1

Packet Error Probability of a Transmit BeamformingSystem with Imperfect Feedback

Yogananda Isukapalli, Student Member, IEEE and Bhaskar D. Rao, Fellow, IEEE

Abstract— Average packet error probability (PEP) is an impor-

tant error statistic for wireless communication system designers.

In this paper we address the problem of analytically quantifying

the effect of channel estimation errors, feedback delay and channel

vector quantization on the PEP of transmit beamforming multiple

input single output (MISO) systems in a spatially independent slow-

fading wireless channel environment. We develop an accurate char-

acterization of estimation errors as well as errors due to feedback

delay, and tools relevant for deriving analytical expressions for

the PEP. The modeling highlights the distinction between errors

that arise due to channel estimation from those that arise due

to feedback delay and represents an important departure from

past work. Analytical expressions are derived for the PEP with

BPSK signaling. The derived approximated closed-form analytical

expression is complemented by simulations.

Index Terms: MISO systems, transmit beamforming, packetfading channel, channel estimation errors, feedback delay, chan-nel quantization, average packet error probability

I. INTRODUCTION

Multiplicative fading is a major source of performance degra-dation in multipath wireless environment. Channel coding andinterleaving can offer some protection from the negative effectsof fading. However, in some wireless systems data has to beorganized into small packets, which are confined to fixed timeslots, with or without interleaving. One popular example of sucha system is the slotted multiple access scheme. It is importantfor the system designers to know the impact of fading on theperformance. An important metric for studying the performanceof a non-interleaved wireless packet data transmission is theaverage packet error probability (PEP) [1]-[16]. Packet errorprobability is also increasingly becoming an important quality ofservice parameter for the wireless networking community sinceit determines how frequently the information packet has to bere-transmitted [17]-[21].

Extensive analytical results quantifying the impact of fadingon average symbol and error probability (SEP/BEP) are availablefor various modulation schemes [22]. However, in slow fadingsituations, there is no mapping between the average SEP/BEPand the average PEP. Consequently knowing average SEP/BEPdoes not help in understanding the average PEP. Analysis ofaverage PEP is a more complicated problem compared to theanalysis of average SEP/BEP. Analytical quantification of packeterror probability has received considerable attention in the litera-ture [1]-[16]. Closed-form expressions for PEP have been derived

Copyright (c) 2008 IEEE. Personal use of this material is permitted. However,permission to use this material for any other purposes must be obtained from theIEEE by sending a request to pubs-permissions@ieee.org. Yogananda Isukapalliand Bhaskar D. Rao are with the Department of Electrical and ComputerEngineering, University of California, San Diego, La Jolla, CA 92092 (e-mail: yoga@ucsd.edu, brao@ece.ucsd.edu). This research was supported inpart by the U. S. Army Research Office under the Multi-University ResearchInitiative (MURI) grant-W911NF-04-1-0224. Part of this work was presentedin [33] and [35].

for the non-coherent FSK modulation. The non-coherent FSK’sSEP, conditioned on the channel, is an exponential function andtaking expectation of the higher powers of conditional SEP w.r.t.the fading random variable is analytically tractable. However,closed-form expressions are not available for coherent BPSKand other constellations. Conditional PEP (conditioned on afunction of the wireless channel) for a scheme such as coherentBPSK results in integer powers of the Gaussian-Q function.This makes the analysis challenging because in order to derivethe average PEP expression, one has to integrate the integerpowers of the Gaussian-Q function w.r.t. the random variablethat captures the fading environment, an analytically difficultexercise. We also note that, to the best of our knowledge,the effect of channel estimation errors on PEP has not beenconsidered in the literature. In this paper we consider the problemof deriving analytical expressions for PEP of a multiple inputsingle output (MISO) system with various forms of practicalfeedback imperfections. We later show that the derived averagepacket error probability expression, captures the analysis of othercommonly interested performance metrics as special cases.

In a MISO system, if the channel state information (CSI) isavailable at the transmitter, one can achieve both the diversityand the array gains with transmit beamforming via maximal ratiotransmission [23]. However, in frequency division duplexingMISO systems the CSI has to be fed back from the receiver tothe transmitter, and practical feedback systems suffer from manyforms of imperfections resulting in performance degradation.The first form of feedback imperfection considered is channelestimation error. Typically the channel is estimated at the receiverwith the help of training symbols. Due to the presence ofthermal noise, channel estimation errors are inevitable in anypractical system. It is now a common practice to model theactual channel and its estimate as a jointly Gaussian randomprocess, with an error term that is orthogonal to the channelestimate ([24], [34], [36], and [49]). The error term associatedwith a particular channel estimate is un-known to the receiver andhence it becomes part of noise when the performance analysisis carried out. If the channel under consideration is varying atsymbol level, or if the performance criteria is average symbol/biterror probability, then the variance of the error term will besimply added (along with the symbol dependency) to the varianceof the receiver noise resulting in an effective noise term withvariance equaling the sum of variance of receiver noise and thevariance of the estimation error term ([24], [36]-[38], and thereferences therein). In this paper, we also follow the standardmodel of joint Gaussianity between the channel and its estimate,but adapt it to the packet fading model. An important differenceis that, in a packet fading model the error term is constantfor the entire packet while each symbol experiences a differentnoise sample, requiring new analytical tools in order to studythe performance.

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The second form of feedback imperfection we address isfeedback delay. In any practical system, delay between construct-ing the beamforming vector at the receiver and using it at thetransmitter is an un-avoidable form of feedback imperfection.Another well accepted formulation [25], [26], [34], and [37]-[43] is to treat the impact of feedback delay in a manner similarto estimation errors, i.e., actual channel and its delayed versionare assumed to be jointly Gaussian with an un-known (to thereceiver) error term that is orthogonal to the delayed version.Since the delay related error term is un-known to the receiver,similar to estimation related error term, during performanceanalysis, it becomes part of noise thus removing any concep-tual distinction between the mismatch in beamforming due tofeedback delay and estimation errors. Though much of the pastwork on feedback delay [25], [26], [34], and [37]-[43] effectivelymake the delay related error term part of receiver noise, alternateoptions were considered (primarily in the context of adaptivemodulation) in [44]-[51]. However, it is important to note thatmuch of the work treated estimation errors and feedback delayin a similar manner, i.e., either both the error terms are assumedto be known or un-known to the receiver. In this paper, based onfeedback system considerations we feel it is appropriate to treatthe errors due to feedback delay to be known at the receiver,while the errors due to estimation errors are un-known at thereceiver. This modeling approach is adopted in this work and itshows that the impact of feedback delay on beamforming MISOsystem performance can be less severe and is also conceptuallyquite different from channel estimation errors. Such modelingof feedback delay has been considered by us in an earlierconference publication, albeit in a SEP context [33].

In frequency domain duplexing systems, the estimated channelhas to be conveyed to the transmitter. In low-rate feedbacksystems, an effective way to send the channel information tothe transmitter is by having a known codebook at both thereceiver and the transmitter so that the receiver can just sendthe index number of the chosen codepoint. This leads to thethird form of feedback imperfection considered in this paper,namely finite-rate quantization of the channel [27]- [32]. Tosummarize, the contributions of this paper are threefold: anaccurate characterization of estimation errors in a packet fadingcontext, a new modeling of feedback delay which shows im-proved performance for a beamforming MISO system and con-ceptually distinguishes it from estimation errors, and derivationof an analytical expression quantifying the impact of channelestimation errors, feedback delay and channel quantization on theaverage packet error probability. All these contributions furtherthe understanding of feedback communication systems. As aside benefit, the analytical tools developed promise to be ofgeneral interest with broad applicability. The rest of this paperis organized as follows. A general framework for the modelingof imperfect feedback in the packet fading context is presentedin Section II. An analytical expression for the average packeterror probability with un-coded BPSK constellation is derivedin Section III. Numerical and simulation results are presented inSection IV. We conclude this paper in Section V.

Notation: Small and upper case bold letters indicate vectorand matrix respectively. E(.), (.)T , (.)H , |.|, (.), and �.� de-note expectation, transpose, Hermitian, absolute value, complexconjugate, and 2-norm respectively. x ∼ p(x) indicates thatthe random variable x is distributed as p(x). x ∼ NC (µ,Σ)

indicates a circularly symmetric complex Gaussian (CSCG)random variable x with mean µ and covariance Σ.

Important Variables: t -number of transmit antennas, � -packetindex, k -symbol index in the packet, N -number of symbolsin a packet, ρe - estimation related correlation co-efficient, ρe -magnitude of ρe, ρd -delay related correlation co-efficient, ρd -magnitude of ρd, B -number of feedback bits, γb -SNR per bit,h� -actual channel of �th packet, �h� -estimated channel of �th

packet.

