On Transmit Beamforming for MISO-OFDM Channels With Finite-Rate Feedback
aa r X i v : . [ c s . I T ] O c t IEEE TRANSACTIONS ON COMMUNICATIONS 1
On Transmit Beamforming for MISO-OFDMChannels With Finite-Rate Feedback
Kritsada Mamat and Wiroonsak Santipach
Member, IEEE,
Abstract —With finite-rate feedback, we propose two feedbackmethods for transmit beamforming in a point-to-point MISO-OFDM channel. For the first method, a receiver with perfectchannel information, quantizes and feeds back the optimal trans-mit beamforming vectors of a few selected subcarriers, which areequally spaced. Based on those quantized vectors, the transmitterapplies either constant, linear, or higher-order interpolation withthe remaining beamforming vectors. With constant interpolation,we derive the approximate sum achievable rate and the optimalcluster size that maximizes the approximate rate. For linearinterpolation, we derive a closed-form expression for the phaserotation by utilizing the correlation between OFDM subcarriers.We also propose a higher-order interpolation that requires morethan two quantized vectors to interpolate transmit beamformers,and is based on existing channel estimation methods. Numericalresults show that interpolation with the optimized cluster size canperform significantly better than that with an arbitrary clustersize. For the second proposed method, a channel impulse responseis quantized with a uniform scalar quantizer. With channelquantization, we also derive the approximate sum achievable rate.We show that switching between the two methods for differentfeedback-rate requirements can perform better than the existingschemes.
Index Terms —Multiple-input single-output (MISO), OFDM,transmit beamforming, feedback, RVQ, beamforming interpo-lation, optimal cluster size, channel quantization.
I. I
NTRODUCTION
Equipping a transmitter and/or a receiver with multiple an-tennas creates a multiantenna wireless channel whose capacitydepends on the channel information available at the transmitterand/or receiver. In multiantenna channels, transmit beamform-ing has been shown to increase an achievable rate by directingtransmit signal toward the strongest channel mode [1]. Withchannel information, the receiver can compute the optimalbeamforming vector that maximizes achievable rate and feedsthe vector back to the transmitter. Due to a finite feedbackrate, the beamforming vector needs to be quantized. Severalquantization schemes and codebooks have been proposed andanalyzed, and the corresponding performance was shown todepend on the codebook design and the number of availablefeedback bits [2], [3, see references therein]. In this work,
This work was supported by the 2010 Telecommunications Research andIndustrial Development Institute (TRIDI) scholarship and joint funding fromthe Thailand Commission on Higher Education, Thailand Research Fund, andKasetsart University under grant MRG5580236.The material in this paper was presented in part at the Electrical Engineer-ing/Electronics, Computer, Telecommunications and Information TechnologyConference (ECTI), Huahin, Thailand, May 2012, and the IEEE InternationalConference on Communications (ICC), Budapest, Hungary, June 2013.The authors are with the Department of Electrical Engineering; Facultyof Engineering; Kasetsart University, Bangkok, 10900, Thailand (email:[email protected]; [email protected]). we consider transmit beamforming for multiple-input single-output (MISO) orthogonal frequency-division multiplexing(OFDM).In MISO-OFDM, a wideband channel is converted intoparallel narrowband subchannels. For each subchannel or sub-carrier, the optimal beamforming vector is different and needsto be quantized at the receiver and fed back to the transmitter.The total number of feedback bits required increases with thenumber of subcarriers, which can be large. References [4]–[18] have proposed to reduce the amount of feedback whilemaintaining performance. Due to high channel correlation intime and frequency domains, feedback of transmit precodingmatrices across time and subcarriers can be compressed.References [16], [17] proposed to compress feedback witheither recursive or trellis-based encodings.In [4], [9], [15], [18], the optimal transmit beamformingvectors of selected subcarriers, which are a few subcarriersapart, are quantized while the remaining ones are approxi-mated to equal the quantized vector of the closest subcarrier.The remaining transmit beamforming vectors are proposedto be linearly interpolated in [5], [12] and spherically inter-polated in [6], [11], [13]. In [10], the authors proposed toquantize the averaged optimal transmit beamformer in eachcluster termed mean clustering. Geodesic-based interpolationof transmit precoding matrices was also proposed in [10]and was extended to multiuser channels in [14]. In [8],each subcarrier cluster uses the same beamforming vector orprecoding matrix, which is searched from a subcodebook thatcontains entries close to the beamformer or precoder in theadjacent cluster. Hence, there is some saving in feedback bits.Most of the works mentioned proposed to use either the sameor interpolated beamforming vectors for a group or cluster ofadjacent subcarriers since subcarriers are highly correlated in afrequency-selective channel. However, none has analyzed theoptimal cluster size and the associated performance.Given a limited feedback rate, we propose to quantize theoptimal beamforming vector at every few subcarriers with therandom vector quantization (RVQ) codebook proposed by [2]and to either use the same quantized vector for the wholesubcarrier cluster or interpolate the remaining beamformingvectors in the cluster from the quantized vectors. For thefirst proposed method termed constant interpolation, we derivean approximate sum achievable rate over all subcarriers. Theanalytical approximation can predict the performance trendwell and the optimal cluster size accurately. The optimalcluster size depends mainly on the available feedback rate,and on how frequency-selective the current channel is.For linear interpolation, we propose a closed-form expres-
IEEE TRANSACTIONS ON COMMUNICATIONS sion for the phase-rotation parameter based on the correla-tion between the transmit beamformers of subcarriers in thecluster. In earlier work by [5], the parameter was exhaus-tively searched. Our modified linear interpolation requiresfewer minimum feedback bits than that in [5]. In [12], theexpression for the phase-rotation parameter was also proposedand is based on a chordal distance between two quantizedbeamforming vectors of the adjacent clusters. However, ourproposed phase rotation combined with the optimized clustersize outperforms the phase rotation proposed by [12].For higher-order interpolation, our method is based onearlier works on comb-type pilot based channel estimationin OFDM [19]–[21]. Three or more quantized beamformingvectors from adjacent clusters are used to interpolate allbeamforming vectors in one cluster. The number of phase-rotation parameters increases with the order of the interpo-lation. The set of phase-rotation parameters that maximizessum achievable rate in a cluster can be searched from thecodebook proposed by [5]. Reference [6] modified the second-order channel estimation in [21] to interpolate transmit beam-forming vectors for subcarriers in a multiple-input multiple-output (MIMO)-OFDM channel. In [13] the authors basedtheir method from the work by [19] to interpolate transmitbeamformers for subcarriers, but without phase rotations. Thelack of phase-rotation parameters degrades significantly theperformance of the method in [13]. Our numerical exampleshows that the higher-order interpolation with our optimizedcluster size results in a good performance in a high feedback-rate regime.When the feedback rate is high, we propose to quantizethe channel impulse response with a uniform scalar quantizerand derive the approximate sum rate for MISO channels.The scalar quantization used in the proposed method is lesscomplex