tuples to be avoided which have symbolsfrom set A. We call this unknown number of ther-tuples to be avoided, the quantity λ, where 0 ≤ λ ≤|A|X .

2. There is an unknown number of r-tuples to be avoidedthat are repeated. The adjustment in (12) still assumesthat all r-tuples to be avoided are unique, in the entire“space” XY r−1. If there are repetitions, this “space”will definitely be reduced. To quantify the number ofrepetitions we define a quantity β, which gives us thenumber of repetitions.

It is, however, difficult, if not impossible, to precisely knowthe λ and β quantities. These quantities are not uniformacross all codewords of a particular code. Moreover, thesequantities will have to be used together taking into accountoverlaps. By overlaps we mean that an r-tuple(s) to beavoided can contain symbols from set A (adding to λ) andalso be a repetition (adding to β).

Considering λ and β, the expression of the upper bound on|A| is

|A|(n− k)(q−|A|)r−1 − γ < (q−|A|)r. (13)

The term on the LHS in (13), γ (= λ+ β), gives the sumof the quantities explained in cases 1 and 2, assuming nooverlaps, 0 ≤ γ < XY r−1.

Using Table 3 in Example 2, we present a pictorialexplanation of the steps involved in arriving at Equation(13) (see Table 4). The table shows the three adjustments to(12) as described above. Firstly, a horizontal line is drawnto cut off the r-tuples that are precisely known to containsymbols in A, resulting to (q−|A|)r−1 = 5 rows. Next, wemark the remaining r-tuples containing symbols from set Awith circles, and their total number is the quantity λ = 18.Finally, ignoring the r-tuples marked with circles, we markrepeated r-tuples with rectangles, and their total number isthe quantity β = 18.

Table 4: Table listing all r-tuples to be avoided, forn− k = 4, r = 2 and A = {5,6,7}. λ = 18 and β = 18,

resulting in γ = 36.

6 01 13 24 37 40 52 65 7

PC3 = 3

v1 v2v1 v2v1 v2 v1 v2

|A|(n−k) = 3×4 = 12

v1 v2 v1 v2v1 v2v1 v2v1 v2v1 v2v1 v2 v1 v2

a1 =6

PC1 = 7 PC

4 = 7PC2 = 4

3 04 16 21 32 45 57 60 7

1 00 13 22 35 44 57 66 7

2 03 10 21 36 47 54 65 7

3 02 11 20 37 46 55 64 7

1 05 12 26 37 43 54 60 7

4 00 17 23 32 46 51 65 7

7 03 14 20 31 45 52 66 7

3 01 17 25 30 42 54 66 7

4 06 10 22 37 45 53 61 7

0 02 14 26 33 41 57 65 7

0 07 15 22 31 46 54 63 7

a1 =5

a1 =6

a1 =7

a1 =5

a1 =6

a1 =7

a1 =5

a1 =6

a1 =7

a1 =5

a1 =7

(q−

|A|)r

−1=

5

qr−1=

8

As it can be seen, Table 2 shows what goes on with justone codeword in which we want to avoid symbols 5, 6 and7. Each codeword of a code, that has undesired symbols,when investigated this way will have its own value of γ,which may not be the same as that of another codeword.This presents a great difficulty in estimating the bound. Todetermine the bound in (13), one has to find the minimumvalue of γ among the codewords investigated.

3. ANALYSIS OF SIMULATION RESULTS

By performing an exhaustive computer search for thebound on |A| in relation to r for several RS codes, weobtained the results in Tables 5, 6 and 7. The resultsshow the limit to possible number of symbols that can beavoided, |A|.

Since the set A is a subset of GF(q) with cardinality |A|,where |A| is the number of symbols to be avoided, thereare

( q|A|)

possible sets with this cardinality. In Tables 5, 6and 7 we list the bounds on |A|, for the cases when all the( q|A|)

sets reach the same upper bound.

From the Tables 5, 6 and 7, we learn the following. Fork ≤ (n − k), we can avoid up to n symbols when r =k − 1 (remember that k = r + k′, then r = k − 1 meansk′ = 1). However, avoiding n symbols will leave only onecodeword in the codebook of the RS code, which is not


a meaningful code. So we shall limit the avoidance ofsymbols to n− 1, which means only a minimum of twosymbols will be remaining in the alphabet since n = q−1.Since the results show that up to n symbols can be avoidedfor r = k− 1, it is obvious that n− 1 symbols can also beavoided.

Table 5: The maximum number of symbols |A| that can beavoided for each r and k for RS codes over GF(7).Symbol avoidance is possible for all the

( q|A|)

sets.

(n − k) = 4 2

|A| ≤ 6

3k = 42

r = 2

r = 1

3

|A| ≤ 6

|A| ≤ 2

|A| ≤ 6

|A| ≤ 2


( q|A|)

sets.

5(n − k) = 8 6

|A| ≤ 10

3k = 4 5 62

r = 2

r = 3

r = 4

r = 1

7

|A| ≤ 1

|A| ≤ 3

|A| ≤ 1

|A| ≤ 4

|A| ≤ 2

|A| ≤ 4

|A| ≤ 5

|A| ≤ 1

4

|A| ≤ 5|A| ≤ 10

|A| ≤ 10 |A| ≤ 10

|A| ≤ 10

4. DECODING

We now know that the code (n, k′, d′)W ′ over GF(q′)is produced by the symbol avoidance operation from thelinear RS code (n, k, d)W over GF(q). The decodingalgorithm of the RS code W is known and well defined.The code W ′ is easily decoded using the decodingalgorithm of the code W , but with some “minor addedoperations” which enhance the performance of W ′. Todescribe these minor added operations, let D ∈ W ′, andD̃ be a received noise corrupted codeword version of D.We assume additive noise which changes one symbol intoanother within the GF(q). Note that D̃ can have undesiredsymbols due to noise corruption in the channel.

Decoding steps of D̃:

1. If the received codeword D̃ has undesired symbols,the undesired symbols are marked as erasures andthen minimum distance decoding is performed.

2. If the minimum distance decoding results in acodeword not in W ′, an error is detected and thecodeword is decoded to the nearest codeword in W ′.


( q|A|)

sets.

7(n − k) = 10 68

|A| ≤ 12

3k = 4 5 62

r = 5

r = 2

r = 3

r = 4

r = 1

9

|A| ≤ 12

|A| ≤ 1

|A| ≤ 3

|A| ≤ 1

|A| ≤ 3

|A| ≤ 1

|A| ≤ 4

|A| ≤ 5|A| ≤ 12 |A| ≤ 5

|A| ≤ 7|A| ≤ 12

|A| ≤ 12

|A| ≤ 2

The code W ′ is highly likely to have better performancethan the original RS code W because of one or acombination of the following:

• d′ ≥ d.

• In some cases, the weight enumerator of W ′ isimproved compared to that of W . This is becausesome codewords of W are excluded in W ′.

5. CONCLUSION

A procedure for avoiding symbols in a Reed-Solomoncode was presented. The procedure created a newReed-Solomon code (W ′) from an original Reed-Solomoncode (W ), but linearity cannot be guaranteed in the newcode. An analysis on how to obtain an upper boundon the number of symbols that can be avoided in thenew Reed-Solomon code was presented. Unfortunately,a final analytic expression for the upper bound was notfound. The expression we came up with could only serveas guidelines towards the upper bound. We resorted tocomputer searches and found numerous results for theupper bound on the number of symbols that can be avoidedin W ′. A decoding procedure for the code W ′ wasdescribed.

REFERENCES

[1] G. Solomon, “A note on alphabet codes and fields ofcomputation,” Information and Control, vol. 25, no. 4,pp. 395–398, Aug. 1974.

[2] B. Sklar, Digital Communications: Fundamentals andApplications. Prentice Hall Inc., 1988.

[3] S. Lin and D. J. Costello Jr., Error control coding:Fundamentals and Applications. Prentice Hall Inc.,1983.

[4] G. Solomon, “Non-linear, non-binary cyclic groupcodes,” in Proceedings of the 1993 IEEE International


Symposium on Information Theory, San Antonio, TX,USA, 1993, pp. 192–192.

[5] M. Hattori, R. J. McEliece, and G. Solomon,“Subspace subcodes of Reed-Solomon codes,” IEEETransactions on Information Theory, vol. 44, no. 5, pp.1861–1880, 1998.

[6] A. Urivskiy, “On subset subcodes of linear codes,” inProblems of Redundancy in Information and ControlSystems (RED), 2012 XIII International Symposiumon, Moscow, Russia, 2012, pp. 89–92.


DETERMINATION OF SPECIFIC RAIN ATTENUATION USING DIFFERENT TOTAL CROSS SECTION MODELS FOR SOUTHERN AFRICA S.J. Malinga*, P.A. Owolawi** and T.J.O. Afullo*** * Dept. of Electrical Engineering, Mangosuthu University of Technology, P. O. Box 12363, Jacobs, Durban,4026, South Africa. E-mail: [email protected] ** Dept. of Electrical Engineering, Mangosuthu University of Technology, P. O. Box 12363, Jacobs, Durban,4026, South Africa. E-mail: [email protected] *** Discipline of Electrical, Electronic and Computer Engineering, University of KwaZulu-Natal, Durban, South Africa E-mail: [email protected] Abstract: In terrestrial and satellite line-of-sight links, radio waves propagating at Super High Frequency and Extremely High Frequency bands through rain undergo attenuation (absorption and scattering). In this paper, the specific attenuation due to rain is computed using different total cross section models, while the raindrop size distribution is characterised for different rain regimes in the frequency range between 1 and 100 GHz. The Method of Moments is used to model raindrop size distributions, while different extinction coefficients are used to compute the specific rain attenuation. Comparison of theoretical results of the existing models and the proposed models against experimental outcomes for horizontal and vertical polarizations at different rain rates are presented. Keywords: Method of moments, microwave attenuation, millimeter wave attenuation, rain attenuation, rainfall regimes, rain types, satellite communication, terrestrial communication, fade margin.

