MULTI-ANTENNA diversity has by now been well established

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1 948 IEEE TRASACTIOS O WIRELESS COMMUICATIOS, VOL. 4, O. 4, JULY 25 Quantifying the Power Loss When Transmit Beamforming Relies on Finite-Rate Feedback Shengli Zhou, Member, IEEE, Zhengdao Wang, Member, IEEE, and Georgios B. Giannakis, Fellow, IEEE Abstract Transmit beamforming achieves optimal performance in systems with multiple transmit antennas and a single receive antenna from both the capacity and the received signalto-noise ratio SR) perspectives but ideally requires perfect channel knowledge at the transmitter. In practical systems where the feedback link can only convey a finite number of bits, transmit beamformer designs have been extensively investigated using either the outage probability or the average SR as the figure of merit. In this paper, we study the symbol error rate SER) for transmit beamforming with finite-rate feedback in a multi-input single-output setting. We derive an SER lower bound that is tight for good beamformer designs. Comparing this bound with the SER corresponding to the ideal case, we quantify the power loss due to the finite-rate constraint across the entire SR range. Index Terms Finite-rate feedback, multi-antenna systems, power loss, transmit beamforming. I. ITRODUCTIO MULTI-ATEA diversity has by now been well established as an effective fading countermeasure for wireless communications. In certain application scenarios, e.g., cellular downlink, the number of receive antennas at the mobile is limited due to size and cost constraints, which motivate deployment of transmit diversity systems. Our attention in this paper is thereby focused on downlink applications dealing with a single receive but multiple transmit antennas. To further improve system performance, the receiver can feed channel state information CSI) back to the transmitter. With perfect CSI, transmit beamforming achieves optimal performance in the multi-input single-output MISO) setting in terms of maximizing both the received signal-to-noise ratio SR) and the mutual information between the channel input and out- Manuscript received February, 24; revised May 22, 24; accepted May 24, 24. The editor coordinating the review of this paper and approving it for publication is T. Duman. The work of S. Zhou was supported by UConn Research Foundation Internal Grant The work of Z. Wang was supported by ational Science Foundation Grant The work of G. B. Giannakis was prepared through collaborative participation in the Communications and etworks Consortium sponsored by the U. S. Army Research Laboratory under the Collaborative Technology Alliance Program, Cooperative Agreement DAAD Part of this work was presented at the 38th Annual Conferences on Information Sciences and Systems, Princeton, J, March 7 9, 24. S. Zhou is with the Department of Electrical and Computer Engineering, University of Connecticut, Storrs, CT 6269 USA shengli@engr. uconn.edu). Z. Wang is with the Department of Electrical and Computer Engineering, Iowa State University, Ames, IA 5 USA zhengdao@iastate.edu). G. B. Giannakis is with the Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, M USA georgios@ece.umn.edu). Digital Object Identifier.9/TWC put 7]. However, in practical wireless systems, the CSI at the transmitter suffers from imperfections originating from various sources such as estimation errors, quantization effects, feedback delay, and feedback errors. These considerations sparked recent research interest towards quantifying and exploiting imperfect or partial) CSI in multi-antenna systems, e.g., ], 2], 7] ], 3] 8], 2], 23], 25] 27]. Partial CSI can be modeled in different ways. One approach is to describe it statistically based on feedback information. For example, CSI at the transmitter can be modeled as Gaussian distributed with local mean and local covariance 7], ], 7], 23]. Under this model, two special forms of partial feedback have been extensively studied. One is the so-called mean feedback 8], ], 5], 7], 23], 26] that assumes knowledge of the channel mean and models the channel covariance as spatially white. The other is termed covariance feedback, where the channel mean is set to zero and the relative geometry of the