Equal Gain Transmission in Multiple-Input Multiple-Output Wireless Systems

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1 TO APPEAR IN IEEE TRANSACTIONS ON COMMUNICATIONS, 003 Equal Gain Transmission in Multiple-Input Multiple-Output Wireless Systems David J. Love, Student Member, IEEE, and Robert W. Heath, Jr., Member, IEEE Abstract Multiple-input multiple-output (MIMO) wireless systems are of interest due to their ability to provide substantial gains in capacity and quality. This paper proposes equal gain transmission (EGT) to provide diversity advantage in MIMO systems experiencing Rayleigh fading. The applications of equal gain transmission with selection diversity combining, equal gain combining, and maximum ratio combining are addressed. It is proven that systems using equal gain transmission with any of these combining schemes achieve full diversity order when transmitting over a memoryless, flat fading matrix channel with independent entries. Since in practice full channel knowledge at the transmitter is difficult to realize, a quantized version of EGT is proposed. An algorithm to construct a beamforming vector codebook that guarantees full diversity order is presented. Monte Carlo simulation comparisons with various beamforming and combining systems illustrate the performance as a function of quantization. Index Terms-Diversity methods, MIMO systems, Rayleigh channels, equal gain transmission. I. INTRODUCTION Antenna diversity has been shown to improve mean signal strength and reduce signal level fluctuations in fading channels []. These benefits are a direct result of the fact that sufficiently spaced antennas encounter approximately independent fading channels. Antenna diversity can be utilized at the transmitter and/or receiver. Receive antenna diversity systems intelligently combine the multiple received copies to provide a higher average receive signal-to-noise ratio (SNR) (see [] [4], and the references therein). Transmit antenna diversity is more difficult to obtain since it requires either channel dependent beamforming or channel independent space-time coding ([5], [6]). Classical wireless research focused on the case where antenna diversity was employed exclusively at either the transmitter or receiver. When multiple antennas are only available at the transmitter, beamforming techniques such as selection diversity transmission (SDT), equal gain transmission (EGT), and maximum ratio transmission (MRT) have been used to exploit the diversity available from the multiple-input singleoutput (MISO) wireless channel. On the other hand, when multiple antennas are only available at the receiver, combining schemes such as selection diversity combining (SDC), equal gain combining (EGC), and maximum ratio combining This work was presented in part at the IEEE Global Telecommunications Conf., Taipei, Taiwan, R.O.C., November 7-, 00. This work was supported by a Microelectronics and Computer Development Fellowship through The University of Texas at Austin. D. J. Love and R. W. Heath, Jr. are with the Wireless Networking and Communications Group, Department of Electrical and Computer Engineering, The University of Texas at Austin, Austin, TX 787 USA. ( {djlove, rheath}@ece.utexas.edu.) (MRC) have been used to obtain diversity advantage from the corresponding single-input multiple-output (SIMO) wireless channel. When antenna diversity is employed at both the transmitter and receiver, the multiple-input multiple-output (MIMO) channel encountered in the memoryless case is a matrix. Beamforming and combining can be used in MIMO communication channels, however the beamforming vector and receive combining vector must now be jointly designed to maximize the receive SNR. MIMO maximum ratio transmission and maximum ratio combining was addressed in [7] and shown to provide full diversity order. Systems using selection diversity transmission and maximum ratio combining were studied in [8] and also shown to provide full diversity order. Designing these vectors is non-trivial and in many cases involves an optimization problem that can not be easily solved in realtime systems. Equal gain transmission has more modest transmit amplifier requirements than maximum ratio transmission since it does not require the antenna amplifiers to modify the amplitudes of the transmitted signals. This property allows inexpensive amplifiers to be used at each antenna as long as the gains are carefully matched. For example, SIMO EGC and MISO EGT have already been considered as low complexity alternatives to MRC and MRT respectively (see [], [9] [] and the references therein). Despite the importance of MIMO communication systems, the application of EGT to these systems has not yet been addressed. In this paper we propose equal gain transmission, combined with either SDC, EGC, or MRC at the receiver, to provide full diversity order in MIMO wireless systems transmitting over memoryless, independent and identically distributed (i.i.d.) Rayleigh fading channels. We jointly solve for the optimal beamforming and combining vectors by maximizing the received SNR. For the cases considered, it is possible to find the optimum combining vector as a function of the beamforming vector; finding the optimum beamforming vector usually requires a nonlinear optimization. We prove that any beamforming and combining system whose set of possible beamforming vectors contains a subset of M t orthogonal vectors and whose set of possible receive combining vectors contains a subset of M r orthogonal vectors, where M t and M r are the number of transmit and receive antennas respectively, provides full diversity order. We use this proof to show that MIMO systems using EGT combined with SDC, EGC, or MRC obtain full diversity order. One problem encountered during implementation of MISO and MIMO beamforming systems is that full channel knowledge is required at the transmitter to design optimal beamform-

