Wireless Pers CommunDOI 10.1007/s11277-012-0953-3

Selected Mapping Technique for PAPR ReductionWithout Side Information Based on m-Sequence

Saber Meymanatabadi · Javad Musevi Niya ·Behzad Mozaffari

© Springer Science+Business Media New York 2012

Abstract Selected mapping (SLM) is a technique for reducing the high peak-to-averagepower ratio (PAPR) in which a suitable signal is selected among a set of alternative signalswhich all indicate alike information. The chief drawback existing in this method is thattransmitter is compelled to send several additional bits called side information (SI) for eachdata block in order that recovering at the receiver side can be possible. In this paper, we presenta novel SLM scheme by using the linear feedback shift register circuit and m-sequence namedMSLM technique by which any side information bit is not explicitly sent. In MSLM, Thebasic idea is to fit the side information into transmitted symbols based upon which somespecial locations in the transmitted data block are expanded, i.e. some transmitted symbolsare extended. In the receiver side, by using some properties of m-sequence the SI bits canbe detected. We present the example of our method for an OFDM system through the use of16-QAM modulation and different m-sequences and finally, concerned results are illustratedfrom the view point of bit error rate, probability of detection failure and PAPR reduction.

Keywords Orthogonal frequency division multiplexing (OFDM) · Peak-to-averagepower ratio (PAPR) · Selected mapping · Side information

1 Introduction

Orthogonal frequency division multiplexing (OFDM) is an interesting method in wirelesscommunications. However, a famous challenge of OFDM is high peak-to-average powerratio (PAPR) in which the waveform of transmitted signals in time-domain indicates a largeamplitude range. High PAPR takes out the high power amplifier from linear region and leadto high Bit Error Rate (BER) and Probability of frame detection failure [1–3]. Plenty oftechniques have been suggested to reduce PAPR the most well-known of which is Selected

S. Meymanatabadi (B)· J. Musevi Niya · B. MozaffariFaculty of Electrical and Computer Engineering, University of Tabriz, Tabriz, Irane-mail:smeymanat@tabrizu.ac.ir

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mapping (SLM) method [4–6]. Because its implementation is easy; also, it can reach betterPAPR by modifying the OFDM signal without distortion [7–10].

In SLM, some phase sequences are predetermined at transmitter then through multiplyingthe input data by phase sequences a set of signals which all shows the same information aregenerated. After that, the one having the lowest PAPR is selected for sending. In order todetect original data at the receiver, transmitter is forced to transmit the selected phase sequenceindex called side information (SI). The transmitted SI bits cause bandwidth efficiency to bedecreased; besides, when SI is detected incorrectly by receiver, all received frame will belost. One of the methods to oppose this problem is using of strong channel code to protectthe side information, as a result of which system complication and reduction of data rate willbe appeared.

For the purpose of transmission data block without side information bits we proposea method by using the m-sequence named MSLM, by which any side information bit isnot explicitly sent. The main idea is to fit the SI into transmitted symbols based on whichsome special locations in the transmitted data block are expanded, i.e. some transmittedsymbols are extended. With the aim of recovering the SI index at the receiver side, itis tried to be discovered the places of the expanded symbols. Because, in our proposedscheme, discovering the locations of expanded symbols is corresponding to finding the SIindex.

This paper is organized as follows, In Sect. 2, maximum length binary sequence (m-sequence) is reviewed. In Sect. 3, we present the proposed SLM in detail. Following that,performance evaluations of MSLM method are exhibited in Sect. 4. Finally, Sect. 5 concludesthe paper.

2 Review of m-Sequence

The linear feedback shift register (LFSR) circuit is a method of generating periodic pseudo-random sequences. In LFSR, the first state value is named the seed and input bit has linearrelation with prior state [11,12]. However, for the goal of producing random sequences with aso long period it is required to use well-chosen feedback in LFSR circuit in Fig. 1. Any LFSRcircuit is associated with a polynomial referred to generator polynomial which determinessituation of feedbacks; accordingly, the well-chosen feedback depends on selecting a suitablegenerator polynomial [13]. A sequence generated by an m-stage LFSR with period 2m − 1 iscalled a maximal length sequence, or m-sequence. To create a LFSR sequence having length2m − 1, maximum length sequence, we need a primitive polynomial of degree m. In thispaper, we specify the generator polynomial by g(α):

g(α) = gmαm + gm−1αm−1 + · · · + g1α + g0 (1)

Fig. 1 Shift register circuit

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The coefficients gi are 1 when feedbacks in Fig. 1 are connected and 0 otherwise. In designof LFSR circuits, the number of LFSR stages is determined by the m-value; furthermore, allalgebraic operations are performed in modulo-2.