propagation paths manifests itself in a generally nonwhite covariance matrix 8], 5], 8], 2], 23], 27]. Another class of CSI models imposes a bandwidth constraint on the feedback channel that is only able to communicate a finite number of bits per block ], 9], ], 3], 4], 6], 7]. In particular, finite-rate power control is investigated in ] to reduce the outage probability that the mutual information falls below a certain rate. Transmit beamforming has been investigated based on the average SR 4], 7] and also on the outage probability criterion 6]. Extension to unitary precoding for high-rate spatial multiplexing systems is pursued in 3]. Subject to finite-rate feedback, optimal transmission is also studied in 3], ] to maximize the average channel capacity, while adaptive modulation together with transmit beamforming has been pursued in 25] to enhance the transmission rate. For transmit beamforming based on finite-rate feedback, a universal lower bound on the outage probability applicable to all beamformers) is derived in 6]. Good beamformers are then constructed to attain this lower bound. On the other hand, beamforming vectors are designed in 4] to minimize the average SR reduction due to finite-rate feedback. Albeit starting from different perspectives, the beamformer design criteria in 4] and 6] are equivalent, leading to the minimization of the maximum correlation between any pair of beamforming vectors. Interestingly, the beamformer design is linked to the line packing problem in Grassmannian manifold, where the minimum chordal distance between any pair of straight lines passing through the origin should be maximized 5]. Different from 4] and 6], we investigate in this paper the symbol error rate SER) performance of transmit beamforming /$2. 25 IEEE

2 ZHOU et al.: QUATIFYIG POWER LOSS WHE TRASMIT BEAMFORMIG RELIES O FIITE-RATE FEEDBACK 949 received sample y = w H hs + η 3) where η is the additive complex Gaussian noise with zero mean and variance. With E s denoting the average symbol energy, the instantaneous SR in 3) is γ = w H h 2 E s. 4) Fig.. System model. with finite-rate feedback. We derive a universal SER lower bound that is applicable to any beamformer design. This bound is tight for the good beamformers constructed in 4], thus serving as a good performance indicator. Furthermore, comparing this lower bound with the SER corresponding to the perfect CSI case enables one to quantify the power loss due to the finite-rate constraint. Our result is valid across the entire SR range. The rest of this paper is organized as follows. We present the system model in Section II, derive the SER lower bound in Section III, and quantify the power loss in Section IV. We then collect numerical results in Section V and draw concluding remarks in Section VI. otation: Bold upper and lower case letters denote matrices and column vectors, respectively; denotes vector norm; ), ) T, and ) H stand for conjugate, transpose, and Hermitian transpose, respectively; E{ } denotes expectation; I stands for an identity matrix of size and M stands for an M all zero matrix; C stands for the -dimensional complex vector space; Cb, Σ) denotes the complex Gaussian distribution with mean b and covariance Σ. II. SYSTEM MODEL Fig. depicts the considered MISO system with t transmit antennas and a single receive antenna. Let h µ denote the channel coefficient between the µth transmit antenna and the single receive antenna, and collect h µ s into the channel vector h := h,...,h t ] T. As in 4] and 6], we assume that h µ s are independent identically distributed i.i.d.) according to a complex Gaussian distribution with zero mean and unit variance; i.e., h C, I t ). ) The extension to correlated channels needs further investigation and goes beyond this paper s scope. We adopt transmit beamforming without temporal power control 4], 6]. Specifically, each information symbol s is multiplied by a beamforming vector w, where w := w,w 2,...,w t ] T 2) has unit norm w =.The t entries of the vector sw are transmitted simultaneously from t antennas, which leads to a As in 4] and 6], we suppose that the receiver is able to feed a finite number of say B) bits back to the transmitter. We assume that the feedback link is error free and delay free. The feedback bits will be used to select the beamforming vector. With B bits, the transmitter has a total of =2 B beamforming vectors to choose from. Let us denote these vectors as w, w 2,...,w and collect them into a matrix the codebook of beamforming vectors) W := w, w 2,...,w ]. 