IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 60, NO. 2, FEBRUARY 2014 1019 Diversity of MIMO Linear Precoding Ahmed Hesham Mehana, Member, IEEE, and Aria Nosratinia, Fellow, IEEE
Abstract— This paper studies multiple-input multiple-output the MF receiver, is interference limited at high SNR but it linear precoding in the high-signal-to-noise-ratio regime under outperforms the ZF precoder at low SNR [4]. The regularized flat fading. The diversity at all fixed rates is analyzed for a ZF precoder, as the name implies, introduces a regularization number of linear precoders. The diversity-multiplexing tradeoffs (DMTs) are also obtained, discovering that for many linear parameter in channel inversion. If the regularization parameter precoders the DMT gives no direct insight into the intricate is inversely proportional to SNR, the RZF of [9] is identical behavior of fixed-rate diversity. The zero-forcing (ZF), regular- to the Wiener filter precoding [4]. Peel et al. [9] introduced a ized ZF, matched filtering, and Wiener filtering precoders are vector perturbation technique to reduce the transmit power of analyzed. It is shown that regularized ZF (RZF) or matched filter the RZF method, showing that in this way RZF can operate (MF) suffers from error floors for all positive multiplexing gains. However, in the fixed rate regime, RZF and MF precoding achieve near channel capacity. full diversity for spectral efficiencies up to a certain threshold This paper analyzes the diversity of MIMO linear precoding, and zero diversity at rates above it. When the regularization with or without linear receivers, under flat fading. We show parameter in the RZF is optimized in the minimum mean square that in a M × N MIMO channel with M ≥ N, the ZF precoder error sense, the structure is known as the Wiener precoder, which has diversity M − N + 1. We show that Wiener precoders in the fixed-rate regime is shown to have diversity that depends not only on the number of antennas, but also on the spectral produce a diversity that is a function of spectral efficiency as efficiency. The diversity in the presence of both precoding and well as the number of transmit and receive antennas. At very equalization is also analyzed. low rates, the Wiener precoder enjoys diversity MN, while at − + Index Terms— MIMO, precoder, equalization, MMSE, zero very high rates it achieves diversity M N 1. These results forcing, diversity. are reminiscent of MIMO linear equalizers [10], even though in general the behavior of equalizers (operating on the receive I. INTRODUCTION side) can be distinct from precoders (operating on the transmit RECODING is a preprocessing technique that exploits side) and the analysis does not directly carry from one to the Pchannel-state information at the transmitter (CSIT) to other. We also show that MIMO systems with RZF and MF match the transmission to the instantaneous channel condi- precoders (together with optimal receivers) exhibit a new kind tions [1]–[4]. In particular, linear precoding is a simple and of rate-dependent diversity that has not to date been observed efficient method that can reduce the complexity of the MIMO or reported, i.e., they either have full diversity or zero diversity receiver; it can also be optimal in certain situations involving (error floor) depending on the operating spectral efficiency R. partial CSIT [5], [6]. We also calculate the DMT for the precoders mentioned Linear precoders include zero-forcing (ZF), matched fil- above. The fact that DMT and the diversity under fixed- tering (MF), Wiener filtering, and regularized zero-forcing rate regime require separate analyses has been established for (RZF). The ZF precoding schemes were extensively stud- MIMO linear equalizers [10], [11]. We find a similar phenom- ied in multiuser systems as the ZF decouples the multiuser enon in MIMO precoding: various fixed rates (spectral effi- channel into independent single-user channels and has been ciencies) result in distinctly different diversities, whereas DMT shown to achieve a large portion of dirty paper coding analysis assigns only a single value of diversity to all fixed capacity [7]. ZF precoding often involves channel inversion, rates (all fixed rates correspond to multiplexing gain zero). = using the pseudo-inverse of the channel or other generalized Remark 1: It may be tempting to substitute r 0inthe ( ) inverses [4]. Matched filter (MF) precoding [8], similarly to DMT expression d r in an attempt to produce the diversity at multiplexing gain of zero d(0), but in fact there is no solid rela- Manuscript received December 22, 2011, revised November 8, 2012 tionship between d(r) and d(0). DMT calculations, as outlined and August 29, 2013, accepted September 7, 2013. Date of publication by Zheng and Tse [12], depend critically on the positivity of r. November 7, 2013; date of current version January 15, 2014. This work was presented in part at the 2012 IEEE International Symposium on Information For example the proof of [12, Lemma 5] depends critically Theory. This work was supported by the National Science Foundation under on r being strictly positive. More importantly, the asymptotic Grant CCF-1219065. outage calculations in [12, p. 1079] implicitly use r > 0and A. H. Mehana was with the University of Texas at Dallas, Dallas, result in the outage region: Richardson, TX 75080 USA. He is now with the Department of Electronics and Electrical Communications Engineering, Cairo University, Cairo 12316, + Egypt (e-mail: [email protected]). A ={α : (1 − αi ) < r} A. Nosratinia is with the Department of Electrical Engineering, The i University of Texas at Dallas, Richardson, TX 75080 USA (e-mail: [email protected]). where αi are the exponential order of the channel eigenvalues, Communicated by A. M. Tulino, Associate Editor for Communications. −α i.e., λi = ρ i .Ifwesetr = 0 this expression implies Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. that the outage region is always empty, which is clearly not Digital Object Identifier 10.1109/TIT.2013.2289860 true. 0018-9448 © 2013 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. 1020 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 60, NO. 2, FEBRUARY 2014
Fig. 1. MIMO with linear precoder.
Fig. 2. MIMO with linear precoder with receive-side equalization.
