Characterizing the Effect of Channel Estimation Error on Distributed Reception with Hard Decision Exchanges

Transcription

1 Characterizing the Effect of Channel Etimation Error on Ditributed Reception with Hard Deciion Exchange Sabah Razavi and D. Richard Brown III Department of Electrical and Computer Engineering Worceter Polytechnic Intitute 100 Intitute Rd, Worceter, A {razavi,drb@wpi.edu Abtract Thi paper conider a ditributed wirele reception ytem with -ary phae hift keyed -PSK modulated ignal in which two or more receiver exchange quantized information about their obervation to approximate a receive beamformer and improve the ignal to noie ratio with repect to ingle-uer reception. To reduce the throughput requirement of ditributed reception, previou work on thi problem conidered the exchange of hard deciion among the node in the receive cluter with a imple form of linear combining called peudo-beamforming. The previou work alo aumed that the node in the receive cluter had perfect channel etimate. Thi paper generalize the previou work by analyzing the performance of peudobeamforming with imperfect channel etimate. While channel etimation error degrade the performance of both ideal receive beamforming no quantization and peudo-beamforming, the aymptotic analyi in thi paper reveal the omewhat urpriing reult that the SNR ratio between ideal receive beamforming and peudo-beamforming doe not depend on the amount of channel etimation error. The SNR ratio with channel etimation error i identical to the previouly derived SNR ratio without channel etimation error. Numerical example with a finite number of receiver are alo preented to confirm the analyi. Index Term Ditributed reception, receiver cooperation, beamforming, channel etimation error, aymptotic analyi. I. INTRODUCTION Thi paper conider a ditributed reception ytem a illutrated in Fig. 1. A ditant tranmitter emit digitally modulated ignal which are received by the N node in the receive cluter. Quantized verion of the received ignal at each node in the receive cluter are then exchanged among the node through a wirele local area network and are ubequently proceed and combined at one or more node in the receive cluter to increae the diverity and ignal to noie ratio [1] [3]. While ditributed reception ha been ued in a variety of context, e.g., aperture ynthei for radio atronomy and enor fuion, the focu of thi paper i on the ue of ditributed reception for improving the reception of digitally modulated ignal. A difficulty in practically implementing ditributed reception ytem, even with a modet number of receive node, i Thi work wa upported by National Science Foundation award CCF and CCF ditant tranmitter forward link Fig. 1. Ditributed reception cenario. fully-connected receive cluter that the throughput requirement of the local area network LAN ued by the receive cluter to exchange quantized verion of the received ignal can be prohibitive. For example, uppoe that the ditant tranmitter employ 8-PSK modulation with an information rate of 1 bit/ and a rate r /3 code. The ymbol are received at each receiver at a rate of 500 Kymbol/. For ideal receive beamforming, auming N 10 receiver and 16-bit quantization of the in-phae and quadrature obervation, the LAN would need to upport a throughput of at leat b/, not including overhead. otivated by thi difficulty, [1] [3] conidered the exchange of hard deciion among the node in the receive cluter. Under the ame parameter a the previou example, the required LAN throughput for exchanging hard deciion i only b/. It ha been hown that thi dramatic reduction in LAN/backhaul throughput requirement can be achieved without a ignificant lo of performance. For the low per-node SNR regime of interet with large receive cluter, aymptotic analyi of a uboptimal combining technique termed peudobeamforming howed that ditributed reception with hard deciion exchange perform within 1- db of ideal receive beamforming, depending on the digital modulation format [3]. A critical aumption in all of the pat work i that the receive cluter ha perfect channel etimate to facilitate combining of the hard deciion. Thi aumption i clearly optimitic, epecially in the low per-node SNR regime of interet with large receive cluter. It i of interet to undertand the effect of channel etimation error on ditributed reception. The effect of channel etimation error on ditributed reception ytem with hard deciion exchange and -ary phae hift