5) The beamforming vector is selected as follows. The receiver is assumed to have perfect knowledge of h and chooses the beamforming vector to maximize the instantaneous SR; i.e., w opt = arg max w {w i } i= w H h 2. 6) The index of w opt is coded into B feedback bits. After receiving the B feedback bits, the transmitter switches to the optimal beamforming vector w opt. With the selection process in 6), the instantaneous SR in 4) becomes { w H γ = max i h 2} E s 7) i that indicates that system performance depends critically on the design of the beamforming vectors {w i } i=. A. Good Beamformers The codebook optimization on {w i } i= has been thoroughly investigated in 4] and 6], where 6] uses the outage probability as the figure of merit and 4] uses the average receive SR. Albeit starting from different perspectives, these two criteria lead to the same beamformer designs. Specifically, with i.i.d. channels in ), it turns out that a good beamformer should minimize the maximum correlation between any pair of beamforming vectors; i.e., W opt = min W C t max i<j w i H w j. 8) In 4], the beamformer design problem is explicitly linked to the Grassmannian line packing problem in 5]. Specifically, w i can be viewed as coordinates of a point on the surface of a hypersphere with unit radius centered around origin. This point dictates a straight line in a complex space C t that passes

3 95 IEEE TRASACTIOS O WIRELESS COMMUICATIOS, VOL. 4, O. 4, JULY 25 through the origin. The two lines generated by w i and w j have a distance defined as dw i, w j ):=sin i,j ) = w i Hw 2 j 9) where i,j denotes the angle between these two lines. The distance dw i, w j ) is known as chordal distance 5]. So the beamformer design in 8) is equivalent to W opt = max W C t min dw i, w j ). ) i<j For illustration purposes, we next list the code design examples from 4] that will also be used in Section V. Example codes from 4]: ) t =2and =4B =2feedback bits). The matrix W is constructed as j j W T j j =. ) j j The achieved maximum correlation is max w H i w j = i<j and the achieved minimum chordal distance is min i<j dw i, w j )=.673 = sin55 ). 2) t =2and =8B =3feedback bits). The matrix W is constructed as j j j j j j W T j j =. 2) j j j j j j j j The achieved maximum correlation is max w H i w j =.8452 i<j and the achieved minimum chordal distance is min i<j dw i, w j )=.298 = sin32 ). Remark Links With Selection Combining): The beamformer design in 8) is challenging only when the codebook size > t. When t, W can be chosen as columns of an arbitrary t t unitary matrix, which leads to the minimal chordal distance as = sin9 ). The resulting beamforming system has identical performance as a selection combining SC) system with diversity branches 4], 6]. Hence, the system error performance can be readily evaluated using the closed-form expression for SC schemes in 4, eq. 36)]; see also 9, eq )] for extensions to generalized selection combining and 22, eq. 8)] for transmit SC combined with receive maximum ratio combining in systems with multiple antennas at both the transmitter and the receiver. III. PERFORMACE AALYSIS Our objective in this section is to evaluate the average performance for beamformed transmissions with finite-rate feedback. Towards this objective, we assume that s is drawn from a phaseshift keying PSK) constellation; similar derivations can be carried out for other constellations, as detailed in Remark 2 later on. Conditioned on the instantaneous SR γ, the SER is 9, eq. 8.22)] SERγ) = π M )π exp where M is the constellation size and g PSK := sin 2 π ) M g PSKγ sin 2 ) d 3) 4) is a constellation-dependent constant. With h as a random vector, the average SER is expressed as SER = E h {SERγ)}. 5) Our objective is to find either exact or tight) approximate expressions for SER. A. SER Formulation We define a normalized channel vector as h = h h 6) such that h = h h and h =. We then decompose the SR in 7) as γ = h 2 E s max h 2 i wh i h = h 2 E s min d 2 w i, h) ] 7) i where d ) is the chordal distance in 9). To simplify the notation, we define the average transmit SR γ := E s / and two random variables such that γ h := h 2 8) Z := min d 2 w i, h) i 9) γ = γ h Z)γ. 2)