Thus, the DMT as calculated by the standard methods This paper also calculates diversity in the presence of both of [12] does not extend to r = 0. The DMT d(r) is sometimes a linear precoder and a linear receiver. We use the notation right-continuous at zero, including e.g. the examples in [12], d A−B to denote a system with a precoder A and receiver B. but continuity at r = 0 does not always hold. There are WFP−ZF = − + . situations where d(0), the diversity at multiplexing gain zero, d M N 1 − 1 is not even uniquely defined, instead diversity takes multiple d RZF ZF = (M − N + 1). values at r = 0 as a function of rate R. This fact has been 2 − 1 observed and analyzed, e.g., in [10], [11], [13]. The work d MFP ZF = (M − N + 1). in the present paper also produces several examples of this 2 WFP−MMSE − R 2 − R phenomenon. d =N2 N + (M − N)N2 N . − − For the convenience of the reader, we now present a catalog d RZF MMSE = d MFP MMSE of the results obtained in this paper. The number of transmit 1 − R 2 − R N2 N + (M − N)N2 N if R > Rth and receive antenna is M and N respectively, with M N, = 2 MN the diversity is denoted with d, spectral efficiency (rate) otherwise with R, and multiplexing gain with r. The type of system where R = N log N . Note the fractional diversities, which is shown with a superscript, including zero-forcing precoding th N−1 are uncommon. (ZFP), regularized-ZF (RZF), matched filter precoding (MFP), This paper is organized as follows. Section II describes Wiener filter precding (WFP), and MMSE receiver. This paper the system model. Section III provides outage analysis of discovers the following precoder diversities in the fixed-rate many precoded MIMO systems. Section IV provides the DMT regime: analysis. The case of joint linear transmit and receive filters is d ZFP = M − N + 1. discussed in Section V. Section VI provides simulations that MN if R < R illuminate our results. d RZF = th 0otherwise MN if R < R II. SYSTEM MODEL d MFP = th 0otherwise A MIMO system with linear precoding is depicted in WFP − R 2 − R Figure 1. This system uses the linear precoder to manage the d =N2 N + (M − N)N2 N interference between the streams in a MIMO system to avoid a N where Rth = N log − . requirement of optimal joint decoding in the receiver, which is N 1 × This paper establishes the following DMTs for MIMO costly. We consider a flat fading channel H ∈ CN M ,where precoders M and N are the number of transmit and receive antennas, r respectively. While M N when using linear precoding d ZFP(r) = (M − N + 1)(1 − ), r ∈ (0, N] N alone, we have N M or M N when using precoding d RZF(r) = 0, r ∈ (0, N] together with receive-side linear equalization depending on d MFP(r) = 0, r ∈ (0, N] whether the precoder is designed for the equalized channel r or the equalizer is designed for the precoded channel (see d WFP(r) = (M − N + 1)(1 − ), r ∈ (0, N] N Figure 2). The input-output system model for flat fading MEHANA AND NOSRATINIA: DIVERSITY OF MIMO LINEAR PRECODING 1021
MIMO precoded channel with M transmit and N receive γˆ, γ˘,andγ¯. Following a well-used notation, we denote antennas is given by f (x) = (g(x)) when there exist two positive constants c1, c2 such that c1g(x) ≤ f (x) ≤ c2g(x) for sufficiently large x. y = HTx + n where T ∈ CM×B is the precoder matrix. Subsequently, we III. PRECODING DIVERSITY will consider the joint effect of precoding and equalization, In this section we analyze a linearly precoded MIMO system where the system model will be where M ≥ N and the number of data streams B is equal to N. For the purposes of the developments in this section, there y = WHTx + Wn (1) is no receive-side equalization. where W ∈ CB×N is the receiver side equalizer. The number of information symbols is B min(M, N), the transmitted A. Zero-Forcing Precoding vector is x ∈ C B×1,andn ∈ C N×1 is the Gaussian noise The ZF precoder completely eliminates the interference at vector. The vectors x and n are assumed independent. the receiver. ZF precoding is well studied in the literature via We aim to characterize the diversity gain, d(R, M, N),as performance measures such as throughput and fairness under a function of the spectral efficiency R (bits/sec/Hz) and the a total (or per antenna) power constraint [15, and references number of transmit and receive antennas. This requires a therein]. Pairwise Error Probability (PEP) analysis which is not directly 1) Design Method I: One approach to design the ZF pre- tractable. Instead, we find the exponential order of outage coder is to solve the following problem [4] probability and then demonstrate that outage and PEP exhibit = E || ||2 identical exponential orders. T arg minT Tx 2 The objective of linear precoding/equalization is to trans- subject to HT = I (6) form the MIMO channel into min(M, N) parallel channels that can be described by The resulting ZF transmit filter is given by √ − × T = β HH (HHH ) 1 ∈ CM N (7) yk = γk xk +˜nk, k = 1,...,B (2) where β is a scaling factor to satisfy the transmit power where γk is the SINR at the k-th receiver output and = ( , ) ˜ constraint, that is [4] B min M N ,andnk are the decision point noise coef- ficients. Following the notation of [14], we define the outage- β2tr TTH ρ (8) type quantities where we assume that the noise power is one and that the Pout(R, N, M) P(I (x; y) We now proceed with a lower bound on outage. The outage The outage probability can then be evaluated as follows probability in (10) can be bounded: N ρ Pout = P log (pk + 1) A lower bound on the outage probability can be given as The quantity in the left hand side of (34) is similar to [13, follows. Eq.(18)], thus the analysis of [13] applies and we obtain N ¯ ¯ ˙ P λ ρ−1 = ρ−(M−N+1). Pout = P log (pk + 1) . Notice that P κα ζμmin = 1sinceκα is vanishing at high MIMO system whose diversity is M − N + 1 independent of SNR and ζ and μmin are positives. We now need to compute rate. P ν = κα and P ν = 1 , or equivalently P αk > 1 ∀k Recall that diversity is the SNR exponent of the probability and its complement. We use one of the results of [10]. of codeword error. In Appendix B, we show that the outage Lemma 1: Let {λn} denotes the eigenvalues of a Wishart exponent tightly bounds the SNR exponent of the error prob- matrix HHH ,whereH is an N × M matrix with i.i.d Gaussian ability. Thus we have the following theorem. log(λ ) α =− n α × entries, and let n log(ρ) .If1 n denotes the number of Theorem 1: For an M N MIMO system that utilizes joint αn that are greater than one, then for any integer s N we spatial encoding and regularized ZF precoder given by (36), have [10, Section III-A] 1 the outage diversity is d RZF = MN if the operating spectral = ( N ) RZF = . −(s2+(M−N)s) efficiency R is less than Rth N log N−1 ,andd 0 P 1α = s = ρ . (57) n if R > Rth . Thus setting s = N (i.e. all αn > 1) in (57) yields Remark 2: Rth is a monotonically decreasing function of N 1 . with the asymptotic value lim →∞ R = ≈ 1.44. Overall P ν = κ = P = = ρ−MN N th ln 2 α 1αn N (58) . ≤ ≤ . we have 1 44 Rth 2, leading to an easily remembered P ν = 1 = (1) (59) rule of thumb that applies to all antenna configurations. Regularized ZF precoders always exhibit an error floor at where (1) is a non-zero constant with respect to ρ. spectral efficiencies above 2 b/s/Hz, and enjoy full diversity Evaluating (56) depends on the values of ζ which is always at spectral efficiencies below 1.44 b/s/Hz. real and positive. If ζ<1thenwehave ν . P ζ = ρ−MN D. Matched Filter Precoding μ (60) min The transmit matched filter (TxMF) is introduced in [4], [8]. because P 1 ζ = 0as1/μ > 1. On the other hand The TxMF maximizes the signal-to-interference ratio (SIR) μmin min if ζ>1then at the receiver and is optimum