2 keyed modulation i particularly intereting ince the channel etimation error have two effect: i channel phae error caue increaed likelihood of hard deciion error and ii channel magnitude error caue combining error. Thi i the problem conidered in thi paper. The effect of channel etimation error on digital communication ytem i a claic problem [4] [6], and continue to be conidered in a variety of context. For example, [7] invetigate the effect of imperfect channel etimation on the bit-error-rate BER performance of amplify-and-forward AF relay-aited cooperative tranmiion. Related to the -PSK focu of thi paper, everal prior paper have invetigated the effect of imperfect carrier phae recovery on the performance of -PSK [8] [16]. Since our focu i on ditributed reception and on combining demodulated and quantized -PSK ignal, our analyi account for the effect of phae and magnitude error in the channel etimate at the receiver. The main contribution of thi paper i an aymptotic analyi of the effect of imperfect channel etimation in a ditributed reception ytem with -ary PSK modulation. Our analyi provide cloed-form SNR expreion for both ideal receive beamforming and a uboptimal hard-deciion combining technique termed peudo-beamforming. Our analyi how the expected effect that ditributed reception performance i degraded by channel etimation error, but alo reveal the omewhat urpriing reult that the SNR ratio between ideal receive beamforming and peudo-beamforming doe not depend on the amount of channel etimation error. Specifically, for QPSK, the SNR ratio i, correponding to a lo of approximately db. For -PSK with, the SNR ratio i 4, correponding to a lo of approximately 1 db. Since thee SNR ratio are independent of the amount of channel etimation error and are identical to the SNR ratio with no channel etimation error, our analyi reveal that channel etimation error caue the identical amount of performance degradation in ideal beamforming and peudo-beamforming ytem depite the fact that the channel etimation error manifet themelve quite differently in both ytem. The ret of the paper i organized a follow. Section II decribe the ytem model for the propoed cenario. In ection III we derive the channel etimation at the receiver. Section IV decribe our aymptotic SNR analyi for each technique with and without channel etimation error. In ection V the numerical reult from imulation i preented and ection VI i about the concluion. The proof for the lemma 1 and corollary 1 are given in Appendix A and B. II. SYSTE ODEL We aume a block tranmiion cenario with block of length n a in [3] and let N denote the number of receive node in the cluter. The complex forward link channel to receive node i in block m i denoted a h i [m] for i 1,..., N and the vector channel for block m i denoted a h[m] [h 1 [m],..., h N [m]]. Over each block, the forward link channel are aumed to be contant but may change block to block. For clarity of expoition and to explore the effect of channel phae and magnitude error on ditributed reception, we aume -PSK modulation in the forward link. The l th ymbol in block m i denoted a X[m, l] for l 1,..., n and i aumed to be drawn equiprobably { from the PSK alphabet X {x 1,, x a, ae j/, ae j4/,, ae j 1/. The energy per tranmitted ignal i denoted a E X[m, l] a. Given an additive white Gauian noie channel AWGN with power pectral denity / in the real and imaginary dimenion, the complex baeband ignal received at the i th receive node for the l th ymbol of block m can be written a U i [m, l] h i [m]x[m, l] + W i [m, l] 1 for i 1,..., N and l 1,..., n where W i [m, l] CN 0, i patially and temporally independent and identically ditributed i.i.d. proper complex Gauian baeband noie. We aume the noie variance