than the vector quantization used in [7]. The proposedscalar quantization of the channel impulse response is shownto perform well with a high feedback rate. Similar results wereobserved by [2] where the optimal beamformer and not thechannel response was scalar quantized. We note that [22] alsoproposed to scalar quantize a channel impulse response, butthe resulting sum rate was not analyzed.Apart from what was presented earlier in [23], [24], herewe show details of all proofs and update the derivation of theachievable rate approximation in Proposition 1. We compareour proposed feedback methods with several existing ones inthe literature and show that selecting the optimal cluster sizethat maximizes the sum rate can significantly improve thesum rate. Higher-order interpolation with quantized transmitbeamforming is also proposed.This paper is organized as follows. Section II describeschannel and feedback models as well as formulates the finitefeedback-rate problem. We propose beamforming interpolationmethods and analyze the optimal cluster size in Section III.Direct quantization of channel impulse response and its per-formance analysis are shown in Section IV. Numerical resultsand conclusions are in Sections V and VI, respectively. Finally,all proofs are in appendices. II. S
YSTEM M ODEL
We consider a point-to-point, discrete-time, MISO-OFDMchannel with N subcarriers. A transmitter is equipped with N t antennas while a receiver is equipped with a singleantenna. We assume that the transmit antennas are placedsufficiently far apart that they are independent. For eachtransmit-receive antenna pair, a transmitted signal propagatesthrough a frequency-selective Rayleigh fading channel withorder L . Applying a discrete Fourier transform (DFT), thefrequency response for the n th subcarrier and the n t th transmitantenna is given by h n,n t = L − X l =0 g l,n t e − j πlnN (1)where g l,n t is a complex channel gain for the l th path betweenthe n t th transmit and receive antenna pairs. Assuming a richscattering, g l,n t for all L paths and all N t transmit antennasare independent complex Gaussian distributed with zero meanand variance E | g l,n t | . In this work, we assume a uniformpower delay profile for which the power of each path is thesame and the total channel power for each transmit-receiveantenna pair is one. Hence, E | g l,n t | = L . Let h n denote an N t × channel vector of the n th subcarrier, whose entry is h n,n t shown in (1). Thus, h n = [ h n, h n, · · · h n,N t ] T . (2)Assuming a transmit beamforming or a rank-one precoding,the received signal on the n th subcarrier is given by r n = h † n v n x n + z n , ≤ n ≤ N, (3)where v n is an N t × unit-norm beamforming vector, x n is atransmitted symbol with zero mean and unit variance, and z n isan additive white Gaussian noise with zero mean and variance σ z . With perfect channel information at the transmitter, theoptimal transmit precoding that maximizes an achievable ratefor MISO channel is rank-one. This fact motivates us touse a rank-one precoding or beamforming. A resulting sumachievable rate over N subcarriers is given by C = N X n =1 E (cid:2) log(1 + ρ | h † n v n | ) (cid:3) (4)where the expectation is over the distribution of h n . Weassume a uniform power allocation for all subcarriers andhence, the background signal-to-noise ratio (SNR) for eachsubcarrier ρ = 1 /σ z .From (4), we note that the sum achievable rate is a functionof transmit beamforming vectors { v , v , . . . , v N } . A receiverwith perfect channel information can optimize the sum achiev-able rate over the transmit beamforming vectors and send theselected beamforming vectors to the transmitter via a feedbackchannel. Since the feedback channel between the receiverand the transmitter has a finite rate, quantizing the transmitbeamforming vectors is required. In this study we apply arandom vector quantization (RVQ) codebook whose entriesare independent, isotropically distributed vectors to quantizea transmit beamforming vector. RVQ is simple, however has AMAT AND SANTIPACH: ON TRANSMIT BEAMFORMING FOR MISO-OFDM CHANNELS WITH FINITE-RATE FEEDBACK 3 been shown to perform close to the optimum codebook [2],[25].We assume B total feedback bits per update. For anequal-bit-per-subcarrier allocation, each beamforming vectoris quantized with B/N bits. Let us denote the RVQ codebookby V = { w , w , . . . , w B/N } with B/N entries. The receiverselects for the n th subcarrier the entry in the codebook thatmaximizes an instantaneous achievable rate as follows: ˆ v n = arg max w ∈V log(1 + ρ | h † n w | ) (5) = arg max w ∈V | h † n w | (6)and the associated achievable rate for the n th subcarrier isgiven by C n = E (cid:2) log(1 + ρ | h † n ˆ v n | ) (cid:3) (7) = E (cid:2) log(1 + ρ k h n k | ¯ h † n ˆ v n | ) (cid:3) . (8)where ¯ h n = h n / k h n k is a unit-norm channel vector thatpoints in the same direction as h n . Evaluating (8) was shownby [25]. We note from (8) that the achievable rate dependson the number of feedback bits per subcarrier, which couldbe small due to a large number of subcarriers in a practicalOFDM system. Hence, this may result in a large quantizationerror, which leads to a substantial performance loss.III. I NTERPOLATING T RANSMIT B EAMFORMING V ECTORS
Feeding back transmit beamforming vectors of all sub-carriers requires quantizing
N N t complex coefficients andthus, a large number of feedback bits. We note that adjacentsubcarriers in OFDM are highly correlated since the number ofchannel taps is much lower than that of subcarriers ( L ≪ N ).The optimal transmit beamformers, which depend on channelmatrices, are also highly correlated. In this section, we proposebeamforming interpolation of different orders to reduce thenumber of feedback bits while maintaining the performance.First we evaluate a squared correlation between normalizedchannel vectors of subcarrier n and n + q defined by E (cid:2) | ¯ h † n ¯ h n + q | (cid:3) = E (cid:20) | h † n h n + q | k h n k k h n + q k (cid:21) . (9)Evaluating (9) is not tractable for a finite-size system. Hence,we approximate the average squared correlation as follows. Lemma 1:
A squared correlation between the n th and n + q th normalized channel vectors is approximated as follows: E (cid:2) | ¯ h † n ¯ h n + q | (cid:3) ≈ L + N t ϕ ( q ) L N t + ϕ ( q ) (10) , ψ ( q, N t ) (11)where ϕ ( x ) = sin( πxLN )sin( πxN ) . (12)The proof of Lemma 1 is shown in Appendix A.As subsequent numerical example in Section V will showthat approximation in Lemma 1 closely predicts the result ofa finite-size system. The correlation in (10) depends on L , N , N t , and most importantly, q , which indicates how far apart the two channel vectors are. When q → , ϕ ( q ) → L andthe squared correlation becomes k ¯ h n k → . We note that thenumber of channel taps L and channel impulse response canbe accurately estimated as shown in [26]. A. Constant Interpolation
In the first method, we group adjacent contiguous subcarri-ers into a cluster and apply the same quantized beamformingvector for all subcarriers in the cluster. We denote the numberof contiguous subcarriers in one cluster by M . Thus, the num-ber of clusters is given by K , ⌊ N/M ⌋ with a possible fewremaining subcarriers. The number of feedback bits allocatedfor each cluster is equal to B/K . All
B/K bits are used toquantize the beamforming vector of the centered subcarrierfor odd M and one subcarrier off the center for even M .Therefore, the beamforming vector used for the k th cluster isgiven by ˆ v kM + m = ( arg max w ∈V | ¯ h † kM + M +12 w | for odd M arg max w ∈V | ¯ h † kM + M w | for even M (13)where ≤ m ≤ M and ≤ k ≤ K − . If N/M is notan integer, then there exist some remaining subcarriers, whichdo not belong in any cluster. We propose to set the transmitbeamforming for these subcarriers to be that of the last clusteras follows: ˆ v KM + q = ˆ v KM for ≤ q ≤ N − KM. (14)With constant interpolation, an achievable rate for eachsubcarrier can be approximated by Proposition 1 .