4 ZHOU et al.: QUATIFYIG POWER LOSS WHE TRASMIT BEAMFORMIG RELIES O FIITE-RATE FEEDBACK 95 Basedon),γ h is Gamma distributed with parameter t and mean E{ h 2 } = t ; hence, its probability density function pdf) is 9] pγ h )= γ t h Γ t ) e γ h 2) where Γ ) is the Gamma function. On the other hand, Z is a random variable within the interval,], i.e., Z, ]. Let pz) and F Z z) denote the pdf and cumulative distribution function cdf) of Z. Due to the i.i.d. assumption in ), γ h is independent of h 6]; thus, γ h and Z are independent. Using the definitions of {γ h,z,γ}, we can write 5) explicitly as SER = γ h = z= SER γ h z)γ] pγ h )pz)dγ h dz. 22) Substituting 3) and 2) into 22) and utilizing the fact that the moment generating function of γ h is E{e sγ h } = s) t 9], we obtain SER = π M )π + g ] t PSKγ sin 2 z) df Z z)d. 23) otice that F Z z) depends on the particular beamformer design W. To obtain the exact SER in 23), one has to find the exact F Z z) for each particular beamformer. Unfortunately, this task is difficult to accomplish. We next develop an SER lower bound that is applicable to all beamformers. Our approach here is analogous to 6], where a lower bound on the outage probability is derived. hypersphere. According to 6, Lemma 2], the surface area of the spherical cap S i z) is A {S i z)} = 2π t z t t )!. 26) otice that F Z z) = Prmin i d 2 w i, h) z] equals F Z z) =Pr d 2 w, h) z, or d 2 w 2, h) z,..., or d 2 w t, h ) ] z 27) that can be interpreted as the probability that h falls in the union of the regions {S i z)} i= denoted by i= S iz)]. Based on the key observation that h is uniformly distributed on the surface of the unit hypersphere, when h is i.i.d 6], we can find F Z z) as F Z z) = A { i= S iz) }. 28) A {S i )} Due to the union operation, it is true that A { i=s i z) } A {S i z)}. 29) i= Based on the area computation of 26), we obtain i= F Z z) A {S iz)} = z t. 3) A {S i )} Taking into account that F Z z), we define an upper bound F Z z) as B. SER Lower Bound To find an SER lower bound, we look for a cdf function F Z z) such that { z F t, z< Z z) =, z ) ) t ) t ). 3) F Z z) F Z z), z. 24) Replacing F Z z) in 23) by F Z z), we define an SER lower bound as SER lb := π M )π + g ] t PSKγ sin 2 z) d F z)d. 25) The proof that SER SER lb is given in Appendix A. We next derive such a function F Z z) under the geometrical framework presented in 6]. Around each beam vector w i,we define a spherical cap on the surface of the hypersphere S i z) ={ h : d 2 w i, h) } z where z. Define A{S i z)} as the area of the cap S i z). Then A{S i )} is the whole surface area of the unit Substituting 3) into 25), we obtain the SER lower bound in 32), shown at the bottom of the next page, where the integration of z can be simply carried out by defining a new variable t =/z. otice that the lower bound is independent of any particular beamformer design. A good beamformer shall try to come as close as possible to this lower bound. This confirms the design guidelines in 8) and ) as follows. Let z o denote the critical value where all the spherical caps S z o ),S 2 z o ),...,S z o ) do not overlap. For a given beamformer matrix W, thevalueofz o is available by reinterpreting the result 6, Lemma 5] as + max w H i w j i<j t z o = 2 max w H i w j i<j t =. 33) 2

5 952 IEEE TRASACTIOS O WIRELESS COMMUICATIOS, VOL. 4, O. 4, JULY 25 For z<z o, because all the spherical caps do not overlap, we have F Z z) = F Z z) =z t, z z o. 34) Hence, FZ z) F Z z) only when z>z o. To minimize the difference between F Z z) and F Z z), a good beamformer design tries to maximize z o as much as possible. If z o reaches the maximum of / ) /t ), the true SER in 23) would coincide with the SER lower bound of 32); e.g., in the case of t =2 and =2. Therefore, the maximum correlation max i<j t wi Hw j should be minimized, agreeing with 8) that was obtained based on either the outage probability or the average SR criterion. umerical results show that the universal lower bound in 32) is tight for good beamformer designs. This is consistent with 6], where the