Performance analysis of PSK systems with phase error in fading channels: A survey

Physical Communication 4 (2011) 63–82

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Physical Communication

journal homepage: www.elsevier.com/locate/phycom

Performance analysis of PSK systems with phase error in fadingchannels: A surveyAniruddha Chandra a,∗, Ananya Patra a, Chayanika Bose b

a Department of Electronics and Communication Engineering, National Institute of Technology, Durgapur-713209, West Bengal, Indiab Department of Electronics and Telecommunication Engineering, Jadavpur University, Kolkata-700032, West Bengal, India

a r t i c l e i n f o

Article history:Received 9 April 2010Received in revised form 30 September2010Accepted 3 December 2010Available online 16 December 2010

Keywords:PSK systemsPhase errorNakagami-m fadingMoment generating functionTikhonov distribution

a b s t r a c t

Communication system designers need to formulate an accurate and thoroughlyreproducible error model for wireless mobile channels in order to assess the qualityof communication with different modulation schemes. The task is relatively easy whenan ideal situation is assumed, and is covered exhaustively in standard text books.However, the random time-varying nature of radio propagation renders estimation ofdifferent channel state information (CSI) very difficult and when these non-idealities(e.g. imperfect phase/frequency/timing information) are considered, the formulationcomplexity increases manifold. In this paper, we have set our attention on phase shiftkeying (PSK), which suffers mostly from phase synchronization error when proper CSI isnot available at the receiver. The article surveys various error modelling methods for aPSK system operating over a slow flat Nakagami-m distributed wireless fading channelperturbed with additive white Gaussian noise (AWGN) in the presence of phase error.The phase distortions are considered to be random, unbiased, i.e. having zero mean,and may be represented by either Gaussian or Tikhonov distribution. We also provide anovel approach to classify these schemes that are surveyed, and summarize the majorcontributions of related works. Further, we identify the method that requires lessermathematical operations and thus proves to be less complex, more stable and accuratethan others. Apart from this, simple alternative approaches for calculating analytical biterror rate (BER) through Hermite’s method of integration for Gaussian distributed phaseerror and throughmoment generating function (MGF) for Tikhonov distributed phase errorhave been proposed. Both of these methods have wider applicability, are able to furnishreproducible results, and show significant improvement in accuracy regarding theoreticalBER calculation as seen from the graphical comparisons. ExtensiveMonte Carlo simulationswere performed to validate the theoretical results.

© 2010 Elsevier B.V. All rights reserved.

1. Introduction

To cope with an ever increasing demand for higherdata rates, M-ary schemes are frequently used in currentwireless systems. The coherent M-ary schemes providebetter error performance or require a lesser signal to noise

∗ Corresponding author. Tel.: +91 343 2755515; fax: +91 343 2547375.E-mail addresses: [email protected],

[email protected] (A. Chandra).

1874-4907/$ – see front matter© 2010 Elsevier B.V. All rights reserved.doi:10.1016/j.phycom.2010.12.001

ratio (SNR) to achieve a target BER when compared totheir non-coherent or differentially coherent counterparts.Out of the coherent schemes, M-ary PSK (MPSK) is oftenpreferred over M-ary frequency shift keying (MFSK) dueto better bandwidth efficiency and overM-ary quadratureamplitude modulation (MQAM) as MPSK has a constantenvelope which facilitates use of efficient non-linearpower amplifiers [1]. Despite these advantages, PSKsystems suffer from various problems, themost prominentone being imperfect phase estimation at the receiver.

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64 A. Chandra et al. / Physical Communication 4 (2011) 63–82

Fig. 1. Generic transmission model for BPSK modulated signal perturbed with AWGN and phase error.

Apart from synchronization errors, the wireless chan-nels are subjected to random multipath fading, andNakagami-m distribution serves as the most general dis-tribution to characterize such fading effects. For PSKtransmission through wireless medium, fading and phaseerror are the two major factors responsible for signaldegradation, and error performance evaluation of PSKwithNakagami-m fading and phase error is of considerable in-terest. This can be substantiated by citing error perfor-mance characterizations of PSK systems with imperfectphase recovery by different researchers in the last fivedecades [2–11].

In our paper, a detailed survey as well as a comparativestudy of the earlier works related to BER calculation of PSKover AWGN/Nakagami-m fading channel with imperfectphase estimation has been presented. Two newly derivedapproaches, to tackle amore generalized form of the phasejitter problem, have also been incorporated. The resultsare derived through Hermite’s method of integration forGaussian distributed phase error and through simple MGFmethod for Tikhonov distributed phase error. Later, theMGF method has been extended through a weighted sumapproach to derive symbol error rate (SER) of generalMPSK systems under phase error. The correspondingBER/SER plots show perfect overlapping with simulationresults. Moreover, the associated percentages of error(with respect to direct numerical integration) are muchlesser compared to previous works.

The rest of the paper is organized as follows. Model ofa system using binary PSK (BPSK) with imperfect phaseestimation and operating over a wireless fading channelis detailed in Section 2. In addition, this section also de-scribes the Gaussian and Tikhonov distribution functionsfor modelling the phase error random variable (RV) andintroduces generic integral forms for calculation of errorrates. In Section 3, the related works in this field are dis-cussed and a comparative study between the correspond-ing approaches is made available in graphical form onthe basis of percentage of error. The overall simulationmethodology, giving more stress on different algorithmsfor generating Tikhonov PDF, is also explained. Two sim-pler approaches for BER calculation with both Gaussianand Tikhonov distributed phase error are presented next,in Sections 4 and 5 respectively. In Section 6, SER of MPSKfor Nakagami-m fading channel models has been derived,

assuming Tikhonov distributed phase error. The paper fi-nally ends in Section 7 with some concluding remarks anda discussion on future scope of the work.

2. Modelling phase error

2.1. Transmission model in the absence of channel fading

PSK is a digital modulation scheme that uses a finitenumber of phases; each assigned to a unique pattern ofbinary bits. An MPSK signal set is defined as

sj (t) =

2EsTs

cos2π fc t + ϕj

;

0 ≤ t ≤ Ts, j = 1, 2, . . . ,M (1)

where fc is the carrier frequency chosen in a way suchthat it becomes an integer multiple of the symbol rateRs (= 1/Ts), and {Es, Ts} denotes energy and durationof any transmitted waveform respectively. Usually M ischosen as a power of 2, i.e. M = 2p

; p = integer.Accordingly, the binary data stream is divided into p-tuplesand each of them is represented by a symbol

sj (t)

Mj=1

with a particular initial phase, ϕj = (2j − 1) π/M .Fig. 1 illustrates block diagram of a communication

system employing BPSK (M = 2) modulation. The effectsof both AWGN and phase error are shown separately. TheBPSK modulator in the transmitter (Tx) operates on binaryinformation ai = {0, 1} and during the signalling intervaliTb ≤ t < (i + 1) Tb produces an output signal

si (t) = ±

2EbTb

cos (2π fc t) ; iTb ≤ t < (i + 1) Tb (2)

where the sign of si (t) is directed by the current inputbinary bit ai, i.e. positive for ‘1’ and negative for ‘0’ andEb, Tb denote the bit energy and bit duration respectively.The transmitter sends the modulated signal s (t) =∑

∞

i=−∞si (t − Tb) through a transmission medium where

it is perturbed with AWGN and a noisy replica of thetransmitted signal, r (t), reaches the receiver (Rx). Atthe receiver, the received signal is first multiplied withthe carrier. Ideally the carrier at receiver should besynchronous with the carrier at transmitter end, i.e. the

A. Chandra et al. / Physical Communication 4 (2011) 63–82 65

time delay between the carrier and reference signal atreceiver is either zero or has a constant value. Two pairsof ideal identical oscillators at the transmitter and receiversides could ensure such synchronization. However, thesignal emitted by a pair of oscillators with the samenominal frequency will start drifting soon from eachother due to various factors, e.g. temperature variation,device non-linearity, ageing, and power supply ripples.In practice, the local phase reference at receiver isreconstructed from a noise-corrupted version of a receivedsignal, and the receiver operates on the transmitted signalwith a phase shifted carrier (phase shift = θ)

cr (t) = ±

2Tb

cos (2π fc t + θ) ; 0 ≤ t < Tb. (3)

The product r (t) cr (t) is then integrated for each bitduration Tb. Output of the integrator (ri) is fed to a decisiondevice where it is compared with a predefined threshold(Th) of zero and accordingly a decision

aiis made.