2 TO APPEAR IN IEEE TRANSACTIONS ON COMMUNICATIONS, 003 ing vectors. In many systems such as those using frequency division duplexing, it is impossible to obtain complete channel information at the transmitter. One solution to this problem is to let the receiver design the beamforming vector and then send the vector to the transmitter ([3], [4]). Since infinite resolution is impossible, it is preferable to quantize the set of possible beamforming vectors into a codebook and then send only the codebook entry of the desired beamforming vector. Quantized maximum ratio transmission for MISO systems was addressed in [3] while MISO quantized equal gain transmission (QEGT) was discussed in [4]. QEGT has also been chosen as one of the closed-loop beamforming techniques in W-CDMA [5]. Due to the difficulty of finding the optimal beamforming vector in beamforming and combining systems, MIMO quantized beamforming represents a much more difficult problem than in MISO systems. Since full channel knowledge is often not available at the transmitter, we propose and study MIMO QEGT. We develop an algorithm for QEGT codebook construction that guarantees full diversity order for memoryless, i.i.d. Rayleigh fading channels given that a sufficient number of bits are allocated for feedback. This minimum number of bits depends directly on the number of transmit antennas. We show that when the number of bits allocated for feedback is equal to log M t then the beamforming scheme performs on average identically to selection diversity transmission. An important side benefit of QEGT is that the optimal beamforming vector can be found through a low complexity brute force search as opposed to a nonlinear optimization. This paper is organized as follows. Section II reviews MIMO communication with beamforming and combining. Basic performance properties of MIMO beamforming and combining systems are presented in Section III. We discuss EGT systems with SDC, EGC, and MRC in Section IV. We propose MIMO QEGT and provide a full diversity codebook design method in Section V. In Section VI we show simulation results that verify the performance analysis of EGT and QEGT systems. We provide some conclusions in Section VII. II. SYSTEM OVERVIEW A MIMO system using beamforming and combining is illustrated in Fig. with M t transmit antennas and M r receive antennas. A symbol s (s C, the field of complex numbers) is multiplied by weight w l (w l C) at the l th ( l M t ) transmit antenna. The signal y k received by the k th ( k M r ) receive antenna is given by ( ) y k = h k,l w l s + n k () l= where h k,l is a memoryless fading channel that is constant over several channel utilizations and distributed according to CN (0, ) and n k is a noise term distributed according to CN (0, N 0 ). We assume that h k,l is independent of h i,j if k i or l j and n k is independent of n i if k i. Note that time dependence has been abstracted from the discussions by assuming that the channel is constant over several transmissions. The data received by the k th receive Fig.. Bit Stream, b(n) Coding and Modulation Weight update s w w w - w h, Feedback h Mr-, Block diagram of a MIMO system. n n y y n Mr- y Mr- z z z Mr- y Mr n Mr z Mr antenna, y k, is multiplied by z k (z k C with denoting conjugation). The weighted output of each of the M r receive antennas is then combined to produce x. This formulation allows the equivalent system to be written in matrix form as x = (z H Hw)s + z H n () with w = [w... w ] T, z = [z... z Mr ] T, n = [n... n Mr ] T, and H denoting the M r M t matrix with coordinate (k, l) equal to h k,l where T denotes transposition and H denotes conjugate transposition. We call z H Hw the effective channel. For optimum performance, w and z should be chosen as a function of the channel to minimize the probability of error. The nearest neighbor union bound on the symbol detection error probability can be stated [6] as ( ) d P e N e Q min γ r (3) where N e is a real constant that is the average number of nearest neighbors per symbol, d min is the minimum distance of the transmit constellation normalized to unit energy, γ r is the receive SNR, and Q is the Gaussian Q-function. Note that N e can be adjusted in order to provide a close approximation to the actual probability of error [6]. Since the Q-function is a monotonically decreasing function and d min is assumed fixed, minimizing the bound requires that we maximize the SNR. It follows from () that γ r = E t z H Hw z N 0 = ( Et w ) z H z H w w N 0 (4) where is the matrix two-norm and E t is the transmitted symbol s energy. Notice that (4) does not vary with z, therefore we can without loss of generality fix z =. We also can see that the transmitter transmits with total energy E t w. Therefore, due to power constraints at the transmitter, we can take w =. With these assumptions, the instantaneous receive SNR, γ r, can be expressed as γ r = E tγ r N 0 (5) where Γ r = z H Hw is the effective channel gain. x