for high signal-to-noise-ratio scenarios [4]. The TxMF is also proposed for non-cooperative ν . P ζ = ρ−MN + P 1 ζ ( ) cellular wireless network [17]. The TxMF is derived by μ μ 1 (61) min . min maximizing the ratio between the power of the desired signal = (1) (62) portion in the received signal and the signal power under the 1 transmit power constraint, that is [4] since P μ ζ is not a function of ρ because μ is indepen- ρ ζ> E ||xH y˜||2 dent of . For the set of rates where 1, equation (62) T = arg max implies that the outage probability in (86) is not a function of T E ||n||2 ρ and thus the diversity is zero, i.e. the system will have error subject to: E||Tx||2 ρ (67) floor. The set of rates for which ζ>1are where y˜ is the noiseless received signal y˜ = Tx. N R > N log Rth. (63) The solution to (67) is given by N − 1 T = βHH (68) This concludes the calculation of a lower bound on the out- age probability. A similar approach will yield a corresponding with upper bound, as follows. Let β = 1 . ∗ 2 H (69) μmax max |ukl u | (64) tr(H H) k =i il We now analyze the diversity for the MIMO system under A lower bound on the SINR is given as TxMF. The received signal is given by ν ˙ γk γ.ˆ (65) y = βHHH x + n = βU UH x + n. (N − 1)μmax The outage probability is bounded as The received signal at the k-th antenna N N ν 2 ˙ yk = β λl |ukl| xk Pout P log(1 +ˆγ) R = P ζ . (66) μmax l=1 k=1 N N ∗ We can evaluate (66) in a similar way as (56), establishing +β λl uklu xi + nk (70) RZF = il that the outage diversity dout MN if the operating spectral i=1,i =k l=1 efficiency R is less than R = N log ( N ),andd RZF = 0if th N−1 out The SINR at k-th receive antenna is R > R . This shows that the performance of RZF precoder th 2 can be much better than that of the conventional ZF precoder β2 ρ N λ | |2 N l=1 l ukl γ = 1Note that [10] analyzes linear MIMO receiver where it is assumed N M. k 2 It can be easily shown that the above Lemma 1 applies for the case considered 2 ρ N N ∗ β = , = = λl uklu + 1 here where M N. N i 1 i k l 1 il 1026 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 60, NO. 2, FEBRUARY 2014 −α Substitute with the value of β and λl = ρ l analysis of [10], [13] for the MIMO MMSE receiver applies 2 for the MIMO Wiener precoding system. It is shown in [10] −α N ρ l | |2 l=1 ukl that this diversity is a function of rate R and number of γk = (71) transmit and receive antennas. We thus conclude the following. 2 M×N N N −α ∗ − N −α Lemma 2: Consider a channel H ∈ C the diversity of ρ l u u + N ρ 1 ρ l i=1,i =k l=1 kl il l=1 the MIMO system under Wiener filter precoding is given by WFP − R 2 − R Observe that (71) is the same as the SINR of the RZF precoded d =N2 N + (M − N)N2 N (79) system given by (44). Hence the analysis in the present case + follows closely that of the outage lower bound of the RZF where (·) = max(·, 0) and ·. precoder, with the following result: the system can achieve full Remark 3: It is commonly stated that MMSE and ZF diversity as long as the operating rate is less than Rth given operators “converge” at high SNR. The developments in this in (63). The pairwise error probability analysis is also similar paper as well as [11] serve to show that although not false, this to that of the RZF precoding system (given in Appendix B) comment is essentially fruitless because the performance of which we omit for brevity. Thus we conclude that Theorem 1 MMSE and ZF at high SNR are very different. This apparent applies for the TxMF precoder. incongruity is explained in the broadest sense as follows: Even though the MMSE coefficients converge to ZF coefficients as ρ →∞ E. Wiener Filter Precoding , the high sensitivity of logarithm of errors (especially at low error probabilities) to coefficients is such that the The transmit Wiener filter TxWF minimizes the weighted convergence of MMSE to ZF coefficients is not fast enough MSE function. for the logarithm of respective errors to converge. − {T,β}=argmin E ||x − β 1y˜||2 T,β subject to E ||Tx |2) ρ. (72) IV. DIVERSITY-MULTIPLEXING TRADEOFF IN PRECODING For increasing sequence of SNRs, consider a corresponding Solving (72) yields sequence of codebooks C(ρ), designed at increasing rates R(ρ) −1 H T = βF H (73) and yielding average error probabilities Pe(ρ). Then define R(ρ) with r = lim ρ→∞ log ρ = H + N F H H I log Pe(ρ) ρ d =− lim . ρ→∞ log ρ 1 β = (74) For each r the corresponding diversity d(r) is defined (with tr(F−2HH H) a slight abuse of notation) as the supremum of the diversities where β can be interpreted as the optimum gain for the over all possible codebook sequences C(ρ). combined precoder and channel [4]. From the viewpoint of definitions, the traditional notion Notice that the TxWF precoding function is similar to that of diversity can be considered a special case of the DMT of the MMSE equalizer [18]. Indeed the SINR of both systems by setting r = 0. However, from the viewpoint of analysis, are equivalent. To see this, we first compute the SINR for the the approximations needed in DMT calculation make use of precoded H ∈ CM×N (with M N) MIMO channel R(ρ) being a strictly increasing function, while for diversity ρ ρβ|( ) |2 analysis R is constant (not strictly increasing function of ). N TH kk γk = (75) Thus, although sometimes DMT analysis may produce results ρβ N |( ) |2 + 2 = N i =k TH ki 1 that are luckily consistent with diversity analysis (r 0), ρ 2 in other cases the DMT analysis may produce results that |(TH)kk| = N (76) are inconsistent with diversity analysis. Certain equalizers and ρ N |( ) |2 + ( −2 H ) N i =k TH ki tr F H H precoders fall into the latter category. In the following, we where we have used the independence of the transmitted signal calculate the DMT of the various precoders considered up to to compute (75). this point. T N×M Now consider a MIMO channel H2 = H ∈ C .The 1) ZF Precoding: Recall that two ZF precoding designs MMSE equalizer for this channel is given by have been considered. For the ZF precoder minimizing power, N given by (7), the outage upper bound in (11) can be written as = ( H + )−1 H . We H2 H2 I H2 (77) ( r −1) ρ Pout P λmin ρ N (80) . −(M−N+1)(1− r ) The received SINR for that system is given by = ρ N (81) ρ 2 |(We H2)kk| = ρ γ MMSE = N . (78) where we substitute R r log to obtain (80), and equa- k ρ N |( ) |2 + ( ) N i =k We H2 ki tr WeWe tion (81) follows in a manner identical to the procedure that led to (13). −2 H Since We H2 = TWFPH and tr(WeWe) = tr(F H H), γ MMSE = γ WFP 2 we conclude that k k . Hence the diversity E.g. the point-to-point MIMO channel with ML decoding. MEHANA AND NOSRATINIA: DIVERSITY OF MIMO LINEAR PRECODING 1027 Similarly the outage lower bound (14) can be written as where (87) follows from Lemma 1, and (88) is true as long P ψ ρ r = ( ) ( r −1) as N 1 , the proof of which is relegated to Pout P z ρ N . −( − + )( − r ) Appendix C. = ρ M N 1 1 N . (82) Since the outage lower bound (88) is not a function of ρ, From (81) and (82) we conclude the system will always have an error floor. In