Proposition 1:
For ≤ n + q ≤ N , the approximate ergodicachievable rate of the ( n + q ) th subcarrier is given by C n + q ≈ C n + q = log(1 + ρN t γ ( n + q, B/K )) (15)where γ ( n + q, B/K ) , ψ ( q, N t ) · (1 − B/K β (2 B/K , N t N t − − ψ ( q, N t )) · (2 B/K β (2 B/K , N t N t − )) N t − (16)and the beta function β ( x, y ) = R t x − (1 − t ) y − d t .The proof of Proposition 1 is shown in Appendix B.With Proposition 1, we obtain the approximate sum achiev-able rate for a single cluster with odd M as follows: C cluster = M − X q = − M − C n + q = log(1 + ρN t γ (0 , B/K ))+ 2 M − X q =1 log(1 + ρN t γ ( q, B/K )) . (17) IEEE TRANSACTIONS ON COMMUNICATIONS
With even M , the sum achievable rate for a single cluster isapproximated by C cluster = log(1 + ρN t γ (0 , B/K ))+2 M − X q =1 log(1+ ρN t γ ( q, B/K ))+log(1+ ρN t γ ( M , B/K )) . (18)We note that the performance of the constant interpolationhas a trade-off between total feedback bits and cluster size andhence, there exists optimal cluster size for a given feedbackbudget. Given B feedback bits and other system parameters,we would like to determine the number of subcarriers M ∗ ,which maximizes the approximate sum achievable rate of all N subcarriers given by M ∗ = arg max ≤ M ≤ NM ∈ Z ( K C cluster + N − KM X r =1 log(1 + ρN t γ ( r + M , B/K )) ) (19)where the first term accounts for the approximate achievablerate of the K clusters and the second term accounts forthe approximate achievable rate of a few remaining subcar-riers. Solving (19) can be accomplished by either integerprogramming for which there exist many available tools or byexhaustive search. Although the optimization in (19) is basedon the approximation of the actual achievable rate, subsequentnumerical examples in Section V show that the solution to (19)accurately predicts the optimal cluster size. We note that thereis no other comparable analysis on the optimal cluster size inthe literature.Besides the sum achievable rate, another important perfor-mance metric is the average received power across subcarriersdefined as follows: η AVE , N N X n =1 ρE (cid:2) | h † n ˆ v n | (cid:3) (20) = ρN t N N X n =1 E (cid:2) | ¯ h † n ˆ v n | (cid:3) . (21)where it is shown in the proof of Proposition 1 that E (cid:2) | ¯ h † n ˆ v n | (cid:3) ≈ γ ( n, B/K ) . (22)Therefore, the average received power can be approximatedas follows. Corollary 1:
For odd M , η AVE ≈ ρN t N ( Kγ (0 , B/K ) + 2 K M − X q =1 γ ( q, B/K )+ N − KM X r =1 γ ( r + M , B/K ) ) , (23) and for even Mη AVE ≈ ρN t N ( Kγ (0 , B/K ) + 2 K M − X q =1 ργ ( q, B/K )+ Kγ ( M , B/K ) + N − KM X r =1 γ ( r + M , B/K ) ) . (24)These analytical expressions give a more accurate approxi-mation than those in Proposition 1 since there is no Jensen’sinequality involved as demonstrated by numerical results inSection V. B. Linear Interpolation
To increase the performance, we propose to modify alinear interpolation proposed by [5]. Similar to the constantinterpolation, all subcarriers are grouped into K clusters. Eachcluster consists of M contiguous subcarriers and a possible lastcluster with a few remaining subcarriers. For each cluster, theoptimal beamforming vector of the first subcarrier is selectedfrom an RVQ codebook with either B/K bits or B/ ( K + 1) bits, depending on the total number of clusters.All other beamforming vectors in a cluster are linearcombinations of the quantized beamforming vector of thefirst subcarrier in the cluster and that in the next cluster asfollows [5]: ˆ v kM + m ( θ m ) , (1 − c m )ˆ v kM + c m e jθ m ˆ v ( k +1) M k (1 − c m )ˆ v kM + c m e jθ m ˆ v ( k +1) M k (25)for ≤ m ≤ M − and ≤ k ≤ K − , where c m = mM (26)is a linear weight and θ m is a phase-rotation parameter. Wenote that for the last cluster, we choose to interpolate with ˆ v instead of ˆ v N to save some feedback bits. Due to DFT, ¯ h issimilar to ¯ h N and hence, ˆ v is also similar to ˆ v N .In [5], θ m is chosen to maximize the sum achievable ratein (4) by performing exhaustive search over the received powerin each cluster as follows. For the k th cluster, θ m = arg max θ ∈ Θ M X i =1 | h † kM + i ˆ v kM + i ( θ ) | (27)where the phase-rotation codebook Θ = (cid:26) , π P , π P . . . , π P − P (cid:27) (28)and P is the number of quantization levels.To avoid search complexity and reduce feedback, here wepropose to determine the phase rotation based on a correlationbetween the optimal beamformers of neighboring subcarriers.We note that the optimal transmit beamforming vector for the n th subcarrier is matched to the normalized channel vector v opt n = ¯ h n . Evaluating a correlation between the optimalbeamformer and the interpolated beamformer that are m subcarriers apart, E | ( v opt kM ) † v kM + m | , follows similar stepsto the proof of Lemma 1. This correlation is most likelyclose to the correlation between the optimal beamformers, AMAT AND SANTIPACH: ON TRANSMIT BEAMFORMING FOR MISO-OFDM CHANNELS WITH FINITE-RATE FEEDBACK 5 which is approximated to be ψ ( m, N t ) in (11). Based onthis assumption, we set E | ( v opt kM ) † v kM + m | to ψ ( m, N t ) andsolve for the phase-rotation parameter given by the followingproposition. Proposition 2:
The phase rotation for the m th subcarrier inthe cluster is given by θ m = arccos U ( m ) V ( m ) (29)where U ( m ) = (1 − c m ) ( ψ ( m, N t ) − N t + 1)+ c m ( N t ψ ( m, N t ) − N t L ϕ ( M ) + 1) (30)and V ( m ) = 2 L (1 − c m ) c m ( N t − N t ψ ( m, N t ) + 1) · ϕ ( M ) cos (cid:18) πM ( L − N (cid:19) . (31)The proof is shown in Appendix C.Finding the optimal cluster size for the linear interpolationis not tractable since the achievable rate expression is notknown. From numerical results, the optimal cluster of theconstant interpolation mostly aligns with that of the linearinterpolation and that of higher-order interpolations as well.We note that computing θ m in Proposition 2 can be performedat the transmitter with the number of channel taps L andcluster size M . For a relatively static environment, L andhence M may not change often [27]. Thus, feedback for theseparameters do not occur often and consists of minimal numberof bits. For [5], the phase-rotation needs to be fed back forevery cluster. The number of additional feedback bits in [5]increases linearly with the number of clusters and can besignificantly larger than that in our method. C. Higher-Order Interpolation
For a better interpolation, more than two quantized trans-mit beamforming vectors should be used to interpolate thebeamforming vectors in the cluster. For instance, the second-order interpolated transmit beamformer in the k th cluster is asfollows: ˆ v kM + m = α − e jθ m ; − ˆ v ( k − M + α ˆ v kM + α e jθ m ;1 ˆ v ( k +1) M k α − e jθ m ; − ˆ v ( k − M + α ˆ v kM + α e jθ m ;1 ˆ v ( k +1) M k . (32)We note that there are 3 quantized beamformers ˆ v ( k − M , ˆ v kM , ˆ v ( k +1) M , which are used for interpolation.This interpolation was modified from the channel interpolationmethods proposed by [19], [20]. The set of constants is givenby [19] α − = 12 c m ( c m − (33) α = − ( c m − c m + 1) (34) α = 12 c m ( c m + 1) . (35) Phase-rotation parameters θ m ; − and θ m ;1 are introduced inthis study to increase the performance of the higher-orderinterpolation. Similar to that in the linear interpolation, theset of the two phase rotations is found by maximizing thesum received power in the k th cluster over the codebook Θ as follows: max { θ m ; − ,θ m ;1 }∈ Θ M X i =1 | h † kM + i ˆ v kM + i | . (36)For order R where R is even and R > , the interpolatedbeamformer in the k th cluster is given by ˆ v kM + m = y k y k (37)where y = − X s = − R α s e jθ m ; s ˆ v ( k + s ) M + α ˆ v kM + R X t =1 α t e jθ m ; t ˆ v ( k + t ) M (38)and { α r } R r = − R is a set of interpolation constants while theset of phase rotations { θ m ; − R , . . . , θ m ; − , θ m ;1 , . . . , θ m ; R } can be found by exhaustive search over the phase codebook Θ .We expect the higher-order interpolation method to performbetter than the previous methods, but the performance gain isobtained at the expense of additional complexity and feedback.Search complexity to locate the optimal set of phase-rotationparameters increases with the number of phase rotations orthe order of the interpolation. Also, the additional number offeedback bits to quantize these phase rotations increases withthe number of clusters and the order of the interpolation. Thesebits are in addition to the number of bits used to quantizetransmit beamformers.IV. Q UANTIZING C HANNEL I MPULSE R ESPONSE
When the available feedback rate is sufficiently high (largerthan 2 bits per complex entry), quantizing the channel impulseresponse directly can perform well [28]. Here we proposeto quantize all channel taps of all transmit-receive antennapairs with a scalar uniform quantizer. A uniform quantizer issimple and performs close to the optimal quantizer when thenumber of quantization bits is high. Real and imaginary partsof all channel taps are quantized independently with the samenumber of bits, which is B N t L . Thus, the quantized l th channeltap for the n t th transmit-receive antenna pair is given by ˆ g l,n t = ˆ g l,n t ,r + j ˆ g l,n t ,i (39) = Q ( g l,n t ,r ) + jQ ( g l,n t ,i ) (40)where g l,n t ,r and g l,n t ,i are real and imaginary parts of g l,n t ,respectively, Q ( · ) is the uniform scalar quantizer with B NtL steps, while variables with hats denote outputs of the quantizer.