3 TO APPEAR IN IEEE TRANSACTIONS ON COMMUNICATIONS, Maximizing Γ r is a multidimensional optimization problem. We will therefore employ standard linear programming terminology in dealing with the maximization. Recall that the set over which a cost function is optimized is called the feasible set [7]. We will denote the set of all possible beamforming vectors as the beamforming feasible set and the set of all possible combining vectors as the combining feasible set. The beamforming feasible set defines the set over which the beamforming vector is chosen. When w can be any unit vector, the beamforming scheme is called maximum ratio transmission (MRT). A beamforming scheme where each transmit antenna is denoted by equal gain transmission (EGT). If w is constrained to be a column of I, the M t M t identity matrix, the beamforming scheme is called selection diversity transmission (SDT). In MIMO systems, the combining vectors need to be chosen in addition to the beamforming vectors, perhaps under different constraints. A receiver where z can be any unit vector is using maximum ratio combining (MRC). An equal gain combining (EGC) receiver constrains each receive antenna l has weight w l with w l = weight z k to have z k = Mr. A receiver where z is a column of I Mr is using selection diversity combining (SDC). In this paper, equal gain transmission and combining are considered. The definition of equal gain transmission allows us to express w as w = e jθ [ ] = e jθ e jθ... e jθ M T t where θ = [θ θ... θ ] T and θ k [0, π). As well EGC vectors can be expressed as z = Mr e jφ where φ = [φ φ... φ Mr ] T and φ i [0, π). It is important to note that uniqueness is not guaranteed for any beamforming and combining scheme. Multiplication of the beamforming vector w by e jξ and the combining vector z by e jϕ with ξ, ϕ [0, π) does not change Γ r. For this reason when optimizing a cost function we will define arg max to return the set of global maximizers. We later exploit this nonuniqueness to reduce the size of the solution set and thus the amount of feedback in the QEGT system. These transmission and combining methods can be intermixed together to suit different system requirements. If beamforming method A is used at the transmitter and combining method B is used at the receiver, we will call this an A/B system. III. SYSTEM PERFORMANCE Given no design constraints on the form of w or z the nearest neighbor union bound tells us that the optimal solutions are the beamforming vector and combining vector that maximize γ r. Since we assume that E t and N 0 are fixed, this simplifies to maximizing the effective channel gain Γ r. Lemma gives a clear upper bound on Γ r. Lemma : The SNR γ r is maximized when z and w are the left and right singular vectors of H respectively corresponding to the largest singular value of H with Γ r = H. The proof of Lemma is given in [8]. Since maximum ratio transmission and maximum ratio combining pose no restrictions other than unit two-norm on the vectors w and z respectively, we can therefore conclude that for any channel matrix H the effective channel gain of an MRT/MRC system is H. Lemma gives us an upper bound on Γ r for EGT systems. It is often difficult to compute meaningful, closed-form expressions for the average probability of symbol error (average taken with respect to the channel [4]) even for much simpler SIMO equal gain combining systems ([4], [0] [], [9]). We will therefore instead use the metrics of diversity order and array gain ([], [3], [4]). A system is said to have array gain A and diversity order D if the average probability of symbol error is inversely proportional to A(E t ) D for E t 0. Lemma : Let Γ r, Γ r be the effective channel gains and D, D be the diversity orders for two different MIMO beamforming and combining system. If Γ r Γ r for all H, then D D. Proof: The nearest neighbor upper bound tells us that for large E t the probability of symbol error is a decreasing function of the effective channel gain. Therefore, if Γ r Γ r for any channel H then the average probability of symbol error for Scheme will always be less than the average probability of symbol error for Scheme. Thus there does not exist a E t 0 such that the average probabilities of symbol error are equivalent. We can therefore conclude that D D. An important corollary that we will use later in upper bounding the diversity order of MIMO equal gain transmission systems follows from this theorem. Corollary : For any M t M r wireless systems using beamforming and combining, the diversity order is always less than or equal to M r M t when transmitting over a memoryless, i.i.d. Rayleigh fading matrix channel. Proof: By Lemma the effective channel gain of MRT/MRC systems, which are known to have diversity order M r M t (see [7], [0]), will be greater than or equal to the effective channel gain of any other M t M r beamforming and combining system. Therefore by Lemma, for any M t M r wireless system using beamforming and combining, the diversity order is always less than or equal to M r M t when transmitting over a memoryless, i.i.d. Rayleigh fading matrix channel. In our diversity advantage proofs we will also lower bound the diversity order. The following lemma provides an important result in the theory