II. SYSTEM MODEL

We consider a multiple input single output system with tantennas at the transmitter and one antenna at the receiver. Leth� be the channel between the transmitter and the receiver forthe �th packet. h� is modeled as a spatially i.i.d frequency-flatRayleigh fading channel which is constant for all the N un-codedsymbols in packet �. The vector valued channel h� ∼ NC (0, I).Furthermore, it is assumed that the channel varies from packetto packet but exhibits significant correlation. The transmitted kth

symbol in the packet � is denoted by s�[k] and E[|s�[k]|2] = Es.Let w� be the unit norm beamforming vector (BV) at thetransmitter for the packet �. Then, the kth received signal inthe packet � is given by

y�[k] = hH

�w� s�[k] + η�[k], k = 1, 2, · · · , N (1)

where η�[k] is the thermal noise that effects the kth symbol of�th packet, η�[k] ∼ NC

�0,σ2

n

�. We now discuss in detail the

three forms of feedback imperfections and develop a generalmodel that captures channel estimation errors, feedback delay,and finite-rate channel quantization for transmit beamformingMISO systems.

A. Channel Estimation Errors - Packet Fading Model

Let �h� be the estimate of h� . We assume that h� and �h� arejointly Gaussian, a reasonable assumption for many practical es-timation techniques ([24], [36]-[38], and the references therein).The jointly Gaussian assumption allows us to relate them asfollows:

h� =ρe√

Λ�h� +

�1− ρ2

eε� e (2)

where ε� e ∼ NC (0, I), �h� ∼ NC (0, ΛI), I denotes the identitymatrix of size t × t, E[|�hi|

2] = Λ, i = 1, · · · , t, and ρe =ρe ejφρe is the complex correlation coefficient that determinesthe degree of accuracy in channel estimation. The closer ρe isto one, the more accurate is the channel estimate. ρe can beassumed to be known at the receiver. Assuming instantaneousfeedback and no channel quantization, the beamforming vectorto be used at the transmitter is given by

w� =�h�

��h��. (3)

The kth received signal of the �th packet with the BV givenin (3) and h� given in (2) is

y�[k] = hH

�w� s�[k] + η�[k]

=�

ρe√

Λ�h� +

�1− ρ2

eε� e

�H �h�

��h��s�[k] + η�[k]

=¯ρe√

��h�� s�[k] +

��1− ρ2

eε� e s�[k] + η�[k]

�(4)

3

where

ε� e =εH

� e�h�

��h��∼ NC (0, 1) .

In the above equation ε� e is un-known to the receiver andhence the receiver can not compensate for the phase rotationcaused due to ε� e. In our previous work ([33] and [34]), theperformance criteria was average SEP/BEP and as a consequencethe estimation error term simply got absorbed into the thermalnoise and increased its variance (along with introducing a symboldependency). In the packet fading model, the estimation errorterm ε� e is constant and impacts the entire packet (while eachsymbol experiences a different noise sample) and hence it cannot be simply lumped into the additive noise term. As will beevident from the next section, not lumping the estimation errorterm into white noise also makes the analysis more complicated.Note that PEP analysis requires that we pay attention to the factthat ε� e is constant for the duration of the packet.

B. Feedback Delay with Imperfect Channel Estimation

In the above discussion, a simplistic assumption of feedbackbeing available instantly was made at the transmitter. In realitythere is a delay and to account for this it is assumed that thebeamforming vector for the current packet � is derived from thechannel estimate of the previous packet (�− 1). Since there is atime lag between forming the BV at the receiver and its use atthe transmitter, the BV w� depends on �h�−1 as opposed to thecurrent channel estimate �h�. Assuming that the channel estimateand its delayed version are jointly Gaussian, we can relate themas follows:

�h� = ρd�h�−1 +

�(1− ρ2

d)Λ ε� d (5)

where �h� ∼ NC (0,ΛI) and ρd = ρd ejφρd is the complexcorrelation co-efficient between �h� and �h�−1, ε� d ∼ NC (0, I)is the error term due to delay and is assumed to be uncorrelatedwith �h�−1. The delay correlation co-efficient ρd is assumed tobe known to the receiver and measures the impact of delay. Withthe help of (2) and (5), the actual channel h� in terms of delayedversion of the estimated channel �h�−1 can be written as

h� = ρe

�ρd√

Λ�h�−1 +

�1− ρ2

dε� d

�+

�1− ρ2

eε� e . (6)

With the inclusion of estimation errors along with the delay, thebeamforming vector (still un-quantized) is given by

w� =�h�−1

��h�−1�. (7)

The beamforming vector w� indicates that the BV is formed withthe help of channel estimate from the previous packet (� − 1)and is used to transmit packet �. With this formulation, the kth

received symbol of the packet � can be written as

y�[k] = hH

�w� s�[k] + η�[k]

=�

ρe

�ρd√

Λ�h�−1 +

�1− ρ2

dε� d

�+

�1− ρ2

eε� e

�H

�h�−1

��h�−1�s�[k] + η�[k]

= ¯ρe

�¯ρd�

�h�−1�√

Λ+

�1− ρ2

dε� d

�s�[k] +

�1− ρ2

eε� e s�[k] + η�[k]� �� �

Un-known to the Receiver

(8)

where

ε� d =εH

� d�h�−1

��h�−1�∼ NC (0, 1) ,

ε� e =εH

� e�h�−1

��h�−1�∼ NC (0, 1) ,

¯ρe and ¯ρd are complex conjugates of ρe and ρd respectively. For aparticular packet both ε� d and ε� e are constant. The distributionsof both ε� d and ε� e indicate their random nature over a numberof packets.

As explained in the previous section, the estimation relatederror term ε� e is un-known to the receiver and hence the phaserotation caused by ε� e can not be compensated. The role of delayrelated error term ε� d, which determines the penalty due to thefeedback delay, will depend on the modeling assumptions. Noticethat if there is no feedback delay, then ρd = 1 and the error termvanishes. If ε� d is also assumed to be un-known, then it can betreated in a manner similar to estimation error. In particular, if theperformance metric is SEP/BEP, or if the channel is varying at asymbol level as opposed to packet level, then ε� d can be mergedinto the receiver noise greatly simplifying the analysis [25], [26],and [37]-[43]. A closer examination, as explained below, indi-cates that there is a distinction between estimation error relatedterm and the delay related term and so lumping them togetheris a questionable simplification. We propose a model/approachto handle this error that we believe more accurately captures theimpact of delay on feedback system performance. The modelis based on the following system assumption: there is a pilotsequence before every packet of data and that the beamformingvector (7) is based on the channel estimate from the previouspacket. Under this assumption, the receiver will be knowing both�h� and �h�−1 and hence it knows the error term due to delayε� d, c.f. 5 (and subsequently ε� d). As shown later, this simplechange in the approach impacts the performance of the systemconsiderably (more performance gain is seen at lower ρd values),and indicates more clearly the conceptual distinction between theimperfections resulting from delay and estimation. However thismodeling also makes the problem of performance analysis morecomplicated. Note that even if the receiver knows ε� d, it can notcompensate for all the loss caused due to delay. Its performancelies in between a system with no feedback delay (ρd = 1), anda system where the receiver does not know ε� d, (0 < ρd < 1)and is lumped into the receiver noise.

C. Quantization of Delayed Version of Channel Estimate

In the previous two sections we assumed that the channel stateinformation is exactly conveyed to the transmitter. In practice,

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the receiver estimates the channel, and quantizes it into one ofC = 2B code words in the codebook which is known to bothtransmitter and receiver. The index, which is represented by Bbits, of the code word corresponding to the channel estimateis fed back to the transmitter. We assume that the feedbackchannel is error free [27]-[32]. The beamforming vector formedby quantizing delayed version of the channel estimate is givenby

w� = Q

��h�−1

��h�−1�

�(9)

where Q is the vector quantization function and here we assumea vector quantization based codebook ([28] and [32]). Using (6)and (9), the kth received signal of the packet � is:

y�[k] = hH

�w� s�[k] + η�[k]

=�

ρe

�ρd√

Λ�h�−1 +

�1− ρ2

dε�d

�+

�1− ρ2

eε�e

�H

�Q

��h�−1

��h�−1�

��s�[k] + η�[k]

= ¯ρe

� ¯ρd√

Λϑ�−1 �

�h�−1�+�

1− ρ2dε� d

�s�[k] +

�1− ρ2

eε� e s�[k] + η�[k] (10)

where

ε� d = εH

� d

�Q

��h�−1/��h�−1�

��,

ε� d ∼ NC (0, 1) ,

ε� e = εH

� e

�Q

��h�−1/��h�−1�

��,

ε� e ∼ NC (0, 1) ,

ϑ�−1 =��h�−1/��h�−1�

�H �Q

��h�−1/��h�−1�

��. (11)

We now take a closer look at the additional changes to thekth symbol in the received packet � because of use of aquantized beamforming vector. The received signal with andwithout quantized BV is given by (12) and (13) respectively

Un-Quantized BV (7)����y�[k] = ¯ρe

� ¯ρd√

Λ��h�−1�+

�1− ρ2

dε� d

�

� �� �Known to the Receiver

s�[k]

+�

1− ρ2eε� e s�[k] + η�[k]� �� �

Un-known to the Receiver

, (12)

Quantized BV (9)����y�[k] = ¯ρe

� ¯ρd√

Λϑ�−1 �

�h�−1�+�

1− ρ2dε� d

�

� �� �ψ�=ψ� e

j φψ� (Known to the Receiver)

s�[k] +�

1− ρ2eε� e s�[k] + η�[k]� �� �

Un-known to the Receiver

. (13)

The above two equations mainly defer in three places. Theeffective error terms with un-quantized BV are ε� d and ε� e andthe effective error terms with quantized BV are ε� d and ε� e. Allthese effective error terms are different in an instantaneous sense,but all of them are CSCG random variables with same meanand variance. The main effect of quantization on the PEP comesfrom ϑ�−1. Note that ϑ�−1 is the inner product between the un-quantized and quantized BV and the statistical characterization

of ∆ � |ϑ�−1|2 will be important for the performance analysis.

