Repetition Code Subblocking for Partial Transmit Sequence OFDM

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 9, NO. 3, MARCH 2010 1

PAPR Reduction of OFDM UsingPTS and Error-Correcting Code Subblocking

Abolfazl Ghassemi, Student Member, IEEE, and T. Aaron Gulliver, Senior Member, IEEE

Abstract—Partial transmit sequence (PTS) is a proven tech-nique to reduce the peak-to-average power ratio (PAPR) inorthogonal frequency division multiplexing (OFDM) systems. Itachieves considerable PAPR reduction without distortion, but thehigh computational complexity of multiple Fourier transforms isa problem in practical systems. To address the complexity, signalsat the middle stages of an 𝑁 -point radix FFT using decimationin frequency (DIF) are employed for PTS subblocking. Weformulate OFDM symbols based on these signals to exploit theperiodic autocorrelation function (ACF) of the vectors in thePTS subblock partitioning. Error-correcting codes (ECCs) areemployed in the subblocking for the PTS radix FFT. This newtechnique significantly decreases the computational complexitywhile providing comparable PAPR reduction to ordinary PTS(O-PTS), even with a small number of stages after PTS parti-tioning. Numerical results are presented which confirm the PAPRimprovements.

Index Terms—Orthogonal frequency-division multiplexing(OFDM), peak-to-average power ratio (PAPR), partial transmitsequence (PTS), radix fast Fourier transform (FFT), decimationin frequency (DIF), error-correcting codes (ECCs).

I. INTRODUCTION

ORTHOGONAL frequency division multiplexing(OFDM) is an effective multicarrier transmission

technique for wireless communications over frequency-selective channels. Using an inverse fast Fourier transform(IFFT) and a fast Fourier transform (FFT) for the basebandmodulation and demodulation, respectively, simplifies thedesign of the transceiver and provides for an efficienthardware implementation. However, the time-domain OFDMsignal can exhibit a large peak-to-average power ratio (PAPR).These peaks can cause nonlinear distortion which introducesspectral spreading, intermodulation, and changes in the signalconstellation. One solution to this problem is to employ anexpensive power amplifier with a large linear range. Othertechniques are based on signal modification.

Numerous techniques have appeared in the literature toreduce the PAPR [1]-[21]. They can largely be classified asdistortion or distortionless techniques. Distortion techniquesare introduced in [1]-[8]. They create in-band distortion [1],peak regrowth [2], or out-of-band radiation [3]-[7]. In [8],a linear nonsymmetrical transform is given that achieves a

Manuscript received December 21, 2006; revised May 31, 2007; acceptedJuly 25, 2007. The associate editor coordinating the review of this paper andapproving it for publication was H. Jafarkhani.

This paper was presented in part at the 2008 IEEE International Conferenceon Communications.

The authors are with the Department of Electrical and Computer Engineer-ing, University of Victoria, P.O. Box 3055, STN CSC, Victoria, BC V8W3P6 Canada (e-mail: {aghassem, agullive}@ece.uvic.ca).

Digital Object Identifier 10.1109/TWC.2010.061099

reasonable tradeoff between PAPR reduction and BER per-formance. Many distortionless techniques have been proposed[9]-[21]. Coding schemes [9]-[12] sacrifice the data rate. Theyrequire memory to store the codewords, and introduce delaydue to the time required to find a low PAPR codeword,particularly when the number of subcarriers is large. Anotherclass of distortionless techniques employ constellation map-ping [13]-[15]. The constellation expansion in [13] requiresa complex optimization process, particularly with a largenumber of subcarriers. Simpler and practical constellationmapping techniques are active constellation extension [14] andtone reservation [15]. Phase optimization techniques achievePAPR reduction with a small amount of redundancy [16]-[21]. With selective mapping (SLM) [16]-[18], multiple se-quences are generated from the original data block and thesequence with the lowest PAPR is selected for transmission.In the partial transmit sequence (PTS) approach [18]-[21],disjoint subblocks of OFDM subcarriers are phase shiftedseparately after the IFFT is computed. If the subblocks areoptimally phase shifted, they exhibit minimum PAPR andconsequently reduce the PAPR of the merged signal. Thenumber of subblocks and their partitioning scheme determinethe PAPR reduction. The search for optimum subblock phasefactors is computationally complex, but this can be reducedwith adaptive PTS [20] or sphere decoding [21]. Typically,the receiver requires side information corresponding to theoptimal phases in PTS and the transmitted sequences in SLM.Techniques for avoiding explicit side information transmissionare presented in [17],[18].

One of the major drawback of PTS arises from the com-putation of multiple IFFTs, resulting in a high complexityproportional to the number of subblocks. In an attempt toreduce this complexity, intermediate signals within the IFFTusing decimation in time (DIT) have been used to obtain thePTS subblocks [22]. The experimental results in [22] showthat the PAPR reduction decreases as the number of stagesafter PTS partitioning decreases. Therefore, to achieve PAPRreduction close to that of original PTS (O-PTS), there shouldbe a substantial number of stages remaining in the IFFT afterthe partitioning into PTS subblocks. Hence, the computationalcomplexity is not significantly reduced. As a consequence,the key question is how to decrease the complexity whilemaintaining a PAPR reduction close to that of O-PTS.

In this paper, we present a solution to the above problem.In particular, we exploit the analysis of the corresponding FFTand formulate OFDM symbols based on the input signalsto each stage of an 𝑁 -point FFT using a decimation infrequency (DIF) radix algorithm. This allows us to construct

1536-1276/10$25.00 c⃝ 2010 IEEE

2 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 9, NO. 3, MARCH 2010

PTS subblocks for the inputs to each stage, and derive theirperiodic autocorrelation function (ACF). The ACF providesa design criteria for PTS subblocking to reduce the PAPRand computational complexity. We also show that pseudo-random [19] and𝑚-sequence subblocking [22] are not optimalas they introduce repeated subcarriers in the subblocks. As aconsequence, we propose a new PTS subblocking techniquebased on error-correcting codes (ECCs). This minimizes thenumber of repeated subcarriers and provides better PAPR re-duction than pseudo-random or 𝑚-sequence subblocking. Thecomputational complexity is reduced as the PAPR reductionis achieved using a small number of stages remaining in theIFFT.

The multiplicative complexity per stage which essentiallydetermines the IFFT computational complexity is obtained foran 𝑁 -point FFT radix algorithm. This enables us to analyzethe multiplicative complexity for multiple transforms. Further,it is used to show that the remaining stages with multipletransforms have significantly lower multiplicative complexitywith DIF compared to DIT [25]. It is important to note thatthe PTS subblock design in this paper can be practicallyimplemented within the FFT of OFDM transceivers. We canuse hardware devices such as a Field Programmable GateArray (FPGA) or a digital signal processor (DSP) [31] toimplement this FFT-based PAPR reduction technique. Theimplementation requires fewer hardware resources than othertechniques, and only a small number of transforms are neededto provide near optimal performance.