i identical at each receive node. The quantity ρ i [m] hi[m] E correpond to the ignal-to-noie ratio SNR at receive node i for ymbol received in block m. To facilitate ditributed reception, it i aumed that the receive cluter ha an etablihed LAN backhaul, either adhoc or through infratructure uch a an acce point, and that LAN tranmiion are reliable. The LAN i alo aumed to upport broadcat tranmiion in which any ingle node can end a meage to all other node imultaneouly. To prevent any interruption in tranmiion over forward link, it i aumed that LAN and forward link operating frequencie differ from each other which enable the receive cluter to end and receive over the LAN, and alo receive ignal from tranmitter at the ame time. The LAN i alo aumed to upport a ufficient throughput for the exchange of hard deciion among all node in the receive cluter. III. CHANNEL ESTIATION Unlike the prior work in [1] [3], we do not aume h i [m] i known perfectly here. To facilitate etimation of h i [m] at receiver i, we aume ome of the ymbol in each tranmitted block are known. Suppoe X[m, 1],..., X[m, P ] are known, where P n. Then node i can etimate h i [m] by computing a leat quare olution to uch that U i [m, 1].. U i [m, P ] X[m, 1]. X[m, P ] h i [m] U i [m] X[m]h i [m] 3 ĥ i [m] XH [m]u i [m] 4

3 Subtituting U i [m] h i [m]x[m, l]+w i [m], we can write ĥ i [m] h i[m] + X H [m]w i [m] h i [m] + XH [m]w i [m] h i [m] + h i [m] 5 where h i [m] CN 0, ρ i a proper complex Gauian random variable with variance ρ in the real and imaginary dimenion. Since the training equence X[m] i known, we can determine ρ by computing var h i [m] E { X H [m]w i [m, l] X[m] X H [m] IN X[m] 0 6 where the lat reult follow from our -PSK aumption and the fact that the length of X[m] i P. IV. ASYPTOTIC SNR ANALYSIS In thi ection, we conider the cae where N and the per-node SNR goe to zero at a rate of 1 N o that the SNR of an ideal receive beamformer combiner i finite and bounded away from zero. We can uppre the block/ymbol indice and conider the calar obervation at receive node i a U i h i X + W i 7 where X i drawn from an -PSK contellation with X E. For our aymptotic analyi, we will aume ignal energy E E 1 /N, i.e., the tranmit power cale a 1/N, where E 1 i the per-ymbol tranmit energy with one receiver. We alo aume P NP 1, i.e., the training ignal length cale with N, where P 1 i the training ignal length with one receiver. Under thi aumption, note that i a contant. Since i alo fixed, the variance of the channel etimation error i contant. The following ubection analyze the performance of ideal ditributed receive beamforming and a uboptimal combining technique called peudo-beamforming with and without channel etimation error. A. Ideal Receive Beamforming: Perfect Channel Etimation The output of ideal receive beamformer at node i i realized by uing unquantized obervation U j and i defined a Y bf Y i ρi U j α j P j P where ρ i hi E E and α. h j U j 8 For the ideal receive beamformer, we have the vector obervation U hx + W. 9 Auming no channel etimation error, the ideal receive beamformer output i given a Y bf h H U h H hx + h H W. 10 The SNR of ideal receive beamforming conditioned on the channel realization can be computed a { E h H hx + h H W X SNR bf var {h H hx + h H W X h 4 E h H E{W W H h h E. 11 If we further aume an i.i.d. Rayleigh fading channel uch that h i CN 0, λ, then aymptotically we have h lim N N λ. The aymptotic SNR i then SNR bf NλE λe 1. 1 B. Ideal Receive Beamformer: Noiy Channel Etimation Now we conider ideal receive beamforming with channel etimate of the form ĥ h + h 13 where h CN 0, ρi. The ideal receive beamformer output with channel etimation error i given a Y bfe ĥh U ĥh hx + W h H h + hx + W h H hx + W + h H hx + W Y bf + Ỹbf. 14 Then, the SNR of ideal receive beamforming with channel etimation error conditioned on the channel realization can be computed a { E Y bf + h H hx + W X SNR bfe { var Y bf + h. 15 H hx + W X Note that h i independent of h and X. Since the channel etimate were generated from different obervation than the one ued in the SNR calculation, h i alo independent of W. Hence, { E Y bf + h H hx + W X E {Y bf X h E 16