Since finding the exact pdf of ∆ is difficult, [28] and [32] upperbounded ∆ by a r.v ∆, whose pdf is given by

p∆(x) = 2B(t− 1)(1− x)t−2, 1− ω < x < 1 (14)

where ω = 2−B/(t−1). In what follows, we use ∆ in place of∆. Because of our modeling assumptions, the receiver knowsρd, ρe, �h�, �h�−1, and Q

��h�−1/��h�−1�

�, so it knows ψ� which

enables coherent detection of the transmitted symbol.

III. AVERAGE PACKET ERROR PROBABILITY

Since the feedback delay related error term and estimationrelated error term are constant for the entire packet, averagepacket error probability, a metric that requires averaging over thepacket index � thereby capturing the effect of imperfect feedbackin transmit beamforming MISO systems, provides a meaningfulway to study performance analysis. Also as pointed out in theintroduction, for slotted multiple access schemes and as a qualityof service parameter for the MAC layer of wireless networks,analytical understanding of PEP is important from system designpoint of view. Existing results primarily are for PEP with non-coherent FSK modulation (limited to single input single outputsystems and under ideal conditions), and so are not applicable tothe present scenario of imperfect feedback with coherent BPSKmodulation.

In this section we present the analysis of the average packeterror probability of an un-coded packet of N BPSK symbolswith imperfect feedback. We first begin with the decision statisticrequired for the detection of kth symbol of �th packet and thenfocus on the PEP. The coherent detection of transmitted symbolis based on r�[k], obtained from y�[k] in (13).

r�[k] = e−j φψ� y�[k] (15)

=�ψ� +

�1− ρ2

eε� e

�s�[k] + η�[k],

ψ� =��� ¯ρe

� ¯ρd√

Λϑ�−1 �

�h�−1�+�

1− ρ2dε� d

� ��� , (16)

ε� e = e−j φψ� ε� e ∼ NC (0, 1) ,

η�[k] = e−j φψ� η�[k] ∼ NC�0,σ2

n

�.

Notice that in the above equation, ψ� and ε� e do not depend on[k] indicating that these two terms are fixed for the entire packet,whereas the notation for the white noise is η�[k] indicatinga different noise sample for each symbol. To highlight thedifferences with our previous work [34], we now briefly contrastthe decision variable (DV) in (15) to that of the DV we usedin [34]. The performance metric in [34] was average SEP/BEPand the DV in [34] was given as

r[k] = κos[k] + ξ[k], (17)

where κo is a complex constant number and ξ is condition-ally (conditioned on both fading and quantization of channel)CSCG random variable. In [34] to derive analytical expressionaverage SEP/BEP, we had to account for the fact that the trans-mitted symbol is scaled and rotated as well as the noise is symboldependent. The DV (15) in this paper is more complicated. Allthe symbols of �th block are scaled by a known random variableψ�. Note that knowledge of ψ� at the receiver is possible dueto the modeling assumptions presented. Also, all the symbols of

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�th block experience the same channel estimation related errorterm ε� e.

Since we are restricting our attention to the BPSK constel-lation, the receiver uses the real part of r�[k] to decode thetransmitted symbol as

r�[k] = Real (r�[k])= κ s�[k] + η�[k]

where

κ = ψ� +�

(1− ρ2e)/2 ε�er , (18)

ψ� > 0, −∞ < κ < ∞,

ε�er and η� are both real random variables with ε�er =Real (ε�e) ∼ N (0, 1) and η�[k] = Real (η�[k]) ∼ N

�0, σ2

n/2

�.

As pointed out earlier ε�er is a fixed constant for a particularpacket, and viewed over a number of packets it is a statisticalquantity. The derivation of an analytical expression for PEP isquite involved and here we outline the important steps in thederivation.

1) Derivation of pdf for κ: The signal scaling random variablein the decision statistic r�[k] is κ. So the pdf of κ isimportant for the analysis of PEP. The derivation of thepdf of κ is complicated by its dependency on the threeforms of imperfection. Details of the derivation can befound in Section III-A.

2) Expectation of the Gaussian Q-function and its higherpowers w.r.t. the random variable κ: Much of the analyticalcomplexity in the performance analysis (PEP) revolvesaround the evaluation of expectations of the Gaussian Q-function and its higher powers w.r.t. the random variableκ in closed-form. For the first two powers of the GaussianQ-function, we are able to evaluate the expectation usingthe exact form of the Gaussian Q-function. For higherpowers (≥ 3), we make use of an approximation of theGaussian Q-function and evaluate its expectation w.r.t. κ.Details are in Section III-B.2.

A. Derivation of pdf for κ

The signal scaling term κ in the decision statistic r�[k] is givenin (18):

κ = ψ� +�

(1− ρ2e)/2 ε�er , ψ� > 0, −∞ < κ < ∞.

Conditioned on ψ�, the conditional pdf of κ is given by

pκ(z|ψ� = x) =1�

π(1− ρ2e)

e−

(z−x)2

1−ρ2e .

and the pdf of κ can be obtained as

pκ(z) =� ∞

−∞

pκ(z, x) dx

=� ∞

−∞

pκ(z|x) pψ�(x) dx (19)

The evaluation of (19) requires the pdf of ψ� which is derivedin Appendix-I and the final expression is given in (64). Notethat if there are no estimation errors in the model, or if theperformance criteria is average symbol/bit error probability [33],or if the performance criteria is average packet error probabilitywith any constellation other than BPSK, then the pdf of ψ�

becomes central to the performance analysis. Here, not only isthe pdf of ψ� required, the extra step discussed in (19) has to becarried out for the pdf of κ. Substituting the pdf of ψ� in (19),we obtain

pκ(z) =n�

l=0

2n−l−1�

p=0

δ�

g=0

�Kp1fa2(l, p, g,L, z) +

Kp2fb2(l, p, g,L, z)�

(20)

where the variable and the corresponding defining equation arelisted as pairs: n - (46), δ - (55), L - (52), Kp1 - (58), andKp2 - (59), and

fb2(l, p, g,L, z) =2Ln−l−g e

−z2

1−ρ2e

ρ2(n−l−g)e Γ(n− l − g)

�π(1− ρ2

e)

� ∞

0x2(n−l−g)−1 e

−x2[ρ2e+L(1−ρ2

e)]ρ2

e(1−ρ2e)

+ 2xz1−ρ2

e dx .

To evaluate the above integral we use the following identity [55]� ∞

0xv−1e−βx

2−γx dx = (2β)−v/2 Γ(v) e

γ28β D−v

�γ√

2β

�. (21)

In the present context

β =ρ2

e+ L(1− ρ2

e)

ρ2e(1− ρ2

e)

,

γ =−2z

1− ρ2e

,

v = 2(n− l − g) .

fb2(l, p, g,L, z) can now be written as

fb2(l, p, g,L, z) = H1 e−z2H2 D−2(n−l−g) (−zH3) , (22)

−∞ < z < ∞

where

H1 =L

v2 (1− ρ2

e) v

2 Γ(v)2( v

2 )−1Γ(v

2 )(ρ2e

+ L(1− ρ2e)) v

2�

π(1− ρ2e), (23)

H2 =ρ2

e+ 2L(1− ρ2

e)

2(1− ρ2e)(ρ2

e+ L(1− ρ2

e))

, (24)

H3 =√

2 ρe�(1− ρ2

e)(ρ2

e+ L(1− ρ2

e))

, (25)

and Dp(l) is the parabolic cylinder function [55]. Finally

fa2(l, p, g,L, z) = fb2(l, p, g, 1, z) , (26)

which completes all the steps required to compute pκ(z) in (20).The analytical expression for pκ(z) is confirmed using simula-tions (histogram approach is used to get the simulated versionof κ’s pdf) in Fig. 1 for number of transmit antennas t ∈ {2, 3},delay correlation co-efficient ρd ∈ {0.98, 0.94}, and estimationerror correlation co-efficient ρe ∈ {0.95, 0.91}. The number offeedback bits B which determines the quantization codebooksize (2B) is fixed at 4.

