performance analysis of precoded wireless ofdm with

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IEEE SYSTEMS JOURNAL 1

Performance Analysis of Precoded Wireless OFDMWith Carrier Frequency Offset

Fatma Kalbat, Member, IEEE, Arafat Al-Dweik , Senior Member, IEEE, Bayan Sharif, Senior Member, IEEE,and George K. Karagiannidis , Fellow, IEEE

Abstract—Precoding is proved to be highly efficient to improvethe bit error rate (BER) of orthogonal frequency division multiplex-ing (OFDM) in frequency-selective fading channels. Nevertheless,most of the work reported in the literature ignores the radio fre-quency impairments such as carrier frequency offset (CFO), whichis known to be of significant impact on OFDM. Consequently, thispaper considers the performance evaluation of precoded OFDM(P-OFDM) in the presence of CFO. The performance of the con-sidered P-OFDM is evaluated in terms of the signal-to-interferenceplus-noise-ratio (SINR) and BER. Various channel models are con-sidered and closed-form analytical expressions are derived for theexact SINR. The obtained analytical results, corroborated by sim-ulation, show that P-OFDM is substantially more sensitive to CFOcompared with conventional OFDM. Generally speaking, if the nor-malized CFO is about 18%, then OFDM will outperform P-OFDMfor most practical channel models. It is also interesting to note thatthe subcarriers in P-OFDM may experience different SINRs, whichis not the case for conventional OFDM. The paper also considersthe SINR of OFDM after the equalization process. The obtainedresults show that ignoring the impact of the equalization processresults in inflated SINR degradation.

Index Terms—Frequency offset, Haar, interference, minimummean square error (MMSE), orthogonal frequency division multi-plexing (OFDM), precoding, synchronization, Walsh–Hadamard,Zadoff–Chu.

I. INTRODUCTION

ORTHOGONAL frequency division multiplexing (OFDM)has received considerable research attention in wireless

communications because of its ability to improve the spec-tral efficiency and immunity to inter-symbol interference (ISI)in frequency-selective multipath fading environments. Further-more, the ISI can be avoided without the need for complicatedequalizers, with the aid of sufficient cyclic prefix (CP), or usinginterference cancellation schemes [1]. Owing to the remarkable

Manuscript received December 23, 2018; revised March 18, 2019 and May26, 2019; accepted June 2, 2019. (Corresponding author: Arafat Al-Dweik).

F. Kalbat and B. Sharif are with the Department of Electrical and Com-puter Engineering, Khalifa University 127788, Abu Dhabi, UAE (e-mail:[email protected]; [email protected]).

G. K. Karagiannidis is with the Department of Electrical and Computer Engi-neering, Aristotle University of Thessaloniki GR-54124, Thessaloniki, Greece(e-mail: [email protected]).

A. Al-Dweik is with the Department of Electrical and Computer Engineer-ing, Khalifa University 127788, Abu Dhabi, UAE, and also with the KU Cen-ter for Cyber Physical Systems and the Department of Electrical and Com-puter Engineering, Western University, London, ON N6A 5B9, Canada (e-mail:[email protected]).

Digital Object Identifier 10.1109/JSYST.2019.2922098

advantages of OFDM, it has been adopted for several wirelesscommunication standards such as digital video broadcasting [2]and digital audio broadcasting [3]. Additionally, OFDM is usedas the transmission scheme in the physical layer for microwaveaccess (WiMAX) [4], long-term evolution (LTE) standards [5],[6], optical wireless communications (OWCs) [7], [8], and itis considered as a promising candidate for the fifth-generationstandards [9]. The channel is typically modeled as frequency-selective for WiMax and LTE, flat for OWC in the presenceof atmospheric turbulences, and additive white Gaussian noise(AWGN) for OWC without atmospheric turbulence.

However, OFDM has critical drawbacks such as sensitivityto synchronization errors due to radio frequency (RF) front-endrelated imperfections, which are often inevitable due to channelvariation, components mismatch, and manufacturing defects.Common examples are carrier frequency offsets (CFO) [10],[11], timing offsets [12], and I/Q imbalance [13], [14]. Generallyspeaking, RF impairments cause loss of orthogonality betweensubcarriers, and hence, inter-carrier interference (ICI) is intro-duced. OFDM also suffers from large peak-to-average powerratio (PAPR) problem [15].

Despite the many advantages of OFDM, its bit error rate(BER) performance is similar to single-carrier (SC) systems infading channels [16]. Therefore, incorporating diversity tech-niques is essential to improve the performance of OFDM infrequency-selective wireless channels. Moreover, reducing thePAPR should be considered to avoid BER degradation due tothe power amplifier nonlinearity. In this context, using precod-ing with OFDM provides considerable diversity gain and ro-bustness against the frequency selectivity of the channel, andcan reduce the PAPR [17]. Particularly, precoded-OFDM (P-OFDM) based on unitary transforms, such as Walsh–Hadamardtransform (WHT), has received significant attention from theresearchers and is considered as energy and spectrally efficientapproach that can drastically enhance the performance of wire-less systems [17]–[19]. Other transforms such as the Haar trans-form [20] and Zadoff–Chu matrix transform (ZCMT) have beenconsidered in the literature as well [21]. More recently, Popescuand Popescu [22] demonstrated that using sub-band P-OFDMmay enhance the immunity of OFDM to narrowband interfer-ence. Furthermore, the performance of P-OFDM over power-line communications with impulse noise was considered in [23]by using various equalization techniques. The presented resultsshow that P-OFDM outperforms conventional OFDM in suchscenarios.