In the next section, the PAPR of an OFDM signal andO-PTS for PAPR reduction are reviewed. The PTS radix𝑁 -point FFT is introduced and its intermediate signals areformulated using a recursive expression in Section III. TheACF of the PTS subblocks and their partitioning using error-correcting codes is presented in Section IV. The computationalcomplexity analysis is given in Section V. Some numericalperformance and complexity results are also provided inSection V. Finally, some conclusions are given in Section VI.

II. PAPR AND THE PARTIAL TRANSMIT SEQUENCE

TECHNIQUE

A. PAPR of OFDM Signals

Let {𝑋(𝑘)}𝑁−1𝑘=0 denote a vector of quadrature-amplitude

modulation (QAM) or phase-shift keying (PSK) complexsymbols, where 𝑁 is the number of IFFT points and 𝑘 is thefrequency index. This vector is transmitted using one OFDMsymbol {𝑥(𝑛)}𝑁−1

𝑛=0 where the discrete-time index is 𝑛. Thediscrete time samples of 𝑥(𝑛) are computed by taking an 𝑁 -point inverse discrete Fourier transform (IDFT)

𝑥(𝑛) =1

𝑁

𝑁−1∑𝑘=0

𝑋(𝑘)𝑇−𝑛𝑘𝑁 (1)

where 𝑇𝑁 = 𝑒−𝑗2𝜋/𝑁 (known as the twiddle factor) and 𝑗2 =−1. In matrix notation, we can express (1) as

𝑥 =1

𝑁[𝑇𝑁 ]∗𝑋 (2)

where [.]∗ denotes complex conjugate and 𝑇𝑁 is the twiddlefactor matrix

𝑇𝑁 =

⎡⎢⎢⎢⎢⎢⎢⎢⎣

1 ⋅ ⋅ ⋅ 1 ⋅ ⋅ ⋅ 1...

. . ....

. . ....

1 ⋅ ⋅ ⋅ 𝑇 𝑛𝑘𝑁 ⋅ ⋅ ⋅ 𝑇 𝑛(𝑁−1)

𝑁...

. . ....

. . ....

1 ⋅ ⋅ ⋅ 𝑇 𝑘(𝑁−1)𝑁 ⋅ ⋅ ⋅ 𝑇 (𝑁−1)2

𝑁

⎤⎥⎥⎥⎥⎥⎥⎥⎦. (3)

To evaluate the variation in the time domain samples 𝑥(𝑛),the discrete peak power to average power ratio (PAPR) of anOFDM symbol 𝑥(𝑛) is defined as

PAPR(𝑥(𝑛)) =max

0≤𝑛<𝐽𝑁−1∣𝑥(𝑛)∣2

𝐸{∣𝑥(𝑛)∣2} (4)

where 𝐸{.} and 𝐽 denote expected value and oversamplingrate, respectively. The continuous-time PAPR is typicallydesired in practice and is well approximated for an OFDMsymbol by oversampling (at a rate 𝐽 ≥ 4), the discrete timePAPR in (4) [23],[24]. This is implemented by adding 𝐽𝑁−𝑁zeros at the end of the 𝑁 -point IDFT [24]. We consider 𝐽 = 4in the remainder of the paper.

In order to evaluate the PAPR reduction, we employ thecomplementary cumulative distribution function (CCDF) ofthe PAPR [27]-[30]

CCDF(PAPR(𝑥(𝑛))) = Pr(PAPR(𝑥(𝑛))) > PAPR0. (5)

This expression represents the probability that the PAPR of asymbol exceeds the threshold level 𝑃𝐴𝑃𝑅0. 𝑋(𝑘) is assumedto be a complex-valued zero-mean signal with variance 𝜎2,and therefore via the central limit theorem for large 𝑁 is wellapproximated as a complex Gaussian random variable [19].

B. Original PTS

With original PTS (O-PTS), the frequency domain vector𝑋(𝑘) is partitioned into 𝑃 disjoint subblocks 𝑋𝑝(𝑘) =[𝑋𝑝(0), . . . , 𝑋𝑝(𝑁 − 1)]𝑇 , 0 ≤ 𝑝 ≤ 𝑃 − 1, so that 𝑋(𝑘) =∑𝑃−1

𝑝=0 𝑋𝑝(𝑘). The combination of these subblocks with ro-tated phase factors 𝑒𝑗𝜃 yields the alternative frequency domainvectors with

𝑋 ′(𝑘) =𝑃−1∑𝑝=0

𝑒 𝑗𝜃𝑝𝑋𝑝(𝑘) (6)

Since each subblock is independently rotated by a phase factor𝜃𝑝, the phase factor multiplication can be performed after theIDFT computation. Hence, we can take the IDFT of (6), andexploit the linearity of the IDFT to obtain

𝑥′(𝑛) =𝑃−1∑𝑝=0

𝑒 𝑗𝜃𝑝 IDFT𝐽𝑁×𝑁 (𝑋𝑝(𝑘)) =𝑃−1∑𝑝=0

𝑒 𝑗𝜃𝑝𝑥𝑝(𝑛) (7)

where 𝑥𝑝(𝑛) = IDFT𝐽𝑁×𝑁 (𝑋𝑝(𝑘)) are the 𝑃 time-domainpartial transmit sequences. IDFT𝐽𝑁×𝑁(𝑋𝑝(𝑘)) is the IDFTof the 𝑁 dimensional vector 𝑋𝑝(𝑘) and results in an 𝑁𝐽dimensional vector 𝑥𝑝(𝑛). The sequence 𝑥′(𝑛) with the small-est PAPR is chosen for transmission based on the followingcriterion

GHASSEMI and GULLIVER: PAPR REDUCTION OF OFDM USING PTS AND ERROR-CORRECTING CODE SUBBLOCKING 3

[𝜃1, . . . , 𝜃𝑃−1] = argmin𝜃1,...,𝜃𝑃−1

{max

0≤𝑛<𝐽𝑁∣𝑥′(𝑛)∣2

}(8)

Assuming 𝑊 is the number of phase values and 𝜃0 = 0,there are (𝑃 − 1) log2𝑊 bits per OFDM symbol requiredfor explicit side information. The search complexity to findthe lowest PAPR sequence is 𝑊𝑃−1. To reduce this com-plexity, we restrict the phase factors to values in the set𝜃𝑝 ∈ {0, 𝜋/2, 𝜋, 3𝜋/2}.

According to (7), the number of IDFT𝐽𝑁×𝑁 transformswhich have to be computed is 𝑃 which is typically in therange 2 to 16. Thus the resulting computational complexitycan be high, particularly when 𝑁 is large.