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−2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 3 3.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t=3, ρd=0.98, ρe=0.95, Simulated

t=3, ρd=0.98, ρe=0.95, Analyitcal

t=2, ρd=0.94, ρe=0.91, Simulated

t=2, ρd=0.94, ρe=0.91, Analytical

Fig. 1. Verification of the pdf for the signal scaling term κ defined in (18).Number of feedback bits B=4.

B. Analytical Expression for Packet Error Probability

Conditioned on κ, the packet error probability (the probabilitythat at least one symbol in the packet is received incorrectly) isgiven by

PB,�(γb, ρe, t, N) = 1− {1− pb,�}N

= 1−N�

m=0

�Nm

�(−1)m (pb,�)m

where pb,� is the error probability of a symbol in the �th packet(calculated with the help of decision statistic r�[k])1 Note thatsince κ is fixed for the entire packet (while the noise sample isdifferent for each symbol), all the symbols have the same errorprobability. The average packet error probability is given by

PB(γb, ρe, t, N) = E� [PB,�(γb, ρe, t, N)] (27)

= 1−N�

m=0

�Nm

�(−1)m E� [(pb,�)m] .

Accounting for the fact that ‘κ’ can be negative with a non-trivialprobability, with BPSK constellation, E� [(pb,�)m] can be writtenas

E� [(pb,�)m] = ApE�

�Qm

��2γb z1

��+

(1−Ap)E�

�1−Q

��2γb z2

��m

= Ap Ez1

�Qm

��2γb z1

��+ (1−Ap)

m�

w=0

(−1)w Ez2

�Qw

��2γb z2

��(28)

where

Q(x) =1√

2π

∞�

u=x

exp�−

u2

2

�du

1Similar to [8], the above equation can be easily modified to the scenario ofa block channel coded system that can correct an arbitrary number of errors.

is the standard Gaussian tail function [22] and Ap is the areaunder the positive side of pdf pκ(z). i.e. Ap =

�∞0 pκ(z)dz.

In (28), γb is the SNR of a symbol in the packet and

z1 = κ2, 0 < κ < ∞,

z2 = κ2, −∞ < κ < 0.

Using transformation of random variables, the pdfs of z1 andz2, needed to evaluate (28), can be shown to be given by (67)and (68) respectively. For readability purpose the pdfs of z1 (67)and z2 (68) are given at the end of Appendix-I. To evaluate(28), we need to find an expression for Ap and evaluate theexpectations of the Gaussian Q-function and its higher powersw.r.t. the random variable κ. These steps are described below.

1) Evaluation of Ap: A closed-form expression for Ap canbe evaluated as

Ap =� ∞

0pκ(z)dz

=n�

l=0

2n−l−1�

p=0

δ�

g=0

�Kp1fap1(l, p, g,L, z) +

Kp2fap2(l, p, g,L, z)�, (29)

fap2(l, p, g,L, z) = 1−H1

� ∞

0e−z

2H2 D−v (H3 z) dz.

where v = 2(n− l − g). Let z =√

x, dz = 12x−

12 dx,

fap2(l, p, g,L, z) = 1−H1 2−(n−l−g+1)√π

Γ(n− l − g + 1)

2F1

�v/2,

12; v/2 + 1;

4H2 −H23

4H2 +H23

�(30)

where 2F1(·, ·; ·; ·) is the hypergeometric function. To evaluatethe above integral we used [55]� ∞

0x−

β2 +1 e−xc D−ν

�2(kx)

12

�dx =

21−β−ν2√

π Γ(β)

Γ�

ν+β+12

�

(c + k)−β2 2F1

�ν

2,β

2;ν + β + 1

2;c− k

c + k

�, (31)

�Re (c + k) > 0, Re

� c

k

�> 0

�.

In (29) fap1(l, p, g,L, z) = fap2(l, p, g, 1, z).2) Evaluating the expectations of the Gaussian Q-functions

in (28): In this subsection we evaluate the expectations of theGaussian Q-function and its higher powers w.r.t. the randomvariable κ. As explained in the previous subsection, the randomvariable κ is now split into two random variables z1 (capturingthe positive part of κ) and z2 (capturing the negative part ofκ) in closed-form. For m ∈ {1, 2} (m being the power ofthe Gaussian Q-function), we evaluate the expectation usingthe exact form of the Gaussian Q-function. For higher pow-ers (m ≥ 3), we make use of an approximation of the GaussianQ-function and evaluate its expectation w.r.t. z1 and z2. Westart with Ap Ez1

�Qm

�√2 γb z1

��, m = 1 and 2, required to

compute (28):

Ap Ez1

�Qm

��2 γb z1

��=

∞�

z1=0

Qm

��2 γb z1

�

[p1(z1) + p2(z1)] dz1. (32)

7

For m = 1, Ap Ez1

�Qm

�√2 γb z1

��can be written as

Ap Ez1

�Q

��2 γb z1

��=

∞�

z1=0

Q��

2 γb z1

�p1(z1) dz1

+∞�

z1=0

Q��

2 γb z1

�p2(z1) dz1

= G1

�π

2

�+ G2

�π

2

�(33)

where G1 (ϕ) and G2 (ϕ) are derived in Appendix-II. For m = 1and 2, in Appendix-II we exploit the fact that the first and secondpowers of the Gaussian Q-function are parameterized by ϕ = π

2and ϕ = π

4 respectively in the function Q(x) given below [22],i.e., Q

�π

2

�= Q (x) and Q

�π

4

�= Q2 (x)

Q(x) =1π

ϕ�

θ=0

e

�−

x2

2 sin2 θ

�

dθ, x ≥ 0. (34)

Following the above steps and Appendix-II, it can be shown that

Ap Ez1

�Q2

��2γbz1

��= G1

�π

4

�+ G2

�π

4

�,

(1−Ap)Ez2

�Q

��2γbz2

��= G1

�π

2

�− G2

�π

2

�,

(1−Ap)Ez2

�Q2

��2γbz2

��= G1

�π

4

�− G2

�π

4

�.

For m ≥ 3, exact expressions for Ap Ez1

�Qm

�√2 γb z1

��and

(1 − Ap)Ez2

�Qm

�√2 γb z2

��are difficult to derive. With the

help of results in [53], the following series representation canbe given for an analytically tractable form for Qm(x). We beginwith the approximated form for Q(x)

Q(x) ≈ e−x2

2

ma�

m=0

cm xm ,

cm =(−1)m+2(A)m+1

B√

π (√

2)m+2 (m + 1)!.

where A = 1.98, B = 1.135, and ma = 10 [53]. Qm(x) can bewritten as

Qm(x) ≈ e−mx2

2

m ma�

k1=0

ck1 xk1 . (35)

The co-efficients ck1 can be calculated in an easy manner usingthe fourier transform properties in a programming language likeMATLAB. The expectations of the Gaussian-Q approximationwith respect to z1 and z2 are carried out in Appendix-III and thefinal expressions are given by (96) and (101).

C. Special Cases of PEP Expression in (27)

In this subsection we briefly discuss a few special cases of thePEP expression given in (27).

1) With block length N = 1 and assuming that the delayrelated error term known at the receiver, the PEP ex-pression (27) coincides with the analytical expression foraverage SEP in [33] (for BPSK constellation). In [33] wederived closed-form average SEP expressions for M1 ×

M2-QAM constellation assuming that the delay relatederror term is known at the receiver. The average SEP

expression for BPSK, based on (27) (i.e., by substitutingN = 1 in (27) and after some simplification) is given by

PSEP = 2G1

�π

2

�+ (1−Ap) ,

G1 (ϕ) is defined in Appendix-II.2) PEP expression in (27) can also be applied to systems

where the delay related error term is not known at thereceiver. In (27) with ρd = 1 and changing the value ofρe such that it captures both estimation related correlationco-efficient and delay related correlation co-efficient2 PEPin (27) gives the performance of a system where the delayrelated error term is not known at the receiver.

3) By making the block length N = 1, and assuming that thedelay related error term is not known at the receiver, thePEP expression (27) coincides with the results presentedin [34] (for BPSK constellation).

4) By making ρe = 1, ρd = 1, and assuming perfectquantization, with t = 1 the results in an approximatedanalytical expression for PEP with BPSK. Except for a fewmodulation schemes (such as non-coherent FSK3), PEP isgenerally studied with the help of computer simulations,and to the best of our knowledge this approximated ana-lytical form (27) is the first available in the literature.

5) With arbitrary t and perfect feedback (ρe = 1, ρd = 1, andB = ∞), (27) gives the average packet error probabilitywith t degrees of diversity.

6) By making appropriate changes to the PEP expressionin (27), one can obtain an analytical understanding intothe effects of fading, i.e., we can study the effects ofchannel estimation errors alone, or delay alone, or channelquantization alone, or other possible combinations of feed-back imperfection. We believe the general framework (andthe closed-form pdfs of random variables) to derive PEPpresented in this paper can be leveraged to analyticallystudy the average packet error probability in other systemsettings as well.