1. INTRODUCTION

The advantages offered by Super High Frequency (SHF) and Extremely High Frequency (EHF) bands such as large bandwidth, small antenna size, and easy installation or deployment have motivated the interest of researchers to study various factors that prevent optimum utilization of these bands. Factors such as cloud, hail, fog, snow, ice crystals, and rain degrade terrestrial and satellite link performance at these frequencies. Rain fade remains the dominant factor in signal fading over satellite and terrestrial links, especially in the tropical and sub-tropical regions like South Africa. In this paper, the focus is on signal attenuation due to absorption and scattering by rain. While other types of hydrometeors such as water vapour, snowfall, and hail are considered secondary deleterious factors to link design at these frequency bands, attenuation due to rain is a fundamental quantity in the estimation of signal degradation in the presence of precipitation for terrestrial and satellite links. As presented in several articles, a simple power-law relationship of specific rain attenuation as given in equation (1) is widely used, and the values of the coefficient parameters k and α are listed in the International Telecommunication Union Recommendation ITU-R P.838.3 [1]. 𝛾𝛾𝛾𝛾 = 𝑘𝑘𝑘𝑘𝑅𝑅𝑅𝑅 (1) Here R (mm/h) is the rain rate and k and α are constants at a given frequency. It is evident from several contributors [2, 3, 4] that k and α vary with the raindrop size distribution (DSD). It is observed as well that the rain DSD is location and climate dependent.

In order to account for the degree of rain attenuation or rain fade in a link, two methods are often considered: the direct method (which refers to direct measurements at the receiver using a spectrum analyzer), and the statistical method (which involves rain rate and raindrop size measurement). The rain rate and raindrop size measurements allow for the estimation of rain attenuation in a cost-effective way as compared with the direct method. Regional and global efforts have been made to obtain suitable distribution functions and related parameters for the raindrop size distribution (DSD). The early DSD models were based on exponential [5] distribution functions, which poorly represented the very small and very large raindrops. The other distribution functions that have been suggested are the Lognormal [2, 6], the Weibull [7, 8], and the gamma [9] distribution functions. There are several ways of fitting measured DSD data, of which the method of moments [10] and maximum likelihood estimation [11] are the most popularly used by many authors. While DSD measurement campaigns have been reported over a considerable period of time in West Africa [2, 12], less attention has been invested in this regard in Southern Africa [13, 14]. One of the aims of this paper is to give a report on the current DSD modelling for the South African region and its application using different extinction cross section models. The resulting fitted DSD is integrated over the scattering cross section to calculate the specific attenuation due to rain. In order to estimate the total cross sectional area of raindrops, the choice of rain shape is the key parameter. Morrison and Cross [15] fitted the drop


shape with a spheroidal model, using a least-squares method. The contribution of Pruppacher and Pitter [16], presented theoretical results of raindrops at different sizes, while Li et al. [17, 18] further simplified Pruppacher and Pitter’s model, with the expression: 𝑟𝑟𝑟𝑟 = 𝑎𝑎𝑎𝑎 1 − 𝑣𝑣𝑣𝑣 𝑓𝑓𝑓𝑓 𝜃𝜃𝜃𝜃 + 𝑓𝑓𝑓𝑓 𝜃𝜃𝜃𝜃

=𝑎𝑎𝑎𝑎 1 − 𝑣𝑣𝑣𝑣 1 + 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝜃𝜃𝜃𝜃 , 0 ≤ 𝜃𝜃𝜃𝜃 ≤ 𝜋𝜋𝜋𝜋/2

𝑎𝑎𝑎𝑎 1 − 𝑣𝑣𝑣𝑣 1 + 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝜃𝜃𝜃𝜃 , 𝜋𝜋𝜋𝜋/2 ≤ 𝜃𝜃𝜃𝜃 ≤ 𝜋𝜋𝜋𝜋 (2)

where, 𝑎𝑎𝑎𝑎 = 1.111582𝑎𝑎𝑎𝑎 (3a)

𝑣𝑣𝑣𝑣 = 1.375447×10 + 6.543960×10 𝑎𝑎𝑎𝑎 (3b)

𝑣𝑣𝑣𝑣 = −7.239211×10 + 1.827561×10 𝑎𝑎𝑎𝑎 (3c)

𝑓𝑓𝑓𝑓 𝜃𝜃𝜃𝜃 = 1 − 𝐻𝐻𝐻𝐻(𝜃𝜃𝜃𝜃 − ) (3d)

𝑓𝑓𝑓𝑓 𝜃𝜃𝜃𝜃 = 1 + 𝐻𝐻𝐻𝐻(𝜃𝜃𝜃𝜃 − ) 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑛𝑛𝑛𝑛 𝜃𝜃𝜃𝜃 (3e) Here a0 denotes the mean raindrop radius, theoretically given as 0.25 mm to 3.00 mm with an incremental interval of 0.25 mm, and H(θ) denotes the step function. Equations (2) to (3) represent the spherical raindrop size model. Another approach with a new formula was presented in reference [19] - in which case the scattered electromagnetic fields were deemed due to spheroidal raindrops. Considering spheroidal raindrop scatterers, the surface of the drops is described by: 𝑟𝑟𝑟𝑟 =

.≈ 𝑎𝑎𝑎𝑎 1 + 𝑣𝑣𝑣𝑣𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝜃𝜃𝜃𝜃 (4)

where 𝑣𝑣𝑣𝑣 = 1 − 𝑎𝑎𝑎𝑎/𝑏𝑏𝑏𝑏 , and a and b represent the raindrop’s minor and major semi-axes respectively, and are measured in centimeters. In [19] Oguchi’s method was used to obtain a and b from the mean drop size radius 𝑎𝑎𝑎𝑎 that was developed to determine the specific rainfall attenuation. Oguchi [20] assumed that: 𝑎𝑎𝑎𝑎𝑏𝑏𝑏𝑏 = 𝑎𝑎𝑎𝑎 ; = 1 − .

.𝑎𝑎𝑎𝑎 (5)

The other method presented in [19] is that of Morrison and Cross [15], which assumes that: 𝑎𝑎𝑎𝑎𝑏𝑏𝑏𝑏 = 𝑎𝑎𝑎𝑎 ; = 1 − 𝑎𝑎𝑎𝑎 (6)

The parameters obtained using the method of Morrison and Cross have shown reasonably consistent results when considering larger raindrops as opposed to that proposed by Oguchi [20]. This paper thus uses the aforementioned total scattering cross section of Morrison and Cross [15], alongside the Pruppacher and Pitter model [16] as well as the Mie model, to compute the specific attenuation due to rain for Southern Africa.

2. DISDROMETER DATA COLLECTION

For the purpose of this submission, two-year DSD data was collected using the Joss-Waldvogel RD-80 disdrometer (JWD) at an integration time of one minute. This equipment is placed at latitude 30058 E and longitude 29052 S atop the Electrical, Electronic, and Computer Engineering building, University of KwaZulu-Natal, at an altitude of 139.7 m above sea-level. It measures raindrop diameters in the range of 0.3 mm to 5 mm in 20 different bins, with the accuracy of ±5%. The expression used to estimate the measured rain rate (mm/hr) by the equipment is presented in the JWD manual as [21]:

𝑅𝑅𝑅𝑅 = 𝐷𝐷𝐷𝐷 𝑛𝑛𝑛𝑛 (7) while the measured DSD, or N(Di), is given by [21]: 𝑁𝑁𝑁𝑁 𝐷𝐷𝐷𝐷 = ×

( )× × ×∆ (8)

Note that S = 5000 mm2, T = 60 s, v(Di) is the terminal velocity, estimated using the Gunn and Kinzer approach, and ni is the number of drops. With the nature of rain distribution in Southern Africa, the rain rate is used to classify the raindrop size distribution models into four classes:

Drizzle (R < 5mm/hr) Widespread (5mm/hr ≤ R < 10 mm/hr) Shower (10 mm/hr ≤ R < 40 mm/hr) Thunderstorm (R ≥ 40 mm/hr)

The rain rate can be estimated from the modelled N(Di) for each class of rain rate type, given by the expression: 𝑅𝑅𝑅𝑅 = 1.8849×10 𝐷𝐷𝐷𝐷 𝑣𝑣𝑣𝑣 𝐷𝐷𝐷𝐷 𝑁𝑁𝑁𝑁(𝐷𝐷𝐷𝐷 )∆𝐷𝐷𝐷𝐷 (9) In the comparative studies of the measured N(Di) and modelled N(Di), the root mean square error (RMSE)% is defined as:

𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 (%) = 1𝑛𝑛𝑛𝑛 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 − 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 ×100

(10)

3. THE METHOD OF MOMENTS

In this paper, the 3rd, 4th, and 6th order moments are employed to determine the three Lognormal DSD parameters because of its simplicity and immediate physical interpretation of the Lognormal parameters. The nth order moment is defined as:


(a)

(b)

(c)

Figure 1: Scattergrams of estimated Lognormal parameters (a) NT (b) µ and (c) 𝜎𝜎𝜎𝜎2, versus the rain rate for the thunderstorm rain type

𝑀𝑀𝑀𝑀 = 𝑁𝑁𝑁𝑁(𝐷𝐷𝐷𝐷)(𝐷𝐷𝐷𝐷) 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 (11) For the Lognormal DSD, the N(Di) model is given as:

𝑁𝑁𝑁𝑁 𝐷𝐷𝐷𝐷 = 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 − (12)

where D is the raindrop diameter, and NT, µ, and σ are the three Lognormal parameters. To simplify equation (11) in the form of equation (12), the integral rain parameters are

approximated by x-th, y-th and z-th moments of the DSD where x, y, and z are non-integer. Then equation (11) is equivalent to: 𝑀𝑀𝑀𝑀 = 𝑁𝑁𝑁𝑁 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥 + 𝑥𝑥𝑥𝑥 𝜎𝜎𝜎𝜎 (13) My and Mz are similar to equation (13). Taking Lx to be the natural logarithm of Mx, the Lognormal DSD parameters are calculated as [10]: 𝑁𝑁𝑁𝑁 = 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 (24𝐿𝐿𝐿𝐿 − 27𝐿𝐿𝐿𝐿 − 6𝐿𝐿𝐿𝐿 )/3 (14a)

𝜇𝜇𝜇𝜇 = −10𝐿𝐿𝐿𝐿 + 13.5𝐿𝐿𝐿𝐿 − 3.5𝐿𝐿𝐿𝐿 /3 (14b)