Since each of the m-stages, si , contains 0 or 1, there are 2m possible states for LFSR. Whenall-zero state happens, the output sequence will be all zero. By deducting the all-zero state,maximum possible period is 2m −1. At ith clock pulse, the LFSR state is a finite-length vectordenoted by: si = (si (m − 1), si (m − 2), . . . , si (0)), so the output sequence is βi = si (0). Bysubstituting αm with βi+m in (1) and g(α) = 0, we have:

βi+m = gm−1βi+m−1 + gm−2βi+m−2 + · · · + g1βi+1 + βi (mod 2). (2)

There are N cyclic shifts for a sequence with period N. It is significant to say thatβ-sequence is an infinite vector with period N. Hence, we consider a period of β-sequence,i.e. β = (β0, β1, . . . , βN−1). A polynomial is irreducible if it is not the multiplication oftwo polynomials of degree greater than zero. An irreducible polynomial of degree m, g(α),is primitive if the powers αi are distinct modulo g(α) for 1 ≤ i ≤ 2m − 1.Note that, any m-sequence contains 2m−1 ones and 2m−1 − 1 zeros [14].

3 Description of Proposed Technique (MSLM)

In this part, we use marking in the shape of B = b(R)r to express a vector B constituted of R

scalar quantities br for r ∈ {1, 2, . . . , R}.3.1 The MSLM Transmitter

Suppose an OFDM system with N subcarriers which N complex data symbols dn aretransmitted concurrently on N subcarriers; in addition, data block is represented in theform of D = d(N )

n . In the conventional SLM technique, to generate a sum of variousOFDM signals, it is needed to multiple the original data block with L phase sequencesQl = q(N )

l,n = [ql,0, ql,1, . . . , ql,N−1], l ∈ {0, 1, . . . , L − 1} which Ql is made up of Ncomplex numbers ql,n defined in order that

∣∣ql,n

∣∣ = 1 where |·| denotes the absolute value

operator. Therefore L variant vectors Dl = d(N )l,n are generated where dl,n = ql,n · dn . After

implementing inverse discrete Fourier transform, corresponding vector D′l with N symbols

d ′l,r , r ∈ {0, 1, . . . , N − 1} is considered as follow:

d ′l,r = 1√

N

N−1∑

n=0

dl,n · e j2πnr/N (3)

Among the L vectors D′l , the one having the lowest PAPR is selected for sending. In other

words, the SLM vector that produces the OFDM waveform with minimum PAPR is selectedfor transmission. Figure 2 depicts a block diagram of the SLM method. To discover theoriginal data block D at receiver, it is required to be sent the number of

⌊

logL2

⌋

SI bits bytransmitter. The SI bits specifies the special phase vector having been minimized the PAPRamong the L phase sequences.

In this paper, we propose a method based on the m-sequence in which some speciallocations in the transmitted data block D are expanded and play the role of side infor-mation. In our proposed scheme, the phase vectors Ql = q(N )

l,n = [ql,0, ql,1, . . . , ql,N−1],l ∈ {0, 1, . . . , L − 1} are such that, for each Ql , the absolute value of some elements ql,n are

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Fig. 2 Block diagram of SLM OFDM system

set to a constant value C >1, called expansion factor, while the other elements ql,n remainhave the absolute value equal to the unit. The phases of elements ql,n , as in conventionalSLM, are set to any random values. Note that the locations whose absolute values are C>1 for each ql,n are not identical. These locations are the same locations in which ones ofm-sequence exist and are represented by λl composed of 2m−1 locations. For example, if asupposed vector Ql is associated with λl = {3, 5, 6, 7}, it indicates that in the vector of Ql

only the elements existing in locations n=3,5,6 and 7 have the absolute value equal to C > 1;that is, λl shows the SI index. So, for a given vector Ql there is one and only one associatedλl and vice-versa.

