OVER the past ten years, research (see the references in

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1 766 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 54, NO. 2, FEBRUARY 2006 Duplex Distortion Models for Limited Feedback MIMO Communication David J. Love, Member, IEEE Abstract The use of limited feedback in multiple-input multiple-output (MIMO) wireless communications has grown in interest over the last few years. Research has shown that feedback can be used to increase achievable data rates and add resilience against fading. Most of the work to this point, however, has ignored the data rate overhead associated with feedback. Models that do not consider feedback overhead are valid in wireless systems with ever present control channels, but they fail to capture the data rate cost of feedback when control channels are not present. In this paper, we present a new system model, called a duplex model, that captures the true loss (or cost) of feedback. A new distortion function for use with codebook design follows from the duplex system model. This system model is used to derive bounds on the amount of feedback needed under different antenna, signal-to-noise ratio, and bandwidth assumptions. Simulation results show the necessity of taking feedback into account. Index Terms Limited feedback, multiple-input multiple-output (MIMO) systems, vector quantization. I. INTRODUCTION OVER the past ten years, research (see the references in [1] and [2]) has shown that multiple-input multiple-output (MIMO) wireless systems can provide large gains in capacity and quality over single antenna wireless systems. Until the past few years, MIMO research could be classified into two situations: perfect knowledge of the channel at the transmitter or zero knowledge of the channel at the transmitter. Systems that adapt the transmitted signal to current channel conditions (often called closed-loop communication systems) using perfect knowledge of the downlink channel can provide improvements in capacity [though the gains decrease as the signal-to-noise ratio (SNR) increases] [3], diversity (see, for example, [4] [7] and the references in [1] and [2]), and mean squared errror [8], [9]. The assumption of perfect channel knowledge, however, is unrealistic in many systems. When frequency-division duplexing (FDD) is used, the forward and reverse links will usually be separated in frequency by an amount larger than the coherence bandwidth of the wireless channel. Because of this frequency separation, the forward and reverse channels will Manuscript received July 21, 2004; revised March 25, This work was supported in part by the SBC Foundation and by the National Science Foundation under Grant CCF This work appeared in part at the 42nd Annual Allerton Conference on Communication, Control, and Computing, Monticello, IL, September 29 October 1, 2004, and the IEEE Fall Vehicular Technology Conference, Dallas, TX, September 25 28, The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Helmut Boelcskei. The author is with the Center for Wireless Systems and Applications, School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN USA ( djlove@ecn.purdue.edu). Digital Object Identifier /TSP have a low correlation and a lack of instantaneous reciprocity. One solution to this problem is the use statistical reciprocity (see the statistical algorithms in [10] [17]) where the forward and reverse channels are assumed to share the same spatial correlation and/or means. Statistical closed-loop communication requires channel long-term statistics to be tracked at the transmitter. When it is impractical to track long-term channel statistics or the assumption of statistical reciprocity is unrealistic, the receiver can send feedback to the transmitter that conveys some form of channel information [18]. Limited feedback has been studied for a variety of closed-loop architectures including beamforming [19] [23], precoding [24] [30], adaptive space-time code design [31], [32], and signal covariance feedback [33], [34]. The general idea with limited feedback is to send bits of feedback each time the channel changes. Often times, the value of is determined by a standardized system specification (see [23]) where the data rate lost on the reverse link to feedback is fixed and uncontrollable. When the feedback rate is not fixed, however, user data rate improvements can be obtained by raising or lowering the number of feedback bits. Systems such as flexible MIMO ad hoc wireless network nodes might be able to adapt the number of feedback bits given operating parameters such as the SNR, bandwidth, coherence time, and antenna configuration. In this paper, we give an overview of a feedback model that takes into account the data rate overhead (or equivalently data rate loss) caused by sending feedback bits. This feedback overhead is nonnegligible and very dependent