Diversity-Multiplexing Tradeoff in Selective-Fading MIMO Channels

Di v ersity-Multiplexing T radeof f in Selecti v e-Fading MIMO Channels Pedro Coronel and Helmut B ¨ olcskei Communication T echnology Laboratory ETH Zurich, 8092 Zurich, Switzerland E-mail: { pco, boelcskei } @nari.ee.ethz.ch Abstract — W e establish the optimal diversity-multiplexing (DM) tradeoff of coherent time, frequency and time-frequency selective-fading MIMO channels and provide a code design criterion for DM-tradeoff optimality . Our results are based on the analysis of the “Jensen channel” associated to a given selective-fading MIMO channel. While the original problem seems analytically intractable due to the mutual information being a sum of correlated random variables, the J ensen channel is equivalent to the original channel in the sense of the DM- tradeoff and lends itself nicely to analytical treatment. Finally , as a consequence of our results, we find that the classical rank criterion for space-time code design (in selective-fading MIMO channels) ensur es optimality in the sense of the DM-tradeoff. I . I N TR O D U C T I O N The div ersity-multiplexing (DM) tradeoff frame work intro- duced by Zheng and Tse [1] allo ws to efficiently characterize the information-theoretic performance limits of communica- tion ov er multiple-input multiple-output (MIMO) fading chan- nels. In addition, the results in [1] hav e triggered significant activity on the design of DM-tradeoff optimal space-time codes. In particular , the non-vanishing determinant criterion [2], [3] on codew ord diffe rence matrices has been shown to constitute a sufficient condition for DM-tradeoff optimality in flat-fading MIMO channels with two transmit and two or more receiv e antennas [3]; this criterion has led to the construction of space-time codes based on constellation rotation [3], [4] and cyclic division algebras [5]. In [6] lattice-based space- time codes hav e been shown to be DM-tradeoff optimal. The DM-tradeoff optimality of appr oximately universal space-time codes was established in [7]. Contributions: While the results mentioned abov e focus on frequency-flat block-fading channels, extensions to frequency- selectiv e channels can be found in [8], [9]. Ho wev er, a general characterization of the optimal DM-tradeoff in time, frequency or time-frequency selectiv e-fading MIMO channels, in the following simply referred to as selectiv e-fading MIMO channels, remains an open problem. The present paper re- solves this problem for the coherent case (i.e., for perfect channel state information (CSI) at the recei ver) and provides a code design criterion guaranteeing DM-tradeoff optimality . The first author was previously with IBM Research, Zurich Research Laboratory , Switzerland. This work was supported in part by the STREP project No. IST -026905 MASCOT within the Sixth Frame work Programme of the European Commission. Our results are based on exponentially tight (in the sense of exhibiting the same DM-tradeoff behavior) upper and lower bounds on the mutual information of (coherent) selectiv e- fading MIMO channels. In particular , we sho w that the DM- tradeoff of this class of channels can be obtained by solving the analytically tractable problem of computing the DM-tradeoff curve corresponding to the associated “Jensen channel”. Notation: M T and M R denote the number of transmit and receiv e antennas, respectiv ely . W e define m := min(M T , M R ) and M := max(M T , M R ) . For x ∈ R , we let [ x ] + := max (0 , x ) . The superscripts T , H and ∗ stand for trans- position, conjugate transposition and complex conjugation, respectiv ely . I n is the n × n identity matrix, A ⊗ B and A  B denote, respectiv ely , the Kronecker and Hadamard products of the matrices A and B , and A  B stands for the positiv e semidefinite ordering. If A has columns a k ( k = 1 , 2 , . . . , m ), v ec( A ) = [ a T 1 a T 2 . . . a T m ] T . For the n × m matrices A k ( k = 0 , 1 , . . . , K − 1 ), diag { A k } K − 1 k =0 denotes the nK × mK block-diagonal matrix with the k th diagonal entry giv en by A k . If S is a set, |S | denotes its cardinality . For index sets S 1 ⊆ { 1 , 2 , . . . , n } and S 2 ⊆ { 1 , 2 , . . . , m } , A ( S 1 , S 2 ) stands for the (sub)matrix consisting of the rows of A indexed by S 1 and the columns of A indexed by S 2 . The eigen values of the n × n Hermitian matrix A , sorted in ascending order , are denoted by λ k ( A ) , k = 1 , 2 , . . . , n . The Kronecker delta function is defined as δ ( m ) = 1 for m = 0 and