IV. SIMULATION RESULTS

In this section we present a sample simulation to verify theaccuracy of the derived analytical expression for the averagepacket error probability and also show the effectiveness of themodeling of feedback delay. Fig. 2 shows the accuracy of derivedanalytical expression (in the form of an infinite series) foraverage packet error probability of transmit beamforming withimperfect feedback of a packet of N ∈ {30, 50} un-coded BPSKsymbols with t ∈ {2, 3} transmit antennas, the estimation errorcorrelation co-efficient ρe ∈ {0.98, 0.95}, the feedback delayrelated correlation co-efficient ρd ∈ {0.96, 0.93}, and numberof feedback bits B ∈ {4, 5}. As pointed out earlier, the firsttwo powers of Q-function are evaluated exactly. For higherpowers (m ≥ 3 in (28)), we calculate the expectation w.r.t.to the tractable approximation of the Q-function given in (35).According to the popular Clark’s model [54], the correlationbetween channel samples with a lag of τ is given by R(τ) =

2As explained in [34], when both estimation and delay related noise termsare absorbed into the receiver noise, then the effective correlation co-efficientbecomes the product of delay only correlation co-efficient and estimation onlycorrelation co-efficient.

3Or other modulation schemes where the conditional BEP/SEP expressionsresult in an exponential function.

8

0 2 4 6 8 10 12 14 16 18 2010−3

10−2

10−1

100

Average received SNR per symbol, dB

Aver

age

Pack

et E

rror P

roba

bilit

y

t=2, B=5, ρe=0.98, ρd=0.96, N=50, Simulated

t=2, B=5, ρe=0.98, ρd=0.96, N=50, Analytical

t=3, B=5, ρe=0.98, ρd=0.96, N=50, Simulated

t=3, B=5, ρe=0.98, ρd=0.96, N=50, Analytical

t=3, B=4, ρe=0.95, ρd=0.93, N=30, Simulated

t=3, B=4, ρe=0.95, ρd=0.93, N=30, Analytical

t=3, B=4, ρe=0.98, ρd=0.96, N=50, Analytical

Fig. 2. Effect of imperfections in feedback (estimation, delay and quantization)on the average packet error probability.

0 2 4 6 8 10 12 1410−4

10−3

10−2

10−1

100

Average received SNR per symbol, dB

Aver

age

Pack

et E

rror P

roba

bilit

y

t=4, ρd=0.985 − constant

t=4, ρd=0.985 − varying

t=4, ρd=0.97 − constant

t=4, ρd=0.97 − varying

t=3, ρd=0.99 − constant

t=3, ρd=0.99 − varying

Fig. 3. Verifying the constant channel assumption.

J0(2πfmτ) where Jk() is the kth order Bessel function of thefirst kind, fm = v/λ, v is the velocity of mobile, and λ is thecarrier wavelength. ρd = 0.96 correspond to a Doppler frequencyof 80 rad, and a delay of 5 milliseconds. In Fig. 2, by fixingN = 50, ρd = 0.98, ρe = 0.96, and B = 5, improvement inperformance can be seen as the number of antennas are increasedfrom t = 2 to t = 3 . Similarly, by fixing t = 3, N = 50,ρd = 0.98, and ρe = 0.96, an improvement in performanceis noticed as the number of feedback bits are increased fromB = 4 to B = 5. In Fig. 3 we verify the constant channel (forthe entire packet) assumption. In Fig. 3 curves with legend‘constant’ implies that the channel is assumed to be constant and‘varying’ implies that the channel is modeled to be varying fromsymbol-to-symbol such that the effective correlation coefficientis ρd. Simulation parameters for Fig. 3: B = 4, N = 30,and ρe = 0.98. Both the curves for all three different sets ofparameters match quite well showing that the assumption ofconstant channel is well justified.

0 5 10 15 20

10−2

10−1

Average received SNR per symbol, dB

Aver

age

Pack

et E

rror P

roba

bilit

y

Delay Term Un−Known

Delay Term Known

Fig. 4. Impact of delay related error term.

As discussed in the modeling of imperfect feedback, one ofthe contributions of this paper is the modeling aspect of feedbackdelay. The impact of the knowledge of delay related error term(at the receiver) on the performance can be seen in Fig.4. Withall system parameters being the same, the solid curve shows theperformance of the system with delay related error term assumedknown to the receiver as in this paper. The dotted curve showsthe performance of the system when the receiver is assumedto not know the delay related error term as has been done inthe past. Apart from being conceptually distinct from estimationerror related error term, the delay related error term as modeledin the paper shows improved system performance. The pastmodeling [34] (and references therein) simplifies the analysis butcan lead to erroneous conclusions on system performance. Thesimulation parameters for Fig.4 are: Number of transmit antennast = 3, packet size N = 30, estimation related correlationco-efficient ρe = 0.97, delay related correlation co-efficientρd = 0.9, and number of feedback bits B = 4.

0 2 4 6 8 10 12 14 16 18 20

10−3

10−2

10−1

Average received SNR per symbol, dB

Aver

age

Pack

et E

rror P

roba

bilit

y

Delay only, ρd=0.95

Estimation error only, ρe=0.95

Fig. 5. Contrast between delay only system and estimation error only system.

Fig.5 contrasts the effect due to feedback delay alone and

9

0 2 4 6 8 10 12 14 16 18 20

10−3

10−2

10−1

Average received SNR per symbol, dB

Aver

age

Pack

et E

rror P

roba

bilit

y

B=2, ρe=0.97B=4, ρe=0.97B=2, ρe=0.99

Fig. 6. Trade-off between channel estimation errors and channel quantization

due to estimation errors alone. The solid dotted curve showsPEP due to estimation errors alone with ρe = 0.95 and thedotted curve shows the PEP due to feedback delay alone withρd = 0.95. Clearly the performance due to estimation errorsalone is worse than performance due to delay alone. In this figurethe feedback delay error term is known at the receiver thus itis able to perform better. If the feedback delay is not modeledin the way explained in this paper, then both curves would besame. The important message from this figure is: if a trade-off is possible it is better to put more resources into reducingestimation errors as opposed to trying to reduce the feedbackdelay. In Fig.5- number of feedback bits B = 4 and the blocklength N = 30. In Fig.6 we consider a possible trade-off betweenthe number of feedback bits and the channel estimation quality.Quality of channel estimation depends on number of pilots andthe pilot SNR [34]. If the forward link budget is constrained thenincreasing the number of feedback bits, which in turn improvesthe quality of channel quantization, can help in achieving aperformance which is equivalant to increasing the pilot SNR.Similarly if the feedback link is constrained then pilot SNRcan be increased to achieve an improvement in performance.Fig.6 illustrates this observation. In Fig.6- number of transmitantennas t = 3, block size N = 30, and delay related correlationcoefficient ρd = 0.97.

V. CONCLUSION

We considered the problem of analyzing the average packeterror probability, an important quality of service parameter, ofclosed loop MISO systems with imperfect feedback. The feed-back imperfections include channel estimation errors, feedbackdelay and finite-rate channel quantization. Modeling of channelestimation errors in the packet fading context makes use of thefact that the channel estimate and the related error term areconstant for the entire packet. The modeling approach distin-guishes between errors due to channel estimation (un-known tothe receiver) from those due to feedback delay (known to thereceiver). Knowledge of delay related error term at the receiverhelps in improved performance compared to a system withoutthe knowledge of delay error term. An approximated analytical

expression for the PEP of an un coded (or a simple blockchannel coded) packet of BPSK symbols is derived. The generalexpressions derived are quite complex and not easily amenableto interpretation. Nevertheless, the steps taken in this paperare necessary and hopefully will prove to be useful for futurework that has greater interpretability. Special cases of interestare discussed. The derived closed-form analytical expression isvalidated by simulations. Simulation results have also been usedto contrast the relative effects of the three forms of feedbackimperfections on the average packet error probability.

ACKNOWLEDGEMENTS

We thank the reviewers for their careful reading of themanuscript and their constructive and detailed comments thatgreatly improved the quality of the paper.

APPENDIX-IIn this appendix we derive the pdf of the random variable ψ�

ψ� =��� ¯ρe

�¯ρd ϑ�−1

��h�−1�√

Λ+

�1− ρ2

dε� d

� ��� .