III. PTS RADIX FFT TECHNIQUE

The IDFT can be computed by taking the complex conju-gate of the input and output sequences while using the samediscrete Fourier transform (DFT) parameters [25]

𝑥(𝑛) =1

𝑁

[𝑁−1∑𝑘=0

𝑋∗(𝑘)𝑇 𝑛𝑘𝑁

]∗(9)

Let 𝑦(.) represents 𝑋∗(.). The expression inside the bracketsin (9) is the DFT of 𝑋∗(.) [25], i.e.

𝑌 (𝑘) =

𝑁−1∑𝑘=0

𝑦(𝑛)𝑇 𝑛𝑘𝑁 (10)

Consequently, we focus here on the DFT computation.An FFT algorithm converts the DFT computation to 𝑟 ×

𝑁/𝑟-point DFTs recurring through 𝑚 = log𝑟 𝑁 stages. As aconsequence, the computational complexity is reduced from𝑂(𝑁2) to 𝑂(𝑁 log𝑟𝑁). The value of 𝑟 corresponds to aradix-𝑟 FFT algorithm using either DIF or DIT. The DIF radix-𝑟 algorithm can be derived from (10) as

𝑌 (𝑟𝑘 + 𝑘0) =

𝑁𝑟 −1∑𝑛=0

((𝑟−1∑𝑖=0

𝑦(𝑛+𝑁

𝑟𝑖)𝑇 𝑖𝑘0

𝑟

)𝑇 𝑛𝑘0

𝑁

)𝑇 𝑘𝑛

𝑁/𝑟

(11)where 𝑘 = 0, . . . , 𝑁/𝑟 − 1, 𝑛 = 0, . . . , 𝑁/𝑟 − 1, and 𝑘0, 0 ≤𝑘0 ≤ 𝑟 − 1, denotes the index of the butterfly outputs. It isassumed that the input sequence is in normal order, and theoutput is in digit-reversed order in DIF and vice-versa for theDIT algorithm. As we consider intermediate signals, i.e. theinputs to stage 𝑣 for PTS subblocking, symbols and indicesfor an intermediate signal are represented by 𝑦 and �̃� for aninput 𝑦 and time index 𝑛, respectively, and 𝑘 for a frequencyindex 𝑘. The expression in (11) can be expanded at a particularstage 𝑣 as

𝑌 𝜂(𝑟𝑘 + 𝑘0) =

𝑁𝑟𝑣−1∑�̃�=0

((𝑟−1∑𝑖=0

𝑦 𝜂(�̃�+𝑁

𝑟𝑣𝑖)𝑇 𝑖𝑘0

𝑟

)𝑇 �̃� 𝑘0

𝑁/𝑟𝑣−1

)𝑇 𝑘𝑣�̃�

𝑁/𝑟𝑣 (12)

where 𝑘 = 0, . . . , 𝑁/𝑟𝑣 − 1, �̃� = 0, . . . , 𝑁/𝑟𝑣 − 1 and 𝜂, 𝜂 =1, . . . , 𝑟𝑣−1, denotes a particular 𝑁/𝑟𝑣−1-point DFT at stage𝑣, 𝑣 = 1, . . . ,𝑚. This decomposition is depicted in Fig. 1.Hence, there are 𝑟𝑣−1 identical 𝑁/𝑟𝑣−1-point DFTs at stage 𝑣

Fig. 1. Recursive reduction to 𝑁/𝑟𝑣-point DFTs with DIF radix-𝑟 at stage𝑣.

and each of these 𝑁/𝑟𝑣−1-point DFTs is individually reducedinto 𝑁/𝑟𝑣-point DFTs in the remaining 𝑚−𝑣 stages. Finally,we can formulate the FFT output corresponding to inputs atstage 𝑣 using (12) as

𝑌 (𝑟𝑘 + 𝑘0) =

𝑟𝑣−1∑𝜂=1

𝑌 𝜂(𝑟𝑘 + 𝑘0) =

𝑟𝑣−1∑𝜂=1

𝑁𝑟𝑣 −1∑�̃�=0((

𝑟−1∑𝑖=0

𝑦 𝜂(�̃�+𝑁

𝑟𝑣𝑖)𝑇 𝑖𝑘0

𝑟

)𝑇 �̃� 𝑘0

𝑁/𝑟𝑣−1

)𝑇

˜𝑘 �̃�𝑁/𝑟𝑣 (13)

The inputs, 𝑦 𝜂(�̃�+ 𝑁𝑟𝑣 𝑖), at stage 𝑣 have dimensions 𝑟𝑣−1 ×

𝑁/𝑟𝑣−1.Similarity, the DIT radix-𝑟 algorithm from (10) is illustrated

in Fig. 2, which shows the DIT reduction to 𝑁/𝑟𝑚−𝑣-pointDFTs. In this case, there are 𝑟𝑚−𝑣 identical 𝑁/𝑟𝑚−𝑣-pointDFTs at stage 𝑣.

To derive the PTS radix FFT, similar to (6), and using (9)and (13), we have

𝑥′(𝑛) =𝑃−1∑𝑝=0

𝑒−𝑗𝜃𝑝1

𝑁

⎡⎣𝑟𝑣−1∑𝜂=1

𝑌 𝜂𝑝 (𝑟𝑘 + 𝑘0)

⎤⎦∗

=

𝑃−1∑𝑝=0

𝑒−𝑗𝜃𝑝𝑥𝑝(𝑛) (14)

where the PTSs are

1

𝑁

⎡⎣𝑟𝑣−1∑𝜂=1

𝑌 𝜂𝑝 (𝑟𝑘 + 𝑘0)

⎤⎦∗

(15)

Subblocks are composed over the inputs 𝑦 𝜂(�̃�+ 𝑁𝑟𝑣 𝑖) in (13)

at stage 𝑣.


Fig. 2. Recursive reduction to 𝑁/𝑟𝑚−𝑣-point DFTs with DIT radix-𝑟 atstage 𝑣.

In order to recover the data at the receiver, we use the samecoefficients for the FFT as the IFFT at the transmitter [25].However, we must take into account the IFFT input orders atthe transmitter. If we assume these inputs are in normal order,the inputs to the FFT at the receiver should be in reverse order.Thus, the FFT computation at the receiver is symmetric to theIFFT computation [25], and if stage 𝑣 is used to obtain thePTS subblocks, the data is recovered at stage 𝑚 − 𝑣 at thereceiver. Hence, the amount of side information remains thesame as that of O-PTS.

IV. SUBBLOCK PARTITIONING IN PTS RADIX FFT

In PTS subblocking [19], a random subcarrier assignmentimproves the ACF properties of the PTS subblocks as itprovides less correlated adjacent time samples compared withother partitioning schemes. This leads to better PAPR reduc-tion with O-PTS. However, as will be shown, it is not thebest approach for the case of PTS radix FFT using DIF (DIF-PTS). Therefore, we use (13) to derive the periodic ACF ofthe PTS subblocks, 𝑥𝑝(𝑛), and use this as a design criteria forthe subblocks. We will see that pseudo-random or𝑚-sequencesubblocking can generate superfluous twiddle factors within asubblock. This increases the magnitude of the ACF vectors.Hence, we employ error-correcting codes for subblocking tominimize the number of repeated subcarriers.

