SEP calculations for coherent M-ary FSK in different fading channels with MRC diversity

INTERNATIONAL JOURNAL OF COMMUNICATION SYSTEMSInt. J. Commun. Syst. 2011; 24:202–224Published online 19 May 2010 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/dac.1150

SEP calculations for coherent M-ary FSK in different fadingchannels with MRC diversity

Aniruddha Chandra1,∗,†, Srinivasa Rao Poram1 and Chayanika Bose2

1Department of Electronics and Communication Engineering, National Institute of Technology, M. G. Avenue,Durgapur-713209, Burdwan, West Bengal, India

2Department of Electronics and Telecommunication Engineering, Jadavpur University, Raja S. C. Mallick Road,

Kolkata-700032, West Bengal, India

SUMMARY

In this paper, the authors derive symbol error probability (SEP) expressions for coherent M-ary frequencyshift keying (MFSK) modulation schemes in multipath fading channels. The multipath or small-scalefading process is assumed to be slow and frequency non-selective. In addition, the channel is also subjectedto the usual degradation caused by the additive white Gaussian noise (AWGN). Different small-scalefading statistics such as Rayleigh, Rician (Nakagami-n), Hoyt (Nakagami-q), and Nakagami-m have beenconsidered to portray diverse wireless environments. Further, to mitigate fading effects through spacediversity, the receiver front-end is assumed to be equipped with multiple antennas. Independent andidentically distributed (IID) as well as uncorrelated signal replicas received through all these antennas arecombined with a linear combiner before successive demodulation. As the detection is coherent in natureand thus involves phase estimation, optimum phase-coherent combining algorithms, such as predetectionmaximal ratio combining (MRC), may be used without any added complexity to the receiver. In the currenttext, utilizing the alternate expressions for integer powers (1�n�4) of Gaussian Q function, SEP valuesof coherent MFSK are obtained through moment generating function (MGF) approach for all the fadingmodels (with or without MRC diversity) described above. The derived end expressions are composed offinite range integrals, which can be numerically computed with ease, dispenses with the need of individualexpressions for different M , and gives exact values up to M =5. When the constellation size becomesbigger (M�6), the same SEP expressions provide a quite realistic approximation, much tighter than thebounds found in previous literatures. Error probabilities are graphically displayed for each fading modelwith different values of constellation size M , diversity order L , and for corresponding fading parameters(K , q , or m). To validate the proposed approximation method extensive Monte-Carlo simulations were alsoperformed, which show a close match with the analytical results deduced in the paper. Both these theoreticaland simulation results offer valuable insight to assess the efficacy of relatively less studied coherent MFSKin the context of the optimum modulation choice in wireless communication. Copyright � 2010 JohnWiley & Sons, Ltd.

Received 13 February 2009; Revised 19 January 2010; Accepted 6 April 2010

KEY WORDS: symbol error probability; M-ary frequency shift keying; Rayleigh fading; Rician fading;Hoyt fading; Nakagami-m fading; predetection maximal ratio combining; Gaussian Qfunction

1. INTRODUCTION

Data traffic in wireless networks had surpassed the voice traffic long ago, almost a decade fromnow. Since then the data-centric networks are trying hard to cope up with new multimedia services

∗Correspondence to: Aniruddha Chandra, Department of Electronics and Communication Engineering, NationalInstitute of Technology, M. G. Avenue, Durgapur-713209, Burdwan, West Bengal, India.

†E-mail: [email protected]

Copyright � 2010 John Wiley & Sons, Ltd.

SEP CALCULATIONS FOR COHERENT M-ARY FSK 203

introduced every other day. For smooth functioning of these killer applications very high-capacitytransmission is required, with the latest application demanding even higher data rate than itspredecessor. Current wireless transceiver designs are resorting to M-ary modulation formats toachieve simultaneous transmission of multiple information bits. Traditionally, M-ary phase shiftkeying (MPSK) or M-ary quadrature amplitude modulation (MQAM) have been preferred overtheir counterparts for the purpose. Both of them realize spectral efficiency at the expense of powerefficiency and work well when the system is bandwidth constrained. However there is a wide rangeof wireless communication systems, deep space, sensor networks, ultra-wideband (UWB), andunderwater communications, which operate in the low-power regime where power consumptionrather than bandwidth is the limiting factor. In such cases, orthogonal signaling schemes likeMFSK is a better choice. In fact the capacity results show [1] that it is possible to make the errorprobabilities of orthogonal signaling arbitrarily small when the constellation size M goes to infinityas long as the normalized signal-to-noise ratio (SNR) is greater than Shannon’s limit, i.e. −1.6dB.For a given SNR in an AWGN channel, coherent MFSK offers minimum symbol error probability(SEP) out of all possible M-ary modulation schemes for M>2. But one has to remember that thispower advantage comes at the expense of asymptotical increment in bandwidth.

MFSK modulation is widely applied in transmitter energy limited space communication systemssuch as deep space probes [2], satellites, and space telemetry where link capacity may be enhancedat the cost of required transmission bandwidth. During the 2010–2020 timeframe, in future NASAMars programs like Competed Scout Mission, Mars Science Orbiter, etc. [3] MFSK continuesto be the modulation choice. MFSK is also suitable for hand-held satellite terminals that requirelow-complexity low-cost receiver structure [4]. Cellular wireless terminals are generallybattery-driven devices and should be handy, which restrict their capability of increasingtransmission power; especially in the reverse link. In addition, small-scale multipath fading inwireless channels contributes to further power penalty. This inspired some researchers [5, 6] toinvestigate whether M-ary orthogonal signaling is a suitable option for CDMA or not. In thisregard, it would not be irrelevant to mention some of the industrial products based on MFSKthat are already available in the market. For example, a low-power narrow-band (140–900 MHz)radio transmission modem MOdems for Radio-based SystEms (MORSE) developed by RACOMs.r.o., Czech Republic [7] uses coherent MFSK. A next-generation paging system called FLEX,based on a wide-band M-ary FSK, has been introduced in many countries including Korea,U.S.A., and Japan [8]. MFSK is also useful for low-power, short-range applications; may it be asensor network or an UWB piconet [9]. Energy-efficient operation is vital in sensor networks tomaintain a reasonable lifetime. On the other hand, power efficiency of high-dimensional orthogonalmodulation makes it very attractive for UWB system design. In UWB, low-power pulses of veryshort duration are used for communication and these wideband pulses must satisfy strict peakpower requirements to avoid interference with the existing systems. Steering our focus from thewireless domain, it is interesting to see that MFSK has found a wide application in other commu-nication domains too. MFSK has been used for underwater acoustic communications successfullyover the past few decades [10] due to simple implementation and reliability in the multipathenvironment and is considered as a viable option for optical communication systems [11, 12].The capability to make the transmissions robust against permanent frequency disturbances andimpulse noise makes MFSK a strong candidate for power line communications [13]. It is alsoquite natural to associate MFSK with frequency division multiplexing (FDM), frequency diversityand frequency-hop spread spectrum (FHSS) systems [14, 15]. With the advent of multiple tone(MT) modulation scheme MT-MFSK [16] for bandwidth constraint systems and multicarrier(MC) technique MC-MFSK [17] for future high-speed wireless systems, MFSK is gaining moreattention of modern researchers and engineers.

