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PAPR Reduction of OFDM Using PTS andError-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 transformsis a problem in practical systems. To address the complexity,signals at the middle stages of an N -point radix FFT usingdecimation in frequency (DIF) are employed for PTS subblocking.We formulate OFDM symbols based on these signals to exploitthe periodic 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 partitioning.the multiple Fourier transforms is greatly reduced. Numericalresults are presented which confirm the PAPR improvements.

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) isan effective multicarrier transmission technique for wirelesscommunications over frequency-selective channels. Using aninverse fast Fourier transform (IFFT) and a fast Fouriertransform (FFT) for the baseband modulation and demodula-tion, respectively, simplifies the design of the transceiver andprovides for an efficient hardware implementation. However,the time-domain OFDM signal can exhibit a large peak-to-average power ratio (PAPR). These peaks can cause nonlineardistortion which introduces spectral spreading, intermodula-tion, and changes in the signal constellation. One solutionto this problem is to employ an expensive power amplifierwith a large linear range. Other techniques are based on signalmodification.

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 areasonable tradeoff between PAPR reduction and BER per-formance. Many distortionless techniques have been proposed[9]-[21]. Coding schemes [9]-[12] sacrifice the data rate. They

Manuscript received December 21, 2006; revised May 31, 2007. This paperapproved by Editor Hamid Jafarkhani.

A. Ghassemi and T. A. Gulliver are with the Department of Electrical andComputer Engineering, University of Victoria, P.O. Box 3055, STN CSC,Victoria, BC V8W 3P6 Canada, email: aghassem,agullive@ece.uvic.ca.

require 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 constructPTS 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-

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random [19] and m-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 and O-PTS for PAPR reduction are reviewed. The PTS radix N -pointFFT is introduced and its intermediate signals are formulatedusing a recursive expression in Section III. The ACF of thePTS subblocks and their partitioning using error-correctingcodes is presented in Section IV. The computational complex-ity analysis is given in Section V. Some numerical performanceand complexity results are also provided in Section 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) =1N

N−1∑k=0

X(k)T−nkN (1)

where TN = e−j2π/N (known as the twiddle factor) and j 2 =−1. In matrix notation, we can express (1) as

x =1N

[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 · · · Tk(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 σ 2,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

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[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) log2 W bits per OFDM symbol required forexplicit side information. The search complexity to find thelowest PAPR sequence is W P−1. To reduce this complexity,we restrict the phase factors to values in the set θp ∈{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 conjugateof the input and output sequences while using the samediscrete Fourier transform (DFT) parameters [25]

x(n) =1N

[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. Asa consequence, the computational complexity is reduced fromO(N2) to O(N logr N). The value of r corresponds to a radix-r FFT algorithm using either DIF or DIT. The DIF radix-ralgorithm 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 n 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∑en=0

((r−1∑i=0

y η(n +N

rvi)T ik0

r

)T en k0

N/rv−1

)T kven

N/rv (12)

where k = 0, . . . , N/rv − 1, n = 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 vand 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∑en=0((

r−1∑i=0

y η(n +N

rvi)T ik0

r

)T en k0

N/rv−1

)T

ek enN/rv (13)

The inputs, y η(n + 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−jθp1N

⎡⎣rv−1∑η=1

Y ηp (rk + k0)

⎤⎦∗

=P−1∑p=0

e−jθpxp(n) (14)

where the PTSs are

1N

⎡⎣rv−1∑η=1

Y ηp (rk + k0)

⎤⎦∗

(15)

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

at stage v.In order to recover the data at the receiver, we use the same

coefficients 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 or m-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.

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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) =1σ2

E{[xp((n + n0)mod N)][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∑en=0

r−1∑i=0

E

{y η

p

(n +

N

rvi

)y ∗η

p

(n +

N

rvi

)}T en n0

N/rv (17)

where n0 = 0, . . . , N − 1. We introduce the variable M ηen,i

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

Rp(n0) =1

σ2N2

rv−1∑η=1

Nrv −1∑en=0

∑i=0

Mηen,i

r−1T enn0

N/rv (18)

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

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

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

rv i) associatedwith subblock p.

• If we increase the number of active η, this increases theautocorrelation function for a particular subblock p andvalue n, i.e. if all inputs over stage v are consideredfor subblocking, the twiddle factor T −enn0

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+ 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 number ofidentical 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(n + 4i) where η = 1, . . . , 4, i = 0, 1, n = 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 within thesubblocks 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 =

⎡⎢⎢⎢⎣S0

S1

...Srv−1−1

⎤⎥⎥⎥⎦ =

⎡⎢⎢⎢⎣s0 s0 . . . s0

s1 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 =

⎡⎢⎢⎢⎣c0

c1

...crv−1−1

⎤⎥⎥⎥⎦ =

⎡⎢⎢⎢⎣U + S0(modP )U + S1(modP )

...U + Srv−1−1(modP )

⎤⎥⎥⎥⎦ (20)

These codeword have maximum Hamming distance N/r v−1

as they differ in all positions.