A. Autocorrelation of the DIF-PTS Subblocks

In order to design DIF-PTS subblocks for the inputs at stage𝑣, we define the normalized periodic ACF of 𝑥𝑝(𝑛) as [19]

𝑅𝑝(𝑛0) =1

𝜎2𝐸{[𝑥𝑝((𝑛+ 𝑛0)mod𝑁)][𝑥𝑝(𝑛)]

∗} (16)

This represents the correlation between 𝑛0-spaced complexsamples in subblock 𝑝. By substituting (13) into (14) and usingthis in (16) and the corresponding FFT, we obtain (see theAppendix)

𝑅𝑝(𝑛0) =1

𝜎2𝑁2

𝑟𝑣−1∑𝜂=1

𝑁𝑟𝑣 −1∑�̃�=0

𝑟−1∑𝑖=0

𝐸

{𝑦 𝜂𝑝

(�̃�+

𝑁

𝑟𝑣𝑖

)𝑦 ∗𝜂𝑝

(�̃�+

𝑁

𝑟𝑣𝑖

)}𝑇 −�̃� 𝑛0

𝑁/𝑟𝑣 (17)

where 𝑛0 = 0, . . . , 𝑁 − 1. We introduce the variable 𝑀𝜂�̃�,𝑖

with value one if �̃� is active in subblock 𝑝, and zero otherwise.Then, we can write (17) as

𝑅𝑝(𝑛0) =1

𝜎2𝑁2

𝑟𝑣−1∑𝜂=1

𝑁𝑟𝑣 −1∑�̃�=0

𝑟−1∑𝑖=0

𝑀𝜂�̃�,𝑖

𝑟−1𝑇 −�̃�𝑛0

𝑁/𝑟𝑣 (18)

We arrive at the following results from (17) and (18).

∙ The ACF vectors 𝑅𝑝(𝑛) are dependent on the active𝑀𝜂

�̃�,𝑖 in subblock 𝑝 and we should have PTS subblockpartitioning based on the inputs 𝑦 𝜂(�̃�+ 𝑁

𝑟𝑣 𝑖) associatedwith subblock 𝑝.

∙ If we increase the number of active 𝜂, this increases theautocorrelation function for a particular subblock p andvalue �̃�, i.e. if all inputs over stage 𝑣 are consideredfor subblocking, the twiddle factor 𝑇 −�̃�𝑛0

𝑁/𝑟𝑣 (which corre-sponds to a particular subcarrier), is repeated in subblock𝑝.

With O-PTS, pseudo-random [19] or 𝑚-sequence [22] sub-blocking has been done over all inputs 𝑦 𝜂(�̃�+ 𝑁

𝑟𝑣 𝑖) The aboveanalysis shows that these techniques can result in repeatedsubcarriers within a subblock. The corresponding ACF can belarge when 𝑚 − 𝑣 is small or 𝑁 is large, i.e. if the numberof identical 𝑁/𝑟𝑣−1-point DFTs per stage is large.

In order to see the effect of repeated subcarriers on the ACFof the subblocks, consider 𝑁 = 32, 𝑣 = 3, 𝑃 = 4, and 𝑟 = 2and the same pseudo-random sequence [01023312] over inputs𝑦 𝜂(�̃� + 4𝑖) where 𝜂 = 1, . . . , 4, 𝑖 = 0, 1, �̃� = 0, . . . , 7, and𝑘 = 0, . . . , 7. In fact, there are 4× 8-point DFTs at this stage.The effect of repeated subcarriers on the ACF is shown in Fig.2. We observe that repeated subcarriers results in a large ACFfor the PTS sequences. This motivates us to propose a newsubblocking technique.

B. Error-Correcting Code Subblocking

We propose a technique using error-correcting codes(ECCs) to minimize the number of repeated subcarriers withinthe subblocks at stage 𝑣. Repetition codes over 𝑍𝑃 , the integerring of 𝑃 elements, are used to generate a set of subblocks.Since the inputs at stage 𝑣 have dimensions 𝑟𝑣−1 ×𝑁/𝑟𝑣−1,the minimum number of subblock should be 𝑃𝑚𝑖𝑛 = 𝑟𝑣−1

in order to avoid repeated subcarriers. Hence, we constructECC subblocks based on the two cases 𝑟𝑣−1 ≤ 𝑁/𝑟𝑣−1 and𝑟𝑣−1 > 𝑁/𝑟𝑣−1.

1)Case 𝑟𝑣−1 ≤ 𝑁/𝑟𝑣−1 : Let the vector 𝑆 =[𝑠0, 𝑠1, . . . , 𝑠𝑟𝑣−1−1] of elements over 𝑍𝑟𝑣−1 , the integersmodulo 𝑟𝑣−1, represent the inputs to the 𝑁/𝑟𝑣−1-point DFTs.


We repeat each element within the sequence 𝑆 𝑁/𝑟𝑣−1 times.Then, we obtain matrix 𝑆

𝑆 =

⎡⎢⎢⎢⎣𝑆0𝑆1...

𝑆𝑟𝑣−1−1

⎤⎥⎥⎥⎦ =

⎡⎢⎢⎢⎣𝑠0 𝑠0 . . . 𝑠0𝑠1 𝑠1 . . . 𝑠1...

.... . .

...𝑠𝑟𝑣−1−1 𝑠𝑟𝑣−1−1 . . . 𝑠𝑟𝑣−1−1

⎤⎥⎥⎥⎦(19)

with dimensions 𝑟𝑣−1×𝑁/𝑟𝑣−1. Next, we generate a pseudo-

random sequence 𝑈 =[𝑢0, 𝑢1, . . . , 𝑢 𝑁

𝑟𝑣−1 −1

]of length

𝑁/𝑟𝑣−1 over 𝑍𝑟𝑣−1 . Finally, we construct the codewords as

𝐶 =

⎡⎢⎢⎢⎣𝑐0𝑐1...

𝑐𝑟𝑣−1−1

⎤⎥⎥⎥⎦ =

⎡⎢⎢⎢⎣𝑈 + 𝑆0(mod𝑃 )

𝑈 + 𝑆1(mod𝑃 )...

𝑈 + 𝑆𝑟𝑣−1−1(mod𝑃 )

⎤⎥⎥⎥⎦ (20)

These codeword have maximum Hamming distance 𝑁/𝑟𝑣−1

as they differ in all positions.