of beamforming and combining wireless systems transmitting over i.i.d. Rayleigh fading MIMO channels. Lemma 3: If the beamforming feasible set and combining feasible set of an M t M r beamforming and combining system contain M t and M r orthogonal vectors respectively then the system has a diversity order of M r M t when transmitting over memoryless, i.i.d. MIMO Rayleigh fading channels. Proof: Let A/B denote a beamforming and combining method satisfying the orthogonality conditions. Corollary tells us that the diversity order is upper bounded by M r M t. Let U t be an M t M t matrix whose columns are the M t orthogonal beamforming vectors and U r be an M r M r matrix whose columns are the M r orthogonal combining vectors. Let Γ rorth be the effective channel gain for a beamforming and combining system that uses only the columns of U t as beamforming vectors and the columns of U r as combining vectors. The orthogonality of the columns and the unit two-

4 TO APPEAR IN IEEE TRANSACTIONS ON COMMUNICATIONS, norm requirement allow us to write that U H t U t = I and U H r U r = I Mr, meaning that U t and U r are both unitary. Let Γ rorig denote the effective channel gain of the original system. Since the columns of U t are contained in the beamforming feasible set of the original system and the columns of U r are contained in the combining feasible set of the original system, we can conclude that Γ rorig Γ rorth for any channel realization H. Therefore, the diversity order of the original system is greater than or equal to the diversity order of the restricted orthogonal system. For any channel realization H, we have that Γ rorth = max m M r = max m M r max (U r ) H n M t mh(u t ) n ( max U H r HU t n M t )m,n (6) with (U t ) n denoting the n th column of U t and ( ) U H r HU t m,n denoting the (m, n) entry of U H r HU t. We assumed that H was a complex Gaussian distributed random matrix whose entries were all independent. By the invariance of Gaussian random matrices to unitary transformation [], H is equivalent in distribution to U H r HU t. Therefore Γ rorth d = max m M r max h m,n (7) n M t with d = denoting equivalence in distribution. The distribution equivalent system defined in (7) is the one that chooses the pair of antennas with the largest gain channel. This is a selection diversity transmission and combining system. These systems are known to provide a diversity order of M r M t [3], [], [3]. We have now upper and lower bounded the diversity order of the A/B system by M r M t. We can conclude that any system using a beamforming feasible set and combining feasible set with M t and M r orthogonal vectors respectively has a diversity of order M r M t. IV. EQUAL GAIN TRANSMISSION In this section we will consider equal gain transmission in conjunction with selection diversity combining, equal gain combining, and maximum ratio combining. We will address the design of the beamforming vectors and the diversity performance for each of the combining schemes. A. Equal Gain Transmission/Selection Diversity Combining It is often convenient to employ a selection diversity combiner at the receiver because of its low complexity implementation. A multi-antenna receiver using selection diversity combining requires only a switch that can choose between M r different antenna outputs and a single radio chain. Selection diversity combining is also the only combining scheme where a general expression for the optimal equal gain transmission vector can be derived. As discussed in Section II, we wish to choose w and z in order to maximize Γ r = z H Hw. When selection diversity combining is employed at the receiver, z is one of the columns of I Mr. Therefore Γ r = max (Hw) m M m M t = max h r m M r m,n w n. (8) n= where (Hw) m is the m th entry of the vector Hw. Substituting in the expression w = e jθ into (8) we find that M t Γ r = max h m M r M t m,n e jθ n. (9) Notice that Γ r is bounded by Γ r max m M r M t = max m M r M t n= ( ) h m,n e jθn n= ( h m,n n= ) = M t H (0) where is the matrix sup-norm. The sup-norm can be rewritten in terms of the rows as H = max m Mr ( H ) T m where is the onenorm and ( H ) T m is the mth row of H. Therefore, the bound in (0) is achievable by letting θ = ξ phase (( H ) ) ( T K where K arg max ) m Mr H T m, ξ [0, π), and the function phase : C [0, π) returns the phase of each entry of a vector. We now have an expression for the optimal equal gain transmission vector when selection diversity combining is employed. In this case with an arbitrary ξ with w = e j(ξ phase((ht ) K )) ( K arg max H T ) m M r m. With this beamformer, the receive SNR γ r is γ r = E tγ r N 0 () = E t H M t N 0. () Using Lemma 3, we can also comment on diversity order. Theorem : The diversity order of a MIMO system using equal gain transmission and selection diversity combining is M r M t when transmitting over memoryless, i.i.d. MIMO Rayleigh fading channels. Proof: Let U be the M t M t point discrete Fourier transform (DFT) matrix where entry (k, l) of U is given by e jπkl/. By our definition, UU H = I so U is unitary. The columns of U are all acceptable equal gain transmission vectors so the beamforming feasible set contains M t orthogonal vectors. The receive combiner uses the columns of I Mr as a feasible set thus it contains M r orthogonal vectors by definition. By Lemma 3, the EGT/SDC system has a diversity order of M r M t.