2.2. Transmission model with channel fading

The immediate effect of the phase error is degrada-tion of the detection performance of coherent systems,which, to some extent, may be mitigated by sending apilot tone [12] with the transmitted signal. In this casethe receiver operates with the unmodulated pilot carrierand thus avoids phase mismatch. However, for wirelessmobile channels, the channel experiences additional ran-dom phase distortions (φ) due to multipath fading. SinceMPSK is a coherent scheme, imperfect estimation of thefading phase (φ), leads to phase error, θ = φ − φ [13].Without losing generality, we may assume that sj=0 (t) =√2Es/Ts cos (2π fc t), corresponding to zero phase shift, is

being transmitted. Hence, one can write the outputs of thein-phase and quadrature correlators of theM-PSK receiver(refer to Fig. 4.11 [14]) as

rI =

∫ Ts

0r (t) exp

−jφ

2Ts

cos (2π fc t) dt

= α exp (jθ)Es + nI; iTs ≤ t < (i + 1) Ts (4a)

rQ =

∫ Ts

0r (t) exp

−jφ

2Ts

sin (2π fc t) dt = nQ ;

iTs ≤ t < (i + 1) Ts (4b)

where α denotes random amplitude variation, n = nI +nQis a zero-mean circularly symmetric complex Gaussian RVwith a variance N0, and the received signal, r (t), is a fadednoisy version of the transmitted signal s (t)

r (t) = α exp (jφ) s (t)+ n (t) (5)

as demonstrated in Fig. 2.A slow and flat fading channel is assumed here, where

the duration of a symbol waveform is sufficiently short sothat the fading variations cause negligible loss of coherencewithin the timeframe of each received symbol. At thesame time, the individual waveform is assumed to besufficiently narrow band (sufficiently long in duration) so

Fig. 2. Transmission model with fading and AWGN.

that frequency selectivity offered by the fading processto different signal spectral components is negligible. Theeffect of such kind of fading on the signal amplitudeis multiplicative distortion characterized by attenuationfactor α. Besides fading, the transmitted signal is alsoperturbed by real bandpass noise having a two-sidedpower spectral density (PSD) of N0/2 which is typicallyassumed to be independent of the fading process. Theequivalent baseband AWGN is therefore complex circularwith a PSD of N0 per quadrature. It may be noted that in acellular system, large scale fading effects (e.g. propagationpath loss and shadowing) are generally compensated bypower control and the attenuation of the received signalis mostly dominated by small scale multipath fading.

A good number of statistical models are available inthe literature [15] to characterize the fading envelopeunder different circumstances. Among them, Nakagami-m distribution is considered most versatile since it candegenerate into Rayleigh distribution and approximateRician and lognormal distributions. Also, it is applicableto a wider range of wireless environments (both indoorand outdoor) and serves as a better modelling toolthan Rayleigh, Rician or lognormal distributions. For aNakagami-m fading channel, the PDF of instantaneous SNRper bit (γb) at the channel output is given by

fγ (γb) =1

Γ (m)

mγb

m

γ m−1b exp

−

mγbγb

;

0 ≤ γb < ∞ (6)

where γb = α2 (Eb/N0) with an average value γb =

α2 (Eb/N0), α2 = Eα2

is the average fading power,and Γ (·) represents gamma function [16, (8.31)]. Theparameter m denotes severity of fading which rangesfrom 1/2 to ∞, i.e. m = 1/2 gives a one-sidedGaussian distribution that manifests most severe fadingwhile m = ∞ corresponds to the no fading case. Form = 1, Nakagami-m distribution reduces to Rayleighdistribution [15].

2.3. Phase error distributions

The fading phase estimation is generally performedwith phase locked loops (PLLs) which work well when themobility of the receiver is limited and the phase distortionsare not severe. It has been shown [9,10] that Tikhonovdistribution [17] matches with the probability densityfunction (PDF) of the carrier phase error for first orderPLLs, and approximates the PDF for second order loops.Sometimes phase jitter is also represented with Gaussiandistribution instead of Tikhonov as demonstrated in [9]and in some contemporary books [18,19].


(a) Gaussian distributed (θm = 0) phase error fordifferent σθ values.

(b) Tikhonov distributed (µ = 0) phase error fordifferent ρ values.

Fig. 3. PDF of Gaussian and Tikhonov distributed phase error.

If Gaussian distribution is assumed for modelling thenthe corresponding PDF may be written as [19]

fθ (θ) =1

σθ√2π

exp[−(θ − θm)

2

2σ 2θ

]; 0 ≤ |θ | < ∞ (7)

where θm is the mean and σθ is the standard deviationof θ . For all practical purposes this error can be assumedas unbiased, i.e. having zero mean (θm = 0) and of smallvariance (σθ = a few degrees).

In probability theory and statistics, the Tikhonov/vonMises distribution (also known as the circular normaldistribution) is a continuous probability distribution onthe circle. It may be thought of as a circular analogue ofthe normal distribution. For systemmodels using Tikhonovdistribution, the phase error PDF is given by

fθ (θ) =1

2π I0 (ρ)exp [ρ cos (θ − µ)] ; 0 ≤ |θ | < π (8)

as derived in [12] or [20] using the Fokker–Plancktechnique. In (8), I0 (·) denotes modified Bessel functionof first kind and order 0 [16, (8.402)], µ is a measure oflocation, and ρ is a measure of concentration. For unbiasedphase error, µ = 0, whereas ρ may be expressed in termsof the PLL parameters as

ρ =P

N0BL(9)

where P is the power in the carrier and BL is the noisebandwidth of the PLL closed-loop transfer function.

Although the von Mises PDF is available in closedform, the cumulative distribution function (CDF) cannotbe achieved by simply integrating the PDF given in (8).Therefore, an alternative PDF expression in terms of Besselfunctions is included hereunder

fθ (θ) =12π

[1 +

2I0 (ρ)

] ∞−j=1

Ij (ρ) cos [j (θ − µ)] ;

0 ≤ |θ | < π. (10)

By integrating (10) over θ , the standard form of the vonMises CDF is attained

Fθ (θ) =1

2π I0 (ρ)

θ I0 (ρ)

+ 2∞−j=0

Ij (ρ) sin [j (θ − µ)]j

; 0 ≤ |θ | < π. (11)

Fig. 3(a) and (b) portrays the PDF of the Gaussiandistribution given by (7) for different values of σθ andthe PDF of Tikhonov distribution given by (8) for differentvalues of ρ respectively. For small ρ values the distributionfθ (θ) approaches uniform distribution U (·, ·) and at ρ =

0, fθ (θ) ≈ U (−π, π). When ρ is large, the distributionbecomes concentrated about the angle µ with ρ beinga measure of the concentration. In fact, as ρ increases,the distribution approaches a Gaussian distribution in θ ,θ ∼ N (µ, σθ ), with mean µ and variance σ 2

θ [11]. Thecorresponding variance, σ 2

θ =θ2fθ (θ) dθ is inversely

related to the Tikhonov parameter ρ [21]

σ 2θ ≈

1ρ

; ρ ≫ 1. (12)

2.4. Calculation of BER with phase error and fading

We begin with the BER expression of BPSK in a pureAWGN channel which is given by

Pe,b (ηb) =12erfc

√ηb

(13)

where ηb = Eb/N0 is the SNR per bit and erfc (·) denotescomplementary error function [22, (7.1.2)]. In presence ofphase error, however, (13) is modified as

Pe,b (ηb, θ) =12erfc

√ηb cos θ

. (14)

This happens as the decision variable in presence of phaseerror is no longer the received signal vector (ri) itself,rather its vertical projection ri cos θ as shown in Fig. 4.Consequently, the corresponding BER expression of BPSK


Fig. 4. Constellation diagram of BPSK with phase error (whereri = received signal vector).

with phase error becomes a function of two variables ηband θ [11].

The average BER in an AWGN channel with phase errormay be found by simply averaging this conditional errorprobability (CEP) over the phase error PDF as

Pe,b =

∫θ

Pe,b (ηb, θ) fθ (θ) dθ (15)

where fθ (θ) is the PDF as mentioned in (7)–(8) andPe,b (ηb, θ) denotes the CEP given in (14).

In fading channel, α2 denotes instantaneous attenua-tion in received signal power and the instantaneous SNRper bit can be defined as γb = α2ηb. Accordingly, the BERexpression of BPSK as given in (14) may be modified as

Pe,b (γb, θ) =12erfc

√γb cos θ

(16)

to incorporate the effect of both fading and phase error. Ina fading channel, however, SNR γb itself becomes a randomvariable and to obtain the average BER one has to perform atwo-fold integration, the first one over the phase error PDFand the second one over the fading SNR PDF. We may thusformulate a generalmathematical equation to calculate theaverage probability of error in presence of fading andphaseerror as

Pe,b =

∫γb

∫θ

Pe,b (γb, θ) fγ (γb) fθ (θ) dθdγb (17)

where Pe,b (γb, θ) is given by (16), fγ (γb) is specified in(6) and the PDF of fθ (θ) is provided by (7) and (8) forGaussian and Tikhonov cases respectively. By substitutingthe expressions for Pe,b (γb, θ), fγ (γb), and fθ (θ) in(17), one may perform a direct numerical integration toobtain the BER values. Nevertheless, a direct evaluationis generally not preferred due to excessive computationtime and numerical complexity. In addition, for generalMPSK systems the CEP is itself given by an integralequationwhich necessitates calculation of a triple integral.In our paper, BER values are obtained through numericalintegration wherever possible, and these values serveas reference points to calculate the percentage of error,defined by

Percentage of error =

Pe,b(Theory) − Pe,b(NI)

Pe,b(NI)× 100% (18)

where Pe,b(Theory) is the theoretical error probabilityobtained for any given approach and Pe,b(NI) is the errorprobability achieved through direct numerical integrationusing (17).