other words the r + DMT is given by d ZFP(r) = (M − N + 1) 1 − . (83) N d RZF(r) = 00< r ≤ B (90) The DMT of the ZF precoder maximizing the throughput, given by (18), is obtained in an essentially similar manner to We saw earlier that in the fixed-rate regime RZF precoding the above, therefore the discussion is omitted in the interest enjoys full diversity for spectral efficiencies below a certain of brevity. threshold, but it now appears that DMT shows only zero 2) Regularized ZF Precoding: We begin by producing an diversity. DMT is not capable of predicting the complex outage lower bound. To do so, we start by the bound on the behavior at r = 0 because the DMT framework only assigns SINR of each stream k obtained in (49), and further bound it a single value diversity to all distinct spectral efficiencies at by discarding some positive terms in the denominator. r = 0. A similar behavior was observed and analyzed for the −α 2 MMSE MIMO receiver [10], [11], [13]. ρ1 min γ¯ = 3) Matched Filter Precoding: The DMT of the MIMO k ∗ 2 ρ1−αmin + ρ1−αmin ⎧ i =k uklu il N system with the TxMF precoder is the same as the DMT −α 2 ⎪ ρ1 min given by (90) due to the similarity in the outage analysis (see ⎪ k = 1 ⎨ ( −α ) ∗ 2 −α Section III-D). We omit the details for brevity. ρ2 1 min u u +Nρ1 min kl 2l −α 2 4) Wiener Filter Precoding: Since the received SINR of the ⎪ ρ1 min ⎩⎪ k > 1 MIMO system using TxWF precoding is the same as that of ( −α ) ∗ 2 −α ρ2 1 min u u +Nρ1 min ⎧ kl 1l MIMO MMSE receiver, we conclude from [13] that the DMT 1 ⎨ ∗ k = 1 for the TxWF precoding system is . |u u |2 = kl 2l 1 r + ⎩ ∗ k > 1 WFP |u u |2 d (r) = (M − N + 1) 1 − . (91) kl 1l N We can now bound the outage probability Similarly to the MIMO MMSE receiver [10], [13], we N observe that DMT for the MIMO system with TxWF does Pout = P log(1 + γk) R not always predict the diversity in the fixed rate regime given k=1 by (79). N ˙ P log(1 +¯γk) R V. E QUALIZATION FOR LINEARLY PRECODED k=1 N 1 TRANSMISSION P N log (1 +¯γ ) R (84) N k The objective of a precoded transmitter is to separate the k=1 N data streams at the receiver. In other words, linear precoding . 1 r = P (1 +¯γk) ρ N (85) is a method of interference management at the transmitter. N k=1 In general, precoded systems do not require interference N . r management at the receiver, however, once a transmitter is = P γ¯ ρ N k designed and standardized (as precoders have been), some k=1 ν N ν standards-compliant receivers may opt to further equalize the ˙ r P ∗ + ∗ ρ N . (86) precoded channel (see Figure 2). This section analyzes the |u u |2 |u u |2 kl 2l k=2 kl 1l equalization of precoded transmissions. where we have used Jensen’s inequality in (84). When the transmit and receive filters can be designed jointly For notational convenience define and from scratch, singular value decomposition becomes an N attractive option whose diversity has been analyzed in [19]. ψ = 1 + 1 . The distinction of the systems analyzed in this section is ∗ 2 ∗ 2 |u u | |u u | that the precoders can be used with or without the receive kl 2l k=2 kl 1l filters, while with the SVD solution neither the transmit nor Then the bound in (86) can be evaluated as follows: the receive filters can operate without each other. r r P νψ ρ N = P νψ ρ N ν = 0 P ν = 0 A snapshot of some of the results of this section is as r +P νψ ρ N ν = P ν = follows. It is shown that equalization at the receiver can 1 1 r r alleviate the error floor that was observed in matched filter = P 0 ρ N P ν = 0 + P ψ ρ N P ν = 1 . − r precoding as well as regularized ZF precoding. It is shown that = ρ MN + P ψ ρ N ( ). 1 (87) MMSE equalization does not affect the diversity of Wiener ρ−MN + ( ) 1 (88) filter precoding, but ZF equalization does indeed affect the = (1) (89) diversity of Wiener filter precoding in a negative way. 1028 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 60, NO. 2, FEBRUARY 2014 Recall that in the system model given in Section II we have Due to the complexity of (96) we proceed to bound the defined the precoder and equalizer matrices T ∈ CM×B and outage from above and below. The upper bound on outage is B×N W ∈ C , respectively, where B is the number of data calculated as follows. Since |ukl| 1, ≤ ( , ) streams, with B min M N . In most wireless systems, the ρ/ γ N equalizer at the receiver is designed to equalize the compound k λ −1 (97) N j N λl +Nρ 2 channel (HT) composed of the precoder and the channel = −1 2 = λ j 1 (λ j +Nρ ) l 1 l (rather than designing the precoder for the equalized channel / = 1 N (WH) although it is possible). In such case we have M N −α −α (98) N ρ1 j N ρ1 l +N 2 = = −α = −α and we set B N. j 1 (ρ1 j +N )2 l 1 ρ1 l A. ZF Equalizer γ.ˆ (99) The ZF equalizer is analyzed when operating together with −α λ = ρ l various precoders, as follows. where we have substituted l in (98). Thus the outage 1) Wiener Filter Precoding: The TxWF precoder is given by probability is bounded as − N 1 N T = β HH H + I HH P = P log(1 + γ ) R ρ out k −1 k=1 H H N = βH HH + IN (92) N ρ R P log(1 +ˆγ) R = P γˆ 2 N − 1 (100) 3 where (92) follows from [20, Fact 2.16.16] . The scalar k=1 coefficient β is given in (74) and, similar to (39), it can be √ Similarly to the analyses of earlier cases, we examine the written as β = 1/ η SINR bound γˆ for different values of αl . Define the set N B ={l | αl > 1} and the event − − λ η = ( + Nρ 1 I) 2 = l tr −1 2 (λl + Nρ ) L ={|B|= } l=1 N (101) The ZF equalizer for the precoder and the channel is we have given by R − P γˆ N − W = (HH H) 1HH (93) Pout 2 1 ZF R R ¯ ¯ The composite channel H is given by = P γˆ 2 N − 1 L P(L) + P γˆ 2 N − 1 L P(L) H = HT. (102) R R ¯ The received signal is given by P γˆ 2 N − 1 L + P γˆ 2 N − 1 L . (103) y = WZFHTx + WZFn. (94) To calculate the first term in (103), we evaluate γˆ when ˜ = The filtered noise n WZFn is is a complex Gaussian αl 1 ∀l vector with zero-mean and covariance matrix Rn˜ given by . 1/N =[HH H]−1 γˆ = (104) Rn˜ N 1−α j N 1 − − − − −1 = ρ = ( −α ) = (HHH + Nρ 1 I) 1(HHH )2(HHH + Nρ 1 I) 1 j 1 l 1 ρ2 1 l − / = ( + ρ−1 )−1 H ( + ρ−1 )−1 H 1 ˙ 1 N U N I U U N I U (105) − − −1 N 1 2 1 2 H ( −α ) = U ( + Nρ I) U l=1 ρ2 1 l . 1 ( −α ) 1 where we have used the eigen decomposition HHH = U UH . = ρ2 1 max = ρ2λ2 min (106) The noise variance of the output stream k is therefore N N −α . ρ1 l + = N −1 2 where (104) follows because N N, (105) follows λl + Nρ −α ( , ) = | |2 N ρ1 j ˙ Rn˜ k k ukl (95) because j=1 1, and (106) follows because the sum λl l=1 in (105) is asymptotically dominated by the largest component. where (95) follows in a similar manner as (31). We can We further bound the first term in (103) compute the signal-to-noise ratio of the ZF filter output: R 1 2 2 R P γˆ 2 N − 1 L ˙ P ρ λ 2 N ρβ2 N min γ = . k ( , ) −1 NRn˜ k k = P λmin ρ (107) ρ/N . −( − + ) = . = ρ M N 1 (108) λ λ + ρ−1 (96) N j N l N 2| |2 j=1 (λ + ρ−1 )2 l=1 λ ukl j N l where (107) is the same as (12), hence (108) follows. 