4 and the numerator of thi expreion i unchanged from the cae with no channel etimation error. A for the denominator, ince Y bf and Ỹbf are independent, we have { var Y bf + Ỹbf X var {Y bf X { hh + var hx + W X h { hh + var hx + W X 17 We can compute the econd term a { hh { hh var hx + W X E hx + W hx + W H h { hh X E hx + W X { hh E hx + W hx + W H h X 18 where the econd equality follow from the fact that h i zero mean and independent of the other term in the expectation. We can further compute { hh { hh var hx + W X E E hh H h X { hh W W H h X + E E h H { h hh E X E h { hh ρ + E W W H h X { hh + E W W H h X h P { hh h + E W W H h X 19 The final expectation can be olved with iterated expectation ince h and W are independent. We can write { hh E W W H h { hh { X E E W W H X, h h X { hh E I h X E { hh h X Nρ N 0 N. 0 Putting it all together, we have { var Y bf + h H hx + W X and hence h + h P + N 0 N. 1 h E SNR bfe + N0 P +. N h Aymptotically, ince P grow proportionally with N and i fixed, the middle term in the denominator vanihe. So for large N with vanihing per-node SNR we can write SNR bfe h E. 3 + N h h oreover, ince lim N N λ, E E1 N, and P NP 1, it can be eaily obtained that SNR bfe 1 + λe 1 λp 1 E 1. 4 The reult in 11 and 3 allow u to compute the penalty of channel etimation error in an ideal receive beamformer a N a P bf SNR bf SNR bfe 1 + λp 1 E 1 C. Peudo-beamforming: Perfect Channel Etimation. 5 Peudo-beamforming i a imple but ub-optimal combining technique where 8 i performed on the hard deciion from each node. Specifically, the peudo-beamformer combiner output i Y pbf Y i j P ρi V j α j P h j V j 6 where V j X for all j and are conditionally independent given the tranmitted ymbol. The aymptotic SNR of peudo-beamforming for variou modulation format wa analyzed in [3]. The main reult are ummarized here. Firt, the conditional mean of -PSK hard deciion can be calculated a ρj in E[V j X x l ] x l. 7 Second, the conditional variance of with -PSK hard deciion in the low per-node SNR regime can be calculated a var[v j X x l ] a. 8 Thee reult allow u to compute the conditional mean and variance of the peudo-beamformer output with -PSK hard deciion. The conditional mean can be computed a E[Y pbf X x l ] α a in h x l. 9 Similarly, the conditional variance of the peudo-beamformer output with -PSK hard deciion in the low per-node SNR regime can be computed a var[y pbf X x l ] α a h 30 where we ued the fact that ρ j hj a N0 and j h j h. Thee reult then imply SNR pbf E[Y pbf X x l ] var[y pbf X x l ] in h E 31 4 in SNR bf. 3 4

5 With QPSK, we have 4 and in 4. Thi then implie SNR QPSK pbf SNR bf. 33 For large, we can ue mall angle approximation which mean we can ay in and it reult in in 4 4. Hence lim SNR PSK pbf 4 SNR bf. 34 D. Peudo-beamforming: Noiy Channel Etimation The effect of channel etimation error on peudobeamforming ha two effect: i channel phae error caue increaed likelihood of hard deciion error and ii channel magnitude error caue combining error. To model the effect of channel etimation error on the deciion variable at an individual receiver, we firt define the perfect and noiy channel etimate, repectively, a h j h j e jθ 35 ĥ j ĥj e j ˆθ. 36 Lemma 1 provide expreion for the conditional mean and variance of hard deciion at an individual receiver with low per-node SNR in preence of channel etimation error. Lemma 1. For a forward link with P SK modulation with 4 and even, at low per-node SNR we have ρj h in E[V j X x l ] E[ ĥ ] x l 37 and the variance i var[v j X x l ] a. 38 A proof for thi lemma i given in Appendix A. A it can be een from the proof, the calculation of mean and variance of hard deciion do not depend on any pecific phae error ditribution. In a low per-node SNR regime and for a large N, ince ρ j become very mall, it i expected that the mean goe to zero. Alo, the variance given in 38 erve a an upper bound for the variance of hard deciion a N get large. The next tep i to find the mean and variance of the peudobeamformer output