lower bound on outage probability is tight. Remark 2 Extension to Arbitrary Constellations): For brevity, we derived the SER lower bound in 32) for the PSK constellation. However, the same procedure can be applied to arbitrary two-dimensional signal constellations. Specifically, we first need to express the conditional SER conditioned on the instantaneous SR γ) as the sum of a small number of finite integrals as described in 6] notice that the SER for PSK in 2) is a special case with only one finite integral). Then, we average the conditional SER over the random variable γ and obtain a lower bound following the same derivation to reach 32). For example, applying this procedure with a square quadrature amplitude modulation QAM) constellation the desired finite integral form of the conditional SER is available in e.g., 9, eq. 8.2)]), we obtain the SER lower bound for QAM in 35), shown at the bottom of this page, where M is the constellation size and g QAM =.5 M. 36) As one can easily verify from the development in the ensuing section, the power loss due to finite-rate as opposed to ideal) feedback for arbitrary two-dimensional constellations will be identical to that for PSK. IV. QUATIFICATIO OF POWER LOSS DUE TO FIITE-RATE FEEDBACK When goes to infinity, the performance of beamformed transmissions with finite-rate feedback should approach the ideal case with perfect channel knowledge at the transmitter. With perfect CSI, Z =in 23) and one can derive the SER for PSK as SER perfect CSI = π M )π + g ) t PSKγ sin 2 d. 37) Therefore, the lower bound in 32) is tight with respect to because it coincides with 37) when =. Comparing 32) with finite against 37) with =, we will be able to quantify the power loss due to the finite-rate constraint. Towards this objective, we assume that AS): the SER lower bound can be achieved by some good beamformer designs. The performance of the beamformers in AS) will depend on t,, and γ; for this reason, we explicitly express the SER as SER lb t,,γ). On the other hand, the SER with perfect CSI depends only on t and γ, denoted as SER perfect CSI t, γ). Due to the finite-rate constraint SER lb t,,γ) SER lb t,, γ) = SER perfect CSI t, γ). 38) SER lb = π = π M )π M )π ) t ) + g ] t PSKγ sin 2 z) t )z t 2 dzd + g ) { ) ] PSKγ t sin 2 + } t g PSK γ sin 2 d 32) SER QAM lb = 4 π M M π 4 + g ) { ) ] QAMγ t sin π M M π 2 π 4 } t g QAM γ sin 2 d + g ) { ) ] QAMγ t sin 2 + } t g QAM γ sin 2 d 35)

6 ZHOU et al.: QUATIFYIG POWER LOSS WHE TRASMIT BEAMFORMIG RELIES O FIITE-RATE FEEDBACK 953 To compensate for this performance loss, one has to increase the transmission power. Suppose that when we increase the average SR from γ to γ new, we arrive at TABLE I POWER LOSS I DECIBELS)DUE TO THE FIITE-RATE COSTRAIT SER lb t,,γ new )=SER lb t,, γ). 39) We define the difference between γ new and γ as the power loss in decibels) due to the finite-rate constraint as L t,,γ) = log γ new log γ. 4) otice that the power loss in 4) is a function of γ and thus varies over the entire SR range. We have the following results on L t,,γ): Proposition : Under AS), the power loss for transmit beamforming due to the finite-rate feedback constraint satisfies ) ] t t L t,, ) = log 4) L t,,γ) L t,, ). 42) Proof: The power loss at high SR γ ) can be easily found. At high SR, we simplify 32) as SER lb t,,γ) G c γ) G d 43) where G d = t is termed as the diversity gain and G c = g PSK π M )π sin ) 2 t d t ) ] t t 44) is referred to as the coding gain see, e.g., 2], 24]). Equation 43) implies that the SER versus average SR curve in fading channels is well approximated by a straight line at high SR when plotted on a log log scale. The diversity gain G d determines the slope of the curve, while G c in decibels) determines the shift of the curve in SR relative to a benchmark SER curve of γ G d. Substituting 43) into 39), we obtain L t,, ) in 4). For the power loss at arbitrary SR, we prove 42) in Appendix B. Proposition testifies that the power loss across the entire SR range is bounded to be less than or equal to L t,, ). In other words, the distance or the horizontal shift) between SER perfect CSI t, γ) and SER lb t,,γ) is no larger than L t,, ) decibels across the entire SR range. When =2 B is large, we use ln + x) x for small x to