3 n×m m×n −1 −1 To calculate the second term in (103), we evaluate γˆ when Let A ∈ C and B ∈ C then (In + AB) A = A(Im + BA) . This fact can be proved via Matrix Inversion Lemma. one or more αl 1. Consider the two summations in the MEHANA AND NOSRATINIA: DIVERSITY OF MIMO LINEAR PRECODING 1029 denominator of (98). The first one can be asymptotically [13, Appendix A] and omitted here for brevity. We evaluated as thus have R N 1−α ˙ P γˆ N − ρ j . 1 −α Pout 2 1 = + ρ1 j 1−α 1−α R R (ρ j + N )2 ρ j = P γˆ N − L P(L) + P γˆ N − L¯ P L¯ j=1 α <1 α >1 2 1 2 1 j j −( −α ) ¯ . ρ 1 max |L|=N R ¯ ¯ = P γˆ 2 N − 1 L P(L) −1+α 1−α ˙ −(1−α ) ¯ (109) max(ρ ,ρ ) ρ max 1 |L| < N . R ¯ α = α α = α = P γˆ 2 N − 1 L (116) where maxα j <1 j and minα j >1 j and (109) fol- − +α −α −( −α ) lows because min(ρ 1 ,ρ1 ) ˙ ρ 1 max . The second . P(L¯) = ( ) summation in the denominator of (98) can be evaluated as where (116) holds since 1 as given by (59). γˆ follows We further bound the outage probability by bounding as follows. Once again consider the two summations in the N −α ρ1 l + N 2 . 1 denominator of (113). For the first summation of (113), we = 1 + ρ1−αl ρ2(1−αl ) have l=1 αl <1 αl >1 N 1−α ¯ ρ j . 1 −α . 1 |L|=N = + ρ1 j = 1−α j 2 1−α j −2(1−α ) ¯ (110) (ρ + N ) ρ ρ max 1 |L| < N j=1 α j <1 α j >1 −( −α ) . ρ 1 max |L¯|= = N γˆ − +α −α −α (117) We now use (109) and (110) to bound (ρ 1 ,ρ1 ) ˙ ρ1 max |L¯| < max 1 N 1−α ¯ ρ max = ρλmin |L|=N where the bound in the second line (117) is true because γˆ ˙ γ¯ (111) ρ2−2αmax = ρ3λ3 |L¯| < 1 N 1 −α −α . −α min + ρ1 j ˙ ρ1 j = ρ1 max −α ρ1 j We thus have α < α > α > j 1 j 1 j 1 R ¯ R ¯ Using (110) and (117) to bound γˆ and substituting back Pout P γˆ 2 N − 1 L P γ¯ 2 N − 1 L in (113) gives: R 1−α ¯ < P γ¯ 2 N − 1 |B|=0 ρ max = ρλ |L|=N γ˘ ˙ min γ˘ (118) 1−αmax ¯ ρ = ρλmin 1 |L| < N R +P γ¯ N − < |B| < 2 1 0 N Thus the outage bound in (116) can be then evaluated as . −1 3 −3 we did for the upper bound = P λmin ρ + P λ ρ . . min = P λ ρ−1 = ρ−(M−N+1). R ¯ min (112) Pout P γˆ 2 N − 1 L This concludes the calculation of outage upper bound. We ˘ R ¯ P γ˘ 2 N − 1 L now proceed with the outage lower bound. Q ={| | ∀ , } ( , ) Define the event akl k l where akl is the k l ˘ R < P γ˘ 2 N − 1 |B|=0 P |B|=0 entry of the unitary matrix U (cf. equation (31)). Define ˘ R /N +P γ˘ N − L¯, < |B¯| < P |L¯| < γ˘ = 1 2 1 0 N N 1−α −α (113) . N ρ j N ρ1 l +N 2 = P λ ρ−1 ( ) + P λ ρ−1 ( ) −α −α min 1 min 1 (119) j=1 (ρ1 j + )2 l=1 ρ1 l N . −1 = P λmin ρ γ>γ˘ | | ∀ , . −( − + ) Notice that because akl k l. = ρ M N 1 . (120) The outage probability is bounded as where (119) follows as a direct result of Lemma 1 (Eq. (57)). N From (112) and (120), we conclude that the diversity of MIMO = P ( + γ ) Pout log 1 k R system using TxWF precoder and ZF equalizer is k=1 WFP−ZF = − + . N d M N 1 P log(1 + γk) R Q P(Q) 2) Regularized Zero Forcing Precoding: The ZF equalizer k=1 H = is given by (93) where the composite channel HT.The N received signal to noise ratio of the k-th output symbol of P ( +˘γ) P(Q) log 1 R (114) the ZF filter as = k 1 2 R ρβ = P γ˘ 2 N − 1 P(Q) (115) γk = NR˜ (k, k) n ρ/N P(Q) = ( ) = . The probability 1 , i.e. non-zero constant λ λ + (121) N j N l N 2| |2 ρ = 2 = λ ukl with respect to . The proof is similar to the one in j 1 (λ j +N ) l 1 l 1030 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 60, NO. 2, FEBRUARY 2014 R The process of obtaining lower and upper bound has many Let C1 = (2 N − 1)N, C2 = C1ξ where ξ is a fixed similarities with the developments of Section V-A.1, therefore positive constant (independent of ρ), we have we omit many of the steps in the interest of brevity by referring ˙ R to the previous developments. Pout P γ˘ 2 N − 1 ˘ R We begin with the outage upper bound, which is developed ˙ P γ˘ 2 N − 1 in a manner similar to (100). ρλ2 ˙ min N P C1 λ j Pout = P log(1 + γk) R ρλ2 k=1 min P C1 λ j ξ P λ j ξ N λ j P log(1 +ˆγ) R P ρλ2 P λ ξ min C2 j = . k 1 = P ρλ2 . R min C2 (128) = P γˆ 2 N − 1 (122) The exponential inequality (128) holds because P λ ξ = where j (1), as proved in Appendix D. We thus conclude: ρ/ γˆ = N λ λ + − 1 N j N l N 2 RZF ZF = ( − + ). j=1 (λ + )2 l=1 λ d M N 1 j N l 2 ρ/N = Remark 4: −α −α We note that the diversity of regularized zero- N ρ j N ρ l +N 2 = −α = −α forcing precoder together with a zero-forcing equalizer can j 1 (ρ j +N )2 l 1 ρ l be fractional. To our knowledge this is the first instance of . ρ/N = −α (123) fractional diversity uncovered in the literature. N ρ j N ρ2αl j=1 l=1 3) Matched Filter Precoding: In this case, the composite ρ/ . ρ/ ˙ N = N . channel is α (124) N ρ2αl ρ2 max l=1 H = HT = βHHH . Thus the outage in (122) can be bounded as R The noise correlation matrix is given by P γˆ N − Pout 2 1 ρ/ H −1 1 H 2 −1 1 2 H −1 N R Rn˜ =[H H] = [(HH ) ] = (U U ) . ˙ P 2 N − 1 β2 β2 ρ2αmax . −0.5 Thus = P(λmin ρ ) . − 1 ( − + ) B = ρ 2 M N 1 . (125) 1 1 2 R ˜ (k, k) = |u | (129) n β2 λ2 kl We now turn to the lower bound, which is obtained in the l=1 l √ same manner as (116): β = / η η The precoder normalization factor 1 ,where is N given by Pout = P log(1 + γk) R N k=1 η = H ]= λ N tr HH l ˙ P log(1 +˘γ) R l=1 k=1 The signal to noise ratio of the k-th symbol of the ZF filter is R = P γ˘ 2 N − 1 (126) ρ γ = k ( , ) where NRn˜ k k ρ/N ρ/N = . γ˘ = N N (130) λ λ + 2 λ 1 | |2 N j N l N j=1 j l=1 λ2 ukl = 2 = λ l j 1 (λ j +N ) l 1 l ρ/ γ γ = N Notice that the SINR k in (130) is similar to the SINR k of −α −α N ρ j N ρ l +N 2 the RZF precoding system with ZF equalizer given by (121). −α −α j=1 (ρ j + )2 l=1 ρ l N The only difference is the term λk + N which, when applying −α . ρ/N λ = ρ k = the transformation of k , has no effect on the diversity −α N ρ j N ρ2αl j=1 l=1 analysis as detailed in the previous section. We then conclude ρ/N that the diversity of the MIMO system applying MF precoder −α for arbitrary j and ZF equalizer is the same as the diversity of the RZF ρ j N ρ2αl l=1 precoder with ZF equalizer. Thus: . ρ/ ρ/Nλ2 = N = min γ.˘ −α α (127) − 1 ρ j ρ2 max λ d MFP ZF = (M − N + 1). (131) j 2 MEHANA AND NOSRATINIA: DIVERSITY OF MIMO LINEAR PRECODING 1031 H 1 H 2 1 2 H B. MMSE Equalizer wherewehaveusedHH = η (HH ) = η U U to {λ } The MMSE equalizer has better performance compared obtain (140), and k are the eigenvalues of the Wishart H η = ( H ) = N λ to ZF and is therefore widely popular. We investigate the matrix HH . The scaling factor tr HH l=1 l . diversity of MIMO systems that deploy different precoders We begin with a hypothetical precoder whose transmit η = at the transmitter and MMSE equalizer at the receiver. power is not normalized, i.e., 1. The outage probability of this un-normalized precoder is similar to that of the MMSE 1) MFTx Precoding: The MFTx precoder, TMFP ,isgiven by (68). The MMSE equalizer for the precoded channel is receiver with no precoding at the transmitter, as given in (136), given by except that the eigenvalues are now squared. Thus similarly to (137), we have the exponential inequality −1 H −1 H = H H + ρ H α − WMMSE N I (132) . ρ2 k 1 α < . 