in order to be able to calculate it SNR performance. The peudo-beamformer output with imperfect channel etimate i given a Y pbfe α j N ĥj V j. 39 Corollary 1 ue the reult obtained from lemma 1 to provide expreion for the conditional mean and variance of the peudo-beamformer output. Corollary 1. Given the channel information h j and input ymbol X x l, the mean and variance of peudo-beamformer output can be computed a E[Y pbfe X x l ] α a in h x l 40 and the variance i var[y pbfe X x l ] α a h + N 41 A proof for thi corollary i provided in Appendix B. With the reult of Corollary 1, we now can compute the SNR of peudo-beamforming with channel etimation error a SNR pbfe E[Y pbfe X x l ] var[y pbfe X x l ] α a in 4 h 4 x l α a h + N0N in h E N h in λe λp 1 E 1 where the final reult aume N with correpondingly vanihing per-node SNR. In light of 4, we can write SNR pbfe in SNR bfe Hence, the SNR gap between peudo-beamforming with channel etimation error and ideal receive beamforming with channel etimation error are identical to the cae without channel etimation error. Uing 31 and 43, the SNR penalty of channel etimation error with peudo-beamforming can be expreed a P pbf SNR pbf 1 + SNR pbfe λp 1 E 1 45 which i identical to 5. Thi how the omewhat urpriing reult that the SNR ratio between ideal receive beamforming and peudo-beamforming doe not depend on the amount of channel etimation error. In other word, the SNR ratio with channel etimation error i identical to the SNR ratio without channel etimation error, a derived in [3]. V. NUERICAL RESULTS In thi ection the reult from the imulation are preented. In thi imulation a QPSK modulation i choen for the forward link between ingle tranmitter and the receive cluter. The number of receive node inide the cluter are N [ ]. The number of iteration for each channel/noie realization i choen to be 1000 and the per-ymbol tranmit energy with one receiver E 1 10 and training ignal length with one receiver P 1 1. The magnitude of each ymbol i a and the number of payload ymbol per block i Q 100. The total noie power 5 and channel variance in real and imaginary dimenion i λ. Fig. how the comparion of the SNR between ideal receive beamforming and peudo-beamforming each with and without channel etimation error.

6 9.5 the hard deciion approache to the upper bound obtained from the theoretical reult. 9 SNR db ideal BF 6 ideal BF with channel etimation error peudo BF peudo BF with channel etimation error number of receiver N Fig.. Comparion of the SNR between ideal receive beamforming and peudo-beamforming with and without channel etimation error. The dotted line are the calculated SNR for large N in each cenario. The reult from Fig. confirm our proof that the ratio of the SNR between ideal receive beamforming and peudobeamforming in both cae of perfect and noiy channel etimation are equal to and the SNR in each cae converge to the calculated limit for large N. Fig. 3 how the comparion of the penaltie between the ideal and peudo beamformer. It can be een that, the penalty term in both cae converge to the ame number ince the SNR ratio in each cae, a hown in equation 5 for an ideal beamformer and 45 for a peudo beamformer are the ame. channel etimation error penalty db number of receiver N ideal BF peudo BF Fig. 3. Comparion of the penaltie between the ideal and peudo beamformer. Fig. 4 how the mean and variance of the hard deciion when there i channel etimation error. The reult from the figure how that, the calculated mean of the hard deciion cloely follow the numerical reult. Alo, the variance of ean Value Variance Value number of receiver N Simulation Reult Theoretical Reult number of receiver N Simulation Reult Theoretical Upper Bound Fig. 4. ean and variance of the hard deciion when there i channel etimation error. VI. CONCLUSION In thi paper we ued theoretical calculation, aymptotic analyi and numerical reult from imulation, to obtain and characterize the effect of imperfect channel etimation in a ditributed reception ytem with -PSK modulation. A mentioned in the paper, channel etimation error had two