obtain L t,, ) ) t 2 B t. 45) ln TABLE II MIIMUM UMBER OF FEEDBACK BITS B FOR POWER LOSS WITHI db Hence, the power loss in decibels due to the finite-rate constraint decays exponentially with the number of feedback bits B for large B. This implies that B does not need to be very large for the system performance to be close to optimal, as confirmed by numerical results. On the other hand, if a system requires the power loss to be within L decibels relative to the perfect CSI case, we can easily identify from 4) the least number of beamforming vectors needed as L t ) t ) t. 46) otice that only high SR analysis is carried out in 6] based on the outage probability. Moreover, the distortion measure defined in 6, eq. 52)] does not translate to power loss. For a given reduction on the average capacity or the average SR, the minimum is also calculated in 4]. However, the calculation in 4] relies on a loose bound on the minimum chordal distance in line packing, which affects the accuracy. Finally, recall that Proposition holds under AS). For practically adopted beamformers with AS) not exactly satisfied, the power loss computation is only approximate. otice that without AS), only asymptotic analysis with ) may be possible, as in 6]. V. UMERICAL RESULTS In this section, we collect some numerical results. Case ) We compute the power loss in 4) for various cases and list the results in Table I, where we are interested in nontrivial configurations with t. On the other hand, if we want to limit the power loss to be within db relative to the perfect CSI case, the minimum number of beamforming vectors can be computed from 46), which leads to the minimum number of feedback bits B; the results are listed in Table II. We clearly see that as t increases, the needed number of feedback bits increases considerably in order to bring the performance close to the optimum. Case 2) We now compare the SER lower bound in 32) with the actual SER of 5), which is obtained via Monte

7 954 IEEE TRASACTIOS O WIRELESS COMMUICATIOS, VOL. 4, O. 4, JULY 25 Fig. 2. Actual SER versus the lower bound t =2, quaternary phase-shift keying QPSK)]. Fig. 3. Actual SER versus the lower bound t =3, QPSK). Carlo simulations. We use 4-PSK constellations in all cases unless otherwise specified. Fig. 2 depicts the results with t =2. The beamformer codebooks for =4, 8 are listed in Section II-A. The codebook with =2 is W = I 2, which corresponds to the selection diversity. As shown in Fig. 2, the SER lower bound for the t =2case is almost identical to the actual SER, even with small. Case 3) We now test the SER lower bound with the actual SER for the t =3case. We use the beamformers listed in 2] with =4, 8, 6. Asshownin Fig. 3, the bound is tight for the given beamformers, although the difference between the bound and the actual SER increases relative to the t =2case. Case 4) We now test the SER lower bound with the actual SER for the t =4case. We use the beamformers listed in 2] with =6, 64. The codebook with =4is W = I 4, corresponding to SC. As shown in Fig. 4, the bound is also tight for the given beamformers, with the difference between the bound and the actual SER being larger than those corresponding to t =2, 3. The increasing difference could be due to either of the following two reasons or both: i) the bound becomes less tight

8 ZHOU et al.: QUATIFYIG POWER LOSS WHE TRASMIT BEAMFORMIG RELIES O FIITE-RATE FEEDBACK 955 Fig. 4. Actual SER versus the lower bound t =4, QPSK). Fig. 5. Actual SER versus the lower bound t =4, 6-QAM). for a large t ; ii) the beamformer currently found for a large t is not as close to the optimum as in the t =2case. Case 5) We now test the SER lower bound in 35) for QAM constellations. Fig. 5 depicts the results for 6-QAM with the same beamformers as in Case 4 where t =4. The same observations can be obtained for QAM constellations as for PSK. Better beamforming codebooks have been reported in 28], whose performance approaches the lower bound provided here. Figs. 2 5 show that the distance between the SER lower bound and the SER with perfect CSI only slightly increases as the average SR increases, yet it stays always bounded by the maximum power loss listed in Table I. This confirms 42) and more importantly shows that high SR analysis is often accurate even in the low and medium SR ranges 24]. VI. COCLUSIO In this paper, we developed an SER lower bound for transmit beamforming systems with finite-rate feedback. This bound applies to all beamformer designs and is tight for well-constructed