3. Related work

In 1960, Viterbi [2] was first to provide a statisticalmodel for the carrier phase error of a first order PLLand Lindsey [3], in 1966, addressed the performance ofBPSK with phase error and AWGN. Later a usable butapproximated expression was provided by Kam et al. [7].In all these works carrier recovery through a first orderPLL was considered, while Tikhonov distribution wasused for statistical modelling of phase error. There is yetanother paper by Rhodes [4] based on the Tikhonov modelwhere the probability of error of the offset QPSK withtwo system imperfections, AWGN and imperfect carriersynchronization, had been analyzed. On the other hand,related works on Gaussian PDF concerns two papers; thefirst one published by Bucket and Moeneclaey [8] wherethey showed that the error rates with Gaussian distributedphase error can be represented with truncated Taylor’sseries, and the second one by Lo and Lam [9] who derivedBER of BPSK with phase error by using an infinite seriesfor error function in Nakagami-m fading channel. Further,Prabhu [5] derived BER performance of BPSK and QPSKassuming the phase noise associated with the recoveredcarrier can have a static component and a Gaussian ora truncated Gaussian-type distribution. In fact he provedthe generality of Gaussian distribution by applying centrallimit theoremand showed that Gaussian PDF can representTikhonov PDF in a limiting sense.

The extension of the above to the fading channel beganwith the work of Weber [6], who studied the performanceof PLLs in Rayleigh, Rician, and lognormal fading environ-ments. Subsequently, performance evaluation over gener-alized fading channels obeying Nakagami-m distributionfollowed. The specific examples include [9] and [10] whereBER of BPSK in Nakagami-m fading and with phase errorhas been derived. Lo and Lam [9] used an infinite series forerror functionwhereas Simon and Alouini [10] approachedthe problem by truncating the CEP expression (a functionof both instantaneous SNR and phase error as given in (16))with McLaurin’s series. In both papers, the carrier recov-ery loop SNR ρ is assumed to be proportional to instanta-neous channel SNR γ , i.e. ρ = Kγ . A more general case,i.e. ρ = Kγ , was studied by Falujah and Prabhu [11]. Theauthors have calculated BER of BPSK and QPSK in the pres-ence of three system imperfections namely phase recoveryerror, slow multipath fading, and AWGN, using an infiniteseries expression to tackle the error function present in theCEP. However, their end expressions involved infinite se-ries and complex mathematical functions, were difficult toevaluate, and not guaranteed to converge.

In summary, although initially the interest was limitedto error probability analysis in pure AWGN channels, withthe rapid advancement in the wireless communicationfield, the research focus gradually shifted to BER evaluationin fading channels. Tikhonov distribution served as thedominant statistical model to characterize the random


phase error due to its effectiveness in phase errormodelling of first (and to some extent second) orderPLL. Moreover, there exist two parallel approaches: first,when the carrier recovery loop SNR is proportional toinstantaneous channel SNR (ρ = Kγ ) [9,10], and second,when the loop SNR is independent of the channel SNR(ρ = Kγ ) [11]. Some authors also vouched in favour ofGaussian distribution for phase error modelling as it canencompass other general cases. Considering the argumentabove, the research work till date may be grouped into thefollowing four categories,

Performance analysis of BPSK:

(a) with Tikhonov distributed phase error and AWGN(b) with Gaussian distributed phase error and fading(c) with Tikhonov distributed phase error and fading

(where ρ = Kγ )(d) with Tikhonov distributed phase error and fading

(where ρ = Kγ ).

Researchworks related to all the four above-mentionedcategories are surveyed in the following subsections.Further, Nakagami-m fading has been used as the standardfadingmodel (although someauthors have calculated errorrates for other fading channels too) as it serves to be themost general distribution for modelling small scale fadingenvironments.

3.1. Performance of BPSK with Tikhonov distributed phaseerror and AWGN

In pure AWGN channels, the error performance of BPSKwith Tikhonov distributed phase error was analyzed byLindsey [3], Kam et al. [7] and Falujah and Prabhu [11].Lindsey [3], in 1966, was the first to address the perfor-mance of BPSK in the presence of imperfect carrier phaserecovery and AWGN. He considered an integration of theform given in (15) and when expressions for Pe,b (ηb, θ)and fθ (θ), from (14) and (8) respectively, are substitutedin (15), the task basically reduces to performing an integra-tion where the integral comprises of an exponential func-tion exp (ρ cos θ) and an complementary error functionerfc

√ηb cos θ

. Using the relation erfc (·) = 1 − erf (·)

[16, (8.25)], and a result given by Luke [23, (p. 174)], viz.

erf (z cos θ) =z2exp

−

z2

2

∞−k=0

(−1)k εkIk

z2

2

×

cos [(2k + 1) θ ]

2k + 1−

cos [(2k − 1) θ ]2k − 1

;

εk =

1 if k = 02 if k > 0 (19)

the term erfc√ηb cos θ

was expanded to form an infinite

series. The term exp (ρ cos θ) can also be decomposed [24]into another series of Bessel functions

exp (ρ cos θ) =

∞−k=0

εkIk (ρ) cos (kθ) . (20)

Further using∫ π

−π

cos (jθ) cos [(2k + 1) θ ] dθ = πδj,k±1;

δj,k =

1 for j = k0 for j = k (21)

and doing a term by term integration, the averageprobability of bit error becomes

Pe,b =12

1 −

2ηbπ

exp−ηb

4

×

∞−k=0

(−1)k εkξk1 − 4k2

Ikηb

4

(22)

where ξk = Ik (ρ) /I0 (ρ). When ηb approaches infinity,(22) reduces to

Pe,b ≈12

1 −

2π

∞−k=0

(−1)k εkξk1 − 4k2

(23)

which shows that the performance of the receiverexhibits an irreducible error probability depending onlyon the SNR in the auxiliary carrier tracking loop. As ρapproaches infinity, i.e., for perfect phase measurement,Pe,b approaches zero.

Later in 1993 ausable but approximated expressionwasprovided by Kam et al. [7]. The following CEP expressionsare used to address a wider range of θ

Pe,b (ηb, θ)

=

12erfc

√ηb cos θ

for |θ | ≤ π/2

1 −12erfc

√ηb cos θ

for |θ | > π/2

(24)

the first one being identical with (14) and is valid for |θ | ≤

π/2 whereas the complement of (14) was used for |θ | >π/2 [25]. For evaluating the BER, the expressions in (24)should be averaged over (8), which requires integrationsover two different ranges of θ . However, the series ofapproximations in [7] resulted in a form like

Pe,b ≈

∫ π

−π

12erfc

√ηb cos θ

fθ (θ) dθ (25)

which is nothing but the form considered in [3] with achanged integration limit of |θ | > π instead of |θ | > π/2.Using a modified form of the Chernoff bound of erfc (·),namely erfc (z) ≈ exp

−z2

/√π , (25) may be written as

Pe,b =1

4π3/2I0 (ρ)

×

∫ π

−π

exp

−ηb

cos θ −

β

2

2

−

β

2

2

dθ (26)

where β = ρ/ηb. Employing an approximation (cos θ −

β/2)2 ≈ (1 − β/2)2+(β − 2) (1 − cos θ) and performingthe integration in (26) we get

Pe,b ≈1

2√π

I0 [ηb (β − 2)]I0 (ρ)

exp (ηb) (27)


where the definition, I0 (ρ) = π−π

exp (ρ cos θ) dθ[22, (9.6.16)], has been used to derive (27). For ηb < ρ/2,as I0 (z) ≈ exp (z) /

√2πz, (27) can be further simplified

to

Pe,b ≈1

2√π

1 −

2ηbρ

−1/2

exp (−ηb) . (28)

For very low SNR (ηb ≪ ρ), considering the first two termsof the binomial expansion of (1 − 2ηb/ρ)−1/2, (28) can bereduced to

Pe,b ≈1

2√π

1 +

ηb

ρ

exp (−ηb) . (29)

This approximation is, of course, not accurate when ηbapproaches ρ/2. For ηb > ρ/2, employing another setof approximation (cos θ − β/2)2 − (β/2)2 ≈ (cos θ)2 −

(β/2)2, they obtained

Pe,b ≈1

2√π I0 (ρ)

I0ηb

2

exp

ρ2

4ηb−ηb

2

(30)

which may be further approximated as

Pe,b ≈

ρ

2πηbexp

−ρ +

ρ2

4ηb

. (31)

Falujah and Prabhu [11], in their paper, used an expansionfor the error function erf (·) in terms of a Fourier series

erf√ηb cos θ

= 2

ηb

πexp

−ηb

2

∞−k=0

(−1)k

2k + 1

× cos [(2k + 1) θ ]Ikηb

2

+ Ik+1

ηb2

(32)

which was originally introduced by Prabhu [5,26]. Theexpansion was inspired by the fact that erf

√ηb cos θ

is

periodic in θ with period 2π and the expansion results in aseries that converges fast [5] for all values ηb. Then, usingthe relation between erfc (·) and erf (·), the CEP in (14) canbe transformed into the following form

Pe,b (ηb, θ) =12

+

ηb

πexp

−ηb

2

∞−k=0

(−1)k

2k + 1

× cos [(2k + 1) θ ]Ikηb

2

+ Ik+1

ηb2

. (33)

Substituting Pe,b (ηb, θ) from (33) in (15) and averaging itover the PDF fθ (θ) they obtained

Pe,b =12

+

ηb

πexp

−ηb

2

∞−k=0

(−1)k+1

2k + 1I2k+1 (ρ)

I0 (ρ)

×

Ikηb

2

+ Ik+1

ηb2

(34)

using the integral representation of modified Besselfunction [22, (9.6.19)]

Ik (z) =1π

∫ π

0exp (−z cos θ) cos (kθ) dθ. (35)

Computation of the average BER of BPSK using the infiniteseries expressions given in (34) converges within a rea-sonable number of terms. The infinite series expressions

have alternating signs and they satisfy the conditions of theLeibnitz theorem for series convergence.

