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LINEAR PRECODERS AND DECODERS DESIGNS FOR MIMO FREQUENCY SELECTIVE CHANNELS A. Scaglione1 , P. Stoica2 , S. Barbarossa3 , H. Sampath4 1

ECE, Cornell University, (NY, USA); 2 Dept. of Systems & Control, Uppsala Univ., (Sweden); 3

Infocom Dept., Univ. of Rome “La Sapienza”, (Italy); 4 Iospan wireless, (CA, USA).

ABSTRACT In this paper we derive and compare designs for the optimal linear precoderes to be used in transmissions over frequency selective multiple-input multiple-output (MIMO) channels. We assume as alternative design constraints the average transmit power and the peak power. The design criteria are scalable with respect to the number of antennas, size of the coding block and transmit average/peak power. The solutions are shown to convert in both cases the MIMO channel with memory into a set of parallel independent flat fading subchannels, regardless of the design criterion, while appropriate power/bits loading on the sub-channels is the specific signature of the different designs.

1. INTRODUCTION Linear precoding is a block transmission technique that uses a pair of linear transformations F (precoder) and G (decoder) of the transmit symbols and receive samples respectively, that operate jointly and linearly on the time and space dimensions. We consider a single user signal model, assuming that the multiple access technique used is TDMA or FDMA. To optimize F and G, the channel state information (CSI) is required at both the transmit and receive sides, as for example in [2], [8] [9]. Compared to [1], that enatails lower complexity and does not require knowledge of the CSI, when applicable the solutions in this paper bring improved performance and also additional flexibility into the design, for [1] impose restrictions on the number of antennas for which the algorithms can be implemented. In all our designs, the paradigm of linear precoding /decoding exploits the channel eigendecomposition in constructing the optimal F , G. The distinct solutions are characterized by how the power is loaded on each channel eigenfunction. Similar results can be found in [6] and [5]: compared to these works, instead of assuming vector coding, we show how optimal linear transceivers naturally result in having this structure from the criterion. We make no assumptions regarding the noise color or the channel matrix size and rank, which are arbitrary. The main results of this paper are the following: 1) optimizing the average mean square error (MSE), equal to the trace of the error covariance matrix, and the determinant of the error covariance matrix under the average power constraint, provides

two different power loading algorithms where the second also maximizes the capacity of the channel; 2) when we constrain the maximum eigenvalue of the covariance matrix (to limit the output peak power rather than the average) the same two minimizations lead to uniform power loading across the eigenvectors; 3) The minimax criterion applied to the minimum distance among symbol vectors leads to the ZF power loading. The proofs are omitted for brevity, and are available in [10]. The different solutions are compared to underline the trade-offs or advantages implied by the various designs. Notation: Boldface letters are vectors (lower case) or matrices (upper case). The tr(A), |A|, λ(A) are the trace, determinant and eigenvalues of A, a = vec(A) is formed stacking vertically the columns of A. Continuous time signals vectors are like a(t) discrete time vector sequences like a[n]. Sequences of vectors obtained by stacking consecutive blocks, such as ai = [a[iM ], . . . , a[iM + M − 1]], are characterized by a suffix i . 1.1. System model The system considered has K transmit and R receive antennas. The complex envelopes of the transmitted signals are the entries of x(t) := (x1 (t), . . . , xK (t))T . We assume that x(t) is obtained by pulse shaping the (coded) symbol vector x[n] and transmitting each pulse at rate 1/T . Correspondingly, z(t) = y(t) + n(t) is the received R × 1 vector which contains the channel output y(t) and additive noise n(t). For a linear (generally time-varying) channel: ∞ X y[k] = H[k, k − n]x[n], (1) n=−∞

is the received noise-free signal vector samples y[k] := y(kT ) in discrete time (DT). If H[k, n] is causal and has finite memory L we can write the I/O relationship (1) in block FIR form. Specifically, with P = M + L, forming the P K × 1 vector xi := vec([x[iP ], .., x[iP +P − 1]]) we precode every P T sec. an N × 1 blocks of symbols si as xi = F si ,

(2)

where N ≤ min(KP, RM ). Forming with M received vectors the M R×1 y i := vec([y[iP +L], .., y[iP+P−1]]),

We start introducing the eigendecomposition:

y i = Hxi ,

(3)

where H is an RM ×KP block-banded matrix. If the channel is also time invariant (LTI), i.e. H[n, l] ≡ H[l] where {H[l]}r,k is the l-th sample channel impulse response between the k-th transmit and the r-th receive antennas, then H in (3) is a block Toeplitz matrix:   H[L] · · · H[0] . . . 0  ..  .. .. .. . (4) H =  ... . . . .  0