on the channel coherence time. We propose a new distortion model, called a duplex distortion model, that encompasses both the data rate gains and losses of limited feedback. This model is fundamentally different than previous feedback models, called simplex feedback models, that do not take into account feedback data rate loss. We also derive bounds on the amount of feedback that can be supported. These analytical bounds are derived using the singular value statistics presented in [35]. This paper represents an interesting relation to the robust signal design work in [36], [37]. Recent research has shown that independent identically distributed (i.i.d.), often called flat, complex Gaussian signaling is robust to any channel model. This robustness was quantified in [36] by Zamir and Erez, who found that there is a universal bound on the capacity difference (or capacity loss) between optimal waterfilling and i.i.d. complex Gaussian signaling. Combining these capacity loss bounds with our duplex feedback model yields necessary conditions on how much feedback should be used. In particular, we show that there is no reason to use feedback if the feedback rate overhead X/$ IEEE

2 LOVE: DUPLEX DISTORTION MODELS FOR LIMITED FEEDBACK MIMO COMMUNICATION 767 Fig. 1. Overview of system under consideration. is greater than the capacity loss encountered from i.i.d. complex Gaussian signaling. In other words, if the capacity gains over flat complex Gaussian signaling obtained by limited feedback are less than the data rate loss incurred, the feedback data rate must be reduced. This paper is organized as follows. Section II presents the system model. Duplex feedback models that incorporate feedback overhead are introduced in Section III. Bounds on the amount of feedback that can be used are given in Section IV. Section V overviews the design of limited feedback codebooks. Comments on some unrealistic assumptions such as the channel model and signal model are discussed in Section VI. Simulation results are presented in Section VII. We conclude in Section VIII. II. SYSTEM MODEL We will consider the system shown in Fig. 1. The system consists of two antenna mobiles that form an MIMO wireless system. The forward and reverse links are of bandwidth, and, where is the coherence bandwidth. This allows the baseband forward and reverse channels, assuming perfect synchronization and sampling, to be modeled as matrices. The forward link baseband model is described as where is the -dimensional transmitted vector, is an -dimensional noise vector with i.i.d. entries distributed according to (0, 1) (the complex normal distribution with i.i.d (0, 1/2) real and imaginary parts), is the matrix that models the channel response, and is the average SNR. The reverse link will be described by using definitions similar to the forward link. We will assume that is independent of and that both matrices have i.i.d. entries. The noise vectors and are independent and take new realizations at each transmission. Both channel matrices will follow a block fading model where they are constant for seconds before changing to an independent realization. Mobile 1 is assumed to have perfect knowledge of and no knowledge of while Mobile 2 is assumed to have perfect knowledge of and no knowledge of. Each time takes a new realization, Mobile 2 (Mobile 1) sends bits of channel feedback to Mobile 1 (Mobile 2). The bits are used to describe only the current channel realization. There- (1) (2) fore, we will restrict to be a nonnegative integer. Note that this integer assumption could be lifted in more realistic nonblock fading models. We will assume that this feedback corresponds to some sort of channel state information and that the reverse and forward links both use the same feedback strategy. Because the noise is Gaussian, the capacity (in bits/second/hertz) given a specific channel realization and side information is given by [34], [38], [39] where is the mutual information between and assuming is known at the receiver, is the base-2 logarithm, denotes conjugate transposition, is the matrix determinant function, and is the identity matrix. In (3), we have maximized over all transmit distributions with side information and, with denoting the expectation conditioned on side information. Thus, the optimal signaling distribution for given feedback is complex Gaussian [34]. Therefore, we will assume that and are zero-mean complex Gaussian vectors. The transmit covariance conditioned on a specific feedback is defined given by for the forward channel and for the reverse channel. We will impose the conditional power constraint that and, where is the matrix trace function. This means that we do not allow the average sum transmit power conditioned on some feedback information to exceed for any feedback possibility. Each feedback information in corresponds to a different covariance matrix. We will thus restrict the forward and reverse covariances to lie in a set (or codebook). The feedback bits on the reverse (forward) link convey the choice of from the codebook. We will assume that, with denoting the identity matrix, when. Because of the symmetry of the link model, we will switch notation and only consider one side of the link. Thus, we will omit the subscript or in the following analysis. Using the above formulation, the feedback capacity in bits/second given a realization is given by [19], [33], [34] Note that in general because of the lack of channel reciprocity. The design techniques in [33] and [34] are aimed at maximizing the ergodic feedback capacity given by (3) (4) (5)