2)Case 𝑟𝑣−1 > 𝑁/𝑟𝑣−1 : We construct the code 𝐶 withdimensions 𝑁/𝑟𝑣−1 × 𝑟𝑣−1 and then take the transpose of𝐶 to obtain the codeword matrix 𝐶 with dimensions 𝑟𝑣−1 ×𝑁/𝑟𝑣−1. Consider the vector 𝑆 =

[𝑠0, 𝑠1, . . . , 𝑠 𝑁

𝑟𝑣−1 −1

]over

𝑍𝑟𝑣−1 and repeat each element within 𝑆 𝑟𝑣−1 times. Thisresults in a matrix 𝑆 with dimensions 𝑁/𝑟𝑣−1 × 𝑟𝑣−1. Apseudo-random sequence 𝑈 = [𝑢0𝑢1 . . . 𝑢𝑟𝑣−1−1] withoutrepeated elements of length 𝑟𝑣−1 is generated over 𝑍𝑟𝑣−1 .From (20), the codeword matrix 𝐶 is obtained with dimensions𝑁/𝑟𝑣−1 × 𝑟𝑣−1. Finally, we obtain 𝐶 by taking the transposeof 𝐶.

As an example, consider DIF radix-2 for a 32-point FFT.We first let 𝑣 = 3, which corresponds to Case 1 as there are4 × 8-point DFTs. The 𝑚-sequence (MS) subblocking from[22] with 𝑃 = 4 is

𝐶𝑀𝑆 =

⎡⎢⎢⎣0 2 3 1 2 1 0 21 0 0 0 2 1 2 12 3 3 1 2 3 1 00 2 3 3 3 3 1 0

⎤⎥⎥⎦ (21)

As seen from (21), there are as many as three repeatedsubcarriers within the subblocks. With our proposed techniqueusing an error-correcting code (ECC) over 𝑍4, we employ𝑈 = [01023312] and

𝑆 =

⎡⎢⎢⎣0 0 0 0 0 0 0 01 1 1 1 1 1 1 12 2 2 2 2 2 2 23 3 3 3 3 3 3 3

⎤⎥⎥⎦to obtain

𝐶𝐸𝐶𝐶 =

⎡⎢⎢⎣0 1 0 2 3 3 1 21 2 1 3 0 0 2 32 3 2 0 1 1 3 03 0 3 1 2 2 0 1

⎤⎥⎥⎦ (22)

which has no repeated subcarriers within the subblocks. Figs.4 and 5 present the absolute value of the ACF vectors forsubblocks 𝑝0 to 𝑝4 with MS and ECC subblocks. The ECCsubblocks show a significant reduction in the ACF with

TABLE ITHE NUMBER OF SUBBLOCKS FOR VARIOUS NUMBERS OF REPEATED

TWIDDLE FACTORS AND FFT SIZES WITH 𝑟 = 2, 4 AND 𝑚− 𝑣 = 2, 3

𝑚− 𝑣 = 2𝑁 𝑟 = 2, 𝑃𝑚𝑖𝑛 𝑟 = 4, 𝑃𝑚𝑖𝑛

𝑏 = 0 𝑏 = 1 𝑏 = 2 𝑏 = 0 𝑏 = 1 𝑏 = 232 8 4 2 2 < 2 < 264 16 8 4 4 2 < 2128 > 16 16 8 8 4 2256 > 16 > 16 16 16 8 4512 > 16 > 16 > 16 > 16 16 81024 > 16 > 16 > 16 > 16 > 16 162048 > 16 > 16 > 16 > 16 > 16 > 16

𝑚− 𝑣 = 3𝑁 𝑟 = 2, 𝑃𝑚𝑖𝑛 𝑟 = 4, 𝑃𝑚𝑖𝑛

𝑏 = 0 𝑏 = 1 𝑏 = 2 𝑏 = 0 𝑏 = 1 𝑏 = 232 4 2 < 2 < 2 < 2 < 264 8 4 2 < 2 < 2 < 2128 16 8 2 2 < 2 < 2256 > 16 16 8 4 2 < 2512 > 16 > 16 16 8 4 21024 > 16 > 16 > 16 16 8 42048 > 16 > 16 > 16 > 16 16 8

𝑛0 ∕= 0 in comparison with MS subblocking. ECC subblockingprovides an ACF which is nearly flat. If 𝑣 = 4 is taken,i.e. 8 × 4-point DFTs, we have Case 2 subblocking. MSsubblocking [22] with 𝑃=8 has

𝐶𝑀𝑆 =

⎡⎢⎢⎣0 2 1 2 6 6 4 76 1 0 5 7 3 6 33 4 0 2 3 1 7 15 2 4 5 5 0 7 4

⎤⎥⎥⎦ (23)

whereas ECC subblocking over 𝑍8 with 𝑢 = [01467352] gives

𝐶𝐸𝐶𝐶 =

⎡⎢⎢⎣0 1 4 6 7 3 5 21 2 5 7 0 4 6 32 3 6 0 1 5 7 43 4 7 1 2 6 0 5

⎤⎥⎥⎦ (24)

The minimum number of subblocks 𝑃𝑚𝑖𝑛 = 𝑟𝑣−1 istypically large when the number of multiple stages, 𝑚 − 𝑣,is small. This increases the number of multiple IFFTs andthe search complexity for the optimal phases. In this case,it is possible to reduce the number of subblocks by using ahigher radix such as radix-4. Table I summarizes the minimumnumber of subblocks to achieve maximum Hamming distancewithin the ECC subblocks for different values of 𝑁 and 𝑟.To eliminate the repeated subblocks, 𝑃 ≥ 32 is required with𝑚 − 𝑣 = 2 and 𝑁 ≥ 512, or 𝑚 − 𝑣 = 3 and 𝑁 ≥ 2048.However, a small number of repeated subcarriers can stillprovide reasonable PAPR reduction.

Let 𝑃 = 𝑟𝑣−1/2𝑏 be the number of subblocks with repeatedsubcarriers where 𝑏 = 1, . . . , (𝑣−1) log2(𝑟−1). There are norepeated subcarriers for 𝑏 = 0.

3)Case 𝑟𝑣−1 ≤ 𝑁/𝑟𝑣−1 and 𝑏 ≥ 1: We define the vector𝑆 over 𝑍𝑟𝑣−1/2𝑏. The matrix 𝑆 is obtained similar to (19).In order to have the sequence 𝑢, we can generate 2𝑏 randomsequences 𝑢 of length 𝑁/𝑟𝑣−1 over 𝑍𝑟𝑣−1/2𝑏. The codewordmatrix 𝐶 is obtained from (20). The number of repeatedelements within vector 𝑆 is 2𝑏−1.


4)Case 𝑟𝑣−1 > 𝑁/𝑟𝑣−1 and 𝑏 ≥ 1: This is the same as thecase without repeated subcarriers (𝑏 = 0), over 𝑍𝑟𝑣−1/2𝑏.