···

H[L]

···

H[0]

RM ×KP

2. OPTIMAL LINEAR DESIGNS The received signal z i = y i + ni contains noise. We will use the notation Rss := E{si sH i }, and assume that a0) The size N of si in (2) is N ≤ rank(H). 2 a1) The transmit symbols are white, i.e., Rss = σss I. The noise ni ∼ N (0, Rnn ) where Rnn > 0 (positive definite) and ni and si are uncorrelated. As the precoder in (2) encodes linearly the symbols, the receiver performs an appropriate inverse mapping G on the vector z i = vec(z[iP + L], . . . , z[iP + P − 1]), estimating ˆi = Gz i . From (3) we have: the symbols as s ˆi = Gz i = GHF si + Gni . s

(5)

A reasonable criterion to design G, for given F and H, is to minimize the mean square error (MSE), which can be computed as the trace of the following matrix: H

M SE(F , G) := E{(ˆ si − si )(ˆ si − si ) }.

(6)

The matrix Gopt which minimizes tr(M SE(F , G)) is the MMSE (Wiener) receiver: Gopt = Rss F H H H (HF Rss F H H H + Rnn )−1 . (7) Plugging Gopt in M SE(F , Gopt ) we get the matrix M SE(F ) := M SE(F , Gopt ) equal to H

2 2 M SE(F ) = σss (I + σss F H

H

−1 R−1 . nn HF )

(8)

We can prove that the difference matrix [M SE(F , G) − M SE(F )] is a positive semidefinite matrix ∀ G 6= Gopt . In the following, we will determine F opt according to different performance measures. Specifically, we will use two alternative constraints: 1) the average power constraint (PC): 2 tr(F F H ) σss = P0 , (9) and 2) the maximum eigenvalue of the transmit vector co2 variance F F H σss : 2 λmax (F F H ) σss = L0 ,

which puts an upper bound on the output peak power.

(10)

¯ ¯ ¯H H H R−1 nn H = V ΛV

(11)

¯ where V¯ may be tall if H H R−1 nn H is rank deficient and Λ H −1 is an Q×Q diagonal matrix, where Q := rank(H Rnn H). We assume that the non-null eigenvalues of H H R−1 nn H, ¯ are in decreasing order (note that a0) re{λqq }Q in Λ, q=1 quires N ≤ Q). For convenience we denote by Λ the N ×N ¯ and diagonal matrix equal to the top left N × N block of Λ, ¯ by V the first N columns of V (eigenvectors paired to the N largest eigenvalues). 2.1. MMSE criterion under average power constraint The MMSE design minimizes the tr(M SE(G, F )) jointly with respect to G and F under the transmit-power constraint. An analogous joint transmit/receive-filter optimization of MIMO channels in the frequency-domain is in [7]. The design that minimizes tr(M SE(G, F )) can be obtained by minimizing tr(M SE(F )) with respect to F . The solution for F opt is given in the following lemma, and Gopt can be obtained by replacing F with F opt in (7). Lemma 1 The solution of the optimization problem: 2 F opt = argmin tr(M SE(F )) , tr(F F H )σss = P0 F (12) is F opt = V Φ , where Φ is N × N diagonal with1 : !+ Ã PN¯ P0 + n=1 λ−1 1 nn −1/2 2 λ − |φii | = (13) PN¯ 2 −1/2 ii λii σss λnn σ2 ss

n=1

¯ ≤ N is the number of where (x) := max(x, 0) and N |φnn |2 > 0. Interestingly, the minimization of the determinant, in lieu of the trace, of the M SE(F ) matrix with respect to F is equivalent to maximizing the information rate, first derived in [2] and, for the MIMO flat fading case, by [3],[8]. Lemma 2 The solution of 2 F opt = arg min |M SE(F )| , tr(F F H ) σss = P0 (14) +

F

is F opt = V Φ, where Φ is N × N diagonal with à !+ PN¯ P0 + k=1 λ−1 1 2 kk |φii | = − , 2 ¯ σ2 λii σss N ss

(15)

¯ ≤ N is the number of positive |φii |2 . For a Gausand N sian input si , the F opt and Gopt also maximize the channel mutual information and (15) corresponds to the so-called “water-filling” solution (see e.g. [4]. The result of Lemma 2 is explained by the fact that, for the G in (7) the mutual information I(ˆ s, x) per block is: 2 I(ˆ s, x) = log |σss M SE

−1

(F )|.