𝜎𝜎𝜎𝜎 = 2𝐿𝐿𝐿𝐿 − 3𝐿𝐿𝐿𝐿 + 𝐿𝐿𝐿𝐿 /3 (14c) Fig. 1 presents logarithmic scattergrams of Lognormal DSD parameters versus rain rate for the thunderstorm rain type. These parameters are derived from equations (13) and (14). For the other rain types such as drizzle, widespread and shower, Table 1 and expressions presented in equation (15) give details of Lognormal DSD parameters and their coefficients for the Southern African region as summarized in the expressions below: 𝑁𝑁𝑁𝑁 = 𝑎𝑎𝑎𝑎 𝑅𝑅𝑅𝑅 (15a) 𝜇𝜇𝜇𝜇 = 𝐴𝐴𝐴𝐴 + 𝐵𝐵𝐵𝐵 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 (15b)

3.1 Application of Rainfall Regimes It has been established that the raindrop size distribution varies considerably for different types of rainfall regimes or classes. These rainfall classes, as defined in section 2, are: drizzle, widespread, shower and thunderstorm. The corresponding parameters of their Lognormal distributions are shown in Table 1. A general distribution model has often been used by many researchers. However, there is a notable difference in the specific attenuation due to rain produced by the general model, hence the need to develop regime-specific rain attenuation models [22, 23]. 3.2. Comparative studies of DSD models between Southern and West Africa The performance test analysis of the proposed Lognormal model and its counterpart from West Africa [22] are evaluated using equation (10). The comparison is done by substituting the coefficients in Table 1a and 1b into equation (15) with measurements carried out in Durban as discussed in section 2 of this paper. The RMSE model performance test is carried out by taking at least three rain rate samples of each rain type, with the average RMSE in percentage then calculated for each rain type. The proposed Lognormal model performs better than its counterparts across all the rain regimes. The test results agree with RMSE results presented in [23].

y = 41.319x0.625

100

1000

40 60 80 100 120

Rain rate (mm/hr)

NT Power (NT)

Log 1

0 N

T (m

-3)

y = 0.063ln(x) + 0.298

0.1

1

40 50 60 70 80 90 100 110 120 130

Rain Rate(mm/hr)

µ

Log 1

0 µ

y = 0.022ln(x) + 0.013

0.01

0.1

1

40 50 60 70 80 90 100 110 120

Rain rate (mm/hr)

σ2 Log. (σ2)

Log

10 σ

2


The slight deviations of RMSE values recorded are due to the optimum best fit function employed (such as linear, power law, and exponential fits) in [23] and the averaged

RMSE calculated for each rain type. The summary of the RMSE performance test is presented in Table 2.

4. SCATTERING PROPERTIES OF DISTORTED RAINDROPS

The investigation and successful prediction of the attenuation of plane electromagnetic waves caused by a rainy medium involves an assumption of a particular physical model for the raindrop shape. The total scattering cross-section (extinction cross-section) of the raindrop depends to a great deal on this physical model. Many researchers have adopted a simple approach that assumes raindrops to be spherical in shape and employed the Mie theory to obtain the specific attenuation due to rain. However, it has been established through photographic measurements [24, 25] that realistic raindrops become oblate spheroidal in shape as the drop size gets larger. Raindrops of fairly large size experience severe distortion, and as such, they are no longer spheroidal in shape but look like hamburgers [26, 27]. This loss of shape is accompanied by a loss of symmetry along their axes. The assumption that raindrops are spherical, as adopted by many researchers, is therefore valid only for small raindrops. 4.1 Scattering Coefficients Scattering coefficients can be obtained by considering different orders of approximations, with the zeroth-order scattering from the spheroid representing the Mie scattering from a sphere [19]. Li et al [19] discuss two approaches for obtaining the first-order scattering approximation. The first approach is to modify the scattering coefficients 𝑆𝑆𝑆𝑆 𝑎𝑎𝑎𝑎 and 𝑆𝑆𝑆𝑆 𝑎𝑎𝑎𝑎 through the use of an effective radius of the spheroid, while the second approach involves the modification of the spherical vector wave functions to the spheroidal ones. The second approach involves lengthy and error-prone calculations hence it is much easier to execute the first approach. This approach suggests that the variability of the raindrop shape merely translates to the variability of the effective raindrop radius. For spherical raindrops, the scattering coefficients are expressed in terms of the mean drop radius of the sphere. The following equation was thus considered to represent the surface of an arbitrarily-distorted scatterer [19]: 𝑟𝑟𝑟𝑟 = 𝑎𝑎𝑎𝑎[1 + 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣 𝜃𝜃𝜃𝜃 ,𝜙𝜙𝜙𝜙 ] (16) where 𝜃𝜃𝜃𝜃 and 𝜙𝜙𝜙𝜙 represent the zenith and the azimuth angles of the incident waves at which reflection occurs. The above equation clearly shows that for small values of v, 𝑟𝑟𝑟𝑟 ≈ 𝑎𝑎𝑎𝑎 . This implies that the second term in the above equation represents a distortion with respect to a sphere of radius a. It can therefore be considered that the term 𝑎𝑎𝑎𝑎 1 + 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣 𝜃𝜃𝜃𝜃 ,𝜙𝜙𝜙𝜙 represents the effective radius 𝑎𝑎𝑎𝑎 of an arbitrarily-shaped scatterer [19].

Table 1a: Coefficients of Lognormal DSD parameters in equation (15) for Southern Africa

Rain Type

a0 b0 Aµ Bµ Aσ Bσ

Drizzle 212.3 0.387 -0.281 0.131 0.086 0.013

Shower 258.3 0.095 -0.321 0.242 0.072 0.005

Wide spread

322.4 0.102 -0.392 0.249 0.083 -0.003

Thunder storm

41.3 0.625 0.299 0.063 0.014 0.022

General model

220.0 0.392 -0.267 0.137 0.077 0.010

Table 1b: Coefficients of Lognormal DSD parameters for West Africa [22]

Rain Type

a0 b0 Aµ Bµ Aσ Bσ

Drizzle 718.00 0.399 -0.51 0.128 0.038 0.013

Shower 137.00 0.370 -0.41 0.234 0.223 -0.03

Wide spread

264.00 -0.23 -0.47 0.174 0.161 0.018

Thunder storm

63.00 0.491 -0.18 0.195 0.209 -0.03

General model

108.00 0.363 -0.20 0.199 0.137 -0.01

Table 2: Error analysis (RMSE in %) of the proposed Lognormal model and West Africa model

Rain Type Lognormal model (West Africa)

Proposed Lognormal Model

Drizzle 8.01 4.03 Shower 9.85 4.53 Widespread 10.98 5.01 Thunderstorm 13.15 6.95 General 11.49 4.14 Table 3: Comparison of rain attenuation for P-P and MC model against the Mie model at different rain rates at 100 GHz

Rain Type Attenuation Percentage Difference (%) between Mie and P-P and MC-model

P-P model MC model

Drizzle 4.36 8.98

Widespread 5.96 10.81

Shower 15.35 17.83

Thunderstorm 30.95 31.47


The scattering coefficients are thus given by [18, 19]: 𝑆𝑆𝑆𝑆 = − (17)

𝑆𝑆𝑆𝑆 = − (18)

and the parameter D is given by: 𝐷𝐷𝐷𝐷 = 𝑘𝑘𝑘𝑘 𝑎𝑎𝑎𝑎[1 + 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣 𝛼𝛼𝛼𝛼, 0 ] (19) where k0 is the free space wave number and α is the incidence angle. The parameter D is analogous to the size parameter commonly used in the Mie calculations. Once the shape of the raindrops is known the scattering coefficients can be determined. The extinction (total) cross-section is then calculated based on the different shapes for different models as follows [15, 20]: 𝑄𝑄𝑄𝑄 = − 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 2𝑛𝑛𝑛𝑛 + 1 [𝑆𝑆𝑆𝑆 𝑎𝑎𝑎𝑎 + 𝑆𝑆𝑆𝑆 𝑎𝑎𝑎𝑎 ] (20)

where 𝑎𝑎𝑎𝑎 is the effective raindrop radius given by [20, 21]: 𝑎𝑎𝑎𝑎 = (21) 4.2 Total Scattering Cross Section In this paper, the total scattering cross-section is evaluated and plotted against the mean raindrop radius for three different models. We consider the Morrison and Cross (M-C) model for spheroidal raindrops; the Pruppacher and Pitter (P-P) model for raindrops of any size and shape; and the Mie model for spherical rain drops. The first two models are polarization dependent while the third model is polarization independent. In order to evaluate the effect of polarization using the three total cross-section models, the following six frequencies are used: 7.8 GHz, 13.6 GHz, 19.5 GHz, 34.8 GHz, 140 GHz, and 245.5 GHz. The corresponding complex refractive indices of water at a temperature of 20 degrees Celsius using the method of Liebe et al for these frequencies are: (8.3614+i1.697), (7.5307+i2.4231), (6.7189+i2.7566), (5.2534+i2.8091), (2.9701+i1.5635), and (2.5945+i1.1046), respectively. Fig. 2 shows that the different models are in agreement in terms of the extinction cross section for small raindrop diameters and lower frequencies. As the drop diameter increases, a noticeable difference is observed among the models. The Mie model gives the lowest values of extinction cross-section while the higher values are obtained from the M-C and P-P models due to the degree of the raindrops’ distortion. The results confirm the inability of the Mie model to realistically represent the raindrop shape, reducing it to a smaller sized sphere in comparison with the realistic spheroid of the M-C and P-

P models. Figure 2 confirms that different polarizations have different effects on the total cross section. The horizontally-polarized wave records a larger value of total cross-section than its vertical counterpart. By using the percentage difference between the P-P model and the Mie model at 7.8 GHz, it confirmed that at lower diameters, the percentage difference is minimal. At a diameter of 2 mm, it is seen that the percentage difference with respect to vertical polarization is 11.76%. In the case of horizontal polarization, a similar trend is observed, with the percentage difference of 18.99% at the same diameter. The lower percentage difference is recorded in the M-C model against the Mie model. At a diameter of 2 mm, the percentage differences between the M-C model and the Mie model are 5.60% and 12.23% for vertical and horizontal polarization, respectively. At a frequency of 245.5 GHz, in Figure 2f, the same trend is observed as was the case for the lower frequency (7.8 GHz). The percentage difference at the 245.5 GHz frequency is approximately twice the values reported at the lower frequency.