Despite the growing interest of MFSK in current wireless systems, relatively few researchersworked on the error performance analysis of coherent MFSK in fading channels. Although thebasic MFSK system was discussed as early as in the fifties [18] followed by a quantitative analysisin Rayleigh fading channel employing diversity by Hahn [19], it took another 20 years for asystematic study of coherent MFSK over fading channels [20–22]. In particular, Abdel-Ghaffar and

Copyright � 2010 John Wiley & Sons, Ltd. Int. J. Commun. Syst. 2011; 24:202–224DOI: 10.1002/dac

204 A. CHANDRA, S. R. PORAM AND C. BOSE

Pasupathy [22] presented, from an asymptotical approach (SNR →∞), the performances of M-aryorthogonal signaling in the Rayleigh and Rician fading channel, whereas the asymptotic (M →∞)performances in the Rayleigh and Nakagami-m fading were examined by Hedge and Stark [20]and Crepeau [21], respectively. A few years later, Kim and Han [23] presented upper bounds for biterror probability (BEP) of coherent MFSK in the Rician and Nakagami-m channels. To the best ofthe author’s knowledge, error performance evaluation of coherent MFSK with MRC diversity wasattempted first by Al-Hussaini et al. [24] and later by Aalo [25]. However both their analysis wereconfined to binary alphabet and Nakagami-m fading channel only. Furthermore, using a parallelapproach based on transformation of co-ordinate, Dong [26] and Dong and Beaulieu [27] have beenable to obtain exact closed-form SEP expressions of coherent MFSK for low values of M(M�4)in the Rayleigh fading. Although the approach was new, a general formulation for arbitrary Mvalue was not possible and further extensions (M>4) seemed quite tedious. Xiao and Dong alsopresented the SEP analysis of MRC diversity-assisted systems operating over the Nakagami-m[28] and Rician [29, 30] fading channels, but the end expression involved integrals with infiniteintegration limit. To make the matter worse when the fading PDF involved complex functions (asin case of Rician) other than simple power or exponentials, their method resulted in infinite seriesas argument of the integrals. One may find a detailed exposition of MGF-based SEP analysis inSimon and Alouini’s book [31] for many modulation schemes operating over generalized fadingchannels with or without diversity. Basically in MGF method, the conditional SEP is transformedinto an exponential-type integral with finite integration limits, so that the averaging over the fadingdistribution can then be represented in terms of an integral of the MGF of the output SNR. ForMFSK, they derived asymptotically tight upper bounds on the average BEP and SEP of 4-ary FSK.In another recent paper, Paris et al. [32] evaluated BEP of coherent MFSK in the Rician channelwith MRC. Their end expressions involve infinite series but contain no integration. Some otherrelated efforts worth mentioning include performance analysis in the Poisson channel [33] anddeduction of probability density functions (PDF) of an M-ary FSK receiver output signal in thepresence of inter-symbol interference (ISI) over the Nakagami-m channel [34] and impulse noiseover the Rayleigh channel [35].

On the other hand, a myriad of research papers are available on the performance of thenon-coherent version [36–42] of MFSK. There are mainly two reasons for this, analyticalintractability of probability of error expressions for coherent MFSK and the wide use ofnon-coherent MFSK due to the absence of costly and complex phase-synchronized receivers.In this context, it is worth to assess the efficacy of coherent MFSK design options so as todetermine the most appropriate choice of modulation method regarding performance, complexity,and implementation constraints.

The major contributions of the present paper are mainly threefold: first, using an MGF-basedapproach exact SEP values (up to M =5) of coherent MFSK are obtained. When the constellationsize becomes bigger (M�6), the same SEP expressions provide a quite realistic approximation,much tighter than the bounds found in the previous literatures. The study was inspired bythe single integral representations of third and fourth powers of the Gaussian Q functionsby Simon [43]. SEP expressions derived in our paper consist of well-behaved integrals, have highnumerical stability, and are applicable to arbitrary signal dimension. A direct calculation of SEPwould involve computation of double integral (the conditional probability itself contains an integraland the second integration is required to average it over the fading PDF) with infinite integrationlimits and an integrand of Q function raised to some integer power, which is intensive andtime consuming, not to mention the difficulty in achieving high accuracy, especially for higherSNR values. For Rician or Hoyt fading case, the problem becomes more severe due to the presenceof Bessel functions. Another novelty of the paper lies in the fact that we have considered all themajor statistical small-scale fading models namely, Rayleigh, Rician, Hoyt, and Nakagami-m forchannel modeling. Neither the SEP evaluation for coherent orthogonal signaling is ever attemptedin the Hoyt channel nor there exists a comprehensive analysis covering all fading models. Apartfrom the non-diversity case, SEP performances of coherent MFSK with MRC diversity for allthese fading models are also derived. Lastly, although not explicitly discussed in this paper, using



the well-known relation between BEP and SEP for coherent detection of FSK, the BEP values forall the above-mentioned scenarios may be found very easily. By considering fewer terms in theSEP expressions, one may have a trade-off between accuracy and complexity too.

The remainder of the paper has the following organization. In Section 2, the channel and receivermodel is described. Next, the performance of coherent MFSK schemes over AWGN, slow flatfading channels, and with MRC diversity receivers is discussed in Section 3. Selected numericaland simulation results are provided in Section 4, including a comparison of the proposed approachwith those results derived from earlier available bounds. Finally, the main points of this paper aresummarized in Section 5.