ψ� has the parameters associated with all three forms of feedbackimperfection. ρe determines the estimation quality, ρd determinesthe effect due to delay,

�1− ρ2

dε� d is the error term due

to feedback delay that is known to the receiver (because ofthe modeling approach presented in section II-B), and ϑ�−1 isthe inner product between the estimated and delayed channeldirection and its quantized version. Because of the analyticalcomplexity involved in deriving pψ� , the pdf of ψ�, we beginwith first writng ψ� as

ψ� = ρe ϕ ,

ϕ =��� ¯ρd ϑ�−1

��h�−1�√

Λ+

�1− ρ2

dε� d

��� . (36)

pdf of ϕ2:

We first derive pϕ2(z), and then use a simple transformationto get the pdf of ψ�. In ϕ,

��h�−1�√

Λ∼

2 x2t−1e−x2

Γ(t), x ≥ 0 , (37)

ε� d ∼ NC (0, 1) , (38)

and the pdf of ∆ = |ϑ�−1|2 (later in the derivation we will be

needing the pdf of ∆ not ϑ�−1) is

p∆(x) = 2B(t− 1)(1− x)t−2, 1− ω < x < 1,

ω = 2−B

t−1 . (39)

Note that ϕ has three random variables (ϑ�−1, ��h�−1�, andε� d), and all three are independent of each other. To begin withassume that ϑ is a constant (we will relax this assumption at alater stage). If X1 and X2 are statistically independent Gaussianrandom variables, each one with same variance σ2 and with thenon-centrality parameter s2 = m2

1 + m22 (m1 and m2 are the

means of X1 and X2 respectively), then the non-central chi-squared distribution, Z = X2

1 + X22 is given by

p(z|s) =1

2σ2e−

(s2+z)2σ2 I0

�√z s

σ2

�(40)

10

where I0(x) is the modified bessel function of 0th order [55].Let

σ =

�1− ρ2

d

2, (41)

s2 � y =ρ2

d∆ ��h�−1�

2

Λ. (42)

After a simple transformation, the conditional pdf of y is givenby

p(y|∆) =e−y/(ρ2

d ∆) yt−1

(ρ2d∆)t Γ(t)

, y ≥ 0. (43)

With σ defined in (41) and y defined in (42), (40) can be appliedto the present context to get pϕ2(z|y, ∆), the conditional pdf ofϕ2

pϕ2(z|y, ∆) =1

2σ2e−

(y+z)2σ2 I0

�√zy

σ2

�.

The conditional pdf of ϕ2 is given by

pϕ2(z|∆) =� ∞

−∞

pϕ2(z|y, ∆) p(y|∆) dy

=� ∞

−∞

12σ2

e−(y+z)2σ2

∞�

k=0

(zy)k

4kσ4k (k!)2

�e−y/(ρ2

d∆)yt−1

(ρ2d∆)t Γ(t)

�dy

= λ e−z

2σ2

∞�

k=0

L(k) zk, z > 0 (44)

where

λ =(1− ρ2

d)t−1

Γ(t) [1− ρ2d(1−∆)]t

,

L(k) =Γ(k + t)ρ2k

d∆k

(1− ρ2d)k [1− ρ2

d(1−∆)]k (k!)2

.

In the above derivation we used the infinite series representationfor I0(x):

Iα(x) =∞�

k=0

(x/2)α+2k

k!Γ(α + k + 1),

and the following identity [55]� ∞

0xne−ωx dx = n!ω−(n+1). (45)

We can further simplify pϕ2(z|∆) given in (44) by writing theinfinte series as

∞�

k=0

L(k) zk =∞�

k=0

(k + t− 1) · · · (k + 1)βk

k!

=dn

dβn

∞�

k=0

βk+n

k!

=dn

dβn

�βneβ

�

= eβ

�n�

l=0

n c ln p l βn−l

�

where

n = t− 1 , (46)

n c l =n!

l!(n− l)!,

n p l =n!

(n− l)!,

β =ρ2

d∆ z

(1− ρ2d) [1− ρ2

d(1−∆)]

.

After some simplification pϕ2(z|∆) can be written as

pϕ2(z|∆) =�1− ρ2

d

�n

Γ(t) [1− ρ2d(1−∆)]n+1 e−

z2σ2 +β

�n�

l=0

n c ln p l βn−l

�

=�1− ρ2

d

�n

e−z/[1−ρ2d(1−∆)]

Γ(t) [1− ρ2d(1−∆)]n+1

n�

l=0

n c l {n p lΓ(l + 1)}zn−l

Γ(l + 1)

�ρ2

d

1− ρ2d

�n−l

=(1− ρ2

d)n

[1− ρ2d(1−∆)]n

n�

l=0

nc lfl(z)�

ρ2d∆

1− ρ2d

�n−l

(47)

where

fl(z) =e−z/[1−ρ

2d(1−∆)] zn−l

[1− ρ2d(1−∆)]n−l+1 Γ(n− l + 1)

, 0 ≤ l ≤ n.

For sanity check the area under the pdf pϕ2(z|∆) can be verifiedto be one.� ∞

0pϕ2(z|∆) dz =

� ∞

0

(1− ρ2d)n

[1− ρ2d(1−∆)]n

n�

l=0

n c l fl(z)�

ρ2d∆

1− ρ2d

�n−l

=(1− ρ2

d)n

[1− ρ2d(1−∆)]n

n�

l=0

n c l

�ρ2

d∆

1− ρ2d

�n−l

=(1− ρ2

d)n

[1− ρ2d(1−∆)]n

�1 +

ρ2d∆

1− ρ2d

�n

= 1.

We now look at the case where ∆ is random. The pdf for ∆ isgiven in (14). We now integrate out the randomness due to ∆ toget pϕ2(z), the pdf of ϕ2.

pϕ2(z) =� ∞

−∞

pϕ2(z|∆)p∆(∆) d∆

=� 1

1−ω

�(1− ρ2

d)n

[1− ρ2d(1−∆)]n

n�

l=0

n c l fl(z)

�ρ2

d∆

1− ρ2d

�n−l�

�2B n (1−∆)n−1

�d∆

= 2Bn(1− ρ2d)n

n�

l=0

n c lzn−l

Γ(n− l + 1)

�ρ2

d

1− ρ2d

�n−l

� 1

1−ω

e−z

1−ρ2d(1−∆) (∆)n−l (1−∆)n−1

[1− ρ2d(1−∆)]2n−l+1

d∆

� �� �J(z)

. (48)

11

In order to evaluate J(z), let 1

[1−ρ2d(1−∆)] = α which implies

∆ = αρ2d−α+1αρ

2d

and d∆ = −dα

α2ρ2d

.

J(z) =� 1

1−ω

e−z/[1−ρ2d(1−∆)] ∆n−l (1−∆)n−1

[1− ρ2d(1−∆)]n−l+1

d∆

=(−1)n

�ρ2

d− 1

�n−l

(ρ2d)2n−l

� 1

11−ρ2

dω

e−zα

�1

ρ2d− 1

+ α

�n−l

(1− α)n−1 dα

=(−1)n

�ρ2

d− 1

�n−l

(ρ2d)2n−l

� 1

11−ρ2

dω

e−zα

2n−l−1�

p=0

cpαpdα . (49)

The co-efficients cp are calculated as follows. Let F (x) be afunction defined as follows:

F (x) = (d + x)s (1− x)r =s+r�

h=0

ch xh,

ch =1h!

dh (F (x))dxh

�����x=0

=1h!

h�

q=0

�hq

�Lr Ls(−1)h−qds−q, (50)

Lr =r!

(r − h + q)!if r ≥ (h− q) else Lr = 0,

Ls =s!

(s− q)!if s ≥ q else Ls = 0.

In the present context,

x = α, s = n− l, r = n− 1, and d =1

ρ2d− 1

.

With the help of the above function, J(z) can now be expressedas

J(z) =(−1)n

�ρ2

d− 1

�n−l

(ρ2d)2n−l

2n−l−1�

p=0

cp Ip (51)

where cp is calculated with the help of (50) and Ip can beevaluated as

Ip =� 1

11−ρ2ω

e−zα αp dα

=

�−e−zα

�αp

z+

p�

g=1

p (p− 1)..(p− g + 1)zg+1

αp−g

��1

L

=p�

g=0

p p g

�e−zLLp−g − e−z

�

zg+1

where

L =1

1− ρ2dω

(52)

where ω is defined in (39). Substituting (51) in (48), the finalclosed for pdf pϕ2(z) is given by

pϕ2(z) =2Bn(1− ρ2

d)n

ρ2n

d

n�

l=0

(−1)l n c l zn−l

Γ(n− l + 1)2n−l−1�

p=0

cp

�p�

g=0

p p g

�e−zL Lp−g − e−z

�

zg+1

�

=n�

l=0

2n−l−1�

p=0

δ�

g=0

�Kp1 f1(l, p, g,L, z) +

Kp2 f2(l, p, g,L, z)�

+Rn,z (53)

where

Rn,z =n�

l=0

2n−l−1�

p=n−l

p�

g=n−l

Kp3(−1)l n c l cp p p g

�e−zL Lp−g − e−z

�

Γ(n− l + 1) zg−n+l+1, (54)

δ =�

p if p ≤ n− l − 1n− l − 1 if p ≥ n− l

, (55)

f2(l, p, g,L, z) =Ln−l−g e−L z zn−l−g−1

Γ(n− l − g), (56)

f1(l, p, g,L, z) = f2(l, p, g, 1, z) , (57)Kp1 = cp 2B n (1− ρ2

d)n (−1)l+1 n c l

Γ(n− l − g) p p g

ρ2n

dΓ(n− l + 1)

, (58)

Kp2 = −Kp1Lp−n+l , (59)

Kp3 =2Bn(1− ρ2

d)n

ρ2n

d

, (60)

and cp’s can be calculated with the help of (50). L is definedin (52).