5 TO APPEAR IN IEEE TRANSACTIONS ON COMMUNICATIONS, B. Equal Gain Transmission/Equal Gain Combining While selection diversity combining is easily implemented, equal gain combining receivers have been shown to improve the average probability of symbol error performance [4]. Equal gain combiners require only moderate hardware complexity because each of the receive antennas weights is restricted to be of magnitude Mr. The gain of the effective channel for an EGT/EGC system can be bounded by z H Hw = M r M r m= ( Mr M r m= e jφm (Hw) m (Hw) m ) = M r Hw (3) where the inequality follows from the equal gain properties of z. The bound in (3) is achievable when z = Mr e j(ϕ+phase(hw)) where ϕ is an arbitrary phase angle. Using the optimal equal gain combining vector, Γ r = M r Hw. This can be rewritten for equal gain transmission as Γ r = He jθ M r M t. Therefore, the optimal phase vector θ is given by θ arg max He jϑ. (4) ϑ [0,π) M t The optimization problem defined by (4) has no known simple, closed-form solution. Again note that the solution defined by (4) also does not have a unique solution. In fact, if e jθ is an optimal equal gain transmission vector then e jξ e jθ is also optimal for any ξ [0, π) because He jθ = He jξ e jθ. Theorem : A MIMO system using equal gain transmission and equal gain combining achieves a diversity order of M r M t when transmitting over memoryless, i.i.d. MIMO Rayleigh fading channels. Proof: We have shown in the proof of Theorem that the equal gain beamforming feasible set contains a set of M t orthogonal vectors. Similarly, let V be the M r M r unitary DFT matrix. Each column of V is a possible equal gain combining vector. Thus the combining feasible set contains a set of M r orthogonal vectors. Lemma 3 tells us that an EGT/EGC system has a diversity order of M r M t. C. Equal Gain Transmission/Maximum Ratio Combining Maximum ratio combining provides the best performance among all combining schemes thanks to the absence of constraints placed on the set of possible combining vectors. The combining vector is designed specifically to maximize the effective channel gain z H Hw. For EGT/MRC systems, the effective channel gain can be upper bounded by z H Hw z Hw = Hw. (5) The upper bound in (5) is achievable if z = Hw/ Hw. Thus the optimum phase vector θ solves θ arg max He jϑ. (6) ϑ [0,π) M t Once again the phase vector θ is not unique because w can be arbitrarily multiplied by any unit gain of the form e jξ with ξ [0, π). Theorem 3: A MIMO system using equal gain transmission and maximum ratio combining has a diversity on the order of M r M t when transmitting over memoryless, i.i.d. MIMO Rayleigh fading channels. Proof: We have already shown in the proof of Theorem that the equal gain beamforming feasible set contains M t orthogonal vectors. Note that each column of I Mr is a possible maximum ratio combining vector. Therefore, the maximum ratio combining feasible set contains M r orthogonal vectors. By Lemma 3, an EGT/MRC system has a diversity of order M r M t. V. QUANTIZED EQUAL GAIN TRANSMISSION In real-world systems equal gain transmission is not an implementable solution for two main reasons, complexity and overhead. First note that the optimization problems in (4) and (6) do not have closed-form solutions for arbitrary M r and M t. Implementation requires an iterative method, costing precious clock cycles. Convergence of such an iterative method to the global maximum is not guaranteed. Second, due to a limited feedback channel in most systems it is impossible to send back high precision phase angles. Wireless systems must always limit control data overhead in order to achieve large user data rates. If high resolution phase angles were sent to the transmitter this control overhead would overwhelm the limited feedback capacity. One solution is to quantize the set of possible θ creating a system called quantized equal gain transmission (QEGT). This quantizes the space of beamforming vectors and eliminates the problem of finding the global maximum by using a brute force search. As we show, the quantization can be quite low, reducing feedback requirements, without much performance sacrifice. Suppose that B bits of quantization are used for each phase. Complete phase vector quantization would require BM t bits of overhead, however, from (), (4), and (6) a term of e jθ can always be factored out without loss of generality. Thus θ can be written in the form θ = [0 (θ θ )... (θ θ )]. Therefore if B bits are used for each phase angle within θ then only B(M t ) bits are necessary to define M t antenna gains. Thus by taking into account the non-uniqueness of the beamforming vector we have reduced the amount of feedback. Let W be the codebook, or set, of all possible quantized equal gain transmission vectors. For B bits of quantization, card(w) = B(M t ) with card( ) denoting cardinality. A brute force search through the possible vectors can be used to solve either (), (4), or (6). We must now turn our attention to the design of the vectors within W. A quantization scheme that does not maintain full diversity order is wasting valuable resources by not making use of

6 TO APPEAR IN IEEE TRANSACTIONS ON COMMUNICATIONS, the full M r M t independently fading channels arising from the multi-antenna system. Therefore when using QEGT, it is imperative to maintain full diversity order for small B. To proceed with the codebook design, note that the proofs of Theorems,, and 3 employ the M t M t unitary DFT matrix. If our codebook always contains the columns of the M t M t DFT matrix then we are guaranteed by Lemma 3 to have full diversity order for selection diversity combining, equal gain diversity combining, and maximum ratio combining. Therefore if B( ) M t and U denotes the M t M t unitary DFT matrix we will require that for all U i, there exists w W such that w = U i. By construction, the beamforming codebook will always contain the columns of the M t M t unitary DFT matrix when B log (M t )/(M t ). We can thus conclude from Lemma 3 that quantized systems such as QEGT/SDC, QEGT/EGC, and QEGT/MRC obtain a diversity advantage of order M r M t if B (log M t ) /(M t ). The proposed algorithm uses a set W made up of the column vectors of the RM t RM t DFT matrix truncated to M t rows and scaled by R where R is an integer that satisfies R ( B( ) )/M t. Since M t divides RM t, the set W will contain the set of column vectors W of the M( t M t unitary) DFT matrix. The codebook then chooses B(M t ) M t vectors from W \W, with \ denoting set difference, that minimize an absolute correlation requirement and adds them to W. W = W can then be used as a full diversity transmitter codebook. The algorithm is given in its entirety below.. Fix a constant R such that RM t B( ).. Construct a matrix A where A consists of the first M t rows of the RM t RM t unitary DFT matrix. Scale this matrix by R to guarantee unit vector columns. 