For creating the set of phase vectors we act as follows:

Step1 For a given phase vector Ql , the absolute value of elements ql,n are set basedon m-sequence as, the first “m” elements of a phase vector Ql , play the role of LFSRstates and by using a primitive polynomial of degree m the other elements ql,n remainare produced. Therefore, in the MSLM scheme, the number of phase vectors Ql is equalto the number of non-zero states of LFSR circuit; that is, (L = 2m − 1). We illustratethis subject via two examples in the cases of m = 3 and m = 4 by Tables 1 and 2.

Table 1 List of L = 7 phase vectors for an OFDM system with N = 7 subcarriers before using the mapping(step1)

Phase vectors (step1)

m = 3

l = 1 0 0 1 0 1 1 1

l = 2 0 1 0 1 1 1 0

l = 3 0 1 1 1 0 0 1

l = 4 1 0 0 1 0 1 1

l = 5 1 0 1 1 1 0 0

l = 6 1 1 0 0 1 0 1

l = 7 1 1 1 0 0 1 0

Phase vectors are created by means of m = 3 and generator primitive polynomial g(α) = α3 + α + 1

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Table 2 List of L = 15 phase vectors for an OFDM system with N = 15 subcarriers before using the mapping(step1)

Phase vectors (step1)

m = 4

l = 1 0 0 0 1 0 0 1 1 0 1 0 1 1 1 1

l = 2 0 0 1 0 0 1 1 0 1 0 1 1 1 1 0

l = 3 0 0 1 1 0 1 0 1 1 1 1 0 0 0 1

l = 4 0 1 0 0 1 1 0 1 0 1 1 1 1 0 0

l = 5 0 1 0 1 1 1 1 0 0 0 1 0 0 1 1

l = 6 0 1 1 0 1 0 1 1 1 1 0 0 0 1 0

l = 7 0 1 1 1 1 0 0 0 1 0 0 1 1 0 1

l = 8 1 0 0 0 1 0 0 1 1 0 1 0 1 1 1

l = 9 1 0 0 1 1 0 1 0 1 1 1 1 0 0 0

l = 10 1 0 1 0 1 1 1 1 0 0 0 1 0 0 1

l = 11 1 0 1 1 1 1 0 0 0 1 0 0 1 1 0

l = 12 1 1 0 0 0 1 0 0 1 1 0 1 0 1 1

l = 13 1 1 0 1 0 1 1 1 1 0 0 0 1 0 0

l = 14 1 1 1 0 0 0 1 0 0 1 1 0 1 0 1

l = 15 1 1 1 1 0 0 0 1 0 0 1 1 0 1 0

Phase vectors are created by means of m = 4 and generator primitive polynomial g(α) = α4 + α + 1

Table 3 List of L = 7 phasevectors for an OFDM systemwith N = 7 subcarriers afterusing the mapping (step2)

Phase vectors are created byusing m = 3, generator primitivepolynomial g(α) = α3 + α + 1and C = c > 1. Just absolutevalue of the elements have beenshown because the phases of theelements are chosen randomly

Phase vectors (step2)

m = 3

l = 1 1 1 C 1 C C C

l = 2 1 C 1 C C C 1

l = 3 1 C C C 1 1 C

l = 4 C 1 1 C 1 C C

l = 5 C 1 C C C 1 1

l = 6 C C 1 1 C 1 C

l = 7 C C C 1 1 C 1

Step2 In each phase vector Ql obtained from previous step, the ones and zeros aremapped to C > 1 and 1, respectively. For a given phase vector Ql the absolute valueof the elements which are set to a constant value C > 1 form a set λl . In fact, the setλl shows the side information that is not clearly sent due to the fact that it is fit intothe vector Ql . The phases of the elements in each Ql are chosen randomly same asconventional SLM. For more explain, in Table 3 the phase vectors for an OFDM systemwith N = 7 subcarriers have been shown. These phase vectors are generated when m = 3,generator primitive polynomial is g(α) = α3 + α + 1 and C = c > 1; moreover, inTable 4 the phase vectors for an OFDM system with N = 15 subcarriers by using m = 4,generator primitive polynomial g(α) = α4 + α + 1 and C = c > 1 have been depicted.