Simplification of pϕ2(z) :

Note that the form of pϕ2(z) given in (53) is separated into twoparts, terms with powers of z in the numerator and terms withpowers of z in the denominator. Rn,z defined in (54) capturesthe terms with powers of z in the denominator. The negativeexponent of z in Rn,z will make the performance analysisintractable. We now take a closer look and analytically provethat Rn,z = 0.

Rn,z = Kp3

n�

l=0

2n−l−1�

p=n−l

p�

g=n−l

(−1)l n c l cp p p g

�e−zL Lp−g − e−z

�

Γ(n− l + 1) zg−n+l+1.

Let g − n + l = e and p− n + l = f .

Rn,z = Kp3

n�

l=0

n−1�

f=0

f�

e=0

(−1)l n c l cf+n−l (n− l + f)!(n− l)! (f − e)!

�e−zLLf−e − e−z

ze+1

�. (61)

12

We now look at the co-efficient cn−l+f , which are evaluatedwith the help of (50)

cn−l+f =1

(n− l + f)!

n−l+f�

q=0

�n− l + f

q

�Lr Ls

(−1)n−l+f−q ds−q

Lr �= 0 if n− 1 ≥ (n− l + f − q) else Lr = 0Ls �= 0 if n− l ≥ q else Ls = 0

The two in-equalities in Lr and Ls in the above equations aresatisfied only if q = n− l, then cn−l+f is

cn−l+f =(n− 1)! (−1)f

f !(n− f − 1)!. (62)

The important point in the above equation is that cn−l+f isindependent of l. We can now write Rn,z in (61) as

Rn,z = Kp3

n�

l=0

(−1)l n c l

n−1�

f=0

(n− l + f)!(n− l)!

T (f)

= Kp3

n�

l=0

(−1)l n c l W(f)

where

T (f) =f�

e=0

cf+n−l

�e−zLLf−e − e−z

�

(f − e)! ze+1,

W(f) =n−1�

f=0

(n− l + f)!(n− l)!

T (f)

=n−1�

f=0

mf lf

where mf ’s are constants that are independent of index l, asshown in the next step we do not need to calculate themexplicitly. Rn,z can now be written as

Rn,z = Kp3

n−1�

f=0

mf

n�

l=0

(−1)l n c l lf

� �� �Cf

.

By using the following property of binomial co-efficientsn�

k=0

(−1)k n c k lb−1 = 0, n ≥ b ≥ 1,

it is clear thatCf = 0, n− 1 ≥ f ≥ 0,

and subsequently Rn,z = 0. This result is important becausethe presence of negative exponents of z in the pdf make theperformance analysis difficult. The final simplified form of pdfpϕ2(z) can now be written as

pϕ2(z) =n�

l=0

2n−l−1�

p=0

δ�

g=0

�Kp1f1(l, p, g,L, z) +

Kp2f2(l, p, g,L, z)�

(63)

where (variable - definition): n - (46), δ - (55), L - (52), Kp1 -(58), Kp2 - (59), f1(, , , , ) - (57), and f2(, , , , ) - (56).

pdf of ψ� :

Note that the random variable that is of interest to us ψ�. Fromearlier discussion ψ� is related to ϕ as ψ� = ρe ϕ. (63) is thepdf of ϕ2. Using transformation of random variables, pψ� , thepdf of ψ� can be shown to be given by

pψ�(x) =n�

l=0

2n−l−1�

p=0

δ�

g=0

�Kp1fa1(l, p, g,L, x) +

Kp2fa2(l, p, g,L, x)�

, x > 0, (64)

fa2(l, p, g,L, x) =2Ln−l−g e

−L x2

ρ2e x2(n−l−g)−1

ρ2(n−l−g)e Γ(n− l − g)

. (65)

fa1(l, p, g,L, x) = fa2(l, p, g, 1, x) (66)

where (variable - definition): n - (46), δ - (55), L - (52), Kp1 -(58), and Kp2 - (59).

pdfs of z1 and z2 :

z1 and z2 are the random variables defined in section.III-B,here we present the expressions for the pdfs of z1 and z2.

pz1(z1) =n�

l=0

2n−l−1�

p=0

δ�

g=0

�Kp1fs1(l, p, g,L, z1) +

Kp2fs2(l, p, g,L, z1)�

, (67)

fs2(l, p, g,L, z1) =H1

2 Ap

z1−

12 e−z1H2

D−2(n−l−g) (−√

z1H3) ,

fs1(l, p, g,L, z1) = fs2(l, p, g, 1, z1) .

pz2(z2) =n�

l=0

2n−l−1�

p=0

δ�

g=0

�Kp1fs3(l, p, g,L, z2) +

Kp2fs4(l, p, g,L, z2)�

, (68)

fs4(l, p, g,L, z1) =H1

2 (1−Ap)z2−

12 e−z2H2

D−2(n−l−g) (√

z2H3) ,

fs3(l, p, g,L, z1) = fs4(l, p, g, 1, z1) .

where the variables and the corresponding defining equation arelisted as pairs: L - (52), Kp1 - (58), Kp2 - (59), H1 - (23),H2 - (24), and H3 - (25), and Dp(l) is the parabolic cylinderfunction. Parabolic cylinder function has many representations,for analytical simplicity we choose to work with the followingrepresentation [55]:

Dp(l) = 2p2 e−

l24

� √π

Γ�

1−p

2

� 1F1

�−

p

2,12;l2

2

�−

√2π l

Γ�−p

2

� 1F1

�1− p

2,32;l2

2

� �(69)

where 1F1(·, ·; ·) is the confluent hypergeometric function ofthe first kind. Using the representation in (69) for the parabolic

13

cylinder function, (67) and (68) can be written as

pz1(z1) =p1(z1) + p2(z1)

Ap

, z1 > 0,

pz2(z2) =p1(z2)− p2(z2)

(1−Ap), z2 > 0,

p1(x) =n�

l=0

2n−l−1�

p=0

δ�

g=0

�Kp1fs5(l, p, g,L, x) +

Kp2fs6(l, p, g,L, x)�

, x > 0, (70)

fs6(l, p, g,L, x) = R1x−

12 e−xRh

1F1

�Rn,

12; xRc

�, (71)

R1 =H1√

π

2n−l−g+1 Γ�

1+2(n−l−g)2

� , (72)

Rh =(4H2 +H2

3)4

, (73)

Rn = n− l − g , (74)

Rc =H2

3

2, (75)

fs5(l, p, g,L, x) = fs6(l, p, g, 1, x) . (76)

p2(x) =n�

l=0

2n−l−1�

p=0

δ�

g=0

�Kp1fs7(l, p, g,L, x) +

Kp2fs8(l, p, g,L, x)�

, x > 0, (77)

fs8(l, p, g,L, x) = R2 e−xRh1F1

�Rn2,

32; xRc

�, (78)

R2 =H1H3

√2π

2n−l−g+1 Γ (n− l − g), (79)

Rn2 = n− l − g +12

, (80)

fs7(l, p, g,L, x) = fs8(l, p, g, 1, x) . (81)

APPENDIX-II

In this appendix we derive the analytical expressions forG1(ϕ) and G2(ϕ), which are used in the evaluation of Ap

E�Qm

�√2 γb z1

� �and (1 − Ap) E

�Qm

�√2 γb z2

��, where

m ∈ {1, 2}.

Derivation of G1(ϕ):

G1(ϕ) =∞�

y=0

Q��

2 γb y�

p1(y) dy, (82)

=n�

l=0

2n−l−1�

p=0

δ�

g=0

�Kp1fc1(l, p, g,L, ϕ) +

Kp2fc2(l, p, g,L,ϕ)�

(83)

where (variable - definition): n - (46), δ - (55), L - (52), Kp1 -(58), Kp2 - (59), Q(x) - (34), and p1(x) - (70).

fc2(l, p, g,L,ϕ) =∞�

y=0

Q��

2 γb y�

fs6(l, p, g,L, y) dy

= R1

∞�

y=0

Q��

2 γb y�

y−12 e−yRh

1F1

�Rn,

12; yRc

�dy

=R1

π

ϕ�

θ=0

dθ

∞�

y=0

y−12 e−y ( γb

sin2 θ+Rh)

1F1

�Rn,

12; yRc

�dy (84)

where (variable - definition): fs6(, , , , ) - (71), R1 - (72), Rh -(73), Rn - (74), and Rc - (75). With yRc = x , (84) becomes

fc2(l, p, g,L, ϕ) =R1

πR32c

ϕ�

θ=0

dθ

∞�

x=0

x−12 e−xS

1F1

�Rn,

12; x

�dx (85)

whereS =

γb

sin2 θRc

+Rh

Rc

. (86)

Notice that S > 1 for ρe < 1. To evaluate the above equation,we use the identity given below [55]:

∞�

r=0

rc−1 e−r d1F1 (a, c; r) dr = Γ(c) d−c

�d− 1

d

�−a

= Γ(c) d−c�1− d−1

�q

, Re c > 0, Re d > 1. (87)

In the present context, in (87),

c =12, d = S, and q = −Rn .