1 = k 0 5 ρ 2 (141) H 1 + λ 1 αk > 0.5. where H = HTMFP = βMFPHH and βMFP is given N k by (69). The analysis of [10] then follows and we have The SINR at the output of the MMSE filter is given by [18] − 1 − R 2 − R ρ ρ 1 d = N2 N + (M − N)M2 N . (142) γ = h I + H HH h 2 k N k N k k k 1 We conclude that the un-normalized matched filter pre- = − −1 1 (133) coding with MMSE receiver results in 50% diversity loss + ρ HH H compared to MMSE receiver with no transmit precoding. I N kk For the normalized precoder, we begin with the outage α α ··· α where Hk is a submatrix of H obtained by removing the k-th probability in (140). Assume 1 2 N ,thesum column, hk. term in (140) is given by The diversity analysis of the precoded system uses some N N results from the un-precoded MMSE MIMO equalizers [10], 1 η ρ = ρ which we quote in the following lemma. 1 + λ2 η + λ2 k=1 Nη k k=1 N k Lemma 3: consider a quasi-static Rayleigh fading MIMO N −α ¯ M×N ρ l channel H ∈ C (M N), the outage probability of the = l −α ρ − α MMSE receiver satisfies ρ l + ρ 2 k k=1 l N ρ . ¯ H ¯ −1 − R N −α P = P tr(I + H H) N2 N (134) . ρ N out N = . (143) ρ−αN + ρ1−2αk N k=1 1 − R = P N2 N (135) −α + ρ λ where we have used the fact that the ρ k is dominated = 1 N k l k 1 by the maximum element at high SNR. It is easy to see . −d MMSE = ρ (136) that the terms of (143) are either one or zero at high SNR, −α − α ρ N ρ1 2 k {λ } ¯ MMSE depending on whether asymptotically dominates where k are the eigenvalues of H and d is given by (79). or vice versa. These two cases are delineated with the threshold −α α ≶ . ( ,α + ) α λ = ρ k k 0 5max 1 N 1 , or, considering that N is positive, Substituting k ,wehave αk ≶ 0.5(αN + 1). Thus at high SNR, the outage probability α −1 1 . ρ k α < 1 is evaluated by counting the ones = k (137) + ρ λ α > 1 N k 1 k 1 N . 1 − R P = P N2 N 1 out + ρ λ2 thus the term +ρλ /N is either zero or one at high SNR, = 1 η k 1 k k 1 N and therefore to characterize the sum in (135) at high SNR . − R we count the number of ones, or equivalently the number of = P 1 N2 N α > α >0.5 (α +1) k 1. Hence the outage probability reduces to [10] k N . . − R = P 1 = L (144) Pout = P 1 = N2 N . (138) αk >0.5 (αN +1) α >1 k − R Now we apply the matched filter precoder. Similarly where L = N2 N . The conversion from inequality to to (134), the outage portability is given by equality in equation (144) follows from arguments developed ρ in [10, Section III-A] . . H −1 − R P = P tr(I + HH ) N2 N (139) Therefore, the outage probability is asymptotically evalu- out N ated by: N 1 − R = P N . ρ N2 (140) = P(α) α 1 + λ2 Pout d (145) k=1 Nη k S+ 1032 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 60, NO. 2, FEBRUARY 2014 − α . P(α) α ··· b ρ ck k (α ) = where is the joint distribution of the ordered 1 In deriving (150) and (151) we have used a d k + N+ −ac αN and the region of integration is defined as S = S ∩R , ρ k [10]. Equations (149) and (152) show that the system where S is given as follows: exhibits two distinct diversity behaviors based on whether − R • If L = N, then we seek the probability that L =N2 N < N. We can solve to find the boundary of α > 1 (α + ) = ,..., R = N N k 2 N 1 for k 1 N, which implies the two regions log N−1 . To summarize: α ∈ (1, ∞). Thus the integration region can be tightly N MFP−MMSE represented as: d − R − R 1 N 2 + ( − ) N > N = 2 N2 M N M2 R N log N−1 . S = αN > 1 , min αk > 0.5(αN + 1) 1≤k α < υ + From. (158). and (159), we see that when k 1then k −1 ρ η = υk = 1. On the other hand, when αk > 1then . −1 2(1−αk ) 1−αl αl −1 υk + ρ η = ρ + ρ + ρ α > α < l 1 l 1 ( −α ) −α −α α − = ρ2 1 k + ρ1 k + ρ1 l + ρ l 1 αl >1 αl <1 = = l k l k . −α −α α − = ρ1 k + ρ1 l + ρ l 1 (160) αl >1 αl <1 l =k l =k . − = ρ 1η (161) where (160) follows because αk > 1. Thus we have ρ−1η . ρ−1ηα< γ = = k 1 k −1 (162) ρ η + υk 1 αk > 1 Fig. 3. ZF and Wiener filtering precoded 2 × 2 MIMO for rates (left to −1 right): R = 1.9, 2.5, and 3 b/s/Hz. and ρ η has negative exponent thus vanishes at high SNR. Observe that (162) is similar to (137) which corresponds to the case of the MMSE-only system (i.e. with no precoding). Thus substituting (162) in the outage probability (157) and repeating the same analysis of the MMSE-only system as in [10], we conclude that the diversity of the MMSE receiver when using WFTx precoding is the same as the diversity of the MMSE receiver with no linear precoding, which is given by (79). 3) RZF Precoding: Using the Regularized Zero Forcing precoding at the receiver results in the composite channel H = HT = βHHH (HHH + c I)−1. where c is a fixed constant, β = 1/η and η is given by (39) N N −α λ ρ l η = l = . 2 −α 2 (163) (λl + c ) (ρ l + c ) l=1 l=1 Similar to (155), the outage probability of RZF precoder Fig. 4. Wiener precoded 3 × 3 MIMO. The diversities are d = 9, 4and1 with MMSE receiver is given by for R = 1.5, 4 and 5 b/s/Hz respectively. N . − R Pout = P γk N2 N υ = 1 λˆ k=1 where we define k N k. We now proceed to express both ρ−1η and υ in terms of {α }, the exponential orders of {λ }. and k k k η γ N − N −α k ρ ρ 1λ ρ1 l η + λ¯ ρ−1η = l = N k (ρ−1λ + )2 (ρ1−αl + )2 = l N = N {λ¯ } HH H l 1 l 1 where k are the eigenvalues of given by . −α α − = ρ1 l + ρ l 1 − α (158) λ2 ρ 2 k k α > α < λ¯ = = , k = ,...,N l 1 l 1 k 2 −α 2 1 (164) (λk + c) (ρ k + c) observe that all the terms in (158) have negative exponent. Notice that at high SNR we have Using (156), N −α − α . ρ l . ρ 2 k − α ¯ ρ 2 k η = λ = . υ = 1 c2 k c2 k −α − = N (ρ k + ρ 1 N)2 l 1 ( −α ) 1 ρ2 1 k Thus the SINR is given by (cf. (143)) = 1−α 2 −α N (ρ k + N) N ρ l −α . l=1 . ρ N γ = = −α − α , k ρ N +ρ1 2 k . 1 αk < 1 N ρ−αl + ρ1−2αk = l=1 2(1−α ) (159) ρ k αk > 1. k = 1,...,N 1034 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 60, NO. 2, FEBRUARY 2014 Fig. 5. MF and regularized ZF precoded 2×2 MIMO. The diversity is d = 4 Fig. 7. Wiener filter precoding and ZF equalization. The diversity is d = 1 for R = 1.9 b/s/Hz and d = 0otherwise. for a 2 × 2MIMOandd = 2fora3× 2MIMOsystem. Fig. 6. Matched filter precoding and ZF equalization. The diversity is d = 0.5 × for a 2 × 2 MIMO system and d = 1fora3× 2MIMOsystem. Fig. 8. Wiener filter precoding and MMSE equalization for 2 2MIMO. The diversity d = 1forR = 3, 4 b/s/Hz, and d = 4forR = 1.5 b/s/Hz. which are the same terms as in (143), implying that the outage probability of the MMSE receiver working with the precoding, the diversity is the same as the one predicted by the regularized zero-forcing precoder is asymptotically the same DMT for high rate (R) values and it departs from the DMT as the outage probability of the MMSE receiver working with for low rate values. A complete diversity characterization is the matched filter precoder. This means: given by (79) which is similar to that of the MMSE MIMO equalizer [10]. Figure 4 shows outage probabilities for a 3 × 3 RZF−MMSE = MFP−MMSE d d MIMO system with Wiener precoding. The diversity for the 1 − R − R N N 2 + ( − ) N > = . , , , = 2 N2 M N M2 R N log N−1 rates R 1 5 4 and 5 b/s/Hz