9 956 IEEE TRASACTIOS O WIRELESS COMMUICATIOS, VOL. 4, O. 4, JULY 25 beamformers. Comparing this bound with the SER corresponding to the ideal case with perfect channel knowledge, we quantified the power loss due to the finite-rate constraint across the entire SR range. Performance analysis in this paper will facilitate future work on adaptive modulation in beamformed transmissions with finite-rate feedback. Especially interesting is the applicability of these results to low or medium SRs, because the target error rate for an uncoded adaptive system is usually not very low and the selected transmission mode will not operate in a high SR regime. APPEDIX A PROOF OF SER SER lb For notational convenience, let us define f, z, γ) := Based on 23) and 25), we have SER SER lb = π M )π π + g ] t PSKγ z) sin 2. 47) f, z, γ) df Z z) df ] Z z) d. 48) } {{ } :=ϕ,γ) Using integration by parts, we can simplify the function ϕ, γ) in 48) as ϕ, γ) =f, z, γ) F Z z) F ] Z z) + F Z z) F ] df, z, γ) Z z) dz. 49) dz The first term in 49) is. Because f, z, γ) is an increasing function of z and F Z z) F Z z)] due to 24), the second term in 49) is always nonnegative for any. Therefore, ϕ, γ), which leads to SER SER lb in 48). APPEDIX B PROOF OF 42) Let us define ξ t,,γ) :=γ new /γ so that L t,,γ) = log ξ t,,γ). 5) Equation 39) implies that SER perfect CSI t, γ) =SER lb t,,γ ξ t,,γ)]. 5) otice that SER lb is a decreasing function of the average SR γ. In order to prove ξ t,,γ) ξ t,, ) 52) we need to show SER perfect CSI t, γ) SER lb t,,γ ξ t,, )]. 53) Define β := g PSK / sin 2 and α := / ) / t ) ] for notational convenience. Substituting 37), 32), and 4) into 53), we need to prove that + βγ) t +βγα t ) +βγα t ) t 54) for each. Taking logarithm on 54), we shall prove ln + βγ) ) ln +βγα t t t + t t ) ln +βγα t. 55) It is easy to verify that the function fx) = ln + βγe x ln α ) is convex because f x) regardless of the values of β, α, and γ. Equation 55) holds true due to f) t f t t ) + t f t ACKOWLEDGMET t ). 56) The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation thereon. REFERECES ] S. Bhashyam, A. Sabharwal, and B. Aazhang, Feedback gain in multiple antenna systems, IEEE Trans. Commun., vol. 5, no. 5, pp , May 22. 2] H. Boche and E. A. Jorswieck, Optimal power allocation for MISO systems and complete characterization of the impact of correlation on the capacity, in Proc. Int. Conf. Acoust., Speech, and Signal Process., Hong Kong, China, Apr. 23, pp ] R. Blum, MIMO with limited feedback of channel state information, in Proc. of IEEE Int. Conf. Acoust., Speech, and Signal Process., Hong Kong, China, Apr. 6, 23, vol. 4, pp ] S. Chennakeshu and J. B. Anderson, Error rates for Rayleigh fading multichannel reception of MPSK signals, IEEE Trans. Commun.,vol.43, no. 2/3/4, pp , Feb./Mar./Apr ] J. H. Conway, R. H. Hardin, and. J. A. Sloane, Packing lines, planes, etc.: Packings in Grassmannian space, Exp. Math.,vol.5,no.2,pp.39 59, ] J. W. Craig, A new, simple and exact result for calculating the probability of error for two-dimensional signal constellations, in Proc. Military Communications Conf., Mclean, VA, ov. 99, vol. 2, pp ] A. Goldsmith, S. A. Jafar,. Jindal, and S. Vishwanath, Capacity limits of MIMO channels, IEEE J. Sel. Areas Commun., vol. 2, no. 5, pp , Jun ] S. A. Jafar and A. Goldsmith, On optimality of beamforming for multiple antenna systems with imperfect feedback, in Proc. IEEE Int. Symp. Information Theory, Washington, DC, Jun. 2, p ] G. Jöngren and M. Skoglund, Utilizing quantized feedback information in orthogonal space-time block coding, in Proc. Global Telecommunications Conf., San Francisco, CA, ov. 27 Dec., 2, vol. 2, pp ] G. Jöngren, M. Skoglund, and B. Ottersten, Combining beamforming and orthogonal space-time block coding, IEEE Trans. Inf. Theory, vol. 48, no. 3, pp , Mar. 22. ] V. K.. Lau, Y. Liu, and T.-A. Chen, On the design of MIMO blockfading channels with feedback-link capacity constraint, IEEE Trans. Commun., vol. 52, no., pp. 62 7, Jan. 24.