Some other related works worth mentioning in-cludes [27–29], where either approximate solutions aregiven or lower/upper bounds have been derived. Detaileddiscussions of the particular approaches are omitted hereas the techniques used by the authors are similar to thoseenlisted above and none of them addressed the problem ofchannel fading.

3.2. Performance of BPSK with Gaussian distributed phaseerror and fading

While the previous subsection considers BER evalua-tion in a pure AWGN channel, in the current subsectionwe extend the idea to incorporate fading effects. Unfortu-nately, the literature is not rich in this regard and when aGaussian PDF is assumed for phase error, there exists onlya single paper by Lo and Lam [9] that discusses the errorperformance of BPSK with noisy phase reference for a flatNakagami-m fading channel. They [9] used yet another in-finite series [22,30] for erfc (·)

erfc√γb cos θ

= 1 −

2π

∞−k=0

(−1)k

(2k + 1)!

×Γ

k +

12

γ k+1/2

1F1

k +

12; 2k + 2; −γb

× cos [(2k + 1) θ ] (36)

where 1F1 (·; ·; ·) is the confluent hypergeometric function[16, (9.21)]. Substituting (7) and (36) in (17), and notingthat

γb

θfγ (γb) fθ (θ) dθdγb = 1, we have

Pe,b =12

−1π

∞−k=0

(−1)k

(2k + 1)!Γ

k +

12

∫∞

0γ

k+1/2b 1F1

×

k +

12; 2k + 2; −γb

fγ (γb)

12πσ 2

θ

×

∫∞

−∞

cos [(2k + 1) θ ] exp

−θ2

2σ 2θ

dθdγb (37)

where the integration over θ may be solved using thefollowing relation [16, (3.896.2)]

12πσ 2

θ

∫∞

−∞

cos [(2k + 1) θ ] exp

−θ2

2σ 2θ

dθ

= exp[−σ 2θ (2k + 1)2

2

]. (38)

When σθ is independent of γbWhen σθ is independent of γb, substituting fγ (γb) from

(6) in (37) and using the result in (38), the probability oferror can be written as

Pe,b =12

−

∞−k=0

(−1)k Γ (k + 1/2)(2k + 1)!Γ (m) π

mγb

m

× exp[−σ 2θ (2k + 1)2

2

] ∫∞

0γ

m+k−1/2b

× exp

−mγbγb

1F1

k +

12; 2k + 2; −γb

dγb. (39)


The integral in (39) may be solved using [16, (7.621.4)]

Pe,b =12

−

∞−k=0

(−1)k Γ (k + 1/2)(2k + 1)!Γ (m) π

mγb

m

× exp[−σ 2θ (2k + 1)2

2

]Γ

m + k +

12

2F1

×

k +

12,m + k +

12; 2k + 2; −

γb

m

(40)

where 2F1 (·, ·; ·; ·) is the Gaussian hypergeometric func-tion [16, (9.10)].When σ 2

θ = 1/ (Kγb)Lo and Lam [9] investigated a more complex case,

i.e., when σ 2θ = 1/ (Kγb), where K is a proportion-

ality constant. When σθ is dependent on γb, the termexp

−σ 2

θ (2k + 1)2 /2becomes a function of γb. Conse-

quently, the term cannot be taken outside the integral overγb. Lo and Lam [9] solved the integral in (37) by first ex-pressing the confluent hypergeometric function in termsof modified Bessel functions Iν (·) and then decomposingeach of these Bessel functions using [22, (16.1.10)]. The fi-nal BER expression that was obtained is as follows

Pe,b =12

−2

πΓ (m)

mγb

m ∞−k=0

(−1)k k!Γ (k + 1/2)(2k + 1)!

×

∞−n=0

142nn!Γ (k + n + 1)

[(2k + 1)2

2KS

]λ/2

×

K−λ (T )+

(2k + 1)2

32KS (k + n + 1)2K−λ−1 (T )

(41)

where λ = m + k + 2n + 1/2, S = m/γb + 1/2, andT = (4k + 2)

√S/ (2K). However, their result is flawed (as

shown in Section 3.6) due to wrong formulation of [9, (5)].A correct result, when σ 2

θ = 1/ (Kγb), is lacking till date.

3.3. Performance of BPSK with Tikhonov distributed phaseerror and fading (where ρ = Kγb)

In this particular subsection the work by Lo and Lam [9]and by Simon and Alouini [10] are discussed in detail.Both of them obtained results for BER of BPSK withTikhonov distributed phase error over a Nakagami-mfading channel (Simon andAlouini [10] obtained results forRayleigh and Rician channels as well, which are omittedhere). Lo and Lam [9] came up with an approximation toavoid infinite sum/numerical integration by employing alinear approximation [31]. They provided an approximateexpression for BER

Pe,b ≈1

2Γ (m)

mγb

m ∫ ∞

0γ m−1b erfc

[√γb

I1 (ρ)I0 (ρ)

]× exp

−

mγbγb

dγb. (42)

Using two additional approximations, erfc (z) ≈ exp−z2

/√π and I1 (z) /I0 (z) ≈ 1− 1/ (2z) [22, (8.39)] can

be simplified as

Pe,b ≈1

2√πΓ (m)

mγb

m ∫ ∞

0γ m−1b

× exp

−γb

mγb

+

1 −

12ρ

2

dγb. (43)

Further algebraic manipulation leads to a simple closed-form expression for the approximated average BER as

Pe,b ≈exp(1/K)

√π (2K)m Γ (m)

m/γb

√1 + m/γb

m

× Km

1 + m/γb

K 2

(44)

where Km (·) is the modified Bessel function of thesecond kind of order m [16, (8.407)]. For higher values ofthe proportionality constant (K ≥ 10), the approximationserves as a tight upper bound which becomes closer forlower values ofm and virtuallymergewith the exact valueswhenm = 1. However, for low K values, the predicted BERvalues (see Section 3.6) fail tomatch the simulation results.

Inspired by the approximation technique as discussedin [32], Simon and Alouini [10] approached the problemby truncating the CEP expression with McLaurin’s series.For BPSKmodulation, adapting the expansion of (16) to thefading channel case and averaging it over the PDF of θ , theyobtained

Pe,b (γb) =12erfc

√γb+

12

γb

πexp (−γb) σ 2

θ

+ higher order terms (45)

where σ 2θ = 1/ρ = [K (Eb/N0)]−1

= (Kγb)−1 and K isa constant of proportionality representing the amount bywhich the loop SNR (in dB) exceeds the energy-per-bit tonoise power spectral density ratio (in dB). The average BERwas solved using an integration of the following form

Pe,b =

∫∞

0Pe,b (γb) fγ (γb) dγb (46)

where Pe,b (γb) is given by (45) and fγ (γb) is simplyNakagami-m PDF given in (6). Solving the integral, thefollowing expression was achieved

Pe,b ≈12

γb

π

mm

(m + γb)m+1/2

Γ (m + 1/2)Γ (m + 1)2

2F1

×

1,m +

12;m + 1;

mm + γb

+

Γ (m − 1/2)2√πγbKΓ (m)

mm

(m + γb)m−1/2 . (47)

It may be noted that (47) is a closed form expression anddoes not contain any infinite series.


3.4. Performance of BPSK with Tikhonov distributed phaseerror and fading (where ρ = Kγb)

Falujah and Prabhu [11], in their work, have alsoevaluated BER of BPSK with Tikhonov distributed phaseerror over Nakagami-m fading channel but there wasno such constraint that the loop SNR should be linearlydependent on the channel SNR. This assumption alsomadethe calculations easier as there is no longer any term in theintegrandwhich is a function of both θ andγb (like the termexp (ρ cos θ) when ρ = Kγb) and integrations over θ andγb can be performed independently.