2. SINGLE AND MULTICHANNEL WIRELESS LINK MODEL

In any M-ary modulation the input binary stream is divided into n-tuples of n = log2 M bits, whereM is the constellation size. If the modulator is following an MFSK scheme, then it converts everysuch n bit message to one of the M possible signals differing in frequency,

si (t)=�{si (t −mTs) exp( j2� fct)}, mTs�t�(m+1)Ts, i ∈{1,2, . . . , M} (1)

where

si (t)=√

2Es

Tsexp

(j�

iht

Ts+�

), 0�t�Ts (2)

is the low-pass complex envelope of the i th signal, fc is the carrier frequency, and Es(=nEb), Eband Ts(=nTb), Tb pairs denote the energy and duration of a symbol and a bit, respectively. Theinitial phase � can be arbitrary but in order to maintain orthogonality h should be an integer, i.e.the minimum separation between two adjacent frequencies should be 1/(2Ts) [44, 45].

In our problem, the transmitted signal passes through a non-ideal channel (corrupted by fadingand AWGN) and reaches the receiver; let us denote the received signal as

r (t)=�{r (t −mTs) exp( j2� fct)}, mTs�t�(m+1)Ts (3)

Now the task of the receiver is to decide, with the help of a demodulator, which signal wasoriginally sent by the transmitter. A coherent demodulator is made of a bank of M correlatorsand the j th correlator produces an output r j proportional to the likeliness of r (t) with s j (t). Atsampling times t =mTs, it compares the M decision variables r j , j ∈{1,2, . . ., M}, and decideson the hypothesis corresponding to the largest of them. To simplify modelling, the modulator andthe demodulator may be combined with the real channel to yield an equivalent complex basebandchannel model as described below.

2.1. Single channel wireless link model

A slow and flat fading channel is assumed here, where the duration of a symbol waveformis sufficiently short so that the fading variations cause negligible loss of coherence within thetimeframe of each received symbol. At the same time, the individual waveform is assumed to besufficiently narrow-band so that frequency selectivity offered by the fading process to differentsignal spectral components is negligible. The effect of such kind of fading on the signal amplitudeis multiplicative distortion characterized by attenuation factor �. Besides fading, the transmittedsignal is also perturbed by real bandpass noise having a two-sided power spectral density (PSD)N0/2, which is typically assumed to be independent of the fading process. The equivalent basebandAWGN is therefore complex circular with PSD N0 per quadrature. As shown in Figure 1, theoverall fading channel model at baseband [46] can be expressed as

r (t)=� exp( j�)si (t)+ n(t) (4)



where � and � denote random amplitude and phase variation and n(t) is a zero-mean circularlysymmetric complex Gaussian process.

As �2 denotes instantaneous attenuation in received signal power, the instantaneous SNR persymbol can be defined as �=�2(Es/N0) with an average value �= E{�2}(Es/N0). With a simplechange of variable the PDF of �, i.e. f�(�), can be obtained from the PDF of �, f�(�). Table I enliststhese PDFs for all the small-scale fading models under consideration [31]. The table also depictsthe MGFs G�(s)=∫∞

0 f�(�) exp(−s�)d�, another important statistical characteristic of fadingchannels.

2.2. Multichannel wireless link model

The communication link model with L branch predetection diversity combiner operating over IIDfading channel is described in Figure 2. There are L available antennas, receiving signals withstatistically independent random amplitudes and random phases.

Figure 1. Baseband equivalent fading channel model.

Table I. PDF and MGF of instantaneous SNR per symbol for some common fading formats.

Channel model PDF and MGF expressions Fading parameter

Rayleigh f�(�)= 2�� exp

(− �2

�

); E{�2}=�, ��0

f�(�)= 1� exp

(− �

�

); E{�}= �, ��0

G�(s)= 11+s �

Rician (Nakagmi-n) f�(�)= ��2 exp

(− �2+s2

2�2

)I0

(�s�2

); E{�2}=s2 +2�2, ��0 0�K <∞

f�(�)= 1+K� exp

[− (1+K )�+K �

�

]I0

[2√

K (1+K )��

];

E{�}= �, ��0 K =s2/2�2

G�(s)= 1+K1+K+s � exp

(− K s �

1+K+s �

)

Hoyt (Nakagami-q) f� (�)=(1+q2)�

q� exp

[−(

1+q2

2q

)2�2

�

]I0

[ (1−q4)�2

4q2�

];

E{�2}=�, ��0 0�q�1

f�(�)= 1+q2

2q � exp

[−(

1+q2

2q

)2��

]I0

[(1−q4)�

4q2 �

]; E{�}= �, ��0

G�(s)=[

1+2s�+(

2qs �1+q2

)2]−1/2

Nakagami-m f�(�)= 2�(m)

(m�

)m �2m−1 exp(−m�2

�

); E{�2}=�, ��0 1

2�m <∞f�(�)= 1

�(m)

(m�

)m�m−1 exp

(−m�

�

); E {�}= �, ��0 m = �2

E{(�2−�)2}G� (s)=

(m

m+s �

)m

In Table I, In(·) denotes nth order modified Bessel function of first kind and �(·) means gamma function.



Under slow flat fading assumption, the received equivalent baseband signal at the kth diversitybranch can be written as

rk(t)=�k exp( j�k)si (t)+ nk(t), k =1,2, . . ., L (5)

where �k and �k being the attenuation factor and the uniformly distributed phase shift of the kthchannel. To each signal, there is a zero mean AWGN component nk(t) that is assumed to beindependent of the signal and uncorrelated with the noise in any other branch.

In case of MRC diversity, the output signal from different receiver antennas are multiplied bythe complex conjugate of their respective fading envelopes, and then summed, i.e. the signal atcombiner output is generated according to the following rule:

r (t)=L∑

k=1wkrk(t) (6)

where wk =�k exp(− j�k) and rk(t) is as defined in (5). Quite naturally at a given instant of time,the instantaneous SNR available at MRC combiner output is simply sum of all the branch SNRs,�MRC =∑L

k=1 �k . To find a suitable statistical description of �MRC, which starts with finding thecorresponding PDF, we need to derive the PDF of sum of L random variables. The PDF expressionsthat appeared in the earlier literatures [31, 47] have been summarized in Table II. As per theIID assumption, each branch has equal average branch SNR �= E{�k}; ∀k and that all diversitychannels have an identical fading parameter.

The first thing one would notice about Table II is that it is not complete; there is no entry forthe Hoyt fading. Also the PDF forms (see Rician) become cumbersome for complex fadingPDF envelopes. Compared with this, the evaluation of the corresponding MGFs G�,MRC(s)

Figure 2. Channel and receiver model employing L order predetection MRC diversity.