1937-9234 © 2019 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

https://orcid.org/0000-0002-3487-3438

https://orcid.org/0000-0001-8810-0345

mailto:[email protected]





2 IEEE SYSTEMS JOURNAL

A. Prior Work

Estimating the CFO and investigating its effect on the perfor-mance of OFDM-based wireless communication systems hasmotivated a vast amount of research [24]–[44]. In most previousresearch, it is common to measure the performance degrada-tion due to CFO in terms of the average signal-to-interferenceplus-noise-ratio (SINR) [24]–[29]. A lower and upper boundson the average SINR in the presence of CFO in multipath fadingchannels are derived in [24] and [25]. The derivation of an ex-act expression for the average SINR requires N -fold numericalintegrations to average out the N correlated random variables,where N is the number of subcarriers. However, using an indi-rect mathematical analysis, the exact expression for the averageSINR is derived by Hamdi [29] in the form of a single integral.The impact of CFO and phase noise (PN) is investigated in [31]and [32], where an exact signal-to-interference ratio expressionfor OFDM in the presence of CFO and PN in double-selectivefading channels is presented in [32].

The analysis of BER deterioration due to CFO is consid-ered widely in the literature [33]–[37]. Approximated BER ofOFDM in the presence of CFO for AWGN and multipath fadingchannels is presented in [33] and [34]. BER analysis of binaryphase shift keying (BPSK) and OFDM with random residualfrequency offset has been considered in [35] where a closed-form BER expressions for BPSK in AWGN, flat, and selectivefading channels are derived. Furthermore, the authors providedexpressions for quadrature phase shift keying (QPSK), but onlyfor AWGN and flat fading channels. However, the expressionsderived in [35] are valid only for small normalized CFO. There-fore, Mahesh and Chaturvedi [36] extended the results of [35]for the BPSK case to be valid for any normalized CFO. Further-more, a closed-form expression for the symbol error rate (SER)of QPSK-OFDM with a frequency offset in Rayleigh selectivefading channels is derived in [37]. In addition, the SER of QPSK-OFDM with frequency offset in flat Rayleigh fading channelsthat is valid for any frequency offset was provided. A formulafor the spectral efficiency of OFDM systems in a frequency-selective multipath fading channel, in the presence of CFO, isderived in [38], whereas the impact of the CFO on the spec-trum sensing of OFDM signals is investigated in [39]. Likewise,the effect of CFO on different types of SC frequency divisionmultiple access (FDMA) systems is demonstrated in [40], [41],whereas an analytical expression for the error vector magnitudemeasure of SC-FDMA system under CFO and joint transmit–receive PN is derived in [42]. The performance degradation dueto CFO in virtual multiple-input single-output systems is high-lighted in [43]. The impact of CFO on multiuser detection andestimation performance of LTE system is demonstrated in [44].Finally, a concatenated P-OFDM system is proposed in [30] toenable channel estimation and mitigate the effects of ICI causedby CFO.

B. Contribution

As can be noted from the aforementioned discussion, the im-pact of CFO on OFDM has been considered extensively in theliterature. However, to the best of the authors’ knowledge, noanalysis has been reported in the open technical literature on the

impact of CFO on P-OFDM wireless systems. Consequently,this paper considers the performance evaluation of P-OFDMsystems in the presence of CFO where WHT is used as theprecoding transform due to its implementation efficiency [19].In particular, the paper focus is to analytically derive the aver-age SINR of P-OFDM in AWGN, flat, and frequency-selectivefading channels. Moreover, the conventional equalized OFDMcase is considered as well. The derived average SINR expres-sions are corroborated by Monte Carlo simulation results. Theobtained analytical and simulation results show that P-OFDMis significantly more sensitive to CFO compared with conven-tional OFDM, which may dilute the P-OFDM transmit diversityadvantage, or even make its performance worse than the conven-tional one. Moreover, P-OFDM has entirely different behavioras compared with OFDM where the SINR can be different foreach subcarrier. Therefore, while certain subcarriers are lightlyaffected by CFO, other subcarriers may experience severe SINRdegradation, which may drive the overall BER of P-OFDM tobecome worse than conventional OFDM. Note that the perfor-mance of P-OFDM is also considered with the Haar and ZCMTs.

C. Structure and Notations

The rest of the paper is organized as follows. Section IIpresents P-OFDM system and channel models in the pres-ence of CFO. In Section III, the average SINR expressionsof P-OFDM in AWGN, flat, and frequency-selective Rayleighfading channels in the presence of CFO are derived. Numer-ical and simulation results with discussions are provided inSection IV, whereas conclusions and closing remarks are pre-sented in Section V.

Notations: Unless otherwise stated, lower and upper case boldletters such as x and X denote vectors and matrices, respec-tively. The matrices I and F denote the identity matrix and thediscrete Fourier transform (DFT) matrix, respectively. Addition-ally, (·)H , (·)∗, (·)T , (·)−1 denote the matrix complex-conjugatetranspose, complex-conjugate, transpose, and matrix inverse op-erations, respectively. The operators E [·] and |·| denote the sta-tistical expectation and the absolute value, respectively.

II. P-OFDM SYSTEM AND CHANNEL MODELS

In P-OFDM, a sequence of complex data symbolsa = [a0,a1,. . ., aN−1]

T , each of which has an average power Ps is appliedto N -point WHT to generate the precoded data sequence

b = Wa (1)

where W is the normalized N -point Walsh–Hadamard matrix.The columns and rows of W are normalized to a unit norm,such that all its elements are equal to±1/

√N andW−1W = I.