In their paper they started with a modified version of(34) applicable to fading channel by expressing it in termsof γb = α2ηb instead of ηb

Pe,b (γb) =12

+

γb

πexp

−γb

2

∞−k=0

(−1)k+1

2k + 1I2k+1 (ρ)

I0 (ρ)

×

Ikγb2

+ Ik+1

γb2

. (48)

Putting the expressions from (48) and (6) in an integralanalogous to (46) they obtained

Pe,b =12

+2 exp (−jmπ)πΓ (m) I0 (ρ)

m

m + γb

m/2

×

∞−k=0

(−1)k+1 I2k+1 (ρ)

(2k + 1)

[Qmk− 1

2

1 +

2mγb

+ Qmk+ 1

2

1 +

2mγb

](49)

where Qµv (.) is the associated Legendre functions of the

second kind [16, (8.703)] of order µ and degree ν. Thesefunctions can be computed using a recursive algorithmdescribed in [33]. Further, (49) can be expressed interms associated Legendre function of the first kind Pµv (.)[16, (8.702)] as

Pe,b =12

+1

Γ (m) I0 (ρ)

γb

mπ

m

m + γb

(m+1/2)/2

×

∞−k=0

(−1)k+1

(2k + 1)I2k+1 (ρ)Ψ (m, k, γb) (50)

where

Ψ (m, k, γb) = Γ

m + k +

12

P−km−

12

×

2m + γb

2√m (m + γb)

+Γ

m + k +

32

P−k−1m−

12

2m + γb

2√m (m + γb)

. (51)

However, both kind of associated Legendre’s functions,Pµv (.) and Qµ

v (.) are not available in common mathe-matical software packages (like Matlab). The BER may beevaluated in terms of common Gauss hypergeometric

functions by integrating (48) over the Nakagami-m fadingPDF to yield

Pe,b =12

+1

√πΓ (m) I0 (ρ)

×

mγb

m ∞−k=0

(−1)k+1

(2k + 1)I2k+1 (ρ) [Int1 + Int2] (52)

where

Int1 =

∫∞

0γ

m−1/2b

× exp[−γb

2m + γb

2γb

]Ikγb2

dγb (53)

and

Int2 =

∫∞

0γ

m−1/2b

× exp[−γb

2m + γb

2γb

]Ik+1

γb2

dγb. (54)

Using∫∞

0zµ−1 exp (−tz) Iν (βz) dz =

βν

2ν tµ+ν

Γ (µ+ ν)

Γ (ν + 1) 2F1

×

µ+ ν + 1

2,µ+ ν

2; ν + 1;

β2

t2

(55)

both the integrals can be expressed with hypergeometricfunction as

Int1 =122k

2γb

2m + γb

m+k+1/2Γ (m + k + 1/2)

Γ (k + 1) 2F1

×

[m + k + 3/2

2,m + k + 1/2

2; k + 1;

γ 2b

(2m + γb)2

](56)

and

Int2 =1

22(k+1)

2γb

2m + γb

m+k+3/2Γ (m + k + 3/2)

Γ (k + 2) 2F1

×

[m + k + 5/2

2,m + k + 3/2

2; k + 2;

γ 2b

(2m + γb)2

](57)

Finally, substituting the values of Int1 and Int2 from(56)–(57) in (52), we get

Pe,b =12

+1

√πΓ (m) I0 (ρ)

mγb

m

×

∞−k=0

(−1)k+1 I2k+1 (ρ)

22k (2k + 1)

2γb

2m + γb

ς1×

Γ (ς1)

Γ (k + 1) 2F1

[ς2

2,ς1

2; k + 1;

γ 2b

(2m + γb)2

]+

γbΓ (ς2)

2 (2m + γb)Γ (k + 2) 2F1

×

[ς3

2,ς2

2; k + 2;

γ 2b

(2m + γb)2

](58)


Fig. 5. Simulation model for BER evaluation of BPSK with phase error (Tikhonov/Gaussian), AWGN and Nakagami-m distributed fading using the MonteCarlo method.

where ς1 = m + k + 1/2, ς2 = m + k + 3/2, andς3 = m + k + 5/2.

The authors whose papers are discussed above includedifferent approaches and assumptions to obtain suitableapproximated closed form/ infinite series expressions. Agraphical comparative study, preceded by a discussion onthe overall simulation methodology, is presented next toclarify their merits/demerits.

3.5. Simulation methodology

Apart from the direct numerical integration MonteCarlo simulations were also performed to assess theaccuracy of different methods described in the previoussubsections. Monte Carlo is a stochastic simulation processthat is used here to estimate the BER by counting theerroneous bits at the receiver and then dividing the countby the total number of bits passed through the system.Usually the number of bits examined at a SNR point is atleast 10 times higher than the inverse of the expected errorrate, i.e. to test a BER of 10−4, 105 bits were examined.Further, an average of 30 individual runs was taken tosmooth any variation about the mean.

The simulation model was realized throughMatlab andits various component blocks are described in Fig. 5. Thefirst block randomly generates binary digits with equalprobability. Next the generated bits aremodulated througha BPSK modulator. The simulations were performed in thebaseband level and thus the modulator does not involveany frequency translation. Themodulated signal is then fedto the channel where three system imperfections, fading,AWGN, and phase error, are introduced. Accordingly, thesimulation process involves generation of three differentkinds of random variables. Gaussian noise may be realizedrather easilywith an in-built function randn (·) available inMatlab. However, realization of a fading envelope whichfollows Nakagami-m distribution is a little bit tricky.For integer/half integer fading parameter (m) values,Nakagami RV may be generated in the following manner.If X1, X2, . . . , X2m denotes a set of Gaussian RV with zeromean and equal variance, i.e. Xl ∼ N

0, σ 2

; ∀l, then a

RV XNak defined as XNak =

X21 + X2

2 + · · · + X22m follows

Nakagami-m distribution. For arbitrary values of m, theRV may be generated by taking square root of samplesof a gamma distributed RV. The root mean square valueof the generated fading RV is normalized to 1. Finally, torealize phase error we need to generate either Gaussianor Tikhonov distributed random samples. Generation of aTikhonov distributed RV is, by far, the most difficult taskencountered in the simulation process and details of thealgorithm used are given later in this section. The output ofthe channel is passed through a basebanddemodulator andits output is compared with the original bits. Subsequentlythe BER values are computed.

The first well-known method for Tikhonov RV gener-ation was developed by Mardia [34] while a faster algo-rithm was derived by Best and Fisher [35], with alternatemethods proposed later by Dagpunar [36], Barabesi [37],Wood [38], Yuan and Kalbeisch [39], Devroye [40], and deAbreu [41].

To generate Tikhonov distributed RV we have testedfour methods given by [34,35,40,41] for a given valueof ρ = 7 dB (µ = 0) and the corresponding graphsare shown in Fig. 6. It is quite evident from the graphsthat the simulated PDFs following Best–Fisher [35] andde Abreu’s [41] algorithm deviate significantly from thetheoretical PDF. Further, although Mardia’s method [34]gives even poorer performance for the given set ofparameters, it has been observed that a good degree ofsimilarity is achieved for small ρ values (ρ < 3 dB).The best result was obtained when we follow the methodproposed by Devroye [40]. It was also verified that thesimilarity is maintained for almost all values of ρ withinthe range of interest. The simulation algorithm followingDevroye’s method is as follows

The only difficulty with this algorithm is that it requiresa Bessel and a beta random variate. Simulating Besseldistribution is generally difficult. Devroye [40] suggested atruncated normal approximation for Bessel RVs when themean is large, and a table sampling procedure when themean is small. On the other hand, generation of a beta RV


Fig. 6. Comparison of the theoretical Tikhonov PDF with simulated PDFsobtained through four different algorithms—Mardia [34], Best–Fisher[35], de Abreu [41], and Devroye [40].

is relatively easy. For any z ≥ 1/2, a beta RV β ∼ (z, 1/2)can be obtained with a one line code

β = 1 −

1 − U

22z−11

cos2 (2πU2) (59)

where U1 ∼ (0, 1) and U2 ∼ (0, 1) are independentuniform RVs.

3.6. Graphical comparison

In this subsection we provide a graphical comparisonof the expressions provided in Sections 3.1–3.4. For thepurpose, direct numerical integration values as well assimulation results are depicted (alongwith the expressionsfrom different authors) in the average BER versus SNRper bit plots. The excellent agreement between numericaland simulation results proves validity of the MonteCarlo simulation model, described in Section 3.5. Further,percentage of error is calculated for every individual caseusing the definition given in (18). This metric is very usefulin differentiating the alternate solutions for a specific caseas spotting the difference from the BER plots with bareeyes is sometimes impossible, especially in the higher SNRrangewhen the BER values are very small and all the curvesoverlap with each other (see Fig. 7).

It is obvious from Figs. 8–12 that the lack of perfectphase recovery in coherent PSK systems leads to extrasystem performance degradation relative to the originalpenalty owing to the existing impairments (channelfading and/or AWGN). All the curves show that the errorprobabilities rise significantly, even when a phase errorof a few degrees is present. To maintain uniformity, thegraphs for Tikhonov distributed phase error are plotted forρ = 10 dB while for Gaussian distributed phase error σθ isassumed to be 20° and the Nakagami-m fading parameteris fixed at m = 2. Inspecting the BER curves one finds thatall of them follow almost exponential decrease, especiallyat high SNR values. Thus,whenplotted in logarithmic scale,

Fig. 7. Generator algorithm for the Tikhonov/von Mises distribution.

the nature of the curves appears linear. Moreover, whilecommunicating over a Nakagami-m fading channel, fora given average SNR per bit, the BER would decrease asfading severity decreases or equivalently m increases. Thevariation with fading parameter is, however, not shown inthese plots.

In Fig. 8, a comparison of three solutions for BER ofBPSK with Tikhonov distributed phase error in AWGNchannel, provided by Lindsey [3], by Kam et al. [7], andby Falujah and Prabhu [11], are depicted. Lindsey’s [3]theoretical plot, given by (22), basically gives an upperbound which is asymptotical in nature for negative SNRvalues but starts drifting away for positive channel SNR. Asa result, the corresponding percentage of error is also large(shoots above 100% just after γb = 0 dB). On the otherhand, the theoretical plot, when the expression given byKam et al. [7] is considered, is much closer. However, noasymptotic behavior has been observed and the plot cutsthe numerical/simulation plot at −4 dB. It may be notedthat two different expressions, (27) and (30) are used toobtain the plot for the two ranges, γb < ρ/2 and γb ≥ ρ/2.Consequently, a kink (sudden change of slope) at γb = 5 dB(ρ = 10 is used for plots) is present in the plot. Thepercentage of error is beyond 100%on the either side of thisthreshold value (γb = 5 dB) but comes down significantlywhen the channel SNR is much higher (or lower). Falujahand Prabhu’s [11] theoretical plot is far better than bothof the earlier solutions giving exact match with numerical/simulation plots with negligible percentage of error.

