2-Norm Condition Number Based SwitchingAlgorithm in MIMO OFDM Systems

Yosra Mlayeh, Fethi Tlili and Adel GhazelCIRTA’COM laboratory

SUP’COM, 2088 Cite technologique des communicationsAriana, Tunisia

Email: y.mlayeh@ttnet.tn, (fethi.tlili,adel.ghazel)@supcom.rnu.tn

Amor NafkhaSCEE-IETR UMR CNRS 6164

SUPELEC, 5 Av. de la Boulaie CS 47601, F-35576Cesson-Sevigne cedex, FranceEmail: amor.nafkha@supelec.fr

Abstract—Performances and capacity of MIMO-OFDM sys-tems are enhanced with adaptive MIMO techniques such asDemmel condition number based switching algorithm. In thispaper we show that the complexity of this algorithm can bereduced by using the 2-norm condition number of the channelmatrix as the selection metric. Simulation results show thatusing this selection criterion, the switching algorithm offers thesame performances obtained when using the Demmel conditionnumber.

Index Terms—MIMO, OFDM, Transmit Diversity, SpatialMultiplexing, Matrix Condition Number.

I. INTRODUCTION

In the last decade, multiple input multiple output (MIMO)communication systems have been extensively studied. Theyhave many promising features such as array gain, spatialmultiplexing gain, diversity gain, interference reduction, andimproved capacity as compared to a single-input single-output(SISO) systems [1]. To cope with severe channel conditions,orthogonal frequency division multiplexing (OFDM) has be-come a popular technique for transmission of signals overwireless channels. Channel equalization is simplified becauseOFDM may be viewed as using many slowly-modulated nar-rowband signals rather than one rapidly-modulated widebandsignal [2].

MIMO-OFDM is currently being considered for a num-ber of developing wireless standards such as digital audiobroadcasting (DAB), digital video broadcasting (DVB-T), theIEEE 802.16e, and has been already used in beyond 3G(B3G) and 4G wireless communications [3]; consequently, thestudy of MIMO-OFDM in realistic environments is of greatimportance.

Multiple antennas can be used to increase data rates throughspatial multiplexing or to improve performance through diver-sity. Diversity techniques use two or more antennas in thetransmitter and/or the receiver side to improve the wirelesslink quality and are not designed to increase the peak data rateof the system [4]. An effective and simple transmit diversityscheme named space time block coding (STBC) was proposedby Alamouti [5]. It encodes the signal through two transmitterantennas and in time to enhance significantly the BER whilepreserving a unit code rate. STBC has been generalized formore than two antennas in [6]. On the other hand, spatial

multiplexing is obtained by exploiting the structure of thechannel gain matrix to obtain independent signaling paths thatcan be used to send independent data to offer higher peakthroughput [7,8].

Moreover, adaptive algorithms, where the transmitterblock’s specifications are adjusted according to the channelstate, are employed in order to make a successful switch-ing between transmit diversity (TD) and spatial multiplexing(SM) and to enhance the spectral efficiency. Many worksare interested to enhance the capacity gains using adapta-tion algorithms, where the selection criterion is based onthe knowledge of the Signal to Noise Ratio (SNR) and thedeterminant of MIMO channel matrix H [9-10]. In order tolimit the quantity of feedback and the number of consideredparameters in switching decision, other works have studiedthe adaptation problem in order to enhance the performancesof MIMO systems in terms of Bit Error Rate (BER) [11-13];The Demmel condition number of the instantaneous channelmatrix was proposed as a parameter based on the minimumEuclidean distance of constellations in order to characterize thesuitability of a given instantaneous channel matrix for spatialmultiplexing compared to transmit Diversity.

The aim of this work is to reduce the complexity ofDemmel condition number based switching algorithm. In thispaper, we show that the regular (2-norm) condition numberof the channel matrix can be used as the metric of selectioninstead of the Demmel condition number. This processingreduces the complexity of the algorithm will keeping theperformances offered by the Demmel condition number basedswitching. In order to keep the same spectral efficiency of thetwo schemes (SM/TD), we used QPSK modulation for thespatial multiplexing and 16-QAM modulation for the transmitdiversity.

The rest of this paper is organized as follows: Section IIpresents the OFDM-based MIMO communication system andan overview of the switching algorithm principle. Then, in Sec-tion III, we define the 2-norm condition number of the channelmatrix and we establish the relationship between the 2-normand the Demmel condition number. In Section IV, we gives thecomputation complexity of the proposed switching algorithm.Simulation results In Section V illustrate the performances interms of Bit Error Rate (BER) of the switching algorithm using

Fig. 1. Block diagram of WiMAX physical Layer

the two selection metrics. Conclusion and perspectives of thework are outlined in Section VI.