With the help of above identity, (85) becomes

fc2(l, p, g,L, ϕ) = M

ϕ�

θ=0

d−c�1− d−1

�q

dθ

where

d−1 = Rd

�sin2 θ

sin2 θ + c1

�,

M =R1 Γ

�12

�

πR32c

,

Rd =Rc

Rh

,

and

c1 =γb

Rh

. (88)

By using the generalized binomial series expansion,

(1− x)q =∞�

n=0

(−1)n pk,n

n!(x)n ,

pk,n = q (q − 1) · · · (q − n + 1).

14

fc2(l, p, g,L,ϕ) can now be written in a series of steps as 4

fc2(l, p, g,L, ϕ) = M

ϕ�

θ=0

∞�

n=0

Rn+ 1

2d

(−1)n pk,ndn+c

n!dθ

= M

ϕ�

θ=0

∞�

n=0

Rn+ 1

2d

(−1)n pk,n

n!�1− (1− d)

�n+c

dθ

= M

ϕ�

θ=0

∞�

n=0

Rn+ 1

2d

(−1)n pk,n

n!

∞�

n=0

(−1)n bk,n

n!

�1− d

�n

dθ

= M

∞�

n=0

Rn+ 1

2d

(−1)n pk,n

n!∞�

n=0

(−1)n bk,n

n!

n�

n1=0

(−1)n1

�nn1

�

D (ϕ, c1, n1) (89)

where

bk,n = (n + c) (n + c− 1) · · · (n + c− n + 1) ,

d =sin2 θ

sin2 θ + c1,

D (ϕ, c1, n1) =

ϕ�

θ=0

d dθ =

ϕ�

θ=0

�sin2 θ

sin2 θ + c1

�n1

dθ (90)

= π

�ϕ

π−

T

π

�c1

1 + c1

n1−1�

k=0

�2kk

�

1[4(1 + c1)]k

−2π

�c1

1 + c1

n1−1�

k=0

k−1�

j=0

�2kj

�(−1)j+k

[4(1 + c1)]ksin[(2k − 2j)T ]

2k − 2j

�,

0 ≤ ϕ ≤ 2π (91)

where

T =12

tan−1

�2�

c1(1 + c1) sin 2ϕ

(1 + 2 c1) cos 2ϕ− 1

�+

π

2�1− 2

�c1(1 + c1) sin 2ϕ

�(1 + 2 c1) cos 2ϕ

2

��.

Equation (89) gives the final expression for fc2(l, p, g,L,ϕ).To complete the calculation of G1(ϕ)in (83) we still needfc1(l, p, g,L,ϕ) which is given by

fc1(l, p, g,L,ϕ) = fc2(l, p, g, 1,ϕ) .

4Note that the convergence of the above series is not a problem since d−1 < 1.

Derivation of G2(ϕ):

G2(ϕ) =∞�

y=0

Q��

2 γb y�

p2(y) dy, (92)

=n�

l=0

2n−l−1�

p=0

δ�

g=0

�Kp1fd1(l, p, g,L,ϕ) +

Kp2fd2(l, p, g,L,ϕ)�

(93)

where (variable - definition): n - (46), δ - (55), L - (52), Kp1 -(58), Kp2 - (59), Q(x) - (34), and p2(x) - (77), and

fd2(l, p, g,L, ϕ) =∞�

y=0

Q��

2 γb y�

fs8(l, p, g,L, y) dy

= R2

∞�

y=0

Q��

2 γb y�

e−yRh

1F1

�Rn2,

32; yRc

�dy

=R2

π

ϕ�

θ=0

dθ

∞�

y=0

e−y ( γbsin2 θ

+Rh)

1F1

�Rn2,

32; yRc

�dy (94)

where (variable - definition): fs8(, , , , ) - (78), R2 - (79), Rh -(73), Rn2 - (80), and Rc - (75). With yRc = x ,

fd2(l, p, g,L,ϕ) =R2

πRc

ϕ�

θ=0

dθ

∞�

x=0

e−xS1F1

�Rn2,

32;x

�dx,

S is defined in (86). fd2(l, p, g,L,ϕ) can be evaluated using thefollowing infinite series expansion for the Gauss hypergeometricfunction

1F1 (a, b; r) dr =∞�

m=0

am rm

bm m!,

am = 1. a1 (a1 + 1) · · · (a1 + m− 1),a1 = Rn2 ,

bm = 1. b1 (b1 + 1) · · · (b1 + m− 1),

b1 =32

.

With the above series representation, fd2(l, p, g,L,ϕ) can nowbe written as

fd2(l, p, g,L,ϕ) =R2

πRc

∞�

m=0

am

bm

ϕ�

θ=0

dθ

∞�

x=0

e−xS xm

m!dx,

=R2

πRc

∞�

m=0

am

bm

ϕ�

θ=0

S−(m+1)dθ,

=R2

πRc

∞�

m=0

amRm+1d

D(ϕ, c1, m + 1)bm

(95)

where D (ϕ, c1, m + 1) is defined in (91) and c1 is definedin (88). Finally fd1(l, p, g,L,ϕ) is given by

fd1(l, p, g,L,ϕ) = fd2(l, p, g, 1,ϕ).

15

APPENDIX-III

In this section, for m ≥ 3, with the help of the ana-lytically tractable approximation of Q(x) given in (35), wederive closed-form expressions for Ap E

�Qm

�√2 γb z1

��and

(1−Ap)E�Qm

�√2 γb z2

��.

Ap E�Qm

��2 γb z1

��=

∞�

z1=0

Qm

��2 γb z1

�p1(z1) dz1

+∞�

z1=0

Qm

��2 γb z1

�p2(z1) dz1

= U1(γb) + U2(γb) (96)

where pz1(z1), p1(z1), and p2(z1) are defined in (67), (70),and (77) respectively and

U1(γb) =n�

l=0

2n−l−1�

p=0

δ�

g=0

�Kp1fe1(l, p, g,L) +

Kp2fe2(l, p, g,L)�

, (97)

fe2(l, p, g,L) = R1

∞�

z1=0

Qm

��2 γb z1

�z1−

12 e−z1Rh

1F1

�Rn,

12; z1Rc

�dz1

= R1

∞�

z1=0

∞�

k1=0

ck1 (2 γb)k12 z

k1−12

1

e−z1(m γb+Rh)1F1

�Rn,

12; z1Rc

�dz1

= R1

∞�

k1=0

ck1(2 γb)k12

F

�Rn,

12

, k1 +12

,Rc, mγb +Rh

�

where (variable - definition): n - (46), δ - (55), L - (52), Kp1 -(58), Kp2 - (59), R1 - (72), Rh - (73), Rn - (74), and Rc - (75),and

F (a, b, α, k, s) =∞�

x=0

xα−1e−s x1F1 (a, b; k x) dx

= Γ(α) s−α2F1

�a,α ; b ; ks−1

�,

[α > 0, |s| > |k|] . (98)

To complete the calculation of U1(γb) in (97) we still needfe1(l, p, g,L) which is given by

fe1(l, p, g,L) = fe2(l, p, g, 1) .

We now focus on U2(γb) in (96):

U2(γb) =n�

l=0

2n−l−1�

p=0

δ�

g=0

�Kp1fe3(l, p, g,L) +

Kp2fe4(l, p, g,L)�

, (99)

fe4(l, p, g,L) =∞�

z1=0

Qm

��2 γb z1

�p2(z1) dz1

= R2

∞�

z1=0

Qm

��2 γb z1

�e−z1Rh

1F1

�Rn2,

32; z1Rc

�dz1

= R2

∞�

z1=0

∞�

k1=0

ck1(2 γb)k12 z

k12

1

e−z1(mγb+Rh)1F1

�Rn2,

32; z1Rc

�dz1

= R2

∞�

k1=0

ck1(2 γb)k12

F

�Rn2,

32

,k1

2+ 1 ,Rc, mγb +Rh

�(100)

where (variable - definition): R2 - (79), and Rn2 - (80). Tocomplete U2(γb) in (99), fe3(l, p, g,L) is given by

fe3(l, p, g,L) = fe4(l, p, g, 1) .

Substituting (97) and (99) in (96) gives the final closed formexpression for Ap E

�Qm

�√2 γb z1

��. It is straightforward to

show that

(1−Ap) E�Qm

��2 γb z2

��= U1(γb)− U2(γb) . (101)

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