effect, channel phae error and channel magnitude error, which our analyi had accounted for both of thee effect in the channel etimation proce at the receiver. In our analyi, phae error did not have a pecific ditribution and our reult are valid for any phae error ditribution. Uing theoretical computation, we derived cloed-form expreion for the SNR of both ideal receive beamforming and peudo-beamforming. A it wa expected, the reult of our analyi how that channel etimation error degrade the performance of ditributed reception with both ideal and peudo-bemforming technique by almot 0.5 db. The intereting outcome of our analyi wa that, the SNR ratio between ideal receive beamforming and peudo-beamforming doe not depend on the amount of channel etimation error and are identical to the SNR ratio with no channel etimation error. So, our analyi how, channel etimation error caue the ame amount of performance degradation in ideal beamforming and peudo-beamforming ytem depite the fact that the channel etimation error manifet themelve quite differently in both ytem. Alo, imulation reult confirmed our calculation for the mean and variance of hard deciion with channel etimation error and alo, our reult for the penalty term in both ideal and peudo-beamforming ytem. APPENDIX A PROOF OF LEA 1 Proof. To be able to find the mean and variance of hard deciion, the ditribution of deciion variable phae at the

7 receiver hould be calculated. The deciion variable with no etimation error would be U j e jθ h j X + e jθ W j 46 and with etimation error would be U je e j ˆθh j X + e j ˆθW j 47 if we define θ e θ ˆθ and replace the h j with it polar format defined in 35 we get U je h j X + e jθ W j e jθe Uj e jθe 48 If we replace the deciion variable with their polar format we get θ U je θ U j + θ e 49 Since we already have the ditribution of θ U j from 11 in [17], we can derive the ditribution of θ U je by convolving the ditribution of θ U j and θ e. So, fθ U je X x 1 fθ U j X x 1 fθ e 1 ρ e ρ j j + coθ U je θ e e ρ j in θ Uje θ e 1 Q ρ j co θ U je θ e fθ e dθ e 50 where fθ e i the ditribution of θ e and could have any ditribution. Uing θ U je ditribution, tranition probability or the probability of deciding V j x m given X x 1, can be expreed a p m,1 m 1 m 3 fθ U je X x 1 dθ U je 51 In a low per-node SNR regime, we can calculate a firt-order Taylor erie expanion of p m,1 at ρ j 0 by computing m 1 p m,1 ρj0 m 3 m 1 m 3 fθ U je X x 1 ρj0 dθ U je [ 1 ] fθ e dθ e dθ U je 1 5 The expreion in the bracket i equal to 1 ince it i the integral of a ditribution. For the econd term we have p m,1 ρ ρj0 j m 1 m 3 m 1 m 3 [ 1 m 1 m 3 fθ ρ U je X x 1 j coθ U je θ e fθ e dθ e dθ U je coθ U je E[coθ e ] ρj0 dθ U je ] + inθ U je E[inθ e ] dθ U je 53 Channel etimation ĥ in polar format can be written a ĥ ej ˆθ h e jθ + h e j θ. Alo, h i given and h CN 0, ρ. Then, from expectation of real and imaginary part of ĥ, repectively, we have h coθ E[coˆθ] E[ ĥ ] 54 E[inˆθ] h inθ E[ ĥ ] 55 Uing θ e θ ˆθ and getting the expectation of coθ e, we would have E[coθ e ] h E[ ĥ ] 56 E[inθ e ] 0 57 Now if we ubtitute thee reult into equation 53 we would have [ ] m 1/ p m,1 ρ 1 ρj0 j m 3/ coθ U je h E[ ĥ ] dθ U je h in [ ] m 1 co 58 E[ ĥ ] So, in a low pre-node SNR regime with ρ j mall, we have p m,1 1 + h in [ ] m 1 co ρ j 59 E[ ĥ ] Under the aumption that 4 i even, we can compute the conditional expectation a follow E[V j X x 1 ] x m p m,1 m1 { ae jm 1/ 1 h in + m1 E[ ĥ ] [ ] m 1 co ρ j aρ j h in E[ ĥ ] / m 1 co m1 ρj h in E[ ĥ ] x 1 60 The conditional variance can be computed imilarly a var[v j X x 1 ] E[ V j X x 1 ] E[V j X x 1 ] a ρj h in E[ ĥ ] a 61 Since ρ j i mall under low per-node SNR aumption, we can dicard the term with ρ j, o we get var[v j X x 1 ] a 6