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Table 4 List of L = 15 phase vectors for an OFDM system with N = 15 subcarriers after using the mapping(step2)

Phase vectors (step2)

m = 4

l = 1 1 1 1 C 1 1 C C 1 C 1 C C C C

l = 2 1 1 C 1 1 C C 1 C 1 C C C C 1

l = 3 1 1 C C 1 C 1 C C C C 1 1 1 C

l = 4 1 C 1 1 C C 1 C 1 C C C C 1 1

l = 5 1 C 1 C C C C 1 1 1 C 1 1 C C

l = 6 1 C C 1 C 1 C C C C 1 1 1 C 1

l = 7 1 C C C C 1 1 1 C 1 1 C C 1 C

l = 8 C 1 1 1 C 1 1 C C 1 C 1 C C C

l = 9 C 1 1 C C 1 C 1 C C C C 1 1 1

l = 10 C 1 C 1 C C C C 1 1 1 C 1 1 C

l = 11 C 1 C C C C 1 1 1 C 1 1 C C 1

l = 12 C C 1 1 1 C 1 1 C C 1 C 1 C C

l = 13 C C 1 C 1 C C C C 1 1 1 C 1 1

l = 14 C C C 1 1 1 C 1 1 C C 1 C 1 C

l = 15 C C C C 1 1 1 C 1 1 C C 1 C 1

Phase vectors are created by using m = 4, generator primitive polynomial g(α) = α4 +α +1 and C = c > 1.Just absolute value of the elements have been shown because the phases of the elements are chosen randomly

In these figures, we show just the absolute value of the elements because the phases ofthe elements are chosen randomly.

Note that the number of the primitive polynomial of degree m is 1m ϕ(2m − 1), where ϕ

is the Euler’s function. The Euler’s function ϕ(γ ) for integer γ is defined as the number ofpositive integers not greater than and co prime to γ . (Two numbers are co prime when theyshare no factor other than 1). Therefore, in our technique, for detecting the phase vectors atthe receiver side, it is necessary that transmitter and receiver arrive at an agreement about thevalue of “m” and a primitive polynomial of degree m. We explain the detection algorithm indetail in Sect. 3.2.

When all the phase vectors Ql were made, our scheme acts as in conventional SLM; thatis, the data block D is multiplied element by element with each phase vector so that L vectorsDl = d(N )

l,n with dl,n = ql,n · dn and also L corresponding D′l are produced. Eventually, the

vector with the lowest PAPR is selected for transmission. All over this paper, this specialvector is associated with the vectors DS and QS where

QS = q(N )s,n , DS = d(N )

s,n , s ∈ {0, 1, . . . , L − 1} (4)

In MSLM method, once the data block D is multiplied with the phase vectors Ql , the averageenergy for each transmitted symbol raises due to the fact that

∣∣dl,n

∣∣ = ∣

∣ql,n∣∣ · |dn |, with

∣∣ql,n

∣∣

equal to one or C > 1, argues that E[∣∣dl,n

∣∣2]

> E[|dn |2], where E[.] is the expectation

operator.

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3.2 The MSLM Receiver

In this paper, we investigate MSLM scheme on a Flat fading channel. The frequency-domainof each transmitted symbol ds,n after passing the Flat fading channel is correspondent to:

ys,n = hn · ds,n + nn n ∈ {0, 1, . . . , N − 1} (5)

where hn , a real sample, indicates the fading affecting the nth subcarrier and nn , a complexsample, exhibits complex Gaussian noise with zero mean and variance σ 2. The receiver,according to the agreement between receiver and transmitter, is aware of the value of mand corresponding primitive polynomial g(α). That is, the receiver side knows that for eachN-sample vector (N = 2m − 1) inside the transmitted data (Ys), some symbols have beenexpanded by a factor C, (2m−1 symbols), and other symbols remain have not been expanded(2m−1 − 1 symbols). By finding the locations of the expanded symbols, the receiver canrecover the SI index so, at the receiver side, it is tried to discover the places of the expandedsymbols to recover the SI index.

Before exhibiting the detection algorithm in the receiver side, it is necessary to explainsome properties of m-sequence [14].