The mth sample, m ∈ {0, 1, . . . , N − 1}, at the output of WHTblock is a linear combination of all data symbols, and can beexpressed as [19]

bm = wma =N−1∑

n=0

Wm,nan (2)


KALBAT et al.: PERFORMANCE ANALYSIS OF PRECODED WIRELESS OFDM WITH CARRIER FREQUENCY OFFSET 3

where wm is the mth row of W

wm = [Wm,0,Wm,1, . . . ,Wm,N−1] (3)

and Wm,n is the element of the mth row and nth column of W,which can be obtained as

Wm,n =1√N

(−1)∑log2(N)−1

r=0 mrnr (4)

with mr and nr are the bit representation of the integer valuesm and n, respectively.

The composite symbols vector b is applied to an N -pointinverse DFT to generate the time domain samples, similar toconventional OFDM, thus

x = FHb (5)

where FH is the Hermitian transpose of the normalized N ×N DFT matrix F. The elements of FH are defined as FH

i,n =√Nej2πinN where i and n denote the row and column numbers

{i, n} ∈ 0, 1,. . ., N − 1, respectively, and N � 1/N . The ithsample of x can be obtained as

xi =√

N

N−1∑

n=0

bnej2πinN , i = 0, 1, . . . , N − 1. (6)

In the case of imperfect frequency synchronization, the time do-main received sequence y = [y0, . . . , yN−1]

T , after discardingthe P CP samples, can be expressed as [10], [11]

y = C(ε)HFHb+ z (7)

where ε � ΔfRS

∈ (−0.5, 0.5) is the normalized CFO, Δf is theactual CFO, RS is the data symbol rate, and C(ε) represents theaccumulated phase shift on the time-domain samples caused bythe normalized CFO. Thus

C(ε) = diag([

ej2πεN×0, ej2πεN×1, . . . , ej2πεN×(N−1)])

.

(8)Given that the channel has Lh + 1 multipath components, thechannel matrix H is an N ×N circulant matrix with h0 onthe principal diagonal and h1, . . ., hLh

on the minor diagonals,z = [z0, z1, . . ., zN−1]

T denotes the system noise where zi ∼CN (

0, σ2z

). The sequence y is applied to the DFT that produces

the sequence r = [r0, r1, . . . , rN−1]T , where

r = FC(ε)HFHb+ Fz (9)

which can be written as

r = Hb+ η. (10)

The mth element of r can be expressed as

rm =N−1∑

p=0

Hm,pbp + ηm (11)

where Hm,p is the element of the mth row and pth column ofH, which can be expressed as

Hm,p = NN−1∑

k=0

Lh∑

l=0

hl ej2πk(p−m+ε)N e−j2πlpN (12)

or, after some algebraic manipulations

Hm,p = αm,pHp (13)

where Hp =∑Lh

l=0 hle−j2πplN denotes the channel frequency

response at subcarrier p and

αm,p =sin (π(p−m+ ε))

N sin(πN (p−m+ ε)

)ejπ(1−N)(p−m+ε). (14)

The diagonal elements of H can be computed by setting m = p,which gives

αm,m � α =sin (πε)

N sin(πNε

)ejπε(1−N). (15)

To extract the vector b from (10) without ICI, the DFT outputvector r should be equalized using the inverse of the channelmatrix H, i.e., H−1. However, such process is prohibitivelyexpensive due to the complexity of estimating H as well asthe complexity of the matrix inversion process. By noting thatthe diagonal elements of H are dominant as compared with theoff-diagonal elements [45], Then, it is more feasible to estimatethe diagonal elements of H and use them to equalize r, which isthe approach used in this paper. Therefore, the minimum meansquare error (MMSE) equalizer output can be expressed as [46]

v = HHb+ η (16)

where η = Hη, H = diag{[

H0, H1, . . . , HN−1

]}, and

Hm =H∗

m,m

|Hm,m|2 + 1SNR

=α∗H∗

m

|αHm|2 + 1SNR

. (17)

In (17), the SNR = Ps/σ2z . As can be noted from (17), the CFO

affects the equalization process, and accurate channel estimatesrequire the knowledge of ε, which might not be available at thereceiver. Nevertheless, the sensitivity of the system to accurateknowledge of α is mostly negligible because α ≈ 1 for ε ≤ 0.2.Moreover, at moderate and high SNR values, the denominatorof (17) is dominated by 1/SNR.

The mth element of v, m ∈ {0, 1, . . . , N − 1}, is given by

vm =N−1∑

p=0

HmHm,pbp + ηm (18)

where ηm = Hmηm. After equalization, the inverseWHT (IWHT) is computed to produce the sequence

d = W−1v = W−1(HHb+ η

), where the kth element

of d is given by

dk = w−1k v

=

N−1∑

m=0

W−1k,mvm

=

N−1∑

m=0

N−1∑

p=0

W−1k,mHmHm,pbp + ζk (19)



where w−1k is the kth row of W−1 and ζk =

∑N−1m=0 W

−1k,mηm.

By extracting the term that corresponds to the kth data element,then

dk = βk,kak +

N−1∑

n=0n�=k

βk,nan + ζk (20)

where

βk,n =

N−1∑

m=0

N−1∑

p=0

W−1k,mWp,nHmHm,p. (21)

III. SINR ANALYSIS OF P-OFDM

Based on (20), the average SINR for the kth subcarrier canbe defined as

SINRk =E[|βk,kak|2

]

E

⎡

⎣

∣∣∣∣∣∣

N−1∑n=0n�=k

anβk,n

∣∣∣∣∣∣

2⎤

⎦+ σ2ζk

(22)

which can be simplified to

SINRk =PsE(|βk,k|2)

Ps

N−1∑n=0n�=k

E[|βk,n|2

]+ σ2

ζk

(23)

where σ2ζk

= E(∑N−1

m=0 |Hm|2)σ2z

N . Substituting (21) in

(23) gives the SINRk in (24) at the bottom of the pagewhere Ψ1

k,m,p,v,i = W−1k,mWp,kW

−1k,vWi,k, Ψ2

k,m,p,n,v,i =

W−1k,mWp,nW

−1k,vWi,n and u � 1

|α|2SNR .