Table II. PDF of instantaneous SNR per symbol at MRC combiner outputfor some common fading formats.

Channel model PDF expressions

Rayleigh f�,MRC(�)= 1(L−1)!�L �L−1 exp

(− �

�

); ��0

Rician (Nakagmi-n) f�,MRC(�)= 1+K�

[(1+K )�

LK�

](L−1)/2exp

(− (1+K )�+LK�

�

)IL−1

(2√

LK(1+K )��

); ��0

Nakagami-m f�,MRC(�)=(

m�

)Lm �Lm−1

�(Lm) exp(−m�

�

); ��0



=∫∞0 f�,MRC(�) exp(−s�)d� is rather easy. The exponential term in the integrand may be

decomposed to a product of L different exponentials as �=�1 +�2 +·· ·+�L . Further, if thebranches are assumed to be independent then the PDF f�,MRC(�) also can be expressed as aproduct of individual PDF f�k

; k ∈{1,2, . . . , L} of L random variables. Thus the single integralbecomes an L-fold integral, G�,MRC(s)=∫∞

0 f�1(�1) exp(−s�1)d�1 . . .

∫∞0 f�L

(�L ) exp(−s�L )d�L ,

where each integral gives the MGF of the corresponding branch, G�,MRC(s)=∏Lk=1 G�k

(s). Nowif we assume that all branches, apart from being independent are identical too (IID assumption),then the MGF with MRC diversity simply becomes the MGF of no-diversity case raised to apower equal to the number of diversity branches (L)

G�,MRC(s)= [G�(s)]L (7)

3. SEP CALCULATION FOR COHERENT MFSK

3.1. AWGN channel

The SEP expression for equal-energy, equiprobable, orthogonal signal set with coherent detectionin AWGN channel is given by [44, 45]

Pe =1− 1√2�

∫ ∞

−∞exp

(− (x −√

2�)2

2

)[1− Q(x)]M−1 dx (8)

where �= Es/N0 is the SNR per symbol and Q(z)=1/√

2�∫∞

z exp(−u2/2)du is the Gaussianprobability integral. An infinite series containing infinite range integrals may be obtained byexpanding the term [1− Q(x)]M−1 in (8) using binomial theorem, out of which first six termsare [48]

Pe ≈ 1− 1√2�

∫ ∞

−∞exp

[− (x −√

2�)2

2

]dx + M −1√

2�

∫ ∞

−∞exp

[− (x −√

2�)2

2

]Q(x)dx

− (M −1)(M −2)

2√

2�

∫ ∞

−∞exp

[− (x −√

2�)2

2

]Q2(x)dx

+ (M −1)(M −2)(M −3)

6√

2�

∫ ∞

−∞exp

[− (x −√

2�)2

2

]Q3(x)dx

− (M −1)(M −2)(M −3)(M −4)

24√

2�

∫ ∞

−∞exp

[− (x −√

2�)2

2

]Q4(x)dx · · · (9)

This truncation was inspired by the fact that we can have alternate representations up to thefourth power of Q function [43]. The second term in (9) merely consists of Gaussian PDF resultingunity and thus cancels out the first term. Also using the identity (1/

√2�)

∫∞−∞ Q(x) exp(−(x −�)2/

(x −�)2/2)dx = Q(�/�√

2) [49] and (A1), the third term may be expressed with an integralcontaining exponential function only. For the rest of the terms, alternate expressions of Qn(x)in terms of finite range integral may be obtained from (A2)–(A4), which renders them to bein the form of double integrals. With a change of order of integration and using the followingresult:

1√2�

∫ ∞

−∞exp

[− (x −√

2�)2

2

]exp

(− x2

2sin2 �

)dx = sin�√

1+sin2 �exp

(− �

1+sin2 �

)(10)



(9) can be written as

Pe = M −1

�

∫ �/2

0exp

(− �

2sin2 �

)d�− (M −1)(M −2)

2�

∫ �/4

0f1(�,�)d�

+ (M −1)(M −2)(M −3)

6�2

∫ �/6

0f1(�,�) f2(�)d�

+ (M −1)(M −2)(M −3)

12�2

∫ sin−1(1/√

3)

0f1(�,�){�− f2(�)}d�

− (M −1)(M −2)(M −3)(M −4)

24�2

∫ �/6

0f1(�,�) f2(�)d� (11)

where f1(�,�)= (sin�/√

1+sin2 �) exp[−�/(1+sin2 �)] and f2(�)=cos−1[{3cos(2�)−1}/{2cos3

(2�)}−1]. A proof of (10) is provided in Appendix B.The expression in (11) gives exact result up to M =5 and for higher values of M it offers a good

approximation. For fixed M , these bounds become increasingly tight as SNR per symbol Es/N0is increased. For completeness, we would like to mention that, neglecting all the terms in (11)except the first, we arrive at Pe ≈ (M −1)Q(

√Es/N0), the standard available SEP approximation

for coherent MFSK [44]. In Figure 3, a comparison of the proposed approximation with the existingupper bound [44] is made. The exact values are also calculated through direct numerical integrationwhile simulation results confirm those values. The figure clearly shows that the gap between exactand approximated values diminishes as more number of terms in (11) are considered. When allthe terms are taken into account, the values obtained from (10) are far superior than to the looseupper bound [44].

Further, utilizing the following relation between BEP and SEP of coherent MFSK [44, 45]:

Pe,bit = 2n−1

2n −1Pe, n = log2 M (12)

an approximated BEP can also be calculated by substituting (11) in (12). Although explicit BEPresults are not provided in this paper, they could be easily calculated from the correspondingSEP expressions while applying (12). It is interesting to note that in AWGN channel, for a

Figure 3. Comparison of the proposed approximation with exact values and earlier bounds for M =10.



given Es/N0, SEP of MFSK increases with M but for a given Eb/N0 the BEP decreases as Mincreases. However, it is not necessary that the fading channel would also follow the same trend.In fact, there might exist some optimum value of M for which Pe,bit is minimum as pointed outby Dong [26].