Property 1- The Window Property:By sliding a window of width m along an m-sequence, the number of 2m − 1 m-foldwindows are extracted, each of which is seen exactly once.Property 2- The Run Property:A run is defined as a sequence of all 1s or a sequence of all 0s. In the m-sequence, a runof 1s of length m happens exactly once.Property 3- Z-locations:Any m-sequence includes 2m−1 − 1 0s. If position of the run of 1s of length m becomedistinguished, the locations of 2m−1 − 1 0s (Z-locations) will become specified.

The detection algorithm in receiver is made of the following stages (I to IV):

I. The average energy of each received symbol in location i is calculated as:

P = pi = |yi |2 − σ 2

(hi )2 i ∈ {1, 2, . . . , K ∗(2m − 1)} (6)

II. According to the property 1, a window of width m is slid along the vector P and 2m − 1successive m-fold windows are extracted.

W = w(m)j j ∈ {1, 2, . . . , 2m − 1} (7)

According to the property 2 (run property), it is tried to find the window to which a runof 1s of length m is attributed.

III. By supposing each of the windows (w(m)j ) as the run of 1s of length m, according to the

property 3, we calculate the average energy of the symbols that exist in the Z-locations(based on the supposed window w

(m)j ) in the N-symbol received frame.

E j = E[

Z − locations energy based on position w(m)j

]

j ∈ {1, 2, . . . , 2m − 1} (8)

Where, E is Expected Value.

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IV. Among the calculated E j , the one having the lowest value is selected. As a result, the

value of j as well as w(m)j is determined and we can find the real location of the run of

1s of length m. After then, by mapping the ones and zeros to C > 1 and 1, respectively,we can guess the phase vector.

Such evaluations may cause erroneous detection of the index “s”; therefore, in this casethe receiver cannot recover the side information correctly. However, the system designer mustbe aware that for a constant m and a larger amount of C , a larger increase of the energy canbe achieved. This increase of energy can causes the error performance to decrease.

4 Example

In this section, to further explain the MSLM scheme we assume an OFDM system with Nsubcarriers and 16-QAM modulation. The LFSR parameters for generating phase sequencesare N = 63, 127, 255, 511, 1,023 and m = 6, 7, 8, 9, 10; plus, Table 5 shows correspondingprimitive polynomials g(α). The phases of ql,n with equal probabilities are set to either 0 orπ . We suppose the Rayleigh Fading channel for transmission and compare the performanceof the MSLM method with NSLM method, a method of transmitting without side informationbits [15], in terms of BER, Pd f and PAPR reduction.

4.1 Probability of SI Detection Failure(

Pd f)

The probability of SI detection failure Pd f declares the probability that receiver is not ableto recover the side information; in other words, an entire received OFDM frame is lost. InFig. 3, we have plotted the probability of SI detection failure Pd f versus expansion factorC for the various numbers of OFDM subcarriers when SNR = 14 dB. For comparisonpurposes, the Pd f curves of an equivalent OFDM system using the NSLM method have beenplotted. In NSLM method, for each sub vector with the length 5 the number of 2 symbolsis expanded. From Fig. 3, it is obvious that the values of N and C much influence Pd f

performance. It is clear that our SI detection algorithm causes more distinction betweenexpanded and non-expanded symbols so, the MSLM scheme especially in higher value of Cacts better. Additionally, whenever N rises, zero locations in each frame are also increased.So, it provides further valid estimates of the average energies in (8). Consequently, in MSLMscheme, the more increase in N, the better result can be got for any value of expansion factorC. Other modulations, such as QPSK or 64-QAM, can be applied in the proposed schemefor the analysis of the probability of SI detection failure

(

Pd f)

compared to 16-QAM. For agiven set of system parameters (such as the number of subcarriers (N), the agreed primitive

Table 5 Primitive polynomialsg(α) corresponding to variousstates of the shift register(m = 6, 7, 8, 9 and 10) used inour technique for an OFDMsystem with N = 63, 127, 255,511 and 1,023 subcarriers

m Primitive polynomial g(α) N = 2m − 1

6 α6 + α + 1 63

7 α7 + α6 + 1 127

8 α8 + α7 + α6 + α + 1 255

9 α9 + α4 + 1 511

10 α10 + α3 + 1 1,023

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Fig. 3 Comparison of Pd f performance of MSLM and NSLM methods, for four various subcarriers andSNR = 14 dB. For each subcarrier, we use 16-QAM modulation and Rayleigh fading channel for transmissionwith perfect CSI

polynomial g(α), etc.) if lower-order signal constellation, for example QPSK, is used theprobability of SI detection failure