5. SPECIFIC AND TOTAL RAIN ATTENUATION

The prediction of rainfall attenuation is achieved through the numerical integration of the following equation:

𝐴𝐴𝐴𝐴 = 4.34×10 𝑄𝑄𝑄𝑄 𝐷𝐷𝐷𝐷 𝑁𝑁𝑁𝑁 𝐷𝐷𝐷𝐷 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 (22) where 𝑄𝑄𝑄𝑄 𝐷𝐷𝐷𝐷 represents the extinction cross section as given in equation (22) above. N(D) is the mean drop size distribution of the rainfall drops given by equations (8) and (9). The impact of distorted raindrops on attenuation due to rain is determined for the P-P model, M-C-model, and Mie model for frequencies up to 100 GHz. The results are plotted on the same set of axes for ease of comparison. As indicated in section 4.2, comparisons are made by using different total scattering cross-section models at circular polarization to determine specific rain attenuation as shown in Figure 3. The choice of circular polarization is considered because of its popularity in satellite communications at SHF and EHF ranges. In order to investigate the variability of specific rain attenuation, a single rain rate has been chosen from each rainfall regime to show the degree of the attenuation impact on the link operating at a given frequency, as shown in Figures 3 (a) to 3(d). To highlight the worst case scenarios, the upper band of rain rate values are used for each rain type, namely: 5 mm/h, 10 mm/h, 40 mm/h, and 100 mm/h for drizzle, widespread, shower, and thunderstorm rain types, respectively. The percentage difference between the specific attenuation due to Mie model and that due to the M-C model is high compared to the difference due to the Mie and the P-P models. The highest average percentage difference in specific attenuation is thus obtained for the thunderstorm rain type with the M-C model, giving a value of 31.47%; while its P-P model counterpart gives a


difference of 30.95%. The lowest difference in specific attenuation is observed with the drizzle rain type, returning values of 9.27% and 15.11% between the Mie and the P-P model and the Mie and the M-C model, respectively. The specific attenuation due to rain in an area gives the radio planner information about the extent to which the transmitted signal will be degraded in the presence of rain. The choice of operating frequency of a radio link is influenced by the rain attenuation and availability stipulations for the service on offer. Figure 3 shows the specific rain attenuation for a radio link operating at frequencies of up to 100 GHz for four different rain types. It is clear that the severity of signal degradation increases with increasing frequency. This translates into higher fade margin requirements at high frequencies to ensure reliable service for radio link users. Of the three models used in this study, the Mie model stipulates lower

fade margin demands at all frequencies, while the M-C model calls for the highest fade margin requirements for all frequencies. The P-P model gives fade margins that lie in between these two. Considering the specific attenuation due to different rain types described in section 2, we can see that the three models are in reasonably good agreement as shown in Fig. 3(a) for the drizzle rain type. The drizzle rain type is characteristically dominated by small rain drops and rain rate of at most 5 mm/h. This is in keeping with the notion that small rain drops are almost spherical in shape. The effective raindrop radius, 𝑎𝑎𝑎𝑎 , of the P-P and M-C models, reduces to nearly that of the spherical model, giving almost the same attenuation levels as those due to the Mie model. However, as the raindrop size increases, the difference in the results from the three models becomes more noticeable. A summary of the percentage

1.E-‐06

1.E-‐05

1.E-‐04

1.E-‐03

1.E-‐02

1.E-‐01

1.E+000 0.5 1 1.5 2 2.5 3 3.5

Ext

inct

ion

Cro

ss s

ectio

n (c

m^2

)

Raindrop mean radius (mm)

Qt(P-‐P-‐H)Qt(P-‐P-‐V)Qt(Mie)Qt(MC-‐V)Qt(MC-‐H)

1.E-‐05

1.E-‐04

1.E-‐03

1.E-‐02

1.E-‐01

1.E+000 0.5 1 1.5 2 2.5 3 3.5

Ext

inct

ion

cros

s se

ctio

n (c

m^2

)



(a) 7.8 GHz (b) 13.6 GHz

1.E-‐05

1.E-‐04

1.E-‐03

1.E-‐02

1.E-‐01

1.E+00

1.E+01

0 0.5 1 1.5 2 2.5 3 3.5

Ext

inct

ion

cros

s se

ctio

n (c

m^2

)


Qt(P-‐P-‐H)

Qt(P-‐P-‐V)

Qt(Mie)

Qt(MC-‐V)

Qt(MC-‐H)1.E-‐04

1.E-‐03

1.E-‐02

1.E-‐01

1.E+000 0.5 1 1.5 2 2.5 3 3.5

Ext

inct

ion

cros

s se

ctio

n (c

m^2

)



(c) 19.5 GHz (d) 34.8 GHz

1.E-‐03

1.E-‐02

1.E-‐01

1.E+000 0.5 1 1.5 2 2.5 3 3.5

Ext

inct

ion

cros

s se

ctio

n (c

m^2

)


Qt(P-‐P-‐H)Qt(P-‐P-‐V)Qt(Mie)Qt(MC-‐V)Qt(MC-‐H) 1.E-‐03

1.E-‐02

1.E-‐01

1.E+00

1.E+01

0 0.5 1 1.5 2 2.5 3 3.5

Ext

inct

ion

cros

s se

ctio

n (c

m^2

)



(e) 140 GHz (f) 245.5 GHz Figure 2: Comparison of extinction cross-sections of MC, P-P, and Mie models at horizontal and vertical polarization versus mean drop radius at (a) 7.8 GHz (b) 13.6 GHz (c) 19.5 GHz (d) 34.8 GHz (e) 140 GHz (f) 245.5 GHz


differences in specific attenuation between the P-P and M-C models on the one hand, with the Mie model on the other, is shown in Table 3. The comparison is made at the frequency of 100 GHz. More importantly, this comparison gives insight into the validity of the widely accepted Mie model. The positive values in the table indicate that the values obtained using the Mie model are less than those of its counterparts. Table 3 clearly indicates that the difference between the values obtained using the Mie model and its counterparts increase with increasing rain rate. The specific attenuation due to rain varies considerably with operating frequency. Figures 4(a) to 4(d) show how the specific rain attenuation varies with rain rate for the three models. For both the M-C and the P-P models, the plotted graphs are presented for both vertical and horizontal polarizations. The effect of raindrop distortion is investigated once more and compared with the spherical model of Mie. The results presented in Table 3 represent the average percentage differences in specific

attenuation between the Mie model and its two counterparts for several frequencies in the range 1 ≤ f ≤ 245.5 GHz. It is seen from Table 4 that the percentage difference of specific attenuation decreases with increasing frequency. The highest percentage difference of 35.90% is obtained between the Mie model and the P-P model, while a difference of 28.85% is obtained between the Mie model and the M-C model at 7.8 GHz. On the other hand, the lowest percentage difference is observed at a frequency of 140 GHz, with values of 0.81% and 1.04 % for the M-C model (vertical polarization) and the P-P model (vertical polarization), respectively. All the values in Table 4 are determined for an extreme rain rate of 100 mm/h. This result is more evident in Fig. 3, where small gaps between the graphs are noticeable at lower frequencies. At high frequencies, the graphs seem to attain close values. This implies that the three models are in reasonably good agreement at high frequencies. The Mie model thus gives a good representation of the drop shape even at high frequencies, while a considerable difference is observed at lower frequencies.

0.0001

0.001

0.01

0.1

1

10

0 20 40 60 80 100

Spec

ific

Atte

nuat

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(dB

/km

)

Frequency (GHz)

drizzle

P-‐P Model

M-‐ C Model

Mie0.0001

0.001

0.01

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10

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Spec

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(dB

/km

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Frequency (GHz)

widespread

P-‐P Model

M-‐C Model

Mie

(a) Drizzle (b) Widespread

0.0001

0.001

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10

100

0 20 40 60 80 100

Spec

ific

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/km

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shower

P-‐P Model

M-‐C Model

Mie0.0001

0.001

0.01

0.1

1

10

100

0 20 40 60 80 100

Spec

ific

Atte

nuat

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(dB

/km

)

Frequency (GHz)

thunderstorm

P-‐P Model

M-‐C Model

Mie

(c) Shower (d) Thunderstorm Figure 3: Specific attenuation due to rain versus frequency for different rain types: (a) Drizzle, (b) Widespread, (c) Shower and (d) Thunderstorm


6. THE 𝒌𝒌𝒌𝒌𝑹𝑹𝑹𝑹𝜶𝜶𝜶𝜶 RAIN ATTENUATION MODEL

The ITU-R recommends the use of the power law relationship given by equation (1) to determine the specific attenuation due to rain, provided the local rain rate is known. In this paper, the results of specific rain attenuation obtained using the Lognormal DSD are modelled using the power law relationship. As mentioned earlier on, the coefficients k and α vary with both frequency and location-dependent rain drop size distribution. These coefficients are determined and presented in Table 5 for Durban, which lies on the east coast of South Africa. In this section circular polarization is assumed. Once again, three raindrop size models are considered, one being the popular spherical drop shape of Mie, while the other two take into account distortion during the drop fall. The power law model parameters are presented in Table 5 with subscripts m, p, and s representing Mie, P-P, and M-C models respectively. Figure 6 shows the parameter α modelled for the frequency range 1 ≤ f ≤ 10 GHz and 10 < f ≤ 100 GHz. The results in Table 6 and 7 can be used to determine the power law coefficients at any desired frequency in the range 1 ≤ f ≤ 100 GHz. The specific rain attenuation can

thus be determined at all frequencies in this range of desired rain rates. The following general model is realized:

𝑘𝑘𝑘𝑘 = 𝐴𝐴𝐴𝐴 𝑓𝑓𝑓𝑓 + 𝐵𝐵𝐵𝐵 𝑓𝑓𝑓𝑓 + 𝐶𝐶𝐶𝐶 𝑓𝑓𝑓𝑓 + 𝐷𝐷𝐷𝐷 𝑓𝑓𝑓𝑓 + 𝐸𝐸𝐸𝐸 (23)

𝛼𝛼𝛼𝛼 = 𝐴𝐴𝐴𝐴 𝑓𝑓𝑓𝑓 + 𝐵𝐵𝐵𝐵 𝑓𝑓𝑓𝑓 + 𝐶𝐶𝐶𝐶 𝑓𝑓𝑓𝑓 + 𝐷𝐷𝐷𝐷 𝑓𝑓𝑓𝑓 + 𝐸𝐸𝐸𝐸 (24)

where A1 and A2 represent the fourth coefficient of the polynomial, B1 and B2 represent the third coefficient, C1 and C2 represent the second coefficient, D1 and D2 represent the first coefficient, and E1 and E2 represent the constant of the polynomial. Substituting the coefficients shown in Table 6 and 7 into equation (23) and (24) enables the determination of k and α for different frequencies.