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Here we select a step size of the quantizer by the existing ruleof thumb for Gaussian input (cf. [29, p. 125]) ∆ = 4 E [( g l,n t ,r ) ]2 B NtL = 1 √ L − B NtL , (41)which changes with the variance of the channel tap and thenumber of quantization bits. Then, the transmitter computes aDFT of the quantized channel impulse response to obtain anapproximate frequency response as follows: ˆ h n,n t = L − X l =0 ˆ g l,n t e − j πlnN , (42)which is the n t th entry of the quantized N t × chan-nel vector for the n th subcarrier denoted by ˆ h n = h ˆ h n, ˆ h n, · · · ˆ h n,N t i T .Based on ˆ h n , the transmitter transmits signal in the directionof the quantized channel vector, namely, ˆ h n / k ˆ h n k and thecorresponding sum rate over all subcarriers is given by C = N X n =1 E " log(1 + ρ | h † n ˆ h n | k ˆ h n k ) (43) = N E " log(1 + ρ | h † n ˆ h n | k ˆ h n k ) (44) ≤ N log(1 + ρE " | h † n ˆ h n | k ˆ h n k ) (45) ≈ N log(1 + ρ E [ | h † n ˆ h n | ] E [ k ˆ h n k ] ) (46)where in (44), we use the fact that the distribution of sub-carriers is identical and in (45) and (46), we apply Jensen’sinequality and approximate an expectation of the quotientby a quotient of the two expectations. The approximationbecomes more accurate as the number of transmit antennasincreases [30]. Consequently, we obtain the approximate sumachievable rate.Since real and imaginary parts of each channel tap areindependent and Gaussian distributed with zero mean andvariance L , we can easily show that E [ k ˆ h n k ] = N t (1 − LE [(ˆ g r − g r ) ]) (47)and E [ | h n ˆ h † n | ] = N t (1 + 1 L − (2 L − E [(ˆ g r − g r ) ]+ 2 LE [ˆ g r g r ] + 4 L ( N t L − E [ˆ g r g r ]) (48)where we have dropped indices n t and l from g l,n t ,r for clarity.The mean squared error is given by E [(ˆ g r − g r ) ] = Z ( Q ( x ) − x ) f g r ( x ) d x (49)and the correlation and its second moment are given by E [ˆ g r g r ] = Z xQ ( x ) f g r ( x ) d x (50) E [ˆ g r g r ] = Z x Q ( x ) f g r ( x ) d x. (51) where f g r ( · ) denotes the probability density function (pdf) of g l,n t ,r .Each term in (49)-(51) can be computed numerically. How-ever, to obtain some insight on how the sum achievable ratedepends on the feedback rate and other channel parameters,we approximate each term in a high feedback-rate regime. Itwas shown that for large B [31], E [(ˆ g r − g r ) ] ≈ ∆
12 = 23 L − BNtL . (52)Applying the property of the optimum quantizer [32], weobtain E [ˆ g r g r ] ≈ L − E [(ˆ g r − g r ) ] . (53)As B → ∞ , ˆ g r → g r . Hence, lim B →∞ E [ˆ g r g r ] = 34 L . (54)Substituting (52) – (54) into (47) and (48), we obtain theapproximate upper bound for a sum achievable rate for theMISO channel with large B as follows C ≈ N log(1 + ρ (1 − L + ( N t L − B + 34 L Ω B )) (55)where Ω B = L − L − BNtL . As the number of feedbackbits per transmit antenna and channel tap, BN t L , increases,the quantization error (52) decreases and the sum rate (55)increases. We note that for as BN t L → ∞ and N t increases,the sum rate in (55) increases as N log( ρN t ) , which isthe sum rate with perfect CSI. With a large feedback rate,quantizing channel impulse response can achieve a largersum rate than beamforming interpolation as will be shownin subsequent numerical examples. To determine at what B toswitch from channel quantization to, for example, the constantbeamforming interpolation, we compare the sum rate obtainedfrom Proposition 1 with that from (55).V. N UMERICAL R ESULTS
To illustrate the performance of the proposed interpolations,Monte Carlo simulation is performed with 3000 channelrealizations. Fig. 1 shows a correlation between subcarriers E | ¯ h † n ¯ h n + q | from simulation results and the analytical ap-proximation in Lemma 1 with N t = 5 , N = 1024 , L =64 and 128, respectively. From this figure, we see that thecorrelation between subcarriers decreases as expected whenthe subcarriers are further apart and note that the analyticalapproximation derived in Lemma 1 predicts the simulationresults quite accurately.Fig. 2 shows the average received power per subcarrier η AVE with constant interpolation for different numbers of feedbackbits B and channel taps L . We set the number of transmitantennas N t = 4 , cluster size M = 32 , and SNR at 10 dB. Wealso place another x-axis on the top of the figure showing thenumber of bits per cluster B/K . In the figure, the solid linesshow the analytical approximation given in Corollary 1 whilethe square and circular markers show the simulation results.Due to search complexity of RVQ, there are no simulationresults for a large-feedback regime.
AMAT AND SANTIPACH: ON TRANSMIT BEAMFORMING FOR MISO-OFDM CHANNELS WITH FINITE-RATE FEEDBACK 7 −20 −15 −10 −5 0 5 10 15 2000.20.40.60.81 N t = 5Subcarrier’s index E [ | h + n h n + q | ] Analytical approx. w/ N = 1024, L = 64Analytical approx. w/ N = 1024, L = 128Simulation w/ N = 1024, L = 64 Simulation w/ N = 1024, L = 128
Fig. 1. Correlation between subcarriers E | ¯ h † n ¯ h n + q | from both simulationand analytical results with N t = 5 , N = 1024 , and L = 64 and . We note from the figure that the average received powerincreases with B as expected and decreases with L . As thechannel becomes more frequency selective, the cluster sizeshould be reduced to maintain the performance. We observethat in this example, the analytical results are very close tothose from the simulation. Unlike the achievable rate analysis,Jensen’s inequality is not used in deriving η AVE . From thefigure, we see that about half a feedback bit per subcarriergives us close to the infinite-feedback performance. A ve r a g e r ece i ve d S NR Analytical approx. w/ L = 30Simulation w/ L = 30Analytixal approx. w/ L = 60Simulation w/ L = 600 5 10 15 20 25 30Feedback bits per cluster (B/K)
Fig. 2. Average received power per subcarrier η AVE with the number of totalfeedback bits B for N = 1024 , N t = 4 , M = 32 , and SNR at dB. Fig. 3 shows the sum achievable rate with constant interpo-lation with cluster size M from both the analytical approxima-tion from Proposition 1 and the simulation results when thenumber of total feedback bits is severely limited at 16 bits.Different plots correspond to different L values. For small M ,more beamforming vectors are quantized and fed back fromthe receiver, but with a smaller number of feedback bits percluster. For large M , the opposite is true. Thus, there exists anoptimal M that maximizes the achievable rate. We can observe from this figure, selecting optimal M = 16 performs 35%better than that for feeding back every subcarrier ( M = 1 )for L = 4 . Comparing the analytical approximation and thesimulation results, we observe that the gap is quite substantial(due to Jensen’s inequality); however, the analytical result stillcan accurately predict the optimal M . For a flat fading channel( L = 1 ), all subcarrier gains are the same and thus, theoptimal M ∗ = 1 . For frequency selective fading ( L > ),subcarriers are less correlated and the optimal M ∗ decreasewith L . We remark that the system size for this figure is set tobe smaller than that in the previous figure. This is due againto computational complexity of RVQ when B is large.