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Madhow, Space-time transmit precoding with imperfect feedback, IEEE Trans. Inf. Theory, vol. 47, no. 9, pp , Sep ] Z. Wang and G. B. Giannakis, A simple and general parameterization quantifying performance in fading channels, IEEE Trans. Commun., vol. 5, no. 8, pp , Aug ] P. Xia, S. Zhou, and G. B. Giannakis, Multi-antenna adaptive modulation with beamforming based on bandwidth-constrained feedback, IEEE Trans. Commun., vol. 53, no. 3, pp , Mar ] S. Zhou and G. B. Giannakis, Optimal transmitter eigen-beamforming and space-time block coding based on channel mean feedback, IEEE Trans. Signal Process., vol. 5, no., pp , Oct ], Optimal transmitter eigen-beamforming and space-time block coding based on channel correlations, IEEE Trans. Inf. Theory, vol. 49, no., pp , Jul ] P. Xia, S. Zhou, and G. B. Giannakis, Achieving the Welch bound with difference sets, IEEE Trans. Inf. Theory, vol. 5, no. 5, pp. 9 97, May 25. Zhengdao Wang M 2) was born in Dalian, China, in 973. He received the B.S. degree in electrical engineering and information science from the University of Science and Technology of China, Hefei, in 996, the M.Sc. degree in electrical and computer engineering from the University of Virginia, Charlottesville, in 999, and the Ph.D. degree in electrical and computer engineering from the University of Minnesota, Minneapolis, in 22. He is currently with the Department of Electrical and Computer Engineering, Iowa State University, Ames. His interests lie in the areas of signal processing, communications, and information theory. Mr. Wang was the corecipient of the IEEE Signal Processing Magazine Best Paper Award in 23 and the IEEE Marconi Paper Prize Award in 24. Georgios B. Giannakis F 97) received the Diploma in electrical engineering from the ational Technical University of Athens, Greece, in 98, and the M.Sc. degree in electrical engineering, the M.Sc. degree in mathematics, and the Ph.D. degree in electrical engineering from the University of Southern California USC), Los Angeles, in 983, 986, and 986, respectively. After lecturing for one year at USC, he joined the University of Virginia, Charlottesville, in 987, where he became a Professor of electrical engineering in 997. Since 999, he has been a Professor with the Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, where he now holds an ADC Chair in Wireless Telecommunications. His general interests span the areas of communications and signal processing, estimation and detection theory, time-series analysis, and system identification subjects on which he has published more than 22 journal papers, 37 conference papers, and two edited books. His current research focuses on transmitter and receiver diversity techniques for single- and multi-user fading communication channels, complex-field and space-time coding, multicarrier ultrawideband wireless communication systems, cross-layer designs, and distributed sensor networks. G. B. Giannakis is the co)recipient of six paper awards from the IEEE Signal Processing SP) and Communications Societies 992, 998, 2, 2, 23, 24). He also received the SP Society s Technical Achievement Award in 2. He served as Editor in Chief for the IEEE SIGAL PROCESSIG LETTERS, as Associate Editor for the IEEE TRASACTIOS O SIGAL PROCESSIG and the IEEE SIGAL PROCESSIG LETTERS, as Secretary of the SP Conference Board, as Member of the SP Publications Board, as Member and Vice-Chair of the Statistical Signal and Array Processing Technical Committee, as Chair of the SP for Communications Technical Committee, and as a Member of the IEEE Fellows Election Committee. He has also served as a Member of the IEEE-SP Society s Board of Governors, the Editorial Board for the PROCEEDIGS OF THE IEEE, and the Steering Committee of the IEEE TRASACTIOS O WIRELESS COMMUICATIOS. Shengli Zhou M 3) received the B.S. and M.Sc. degrees in electrical engineering and information science from the University of Science and Technology of China, Hefei, in 995 and 998, respectively, and the Ph.D. degree in electrical engineering from the University of Minnesota, Minneapolis, in 22. He has been an Assistant Professor at the Department of Electrical and Computer Engineering, University of Connecticut, Storrs, since 23. His research interests lie in the areas of communications and signal processing, including channel estimation and equalization, multi-user and multicarrier communications, space-time coding, adaptive modulation, and cross-layer designs. Dr. Zhou has been an Associate Editor for IEEE TRASACTIOS O WIRELESS COMMUICATIOS since February 25.