3.2. Fading channel (no-diversity)

For a digital receiver that makes its decision based on an undistorted symbol waveform corruptedby stationary AWGN only, it is known that the SEP depends only on the instantaneous SNREs/N0 associated with each symbol. The SEP in a fading channel, on the other hand, becomesa conditional error probability (CEP) Pe(�) conditioned by � and the corresponding average SEPcan be found by averaging the CEP over � as

Pe =∫ ∞

0Pe(�) f�(�)d� (13)

where f�(�) is the PDF of � for a specified fading environment as described in Table I. This methodis popularly referred to as the standard PDF approach in the literature. If the CEP can be expressedin an exponential form Pe(�)=C1 exp(−C2�), the average SEP Pe may be found easily throughthe MGF method as Pe =C1G�(C2), avoiding the need of direct integration provided that the MGFG�(s) is available in closed-form.

For coherent MFSK, our proposed approximation in (11) contains a sum of integrals havingonly exponential functions (with respect to fading SNR �) as integrands. From (11) and (13) wemay write the expression for average error probability as

Pe = M −1

�

∫ ∞

0

∫ �/2

0exp

(− �

2sin2 �

)f�(�)d�d�− (M −1)(M −2)

2�

∫ ∞

0

∫ �/4

0f1(�,�) f�(�)d�d�

+ (M −1)(M −2)(M −3)

6�2

∫ ∞

0

∫ �/6

0f1(�,�) f2(�) f�(�)d�d�

+ (M −1)(M −2)(M −3)

12�2

∫ ∞

0

∫ sin−1(1/√

3)

0f1(�,�){�− f2(�)} f�(�)d�d�

− (M −1)(M −2)(M −3)(M −4)

24�2

∫ ∞

0

∫ �/6

0f1(�,�) f2(�) f�(�)d�d� (14)

The double integrals that is to be calculated when the PDF approach is adopted can be reduced tosingle integrals with the equivalent MGF approach as

Pe = M −1

�

∫ �/2

0G�

(1

2sin2 �

)d�− (M −1)(M −2)

2�

∫ �/4

0

sin�√1+sin2 �

G�

(1

1+sin2 �

)d�

+ (M −1)(M −2)(M −3)

6�2

∫ �/6

0

sin�√1+sin2 �

G�

(1

1+sin2 �

)f2(�)d�

+ (M −1)(M −2)(M −3)

12�2

∫ sin−1(1/√

3)

0

sin�√1+sin2 �

G�

(1

1+sin2 �

){�− f2(�)}d�

− (M −1)(M −2)(M −3)(M −4)

24�2

∫ �/6

0

sin�√1+sin2 �

G�

(1

1+sin2 �

)f2(�)d� (15)

For deriving (15) a change of integral has been performed, i.e. the integration over � has beenexecuted first.



3.2.1. Rayleigh fading channel. For Rayleigh fading the MGF is G�(s)=1/(1+s�) (see Table I).Putting this value in (15), we can obtain the average error rate as

Pe = M −1

�

∫ �/2

0

2sin2 �

�+2sin2 �d�− (M −1)(M −2)

2�

∫ �/4

0f3(�)d�

+ (M −1)(M −2)(M −3)

6�2

∫ �/6

0f3(�) f2(�)d�

+ (M −1)(M −2)(M −3)

12�2

∫ sin−1(1/√

3)

0f3(�){�− f2(�)}d�

− (M −1)(M −2)(M −3)(M −4)

24�2

∫ �/6

0f3(�) f2(�)d� (16)

where f3(�)= (sin�√

1+sin2 �)/[�+(1+sin2 �)]. The same result also can be obtained directly(PDF method) by performing term by term integration of the Rayleigh PDF. The averaging operationinvolves integration of simple exponential terms in the form of

∫∞0 exp(−ax)dx =1/a.

3.2.2. Rician (Nakagami-n) fading channel. In case of a Rician channel, a similar substitution ofG�(s) from Table I results in

Pe = (M −1)

�

∫ �/2

0f4(�) f5(�)d�− (M −1)(M −2)

2�

∫ �/4

0f6(�) f7(�)d�

+ (M −1)(M −2)(M −3)

6�2

∫ �/6

0f6(�) f2(�) f7(�)d�

+ (M −1)(M −2)(M −3)

12�2

∫ sin−1(1/√

3)

0f6(�){�− f2(�)} f7(�)d�

− (M −1)(M −2)(M −3)(M −4)

24�2

∫ �/6

0f6(�) f2(�) f7(�)d� (17)

where f4(�)=[2(1+K ) sin2 �]/[�+2(1+K ) sin2 �], f5(�)=exp[−K �/{�+2(1+K ) sin2 �}], f6(�)

=(1+K ) sin�√

1+sin2 �/[�+(1+K )(1+sin2 �)] and f7(�)=exp[−K �/{�+(1+K )(1+sin2 �)}].For K = 0, as f6(�) becomes equal to f3(�) and f5(�)= f7(�)=1, (17) reduces to the SEPexpression for the Rayleigh channel in (16) as expected. Direct evaluation from the PDF is alsopossible by using the following result:∫ ∞

0exp(−�x)I0(2

√x)dx = 1

�exp

(

�

)(18)

and performing integration of all five terms. A proof of (18) is provided in Appendix B.

3.2.3. Hoyt (Nakagami-q) fading channel. For Hoyt fading, the SEP takes the form

Pe = (M −1)

�

∫ �/2

0f8(�)d�− (M −1)(M −2)

2�

∫ �/4

0f9(�) f10(�)d�

+ (M −1)(M −2)(M −3)

6�2

∫ �/6

0f9(�) f2(�) f10(�)d�



+ (M −1)(M −2)(M −3)

12�2

∫ sin−1(1/√

3)

0f9(�){�− f2(�)} f10(�)d�

− (M −1)(M −2)(M −3)(M −4)

24�2

∫ �/6

0f9(�) f2(�) f10(�)d� (19)

where f8(�)= [1+ �/sin2 �+q2�2/{(1+q2)sin2 �}2]−1/2, f9(�)=sin�/√

1+sin2 �, and f10(�)=[1+2�/(1+sin2 �)+4q2�2/{(1+q2)(1+sin2 �)}2]−1/2.