7. CONCLUSION This paper presents realistic raindrops based on their extinction total cross section and the specific attenuation due to rain using three models, namely, the M-C (or spheroidal) model, the P-P model, and the Mie (or spherical) model. The paper further investigates the validity of the Mie spherical raindrops model that is used

0.001

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km)

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f=7.8 GHzM-‐C V

M-‐C H

P-‐P-‐H

P-‐P-‐V

Mie 0.01

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f=19.5 GHzM-‐C V

M-‐C H

P-‐P-‐H

P-‐P-‐V

Mie

(a) 7.8 GHz (b) 19.5 GHz

0.01

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f=34.8 GHz M-‐C V

M-‐C H

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P-‐P-‐V

Mie0.1

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10

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0 20 40 60 80 100

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f=140 GHzM-‐C V

M-‐C H

P-‐P-‐H

P-‐P-‐V

Mie

(c) 34.8 GHz (d) 140 GHz Figure 4: Specific rain attenuation against rain-rate for (a) 7.8 GHz (b) 19.5 GHz (c) 34.8 GHz and (d) 140 GHz


by many researchers. It is observed that while the extinction cross-section varies considerably with raindrop size, the three models show a close agreement for small raindrop diameters but differ as the drop size increases. The Mie model thus remains accurate for fade margin determination for small raindrop diameters; this model

therefore gives reliable specific rain attenuation predictions in regions dominated by drizzle rainfall, but will considerably underestimate the attenuation in regions dominated by shower and thunderstorm rain types. The results also show that the raindrop extinction cross section for the three models agree at the lower frequencies while as the frequency increases, a noticeable difference is observed. The effect of polarization is noticed in the M-C model and the P-P model, with the horizontal polarization returning the higher percentage difference compared to the Mie model. The specific attenuation due to rain for different total cross-sections (P-P, M-C and Mie models) with circular polarization for different rain types shows a progressive increase in the average percentage differences (with respect to the Mie model) from drizzle through widespread to thunderstorm. The average specific attenuation percentage difference for the P-P model is low compared to the M-C model. The results further show that the power law model parameters, k and α, can be modelled using the fourth order polynomial as shown by equations (23) and (24). The specific rain attenuation can thus be determined using the power law model for frequencies up to 100 GHz and rain rate up to 100 mm/h. In regions where there is a high presence of large raindrops (as is the case in tropical and sub-tropical regions like South Africa), it is inadequate to employ the Mie model in the determination of local rain fade margins. However, this model is acceptable for regions dominated by the drizzle rain type due to a high presence of small drops.

8. REFERENCES

[1] International Telecommunication Union Radio communication (ITU-R), “Specific attenuation model for rain for use in prediction methods”, Recommendation ITU-R P.838.3, International Telecommunication Union, Geneva, Switzerland, 2005.

[2] G.O. Ajayi and R.L.Olsen, “Modelling of a tropical

raindrop size distribution for microwave and millimeter wave applications,” Radio Science, Vol. 20, Issue 2, pp. 193–202, March-April 1985.

[3] F. Moupfouma, “A new theoretical formulation for

calculation of the specific attenuation due to precipitation particles on terrestrial and satellite radio links,” International Journal of Satellite Communications, Vol. 15, pp. 89-99, 1997.

[4] I. Seishiro, “Rain attenuation at millimeter

wavelength of 1.33 mm,” International Journal of Infrared and Millimeter waves, Vol. 25, No.10, pp. 1495-1501, October 2004.

Table 4: Comparison of the P-P and M-C models to the Mie model at different frequencies

Frequency

(GHz)

Average Attenuation Percentage difference (%)

P-P V

Model

P-P H

Model

M-C V

Model

M-C H

Model

7.8 17.36 35.90 10.20 28.85

13.6 9.19 18.00 10.25 19.04

19.5 7.26 15.80 4.79 13.32

34.8 10.99 22.59 5.67 17.25

140 1.04 10.47 0.81 8.46

245.5 2.47 13.55 1.71 9.18

Table 5: Power law model parameters for the three drop-size models

Frequency

(GHz)

Mie Model P-P Model M-C Model

𝑘𝑘𝑘𝑘 𝛼𝛼𝛼𝛼 𝑘𝑘𝑘𝑘 𝛼𝛼𝛼𝛼 𝑘𝑘𝑘𝑘 𝛼𝛼𝛼𝛼

1 0.00006 0.7595 0.00006 0.7759 0.00006 0.7758

2 0.0002 0.7836 0.0002 0.8043 0.0002 0.8040

4 0.001 0.8663 0.001 0.9064 0.001 0.9039

5 0.0016 0.9239 0.0016 0.9776 0.0017 0.9779

6 0.0025 0.9859 0.0025 1.0500 0.0026 1.0448

8 0.0052 1.0797 0.0053 1.1319 0.0056 1.1291

10 0.0099 1.1168 0.0103 1.1574 0.0109 1.1562

12 0.0172 1.1180 0.0179 1.1442 0.019 1.1424

20 0.0714 1.0471 0.0736 1.0667 0.0776 1.0636

30 0.1914 0.9692 0.1975 0.9835 0.2088 0.9807

40 0.382 0.9145 0.3928 0.9239 0.4137 0.9208

50 0.6226 0.8427 0.6364 0.8489 0.6682 0.8441

60 0.929 0.7638 0.9477 0.7687 0.9951 0.7637

80 1.6774 0.6631 1.7026 0.6692 1.7705 0.6672

90 1.9331 0.6412 1.954 0.6471 2.014 0.6459

100 2.053 0.6266 2.0686 0.6325 2.1186 0.6319

Table 6: Modelled coefficient k

Fit Model Mie P-P M-C

Fourth Order

Polynomial

1 ≤ f ≤ 100 GHz

4th -7E-8 -7E-8 -7E8

3rd 1E-5 1E-5 1E-5

2nd -1E-4 -1E-4 -1E-4

1st 3.3E-3 3.3E-3 3.2E-3

constant -0.0114 -0.0115 -0.0113


[5] J.O. Laws and D.A. Parsons, “The relation of raindrop size to intensity,” Transactions, American Geophysical Union, Volume 24, Issue 2, pp. 452-460, 1943.

[6] K.I. Timothy, J.T. Ong and E.B.L. Choo, “Raindrop

Size Distribution Using Method of Moments for Terrestrial and Satellite Communication

Applications in Singapore,” IEEE Transactions on Antennas and Propagation, Vol. 50, No.10, pp. 1420-1424, October 2002.

[7] H. Jiang, M. Sano, and M. Sekine, “Weibull

raindrop size distribution and its application to rain attenuation,” IEE Proc. Microwave and Antennas Propag. Vol. 144, No.3, pp. 197-200, June 1997.

[8] M. Sekine, C. Chen, and T. Musha, “Rain

attenuation from log-normal and Weibull raindrop-size distributions,” IEEE Trans. Antenna and propagation, Vol. AP-35, No.3, pp. 358-359, March 1987.

[9] D.A. De Wolf, “On the Law-Parsons distribution of

raindrop sizes,” Radio Science, Vol. 36, No.4, pp. 639-642, June/August 2001.

[10] T.Kozu, Estimation of raindrop size distribution

from space borne radar measurements. Ph.D. dissertation, Univ.of Kyoto, Japan, 1991.

[11] P.W. Mielke Jr., “Simple iterative procedures for

two parameter gamma distribution maximum likelihood estimates,” J. Appl. Meteor. Vol. 15, pp. 181-183, 1976.

[12] S. Moumouni, M. Gosset, and E. Houngninou,

“Main features of rain drop size distributions observed in Benin, West Africa, with optical distrometers,” Geophysical Research Letters, Vol. 35, No. 23, L23807, doi: 10.1029/2008GL035755, December 2008.

[13] P.A. Owolawi, “Raindrop Size Distribution Model

for the Prediction of Rain Attenuation in Durban,” PIERS, Vol. 7, No.6, pp. 51-523, 2011.

[14] T.J.O. Afullo, “Raindrop Size Distribution

Modelling for Radio Link Design along the Eastern Coast of South Africa,” PIER B, Vol.34, pp. 345-366.

[15] J.A Morrison and M.J. Cross, “Scattering of a plane

electromagnetic wave by axisymmetric raindrops,” Bell Syst. Tech, J., Vol. 53, pp. 955-1019, July-August, 1974.

[16] H.R. Prupacher, and R.L. Pitter, “A semi-empirical

determination of shape of cloud and rain drops,” J. Atmos. Science, Vol. 28, pp. 86-94, 1971.

[17] L.W. Li, P.S. Kooi, M.S. Leong, , T.S. Yeo, and

M.Z. Gao, “Microwave attenuation by realistically distorted raindrops: Part I-Theory,” IEEE Trans. Antenna Propagation, Vol. 43, No.8., pp. 811-821, 1995.

Figure 5: Coefficient k against frequency (Note: the Spheroidal model is the M-C model)

Figure 6: Coefficient α against frequency (Note: the Spheroidal model is the M-C model) Table 7: Modelled coefficient α

Polynomial Fit Mie P-P M-C

4th Order

Polynomial

1 ≤ f ≤ 10 GHz

4th - 8E-5 8E-5

3rd -9E-4 -2.8E-

-2.7E-3

2nd 1.43E-

2.81E-

2.76E-

1st -1.65E-

-4.4E-

-4.28E-

constant 0.7638 0.7964 0.7954

3rd Order

Polynomial

10 < f ≤ 100 GHz

3rd 5E-7 4E-7 4E-7

2nd -4E-5 -3E-5 -2E-5

1st -6.3E-3 -7.6E-

-7.7E-3

constant 1.1923 1.232 1.232

0.0

0.5

1.0

1.5

2.0

2.5

0 20 40 60 80 100

Coe

ffici

ent,

k

Frequency (GHz)

Mie Model

P-‐P Model

Spheroidal Model

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0 20 40 60 80 100

Coe

ffici

ent, α

Frequency (GHz)

Mie Model P-‐P Model Spheroidal Model


[18] L. W. Li, P. S. Kooi, M. S. Leong, and T. S. Yeo, “Integral Equation Approximation to Microwave Specific Attenuation by Distorted Raindrops: The Pruppacher-and-Pitter Model,” Proceedings of IEEE Singapore International Conference, 3-7 Jul 1995, pp. 600-604, ISBN: 0-7803-2579-6.