is 9 4 and 1 respectively. MN otherwise. Figure 5 shows an error floor for the regularized ZF and × (165) matched filtering precoded 2 2 system at high rates. However we observe that the maximum diversity is achieved for any rate R < 2 (cf. Equation (63)). Figure 6 shows outage VI. SIMULATION RESULTS probabilities for a 2 × 2anda3× 3 MIMO system with This section produces numerical results for the outage matched filter precoding and ZF equalization. The observed probabilities of ZF, regularized ZF (RZF), matched filter (MF) diversity values are consistent with Eq. (131). Figure 7 shows and Wiener precoding systems. Figure 3 shows the outage outage probabilities for a 2 × 2anda3× 3MIMOsystem probabilities of the ZF and Wiener-filter precoded 2×2MIMO with Wiener filter precoding and ZF equalization. Figure 8 and systems. The diversity in the case of the ZF case is the same Figure 9 show outage probabilities for a 2 × 2anda3× 3 as the one predicted by the DMT. In the case of Wiener MIMO system, respectively, with Wiener filter precoding and MEHANA AND NOSRATINIA: DIVERSITY OF MIMO LINEAR PRECODING 1035 independent of the signal-to-noise ratio) or matched filter (MF) precoder is used, we have d(r) = 0forallr, implying an error floor under all conditions. It is also shown that in the fixed rate regime RZF and MF precoding achieve full diversity up to a certain spectral efficiency, while at higher spectral efficiencies they produce an error floor. If the regularization parameter in the RZF is optimized in the MMSE sense, the RZF precoded MIMO system exhibits a complex rate-dependent behavior. In particular, the diversity of this system (also known as Wiener − R filter precoding) is characterized by d(R) =N2 N 2 + − R (M − N)N2 N where M and N are the number of transmit and receive antennas. This is the same behavior observed in linear MMSE MIMO receivers [10]. Various results for the diversity in the presence of both precoding and equalization have also been obtained. Fig. 9. Wiener filter precoding and MMSE equalization for 3×3MIMO.The diversity mimics that of Wiener precoding without equalization (Eq. (79)). APPENDIX A ASYMPTOTIC MARGINAL DISTRIBUTION OF SMALLEST EIGENVALUE OF WISHART MATRIX Define a Wishart matrix W using the Gaussian matrix H. HHH M > N W = HH H N N. Let m = max(M, N) and n = min(M, N). The matrix W is m×m random non-negative definite that has real, non-negative eigenvalues with λ1 ··· λn 0, where for emphasis we denote λn = λmin. The joint density of the ordered eigenvalues is [21] − − λ − (λ) = 1 i i λn m (λ − λ )2. f Km,ne i i j (166) i i< j Define λ α =− k Fig. 10. MF precoding and MMSE equalization 2 × 2 MIMO. The diversity k log ρ (167) is given by Eq. (153). Using (166) and (167), the joint distribution of α n n MMSE equalization. The diversity for the 3 × 3 system is the −1 −α n −(m−n+1)α f (α) = K , exp − ρ i (log ρ) ρ i same as the diversity of the Wiener filtering precoding-only m n i=1 i=1 (cf. Figure 4). −α −α 2 Figure 10 shows the outage probability of a 2 × 2MIMO × |ρ i − ρ j | system with matched filter precoding and MMSE equalization, i< j which is consistent with Eq. (153). We also plot the outage Define the event A ={αk : αk 1}. We now compute the probability of the MMSE MIMO equalizer (without any pre- −1 probability that λmin <ρ . coding) for comparison. −1 −1 P λmin ρ = 1 − P λmin ρ VII. CONCLUSION −1 = 1 − P λi ρ , ∀i Linear precoders provide a simple and efficient process- −α − = 1 − P ρ i ρ 1, ∀i ing, and have been shown to be optimal in some scenarios [5]–[7]. This paper studies the high-SNR performance of linear = 1 − P αi 1, ∀i precoders. It is shown that the zero-forcing precoder under two common design approaches, maximizing the throughput and = 1 − f (α) dα (168) A minimizing the transmit power, achieves the same DMT as n that of MIMO systems with ZF equalizer. When a regularized Define B = A ∩ R+ ={αk : 0 αk 1}. Following the ZF (RZF) precoder (for a fixed regularization term that is same analysis as in [10], [12], the integral in (168) can be 1036 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 60, NO. 2, FEBRUARY 2014 asymptotically evaluated as APPENDIX B n . −( − + − )α PAIRWISE ERROR PROBABILITY (PEP) ANALYSIS f (α) dα = ρ 2i 1 m n i dα A In this section we perform PEP analysis for the zero- B i=1 forcing (ZF) and the regularized ZF (RZF) precoding systems. n −( − + − ) = 1 − ρ 2i 1 m n (169) The presented analysis can be easily extended to all other precoding systems. The basic strategy is to show the SNR i=1 n exponent of outage probability bounds the SNR exponent of −( − + − ) = 1 − ρ 2i 1 m n + G(ρ) (170) PEP from both sides The PEP analysis follows from [10], i=1 [14], with careful attention to the system model given by Equation (1). where (170) expands the product in (169), and G(ρ) col- The lower bound immediately follows from [14, Lemma 3] lects all the higher-order terms. It can be easily seen that by recognizing that although it was developed for SISO block n ρ−(2i−1+m−n) G(ρ) i=1 dominates at high SNR, thus we equalization, nowhere in its development does it depend on have the number of receive antennas, therefore we can directly use n n −( − + − ) . −( − + − ) it for our purposes: 1− ρ 2i 1 m n +G(ρ)=1 − ρ 2i 1 m n (171) ˙ i=1 i=1 Perr Pout. (175) ρ−(m−n+1) Moreover, the term dominates all other terms of The upper bound on PEP for the ZF/RZF precoding systems n ρ−(2i−1+m−n) i=2 at high SNR, i.e. receiver is developed using the union bound. Denote the n channel outage event by O and the error event by E.The −( − + − ) . −( − + ) ρ 2i 1 m n = ρ m n 1 (172) PEP is given by i=1 ¯ Perr = P(E|O) Pout + P(E, O) which yields the following ¯ Pout + P(E, O). (176) . −( − + ) f (α) dα = 1 − ρ m n 1 (173) A In order to show that Pout dominates the right hand side of (176), it is shown in [10] that the probability P(E, O¯ ) can be We now evaluate (168) using (173) bounded as follows using the union bound P λ ρ−1 = − (α) α − ρ/N n 1 f d σ2( ) − P( , ¯ ) ˙ Rl ˜ k ˙ ρ MN A E O 2 e n (177) . −( − + ) = 1 − 1 − ρ m n 1 2 −( − + ) where l is the codeword length and σ (k) is the variance of the = ρ m n 1 (174) n˜ interference plus noise signal n˜ in the k-th receive stream4.The which concludes the proof. proof of [14] does not depend on the codeword length for both Remark 7: The result proven in this appendix, namely upper and lower PEP bounds. The bound are tight and were (λ ) ∝ λm−n λ < confirmed by simulations for outage and error probabilities. f min min for min , has been used earlier in the literature [13], [19] with a simple reference to the seminal We now show that a similar proof holds for regularized work of Telatar [21] but as far as we know a detailed proof zero-forcing (RZF). Recall that the outage probability of the has not been available until now. Indeed a direct proof using RZF can be upper bounded by (66) Telatar’s result is possible and is sketched as follows. Using ν P ζ b Telatar’s joint distribution of unordered eigenvalues, it is easy Pout Pout (178) μmax to see that the marginal distribution of an unordered eignevalue (λ ) ∝ λm−n λ → b ( , ¯ ) is in fact f i i as i 0. To complete the proof We will use Pout to further bound (176). Moreover P E O λ can be upper bounded by bounding the noise variance σ 2(k) for min it remains to be shown that close to the. origin, n˜ −1 f (λmin) ∝ f (λi ), or equivalently P(λmin <ρ ) = P(λi < in (177) ρ−1). This can be accomplished by noting {λ <ρ−1}⊂ min 2 −1 σ (k) = P + P < P + 1 (179) {λi <ρ )}, then showing the difference of the two events n˜ I n T constitutes a volume in the eigenvalue space that vanishes where we have used the noise power P = 1, and bound the sufficiently fast with ρ →∞so that its probability can be n interference power by the total received power P . We will bounded (using boundedness of the joint distribution), and T first consider the case of RZF precoding since the case of ZF thus the SNR exponent of the two probabilities remain equal. precoding can be easily deduced from RZF by substituting the Details of this alternative proof are omitted for brevity. regularization parameter c = 0. For the RZF precoding system For the proof in this appendix, we have taken a different approach based on the exponential order of the eigenvalues, 4 [14] analyzes linear receivers so n˜ is the k-th output filtered interference which is by now a well-established tool in diversity analysis, plus noise signals. By symmetry assumption all the equalizer outputs have and seems better-suited to the tone and technique of this paper. equal noise variance. MEHANA AND NOSRATINIA: DIVERSITY OF MIMO LINEAR PRECODING 1037 we use the PT given by (40) which can be simplified in a way Observe that all the terms of ψb are distinct except for the similar to earlier sections first two. P ψ ρ r β2ρ N λ2 We now bound the probability N . P = l T (λ + )2 r r N = l c P ψ ρ N P ψ ρ N | ψ REFERENCES [16] D. Gore, R. W. Heath, and A. Paulraj, “On performance of the zero forcing receiver in presence of transmit correlation,” in Proc. IEEE Int. [1] E. Biglieri, J. Proakis, and S. Shamai, “Fading channels: Information- Symp. Inf. Theory, Jun. 2002, p. 159. theoretic and communications aspects,” IEEE Trans. Inf. Theory, vol. 44, [17] T. Marzetta, “Noncooperative cellular wireless with unlimited numbers no. 6, pp. 2619–2692, Oct. 1998. of base station antennas,” IEEE Trans. Wireless Commun., vol. 9, no. 11, [2] A. Scaglione, P. Stoica, S. Barbarossa, G. Giannakis, and H. Sampath, pp. 3590–3600, Nov. 2010. “Optimal designs for space-time linear precoders and decoders,” IEEE [18] S. Verdu, Multiuser Detection. Cambridge, U.K.: Cambridge Univ. Press, Trans. Signal Process., vol. 50, no. 5, pp. 1051–1064, May 2002. 1998. [3] S. Jayaweera and H. Poor, “Capacity of multiple-antenna systems with [19] E. Sengul, E. Akay, and E. Ayanoglu, “Diversity analysis of single both receiver and transmitter channel state information,” IEEE Trans. and multiple beamforming,” IEEE Trans. Commun., vol. 54, no. 6, Inf. Theory, vol. 49, no. 10, pp. 2697–2709, Oct. 2003. pp. 990–993, Jun. 2006. [4] M. Joham, W. Utschick, and J. Nossek, “Linear transmit processing in [20] D. S. Bernstein, Matrix Mathematics: Theory, Facts and Formula. MIMO communications systems,” IEEE Trans. Signal Process., vol. 53, Princeton, NJ, USA: Princeton Univ. Press, 2009. no. 8, pp. 2700–2712, Aug. 2005. [21] I. E. Telatar, “Capacity of multi-antenna Gaussian channels,” Eur. Trans. [5] G. Caire and S. Shamai, “On the capacity of some channels with Telecommun., vol. 10, pp. 585–595, Nov./Dec. 1999. channel state information,” IEEE Trans. Inf. Theory, vol. 45, no. 6, pp. 2007–2019, Sep. 1999. [6] M. Skoglund and G. Jongren, “On the capacity of a multiple-antenna communication link with channel side information,” IEEE J. Sel. Areas Commun., vol. 21, no. 3, pp. 395–405, Apr. 2003. [7] T. Yoo and A. Goldsmith, “On the optimality of multiantenna broad- Ahmed Hesham Mehana (S’10–M’13) is currently with the Department of cast scheduling using zero-forcing beamforming,” IEEE J. Sel. Areas Electronics and Electrical Communications Engineering at Cairo University. Commun., vol. 24, no. 3, pp. 528–541, Mar. 2006. He received his B.Sc. and M.Sc. degrees from Cairo University in 2004 and [8] R. Esmailzadeh and M. Nakagawa, “Pre-RAKE diversity combination 2007, respectively, both in Electrical Engineering, and his Ph.D. degree in for direct sequence spread spectrum communications systems,” in Proc. Electrical Engineering at the University of Texas at Dallas in 2012. He was IEEE ICC, vol. 1. May 1993, pp. 463–467. with Samsung Telecommunication America in 2013, and an intern at Research [9] C. Peel, B. Hochwald, and A. Swindlehurst, “A vector-perturbation in Motion Co. Ltd in 2010. His current interests include MIMO precoding, technique for near-capacity multiantenna multiuser communication— linear receivers, and interference management. part I: Channel inversion and regularization,” IEEE Trans. Commun., vol. 53, no. 1, pp. 195–202, Jan. 2005. [10] A. H. Mehana and A. Nosratinia, “Diversity of MMSE MIMO receiver,” IEEE Trans. Inf. Theory, vol. 58, no. 11, pp. 6788–6805, Nov. 2012. [11] A. Hedayat and A. Nosratinia, “Outage and diversity of linear receivers in flat-fading MIMO channels,” IEEE Trans. Signal Process., vol. 55, Aria Nosratinia (S’87–M’97–SM’04–F’10) is Erik Jonsson Distinguished no. 12, pp. 5868–5873, Dec. 2007. Professor and associate head of the Electrical Engineering Department at [12] L. Zheng and D. N. C. Tse, “Diversity and multiplexing: A fundamental the University of Texas at Dallas. He received his Ph.D. in Electrical and tradeoff in multiple-antenna channels,” IEEE Trans. Inf. Theory, vol. 49, Computer Engineering from the University of Illinois at Urbana-Champaign no. 5, pp. 1073–1096, May 2003. in 1996. He has held visiting appointments at Princeton University, Rice [13] K. R. Kumar, G. Caire, and A. L. Moustakas, “Asymptotic performance University, and UCLA. His interests lie in the broad area of information theory of linear receivers in MIMO fading channels,” IEEE Trans. Inf. Theory, and signal processing, with applications in wireless communications. He was vol. 55, no. 10, pp. 4398–4418, Oct. 2009. the secretary for the IEEE Information Theory Society in 2010-2011 and was [14] A. Tajer and A. Nosratinia, “Diversity order in ISI channels with single- the treasurer for ISIT 2010 in Austin, Texas. He has served as editor for the carrier frequency-domain equalizer,” IEEE Trans. Wireless Commun., IEEE TRANSACTIONS ON INFORMATION THEORY, IEEE TRANSACTIONS vol. 9, no. 3, pp. 1022–1032, Mar. 2010. ON WIRELESS COMMUNICATIONS, IEEE SIGNAL PROCESSING LETTERS, [15] A. Wiesel, Y. Eldar, and S. Shamai, “Zero-forcing precoding and IEEE TRANSACTIONSON IMAGE PROCESSING,and IEEE WIRELESS COM- generalized inverses,” IEEE Trans. Signal Process., vol. 56, no. 9, MUNICATIONS (MAGAZINE). He has been the recipient of the National pp. 4409–4418, Sep. 2008. Science Foundation career award, and is a fellow of IEEE.