Figs. 9 and 10 shows the theoretical plots with Gaussiandistributed phase error as mentioned in (40) and (41).Fig. 9 depicts the case when the Gaussian phase errorparameter σθ is independent of average channel SNRγb. The analytical solution appears to be a lower boundwhich becomes tighter as the SNR decreases. For largerchannel SNR, the variation between the estimated BERand the numerical values become non-negligible andthe corresponding percentage of error starts increasingexponentially. Lo and Lam’s [9] solution for the case σ 2

θ =

1/ (Kγb), is however, seems flawed (the reason is describedin Section 3.2). As seen from Fig. 10, the predicted BERvalues are nowhere near the numerical/simulated plots forK = 0.1, and the percentage of error reaches 100% atγb = 11 dB.


(a) Bit error probability. (b) Percentage of error.

Fig. 8. Error performance of BPSK with Tikhonov distributed phase error and AWGN.


Fig. 9. Error performance of BPSK with Gaussian distributed phase error, AWGN, and Nakagami-m fading (when σθ is independent of γb).


Fig. 10. Error performance of BPSK with Gaussian distributed phase error, AWGN, and Nakagami-m fading (when σ 2θ = 1/(Kγb)).

The comparison between Lo and Lam [9] and Simonand Alouini’s [10] approach for evaluating BER of BPSKover Nakagami-m fading channel with imperfect phaserecovery (Tikhonov distributed, ρ = Kγb) is displayedin Fig. 11. Again the nature of the curve at K = 0.1is completely different from the actual values for Lo and

Lam’s solution, given by (44). Although their solutiongives a loose upper bound for high K (≥10) values, thepercentage of error rise linearly with channel SNR.

A much better solution was given by Simon andAlouini [10], almost following the numerical integra-tion/simulation plot at high K values. The corresponding



Fig. 11. Error performance of BPSK with Tikhonov distributed phase error, AWGN, and Nakagami-m fading (when ρ = Kγb).


Fig. 12. Error performance of BPSK with Tikhonov distributed phase error, AWGN, and Nakagami-m fading (when ρ = Kγb).

expression, given by (47), serves as a much tighter upperbound while the percentage of error remains invariable tochannel SNR fluctuations. For K values lower than 1, how-ever, their solution also fails to track the BER variation in areliable manner.

The average BER of BPSK over Nakagami-m fadingchannel with Tikhonov distributed imperfections in phaseestimation (ρ is independent of γb) and the correspondingpercentage of error is plotted in Fig. 12 following Falujahand Prabhu’s method [11]. Actually a variant of their basicexpression (the original expression contains associatedLegendre’s functions), in terms of common Gaussianhypergeometric functions and given by (58), is plotted. TheBER plot exactly follows numerical/simulation plot and thepercentage of error is also very low till the SNR reaches12 dB, after which an exponential increment has beenobserved.

4. Simpler approach for BER calculation with Gaussiandistributed phase error

After discussing the merits and demerits of thesolutions provided by earlier researchers, in this sectionand in next section, we propose two simple alternative

approaches for calculating analytical bit error rate throughHermite’s method of integration (when the phase errorassumes Gaussian distribution) and through MGF method(when phase error is Tikhonov distributed). Onemay arguethat instead of adoptingHermite’smethod orMGFmethod,a direct numerical integration of (17) through powerfultools like Mathematica would yield the same result andthat too with lesser complexity. But interestingly this isnot true. The infinite limits of the integrals often leadto numerical instability. The authors have verified thatthis instability results in considerable inaccuracy whencompared to values that are found by simulation runs.

The present subsection is devoted to BER performanceanalysis of BPSK over Nakagami-m fading channel, assum-ing relatively less studied Gaussian model for phase errordistribution. The novelty lies in the treatment as we [42,43], unlike others, have averaged the CEP over the fadingchannel first and then used numerical summation to findBER values. Compared to earlier methods, this techniqueresulted in a better match with the simulated values.

4.1. Hermite’s method

Hermite’s method [22, (25.4.46)] is a well-knowntechnique for numerical evaluation of integrals where the


integrand consists of an exponential term of the formexp

−z2

along with some other function of z and where

the integral is defined over−∞ < z < ∞.Mathematically∫∞

−∞

exp−z2

g (z) dz ≈

n−i=1

κig (χi) (60)

where χi is the ith zero of the Hermite polynomial Hn (·), nis the order of the polynomial and weight κi is defined as

κi =2n−1n!

√π

n2 [Hn−1 (χi)]2. (61)

A detailed list of values of the pair {χi, κi} for different n isavailable in [22, Table 25.10].

4.2. BER for Nakagami-m fading channel

We proceed by interchanging the order of integral in(17) and solving the integral over γb first

Pe,b =

∫∞

θ=−∞

[∫∞

γb=0Pe,b (γb, θ) fγ (γb) dγb

]fθ (θ) dθ

=

∫∞

θ=−∞

Pe,b (γb, θ) fθ (θ) dθ (62)

where, for Nakagami-m fading channel, Pe,b (γb, θ) is givenby

Pe,b (γb, θ) =Γ (2m)

Γ (m + 1)Γ (m)

×

m

4m + γb cos2 θ

mγb cos2 θ

m + γb cos2 θ2F1

×

1,m +

12;m + 1;

mm + γb cos2 θ

(63)

which may be obtained by replacing γb with γb cos2 θ inthe error probability expression of BPSK over Nakagami-mfading channel without phase error

Pe,b (γb) =Γ (2m)

Γ (m + 1)Γ (m)

[m

4 (m + γb)

]m×

γb

m + γb2F1

1,m +

12;m + 1;

mm + γb

. (64)

Eq. (64) may be obtained by directly averaging Pe,b (γb) =

(1/2) erfc√γbover (6) after expressing the complemen-

tary error function erfc (·) in terms of complementary in-complete gamma function Γ (·, ·) [16, (8.350.2)] and thenusing [16, (6.455.1)].

The basic idea stems from the fact that the projectionof the received signal vector (ri) with phase error (as seenfrom Fig. 4) is ri cos θ which is used in the decision makingprocess. Hence, the effective SNR per bit with phase errormay be written as γb cos2 θ , where γb denotes the averagechannel SNR per bit under perfect phase estimation.

Substituting Pe,b (γb, θ) from (63) and fθ (θ) from (7) in(62) we obtain

Pe,b =1

2πσ 2θ

Γ (2m)Γ (m + 1)Γ (m)

×

∫∞

−∞

m

4m + γb cos2 θ

mγb cos2 θ

m + γb cos2 θ

× exp

−θ2

2σ 2θ

2F1

1,m +12;m + 1;

mm + γb cos2 θ

dθ. (65)

It may be noted that (65) is an exact result and doesnot suffer from any approximation or series truncation asfound in previous literature [9].

Unfortunately there are no known formulas to solve(65) in closed form. Nevertheless, the need of directintegration may be avoided by resorting to numericaltechniques. We invoke Hermite’s method, described in theprevious subsection, for the purpose. From (60) and (65),with the substitution χ = θ/

√2σθ

, we get the BER in

Nakagami-m fading channel as [43]

Pe,b ≈

n−i=1

κigNak (χi) (66)

where

gNak (χ) =Γ (2m)

√πΓ (m + 1)Γ (m)

×

[m

4{m + γb cos2(√2σθχ)}

]m×

γb cos2(

√2σθχ)

m + γb cos2(√2σθχ)

2F1

×

[1,m +

12;m + 1;

m

m + γb cos2(√2σθχ)

]. (67)

As a double check, by putting m = 1 in (67) and using[22, (15.1.14)], it can be easily verified that (67) reducesto the Rayleigh fading case

gRay (χ) =1

2√π

1 −

γb cos2√

2σθχ

1 + γb cos2√

2σθχ (68)

as mentioned in [42, (15)].

4.3. BER plot and related discussions

Using the expression given in (66) and (67), BER valuesare plotted in Fig. 13 with the following set of parameters,σθ = {10°, 20°}, m = {2, 3}, and n = 5. The percentage oferror curves are also included to compare the results withthe earlier approach demonstrated in Section 3.2.

From Fig. 13, it is evident that the numerical/simulationresults almost coincide with the calculated analytical



Fig. 13. Error performance of BPSK with Gaussian distributed phase error, AWGN, and Nakagami-m fading for different fading parameter (m = 2 and 3)and phase error standard deviation (σθ = 10° and 20°) values.

values. Further, the percentage of error value is also smallwhen compared to the analytical result presented in Fig. 9(Gaussian phase error, σθ is independent of γb). At an SNRvalue of 20 dB (m = 2, σθ = 20°), the Hermite’s methodyield a percentage error of 1.46%, while the percentageof error was 11.40% with the earlier analytical approach,showing improvement of about an order. Also, for σθ =

10°, the error percentage is almost down to zero. Forlarger n values (n > 5), at a price of slight increase in thecomputation time, this errormay be further reduced givingmore accurate results.