(

Pd f)

will be smaller. But by replacing 64-QAM or 256-QAM with 16-QAM this probability will be worsen. As a conclusion, if the probability of SIdetection failure can be made small enough (e.g., by increasing the expansion factor C and/orthe number of subcarriers), this probability in case QPSK will be smaller than higher-ordersignal constellation.

4.2 BER Performance Evaluation

In Fig. 4, we show the bit error rate (BER) vs. SNR values for our proposed scheme havingsupposed parameters in this section with N = 255 subcarriers and various values of constantC. For comparison purposes, the BER curves of an equivalent OFDM system using the NSLMmethod have been plotted. In NSLM method, for each sub vector with length 5, the number of2 symbols is expanded. As it is seen from Fig. 4, increasing the average energy of transmittedsymbol has a great effect on BER performance. Also, Fig. 4 confirms that operation of ourscheme is relatively satisfactory, for example, by targeting a BER of 10−2 in MSLM curves,it is observed that C = 1.3 like C = 1.4 and 1.5 give the acceptable result; in addition, itapproximates the classical SLM technique with perfect SI specially in C = 1.4 and 1.5.

4.3 PAPR Reduction Performance

In Fig. 5, the complementary cumulative distribution function (CCDF) of the PAPR hasbeen shown. We have considered performance of PAPR reduction, CCDF, with supposedparameters in this section, C = 1.4 as well as N = 63, 127, 255, and 511 subcarriers. Forcomparison aim, the PAPR curves obtained with the NSLM method, in which for each subvector with length 5 the number of 2 symbols is expanded, have been plotted. These results

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Fig. 4 Comparison of BER performance of proposed technique and NSLM method as a function of SNR forN = 255. For each subcarrier, we use 16-QAM modulation and Rayleigh fading channel for transmission withperfect CSI

Fig. 5 CCDF of the PAPR of MSLM technique for different values of subcarriers. For comparison aim, thePAPR curves of NSLM method have been plotted. We use 16-QAM modulation and oversampling factor equalto 4

have been achieved by application of an oversampling factor equal to 4 [16]. According tothe results obtained in Fig. 5, the PAPR reduction performance of the MSLM scheme, for allN values, is better than the NSLM scheme.

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5 Conclusion

In this paper, we have proposed a new SLM scheme named MSLM to reduce the PAPR withouttransmitting the explicit SI bits. In designing of MSLM method, we have used LFSR circuitand m-sequence which causes the phase coefficient vector to be random. Our examinationsperformed via considering an OFDM system by using 16-QAM modulation. In comparisonwith NSLM, a method without transmitting the SI bits explicitly, we found that proposedtechnique in PAPR reduction for all subcarrier numbers works better. In addition, it could beconcluded from the results that BER performance and probability of detection failure of sideinformation in MSLM method have satisfactory results.

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Author Biographies

Saber Meymanatabadi received the B.S. degree and the M.S. degreein Electrical Engineering in 2003 and 2007, respectively. He is cur-rently pursuing the Ph.D. degree in Electrical Engineering at the Uni-versity of Tabriz, Iran. His research interests are signal processingin communications, in particular multicarrier modulation (OFDM) forwireless communication systems.

J. Musevi Niya received his B.S. degree from University of Tehran, hisM.S. degree from Sharif University of Technology and his Ph.D. degreein Electrical Engineering from University of Tabriz. He is working asan academic member of the Faculty of Electrical and Computer Engi-neering, University of Tabriz. His research interests are wireless com-munication systems, and signal processing.

Behzad Mozaffari received his B.S. degree in 1993 from Universityof Tabriz and his M.S. degree in 1996 from K. N.Toosi University ofTechnology and his Ph.D. in Electrical Engineering degree from Uni-versity of Tabriz. He is working as an academic member of the Fac-ulty of Electrical and Computer Engineering, University of Tabriz. Hisresearch interests are signal processing and speech processing.

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