Consider the above example for 𝑏 = 1. With 𝑣 = 4 and𝑏 = 1 (Case 4), the number of subblocks is reduced from 8to 4. Matrix 𝐶 over 𝑍4 is obtained as

𝐶𝐸𝐶𝐶 =

⎡⎢⎢⎣0 3 1 0 2 1 3 21 3 2 0 3 1 0 20 1 1 2 2 3 3 02 2 3 3 0 0 1 1

⎤⎥⎥⎦𝑇

(25)

The magnitude of the ACF vectors for the ECC subblocksis depicted in Fig 6. This shows that even with repeatedsubcarriers (𝑏 = 1), the proposed ECC subblocking provideslow autocorrelation PTS sequences. If we consider Case 3with 𝑣 = 3, the number of subblocks is reduced from 4 to2. The ECC subblocking over 𝑍2 with two random sequences𝑢1 = [10011100] and 𝑢2 = [00111001] is

𝐶𝐸𝐶𝐶 =

⎡⎢⎢⎣1 0 0 1 1 1 0 00 1 1 0 0 0 1 10 0 1 1 1 0 0 11 1 0 0 0 1 1 0

⎤⎥⎥⎦𝑇

(26)

With MS subblocking [22], we have

𝐶𝑀𝑆 =

⎡⎢⎢⎣0 1 1 0 1 0 0 10 0 0 0 1 0 1 01 1 1 0 1 1 0 00 1 1 1 1 1 0 0

⎤⎥⎥⎦𝑇

(27)

From (26) and (27), the number of repeated subcarriers withinsubblocks using ECC is reduced compared to MS subblocking.

V. PERFORMANCE RESULTS

In this section, we first obtain the multiplicative complexityfor the DIF and DIT algorithms. Then, we discuss the tradeoffsbetween the number of subblocks, multiplicative complexity,number of repeated subcarriers, and PAPR reduction. Weexamine the performance of a PTS radix FFT using ECCsubblocking (PTS-ECC). Finally, the proposed technique iscompared with O-PTS [19] and PTS using𝑚-sequences (PTS-MS) [22] in terms of PAPR reduction and multiplicativecomplexity.

A. Multiplicative Complexity Analysis

We define the multiplicative complexity of the DIF andDIT algorithms as the number of complex multiplications bynontrivial twiddle factors 𝑇 �̃�𝑘0

𝑁/𝑟𝑣−1 and 𝑇 �̃�𝑘0

𝑁/𝑟𝑚−𝑣 as shownin Figs. 1 and 2, respectively. The twiddle factors 𝑇 𝑖𝑘0

𝑟 aretrivial (±1,±𝑗) at each stage as we consider only radix-2and radix-4. Thus, they do not introduce any multiplicativecomplexity. A small number of twiddle factors 𝑇 �̃�𝑘0

𝑁/𝑟𝑣−1 and

𝑇 �̃�𝑘0

𝑁/𝑟𝑚−𝑣 when �̃� ∕= 0 and 𝑘0 ∕= 0 are trivial [26]. However,without loss of generality and to simplify our complexitycalculations, we do not consider these as trivial twiddle factors.In fact, they do not significantly affect the multiplicativecomplexity comparisons of O-PTS, PTS-MS and PTS-ECC asthe numbers are approximately the same in all cases. Let 𝑎𝐷𝐼𝐹

𝑣

denote the number of twiddle factors 𝑇 �̃�𝑘0

𝑁/𝑟𝑣−1 at stage 𝑣 for

the DIF algorithm. Summing over all dimensions �̃�, butterflyoutputs 𝑘0, and 𝑟𝑣−1 𝑁/𝑟𝑣−1-point DFTs, we obtain

𝑎𝐷𝐼𝐹𝑣 = 𝑟𝑣−1(𝑟 − 1)[(𝑁/𝑟𝑣)− 1] (28)

The ratio of successive values of 𝑎𝐷𝐼𝐹 is

𝑎𝐷𝐼𝐹𝑣 /𝑎𝐷𝐼𝐹

𝑣+1 = [(𝑁/𝑟𝑣)− 1]/[(𝑁/𝑟𝑣)− 𝑟] (29)

which is obviously greater than one. Similarity for the DITalgorithm, we can obtain the number of twiddle factors𝑇 �̃�𝑘0

𝑁/𝑟𝑚−𝑣 at stage 𝑣 as

𝑎𝐷𝐼𝑇𝑣 = 𝑟𝑚−𝑣(𝑟 − 1)[(𝑁/𝑟𝑚−𝑣+1)− 1] (30)

The ratio of successive values of 𝑎𝐷𝐼𝑇 is

𝑎𝐷𝐼𝐹𝑣 /𝑎𝐷𝐼𝐹

𝑣+1 = [(𝑁/𝑟𝑚−𝑣+1)−1]/[(𝑁/𝑟𝑚−𝑣+1)−𝑟−1] (31)

which is less than one. Thus, DIF introduces less multiplica-tive complexity for the 𝑚 − 𝑣 final stages than DIT. As aconsequence, we choose DIF for the PTS radix FFT. From(28), the overall multiplicative complexity for the PTS radixFFT using DIF is

𝑀𝐷𝐼𝐹𝑡𝑜𝑡𝑎𝑙 =

𝑣−1∑𝛽=1

𝑎𝐷𝐼𝐹𝛽 + 𝑃

𝑚∑𝛽=𝑣

𝑎𝐷𝐼𝐹𝛽 (32)

Similarity, from (30), the overall multiplicative complexityfor DIT is obtained using 𝑎𝐷𝐼𝑇

𝑣 in (32). Finally, in or-der to compare the computational complexity between twoPAPR techniques, we define the complexity reduction ratio as𝑅𝑚𝑢𝑙 = 1− (𝑀1

𝑡𝑜𝑡𝑎𝑙/𝑀2𝑡𝑜𝑡𝑎𝑙).

B. Tradeoffs in a PTS radix FFT

A high number of subblocks increases the search complex-ity for the optimal phases with PTS and also the number oftransforms. Therefore, we limit this number to 𝑃𝑚𝑎𝑥 = 16.As seen in Table I, we can reduce this number by usinga higher value of 𝑟 while still minimizing the number ofrepeated subcarriers. Another advantage of a higher radix islower multiplicative complexity compared to using a smallerradix. If we employ radix-8, the number of subblocks issignificantly reduced, but they become complex to implement.As a consequence, we choose 𝑟 = 4 which provides lowmultiplicative complexity, low hardware complexity, and asuitable number of subblocks, i.e. a sufficiently low numberof transforms. If very low complexity for the transforms isrequired, such as with 𝑚−𝑣 = 2, PTS sequences with perfectACF properties (𝑏 = 0) can be obtained for 32 ≤ 𝑁 < 512.However, for 𝑁 ≥ 512, there is a tradeoff between PAPRreduction, number of transforms, and number of subblocks.This will be examined next.