II. SYSTEM OVERVIEW

Combining MIMO processing with OFDM is the key en-abling technology for several current and future broadbandwireless access systems and standards (LTE, WiMAX, etc).

Let us assume a MIMO-OFDM system with 𝑁𝑡 transmitantennas and 𝑁𝑟 receive antennas as depicted in Figure 1. Dif-ferent blocs are defined following IEEE 802.16e specifications[3]. First, data are encoded using convolutional codes. Afterbitwise interleaving, the remaining R=𝑙𝑜𝑔2(M) bits are thenmapped to a symbol using ordinary 𝑀 -QAM constellation.The generated symbols are passed through MIMO encoder.Finally, for each encoded signal, an inverse fast fourier trans-form (IFFT) of size 𝐾 is performed, and the cyclic prefix (CP)is added to mitigate the residual intersymbol interference (ISI)due to previous OFDM symbol.

The MIMO module can perform spatial multiplexing, trans-mit diversity, or beamforming to achieve a constant high datarate and a small outage probability. In this paper, the switchingalgorithm is used to select between spatial multiplexing andtransmit diversity. In the next subsection, we briefly describethe TD and SM MIMO techniques in the OFDM context.

A. Transmit diversity

The main idea behind transmit diversity techniques is toproduce different replicas of the transmitted signal to thereceiver. If these replicas are sent over the propagation channelsuch that their statistics are independent, when one of themfades, it is less likely that the other copies of the transmittedsignal will be in deep fade simultaneously. Thanks to thisredundancy, the receiver can decode the transmitted signaleven in fading conditions, as long as they all do not fadesimultaneously. Receive diversity may be implemented viatwo rather different combining methods: the first method isthe selection combining when the combiner selects the branchwith the highest SNR among the 𝑁𝑟 received signals, whichis then used for detection. The second method is the gaincombining when the signal used for detection is a linearcombination of all branches. Note that we assume in the

following that the receiver is able to acquire perfect knowledgeof the channel.

A significant difference with receiver diversity at that thetransmitter might not have the knowledge of the MISO chan-nel. In this work, MIMO Diversity employs the Alamouti code[5] at the transmitter and the selective combining diversityin the receiver. In OFDM systems, this technique could beapplied per subcarrier or per OFDM symbol. We remind thatsymbols are derived from a constellation with 𝑅 bits persymbol.

B. Spatial multiplexing

When employed at both sides of the link, multiple anten-nas may also be used to increase the transmission rate (orthe capacity) of communication systems. The idea behindspatial multiplexing is that multiple independent streams canbe transmitted in parallel over multiple antennas and canbe separated at the receiver using multiple reception chainsthrough appropriate signal processing. This can be done aslong as multipath channels seen by the various antennas aresufficiently decorrelated, as would be the case in a scattering-rich environment. Spatial multiplexing provides data rate andcapacity gains proportional to the number of antennas used.

At the receiver side, SM streams can be demodulatedusing optimal detector such maximum likelihood detectors,sphere decoding or sub-optimal detector such as zero forcing(ZF),minimum mean square error (MMSE) detectors [14]. TheZF detector inverts the channel matrix to detect the transmittedsymbols. Even though it suffers from poor performances atlow SNR, it is employed in this work as the detection schemebecause it has a very small complexity that does not depend onthe modulation type. The SM offers increasing throughput ata given SNR when the channels matrix are well conditioned.The base station transmits independent data streams from eachtransmit antenna to the subscriber’s stations. Thus, duringevery symbol period, the encoder multiplexes 𝑁𝑡 OFDMsymbols. As well as the TD technique, the spatial processingcould be done per subcarrier inside one OFDM symbol. Toensure that a rate of 𝑅 bits per codeword is maintained, thesymbols are derived from a constellation with 𝑅/𝑁𝑡 bits persymbol.

III. SWITCHING ALGORITHM BASED ON DEMMEL

CONDITION NUMBER

The switching algorithm proposed in [11] is based onthe Demmel condition number 𝐾𝑑 criterion. This metric canprovide information about the invertibility of the channelwhich provides knowledge its ’goodness’ for use in spatialmultiplexing or transmit diversity operational modes [15].