3. Construct a set of vectors W where the members of W are the columns of A. 4. Let the set W be the columns of the M t M t unitary DFT matrix. 5. Choose the vector w W \W such that v W \W, f(v) f(w) where f is defined as f(w) = max x W x H w. (7) Set W = W {w}. 6. Repeat 5 until card(w ) = B( ). The intuition behind this algorithm is to begin with a codebook of only M t orthogonal vectors and then add vectors one-by-one to this codebook such that the vector added at each step is distant from the current codevectors. We have shown in Section III that for any ξ [0, π), the beamforming vectors w and e jξ w provide the same receive SNR. We will therefore try to maximize the phase-invariant distance d between any two vectors defined by d(w, w ) = min w e jξ w = w Hw ξ [0,π) (8) where w and w are unit vectors. Thus f(w) returns the absolute correlation corresponding to the phase-invariant distance of the closest vector in W to w [4], [5]. Two points are imperative to note about this algorithm. First, as R grows large W approaches an optimal equal gain codebook in terms of the cost function. Second, as B increases it is possible to approach a true EGT system since for any phase vector entry θ i, k can be chosen given RM t and B such that the error of e jθ i e jπk/(rm t) goes to zero as B (and thus R) grows large. This shows that QEGT can perform arbitrarily close to EGT. Selecting an optimal B requires making tradeoffs between the amount of tolerable feedback and array gain. As a rule of thumb we have found that QEGT using a total feedback of at least M t bits, or rather B M t /(M t ), provides performance almost identical to unquantized EGT. Notice that when B = (log M t )/(M t ), card(w ) = M t. In this case the beamforming feasible set will contain exactly M t orthogonal vectors. The following observation gives an exact performance analysis for this case. Observation If B = (log M t )/(M t ) the system is equivalent in distribution to an SDT system with the same combining scheme. The proof of this follows easily from the distribution invariance of memoryless, i.i.d. Rayleigh fading matrices to multiplication by unitary matrices. The implication of this observation is that when B = (log M t )/(M t ) our algorithm becomes a modified selection diversity beamformer. The only difference is that the beamforming vectors have been rotated by the unitary DFT matrix. VI. SIMULATIONS For this section we simulated the average probability of symbol error with various antenna configurations and beamforming schemes. All simulations used i.i.d. Rayleigh fading with h k,l distributed according to CN (0, ). Monte Carlo simulations ran over.5 million iterations per SNR point. Experiment : We considered an M t = M r = MIMO QEGT/EGC system with various values of B and R = B( ) /M t so W \W was the empty set at the conclusion of the algorithm. Unquantized EGT/EGC, SDT/MRC, SDT/SDC, and MRT/MRC systems were also simulated. Each simulated system used binary phase shift keying (BPSK) modulation. Fig. shows the results from this experiment. Notice that all the quantized curves have a diversity order of 4. The array gain between and bits of quantization is approximately 0.6dB. However, the array gain between and 3 bits of quantization is only about 0.08dB. This is indicative that QEGT approaches EGT performance as B increases. Experiment : This experiment considered an M t = M r = beamforming and combining systems using QEGT/MRC system with various values of B, unquantized EGT/MRC, SDT/MRC, and MRT/MRC systems. All simulations used BPSK modulation. Fig. 3 shows the performance. It is important to note that for B = the average symbol error rate curve for a QEGT/MRC system is on top of the average symbol error rate curve for an SDT/MRC system. This verifies the observation presented for this special case. We took R = B( ) /M t once again. The array gain between and bits of quantization is approximately db. Once again the gain of

7 TO APPEAR IN IEEE TRANSACTIONS ON COMMUNICATIONS, Performance for a Transmit and Receive System Using BPSK bit QEGT/EGC bit QEGT/EGC 3 bit QEGT/EGC EGT/EGC SDT/SDC MRT/MRC 0 0 Performance for a Transmit and 3 Receive System Using BPSK 4 bit QEGT/EGC 4 bit QEGT/MRC SDT/SDC SDT/MRC MRT/MRC E b E b Fig.. Average symbol probability of error for systems using QEGT/EGC with various weight quantizations, EGT/EGC, SDT/SDC, and MRT/MRC. 0 Performance for a Transmit and Receive System Using BPSK bit QEGT/MRC bit QEGT/MRC 3 bit QEGT/MRC EGT/MRC SDT/MRC MRT/MRC Fig. 4. Average symbol probability of error for 3 systems using QEGT/EGC with 4 bits of feedback per weight with bounds, QEGT/MRC with 4 bits of feedback per weight with bounds, SDT/SDC, SDT/MRC, and MRT/MRC. 0 Performance for a 3 Transmit and 3 Receive System Using BPSK 4 bit QEGT/EGC 4 bit QEGT/MRC SDT/SDC SDT/MRC MRT/MRC E b Fig. 3. Average symbol probability of error for systems using QEGT/MRC system with various weight quantizations, EGT/MRC, SDT/MRC, and MRT/MRC. around 0.05dB between and 3 bits quantization is much smaller. The diversity order is once again seen to be 4 as expected. Experiment 3: In the third experiment we took M t =, M r = 3, and transmitted BPSK symbols. We considered QEGT/EGC and