3 768 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 54, NO. 2, FEBRUARY 2006 when the channel is an i.i.d. complex Gaussian matrix [3], [40]. We will also define a duplex distortion. The duplex distortion is given by (9) Fig. 2. Illustration of feedback overhead or cost. assuming has i.i.d. complex normal entries and is fixed. Maximizing (5) is equivalent to minimizing the loss (or distortion) where is the capacity assuming perfect knowledge of at both the transmitter and the receiver that is obtained via waterfilling. Capacity loss distortion has recently been studied with respect to robust input design. Both [36] and [37] studied the distortion when. This corresponds to the case of no feedback and a fixed covariance. Designing to minimize (6) assuming a fixed is a simplex design. This kind of design ignores the data rate loss associated with the feedback. Thus, we will denote as the simplex feedback capacity and as the simplex distortion. As mentioned earlier, modeling as a given constant is only reasonable when using a preallocated control channel. III. DUPLEX LIMITED FEEDBACK MODELING When a feedback of bits is transmitted per coherence time, a feedback rate of bits per second is needed. This rate can be nonnegligible in many situations. Therefore, the feedback rate must be taken into account when doing feedback designs and optimizations. To graphically illustrate this, consider Fig. 2. While feedback on the reverse link can increase the total data rate (which graphically corresponds to the total pipe diameter) of the forward link, it requires that much of the achievable data rate be used for transmitting feedback information. Thus, the user will only see a reduced portion of the possible data rate. This means that feedback is not free. For example, if we have a forward rate that supports 1 Mbits/s and a feedback rate of 200 kbits/s, the user will only see a data rate of at most 800 kbits/s. To take this rate into account in our model, we will define the duplex user rate as where. Unlike the feedback capacity in (5), the duplex user rate is not an increasing function of. When for any codebook.as well, the simplex feedback capacity and duplex user rate when bits (known as the uninformed transmitter capacity) are both given by (6) (7) (8) Unlike a simplex model, a duplex model has an optimal number of feedback bits. This is defined as (10) where is the set of nonnegative integers, argmin returns a global minimizer, and card and (11) with card denoting the set cardinality function, denoting the number at position of and denoting the nonnegative reals. A duplex model raises an important relationship between the amount of feedback needed and the performance when no channel knowledge is present at the transmitter. Feedback Design Property: When the rate loss incurred from feedback, is larger than the capacity loss obtained using an uninformed transmitter, feedback will only degrade performance. This follows from the fact that (12) when. Thus, the performance of the transmitter with no channel knowledge must always be taken into account when characterizing the optimal feedback amount. IV. FEEDBACK BOUNDS Given the duplex feedback model proposed in Section III, we can now determine bounds on the amount of feedback that can be supported. A. Universal Feedback Bound While the optimal feedback amount is inherently dependent on the chosen SNR, a universal bound can be derived that holds for all SNRs. We will combine work from [36] with the feedback design property to bound the amount of feedback that can provide capacity benefits over the uninformed transmitter. We will first review a result by Zamir and Erez regarding the performance loss incurred by flat (or i.i.d.) complex Gaussian signaling. It was shown in [36] and [41] that (13)