10 20 30 40 50 60100120140160180200220240260 Cluster size (M) S u m ac h i eva b l e r a t e ( n a t) Analytical approx. w/ L = 1Analytical approx. w/ L = 4Analytical approx. w/ L = 12Simulation w/ L = 1Simulation w/ L = 4Simulation w/ L = 12
Fig. 3. Sum achievable rate with different cluster size M and differentnumber of channel taps L for N = 64 , N t = 4 , B = 16 , and SNR at 10dB. Fig. 4 shows the optimal number of subcarriers per cluster M ∗ obtained from the analytical bound approximation withdifferent numbers of channel taps and total feedback bits. Inthis figure, we observe that M ∗ decreases when L increases.In other words, when the channel becomes more frequencyselective, cluster size should be reduced. Furthermore, withmore available feedback bits, cluster size should also bereduced. The explanation is as follows. As shown in Fig. 2, anincrease in the number of quantization or feedback bits beyonda certain point will give diminishing rate return. Therefore, toextend a rate increase, cluster size should be reduced for abetter interpolation of transmit beamforming.In Fig. 5, we compare the sum achievable rate of allinterpolation methods proposed in the study with either theoptimized cluster size obtained from (19) or fixed cluster size M = 16 . For this figure, we set N = 256 , L = 24 , N t = 3 ,and SNR at 10 dB. We observe that with a low feedback, theperformance of interpolation with optimal cluster size and thatwith M = 16 do not differ much. However,the performancegap between interpolation with or without the optimizedcluster size widens significantly as available feedback becomeslarger. The gain on the performance gain could be as high as15% for the constant interpolation. The solid line on the topof the figure shows the infinite feedback performance. We seethat with only two bits per subcarriers, our proposed methods IEEE TRANSACTIONS ON COMMUNICATIONS L M * N = 1024; N t = 4; SNR = 10 dB B = 64B = 96B = 128B = 160
Fig. 4. The optimal M ∗ shown with L and B for N = 1024 , N t = 4 , andSNR at 10 dB. achieve near optimal performance. We note that the second-order interpolation performs worse than the constant or thefirst-order interpolation in low or moderate B regimes. Thisis due to the extra feedback bits required to feed back the twophase-rotation parameters θ m ; − and θ m ;1 by the second-ordermethod. The additional feedback bits can be significant. For afixed M = 16 (hence, K = 16 ), 8 bits per cluster or total 128bits are used to feed back the two phase-rotation parameters. t = 3; SNR = 10 dBTotal feedback bits (B) S u m ac h i eva b l e r a t e ( n a t) Optimal BeamformingConstant w/ M = 16Constant w/ Optimal M * nd −order w/ M = 162 nd −order w/ Optimal M * Linear w/ M = 16Linear w/ Optimal M * Fig. 5. Comparison of the sum achievable rate with different interpolationmethods and with either optimized or fixed cluster sizes for N = 256 , L =24 , N t = 3 , and SNR at dB. In Fig. 6, we compare our linear interpolation method withthe optimized subcarrier cluster size from Section III anddirect channel quantization from Section IV with existingmethods [5], [7], [12], [21]. References [5], [12], [21] proposea beamforming interpolation in frequency domain while [7]proposes to vector-quantize channel impulse response. Inboth [5], [21], a single phase-rotation parameter is used forthe whole cluster. Hence, the number of phase rotations tobe quantized and fed back in [5], [21] equals the number of clusters. The main difference between [5] and [21] is linearweight c m . In our method, the phase rotation θ m differs fordifferent subcarriers in the same cluster and can be determinedat the transmitter with just the number of channel taps L andcluster size M fed back to the transmitter. Thus, the additionalnumber of feedback bits in our method is minimal while thosein [5], [21] increase linearly with the number of clusters K .For this figure, methods proposed by [5], [21] require 64additional bits. In [12], phase rotation is based on the chordaldistance between the two adjacent quantized beamformers andis the same for all subcarriers in the cluster. References [5],[12], [21] do not optimize cluster size and in this figure, it isfixed at 16. For [7], magnitudes and phases of all channel tapsare vector-quantized. The method proposed in [7] performsworse than our channel quantization for small B . However, weexpect the performance of the two methods to be comparablewhen B is large.From Fig. 6, we remark that the combination of our methods(linear interpolation in a low feedback-rate regime and directchannel quantization in a high feedback-rate regime) domi-nates all mentioned works in all feedback range. Also fromthis figure, we can conclude that with roughly one feedback bitper subcarrier, the direct channel-tap quantization is preferred,and with fewer than one bit per subcarrier, interpolation fromquantized transmit beamformers is preferred. t = 3; SNR = 10 dBTotal feedback bits (B) S u m ac h i eva b l e r a t e ( n a t) Optimal BeamformingQuantizing Channel tapsChoi and Heath w/ M = 16 [5]He et al. w/ M = 16 [21]Kim et al. w/ M = 16 [12]Linear interpolation w/ optimal M * Huang et al. [7]