As can be noted from (24), evaluating SINRk for an ar-bitrary H � diag {[H0, H1, . . . , HN−1]} is infeasible becausecomputing the expected values in the numerator and denom-inator of (24) requires averaging over the joint probabilitydensity function (PDF) of the channel coefficients Hm, m =[0, 1, . . . , N − 1], which requires using N -fold integrals to av-erage out the N correlated random variables for each sub-carrier [29], which is prohibitively expensive. Therefore, (24)can be used to compute SINRk semi-analytically by gener-ating large number of realizations of H, and then computing

E[

H∗mHp

|Hm|2+uHvH

∗i

|Hv |2+u

]∀ {k,m, p, v, i}. As the elements of H are

identically distributed, then, without loss of generality, we can

write E[

|Hm|2(|Hm|2+u)2

]= E

[|H0|2

(|H0|2+u)2

]and, hence

N

SNR

N−1∑

m=0

E

[ |Hm|2(|Hm|2 + u)2

]=

1

SNRE

[ |H0|2(|H0|2 + u)2

].

(25)Although it is intractable to obtain a closed-form expressionfor the SINRk in frequency-selective channels, it is shown inthe following two sections that closed-form expressions can bederived for the special cases of AWGN and flat fading channels.

A. AWGN Channels

For AWGN channels, Hp = 1 ∀p. Consequently, (24) isreduced to

SINRk =

∣∣∣∣∣N−1∑m=0

N−1∑p=0

W−1k,mWp,kαm,p

∣∣∣∣∣

2

N−1∑n=0n�=k

∣∣∣∣∣N−1∑m=0

N−1∑p=0

W−1k,mWp,nαm,p

∣∣∣∣∣

2

+ 1SNR

. (26)

Moreover, the output of the IWHT at the receiver side can beexpressed as

d = WHF(C(ε)IFHWa+ z

)

= Ma+ ζ (27)

where M is given by (28) at the bottom of the next page, ζ =[ζ0, ζ1, . . . , ζN−1]

T where ζk = α∗|α|2+ 1

SNR

∑N−1m=0 W

−1k,mηm and

E[|ζk|2

]� σ2

ζkcan be expressed as

σ2ζk

=

∣∣∣∣α∗

|α|2 + 1SNR

∣∣∣∣2

σ2z . (29)

By noting that M is a block matrix, then not all subcarrierscontribute to the ICI. For example, subcarriers d0 and d1 canbe expressed as d0 = β0,0a0 + ζ0 and d1 = β1,1a1 + ζ1, and,hence, d0 and d1 do not suffer from ICI. Moreover, the atten-uation factors β0,0 and β1,1 can be written as α∗

|α|2+ 1SNR

andα∗

|α|2+ 1SNR

ejπε, respectively, which implies that |β0,0| = |β1,1|.Therefore, SINR0 = SINR1 = SNR. It is worth noting thatsimplifying the values β0,0 and β1,1 is based on the fact thatthe elements of the first row and column of W and W−1 areall ones, and hence, W−1

0,mWp,0 = 1N ,∑N−1

m=0

∑N−1p=0 αm,p = N

∀ε, and∑N−1

m=0

∑N−1p=0 W−1

1,mWp,1αm,p = ejπε.As another example, consider subcarrier d2, which can be

expressed as

d2 = β2,2a2 + β2,3a3 + ζ2. (30)

SINRk =

N−1∑{v,i,m,p}=0

Ψ1k,m,p,v,iαm,pα

∗v,iE

[H∗

mHp

|Hm|2+uHvH

∗i

|Hv |2+u

]

N−1∑{n,v,i,m,p}=0

n�=k

Ψ2k,m,p,n,v,iαm,pα∗

v,iE[

H∗mHp

|Hm|2+u

HvH∗i

|Hv |2+u

]+ N

SNR

N−1∑m=0

E[

|Hm|2(|Hm|2+u)2

] . (24)



Therefore, only a3 contributes to the ICI. Moreover, because|β2,2| = |β3,3| and |β2,3| = |β3,2|, the SINRs of d2 and d3 areidentical.

B. Flat Fading Channels

In flat Rayleigh fading, all channel coefficients are equal, andhence, Hk = H ∀k, Hk = H = H∗

|H|2+ 1SNR

. By defining λ1 =

|H|4[|H|2+u]2

and λ2 = |H|2[|H|2+u]2

, then (24) reduces to

SINRk =

∣∣∣∣∣N−1∑m=0

N−1∑p=0

W−1k,mWp,kαm,p

∣∣∣∣∣

2

N−1∑n=0n�=k

∣∣∣∣∣N−1∑m=0

N−1∑p=0

W−1k,mWp,nαm,p

∣∣∣∣∣

2

+ 1SNR

E[λ2]E[λ1]

.