QEGT/MRC with B = 4 which provide a close approximation to EGT/EGC and EGT/MRC performance respectively. Here R was taken to be 4 = 8. SDT/SDC, SDT/MRC, and MRT/MRC were also simulated. The results are shown in Fig. 4. This plot shows that using MRC instead of EGC at the receiver with EGT gives around a 0.8dB gain. The diversity order for all of the curves is seen to be 6 as one would expect. Experiment 4: This experiment used M t = M r = 3 with BPSK. The results are shown in Fig. 5. We considered QEGT/EGC and QEGT/MRC with B = 4 which again provide a close approximation to EGT/EGC and EGT/MRC performance respectively. SDT/SDC, SDT/MRC, and MRT/MRC were simulated for comparison. Here we took R = 43. This E b Fig. 5. Average symbol probability of error for 3 3 systems using QEGT/EGC with 4 bits of feedback per weight, QEGT/MRC with 4 bits of feedback per weight, SDT/SDC, SDT/MRC, and MRT/MRC. value of R led to a non-empty W \W when the algorithm was completed. The array gain difference between receive MRC and EGC with EGT is around 0.6dB. The diversity order for all of the plotted curves is 9. Experiment 5: This experiment shows that the performance of QEGT systems is independent of the modulation scheme. Fig. 6 shows the average probability of symbol error for an MRT/MRC system using four point quadrature amplitude modulation (4-QAM), a QEGT/MRC system with B = 3 using 4-QAM, an MRT/MRC system using 6-QAM, and a QEGT/MRC system with B = 3 using 6-QAM. We used R = 0 3. All systems used M t = and M r = 4. For both modulations schemes, the MRT/MRC system has an array gain of approximately 0.4dB over the QEGT/MRC system. Experiment 6: This experiment illustrates the benefits of employing transmit and receive antenna diversity over simply receive diversity. In Fig. 7, the average probability of symbol error curves are shown for an M t = and M r = 4 EGC

8 TO APPEAR IN IEEE TRANSACTIONS ON COMMUNICATIONS, Performance for a Transmit and 4 Receive System with Various Constellations bit QEGT/MRC (6 QAM) MRT/MRC (6 QAM) 3 bit QEGT/MRC (4 QAM) MRT/MRC (4 QAM) E s Fig. 6. Bound and average symbol probability of error for a 4 QEGT/MRC system with 3 bits of feedback per weight with various modulations. 0 0 Performance for Various Systems with 4 QAM 4 EGC 4 QEGT/EGC 8 EGC or maximum ratio combining. We showed that in each of these cases the beamforming and combining system obtains full diversity order. We propose a quantized version of equal gain transmission for systems without transmitter channel knowledge. We presented a codebook design method for quantized equal gain transmission that guarantees full diversity order. The primary performance limitation of QEGT derives from the equal gain assumption. In other work [4], [5] we show that quantized maximum ratio transmission provides further performance improvement at the expense of a signal peakto-average ratio increase. A thorough probabilistic analysis of Rayleigh fading MIMO channels is needed in order to understand the performance of quantized beamforming systems [4], [5]. Another point of future interest is the derivation of exact expressions for the average probability of error for MIMO equal gain systems. Many papers have derived closed-form probability of error expressions for the SIMO equal gain case ([4], [0] [], [9]), but there has been little work on deriving exact probability of error expressions for MIMO equal gain transmission systems E s Fig. 7. Average symbol probability of error for a 4 QEGT/EGC system with 3 bits of feedback per weight, a 4 receive EGC system, and a 8 receive EGC system. system, an M t = and M r = 4 QEGT/EGC system with B = 3 and R = 0 3, and an M t = and M r = 8 EGC system. Each simulation used 4-QAM. The 4 QEGT/EGC system outperforms the 4 EGC system by approximately 3.4dB at an error rate of 0 3. The 4 QEGT/EGC system also provides 8 th order diversity compared with 4 th order diversity of the 4 EGC system. Thus adding another transmit antenna provides substantial performance gains. The 8 EGC system also provides 8 th order diversity and provides approximately a.5db array gain over to the 4 QEGT/EGC system. This performance increase comes at a great cost because the 8 system requires three more antennas than the 4 system. VII. CONCLUSION In this paper, we examined equal gain transmission for MIMO wireless systems operating in memoryless, MIMO Rayleigh fading channels. We specifically examined the design and performance of equal gain transmission when used with receive selection diversity combining, equal gain combining, REFERENCES [] J. Bach Anderson, Antenna arrays in mobile communications: gain, diversity, and channel capacity, IEEE Antennas and Propagation Magazine, vol. 4, pp. 6, April 000. [] D. G. Brennan, Linear diversity combining techniques, Proc. IRE, vol. 47, pp , June 959. [3] J. G. Proakis, Digital Communications. Boston: McGraw Hill, 995. [4] M. K. Simon and M. S. Alouini, Digital Communications over Fading Channels. New York: John Wiley & Sons, 000. [5] V. Tarokh, N. Seshadri, and A. R. Calderbank, Space-time codes for high data rate wireless communication: Performance criterion and code construction, IEEE Trans. Inf. Th., vol. 44, pp , March 998. [6] V. Tarokh, H. Jafarkhani, and A. R. Calderbank, Space-time block codes from orthogonal designs, IEEE Trans. Inf. Th., pp , July 999. [7] T. K. Y. Lo, Maximum ratio transmission, IEEE Trans. Comm., vol. 47, pp , Oct [8] S. Thoen, L. Van der Perre, B. Gyselinkx, and M. Engels, Performance analysis of combined transmit-sc/receive-mrc, IEEE Trans. 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Paulraj, A simple scheme for transmit diversity using partial channel feedback, in Proc. 3nd Asilomar Conf. on Sig., Sys., and Comp., Nov [5] H. Holma and A. Toskala, eds., WCDMA for UMTS: Radio Access for Third Generation Mobile Communications, Revised Edition. New York: John Wiley & Sons, March 00. [6] J. Cioffi, Digital transmission: Volume I. Stanford University EE 379A: [7] S. G. Nash and A. Sofer, Linear and Nonlinear Programming. New York: McGraw-Hill, 996.