As a check, one may verify that for q =1, (19) reduces to (16). SEP calculation in the Hoytfading through PDF method is similar to the Rician case in that it also involves integrationscontaining exponential and the Bessel functions but of the form

∫∞0 exp(−ax)I0(bx)dx . Solution

to such kind of integral may be found using a modified form of (6.611.4) [50] namely∫ ∞

0exp(−ax)I0(bx)dx = (a2 −b2)−1/2, �(a)>|�(b)| (20)

3.2.4. Nakagami-m fading channel. Lastly, the average SEP of coherent MFSK over the Nakagami-m fading channel may be found by substituting G�(s) from Table I in (15) to yield

Pe = (M −1)

�

∫ �/2

0f11(�)d�− (M −1)(M −2)

2�

∫ �/4

0f9(�) f12(�)d�

+ (M −1)(M −2)(M −3)

6�2

∫ �/6

0f9(�) f2(�) f12(�)d�

+ (M −1)(M −2)(M −3)

12�2

∫ sin−1(1/√

3)

0f9(�){�− f2(�)} f12(�)d�

− (M −1)(M −2)(M −3)(M −4)

24�2

∫ �/6

0f9(�) f2(�) f12(�)d� (21)

where f11(�)= [2m sin2 �/(�+2m sin2 �)]m and f12(�)= [m(1+sin2 �)/{�+m(1+sin2 �)}]m . LikeRician and Hoyt fading, the SEP expression in (21) also reduces to (16) for a particular value(m =1) of the fading parameter. Again, from the generalized error expression in (13) and theNakagami-m PDF in Table I, we find that the averaging requires solution of integrals containingpowers and exponential of the dependent variable. Now using the identity,

∫∞0 xb−1 exp(−ax)dx =

a−b�(b) and performing the same integration for all five terms in (11) we finally get (21) onceagain.

3.3. Fading channel with MRC diversity

It has been already pointed out in Section 2.2 that although the SEP expressions can be obtainedthrough the PDF approach without much difficulty for the no-diversity (fading only) case, thesituation becomes a bit difficult for receivers with MRC diversity, primarily due to the complexforms of PDF of the combiner output SNR. This fact also explains our SEP analysis in Section 3.2with MGF method; the goal was to maintain uniformity. Now for SEP evaluation with MRC westart with (7) and by modifying (15) we can write the following general equation:

Pe = M −1

�

∫ �/2

0

[G�

(1

2sin2 �

)]L

d�− (M−1)(M−2)

2�

∫ �/4

0

sin�√1+sin2 �

[G�

(1

1+sin2 �

)]L

d�

+ (M −1)(M −2)(M −3)

6�2

∫ �/6

0

sin�√1+sin2 �

[G�

(1

1+sin2 �

)]L

f2(�)d�



+ (M −1)(M −2)(M −3)

12�2

∫ sin−1(1/√

3)

0

sin�√1+sin2 �

[G�

(1

1+sin2 �

)]L

{�− f2(�)}d�

− (M−1)(M−2)(M−3)(M−4)

24�2

∫ �/6

0

sin�√1+sin2 �

[G�

(1

1+sin2 �

)]L

f2(�)d� (22)

to calculate SEP performance over different fading channels for a coherent MFSK receiveremploying MRC diversity.

3.3.1. Rayleigh fading channel with MRC diversity. Inserting the MGF from Table I in (22), weobtain the error performance of a MRC diversity receiver with coherent MFSK in the Rayleighchannel as

Pe = M −1

�

∫ �/2

0g1(�)d�− (M −1)(M −2)

2�

∫ �/4

0g2(�)d�

+ (M −1)(M −2)(M −3)

6�2

∫ �/6

0g2(�) f2(�)d�

+ (M −1)(M −2)(M −3)

12�2

∫ sin−1(1/√

3)

0g2(�){�− f2(�)}d�

− (M −1)(M −2)(M −3)(M −4)

24�2

∫ �/6

0g2(�) f2(�)d� (23)

where g1(�) = [2sin2 �/(� + 2sin2 �)]L and g2(�) = (sin�/√

1 + sin2 �)[(1 + sin2 �)/{�+(1+sin2 �)}]L . As expected, by putting L =1 in (26) we get back (16). Equations (22)–(23) demon-strate that MGF approach is quite useful for extending the single-channel SEP expressions tomulti-channel reception without much effort. It is interesting to note that the method may beadvantageous to other diversity combining techniques too. For example with L branch selectioncombining (SC) over IID Rayleigh fading channels, the PDF of instantaneous SNR at combineroutput can be expressed as [51]

f�,SC(�)= L

�

[1−exp

(−�

�

)]L−1

exp

(−�

�

), ��0 (24)

Utilizing the binomial series expansion of (1−x)L =∑Lj=0

LC j (−1) j x j , the PDF can be writtenin a more suitable form

f�,SC(�)=L∑

k=1

LCk(−1)k−1 k

�exp

(−k�

�

), ��0 (25)

with the corresponding MGF being

G�,SC(s)=L∑

k=1

LCk(−1)k−1 k

k+s�(26)

Now from (15) and (26), it is easy to find the average error rate as

Pe =L∑

k=1

LCk(−1)k−1

{M −1

�

∫ �/2

0

2k sin2 �

�+2k sin2 �d� − (M −1)(M −2)

2�

∫ �/4

0

k sin�√

1+sin2 �

�+k(1+sin2 �)d�

+ (M −1)(M −2)(M −3)

6�2

∫ �/6

0f2(�)

k sin�√

1+sin2 �

�+k(1+sin2 �)d�



+ (M −1)(M −2)(M −3)

12�2

∫ sin−1(1/√

3)

0{�− f2(�)}k sin�

√1+sin2 �

�+k(1+sin2 �)d�

− (M −1)(M −2)(M −3)(M −4)

24�2

∫ �/.6

0f2(�)

k sin�√

1+sin2 �

�+k(1+sin2 �)d�

}(27)

3.3.2. Rician (Nakagami-n) fading channel with MRC diversity. Substituting the MGF for theRician channel in (22), we get the SEP of coherent MFSK for the Rician channel with MRCdiversity as

Pe = M −1

�

∫ �/2

0g3(�)d�− (M −1)(M −2)

2�

∫ �/4

0g4(�)d�

+ (M −1)(M −2)(M −3)

6�2

∫ �/6

0g4(�) f2(�)d�

+ (M −1)(M −2)(M −3)

12�2

∫ sin−1(1/√

3)

0g4(�){�− f2(�)}d�

− (M −1)(M −2)(M −3)(M −4)

24�2

∫ �/6

0g4(�) f2(�)d� (28)

where g3(�)= [2(1 + K ) sin2 �/{� + 2(1 + K ) sin2 �}]L exp[− L K �/{� + 2(1 + K ) sin2 �}] and

g4(�) = (sin�/√

1+sin2 �) [(1+K ) (1+sin2 �)/{� + (1+K )(1+sin2 �)}]exp[−L K �/{�+ (1+K )(1+sin2 �)}]L .