(31)Assuming that H ∼ CN (

0, σ2H

), then |H|2 � θ ∼ χ2, and by

considering that σ2H = 1, the PDF of θ is given by

fΘ(θ) = e−θ, θ > 0. (32)

Consequently, the values of E [λ1] and E [λ2] can be evaluatedas

E [λ1] =

∫ ∞

0

θ2

(θ + u)2e−θdθ. (33)

Using the change of variables ρ = θ + u and expanding the re-sultant terms, then E [λ1] can be rewritten as

E [λ1] = eu∫ ∞

u

e−ρ − 2ue−ρ

ρ+ u2 e

−ρ

ρ2dρ. (34)

Using [47, p. 133, eq. (44)] and after some straightforward ma-nipulations, the integral in (34) can be evaluated as

E [λ1] = (1 + u) + (2 + u)ueu Ei(−u) (35)

where Ei(·) is the exponential integral.Similarly

E [λ2] =

∫ ∞

0

θ

(θ + u)2e−θdθ. (36)

Using the change of variablesρ = θ + u, expanding the resultantterms and evaluating the integral gives

E [λ2] = eu∫ ∞

u

e−ρ

ρ− u

e−ρ

ρ2dρ

= −Ei(−u)(u+ 1)eu − 1. (37)

Therefore, a closed-form expression for the SINRk in flatRayleigh fading channels can be obtained by substituting E [λ1]and E [λ2] in (31), which are given in (35) and (37), respectively.

It is also worth noting that the output structure of the IWHTat the receiver side is similar to the AWGN case, which can bewritten as depicted in (27), where

βk,n =

(α∗|H|2

|αH|2 + 1SNR

) N−1∑

m=0

N−1∑

p=0

W−1k,mWp,nαm,p (38)

and ζk = α∗H∗|αH|2+ 1

SNR

∑N−1m=0 W

−1k,mηm whose variance is given

by

σ2ζk

= E

(∣∣∣∣α∗H∗

|αH|2 + 1SNR

∣∣∣∣2)σ2z

= E (λ1)σ2z . (39)

As a special case, the first two diagonal elements in the ma-trix M can be simplified to β0,0 = α∗|H|2

|αH|2+ 1SNR

and β1,1 =

α∗|H|2|αH|2+ 1

SNR

ejπε. Therefore, similar to the AWGN caseSINR0 =

SINR1, where

SINR0 =E(|β0,0|2

)Ps

σ2ζkσ2z

= SNRE (λ2)

E (λ1).

By noting the values of E (λ1) and E (λ2) given in (35) and(37), respectively, it can be numerically demonstrated thatE(λ2)E(λ1)

< 1 for SNR � 0 dB, which implies that SINR0|Flat <

SINR0|AWGN.It is also worth noting that SINR of the equalized conventional

OFDM can be derived as a special case of the P-OFDM byreplacing the WHT/IWHT matrices by the identity matrix, i.e.,

M =

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

β0,0 0 0 · · · 0 0 0 0 0

0 β1,1 0 · · · 0 0 0 0 0

0 0 β2,2 β2,3 0 0 0 0 0

0 0 β3,2 β3,3 0 0 0 0 0

......

......

. . . 0 0 0 0

0 0 0 0 0 βN2 ,N2

βN2 ,N2 +1 · · · βN

2 ,N−1

0 0 0 0 0 βN2 +1,N2

βN2 +1,N2 +1 · · · βN

2 +1,N−1

0 0 0 0 0...

......

...

0 0 0 0 0 βN−1,N2βN−1,N2 +1 · · · βN−1,N−1

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

. (28)



Fig. 1. Analytical, semi-analytical, and simulated average SINR per subcarrier, ε = 0.1 and , = 20 dB. (a) AWGN. (b) Flat fading. (c) Frequency-selective.

Wn,m = 1 forn = m, and 0 otherwise. As shown in Appendix I,the SINR for the AWGN case is given by

SINR =|α|2

1− |α|2 + 1SNR

. (40)

By comparing the SINR with and without MMSE equalization,which are given by (40) and [25, Eq. (20)], respectively, it canbe concluded that both formulas are identical. Therefore, theequalization has no impact on the average SINR in the AWGNchannel case.

For the flat Rayleigh fading, the SINR for equalized conven-tional OFDM can be expressed as

SINR =|α|2

1− |α|2 + 1SNR

−Ei(−u)(u+1)eu−1(u+1)+(u+2)ueu Ei(−u)

. (41)

As it can be noted from (40) and (41), the subcarrier in-dex is dropped because SINRm is the same for all valuesof m. Moreover, both formulas are simplified using the factthat

∑N−1p=0p�=m

|αm,p|2 = 1− |α|2. It is straightforward to show

that when SNR → ∞, SINR for both the AWGN and flatfading converges to |α|2

1−|α|2 . Furthermore, without equalizationE [λ2] = E [λ1], the average SINR in (41) becomes equal to theSINR of the AWGN case.

Although the main focus of this paper is P-OFDM using WHT,applying the obtained results to any other precoding transform isstraightforward, and can be achieved by replacing the transformmatrix W by the desired transform matrix, and then use (24) tocompute SINRk, k = 0, 1,. . ., N − 1.

IV. NUMERICAL RESULTS

This section presents the performance evaluation of P-OFDMsystems in terms of SINR and BER over AWGN, flat Rayleigh

fading, and frequency-selective Rayleigh fading channels. TheOFDM symbol structure is generally based on the LTE [48]configuration except that the number of subcarriers is N =64. The data part of the OFDM symbol has a duration ofTu = 66.68 μs and the subcarrier spacing is 15 kHz. The num-ber of CP samples P = 16, with a duration of Tcp = 16.67 μs.The data symbols are drawn from QPSK constellation and thetotal OFDM symbol period is Tt = 83.35μs.