[19] L. W. Li, P. S. Kooi, M. S. Leong, and T. S. Yeo,

“Integral Equation Approximation to Microwave Specific Attenuation by Distorted Raindrops: The Spheroidal Model,” Proceedings of IEEE Singapore International Conference, 3-7 Jul 1995, pp. 658-662, ISBN: 0-7803-2579-6.

[20] T. Oguchi, “Attenuation of electromagnetic wave

due to rain with distorted raindrops,” J. Radio Res. Labs., Vol. 7, pp. 467-485, 1960.

[21] Distromet Ltd., Distrometer RD-80 instructional

manual, Basel, Switzerland, June 20, 2002.

[22] I.A. Adimula and G.O. Ajayi, “Variation in drop size distribution and specific attenuation due to rain in Nigeria,” Annales Des Télécommunications, Vol. 51, Issue 1-2, pp. 87-93, Jan-Feb 1996.

[23] S.J. Malinga and P.A. Owolawi, “Obtaining raindrop size model using method of moment and its applications for South Africa radio systems,” PIER B, Vol.46, pp. 119-138, 2013.

[24] T.J. Schuur, A.V. Ryzhkov, D.S. Zrnic, and M

Schonhuber, “Drop Size Distribution Measured by a 2D Video Disdrometer: Comparison with Dual-Polarization Radar Data,” Journal of Applied Meteorology, Vol. 40, pp. 1019 – 1034, June, 2001.

[25] H. R. Pruppacher and K. V. Beard, “A wind tunnel

investigation of the internal circulation and shape of water drops falling at terminal velocity in air,” Quart. J. R. Met. Soc., Vol. 96, pp. 247-256, April 1970.

[26] T. Oguchi, “Electromagnetic wave propagation and

scattering in rain and other hydrometeors,” Proc. IEEE, Vol. 71, Issue 9, pp. 1029-1078, Sept. 1983.

[27] L. W. Li, P. S. Kooi, M. S. Leong, and T. S. Yeo,

“On the simplified expression of realistic raindrop shapes,” Microwave & Optical Technol. Letters, Vol. 7, Issue 4, pp. 201–205, March 1994


THE INFLUENCE OF DISDROMETER CHANNELS ON SPECIFIC ATTENUATION DUE TO RAIN OVER MICROWAVE LINKS IN SOUTHERN AFRICA

O. Adetan* and T. Afullo** * Discipline of Electrical, Electronic and Computer Engineering School of Engineering, University of KwaZulu-Natal, Private Bag X54001, Durban 4041, South Africa E-mail:[email protected], or [email protected] ** Discipline of Electrical, Electronic and Computer Engineering, School of Engineering, University of KwaZulu-Natal, Private Bag X54001, Durban 4041, South Africa E-mail: [email protected] Abstract: The influence of raindrop channels on the measured rain rate and specific rainfall attenuation in Durban (29o52'S, 30o 58'E), South Africa for six rain rate values: 1.71, 4.46, 22.97, 64.66, 77.70 and 84.76 mm/hr selected from the rain events that occurred on the 27th December 2008 is analyzed in this work. Consecutive and alternate drop channels are removed to study the influence of raindrop diameters on the overall measured rain rate. The three-parameter gamma and lognormal models are used for the purpose of analysis. In general and for the two models considered, the lower channels (channels 1 to 5) do not alter significantly the overall rain rate and equally produce minimal attenuation. The critical raindrop channels contributing the larger percentage to the overall rain rate are created by D ≥ 0.771mm. The deviations of the measured and estimated rain rates are very small for all the rainfall rates selected. The estimation of the specific rain attenuation at frequency of 19.5 GHz is also discussed. The error estimation using the two models are also analysed. Keywords: Raindrop size distribution, raindrop channels, specific rainfall attenuation, root mean square error.

1. INTRODUCTION The failure and inability of service providers to constantly meet the design target of 99.99 % availability of the line-of-sight (LOS) microwave links has caused several rifts between the operators and the consumers. The non-availability of the links is predominantly due to propagation impairments along the link. These propagation effects include cloud, fog, gas attenuation, rain and atmospheric scintillation. Of these impairments, rain attenuation is the dominant and most disturbing atmospheric impairment for satellite communications systems operating at any frequency above 10 GHz. Rain attenuation may be due to rain at any point along the propagation path [1-4]. According to Safaai-Jazi et al. [5], attenuation below 6 dB is caused mainly by stratiform rain, while attenuation more than 6 dB may largely be attributed to thunderstorms in the frequency range of 12-30 GHz. A good knowledge of the raindrop size distribution (DSD) is essential in the estimation of the rain attenuation at the radio frequency band because it governs all the microwave and rainfall integral relations. It is a known fact that modeling of DSD varies from one region to another. Drop size

distribution modeling in temperate region; characterized by moderate rainfall is well suited with models such as proposed by Marshall and Palmer [6], Laws and Parsons [7] and similar negative exponential models [8]. The globally accepted Ajayi and Olsen [9] and the Ajayi and Adimula [10] lognormal distribution models better describe the drops size distributions modeling in the tropical regions, characterized by heavy rainfall. In this work, we analyze the specific contribution of individual raindrop channels to the overall measured rainfall rate and specific rain attenuation in southern Africa. The estimation of the specific rainfall attenuation at 19.5 GHz is also discussed. Six (6) rain rate values are selected from the rain events that took place in Durban (29o52'S, 30o 58'E) on the 27th of December 2008 using the Joss-Waldvogel RD-80 disdrometer measurement. The rain rates are 1.71, 4.46, 22.97, 64.66, 77.70 and 84.76 mm/hr representing stratiform and convective rainfall types as defined by [1]. The three-parameter gamma and lognormal DSD models are used for the purpose of analysis. The paper is divided into five sections. Related works done in South Africa and other regions is discussed in section 2. Data analysis, DSD models and specific attenuation


are covered in section 3. The results and discussion on the influence (contributions) of individual channels to overall measured rain rate and the estimation of the specific attenuation at 19.5 GHz are discussed in section 4. Finally, some conclusions based on the results are drawn in section 5.

2. OVERVIEW OF RELATED WORKS IN DURBAN AND OTHER REGIONS

In Durban (29o52'S, 30o 58'E), South Africa,

researchers have studied rainfall attenuation and drop size distribution and established the suitability of the lognormal and gamma models for DSD modeling in the region. Odedina and Afullo [11], proposed the lognormal and the modified gamma distribution models as best fit for the measurements of rainfall in Durban. Afullo [12], while studying the rain drop size distribution model for the eastern coast of South Africa established that the optimized lognormal and gamma DSD models compete favorably well with the DSD obtained for same tropical regions using the Biweight kernel estimation technique.

While using the maximum likelihood estimation

(MLE) method, Owolawi [13], also proposed the lognormal model as best fit for DSD modeling in Durban. In their study, Alonge and Afullo [14], considered the DSD for different seasons- summer, autumn, winter and spring; they established that the lognormal model is suitable for summer and autumn; the modified gamma for winter and Weibull distribution is best for the spring season in Durban. The values of R0.01 for the different season were also estimated. Akuon and Afullo [15], derived the rain cell sizes the southern Africa and estimated the overall R0.01 for Durban as 60 mm/hr. The method of moments (MoM) and the maximum likelihood estimation (MLE) techniques were compared to estimate the parameters of the lognormal distribution model by Adetan and Afullo [16], using the DSD measurements in Durban for microwave propagation in South Africa. They concluded that the method of moment provides a better estimate of the DSD parameters in southern Africa with the lognormal model giving the best fit. Adetan and Afullo [17], showed that the three-parameter lognormal DSD gives a better fitting and performance when compared with the gamma distribution model. However, the gamma distribution model is also suitable as the error deviation is minimal at low rain rates.

Recently, Adetan and Afullo [18], analyzed the

critical diameters for rainfall attenuation in southern Africa and concluded that the highest contribution of raindrops diameters to the specific rain attenuation was created by drop diameters not exceeding 2 mm, especially at higher frequencies in Durban. In their findings, the total percentage fraction created by raindrops in the diameter range 0.5 mm ≤ D ≤ 2.5 mm

and 1 mm ≤ D ≤ 3 mm to the total specific attenuation is found to be critical for the overall and seasonal rainfall attenuation at 2.5 GHz-100 GHz in Durban. Similarly, within and outside the tropical region, a good number of researchers have also investigated the particular contributions of certain drop diameters to the specific rain attenuation. These include: Fiser [19], in the Czech Republic, Lakshmi et al. and Lee et al. [20,21], in Singapore, Marzuki et al. [22], in the Equatorial Indonesia and Lam et al. [23], in Malaysia. They concluded that small and medium-size drops contributed more to the rainfall attenuation as frequency increases.

3. DSD MEASUREMENTS, MODELS AND SPECIFIC ATTENUATION

3.1 Drop Size Distribution Measurements

The data used in this work is obtained from the Joss-

Waldvogel (JW) RD-80 disdrometer mounted at the roof top of Electrical, Electronic and Computer Engineering building of the University of KwaZulu Natal South Africa. Although over 80000 data samples have been gathered between 2008 and 2010 since the installation of the instrument in September 2008, only the rain events of 27th December 2008 was used for the purpose of determining the contributions of specific channels to the overall rainfall rate and specific attenuation due to rain in Durban. The J-W RD-80 disdrometer converts the momentum of each of falling raindrop impacting on the sensor’s surface into an electrical pulse of commensurate voltage [24]. The detectable diameter range is divided into 20 channels of varying diameters between 0.359-5.373 mm. The sampling area, A of the disdrometer is 50cm2 (0.005 m2) and sampling time, T is 60s. It is needful to mention that rainfall samples with overall sum of drops less than 10 were ignored from the data samples to compensate for the dead- time errors. The instrument is located at an altitude of 139.7m above sea level. The location site is shielded from abnormal winds and free of undue noise. The raindrop size distribution N (D) in m-3 mm-1 is computed from the disdrometer measurements using (1) as given by [24, 25]:

𝑁𝑁𝑁𝑁 𝐷𝐷𝐷𝐷 = 𝑛𝑛𝑛𝑛

𝑣𝑣𝑣𝑣 𝐷𝐷𝐷𝐷 ∗ 𝐴𝐴𝐴𝐴 ∗ 𝑇𝑇𝑇𝑇 ∗ 𝑑𝑑𝑑𝑑𝐷𝐷𝐷𝐷 (1)

where ni is the number of drops per channel, v (Di) is the Gun-Kinzer [26] terminal velocity of water droplets and dDi is the change in diameter of the channel in mm. Table 1 shows the threshold of the drop size channels and the measured drops for the selected rain rates. The asterisks (*) in the table represent lack of rain drops in the channel. This lack of raindrops appears to be noticeable at lower channels as the rain rate increases. This will be discussed in more details in section 4.