Fig. 6. Sum achievable rate of a × OFDM channel with various feedbackschemes plotted with the total number of feedback bits B and N = 256 , L = 24 , and SNR at 10 dB. In Fig. 7, we compare an achievable rate per subcarrier ofa × channel obtained from simulation and the approxima-tion (55) for a direct quantization of channel taps. A numberof channel taps L varies between 32 and 128. From the figure,the approximate sum rate exhibits the same performance trendas the simulation results and the gap between the two isabout 10%. Again we can attribute the gap between the tworesults to Jensen’s inequality. Although the approximation isderived for a large feedback rate, it seems to predict well thesimulation result even with relatively small B . In addition, The expression of c m in (26) was also used by [5]. AMAT AND SANTIPACH: ON TRANSMIT BEAMFORMING FOR MISO-OFDM CHANNELS WITH FINITE-RATE FEEDBACK 9 we observe from the simulation results that approximately 3bits per real coefficient are needed to achieve close to themaximum achievable rate. While the number of fading paths L increases, B also increases to achieve close to the maximumrate. S u m ac h i eva b l e r a t e ( n a t) Quantizing channel taps; N = 1024; N t = 3; SNR = 10 dB Analytical approx. w/ L = 32Analytical approx. w/ L = 64Analytical approx. w/ L = 128Simulation w/ L = 32Simulation w/ L = 64Simulation w/ L = 128
Fig. 7. Comparison between achievable rate obtained from simulation andthe analytical approximation for channel-tap quantization with N = 1024 , N t = 3 , and SNR at 10 dB. In Figs. 8 and 9, we compare sum rate of our linear inter-polation method with those of perfect transmit beamforming(infinite feedback) and random transmit beamforming (zerofeedback). In Fig. 8, we plot sum achievable rates with thenumber of transmit antennas for N = 256 , L = 24 , and SNRat 10 dB. We see that the sum rates of the perfect beamformingand the linear interpolation with B = 128 bits increase with N t . The gap between the perfect beamforming and our methodgrows larger as N t increases. To close the gap, more feedbackbits are needed for quantizing transmit beamformers. Number of transmit antennas (N t ) S u m ac h i eva b l e r a t e ( n a t) N = 256; L = 24; SNR = 10 dB
Optimal beamforming (B = ∞ )Linear interpolation w/ B = 128Random beamforming (B = 0) Fig. 8. Sum achievable rates of the proposed linear interpolation method,the perfect transmit beamforming, and random beamforming are plotted with N t for N = 256 , L = 24 and SNR = 10 dB. In Fig. 9, sum rates are plotted with SNR while N t = 3 . As expected, all sum rates increase with SNR. We also add theperformance of channel quantization with B = 288 or 1.125bits per subcarrier, which is close to that of the perfect transmitbeamforming. The linear interpolation with only 32 bits or0.125 feedback bits per subcarrier can significantly outperformrandom beamforming or a system with zero feedback. −10 −5 0 5 10 15 20 25 3005001000150020002500 SNR (dB) S u m ac h i eva b l e r a t e ( n a t) N = 256; L = 24; N t = 3 Linear interpolation w/ B = 32Quantizing channel taps w/ B = 288Random beamforming (B = 0)Optimal beamforming (B = ∞ ) Fig. 9. Sum achievable rates from various methods are plotted with SNRfor N = 256 , L = 24 and N t = 3 . VI. C
ONCLUSIONS
We have proposed feedback methods for MISO-OFDMchannels. Beamforming interpolation with RVQ performs wellwith limited feedback while direct quantization of the channelimpulse response performs well with large feedback. Thus,switching between the two methods for different feedbackrates is recommended. We analyzed the sum achievable ratewith constant interpolation and RVQ and showed that theanalytical results can predict the performance trend and ac-curately predict the optimal cluster size. From numericalexamples shown, operating at the optimal cluster size can givea significant rate gain over an arbitrary size. For a relativelystatic channel in which the number of channel taps or SNR donot change often, the cluster size does not have to be updatedfrequently as well.For linear interpolation, we have derived a closed-formexpression for a phase rotation to avoid exhaustive search andadditional number of feedback bits in quantizing the phaserotation. We also considered the higher-order interpolationinspired from the OFDM channel estimation problem. Bothlinear and higher-order interpolations are improved signifi-cantly with the optimized cluster size derived for the constantinterpolation. Furthermore, we have analyzed the achievablerate with direct quantization of channel taps, which dependson the feedback rate and the number of antennas and channeltaps.Future work can take different directions. In the problemconsidered, the MISO channel was investigated. Extending ourresults to MIMO beamforming is not straightforward and thus,MIMO beamforming could be a good problem to consider. In addition, here we considered channels with a uniform powerdelay profile. Other practical channel models might be ofinterest. A
PPENDIX
A. Proof of Lemma 1
We approximate E (cid:2) | ¯ h † n ¯ h n + q | (cid:3) ≈ E (cid:2) | h † n h n + q | (cid:3) E [ k h n k k h n + q k ] . (56)First, we evaluate E (cid:2) | h † n h n + q | (cid:3) as follows E (cid:2) | h † n h n + q | (cid:3) = E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N t X m =1 h ∗ n,m h n + q,m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (57) = N t X m =1 E (cid:2) | h n,m | | h n + q,m | (cid:3) + N t X m =1 N t X m =1 m = m E [ h ∗ n,m h n + q,m ] E [ h n,m h ∗ n + q,m ] (58) = N t E (cid:2) | h n,m | | h n + q,m | (cid:3) + N t ( N t − | E (cid:2) h ∗ n,m h n + q,m (cid:3) | (59)where we apply the assumption that channel gains acrossantennas are i.i.d .Next we evaluate each term in (59) by substituting (1). E (cid:2) | h n | | h n + q | (cid:3) = L X l ,l ,l ,l =1 E h g l g ∗ l g l g ∗ l e − j π { ( l − l l − l n +( l − l q } N i (60)where we omit the antenna subscript m for brevity. It isstraightforward to show that E [ g l g ∗ l g l g ∗ l e − j π { ( l − l l − l n +( l − l q } N ]= L : l = l = l = l , − L : ( l = l ) = ( l = l ) , L ( ϕ ( q ) −