5. Simpler approach for BER calculation with Tikhonovdistributed phase error

Nextwewill be deducing themathematical expressionsfor BER of BPSK in Nakagami-m fading channel, whenaffected by Tikhonov distributed phase error. For thepurpose, a simpler approach to evaluate the average BER isadopted which involves MGF of the fading SNR γb, Mγ (s).

Although the BER values can be computed via directnumerical integration, the MGF method performs farbetter in terms of accuracy, applicability, and computationtime. The direct integration often suffers from numericalinstability due to the presence of infinite integration limitswhich leads to inaccuracy. Regarding applicability, whendiversity combiners are applied to mitigate the effectof fading, the direct integration of PDF fails in case thePDF of instantaneous SNR is not available or containscomplex mathematical functions. However, calculation ofthe correspondingMGF is rather easy and theMGFmethodmay still provide the BER values. Finally, the averagecomputation time is also less withMGFmethod (describedin detail at the end of this section).

5.1. MGF method

Utilizing alternate representation of erfc (·) [15, (4.2)],the CEP of BPSK with phase error and fading, given by (16),can be expressed as

Pe,b (γb, θ) =1π

∫ π/2

φ=0exp

−γb cos2 θsin2 φ

dφ. (69)

We proceed by substituting (69) in (17), which renders(17) to be in a form of triple integral now, and thereafterinterchange the order of integrals to obtain

Pe,b =1π

∫ π

θ=−π

∫ π/2

φ=0

[∫∞

γb=0exp

−γb cos2 θsin2 φ

× fγ (γb) dγb

]fθ (θ) dφdθ (70)

which is possible if we consider the loop SNR ρ to beindependent of the channel SNR γb (ρ = Kγb), and as aconsequence, fθ (θ) does not contain any term dependenton the RV γb. The origin of the assumption ρ = Kγb,comes from the fact that the PLL (used as the phaseestimator) carrier power term, P , in (9) may be expressedas, P = α2Eb/Tb, where γb = α2Eb/N0 as describedearlier. However, generally the fade bandwidth is muchsmaller than the loop bandwidth of the PLLs so that thePLL can attain steady state in every tracking interval. Thus,for a slow fading channel, P may be treated as beingconstant over many bit durations and the phase error θcan be treated as independent of the fading envelope α, orequivalently the SNR term γb [13,44,45].

From the definition of MGF Mγ (s) =

∞

0 fγ (γ ) exp(−sγ ) dγ , we may replace the term

∞

γb=0 exp(−γb cos2 θ

/ sin2 φ)fγ (γb) dγb in (70) by Mγ

cos2 θ/ sin2 φ

. Further

substituting fθ (θ) from (8) in (70) we have

Pe,b =1

2π2I0 (ρ)

∫ π

θ=−π

∫ π/2

φ=0Mγ

cos2 θsin2 φ

× exp (ρ cos θ) dφdθ. (71)

A limitation of the MGF method is, it cannot provideclosed-form solutions. The end expressions always consistof double integrals. Nevertheless, both the integrals in (71)are having finite integration limits and their evaluationis generally free from any numerical instability. This alsoexplains why we have not opted for the MGF methodinstead of Hermite’s formula in the Gaussian case. Thelimits for θ is infinite for the Gaussian case and the endresults with MGF method would have infinite integrationlimits, evaluation of which may suffer from convergenceproblems.



Fig. 14. Error performance of BPSK with Tikhonov distributed phase error, AWGN, and Nakagami-m fading for different fading parameter (m = 2 and 3)and Tikhonov phase parameter (ρ = 9 dB and 10 dB) values.

5.2. BER for Nakagami-m fading channel

Using the expression of MGF of SNR γb in Nakagami-mfading channel from [15, (2.22)]

Mγ (s) =

m

m + sγb

m

(72)

the final expression for average BER of BPSK becomes

Pe,b =1

2π2I0 (ρ)

∫ π

−π

∫ π/2

0

m

m + γbcos2 θ/ sin2 φ

m

× exp (ρ cos θ) dφdθ. (73)

For detailed calculations the readers are referred to [46,47],the former one discussing BER over Nakagami-m fadingchannel whereas, using the same approach, results forRayleigh, Rician and Hoyt fading are reported in the later.

5.3. BER plot and related discussions

Assuming ρ = 9 dB and 10 dB and a Nakagami-mparameter of m = {2, 3}, the final average BER expres-sion in (73) is plotted in Fig. 14. The theoretical valuesobtained through MGF method follows Monte Carlo sim-ulation/direct numerical integration curves with a highdegree of accuracy, which in turn, validates our analy-sis. When compared to the earlier work by Falujah andPrabhu [11], one may find from Fig. 12 that the percent-age error obtained for MGF approach is much lesser. Up toa channel SNR of 7dB the error percentage is almost zeroand at an SNR of 20dB, the error rises to a mere value of5.16% for m = 2, ρ = 10 dB. On the other hand, thepercentage of error with Falujah and Prabhu’s approachincreases exponentially with SNR and attains a value of24.51% (ρ = 10 dB) at an SNR of 20 dB.

5.4. Computation time

Regarding the earlier developments, we would nowlike to compare the computational time required for MGFmethod against the time required for direct numericalintegration [47]. For the purpose, the total computational

time is calculated for different SNR (γb) values and isdisplayed in Table 1. It is quite evident that time takenwithMGF method (t2) is 5–10 times less than the numericalintegration time (t1).

6. SER of M-ary PSK with Tikhonov distributed phaseerror and fading

In this section, the MGF approach described in the pre-vious section has been extended to calculate SER of generalM-ary PSK systems in fading channels when they are af-fected with Tikhonov distributed phase error. A great dealof research work [48–52] may be found in the recent lit-erature discussing SER performance of PSK constellations.In all these papers perfect carrier phase recovery and sym-bol timing synchronization have been assumed. However,phase and timing synchronization errors can have a signif-icant effect on the receiver’s capability to make the correctdecisions. The problem gets more severe as data rates in-crease and the corresponding symbol durations decrease.To the best of author’s knowledge, the error rate analy-sis of MPSK under imperfect carrier synchronization or bittiming error is limited so far to binary signals (see specifi-cally [53], Sections 4.3.2 and 4.3.3 and references therein)only.

6.1. Calculation of SER by MGF method

The exact formula for the SER of MPSK was first givenby Pawula [54, (2)], which was derived assuming perfectphase synchronization between transmitter and receiver.In practice, we have to consider a coherent MPSK systemwith a carrier tracking loop for synchronization purpose.Any inaccuracy of the estimator is manifested as phaseerror, characterized by θ = φ − φ.

The vector representation of an MPSK system consistsof points uniformly distributed on a circle of radius

√Es

as demonstrated in (1). Fig. 15 represents the geometryfor correct decision region when the symbol waveforms0 (t) (corresponding to the signal point s0 = −

√Es

is transmitted), where the origin of coordinates hasbeen shifted to the signal point and for convenience the


Table 1Computation time for direct numerical integration (t1) and with MGF method (t2) for Nakagami-m (m = 2) fading channel.

Avg SNR per bit (γb), in dB Time required for numerical integration (t1), in seconds Time required for MGF method (t2), in seconds

0 0.1489 0.08931 0.0212 0.07962 0.1913 0.05763 0.1305 0.04024 0.1009 0.04925 0.0873 0.03266 0.0766 0.02547 0.0650 0.02478 0.0507 0.02359 0.0458 0.0208

10 0.0180 0.017511 0.0179 0.017412 0.0184 0.016413 0.0203 0.014014 0.0183 0.014115 0.0191 0.013816 0.0188 0.014017 0.0184 0.014718 0.0188 0.014219 0.0182 0.013820 0.0182 0.0140

Fig. 15. Geometry for correct decision region for s0(t) [55].

coordinate system has been rotated by θ radians. EE ′ isthe boundary region for the waveform and R1,2 denote thedistances from the signal point to the boundary points Eand E ′ (in general, a function ofΦ , the noise phase).

Using bivariate Gaussian PDF which represents thenoise vector in polar coordinates and applying the lawof sines to triangles ∆OO′E and ∆OO′E ′, the final desiredresult for the conditional SER for MPSK with phase errorcan be written as [55]

Pe (η, θ) =12π

∫ ψ1

φ=0exp

[−η

sin2 (π/M + θ)

sin2 φ

]dφ

+12π

∫ ψ2

φ=0exp

[−η

sin2 (π/M − θ)

sin2 φ

]dφ (74)

where ψ1,2 = (M − 1) π/M ∓ θ . In fading channel, theconditional SEP expression of MPSK as given in (74) maybe modified as

Pe (γ , θ) =12π

∫ ψ1

0exp

[−γ

sin2 (π/M + θ)

sin2 φ

]dφ

+12π

∫ ψ2

0exp

[−γ

sin2 (π/M − θ)

sin2 φ

]dφ (75)

to incorporate the effect of both fading and phase error.Further, it may be noted γ = α2Es/N0 denote the symbolSNR and forM = 2 (BPSK), (75) reduces to (16).