The rainfall rate, R (mm/hr) is calculated from the disdrometer data using (2) given by [25]:

𝑅𝑅𝑅𝑅 = 𝜋𝜋𝜋𝜋6𝐷𝐷𝐷𝐷 𝑛𝑛𝑛𝑛

3600𝐴𝐴𝐴𝐴.𝑇𝑇𝑇𝑇

(2)

A well-known integral equation of the rain rate R as computed from the DSD model is given by [26] with the associated discrete form as (3):

𝑅𝑅𝑅𝑅 = 6𝜋𝜋𝜋𝜋×10 𝐷𝐷𝐷𝐷 . 𝑣𝑣𝑣𝑣 𝐷𝐷𝐷𝐷 .𝑁𝑁𝑁𝑁 𝐷𝐷𝐷𝐷 𝑑𝑑𝑑𝑑𝐷𝐷𝐷𝐷

= 1.88496×10 𝐷𝐷𝐷𝐷 . 𝑣𝑣𝑣𝑣(𝐷𝐷𝐷𝐷 ).𝑁𝑁𝑁𝑁(𝐷𝐷𝐷𝐷 )𝑑𝑑𝑑𝑑𝐷𝐷𝐷𝐷

(3)

Table 1: Drop size channels and measured drops from disdrometer measurements for the selected rain rates.

Rain rate (mm/hr)/Time ((min.)

20:53 20:57 21:01 21:10 21:05 21:07

Channels i

Mean Diameter

Di 1.71 4.46 22.97 64.66 77.70 84.76

1 0.359 1 11 20 * * * 2 0.455 4 10 47 1 * * 3 0.551 * 10 45 1 5 4 4 0.656 9 29 41 10 19 3 5 0.771 9 47 41 14 19 15 6 0.913 6 41 44 56 60 47 7 1.112 16 40 75 124 123 143 8 1.331 15 61 64 120 118 138 9 1.506 5 36 44 114 98 116

10 1.656 5 26 34 99 86 148 11 1.912 10 20 108 171 168 225 12 2.259 8 4 63 170 124 144 13 2.584 * 1 29 94 87 103 14 2.869 * * 16 61 60 49 15 3.198 * * 11 53 51 44 16 3.544 * * 4 10 34 34 17 3.916 * * 2 9 28 24 18 4.35 * * * * 9 10 19 4.859 * * * * * 4 20 5.373 * * * * * * Sum of drops 88 336 688 1107 1089 1251

This computation is necessary in order to compare the measured rain rates and the observed contribution of individual channels on the overall rain rate and on the modeled N (D).

3.2.1 Gamma DSD Model

The three parameter gamma DSD model is expressed by [8] as:

𝑁𝑁𝑁𝑁(𝐷𝐷𝐷𝐷) = 𝑁𝑁𝑁𝑁 𝐷𝐷𝐷𝐷 exp(−Ʌ𝐷𝐷𝐷𝐷 ) [𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ] (4)

where 𝜇𝜇𝜇𝜇 is the shape parameter, No (m-3 mm-1-µ)

is the scale parameter in mm-1. The distribution is

particularly useful in tropical regions where the exponential distribution has been found on numerous occasions to be inadequate (see, for example [8]). The shape parameter is assumed to be 2 while No and Ʌ are estimated as [17]:

𝑁𝑁𝑁𝑁 = 78259𝑅𝑅𝑅𝑅 .

Ʌ = 6.3209 𝑅𝑅𝑅𝑅 .

𝜇𝜇𝜇𝜇 = 2 (5)

3.2.2 Lognormal DSD Model The lognormal distribution model was proposed by [9,10] was primarily for modeling DSD in the tropical region and expressed in the form of (6) as:

𝑁𝑁𝑁𝑁 𝐷𝐷𝐷𝐷 = 𝑁𝑁𝑁𝑁

𝜎𝜎𝜎𝜎𝜎𝜎𝜎𝜎 2𝜋𝜋𝜋𝜋𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 −0.5

ln 𝐷𝐷𝐷𝐷 − 𝜇𝜇𝜇𝜇𝜎𝜎𝜎𝜎

(6)

where µ is the mean of ln 𝐷𝐷𝐷𝐷 ,σ is the standard deviation which determines the width of the distribution and 𝑁𝑁𝑁𝑁 depends on climates, geographical location of measurements and rain types. The lognormal distribution model is more advantageous in the sense that each of its parameters have a clear physical significance. Moreover, the parameters (NT, µ and σ2) are linearly related to the moments of drop size distribution. Therefore, it is considered adequate for describing the DSD in this region [27]. The three parameters in (6) are related to R as:

𝑁𝑁𝑁𝑁 = 𝑎𝑎𝑎𝑎 𝑅𝑅𝑅𝑅𝜇𝜇𝜇𝜇 = 𝐴𝐴𝐴𝐴 + 𝐵𝐵𝐵𝐵 ln𝑅𝑅𝑅𝑅𝜎𝜎𝜎𝜎 = 𝐴𝐴𝐴𝐴 + 𝐵𝐵𝐵𝐵 ln𝑅𝑅𝑅𝑅

(7)

where 𝑎𝑎𝑎𝑎 ,𝑏𝑏𝑏𝑏 , 𝐴𝐴𝐴𝐴𝝁𝝁𝝁𝝁, 𝐵𝐵𝐵𝐵𝝁𝝁𝝁𝝁, 𝐴𝐴𝐴𝐴 and 𝐵𝐵𝐵𝐵 are coefficients of moment regression determined using the least squares method of regression technique. These parameters have been estimated for Durban using the method of moments in [17] and represented in (8) as:

𝑁𝑁𝑁𝑁 = 268.07𝑅𝑅𝑅𝑅 .

𝜇𝜇𝜇𝜇 = −0.3104 + 0.1331 ln𝑅𝑅𝑅𝑅𝜎𝜎𝜎𝜎 = 0.0738 + 0.0099 ln𝑅𝑅𝑅𝑅

(8)

3.3 Specific Rain Attenuation

The estimation or prediction of rainfall attenuation in both satellite and terrestrial communication applications is very necessary. The specific rain attenuation 𝛾𝛾𝛾𝛾 (dB/km) due to rain, R (mm/hr.) is given by [28] as (9):


𝛾𝛾𝛾𝛾 = 4.343×10−3 𝑄𝑄𝑄𝑄𝑡𝑡𝑡𝑡 𝐷𝐷𝐷𝐷 .𝑁𝑁𝑁𝑁 𝐷𝐷𝐷𝐷 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 (9)

∞

0

where 𝑄𝑄𝑄𝑄 (𝐷𝐷𝐷𝐷) is the extinction cross section which depends on the raindrop diameter D, the wavelength 𝜆𝜆𝜆𝜆, and the complex refractive index of water drop m, 𝑁𝑁𝑁𝑁 𝐷𝐷𝐷𝐷 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 is the number of raindrops per unit volume where the diameter of the equivalent volume spherical drop is between D and D+dD. The extinction cross section 𝑄𝑄𝑄𝑄 (𝐷𝐷𝐷𝐷) can further be expressed by [29, 30] as:

𝑄𝑄𝑄𝑄 (𝐷𝐷𝐷𝐷) = 𝜆𝜆𝜆𝜆2𝜋𝜋𝜋𝜋

(2𝑛𝑛𝑛𝑛 + 1)Re[𝑎𝑎𝑎𝑎 + 𝑏𝑏𝑏𝑏 ] (10)

where an and bn are the Mie scattering coefficients. The extinction cross section is calculated by applying the classical scattering theory of Mie for a plane wave impinging upon a spherical absorbing particle. The Mie scattering theory is applied under the assumption that each spherical raindrop illuminated by a plane wave is uniformly distributed in a rain filled medium. Similarly, it is assumed that the distance between each drop is large enough to avoid any interaction between them. The extinction cross section, 𝑄𝑄𝑄𝑄 (𝐷𝐷𝐷𝐷) provided by [11] as a frequency-dependent, power law function with coefficients κ and α is given in (11) and adopted in this work. It should be mentioned that the absorption and scattering can be modelled by different methods depending on the frequency of the signal and the shape of the raindrop. Examples of such methods include the use of Rayleigh theory, Mie theory and the T-matrix technique. The Rayleigh approximation applies to frequencies lower than the frequency considered in this work. The Mie theory assumes that the shape of the shape of the raindrop is spherical, which is not the case for raindrops above 2 mm in diameter. The T-matrix can model axisymmetric particles, which improves the representation of a raindrop and hence increases the accuracy of attenuation estimates.

𝑄𝑄𝑄𝑄 𝐷𝐷𝐷𝐷 = κ𝐷𝐷𝐷𝐷2

(11) where κ and α are the coefficients that depend on a number of parameters such as temperature, raindrop size distribution, raindrop shape, frequency of transmission and polarization to a certainty[31]. The Mätzler’s MATLAB [32] functions are used to compute κ and α values at f = 19.5 GHz. The computed values of κ and α at f = 19.5GHz are 1.6169 and 4.2104 respectively with the complex refractive index of water drop, m = 6.7332+j2.7509 of the scattering amplitudes.

However, Olsen et al. [33] further approximated (9) in a power law as a function of rain rate R, given as:

𝛾𝛾𝛾𝛾 = κ𝑅𝑅𝑅𝑅 (12)

In this work, we analyze the specific contribution of individual raindrop channels to the overall measured rainfall rate and specific rainfall attenuation in southern Africa. The Mie scattering functions at 20oC is employed to calculate the scattering functions at frequency f = 19.5 GHz with the assumption that the raindrops are spherical in shape. Each particle is assumed to be a homogeneous sphere and the incident field is assumed to be a plane wave. The three parameter lognormal and gamma DSD models as determined in [17] for Durban are used to characterize the measured drop size distribution N (D). The three-parameter model ensures high representability of the experimented measurements.