4 LOVE: DUPLEX DISTORTION MODELS FOR LIMITED FEEDBACK MIMO COMMUNICATION 769 for all that that are full rank. Thus, it easily follows where is the minimum (maximum) eigenvalue of. This follows because (14) for any using the simplex distortion defined in (6). This allows the following lemma to be proven. Lemma 1: In order to provide a duplex capacity greater than or equal to that provided by i.i.d. complex Gaussian signaling, the number of feedback bits must always satisfy where is the maximum power allocated to any of the channel eigenvalues from waterfilling. From the waterfilling algorithm [1, p. 68], it is easily shown that Equation (19) implies that Proof: We would like to bound such that or equivalently (15) (20) Noting that waterfilling with perfect knowledge of transmitter is optimal yields (16) at the Note that the bound in (20) goes to zero as because goes to infinity (21) By the Zamir and Erez bound in (14) (17) (18) This reverifies that the capacity of the uninformed transmitter approaches the perfect channel knowledge channel capacity for high SNR. It was shown in [35] that the probability density function (pdf) for (denoted as ) is given by 1 (22) This bound leads to an important corollary that gives a universal bound on coherence times that can support feedback. Corollary 1: If, feedback does not yield any capacity gain. This corollary establishes an important bound on the coherence time. Feedback is guaranteed to degrade the capacity if. To put some numbers to this, if khz and, then must be larger than s. Thus, for some mobile speeds and bandwidths, feedback cannot be supported for any SNR. B. SNR Dependent Feedback Bounds While the last section established a universal upper bound on the amount of necessary feedback, it ignored the important dependence of system performance on the SNR. It is well known that the mutual information achieved by open-loop MIMO with i.i.d. complex Gaussian signaling approaches the mutual information achieved by waterfilling as the SNR grows large [2]. SNR dependencies such as this must be taken into account when understanding feedback allocation. We will analytically characterize the simplex capacity loss incurred by i.i.d. complex Gaussian signaling at high SNR and low SNR. We will use random matrix results from the theory of Gaussian matrices to characterize the ergodic capacity loss. 1) High SNR Bound: Assume that is full rank. This assumption is valid because is full rank with probability one. It was shown in [37] that (19) Equation (20) can be rewritten as (23) Let denote the natural logarithm. Using the pdf of the minimum eigenvalue (24) where Ei is the exponential integral function defined by [42] and (24) follows from [42, p. 573]. The second term in (23) is simplified to (25) 1 The minimum eigenvalue pdf in [35] was given forg G; whereghas i.i.d. CN(0; 2) entries.

5 770 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 54, NO. 2, FEBRUARY 2006 where is the Euler constant given by [42] and (25) follows from an integral evaluation in [42, p. 573]. Collecting the results in (20), (24), and (25) gives Thus we must have that (26) (27) using the fact that. 2) Low SNR Bound: Waterfilling uses a covariance matrix formed as, where is the right singular vector matrix of and is a diagonal positive real matrix corresponding to the power levels on the different eigenmodes of. Using this covariance (28) Fig. 3. Normalized feedback bounds for a system. Therefore, the feedback must satisfy where again we used that. (34) where is the th entry of and is the th largest eigenvalue of. To satisfy tr, we must have that Therefore, for. Plugging this bound on in (28) gives Using this result (29) (30) C. Discussion The universal, high SNR, and low SNR feedback bounds give an interesting analysis of possible feedback rates. Fig. 3 demonstrates the feedback bounds in bits (normalized by ) for a 4 4 MIMO system as a function of the SNR. Note that for all SNRs not in the range of 12.3 to 25 db, feedback cannot be supported when. The normalized bounds for a 2 2 system are shown in Fig. 4. Unlike the first case, feedback cannot be supported outside of the 4.5 to 14.9 db range when. Thus, the number of antennas plays a major role in the feedback rate that can be supported. Both plots clearly show the tightness of the bounds at high and low SNR. This follows because (34) goes to zero as goes to zero and (27) goes to zero as goes to infinity. It must also be pointed out that these are only bounds on the amount of feedback. They only show that feedback might be applicable. (31) (32) where (31) follows from the fact that for and is the Frobenius norm of. Taking the expectation of (32) gives (33) V. VECTOR QUANTIZATION CODEBOOK DESIGN The design of the covariance codebook is directly related to traditional vector quantization (VQ) techniques [43]. The goal of a simplex codebook design is to choose to minimize (6) for a given SNR. The key difference between simplex distortion codebook design and traditional VQ is that the distortion is not of the form vec vec where is some sort of normed distance and vec is the vector obtained by stacking the columns of. A codebook designed using a simplex distortion is chosen to minimize the average capacity loss rather than obtain a better reconstruction of the