Now substituting (75) in the generic equation Pe =γ

θPe (γ , θ) fγ (γ ) fθ (θ) dθdγ , we get

Pe =12π

∫∞

0

∫ π

−π

[∫ ψ1

0exp (−γ β1) dφ

]× fγ (γ ) fθ (θ) dθdγ

+12π

∫∞

0

∫ π

−π

[∫ ψ2

0exp (−γ β2) dφ

]× fγ (γ ) fθ (θ) dθdγ (76)

where β1,2 = sin2 (π/M ± θ) / sin2 φ. Eq. (76) clearlyshows that a direct numerical integration would requireevaluation of triple integrals. Further, the innermostintegrals result in functions of θ (as the upper limits ψ1,2are both functions of θ ), rendering the triple integral to bea mixture of numerical and symbolic integration which isnot supported in popular mathematical software packageslike Matlab.

Interchanging the order of integrals in (76), we obtain

Pe =12π

∫ π

−π

∫ ψ1

0

[∫∞

0exp (−γ β1)

× fγ (γ ) dγ]dφfθ (θ) dθ

+12π

∫ π

−π

∫ ψ2

0

[∫∞

0exp (−γ β2)

× fγ (γ ) dγ]dφfθ (θ) dθ. (77)

From the definition of MGF, Mγ (s) =

∞

0 fγ (γ ) exp(−sγ ) dγ wemay replace the terms

∞

0 exp (−γ βi) fγ (γ )dγ in (77) byMγ (βi) ; i ∈ {1, 2}. Further substituting fθ (θ)


from (8) in (77), we finally get an expression

Pe =1

4π2I0 (ρ)

∫ π

−π

∫ ψ1

0Mγ (β1) exp (ρ cos θ) dφdθ

+1

4π2I0 (ρ)

∫ π

−π

∫ ψ2

0Mγ (β2) exp (ρ cos θ) dφdθ (78)

involving double integrals.

6.2. Simplification with weighted summation [56]

From (78), it is evident that the use of MGF methodresults into a double integral, each having a finiteintegration limit. When compared to the original SERexpression containing triple integral with infinite limitsgiven in (76), (78) proves to be more usable and easier toevaluate. For further simplification, (78) may be written as

Pe =12π

[∫ π

−π

Int1 (θ) fθ (θ) dθ

+

∫ π

−π

Int2 (θ) fθ (θ) dθ]

(79)

where

Int1,2 (θ) =

∫ ψ1,2

0Mγ

β1,2

dφ (80)

However, the problem of symbolic integration is yet un-resolved which we overcome with a weighted summationtechnique described next.

The double integral in (78) can be reduced to asingle integral with a weighted summation method in thefollowing manner. The expression in (79) is in the formx g (x) f (x) dx, with g (x) being a function of the RV x and

f (x) denoting the PDF of x. When the integration limitsspan thewhole range of the RV x, the integral gives nothingbut the mean, g (x) = E {g (x)}. To avoid the integration,the expectation g (x) can be approximated in terms of thefollowing summation

g (x) ≈

∑ig (xi) f (xi)∑if (xi)

(81)

where the denominator∑

i f (xi) is used to normalize thesum. Applying (81), (79) can be modified as

Pe ≈12π

π∑

θj=−π

Int1θjfθθj

π∑θj=−π

fθθj

+

π∑θj=−π

Int2θjfθθj

π∑θj=−π

fθθj

(82)

involving only single integrals (Int1 and Int2). An incrementof θj in steps of π/10 radians is enough to ensure anaccuracy of SER values up to 4digits after the decimal point.

6.3. SER for Nakagami-m fading channel

In order to evaluate Int1θjand Int2

θj, we require

MGF expression of Nakagami-m fading channel which isgiven by (72). The final SER expression for MPSK withphase error in Nakagami-m fading channel is

Pe ≈

π∑θj=−π

fθθj [ ψ ′

10 G1

φ, θj

dφ +

ψ ′2

0 G2φ, θj

dφ]

2ππ∑

θj=−π

fθθj (83)

where

G1,2φ, θj

=

m

m + γ β ′

1,2

m

(84)

and γ = E {γ }, β ′

1,2 = β1,2|θ=θj , ψ′

1,2 = ψ1,2|θ=θj . Sideby side, let us compare the expression that would be usedfor numerical evaluation, i.e. by directly inserting the PDFsfγ (γ ) and fθ (θ) in Pe =

γ

θPe (γ , θ) fγ (γ ) fθ (θ) dθdγ

Pe =1

4π2γ I0 (ρ)Γ (m)

mγ

m

×

[∫∞

0

∫ π

−π

∫ ψ1

0H1 (φ, θ, γ ) dφdθdγ

+

∫∞

0

∫ π

−π

∫ ψ2

0H2 (φ, θ, γ ) dφdθdγ

](85)

where H1,2 (φ, θ, γ ) = γ m−1 exp(−γ β1,2 − mγ /γ+ ρ cos θ) and a quick view reveals sufficient reduction incomplexity in the former expression.

6.4. SER plot

Fig. 16 plots SER values for the Nakagami-m fadingchannel using the weighted summation approach. Excel-lent agreement between analytical and simulation valueshave been observed.

7. Conclusions and future scope

In this article we have provided a comprehensivesurvey of BER/SER calculation of PSK modulation, whenperturbed with erroneous phase estimate in Nakagami-mfading channel, and have proposed a detailed classificationthereof. A graphical comparison of BER curves for BPSK suf-fering from phase synchronization problem is displayed.Apart from surveying the existing methodologies, we havealso introduced two novel approaches for calculating biterror probabilities of BPSK, using Hermite’s method of in-tegration for Gaussian distributed phase error and MGFmethod for Tikhonov distributed phase error. Later, ex-tending the MGF approach, we have calculated simple SERexpressions for the general MPSK case. Extensive simula-tions were also performed to authenticate the theoreticalresults.

An extension of the approaches described here toother wireless fading models (Rayleigh, Rician/Nakagami-n, Hoyt/Nakagami-q, Weibull, or generalized Gamma) is


Fig. 16. SER of MPSK with Tikhonov distributed phase error (ρ = 20 dB)in Nakagami-m (m = 2) fading channel for different constellation sizes(M = 4, 8, and 16).

quite straightforward [47,56]. Another scope of futureresearch is to see how coherent PSK schemes with phaseerror will perform in a multi-antenna system employingsome sort of diversity combining. There are already someliterature available in this regard. Specifically results forspace diversity with equal gain combining (EGC) werepresented by Najib and Prabhu [57] whereas Ziemeret al. [44] considered the effects of imperfect carriertracking on RAKE reception.

Acknowledgements

The authors would like to thank the anonymousreviewers, Dr. Manas Kr. Bose, Ansaldo STS, Newcastle,Australia, and Ms. Dipanwita Biswas, Bengal College ofEngineering and Technology, Durgapur, WB, India for theirvaluable inputs.

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[57] M.A. Najib, V.K. Prabhu, Analysis of equal-gain diversity withpartially coherent fading signals, IEEE Transactions on VehicularTechnology 49 (3) (2000) 783–791.

Aniruddha Chandra received his B.E. (Hons.)degree in Electronics and Communication En-gineering and M.E. degree in Communica-tion Engineering from Jadavpur University (JU),Kolkata, India in 2003 and 2005 respectively andis currently pursuing a Ph.D. there.

He joined Department of Electronics andCommunication Engineering, National Instituteof Technology (NIT) Durgapur in 2005 as aLecturer and currently serving as an AssistantProfessor there. His research interests include

wireless fading channelmodelling, diversity combining,modulation tech-niques, broadband access over powerline, and underwater communica-tion.

Mr. Chandra has co-authored a book titled Analog Electronic Circuitsand published 30 research papers in reputed journals and peer-reviewedconferences. He has also delivered several invited lectures including IEEEComSoc lecture meetings at JU, Kolkata. He received the prestigious NTSEand EFIP scholarships from NCERT andMHRD, Govt. of India respectively.He is a member of IEEE ComSoc, IAENG and has also served as a reviewerfor IEEE Transactions on Vehicular Technology, IEEE Communication Letters,IEEE Potentials, Computer and Electrical Engineering (Elsevier), Journal of theFranklin Institute(Elsevier), and International Journal of Electronics(Taylorand Francis).

Ananya Patra was born in Burdwan, WestBengal, India. She received her B.Tech. degreein Electronics and Communication Engineer-ing from College of Engineering and Manage-ment, Kolaghat, WB in 2008, M.Tech. degree inTelecommunication Engineering from NationalInstitute of Technology, Durgapur, and presentlypursuing her Ph.D. at Indian Institute of Technol-ogy, Guwahati, India.

Her research areas include digital communi-cation, M-ary modulations, fading and diversity

concepts. She is a member of IEEE.

Chayanika Bose received her B.Tech., M.Tech.and Ph.D. degrees in radio physics and electron-ics from the University of Calcutta in 1981, 1983and 1990.

She served as Scientist B in Training Programin Millimeter Wave Technology, DOE at Instituteof Radio Physics and Electronics, Calcutta Uni-versity and subsequently joined as post-doctoralfellow (UGC) in the department of Electronicsand Telecommunication Engineering, JadavpurUniversity. Presently she is a Reader in the same

department. Her research interests involve semiconductor nanostruc-tures and radio wave propagation.

Dr. Bose has published more than 40 papers in referred journals. Sheis a member of IEEE Electron Device Society and IEEE Communication So-ciety since 1999 and became a Senior Member in 2005.

Performance analysis of PSK systems with phase error in fading channels: A survey

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