3.3.3. Hoyt (Nakagami-q) fading channel with MRC diversity. Like the Rayleigh and Rician case,the SEP for the Hoyt channel also can be found easily

Pe = M −1

�

∫ �/2

0g5(�)d�− (M −1)(M −2)

2�

∫ �/4

0g6(�)d�

+ (M −1)(M −2)(M −3)

6�2

∫ �/6

0g6(�) f2(�)d�

+ (M −1)(M −2)(M −3)

12�2

∫ sin−1(1/√

3)

0g6(�){�− f2(�)}d�

− (M −1)(M −2)(M −3)(M −4)

24�2

∫ �/6

0g6(�) f2(�)d� (29)

where g5(�)= [1+ �/sin2 �+q2�2/{(1+q2)sin2 �}2]−L/2 and g6(�)= (sin�/√

1+sin2 �)[1+2�/(1+sin2 �)+4q2�2/{(1+q2)(1+sin2 �)}2]−L/2.

3.3.4. Nakagami-m fading channel with MRC diversity. Finally for the Nakagami-m channel, theSEP is

Pe = M −1

�

∫ �/2

0g7(�)d�− (M −1)(M −2)

2�

∫ �/4

0g8(�)d�

+ (M −1)(M −2)(M −3)

6�2

∫ �/6

0g8(�) f2(�)d�



+ (M −1)(M −2)(M −3)

12�2

∫ sin−1(1/√

3)

0g8(�){�− f2(�)}d�

− (M −1)(M −2)(M −3)(M −4)

24�2

∫ �/6

0g8(�) f2(�)d� (30)

where g7(�)= [2m sin2 �/(�+2m sin2 �)]Lm and g8(�)= (sin�/√

1+sin2 �)[m(1+sin2 �)/{�+m(1+sin2 �)}]Lm . As a double check, it can be easily verified that putting L =1 in (28)–(30), weget back (17), (19), and (21) respectively.

4. RESULTS AND DISCUSSIONS

In the result section, at first we would consider the coherent MFSK SEP over different fadingchannels. In Figure 4–7, SEP of coherent MFSK is depicted for various values of modulation orderM (=2, 4, and 10). As our proposed method gives exact results up to M =5, the plots for M =2and M =4 are free from any approximation. The plot for M =10 in each figure is supposed toshow the inaccuracy due to the approximation process but it is hard to find any such effect, at leastwith bare eyes. The general characteristics denote that at a particular SNR value, error probabilityrises with constellation size M when the fading parameter (K , q , or m) is kept constant. On theother hand, keeping the SNR fixed, if the fading severity increases (realized by a reduced valueof K , q , or m) the SEP value tends to increase. Also, all the SEP curves follow almost lineardecrease especially at high SNR values.

Specifically, Figure 4 gives a graphical version of (16) derived for the Rayleigh channel,whereas Figure 5 demonstrates the plot of (17), i.e. SEP of coherent MFSK over the Rician(Nakagami-n) fading channel for various values of the specular to scatter signal strength ratio K(=3 and 9 dB). Analysis of coherent MFSK schemes over the Hoyt (Nakagami-q) fading channelis shown in Figure 6 using (19) for two different fading parameters (q =0.4 and 0.8). Figure 7 plotsthe analytical results from (21) giving symbol error rates for coherent MFSK over the Nakagami-mfading channel. Two different fading parameters (m =0.5 and 2.0) are considered.

For simulation, the Monte Carlo technique, a stochastic simulation process, is used to estimatethe SEP by counting the erroneous symbols at the receiver and dividing the count by the total

Figure 4. Analytical and simulation SEP of coherent MFSK as a function of the average SNR per symbolin the Rayleigh channel with constellation size M =2, 4, and 10.



Figure 5. Analytical and simulation SEP of coherent MFSK as a function of the average SNR per symbolin the Rician channel with fading parameter K =3 and 9 dB and constellation size M =2, 4, and 10.

Figure 6. Analytical and simulation SEP of coherent MFSK as a function of the average SNR per symbolin the Hoyt channel with fading parameter q =0.4 and 0.8 and constellation size M =2, 4, and 10.

number of symbols passed through the system. An average of 20 different iterations, each with 107

symbols, was taken for estimating SEP at every SNR value. The simulated points are superimposed(shown by asterisk mark) on the analytical values obtained in the paper. Since the values obtainedfrom analytical expressions almost coincide with their simulation counterparts, accuracy of ourderivations is validated. However, it was also observed that for relatively higher error probability(in the low SNR region, or when M becomes large, or for low K , q , or, m values) the approximatedvalues deviate slightly from the values predicted by simulation.

Instead of (11) if we assume the crude upper bound [44] and calculate the SEP for fadingchannels based on it, we would end up with the first terms present in (16), (17), (19), or (21).



Figure 8 compares these bounds with those presented in our paper. A quick examination revealsthat these bounds are not only far away from the actual values (simulated points) but also provideus with unrealistic values under some circumstances. For example, in the low SNR region the SEPpredicted by these crude bounds may be more than 1, i.e. more number of erroneous symbols thanactually transmitted.

Next we will be concentrating on the coherent MFSK error rates over different fading channelswith MRC diversity. The pattern of SEP variation with modulation order M or fading parameter(K , q , or m) is well studied in non-diversity cases and hence is not repeated for the MRC

Figure 7. Analytical and simulation SEP of coherent MFSK as a function of the average SNR per symbolin Nakagami-m channel with fading parameter m =0.5 and 2 and constellation size M =2, 4, and 10.

Figure 8. Comparison of the proposed approximation (numerical) and conventional approximation (upperbound) with the simulated SEP of coherent MFSK (M =8) for different fading channels.



Figure 9. Analytical and simulation SEP of coherent MFSK (M =8) as a function ofthe average SNR per symbol in the Rayleigh channel with MRC and SC diversity for

different diversity orders L =2 and 4.

Figure 10. Analytical and simulation SEP of coherent MFSK (M =8) as a function ofthe average SNR per symbol in the Rician channel (K =6dB) with MRC diversity for

different diversity orders L =2, 3, and 4.

case. Keeping the constellation size (M =8) and fading parameters (K =6dB, q =0.4, m =2),fixed SEP values are depicted for various diversity orders (L =2, 3, and 4) in Figures 9–12. Forthe Rayleigh fading case, the SEP values with SC is also shown along with the MRC values.Figure 13 compares the upper bound derived from the conventional approximation with ours andit shows that the nature of the curves is entirely different for two approximations in low SNRvalues.