The frequency-selective fading channel considered in this pa-per has five taps with normalized delays of [0, 1, 3, 5, 10] samplesand average taps’ gains of [0.47, 0.29, 0.12, 0.07, 0.05]. There-fore, the root mean-square delay spread of the channel σ(τ) =2.52 μs given that the sampling period is Tt/80 ≈ 1.04 μs. Themultipath components’ gains are generated as complex indepen-dent Gaussian random variables. The channel gains remain con-stant for a given OFDM symbol period, but changes randomlyover different symbols, which corresponds to a quasi-static chan-nel. All the presented results are obtained using WHT unless itmentioned otherwise.

Fig. 1 depicts SINRk versus the subcarriers’ indices for theOFDM and P-OFDM over AWGN, flat, and frequency-selectivefading channels using ε = 0.1 and SNR = 20 dB. The analyti-cal results correspond to the AWGN and flat fading, whereas thesemi-analytical is derived for the frequency-selective case. Asdepicted in the figure, the SINRs of all subcarriers in OFDMare identical. On the contrary, the SINR of P-OFDM variesas a function of the subcarrier index. Such results imply thatCFO may cause severe BER degradation for P-OFDM becausethe overall system BER is dominated by subcarriers with lowSINRs. The figure also shows that SINR is different for OFDMand P-OFDM in AWGN and flat fading, yet it is the samefor frequency-selective fading. Moreover, the obtained simu-lation results match the analytical/semi-analytical (A/S) resultsvery well, where the semi-analytical results correspond to the



Fig. 2. Analytical, semi-analytical, and simulated average SINR versus CFO, SNR = 20 dB. (a) AWGN. (b) Flat fading. (c) Frequency-selective.

frequency-selective channel obtained using (24). It is worth not-ing that d0 and d1 are immune to CFO in AWGN channel whereSINR0 = SINR1 = SNR.

Fig. 2 shows SINRk for selected subcarriers versus ε for P-OFDM in AWGN, flat, and frequency-selective fading channels,and it also shows SINR for OFDM. As can be noted from thefigure, not only the SINR degradation depends on the subcar-rier index in the P-OFDM, but also the sensitivity to CFO. Forexample, d16 has the worst SINR and it is the most sensitive toCFO among the considered subcarriers, regardless of the channelconditions. The SINR for any subcarrier is determined by βk,k

given in (21), where increasing βk,k increases the signal powerand reduces the interference, and vice versa. For d16, β16,16 hasthe lowest value for the frequency-selective channel case, andthus, it has the largest interference and lowest SINR. Moreover,it is worth noting that increasing the number of subcarriers Ndoes not affect the SINR for any subcarrier in AWGN and flatfading channels due to the block nature of the matrix M,whichmaintains βk,k and the number of interfering terms for the kthsubcarrier fixed regardless the value ofN . In frequency-selectivechannels, it can be demonstrated numerically using (24) that in-creasing N may improve SINRk for certain values of k, anddecrease it for other values.

As can be noted from the Fig. 2, SINR = SNR when ε = 0for the AWGN channel, whereas it is not the case for the flat andfrequency-selective channels, which is due to the equalizationprocess. More specifically,SINR ≤ SNR for flat and frequency-selective channels when ε = 0. As confirmed by the analysis,the SINR of both OFDM and P-OFDM does not depend onthe equalization process in AWGN channels. The SINR differ-ence between P-OFDM and OFDM, and the difference betweenSINRk for each subcarrier is due to the structure of the WHT

transform, which is represented by Ψ1k,m,p,v,i and Ψ2

k,m,p,n,v,i

in (24). As sign{Ψ1, Ψ2

} ∈ ±1, the summations in the numer-ator and denominator of (24) may increase or decrease based onthe specific value of k. Interestingly, the interference terms in theAWGN and flat fading channels cancel each other for k = 0, 1.In OFDM, sign

{Ψ1, Ψ2

}= 1, the summations of the numer-

ator and denominator of (24) are independent of k, and thus,the OFDM is a special case of the P-OFDM as described inAppendix I.

The average BER versus SNR of P-OFDM for selected sub-carriers and the average BER for OFDM and P-OFDM systemsfor the three considered channels with ε = 0.1 are shown inFigs. 3 and 4. The BER results shown in the figures confirmthe results in Fig. 1, where the P-OFDM subcarriers that haveSINR higher than the conventional OFDM have a better BERperformance. It is noteworthy that the average BER of P-OFDMin frequency-selective channels is better than the conventionalOFDM due to the frequency diversity gain provided by the pre-coding process. Since such gain does not exist in AWGN and flatfading channels, the BER of the conventional OFDM is betterthan P-OFDM, which is more sensitive to CFO.

Fig. 5 shows the average BER versus SNR of both conven-tional OFDM and P-OFDM over AWGN, flat, and frequency-selective channels for different values of εwhere SNR = 20 dB.It can be noticed from the figure that the BER of both systemsdegrades severely for high values of ε regardless of the chan-nel model, particularly for ε ≥ 0.05. However, the relative BERperformance between the conventional and P-OFDM dependssignificantly on the channel model. For example, the BER re-sults over AWGN and flat fading channels presented in Fig. 5(a)and (b) show that conventional OFDM consistently outperformsP-OFDM. Such behavior can be justified by noting that P-OFDM



Fig. 3. Average BER of conventional OFDM and P-OFDM systems in AWGNand Rayleigh flat fading channels, ε = 0.1.