6 LOVE: DUPLEX DISTORTION MODELS FOR LIMITED FEEDBACK MIMO COMMUNICATION 771 Lemma 2: If there exists such that tr with then, where is the codebook constructed from by replacing with. Proof: Suppose there exists such that tr and that is nonzero. Let tr and be the codebook obtained with replaced by. Using these (36) (37) Fig. 4. Normalized feedback bounds for a system. optimal covariance matrix. Note, however, that the simplex distortion is guaranteed to decrease as increases. In contrast, the duplex distortion is a function of the feedback rate. Increasing the feedback rate by increasing may actually increase the duplex distortion. Both the simplex and duplex distortions can use the same codebook design techniques given a feedback rate. The difference is that the duplex model has an optimal feedback rate. This rate can be determined numerically by designing codebooks for various rates and choosing the rate and codebook that minimize the duplex distortion. The Lloyd vector quantization algorithm was shown in [34] to yield large ergodic simplex feedback capacities. The Lloyd algorithm works by: 1) generating a large number of test channels and an initial codebook; 2) partitioning the test data into Voronoi regions; 3) calculating a new codebook by finding the optimal codebook matrix from a distortion perspective for each Voronoi region of test channels; 4) repeating steps 2) and 3) until the codebook converges to a local minimum [43]. For a given Voronoi region it was shown in [34] that the calculation of the optimal codebook matrix can be approximately obtained by using the waterfilling covariance solution obtained from. To obtain a codebook that is approximately equal to the globally optimal solution, the Lloyd algorithm should be run multiple times with independently generated test channels and a different initial codebook. Lau et al. formulated their distortion function as (35) where is a Lagrange multiplier for the transmit power. The following lemma shows that the power penalty term in (35) can be omitted when using our power constraint. where is the th largest eigenvalue of. Therefore, it follows that the ergodic feedback capacity using the codebook is larger than with. Lemma 2 greatly simplifies the codebook design problem. We know that we only need to concentrate on positive semidefinite Hermitian matrices in the set. By restricting the feasible set for codebook design, (35) is equivalent to (6) with an added capacity term. Thus, the Lau et al. design techniques can be successfully employed for the design of limited feedback codebooks with the simplex and duplex distortion models. There is a further property of codebooks that follows specifically because of the model for. Lemma 3: For any unitary matrix, where. Proof: For a fixed unitary matrix has entries that are i.i.d. (0, 1) when has i.i.d. (0, 1) entries [44]. Therefore (38) Thus, there does not exist a unique optimal codebook for either simplex or duplex distortion. It is similar to the unitary property used in [19] [21] for the design of limited feedback beamforming vectors. This property can be used to reformulate the Lloyd search and avoid additional computational complexity. VI. COMMENTS ON FEEDBACK IMPLEMENTATION When a limited feedback system is implemented, a number of effects will change the analysis performed in this paper. We will address the effect of channels other than spatially uncorrelated complex Gaussians. While the analysis in these papers is for an i.i.d. complex Gaussian channel matrix, several of the results extend to arbitrary channel distributions. Equation (13) is true for any channel

7 772 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 54, NO. 2, FEBRUARY 2006 Fig. 5. Duplex capacity loss for an system as a function of the SNR. Fig. 6. Capacity comparison as a function of the SNR of various transmissions. realization that is full rank. Thus the bound in (18) will hold for any channel that is rank-deficient with probability zero. The high SNR and low SNR analysis, however, will greatly depend on the channel distribution. While it must be true that the feedback rate must still go to zero as the SNR goes to zero, the behavior of the feedback bound at low SNR will be channel dependent. It may not be true, however, that the feedback amount goes to zero as the SNR goes to infinity depending on the level of correlation. If a channel was rank deficient, the feedback bounds would have to be readdressed to incorporate the loss in performance of the zero channel state information (CSI) scenario. Note that the analysis throughout this paper applies primarily to i.i.d. Gaussian signaling on each antenna. Thus this analysis is primarily for precoded spatial multiplexing systems [2]. The analysis, however, can be adapted to more general space-time block codes (e.g., [45]) if the space-time transmission framework