C. Numerical Results

The CCDF of the PAPR of 16 QAM-modulated OFDMsignals is considered with 𝑃 = 16. When 𝑁 ∕= 4𝑚, a mixed-radix of 2 and 4 is used. The phase factors are values from theset 𝜃𝑝 ∈ {0, 𝜋/2, 𝜋, 3𝜋/2} with 𝑊 = 2. The first subblock isnot rotated so only 𝑃−1 phases are optimized. For simplicity,we consider only 256 ≤ 𝑁 ≤ 2048, as the results for 32 ≤𝑁 < 256 are similar to the case with 𝑁 = 256. For 𝑁 >


0 5 10 15 20 25 300

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1pop1p2p3

AB

S[R

(no)

]

no

Fig. 3. The autocorrelation of PTS sequences in a PTS radix FFT for𝑁 = 32, 𝑣 = 3, 𝑃 = 4 and 𝑟 = 2 with repeated subcarriers within eachsubblock.

0 5 10 15 20 25 300

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1po MS subblockingpo EEC subblocking b=0p1 MS subblockingp1 EEC subblocking b=0

AB

S[R

(no)

]

no

Fig. 4. The autocorrelation of PTS sequences in a PTS radix FFT for𝑁 = 32, 𝑣 = 3, 𝑃 = 4, 𝑟 = 2, 𝑏 = 0, 𝑚-sequence and ECC subblocks, 𝑝0and 𝑝1.

0 5 10 15 20 25 300

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1p2 MS subblockingp2 ECC subblocking b=0p3 MS subblockingp3 ECC subblocking b=0

AB

S[R

(no)

]

no

Fig. 5. The autocorrelation of PTS sequences in a PTS radix FFT for𝑁 = 32, 𝑣 = 3, 𝑃 = 4, 𝑟 = 2, 𝑏 = 0, 𝑚-sequence and ECC subblocks, 𝑝2and 𝑝3.

TABLE IICOMPARISON OF PTS-ECC, O-PTS [19] AND PTS-MS [22] IN TERMS

OF PAPR IMPROVEMENT AND COMPUTATIONAL COMPLEXITYREDUCTION FOR 256 ≤ 𝑁 ≤ 2048, CCDF=10−4, AND 𝑚− 𝑣 = 2, 3

𝑚− 𝑣 = 2PTS-ECC compared to PTS-MS PTS-ECC compared to O-PTS

𝑁 𝑅𝑚𝑢𝑙 PAPR improvement 𝑅𝑚𝑢𝑙 PAPR loss(%) (dB) (%) (dB)

256 41 2 78 1.2512 65 1.91 89 1.31

1024 41 1.51 83 1.62048 62 ∗ 89 ∗

𝑚− 𝑣 = 3PTS-ECC compared to PTS-MS PTS-ECC compared to O-PTS

𝑁 𝑅𝑚𝑢𝑙 PAPR improvement 𝑅𝑚𝑢𝑙 PAPR loss(%) (dB) (%) (dB)

256 16 2.2 56 0.1512 56 1.6 80 0.21024 17 1.61 66 0.32048 54 1.5 81 0.2

1 𝑏 = 1∗ 𝑃 > 16

0 5 10 15 20 25 300

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1p0 ECC subblocking b=1p1 ECC subblocking b=1p2 ECC subblocking b=1p3 ECC subblocking b=1

AB

S[R

(no)

]

no

Fig. 6. The autocorrelation of PTS sequences in a PTS radix FFT for𝑁 = 32, 𝑣 = 4, 𝑃 = 4, 𝑟 = 2, 𝑏 = 1, and ECC subblocks, 𝑝0, 𝑝1, 𝑝2, and𝑝3.

2048, we require 𝑃 > 16 in order to improve the PAPR. Wecompare the CCDF of O-PTS [19], with PTS-MS [22] andPTS-ECC.

Fig. 7 presents the PAPR performance for 𝑁 = 256. Thisshows that PTS-ECC improves the PAPR performance byapproximately 2 dB for 𝑚−𝑣 = 2 and 2.2 dB for 𝑚−𝑣 = 3,compared to PTS-MS [22]. As shown in Table II, our proposedtechnique achieves a multiplicative reduction of 41% and 78%for𝑚−𝑣 = 2, and 16% and 56% for𝑚−𝑣 = 3, over PTS-MSand O-PTS, respectively.

The results for 𝑁 = 512 are shown in Fig. 8. With 𝑃 = 16for𝑚−𝑣 = 2 stages, there is a negligible degradation in PAPRperformance with 𝑏 = 1 compared to the case for 𝑏 = 0. ThePAPR improvement is 1.8 dB for 𝑚 − 𝑣 = 2 and 1.6 dB for𝑚 − 𝑣 = 3 compared to PTS-MS. In terms of multiplicativecomplexity for 𝑁 = 512, from Table II, PTS-ECC with thesevalues results in a reduction of 65% and 89% for 𝑚− 𝑣 = 2,and 56% and 80% for 𝑚− 𝑣 = 3, over PTS-MS and O-PTS,respectively.


4 5 6 7 8 9 10 11 1210

−4

10−3

10−2

10−1

100

Original OFDMO−PTSPTS−MS m−v=2PTS−ECC m−v=2 , b=0PTS−MS m−v=3PTS−ECC m−v=3 , b=0

PAPRo (dB)

CC

DF

Fig. 7. CCDF of PTS-ECC in comparison with PTS-MS and O-PTS with𝑚− 𝑣 = 2 and 𝑚− 𝑣 = 3 for 𝑁=256.

5 6 7 8 9 10 11 12 1310

−4

10−3

10−2

10−1

100

Original OFDMO−PTSPTS−MS m−v=2PTS−ECC m−v=2 , b=0PTS−ECC m−v=2 , b=1PTS−MS m−v=3PTS−ECC m−v=3 , b=0

CC

DF

PAPRo (dB)


5 6 7 8 9 10 11 12 1310

−4

10−3

10−2

10−1

100

Original OFDMO−PTSPTS−MS m−v=2PTS−ECC m−v=2 , b=0PTS−ECC m−v=2 , b=1PTS−ECC m−v=2 , b=2PTS−MS m−v=3PTS−ECC m−v=3 , b=0

CC

DF

PAPRo (dB)


5 6 7 8 9 10 11 12 13 1410

−4

10−3

10−2

10−1

100

Original OFDMO−PTSPTS−MS m−v=3PTS−ECC m−v=3 , b=1

PAPRo (dB)

CC

DF

Fig. 10. CCDF of PTS-ECC in comparison with PTS-MS and O-PTS with𝑚− 𝑣 = 3 for 𝑁=2048.

The performance with 𝑁 = 1024 is given in Fig. 9. Inthis case, the PAPR performance with 𝑏 = 1 and 𝑚 − 𝑣 =2 is similar to that with 𝑏 = 0 and 𝑚 − 𝑣 = 2. There isalso 1.5 to 1.6 dB PAPR improvement compared to PTS-MS.However, with 𝑏 = 2, the performance improvement becomesinsignificant. The multiplicative complexity reduction is 41%and 83% for 𝑚− 𝑣 = 2, and 18% and 66% for 𝑚− 𝑣 = 3.