Let us denote by H𝑛 the MIMO channel matrix of the 𝑛𝑡ℎ

subcarrier, then its Demmel condition number is expressedsuch that:

𝐾𝑑(H𝑛) =∥H𝑛∥𝐹𝜆𝑚𝑖𝑛

(1)

The notation ∥.∥𝐹 refers to the Frobenius norm. The de-nominator of (1) is the minimum singular value of the 𝑛𝑡ℎ

subcarrier channel matrix. The Demmel condition numbermeasures how ill posed, that is, how invertible, a given matrixis. Physically, in [11] it was shown that this metric pro-vides a comparison between the minimum signal constellationdistance needed to support spatial multiplexing and spatialdiversity modes of operation for a given channel.

Considering the expression of the Demmel condition num-ber given by equation (1), one sufficient condition that multi-plexing will be better than diversity for a given channel matrixis given by [11]:

𝐾𝑑(H𝑛) ≤ 𝑑𝑚,𝑆𝑀

𝑑𝑚,𝑇𝐷(2)

Where 𝑑𝑚,𝑆𝑀 and 𝑑𝑚,𝑇𝐷 are the square-root minimum Eu-clidean distance of spatial multiplexing and transmit diversityconstellations at the transmitter side, respectively. They aredefined as:

𝑑2𝑚,𝑆𝑀 = min𝑠,𝑐∈𝜉, 𝑠 ∕=𝑐

∥𝑠− 𝑐∥22𝑑2𝑚,𝑇𝐷 = min

𝑠,𝑐∈𝜉, 𝑠 ∕=𝑐∥G(𝑠− 𝑐)∥22

Where 𝑠, 𝑐 are two different uncoded code-words, 𝜉 is theconstellation set, and G is the space-time coded matrix.

IV. PROPOSED SWITCHING ALGORITHM

To choose the suitable technique, the Demmel conditionnumber has to be calculated for every subcarrier. Hence,for OFDM systems with high number of subcarriers, thisprocessing needs high computational capability. Aiming toreduce this complexity, we show that the 2-norm conditionnumber could be used as the selection metric of switchingalgorithm instead of the Demmel condition number. Aftergiving the definition of the regular condition number weaddress the new adapted selection threshold.

A. 2-norm Condition number

Let A be an 𝑚 × 𝑛 matrix. If 𝜆𝑚𝑎𝑥 = 𝜆1 ≥ 𝜆2 ≥ ... ≥𝜆𝑟 = 𝜆𝑚𝑖𝑛 are the 𝑟, 𝑟 = 𝑚𝑖𝑛(𝑚,𝑛) , singular values of A.Then the 2-norm condition number of A is [16]:

𝐾2(A) =∥A∥2

∥A−1∥2=

𝜆𝑚𝑎𝑥

𝜆𝑚𝑖𝑛(3)

If the condition number is large, the matrix tends to be ill-conditioned with the convention that 𝐾2 for a singular matrix.An ill-conditioned matrix has very bad numerical stability inmatrix inverse operations.

B. Threshold selection using a 2-norm condition number

First, we remind the methodology adopted by [14] todefine this criterion based on a comparison of the minimumEuclidean distance calculated at the receiver for each MIMOtechnique.

For transmit diversity, the squared minimum Euclideandistance in the receiver 𝑙𝑚,𝑇𝐷 for the full rate space-time codesis upper bounded by the following expression [11].

𝑙2𝑚,𝑇𝐷 ≤ ∥H𝑛∥2𝐹 𝑑2𝑚,𝑇𝐷

𝑁𝑡(4)

For spatial multiplexing, the squared minimum Euclideandistance 𝑙𝑚,𝑆𝑀 is bounded by the following expression:

𝜆2𝑚𝑖𝑛𝑑

2𝑚,𝑆𝑀

𝑁𝑡≤ 𝑙2𝑚,𝑆𝑀 ≤ 𝜆2

𝑚𝑎𝑥𝑑2𝑚,𝑆𝑀

𝑁𝑡(5)

So to guarantee that the minimum Euclidean distance in thereceiver side of the spatial multiplexing is greater than thatof the MIMO diversity, it is sufficient to verify the conditionexpressed by equation (2).