(16)

1 Note that only the amplitude of φ is fixed while the phase is arbitrary ii thus φii can be a real number.

2.2. MMSE criterion under λmax (F F H ) constraint Lemma 3 The solution of both

Lemma 5 All optimal designs have the form: F opt = V Φ,

Gopt = ΓΛ−1 V H H H R−1 nn , (25)

F opt

=

arg min tr(M SE(F )) ,

(17)

F opt

=

arg min |M SE(F )| ,

(18)

where Φ and Γ are diagonal matrices. The MIMO channel seen between the matrices Φ and Γ is an I and the noise covariance matrix is Λ−1 .

As with Lemma 2, because of (16), the solution in Lemma 3 also provides the maximum information rate under (10).

Thus, the N subchannels are decoupled and symbols are multiplied individually to the diagonal elements of ΓΦ, and the noise components {β i }k , k = 1, . . . , N , are uncorrelated with variance λ−1 kk .

F

F

2 under λmax (F F H ) σss = L0 is F opt =

p

2 V. L0 /σss

2.3. The max λmin (SN R) under λmax (F F H ) constraint Designs minimizing the probability of error are difficult to deal with because they are rarely solvable in closed form, they depend on the symbol alphabet and on the detection rule. We adopt a minimax criterion based on the distance Dh,k := [s(Hh ) − s(Hk )]H SN R(F , G)[s(Hh ) − s(Hk )] (19) where the SNR-like matrix is: 2 SN R(F , G) = F H H H GH (GRnn GH )−1 GHF σss ,

which, interestingly, allows us to extend to MIMO case the capacity formula of the SISO AWGN channel as: I(ˆ s, x) = log |I + SN R(F , G)|.

(20)

Instead of maximizing (19), because λmin (SN R(F , G)) min ks(Hh ) − s(Hk )k2 . 2 h6=k σss (21) we obtain a criterion that is alphabet independent by maximizing the lower bound, i.e. λmin (SN R(F , G)). min Dh,k ≥ h6=k

Lemma 4 The solution of (F opt , Gopt ) = argmax λmin (SN R(F , G)) F ,G

(22)

2 2 For tr(F F H )σss = P0 and λmax (F F H )σss = L0 is F opt = V Φ with Φ diagonal N × N with, respectively

|φii |2

=

2 P0 /(σss

X

−1 λ−1 kk ) λii ,

(23)

k

|φii |2

=

2 L0 /(λN N σss )) λ−1 ii ,

(24)

˜ H H H R−1 . Both solutions are zero forcand Gopt = ΓV nn ing (ZF), i.e. SN R(F opt , Gopt ) ∝ I. 2.4. Equivalent independent subchannels As mentioned before, all optimal designs lead invariably to loading the power across the eigenvectors of H H R−1 nn H.

3. NUMERICAL EXAMPLES AND DISCUSSION Channel Model The complex channel taps hk,r (l) ∼ N (0, σ 2 (l)) (Rayleigh fading) are uncorrelated and σ 2 (l) follows the channel power-delay profile named “Channel A”, which represents a typical indoor multipath channel for HIPERLAN/2 with carrier 5.2 GHz and bandwith B= 200 M Hz (T = 10 nsec). The channel order is L ≈ 19, ( the delay spread is ≈ 190 nsec.). The results are averaged over 100 random channels. The AWGN variance for 2 each antenna is N0 and the SNR is SN R := tr(F F H )σss /N0 , and does not include possible gain/attenuation of the channel.

Example 1 Figs. 1 compare the designs in Lemmas 1÷4 in terms of MSE, BER, mutual information. Lemma 1, which minimizes the MSE subject to the average power constraint, provides perhaps the best compromise between BER and information rate. In Figs. 1 the design in Lemma 3, that minimizes the MSE subject to the peak power constraint, and leads to Φ = KI performs very similar to that in Lemma 2, that requires Φ 6= I, confirming that most of the water-filling gain derives from the bit loading. The designs in Lemma 4, maximizing the minimum distance under the power and maximum eigenvalue constraints, are ZF and have lower BER than the other criteria. In Fig. 1 (e) and (f) are shown the MSE and information rate curves of all designs, versus the ratio of the maximum eigenvalue (which is an upperbound for the peak power) over the noise variance. Example 2 Designs like the one in [1] cannot be generalized to an arbitrary number of antennas and do not make use or take advantage of the CSI, if available. Methods such as [1], operate as diversity schemes, that exploit multiple antennas to increase the symbol SNR but not the symbol rate and thus the number of transmitted symbols is always N = M . Setting the parameters to match the values required by the design in [1], for example K = 2, R = 2, N = M symbols can be transmitted over M orthogonal subcarriers (space-time block coded OFDM). In this case, it can be shown that the SN Rk of the symbols received in P2each frequency2bin for the2scheme of [1] is: SN Rk = k,r=1 |Hk,r (k)| P0 /(2M σnn ), where Hk,r (k) = F F T [hk,r (l)] is the k, r channel transfer function at frequency bin k/M . We compare the BER of our scheme with that of [1] in Fig. 2.