9 TO APPEAR IN IEEE TRANSACTIONS ON COMMUNICATIONS, [8] C.-H. Tse, K.-W. Yip, and T.-S. Ng, Performance tradeoffs between maximum ratio transmission and switched-transmit diversity, in Proc. th IEEE Int. Symp. on Personal Indoor and Mobile Radio Comm., vol., pp , 000. [9] X. Qi, M.-S. Alouini, and Y.-C. Ko, Closed-form analysis of dualdiversity equal-gain cominers over Rayleigh fading channels, in Proc. of Veh. Tech. Conf., Spring 00. [0] P. A. Dighe, R. K. Mallik, and S. S. Jamuar, Analysis of a transmitreceive diversity in Rayleigh fading, in Proc. of IEEE Globecom, vol., pp. 3 36, November 00. [] A. T. James, Distributions of matrix variates and latent roots derived from normal samples, Annals of Mathematical Statistics, vol. 35, pp , June 964. [] S. Chennakeshu and J. B. Anderson, Error rates for Rayleigh fading multichannel reception for MPSK signals, IEEE Trans. Comm., vol. 43, pp , Feb/March/April 995. [3] J. Lu, T. T. Tjhung, and C. C. Chai, Error probability performance of L-Branch diversity reception of MQAM in Rayleigh fading, IEEE Trans. Comm., vol. 46, pp. 79 8, Feb [4] D. J. Love, R. W. Heath, Jr., and T. Strohmer, Grassmannian beamforming for multiple-input multiple-output wireless systems. Submitted to IEEE Trans. Inf. Th. in Oct. 00. [5] D. J. Love, R. W. Heath, Jr., and T. Strohmer, Quantized maximum ratio transmission codebook design for multiple-input multiple-output wireless systems, in Proc. 36th Asilomar Conf. on Sig., Sys., and Comp., Nov. 00. PLACE PHOTO HERE David J. Love (S 98) was born in Garland, Texas on May 8, 979. He received the B.S. (highest honors) and M.S.E. degrees in electrical engineering from The University of Texas at Austin in 000 and 00, respectively. During the summers of 000 and 00, he was a summer research intern at the Texas Instruments DSPS R&D Center in Dallas, Texas. At Texas Instruments, he performed research on physical layer system design for next generation wireless systems employing multiple antennas. He is currently a Ph.D. student in the Department of Electrical and Computer Engineering at The University of Texas at Austin. He is a member of Tau Beta Pi and Eta Kappa Nu. His current research interests are in the design and analysis of multiple antenna wireless systems. PLACE PHOTO HERE Robert W. Heath, Jr. (S 96-M 0) received the B.S. and M.S. degrees from the University of Virginia, Charlottesville, VA, in 996 and 997 respectively, and the Ph.D. from Stanford University, Stanford, CA, in 00, all in electrical engineering. From 998 to 999, he was a Senior Member of the Technical Staff at Iospan Wireless Inc, San Jose, CA where he played a key role in the design and implementation of the physical and link layers of the first commercial MIMO-OFDM communication system. From 999 to 00 he served as a Senior Consultant for Iospan Wireless Inc. In January 00, he joined the Department of Electrical and Computer Engineering at The University of Texas at Austin where he serves as an Assistant Professor as part of the Wireless Networking and Communications Group. His research lab, the Wireless Systems Innovations Laboratory, focuses on the theory, design, and practical implementation of wireless communication systems. Dr. Heath s current research interests include interference management in wireless networks, sequence design, and coding, modulation, equalization, and resource allocation for MIMO communication systems.