Using the definition of the Frobins norm, and under thecondition that 𝑁𝑡 ≤ 𝑁𝑟, the expression (4) can be written as:

𝑙2𝑚,𝑇𝐷 ≤∑𝑁𝑡

𝑘=1 𝜆2𝑘𝑑

2𝑚,𝑇𝐷

𝑁𝑡≤ (𝜆2

𝑚𝑖𝑛 + (𝑁𝑡 − 1)𝜆2𝑚𝑎𝑥)𝑑

2𝑚,𝑇𝐷

𝑁𝑡(6)

As it has been mentioned before, in order to select the spa-tial multiplexing technique, we need to have 𝑙2𝑚,𝑆𝑀 ≥ 𝑙2𝑚,𝑇𝐷.To satisfy this condition, we need to verify the followingequation:

(𝜆2𝑚𝑖𝑛 + (𝑁𝑡 − 1)𝜆2

𝑚𝑎𝑥)𝑑2𝑚,𝑇𝐷

𝑁𝑡≤ 𝜆2

𝑚𝑖𝑛𝑑2𝑚,𝑆𝑀

𝑁𝑡

1 + (𝑁𝑡 − 1)𝜆2𝑚𝑎𝑥

𝜆2𝑚𝑖𝑛

≤ 𝑑2𝑚,𝑆𝑀

𝑑2𝑚,𝑇𝐷

(7)

Then, the spatial multiplexing will be selected if the 2-norm condition number of the matrix verify the followingexpression:

𝐾2(H𝑛) ≤√

1

𝑁𝑡 − 1(𝑑2𝑚,𝑆𝑀

𝑑2𝑚,𝑇𝐷

− 1) (8)

V. COMPUTATION COMPLEXITY

The Frobenius norm, used to calculate the Demmel con-dition number, can be exactly evaluated using the SingularValue Decomposition (SVD) of the channel matrix. Moreover,the SVD decomposition is an iterative algorithm that is usedto calculate all eigenvalues. By using this approach, thecomputation complexity of the switching algorithm is high.Our proposed algorithm only need the highest and the lowesteigenvalues of the channel matrix. The basic idea to use ouralgorithm is to use parallel Lanczos method to evaluate theextreme eigenvalues (𝜆𝑚𝑎𝑥, 𝜆𝑚𝑖𝑛) of the channel matrix. TheLanczos method is well suited to the task of computing a

Fig. 2. Performances of Demmel condition number based switchingalgorithm

few eigenvalues and eigenvectors of a large 𝑛 × 𝑛 matrix.The wanted eigenvalues may be at either, or both, ends of thespectrum of the given matrix.

VI. SIMULATION CONTEXT & PERFORMANCES

EVALUATION

In order to evaluate the performances of the proposedalgorithm, we reproduce the same physical layer as specifiedin 802.16e standard. Simulations setting are described in tableI.

Channel bandwidth 5 MhzSub-carriers spacing 10.94 KhzNumber of sub-carriers 512Cyclic prefix 22.8 𝜇 sOFDM symbol period 91.4 𝜇 sModulation for SM / TD 4-QAM / 16-QAMPower delay profile SUI4 channel modelSampling frequency 5.6 Mhz

TABLE ISIMULATIONS SETTING

Moreover, to ensure a fixed bit rate, the switching moduleswitches between the Alamouti MIMO diversity scheme with16-QAM modulation and spatial multiplexing with 4-QAMmodulation. In each channel realization, we assume having aconstant Rayleigh fading channel for every block of 4 OFDMsymbols. Simulations are done such as Bit Error Rates (BER)is averaged for every 300 channel realizations. The SUI4channel model is adopted to generate channels used in thesimulations.

Curves of Figure 2 illustrate the performances of theswitching technique, using Demmel condition number as theselection metric, in comparison with spatial multiplexing andtransmit diversity. Results show a signal to noise ratio gain of2 dB at BER=10−3.

Curves in Figure 3 illustrate the performances of switchingalgorithm using the 2-norm condition number as the metric ofselection.

Fig. 3. Performances of 2-norm condition number based switching algorithm

Results show that as well as the Demmel condition number,the 2-norm condition number could be used as a selectionmetric in the switching algorithm. In fact The performancesoffered by the switching algorithm are kept wile reducing thecomplexity of the algorithm.

VII. CONCLUSION

The main contribution of this work is the reduction ofcomplexity of the switching algorithm based on Demmelcondition number criterion. We developed a new switchingalgorithm with reduced complexity. This algorithm use the 2-norm condition number as the selection metric. Simulations re-sults show that the 2-norm condition numbed based switchingalgorithm offer better performances than the transmit diversityor the spatial multiplexing considered separately.

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