The CCDF of the OFDM signals using 𝑁 = 2048 with𝑏 = 1 and 𝑚−𝑣 = 3 is shown in Fig. 10. The PAPR reductionis 1.5 dB better than with PTS-MS, and Table II shows amultiplicative complexity reduction of 54% and 81% for 𝑚−𝑣 = 3 over PTS-MS and O-PTS, respectively. For 𝑚 − 𝑣 =2, the number of subblocks is 𝑃 > 16, so this case is notconsidered.

The numerical results presented verify the analysis inSection III-B. ECC subblocking with 𝑏 = 1 provides thebest trade-off between PAPR reduction, number of subblocks,and number of transforms for 512 ≤ 𝑁 < 4095. For32 ≤ 𝑁 < 512, the best PAPR considering the number oftransforms is achieved with 𝑏 = 0.

VI. CONCLUSIONS

In this paper, we considered partial transmit sequence (PTS)in OFDM systems. One of the main drawbacks of PTS is thecomputational complexity due to the calculation of multipletransforms. The construction of the OFDM symbols was con-sidered based on the inputs to each FFT stage. This enables usto compose PTS subblocks over the inputs to each stage, andderive their periodic autocorrelation function (ACF). The ACFwas used to develop a new PTS subblocking technique usingerror-correcting codes (ECCs). This technique minimizes thenumber of repeated subcarriers within a subblock and providesbetter PAPR reduction than pseudo-random or 𝑚-sequencesubblocking. We also presented a computational complexityanalysis and showed that using decimation in frequency (DIF)provides a lower multiplicative complexity than DIT. Finally,we presented a PAPR reduction and multiplicative complexitycomparison between our proposed technique and previous


approaches. This shows that the new PTS subblocks providesignificant PAPR reduction with low complexity.

APPENDIX ATHE NORMALIZED PERIODIC AUTOCORRELATION OF PTS

RADIX FFT SUBBLOCKS

By inserting (13) into (14) and using this in (16) we obtain

𝑅𝑝(𝑛0) =1

𝜎2𝑁2𝐸

⎧⎨⎩⎡⎣ 𝑟𝑣−1∑

𝜂 1=1

𝑌 ∗ 𝜂 1𝑝

(𝑟𝑘1 + 𝑘0

)⎤⎦⎡⎣ 𝑟𝑣−1∑

𝜂 2=1

𝑌 𝜂 2𝑝

(𝑟𝑘2 + 𝑘0

)⎤⎦⎫⎬⎭ (33)

Since we can obtain the IFFT output based on the correspond-ing FFT, �̃� is equivalent to 𝑘, and again using (13) we have

𝑅𝑝(𝑛0) =1

𝜎2𝑁2𝐸

⎧⎨⎩⎡⎣𝑟𝑣−1∑

𝜂1=1

𝑁𝑟𝑣

−1∑˜𝑘1=0

((𝑟−1∑𝑖1=0

𝑦∗𝜂1𝑝

(𝑘1 +

𝑁

𝑟𝑣𝑖1

)𝑇 −𝑖1𝑘0

𝑟

)𝑇−˜𝑘1𝑘0

𝑁/𝑟𝑣−1

)𝑇−𝑛1

˜𝑘1𝑁/𝑟𝑣

]⎡⎣𝑟𝑣−1∑

𝜂2=1

𝑁𝑟𝑣

−1∑˜𝑘2=0

((𝑟−1∑𝑖2=0

𝑦𝜂2𝑝

(𝑘2 +

𝑁

𝑟𝑣𝑖2

)𝑇 𝑖2𝑘0

𝑟

)

𝑇˜𝑘2𝑘0

𝑁/𝑟𝑣−1) 𝑇𝑛2˜𝑘2

𝑁/𝑟𝑣

]}(34)

Letting �̃�1 = �̃� and �̃�2 = �̃�+ 𝑛0, (34) can be written as

𝑅𝑝(𝑛0) =1

𝜎2𝑁2

𝑟𝑣−1∑𝜂1=1

𝑟𝑣−1∑𝜂2=1

𝑁𝑟𝑣 −1∑˜𝑘1=0

𝑁𝑟𝑣 −1∑˜𝑘2=0

𝑟−1∑𝑖1=0

𝑟−1∑𝑖2=0

𝐸

{𝑦∗ 𝜂1𝑝

(𝑘1 +

𝑁

𝑟𝑣𝑖1

)𝑦 𝜂2𝑝

(𝑘2 +

𝑁

𝑟𝑣𝑖2

)}𝑇 (𝑖2−𝑖1)𝑘0

𝑟 𝑇(˜𝑘2−˜𝑘1) 𝑘0

𝑁/𝑟𝑣−1 𝑇𝑛1(˜𝑘1−˜𝑘2)−˜𝑘1𝑛0

𝑁/𝑟𝑣 (35)

Assuming that the twiddle factor amplitudes are uncorrelatedfor 𝑘1 ∕= 𝑘2 and 𝑖1 ∕= 𝑖2, we can write (35) as

𝑅𝑝(𝑛0) =1

𝜎2𝑁2

𝑟𝑣−1∑𝜂=1

𝑁𝑟𝑣 −1∑�̃�=0

𝑟−1∑𝑖=0

𝐸

{𝑦∗ 𝜂𝑝

(𝑘 +𝑁

𝑟𝑣𝑖

)𝑦 𝜂𝑝

(𝑘 +𝑁

𝑟𝑣𝑖

)}𝑇 −˜𝑘 𝑛0

𝑁/𝑟𝑣 (36)

Finally, considering (14) and using the IFFT output, we obtain(17).

ACKNOWLEDGMENT

The authors would like to thank the anonymous reviewersfor their helpful comments.

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A. Ghassemi received the M.A.Sc. degree in Electri-cal Engineering from the University of Victoria, Vic-toria, BC, Canada in 2003. He is currently workingtowards the Ph.D. degree in Electrical Engineering atthe University of Victoria, BC, Canada. His researchinterests are signal processing in communications,in particular multicarrier modulation (OFDM) forwireless communication systems.

T. Aaron Gulliver received the Ph.D. degree inElectrical Engineering from the University of Vic-toria, Victoria, BC, Canada in 1989. From 1989to 1991 he was employed as a Defence Scientistat Defence Research Establishment Ottawa, Ottawa,ON, Canada. He has held academic positions atCarleton University, Ottawa, and the University ofCanterbury, Christchurch, New Zealand. He joinedthe University of Victoria in 1999 and is a Pro-fessor in the Department of Electrical and Com-puter Engineering. In 2002, he became a Fellow of

the Engineering Institute of Canada. He is currently an Editor for IEEETRANSACTIONS ON WIRELESS COMMUNICATIONS. From 2000-2003, hewas Secretary and a member of the Board of Governors of the IEEEInformation Theory Society. His research interests include information theoryand communication theory, algebraic coding theory, MIMO systems and ultrawideband communications.

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