4. RESULTS AND ANALYSIS

4.1 Rain rate, drop size distributions N(D) and raindrop diameters

The drops size distribution is an essential parameter for the evaluation of the rainfall attenuation at microwave and millimeter wave frequencies. Fig. 1 shows the measured drop size distribution, N (D) from the disdrometer data using (1) and the drop diameters for the selected rain rates. The drop diameter increases as the rain rates increases. Fig. 2 shows the modeled and measured N (D) with respect to the drop diameters for the gamma and lognormal DSD models using the parameters in (5) and (8) and rain rate of 84.76 mm/hr. Although, the two models overestimate the measured DSD at the lower and larger diameters, the gamma model overestimates more at lower diameters than the lognormal model. This may be due to the parameter estimation.

Figure 1: Measured raindrop size distributions for the selected rain rate values [using (1)].

1

10

100

1000

10000

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Mea

sure

d D

SD [

m-3

mm

-1]

Raindrop diameters [mm]

R =1.71 mm/hr R = 4.46 mm/hr R = 22.97 mm/hr R = 64.66 mm/hr R = 77.70 mm/hr R = 84.76 mm/hr


Figure 2: Comparison of the measured and modeled N (D) with drop diameters.

In Fig. 3, a comparison of the modeled values of the rain rates with the actual disdrometer measured values is shown for gamma and lognormal models. A good agreement is observed between the measured and estimated values of the rain rates.

(a)

(b)

Figure 3: Comparison between measured and modeled rain rate values for (a) gamma and (b) lognormal models.

4.2 Estimation of the specific rainfall attenuation and raindrop channels

The specific rainfall attenuation is calculated using (4, 6, 9, and 11). The computed values of κ and α are used. There is an agreement between the two models with respect to rain rate as shown in Fig. 4. The contributions of individual channels and removal of consecutive channels to the specific rain attenuation is illustrated in Tables 2 to 4 for the two models at f = 19.5 GHz. It can be observed that the roles of smaller raindrop diameters to the specific attenuation are insignificant (negligible). Their removals produce least attenuation when compared to the larger diameters (Tables 3 and 4).

4.3 Contributions of raindrop channels to overall rain rates

The range of raindrop channels is studied by removing the individual channels and the effect on the measured rain rate observed. Fig. 5 shows the difference between the measured rain rates with the individual channels containing raindrops removed. Firstly, the individual channel with raindrops is removed consecutively and the effect on the overall rain rate is observed. That is, channel 1 is removed, and then channels 1 and 2, channels 1 to 3 and so on. Thereafter, the channels are removed (with or without raindrops) in an alternate manner such that channel 1 is removed first, then channels 1 and 20, channels 1,2 and 20, then channels 1,2, 19 and 20 and so on and the effect on the overall rain rate is observed.

In general and for the two models considered, the removal of smaller diameters (channels 1 to 5) does not alter significantly the total rain rates. As the number of channels removed increases, the deviation tends to increase sharply as observed from Fig.4 for all the rain rates. The critical raindrop channels lies at the center of the disdrometer channel for D ≥ 0.771mm. Similarly, as observed from Table 1, the largest raindrops for the lower rain rate are observed in channels 7 and 8 and as the rain rate increases, there seems to be no raindrops in the first 2 channels. At high rain rates, the largest (maximum) raindrops occur in channel 11 (1.912mm). This further confirms that the roles of the channels in the diameter range D ≥ 0.771mm are critical in the overall rain rate. Hence, raindrops size distribution and the drop diameters (channels) are significant to the overall rain rate estimation.

1

10

100

1000

10000

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

N(D

) [m

-3 m

m-1

]

Raindrop diameter [mm]

Measured data gamma model lognormal model

y = 0.9052x R² = 0.9996

0

20

40

60

80

100

0 20 40 60 80 100

Mod

eled

val

ues [

mm

/hr]

Measured values with disdrometer [mm/hr]

Rainfall rate [mm/hr]

y = 1.1127x R² = 0.99966

0

20

40

60

80

100

0 20 40 60 80 100

Mod

eled

val

ues [

mm

/hr]

Measured values with disdrometer [mm/hr]



4.4 Root Mean Square Error (RMSE)

The root mean square (RMS) error used for assessing the deviation of the estimated rain rate from the measured is given as (13):

𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 =1𝑛𝑛𝑛𝑛

𝐹𝐹𝐹𝐹(𝑋𝑋𝑋𝑋 ) − 𝐹𝐹𝐹𝐹(𝑋𝑋𝑋𝑋 ) (13)

where 𝐹𝐹𝐹𝐹(𝑋𝑋𝑋𝑋 ) is the estimated rain rate values obtained from the model using (3), 𝐹𝐹𝐹𝐹(𝑋𝑋𝑋𝑋 ) is the measured rain rate using (2) and n is the number of channels containing raindrops. It was observed that the values of root mean square deviation (RMS) error using the two models are low even at high rain rates. This is shown in Table 5. They are slightly different from each other and this further

confirms the suitability of these models for DSD modeling in Durban.

Table 2: Specific rain attenuation (dB/km) with channels (containing raindrops) removed at f =19.5GHz.

Gamma Model(G-M) Lognormal Model (Ln-M) Channel removed Channel removed

R (mm/hr)

With no channels removed

1

2

3

4

5

With no channels Removed

1

2

3

4

5

1.71 0.09 0.09 0.09 0.09 0.08 0.08 0.09 0.09 0.09 * 0.08 0.08 4.46 0.26 0.26 0.26 0.26 0.25 0.25 0.26 0.26 0.26 0.26 0.25 0.24

22.97 1.46 1.46 1.46 1.46 1.45 1.44 1.44 1.44 1.44 1.44 1.44 1.43 64.66 4.35 * 4.35 4.35 4.34 4.32 4.29 * 4.29 4.29 4.29 4.28 77.70 5.28 * * 5.27 5.26 5.25 5.21 * * 5.21 5.21 5.20 84.76 5.78 * * 5.78 5.77 5.75 5.71 * * 5.71 5.71 5.70

Table 3: Specification rain attenuation (dB/km) with consecutive channels removed starting with lower channels at f =19.5 GHz.

Gamma Model(G-M) Lognormal Model (Ln-M) Channels removed Channels removed

R (mm/hr)

With no channels removed

1 1-2 1-3 1-4 1-5 With no channels removed

1 1-2 1-3 1-4 1-5

1.71 0.09 0.09 0.09 0.89 0.08 0.07 0.09 0.09 0.09 0.09 0.08 0.07 4.46 0.26 0.26 0.26 0.25 0.24 0.23 0.26 0.26 0.26 0.25 0.25 0.24

22.97 1.46 1.46 1.46 1.45 1.44 1.42 1.44 1.44 1.44 1.44 1.44 1.43 64.66 4.35 4.35 4.35 4.34 4.33 4.30 4.29 4.29 4.29 4.29 4.29 4.28 77.70 5.28 5.28 5.28 5.27 5.25 5.22 5.21 5.21 5.21 5.21 5.21 5.20 84.76 5.78 5.78 5.78 5.77 5.76 5.72 5.71 5.71 5.71 5.71 5.70 5.69

Table 4: Specific rain attenuation (dB/km) with consecutive channels removed starting with larger channels using the lognormal model at f =19.5GHz

Channels removed R

(mm/hr) With no channels

removed 20

19-20

18-20

17-20

16-20

1.71 0.09 0.09 0.09 0.09 0.08 0.09 4.46 0.26 0.26 0.26 0.26 0.26 0.26

22.97 1.44 1.44 1.44 1.43 1.43 1.42 64.66 4.29 4.29 4.26 4.22 4.12 3.99 77.70 5.21 5.20 5.16 5.09 4.95 4.76 84.76 5.71 5.69 5.65 5.56 5.39 5.17


Figure 4: Specific rain attenuation at f = 19.5GHz for the models.

(a)

(b)

Figure 5: Normalized deviation for channels removal (a) Gamma (b) lognormal models

0

1

2

3

4

5

6

7

0 20 40 60 80 100 Sp

ecifi

c at

tenu

atio

n [d

B/km

]


G-M

Ln-M

0

20

40

60

80

100

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Nor

mal

ized

dev

iatio

n [%

]

Chnannels removed

R =1.71 mm/hr R = 4.46 mm/hr R = 22.97 mm/hr R = 64.66 mm/hr R = 77.70 mm/hr R = 84.76 mm/hr

0

20

40

60

80

100

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Nor

mal

ized

dev

iatio

n [%

]

Channels removed

R = 1.71 mm/hr R = 4.46 mm/hr R = 22.97 mm/hr R = 64.66 mm/hr R = 77.70 mm/hr R = 84.76 mm/hr


Table 5: Estimation of root mean square deviation between measured and modeled rain rate.

R (mm/hr.) Gamma Lognormal 1.71 0.01 0.02 4.46 0.01 0.02

22.97 0.29 0.24 64.66 1.53 1.42 77.70 2.00 1.88 84.76 2.21 2.08

5. CONCLUSION

The paper presents the contribution of individual raindrop channels to the overall rain rate and specific rainfall attenuation in southern Africa. The six measured rain rate values recorded per minute using the J-W RD-80 disdrometer was used in analysis. These rain rate values also represent the stratiform and convective types of rainfall. Smaller channels were found to contribute little to the overall measured rain rates and produce least rain attenuation with the removal of the bins. The amount of attenuation produced with the removal of the larger diameters is significant. A comparative analysis of the error deviation shows that the lognormal DSD model gives a better performance as the rain rate increases from stratiform to convective. The removal of lower channels does not alter the attenuation and can be ignored while estimating the specific rain attenuation. However, larger diameters produce significant effect on the specific rain attenuation especially at very high rain rates. We conclude therefore that the smaller channels (1-4) can be ignored while estimating the rain rate and specific attenuation as they do not alter significantly the estimated values. The major contribution to the overall measured rain rate is formed by diameter D ≥ 0.771mm. These critical diameters and attenuation statistics are essential to properly design adequate fade margin levels and for the purpose of link budget design by service providers and system engineers in the region.

6. REFERENCES

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APPENDIX

The normalized deviation of Figs. 5 (in %) as used in [27] is adopted in this work. It is calculated using the true measured rain rate and the rain rate with some of the bins (channels) removed. The equation is given as:

𝑁𝑁𝑁𝑁 𝐷𝐷𝐷𝐷 % = 𝑅𝑅𝑅𝑅 − 𝑅𝑅𝑅𝑅

𝑅𝑅𝑅𝑅×100 (1)

where 𝑅𝑅𝑅𝑅 is obtained from (2) and 𝑅𝑅𝑅𝑅 is also obtained from (2) but with some of the bins removed.


NOTES


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