1) : ( l = l ) = ( l = l ) , otherwise . (61)Substituting (61) into (60) gives E (cid:2) | h n | | h n + q | (cid:3) = 1 + 1 L ϕ ( q ) (62) Also, E [ h ∗ n h n + q ] = L X l =1 E | g l | e − j πl qN + L X l =1 L X l =1 l = l E [ g ∗ l g l ]e − j πl nN e − j πl n + q ) N (63) = L X l =1 L e − j πl qN (64) = 1 L e − j π ( L − qN ϕ ( q ) . (65)where the second term in (63) is equal to zero.Substituting (62) and (65) into (59) gives E (cid:2) | h † n h n + q | (cid:3) = N t + N t L ϕ ( q ) . (66)Following similar steps as the above evaluation of E (cid:2) | h † n h n + q | (cid:3) , we can show that E (cid:2) k h n k k h n + q k (cid:3) = N t + N t L ϕ ( q ) . (67)Substituting (66) and (67) into (56) yields the Lemma. B. Proof of Proposition 1
From (8), C n + q = E h log(1 + ρ k h n + q k | ¯ h † n + q ˆ v n + q | ) i (68) ≤ log(1 + ρE [ k h n + q k | ¯ h † n + q ˆ v n + q | ]) (69) = log(1 + ρE k h n + q k E | ¯ h † n + q ˆ v n + q | ) (70) = log(1 + ρN t E | ¯ h † n + q ˆ v n + q | ) (71)where Jensen’s inequality is applied in (69). Eq (70) is due tothe fact that k h n + q k and | ¯ h † n + q ˆ v n + q | are independent [25].In addition, E k h n + q k = N t since each element in h n + q hasunit variance. Jensen’s inequality is tighter when the numberof transmit antennas increases.To derive the upper bound on C n + q in (71), we need todetermine E | ¯ h † n + q ˆ v n + q | . With constant interpolation, ˆ v n + q is set to equal the representative beamforming of a cluster,which is q subcarriers away. Therefore, we would like to eval-uate E | ¯ h † n + q ˆ v n | . To accomplish this goal, we project ¯ h n + q onto ¯ h n and its N t − -dimensional orthogonal complementdenoted by ¯ h ⊥ n .Let { u , u , . . . , u N t − } be a basis of ¯ h ⊥ n . Hence, we canwrite ¯ h n + q as a linear combination of its projection onto ¯ h n and the basis of ¯ h ⊥ n as follows. ¯ h n + q = (¯ h † n ¯ h n + q )¯ h n + N t − X i =1 ( u † i ¯ h n + q ) u i . (72) AMAT AND SANTIPACH: ON TRANSMIT BEAMFORMING FOR MISO-OFDM CHANNELS WITH FINITE-RATE FEEDBACK 11
With (72), we have E h | ¯ h † n + q ˆ v n | i = E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (¯ h † n + q ¯ h n )(¯ h † n ˆ v n ) + N t − X i =1 (¯ h † n + q u i )( u † i ˆ v n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (73) = E h | ¯ h † n + q ¯ h n | | ¯ h † n ˆ v n | i + N t − X i =1 E h | ¯ h † n + q u i | | u † i ˆ v n | i + 2 E ℜ ( (¯ h † n + q ¯ h n )(¯ h † n ˆ v n ) N t − X i =1 (¯ h † n + q u i )( u † i ˆ v n ) ) (74)where ℜ { x } is the real part of x . Similar to [25], it can beshown that | ¯ h † n + q ¯ h n | and | ¯ h † n ˆ v n | are independent. In [25], E | ¯ h † n ˆ v n | was also analyzed while E | ¯ h † n + q ¯ h n | ≈ ψ ( q, N t ) from Lemma 1. Thus, E [ | ¯ h † n + q ¯ h n | | ¯ h † n ˆ v n | ]= E | ¯ h † n + q ¯ h n | E | ¯ h † n ˆ v n | (75) ≈ ψ ( q, N t ) (cid:18) − B/K β (2 B/K , N t N t − (cid:19) . (76)For the second term on the right-hand side of (74), we havethat similar to the first term, E h | ¯ h † n + q u i | | u † i ˆ v n | i = E | ¯ h † n + q u i | E | u † i ˆ v n | . (77)We can evaluate the second term in (77) as follows. Similarto (72), we can write ˆ v n as a linear combination of itsprojection onto basis { ¯ h n , u , . . . , u N t − } as follows: ˆ v n = (¯ h † n ˆ v n )¯ h n + N t − X i =1 ( u † i ˆ v n ) u i . (78)Evaluating (ˆ v † n ˆ v n ) with (78) and applying the fact that k ˆ v n k = 1 results in | ¯ h † n ˆ v n | + Nt − X i =1 | u † i ˆ v n | = 1 . (79)We take expectation on both sides and substitute a closed-form expression of E | ¯ h † n ˆ v n | from [25]. Also, E | u † i ˆ v n | isthe same for all ≤ i ≤ N t − due to identical distributions.Thus, from (79), we have E | u † i ˆ v n | = 2 B/K β (2 B/K , N t N t − ) N t − . (80)Similar to the steps that derive (80), we can show that E | ¯ h † n + q u i | ≈ − ψ ( q, N t ) N t − . (81)Applying (80) and (81), we have N t − X i =1 E h | ¯ h † n + q u i | | u † i ˆ v n | i ≈ (1 − ψ ( q, N t )) · B/K β (2 B/K , N t N t − ) N t − (82) Evaluating the final term of the right-hand side of (74) isnot tractable. However we note that for both small and largefeedback, the term is close to zero due to ¯ h † n ˆ v n and u † i ˆ v n ,respectively. Thus, we approximate E ℜ ( (¯ h † n + q ¯ h n )(¯ h † n ˆ v n ) N t − X i =1 (¯ h † n + q u i )( u † i ˆ v n ) ) ≈ . (83)Finally, substituting (76), (82), and (83) in (74) yieldsProposition 1. C. Proof of Proposition 2
Applying the linear interpolation (25) and assuming optimal, unquantized beamforming, we have E | ( v opt kM ) † v kM + m | ≈ E (cid:12)(cid:12)(cid:12) h † kM (cid:8) (1 − c m ) h kM + c m e jθ m h ( k +1) M (cid:9)(cid:12)(cid:12)(cid:12) E k (1 − c m ) h kM + c m e jθ m h ( k +1) M k . (84)Here we propose to set phase rotation θ m by solving E (cid:12)(cid:12)(cid:12) h † kM (cid:8) (1 − c m ) h kM + c m e jθ m h ( k +1) M (cid:9)(cid:12)(cid:12)(cid:12) E k (1 − c m ) h kM + c m e jθ m h ( k +1) M k = ψ ( m, N t ) . (85)where ψ ( m, N t ) is defined in Lemma 1.Similar to steps shown in the proof of Lemma 1, we canshow that E (cid:12)(cid:12)(cid:12) h † kM (cid:8) (1 − c m ) h kM + c m e jθ m h ( k +1) M (cid:9)(cid:12)(cid:12)(cid:12) = (1 − c m ) ( N t + 1) + c m (cid:18) N t L ϕ ( M ) + 1 (cid:19) +2(1 − c m ) c m cos θ m cos (cid:18) πM ( L − N (cid:19) (cid:18) N t + 1 L (cid:19) ϕ ( M ) (86)and E k (1 − c m ) h kM + c m e jθ m h ( k +1) M k = (1 − c m ) ( N t + 1) + c m N t + 2 N t L (1 − c m ) c m cos θ m cos (cid:18) πM ( L − N (cid:19) ϕ ( M ) . (87)Substituting (86) and (87) into (84) and solving for θ m givesProposition 2. R EFERENCES[1] T. K. Y. Lo, “Maximum ratio transmission,”
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