Fig. 4. Average BER of conventional OFDM and P-OFDM systems infrequency-selective channel, ε = 0.1.

does not offer any diversity advantage over AWGN and flat fad-ing channels, while it is more sensitive to CFO. In frequency-selective channels, as depicted in Fig. 5(c), the diversity gain ofP-OFDM system dominates the BER performance at low val-ues of ε � 0.1, and hence, the P-OFDM outperforms the con-ventional OFDM. For high ε values, the excessive sensitivity ofP-OFDM will dominate the performance and may actually elim-inate the diversity gain advantage of the precoding process. Con-sequently, at high CFO values, conventional OFDM may out-perform the P-OFDM even in frequency-selective channels.

The SINR versus the normalized CFO for conventionalOFDM with MMSE is shown in Fig. 6. As can be noted from thefigure, the equalization process reduces SINR even for ε = 0,which can be justified by noting that

SINRk|ε=0 = SNRE [λ1]

E [λ2](42)

where E[λ1] < E[λ2] given that E[|Hk|2] = 1 and SNR � 1.46dB. For example, in Rayleigh fading channels with SNR = 20dB, SINR|ε=0 = 10 log10 (SNR)− 5.25 = 14.73 dB. Conse-quently, the SINR degradation becomes less sensitive to ε. Toclarify this point, consider the case where ε = 0.05, which givesSINR degradation of about 13.2%without equalization, while itis 8.7% for the equalized system. Therefore, analyzing theSINRwithout equalization might be misleading because it amplifiesthe impact of the CFO on OFDM.

Figs. 7 and 8 present the simulated SINRk and BER usingthe Haar transform and ZCMT. The semi-analytical results us-ing (24) matches the simulation results very well, but omitted toavoid figure congestion. As can be noted from Fig. 7, SINRk forboth transforms is not uniform for all subcarriers, however, thedifferences for the Haar transform case are very small. More-over, the distribution of SINRk for the ZCMT is different fromthe WHT. Although the WHT and ZCMT SINRk distributionis different, the BER results in Fig. 8 show that their BER sen-sitivity to CFO is equivalent. As expected, the BER of the Haartransform at ε = 0 is higher than the WHT and ZCMT, how-ever, it is more immune at high values of ε, nevertheless, itsBER advantage is insignificant.

V. CONCLUSION

This paper presented the SINR analysis of P-OFDM in thepresence of CFO and evaluated its impact on the BER. Theconsidered P-OFDM system is based on the WHT, the conven-tional OFDM was used as a benchmark. The obtained analysisfor the WHT was then generalized for the Haar and ZCMT.The average SINR for P-OFDM is analytically evaluated for theconsidered wireless channel models. Semi-analytical averageSINR expression is obtained for the frequency-selective fadingchannel models, whereas exact expressions are presented for theflat Rayleigh fading and AWGN channels. The validity of ouranalysis was shown through the perfect match of the analyticaland the simulated average SINR. The simulation results showedthat the average SINR of the P-OFDM is significantly differentin the presence of CFO, where the average SINR per subcarrieris dependent on the subcarrier index. The variable SINR per in-dividual subcarrier in the P-OFDM system can be mitigated byconsidering adaptive bit loading to reduce the number of bits forsubcarriers with low SINR given that the CFO value is knownat the transmitter. The obtained results also showed that each ofthe considered precoding transforms have different sensitivitiesto CFO.



Fig. 5. Average BER of OFDM and P-OFDM over AWGN, flat, and frequency-selective fading channels. (a) AWGN. (b) Flat fading. (c) Frequency-selective.

Fig. 6. Average SINR for an equalized conventional OFDM.

APPENDIX ISINR ANALYSIS OF CONVENTIONAL OFDM

In a frequency-selective fading channel, the instantaneousSINR of the mth subcarrier can be defined as

SINRm =

∣∣∣HmHm,mam

∣∣∣2

∣∣∣∣∑N−1

p=0p�=m

HmHm,pap

∣∣∣∣2

+ |ηm|2. (43)

Fig. 7. SINRk using the Haar transform and ZCMT in frequency-selectivechannel, ε = 0.1 and SNR = 20 dB. (a) Haar. (b) ZCMT.

For the AWGN channel case, the channel frequency responseHm = Hp = 1∀ m. Thus, SINRm can be simplified to

SINRm =|α|2 |am|2

∣∣∣∣∑N−1

p=0p�=m

αm,pap

∣∣∣∣2

+ |ηm|2. (44)



Fig. 8. BER of precoded and conventional OFDM using different precodingtransforms in frequency-selective channel.

Therefore, the average SINR can be expressed as

SINRm =|α|2E

[|am|2

]

E

[∣∣∣∣∑N−1

p=0p�=m

αm,pap

∣∣∣∣2]+ E

[|ηm|2

] (45)

=|α|2 Ps

Ps

∑N−1p=0p�=m

|αm,p|2 + σ2z

(46)

=|α|2

∑N−1p=0p�=m

|αm,p|2 + 1SNR

. (47)

However, by noting that the average power per subcarrier isPs, then the sum of the signal and interference powers can beexpressed as

Ps = |α|2Ps + Ps

N−1∑

p=0p�=m

|αm,p|2 . (48)

Moreover, since the interference power can be written as

N−1∑

p=0p�=m

|αm,p|2 = 1− |α|2 (49)

then the average SINR for OFDM in AWGN can be expressedas

SINRm =|α|2

1− |α|2 + 1SNR

=

∣∣∣∣sin(πε)

N sin( πN ε)

∣∣∣∣2

1−∣∣∣∣

sin(πε)

N sin( πN ε)

∣∣∣∣2

+ 1SNR

(50)

which is similar to the results obtained in [25, eq. (28)].In flat fading channels, the average SINR can be written as

SINRm =|α|2 E [λ1]E

[|am|2

]

E [λ1]E

[∣∣∣∣∑N−1

p=0p�=m

αm,pap

∣∣∣∣2]+ E [λ2]E

[|ηm|2

]

(51)which after some straightforward manipulations can be writtenas

SINRm =

∣∣∣∣sin(πε)

N sin( πN ε)

∣∣∣∣2

E [λ1]

E [λ1]∑N−1

p=0p�=m

|αm,p|2 + 1SNRE [λ2]

(52)

where E [λ1] and E [λ2] are given in (35) and (37), respectively.