M=32

0

10

min(tr( MSE))−CP min(| MSE|)−CP min(tr( MSE))−Cλ min(λ ( SNR))−CP min min(λmin( SNR))−Cλ

−2

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ST−map −2

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BER

10

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min(tr( MSE))−CP min(| MSE|)−CP min(tr( MSE))−Cλ min(λmin( SNR))−CP min(λmin( SNR))−Cλ

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20

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30

SNR (dB)

0

(a)

5

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15 SNR (dB)

20

25

30

Fig. 2. BER obtained with the designs in Lemma 1÷5 K = R = 2, M = 32, N = 32 and the design in [2](ST-map).

0

10

min(tr( MSE))−CP min(| MSE|)−CP min(tr( MSE))−Cλ min(λmin( SNR))−CP min(λmin( SNR))−Cλ

−1

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4. REFERENCES −3

BER

10

[1] S. M. Alamouti, “A simple transmit diversity technique for wireless communications,” IEEE Journ. on Selected Areas in Commun., Vol. 16, No. 8, pp.1451–1458, Oct. 1998.

−4

10

−5

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[2] L. H. Brandenburg and A. D. Wyner, “Capacity of the Gaussian channel with memory: The multivariate case,” Bell Syst. Tech. Journ., Vol. 53, pp. 745-778, May/June 1974.

−6

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−7

10

0

(b)

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15 SNR (dB)

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25

30

22 min(tr( MSE))−CP min(| MSE|)−CP min(tr( MSE))−Cλ min(λ ( SNR))−CP min min(λmin( SNR))−Cλ

20 18

[4] R. Gallager, Elements of Information Theory, Section 8, New York: Wiley, 1968.

16

I(x,s), bits/(sec X Hz)

[3] G.J. Foschini, “Layered space-time architecture for wireless communication in a fading environment when using multielement antennas,” Bell Labs. Tech. Journ., Vol. 1, No. 2, pp. 41–59, 1996.

14

[5] S. Kasturia, J. T. Aslanis, J. M. Cioffi, “Vector coding for partial response channels,” IEEE Trans. on Info. Theory, Vol. 36, pp. 741–761, July 1990.

12 10 8

[6] J. W. Lechleider, “The optimum combination of block codes and receivers for arbitrary channels,” IEEE Trans. on Com., pp. 615-621, May 1990.

6 4 2 0

(c)

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min(tr( MSE))−CP min(| MSE|)−CP min(tr( MSE))−Cλ min(λmin( SNR))−CP min(λmin( SNR))−Cλ

[8] T.L. Marzetta, B. H. Hochwald, “Capacity of a mobile multiple-antenna communication link in Rayleigh flat fading”, IEEE Trans. on Info. Theory, Vol. 45, No. 1, pp.139157, Jan. 1999.

−3

MSE

10

[9] G. G. Raleigh, J. M. Cioffi, “Spatiotemporal coding for wireless communications,” IEEE Trans. on Com., Vol. 46, No. 3, pp. 357–366, March 1998.

−4

10

−10

(d)

[7] J. Yang and S. Roy, “On joint transmitter and receiver optimization for multiple-input-multiple-output (MIMO) transmission systems,” IEEE Trans. Com., vol. 42, pp. 3221– 3231, Dec. 1994.

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Average λmax( F FH)/σ2 (dB) v

Fig. 1. Comparison between the designs in Lemma 1 ÷ 4M = 32, K = R = 4, N = 112, P0 = 32: (a)-(d) ˆ). MSE=tr(M SE(F )); (b) average BER; (c) I(x, s

[10] A. Scaglione, P. Stoica, S. Barbarossa, G. B. Giannakis, H. Sampath, “Optimal Designs for Space-Time Linear Precoders and Decoders,” IEEE Trans. on Sig. Proc., accepted Aug. 2001. (http://people.ece.cornell.edu/scaglione)