2) Case rv−1 > N/rv−1 : We construct the C with dimensionsN/rv−1×rv−1 and then take the transpose of C to obtain thecodeword 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. This results in a matrix Swith dimensions N/rv−1 × rv−1. A pseudo-random sequenceU = [u0u1 . . . urv−1−1] without repeated elements of lengthrv−1 is generated over Zrv−1 . From (20), the codeword matrixC is obtained with dimensions N/rv−1 × rv−1. Finally, weobtain C by taking the transpose of 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

⎤⎥⎥⎦

5

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 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 2 b 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 8 to4. 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 ECC subblocks isdepicted in Fig 6. This shows that even with repeated sub-carriers (b = 1), the proposed ECC subblocking provides lowautocorrelation PTS sequences. If we consider Case 3 withv = 3, the number of subblocks is reduced from 4 to 2.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 using m-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 enk0

N/rv−1 and T enk0N/rm−v as shown

in Figs. 1 and 2, respectively. The twiddle factors T ik0r are

trivial (±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 enk0

N/rv−1 and

T enk0N/rm−v when n �= 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 enk0N/rv−1 at stage v for

the DIF algorithm. Summing over all dimensions n, 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)

6

which is obviously greater than one. Similarity for the DIT al-gorithm, we can obtain the number of twiddle factors T enk0

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 complexityfor 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 are 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 > 2048,we require P > 16 in order to improve the PAPR. We comparethe CCDF of O-PTS [19], with PTS-MS [22] and PTS-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 proposed

technique achieves a multiplicative reduction of 41% and 78%for m−v = 2, and 16% and 56% for m−v = 3, over PTS-MSand O-PTS, respectively.

The results for N = 512 are shown in Fig. 8. With P = 16for m−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.

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 in Sec-tion III-B. ECC subblocking with b = 1 provides the besttrade-off between PAPR reduction, number of subblocks, andnumber of transforms for 512 ≤ N < 4095. For 32 ≤ N <512, the best PAPR considering the number of transforms isachieved 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 previousapproaches. This shows that the new PTS subblocks providesignificant PAPR reduction with low complexity.

7

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

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

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 for N =32, v = 3, P = 4 and r = 2 with repeated subcarriers within each subblock.

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 for N =32, v = 3, P = 4, r = 2, b = 0, m-sequence and ECC subblocks, p0 andp1.

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 for N =32, v = 3, P = 4, r = 2, b = 0, m-sequence and ECC subblocks, p2 andp3.

8

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 for N =32, v = 4, P = 4, r = 2, b = 1, and ECC subblocks, p0, p1, p2, and p3.

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)

Fig. 8. CCDF of PTS-ECC in comparison with PTS-MS and O-PTS withm − v = 2 and m − v = 3 for N=512.

TABLE I

THE 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

TABLE II

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

OF PAPR IMPROVEMENT AND COMPUTATIONAL COMPLEXITY

REDUCTION 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

APPENDIX ITHE 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)

9

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)

Fig. 9. CCDF of PTS-ECC in comparison with PTS-MS and O-PTS withm − v = 2 and m − v = 3 for N=1024.

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.

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

Rp(n0) =1

σ2N2E

⎧⎨⎩⎡⎣rv−1∑

η1=1

Nrv −1∑fk1=0

((r−1∑i1=0

y∗η1p

(k1 +

N

rvi1

)T −i1k0

r

)T−fk1k0

N/rv−1

)T−fn1fk1

N/rv

]⎡⎣rv−1∑

η2=1

Nrv −1∑fk2=0

((r−1∑i2=0

yη2p

(k2 +

N

rvi2

)T i2k0

r

)

Tfk2k0N/rv−1) T fn2fk2

N/rv

]}(34)

Letting n1 = n and n2 = n + n0, (34) can be written as

Rp(n0) =1

σ2N2

rv−1∑η1=1

rv−1∑η2=1

Nrv −1∑fk1=0

Nrv −1∑fk2=0

r−1∑i1=0

r−1∑i2=0

E

{y∗ η1

p

(k1 +

N

rvi1

)y η2

p

(k2 +

N

rvi2

)}T (i2−i1)k0

r T(fk2−fk1) k0

N/rv−1 Tfn1(fk1−fk2)−fk1n0N/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∑en=0

r−1∑i=0

E

{y∗ η

p

(k +

N

rvi

)y η

p

(k +

N

rvi

)}T −ek 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 Electrical Engineering fromthe University of Victoria, Victoria, BC, Canada in 2003. He is currentlyworking towards the Ph.D. degree in Electrical Engineering at the Universityof Victoria, BC, Canada. His research interests are signal processing incommunications, in particular multicarrier modulation (OFDM) for wirelesscommunication systems.

T. Aaron Gulliver received the Ph.D. degree in Electrical Engineering fromthe University of Victoria, Victoria, BC, Canada in 1989. From 1989 to 1991he was employed as a Defence Scientist at Defence Research EstablishmentOttawa, Ottawa, ON, Canada. He has held academic positions at CarletonUniversity, Ottawa, and the University of Canterbury, Christchurch, NewZealand. He joined the University of Victoria in 1999 and is a Professorin the Department of Electrical and Computer Engineering. In 2002, hebecame a Fellow of the Engineering Institute of Canada. He is currentlyan Editor for IEEE Transactions on Wireless Communications. From 2000-2003, he was 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.