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Fatma Kalbat (M’11) received the B.Sc. degree inelectrical engineering from the United Arab EmiratesUniversity, Al Ain, United Arab Emirates, in 2007and the M.Sc. degree in electrical engineering fromKhalifa University, Abu Dhabi, United Arab Emi-rates, in 2013, where she is currently working towardthe Ph.D. degree with the Department of ElectricalEngineering.

She is currently a Research Assistant. Her cur-rent research interests include orthogonal frequency-division multiplexing systems, synchronization, and

channel estimation techniques.

Arafat Al-Dweik (S’97–M’01–SM’04) received theM.S. (Summa Cum Laude) and Ph.D. (Magna CumLaude) degrees in electrical engineering from Cleve-land State University, Cleveland, OH, USA, in 1998and 2001, respectively.

He was with Efficient Channel Coding, Inc.,Cleveland-Ohio, from 1999 to 2001. From 2001 to2003, he was the Head of the Department of Informa-tion Technology at the Arab American University inPalestine. Since 2003, he has been with the Depart-ment of Electrical Engineering, Khalifa University,

Abu Dhabi, United Arab Emirates. He was an Associate Professor with the Uni-versity of Guelph, Guelph, ON, Canada, from 2013 to 2014. He is a VisitingResearch Fellow with the School of Electrical, Electronic, and Computer En-gineering, Newcastle University, Newcastle upon Tyne, U.K., and a ResearchProfessor with Western University, London, ON, Canada and with the Univer-sity of Guelph.

Dr. Al-Dweik has extensive editorial experience as he has served as an As-sociate Editor at the IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY andthe IET Communications. He has extensive research experience in various ar-eas of wireless communications that include modulation techniques, channelmodeling and characterization, synchronization and channel estimation tech-niques, OFDM technology, error detection and correction techniques, MIMO,and resource allocation for wireless networks. He is a member of Tau Beta Piand Eta Kappa Nu. He received the Fulbright Scholarship from 1997 to 1999.He received the Hijjawi Award for Applied Sciences in 2003, the FulbrightAlumni Development Grant in 2003 and 2005, and the Dubai Award for Sus-tainable Transportation in 2016. He is a Registered Professional Engineer in theProvince of Ontario, Canada.

https://dx.doi.org/10.1109/ACCESS.2018.2870278

https://dx.doi.org/10.1109/ACCESS.2018.2865596



Bayan Sharif (M’92–SM’01) received the B.Sc. de-gree from Queen’s University, Belfast, U.K., in 1984,the Ph.D. degree from Ulster University, Coleraine,U.K., in 1988, and the D.Sc. degree from NewcastleUniversity, Newcastle upon Tyne, U.K., in 2013.

From 1990 to 2012, he was with the Faculty ofNewcastle University, where he held a number ofappointments including a Professor of digital com-munications, the Head of communications and sig-nal processing research, and the Head of the Schoolof Electrical, Electronic, and Computer Engineering.

He is currently the Dean of the College of Engineering, Khalifa University, AbuDhabi, United Arab Emirates. His main research interests include digital com-munications with a focus on wireless receiver structures and optimization ofwireless networks.

Dr. Sharif is a Chartered Engineer and a Fellow of the IET, U.K.

George K. Karagiannidis (M’96–SM’03–F’14) wasborn in Pithagorion, Samos, Greece. He receivedthe Diploma and Ph.D. degrees in electrical andcomputer engineering from the University of Patras,Patras, Greece, in 1987 and 1999, respectively.

From 2000 to 2004, he was a Senior Researcherwith the Institute for Space Applications and RemoteSensing, National Observatory of Athens, Athens,Greece. In 2004, he joined the Faculty of Aristo-tle University of Thessaloniki, Thessaloniki, Greece,where he is currently a Professor with the Depart-

ment of Electrical and Computer Engineering and the Director of the DigitalTelecommunications Systems and Networks Laboratory. He is also an Hon-orary Professor with South West Jiaotong University, Chengdu, China. His re-search interests include the broad area of digital communications systems andsignal processing, with emphasis on wireless communications, optical wirelesscommunications, wireless power transfer and applications, molecular commu-nications, communications and robotics, and wireless security. He is an authoror co-author of more than 400 technical papers published in scientific journalsand presented at international conferences. He is also an author of the Greekedition of a book: Telecommunications Systems (Tziolas Publications, March2017) and a co-author of the book: Advanced Optical Wireless CommunicationsSystems (Cambridge Publications, 2012). He has been involved as the GeneralChair, the Technical Program Chair, and a Technical Program Committee mem-ber in several IEEE and non-IEEE conferences. He was an Editor of the IEEETRANSACTIONS ON COMMUNICATIONS, a Senior Editor of the IEEE COMMU-NICATIONS LETTERS, an Editor of the EURASIP Journal of Wireless Commu-nications and Networks and several times Guest Editor of the IEEE SELECTED

AREAS IN COMMUNICATIONS. From 2012 to 2015, he was the Editor-in Chiefof the IEEE COMMUNICATIONS LETTERS. He is a highly cited author across allareas of electrical engineering, and was recognized as a 2015 and 2016 ThomsonReuter’s Highly Cited Researcher.

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