can be expressed as an effective input output relationship (39) The problem with this is that the is usually no longer spatially uncorrelated (or possibly not even complex Gaussian). In this case, non-gaussian channel analysis techniques must be employed. VII. SIMULATIONS To understand actual performance of the ergodic duplex user rate, we simulated a feedback system using codebooks designed from the algorithm in [34] with the restriction that for all. We considered an 8 8 MIMO system with and simulated the duplex distortion (or duplex capacity loss) given in (9) normalized by the bandwidth. The results are shown in Fig. 5. One-, four-, and eight-bit codebooks were simulated. For comparison, i.i.d. complex Gaussian signaling with a codebook of was simulated. From 10 to 3 db, eight bits of feedback performs better than no feedback. Outside of this range, it is better to use no feedback. This plot demonstrates that at low Fig. 7. Optimal number of feedback bits as a function of the SNR. and high SNR, feedback is unnecessary and possibly degrades performance. Fig. 6 shows the duplex user rate (in bits/second/hertz) of a limited feedback 2 2 system along with capacities of perfect channel knowledge waterfilling and i.i.d. complex Gaussian signaling. It was assumed that, and the duplex user rate was calculated as in (7). The feedback amounts were optimized to minimize the duplex distortion. Note that the 2 2 case offers only modest gains from feedback when the feedback rate overhead is subtracted out. For SNRs outside of the pictured range, limited feedback designed from a duplex distortion model optimizes to bits. Note that the optimal number of feedback bits is a direct function of the SNR. Fig. 7 shows the optimal amount of feedback for a 2 2 system when Lloyd designed codebooks are used for various values of. The number of feedback bits scales very slowly with the coherence time. Even at, all the tested SNRs had optimal feedback levels less than five bits. Interestingly, when, no SNRs required feedback. This

8 LOVE: DUPLEX DISTORTION MODELS FOR LIMITED FEEDBACK MIMO COMMUNICATION 773 An additional area of research is the application of feedback when more than two users are present. Obviously, it is impractical for each mobile (or node) to send back channel data to every other mobile. In this case, feedback overhead would quickly overwhelm the total network capacity. It is of interest to develop smart feedback strategies for multiuser systems. REFERENCES Fig. 8. Duplex capacity loss for a system as a function of the SNR. is indicative of the fact that feedback cannot be used when the channel is varying quickly. The final simulation, shown in Fig. 8, shows the duplex distortion (in bits/second/hertz) for a 3 3 MIMO system with. One-, two-, and three-bit codebooks were simulated. As well, i.i.d. complex Gaussian signaling (or equivalently zero feedback bits) was simulated. Three bits of feedback is optimal from 11.2 to 8.25 db. Note that between 15 and 11 db, there exist SNRs where one and two bits of feedback are, respectively, optimal. VIII. CONCLUSION In this paper, we studied the design of MIMO feedback schemes when the feedback overhead is taken into account. We proposed a new duplex feedback model that incorporates the data rate loss from feedback into the capacity expressions. We found that there is no reason to use feedback if the feedback rate is greater than the capacity loss associated with i.i.d. complex Gaussian signaling. In particular, if a rate loss of bits/second is incurred by using i.i.d. complex Gaussian signaling instead of optimal waterfilling, there is no reason to have a feedback rate larger than. This basic fact led to a derivation of a universal (i.e., applicable to all SNR values) feedback bound following the results in [36]. We also derived two feedback bounds that are tight at low and high SNR, respectively. Simulations show that feedback can be detrimental if the coherence time is not sufficiently large. Future research in duplex distortion abounds. First, a more realistic statistical channel evolution model other than the block fading model is needed. In some sense, the block fading model is a best case feedback model. The channel will evolve, creating a mismatch between the designed covariance matrix and the actual channel. Additionally, it may be possible to analytically characterize the optimal amount of feedback given the coherence time, bandwidth, number of antennas, and SNR. This paper presented bounds on this feedback amount, but simulations show these bounds can be quite loose. [1] A. Paulraj, R. Nabar, and D. Gore, Introduction to Space-Time Wireless Communications. New York: Cambridge Univ. Press, [2] E. G. Larsson and P. Stoica, Space-Time Block Coding for Wireless Communications. New York: Cambridge Univ. Press, [3] I. E. 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