8 APPENDIX B PROOF OF COROLLARY 1 Proof. By having the mean and variance of hard deciion we can now calculate the mean and variance of peudobeamformer output. The peudo-beamformer ue the etimated channel magnitude to compute the combiner output Y pbfe α j N ĥj V j 63 Therefore the mean of the peudo-beamformer output i E[Y pbfe X x l ] α j N E[ ĥj X x l ]E[V j X x l ] by replacing ρ j : hj a N0 would have α in ρ j h j x l 64 j N and etting j N h j h we E[Y pbfe X x l ] α a in h x l 65 Alo, the variance of the peudo-beamformer output can calculate a follow var[y pbfe X x l ] α j N var[ ĥj V j X x l ] α j N E[ ĥj ]E[V j X x l ] E[ ĥj ] E[V j X x l ] 66 the econd term can be et equal to zero ince in a low pernode SNR regime E[V j X x l ] 0. Then we would have var[y pbfe X x l ] α a j N E[ ĥj ] 67 To obtain E[ ĥj ] we have to ue the fact that ĥj h j coθ j + h j co θ j 68 + h j inθ j + h j in θ j 69 After implifying the above equation and getting the expectation of both ide, we have E[ ĥj ] h j + E[ h j ] h j + 70 by replacing it back in the equation 67 the variance of peudo-beamformer output i obtained. var[y pbfe X x l ] α a h + N 0N 71 REFERENCES [1] D. R. Brown,. Ni, U. adhow, and P. Bidigare, Ditributed reception with coarely-quantized obervation exchange, in Information Science and Sytem CISS, th Annual Conference on, arch 013, pp [] R. Wang, D. R. Brown,. Ni, U. adhow, and P. Bidigare, Outage probability analyi of ditributed reception with hard deciion exchange, in Signal, Sytem and Computer, 013 Ailomar Conference on, Nov 013, pp [3] D. Brown, U. adhow,. Ni,. Rebholz, and P. Bidigare, Ditributed reception with hard deciion exchange, Wirele Communication, IEEE Tranaction on, vol. 13, no. 6, pp , June 014. [4]. Stojanovic, J. G. Proaki, and J. A. Catipovic, Analyi of the impact of channel etimation error on the performance of a deciionfeedback equalizer in fading multipath channel, IEEE Tranaction on Communication, vol. 43, no. /3/4, pp , Feb [5] X. Tang,. S. Alouini, and A. J. Goldmith, Effect of channel etimation error on m-qam ber performance in rayleigh fading, IEEE Tranaction on Communication, vol. 47, no. 1, pp , Dec [6]. edard, The effect upon channel capacity in wirele communication of perfect and imperfect knowledge of the channel, IEEE Tranaction on Information Theory, vol. 46, no. 3, pp , ay 000. [7] S. Han, S. Ahn, E. Oh, and D. Hong, Effect of channel-etimation error on BER performance in cooperative tranmiion, IEEE Tranaction on Vehicular Technology, vol. 58, no. 4, pp , ay 009. [8] A. Chandra, D. Biwa, and C. Boe, BER performance of coherent PSK in rayleigh fading channel with imperfect phae etimation, in Recent Trend in Information, Telecommunication and Computing ITC, 010 International Conference on, arch 010, pp [9] V. K. Prabhu, PSK performance with imperfect carrier phae recovery, IEEE Tranaction on Aeropace and Electronic Sytem, vol. AES-1, no., pp , arch [10] W. C. Lindey, Phae-hift-keyed ignal detection with noiy reference ignal, IEEE Tranaction on Aeropace and Electronic Sytem, vol. AES-, no. 4, pp , July [11] P. Y. Kam, S. K. Teo, Y. K. Some, and T. T. Tjhung, Approximate reult for the bit error probability of binary phae hift keying with noiy phae reference, IEEE Tranaction on Communication, vol. 41, no. 7, pp , Jul [1]. K. Simon and. S. Alouini, Simplified noiy reference lo evaluation for digital communication in the preence of low fading and carrier phae error, IEEE Tranaction on Vehicular Technology, vol. 50, no., pp , ar 001. [13] I. A. Falujah and V. K. Prabhu, Performance analyi of PSK ytem in the preence of low fading, imperfect carrier phae recovery, and AWGN, in Electrical and Computer Engineering, 005. Canadian Conference on, ay 005, pp [14] K. Bucket and. oeneclaey, Effect of random carrier phae and timing error on the detection of narrowband -PSK and bandlimited DS/SS -PSK ignal, IEEE Tranaction on Communication, vol. 43, no. /3/4, pp , Feb [15] C.. Lo and W. H. Lam, Error probability of binary phae hift keying in nakagami-m fading channel with phae noie, Electronic Letter, vol. 36, no. 1, pp , Oct 000. [16] F. Fan and L.-. Li, Effect of noiy phae reference on coherent detection of band-limited offet-qpsk ignal, IEEE Tranaction on Communication, vol. 38, no., pp , Feb [17]. Guroy, On the low-snr capacity of phae-hift keying with harddeciion detection, in Information Theory, 007. ISIT 007. IEEE International Sympoium on, June 007, pp