Figure 11. Analytical and simulation SEP of coherent MFSK (M =8) as a functionof the average SNR per symbol in the Hoyt channel (q =0.4) with MRC diversity for


Figure 12. Analytical and simulation SEP of coherent MFSK (M =8) as a function ofthe average SNR per symbol in the Nakagami-m channel (m =2) with MRC diversity for


5. CONCLUSIONS

The error performance of coherent MFSK schemes has been studied using single integral represen-tation of powers of Q(x) for wireless environments subjected to multipath fading. Different types offading distributions, namely Rayleigh, Rician, Hoyt, and Nakagami-m, have been considered. Forall the models, error rates for both single and multi-channel (MRC) reception have been presented.In addition, SC error rates for the Rayleigh channels have been found. An MGF-based approach is



Figure 13. Comparison of the proposed approximation (numerical) and conventional approxi-mation (upper bound) with the simulated SEP of coherent MFSK (M =8) for different fading

channels with MRC (L =2) diversity.

used for the deduction of error probabilities, which are sufficiently simple to calculate numericallyand provide either exact values (up to M =5) or tight realistic bounds (M�6). The analytical SEPmatches exactly with the simulated SEP for all the fading channels (with or without diversity) understudy. By dropping a few terms in the end expressions, a trade-off between accuracy and complexitymay be achieved. Further, a trivial extension of the paper would yield BEP values for all the above-mentioned scenarios by exploiting the well-known relation between BEP and SEP. Results presentedin this paper would definitely help wireless engineers to assess the viability of relatively less-studiedcoherent MFSK.

APPENDIX A: ALTERNATE REPRESENTATION OF INTEGRAL POWERSOF Q FUNCTION

In a recent paper, Simon [43] the powers of Q function up to 4 with the help of finite rangeintegrals. These forms along with Craig’s [52] basic form are as follows:

Q(z) = 1

�

∫ �/2

0exp

(− z2

2sin2 �

)d�; z�0 (A1)

Q2(z) = 1

�

∫ �/4

0exp

(− z2

2sin2 �

)d�; z�0 (A2)

Q3(z) = 1

�2

∫ �/6

0cos−1

(3cos2�−1

2cos3 2�−1

)exp

(− z2

2sin2 �

)d�

+ 1

2�2

∫ sin−1(1/√

3)

0

{�−cos−1

(3cos2�−1

2cos3 2�−1

)}exp

(− z2

2sin2 �

)d� (A3)

Q4(z) = 1

�2

∫ �/6

0cos−1

(3cos2�−1

2cos3 2�−1

)exp

(− z2

2sin2 �

)d� (A4)



APPENDIX B: PROOF OF (10) AND (18)

Let

I1 = 1√2�

∫ ∞

−∞exp

[− (x −√

2�)2

2

]exp

(− x2

2sin2 �

)dx (B1)

Now the integral may also be written in the following form:

I1 = 1√2�

exp

(− �

1+sin2 �

)∫ ∞

−∞exp

⎡⎣−1+sin2 �

2sin2 �

(x −

√2�sin2 �

1+sin2 �

)2⎤⎦ dx . (B2)

A close inspection reveals that the integral in (B2) consists of PDF of a Gaussian random

variable with mean√

2�[sin2 �/(1+sin2 �)] and variance sin�/√

1+sin2 � which when evaluatedresults in unity. Thus

I1 = sin�√1+sin2 �

exp

(− �

1+sin2 �

)(B3)

Let us define another integral,

I2 =∫ ∞

0exp(−�x)I0(2

√x)dx (B4)

Using (6.614.3), (9.220.2), and (9.215.1) [50], we get

I2 = 1√�

exp

(

2�

)M−1/2,0

(

�

)

= 1

�1 F1

(1;1;

�

)

= 1

�exp

(

�

)(B5)

where M,�(·) denotes Whittaker’s function and 1 F1(·; ·; ·) means confluent hypergeometricfunction.

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AUTHORS’ BIOGRAPHIES

Aniruddha Chandra received his BE (Hons) degree in Electronics and CommunicationEngineering and ME degree in Communication Engineering from the Jadavpur University(JU), Kolkata, India, in 2003 and 2005, respectively, and is currently pursuing a PhDthere.

He joined Electronics and Communication Engineering Department, National Insti-tute of Technology (NIT), Durgapur, in 2005 as a lecturer and is currently servingas an assistant professor there. His research interests include wireless fading channelmodelling, diversity combining, modulation techniques, broadband access over power-line, and underwater communication.

Mr Chandra has co-authored a book titled Analog Electronic Circuits and published 30research papers in reputed journals and peer-reviewed conferences. He has also deliveredseveral invited lectures including IEEE comsoc lecture meetings at JU, Kolkata. He

received the prestigious NTSE and EFIP scholarships from NCERT and MHRD, Govt. of India, respectively.He is a member of IEEE (Communication Society), IAENG, and has also served as a reviewer for IEEETransactions on Vehicular Technology, IEEE Communication Letters, IEEE Potentials and Computer andElectrical Engineering (Elsevier).

Srinivasa Rao Poram was born in Vizianagaram in Andhra Pradesh, India. He receivedhis B. Tech. degree in Electronics and Communication Engineering from JNTU, Hyder-abad in 2004 and MTech degree in Telecommunication Engineering from the Departmentof Electronics and Communication Engineering, NIT, Durgapur in 2008.

He is now with Reliance Energy Ltd. His research areas include digital communi-cation, M-ary modulations, fading, and diversity concepts.



Chayanika Bose received her BTech, MTech, and PhD degrees in radio physics andelectronics from the University of Calcutta in 1981, 1983, and 1990.

She served as Scientist B in Training Program in Millimeter Wave Technology, DOEat Institute of Radio Physics and Electronics, Calcutta University and subsequently joinedas post-doctoral fellow (UGC) in the department of Electronics and TelecommunicationEngineering, Jadavpur University. Presently, she is a reader in the same department. Herresearch interests involve semiconductor nanostructures and radio wave propagation.

Dr Bose has published more than 35 papers in referred journals. She is a memberof IEEE electron device society and communication society since 1999 and became asenior member in 2005.


SEP calculations for coherent M-ary FSK in different fading channels with MRC diversity

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