zero otherwise. If X and Y are random variables (R Vs), X ∼ Y denotes equality in distribution and E X is the expectation operator with respect to (w .r .t.) the R V X . The random vector x ∼ C N ( 0 , C ) is multiv ariate circularly symmetric zero-mean complex Gaussian with E  xx H  = C . f ( x ) and g ( x ) are said to be exponentially equal, denoted by f ( x ) . = g ( x ) , if lim x →∞ log f ( x ) log x = lim x →∞ log g ( x ) log x . Exponential inequality , denoted by ˙ ≥ and ˙ ≤ , is defined analogously . I I . C H A N N E L A N D S I G NA L M O D E L The input-output relation for the class of MIMO channels considered in this paper is giv en by y n = r SNR M T H n x n + z n , n = 0 , 1 , . . . , N − 1 (1) where the index n corresponds to a time, frequency or time- frequency slot and SNR denotes the signal-to-noise ratio at each receiv e antenna. The vectors y n , x n and z n denote, respectiv ely , the corresponding M R × 1 receiv e signal vector , M T × 1 transmit signal vector , and M R × 1 zero-mean circularly symmetric complex Gaussian noise vector satisfying E  z n z H n 0  = δ ( n − n 0 ) I M R . W e restrict our analysis to spatially uncorrelated Rayleigh f ading channels so that, for a giv en n , H n has i.i.d. C N (0 , 1) entries. W e do allow , howe ver , for correlation across n , assuming, for simplicity , that each scalar subchannel has the same correlation function, i.e., E { H n ( i, j )( H n − m ( i, j )) ∗ } = r H ( m ) , ( i = 1 , 2 , . . . , M R , j = 1 , 2 , . . . , M T ) . Defining H = [ H 0 H 1 . . . H N − 1 ] , we therefore hav e E  v ec( H ) (vec( H )) H  = R H ⊗ I M T M R (2) where the covariance matrix R H ( i, j ) = r H ( i − j ) ( i, j = 0 , 1 , . . . , N − 1) follo ws from the channel’ s scattering func- tion [10]. In the purely frequency-selectiv e case, e.g., assuming an orthogonal frequency-di vision multiplexing (OFDM) system [11] with N tones and hence H n = P L − 1 l =0 H ( l ) e − j 2 π N ln , where the uncorrelated (across l ) matrix- valued taps H ( l ) hav e i.i.d. C N  0 , σ 2 l  entries, we obtain r H ( m ) = P L − 1 l =0 σ 2 l e − j 2 π N lm ( m = 0 , 1 , . . . , N − 1) . In the remainder of the paper , we use the definition ρ := rank( R H ) . I I I . D I V E R S I T Y - M U LT I P L E X I N G T R A D E O FF A. Pr eliminaries Assuming perfect CSI in the receiv er , the mutual informa- tion of the channel in (1) is giv en by I( SNR ) = 1 N N − 1 X n =0 log det  I M R + SNR M T H n C n H H n  (3) where the transmit signal vectors are uncorrelated across n and satisfy x n ∼ C N ( 0 , C n ) with power constraint T r ( C n ) ≤ M T , n = 0 , 1 , . . . , N − 1 . The DM-tradeoff realized by a fam- ily (w .r .t. SNR) of codes C r with rate R ( SNR ) = r log SNR , where r ∈ [0 , m] , is giv en by the function d C ( r ) = − lim SNR →∞ log P e ( r , SNR ) log SNR where P e ( r , SNR ) is the error probability obtained through ML detection. At a gi ven SNR, the corresponding codebook C r ( SNR ) contains SNR N r codew ords X = [ x 0 x 1 . . . x N − 1 ] . W e say that such a family of codes C r operates at multiplexing rate r . The optimal tradeoff curve d ? ( r ) = sup C r d C ( r ) , where the supremum is taken over all families of codes satisfying R ( SNR ) = r log SNR , quantifies the maximum achiev able div ersity gain as a function of r . Since the outage probability P O ( r , SNR ) is a lower bound to the error probability [1], we hav e d ? ( r ) ≤ d O ( r ) = − lim SNR →∞ log P O ( r , SNR ) log SNR . Extending the arguments that lead to [1, Eq. (9)] to the case N > 1 , we can conclude that setting C n = I M T ( n = 0 , 1 , . . . , N − 1 ) in (3) does not alter the exponential behavior of mutual information. Hence P O ( r , SNR ) . = P 1 N N − 1 X n =0 log det  I M R + SNR H n H H n  < r log SNR ! (4) where we used the fact that the factor 1 / M T in (3) can be neglected in the scale of interest. Let µ ( n ) := [ µ 1 ( n ) µ 2 ( n ) . . . µ m ( n )] ( n = 0 , 1 , . . . , N − 1 ), with the singularity lev els defined as µ k ( n ) = − log λ k ( H n H H n ) log SNR , k = 1 , 2 , . . . , m and note that [1] P O ( r , SNR ) . = P ( O ( r )) (5) where O ( r ) =  µ ( n ) ∈ R m + , n = 0 , 1 , . . . , N − 1 : 1 N N − 1 X n =0 m X k =1 [1 − µ k ( n )] + < r  (6) and R m + denotes the nonnegati ve orthant. Unlike the frequency- flat fading case treated in [1], characterizing d O ( r ) for the selectiv e-fading case seems analytically intractable with the main difficulty stemming from the fact that one has to deal with the sum of correlated (recall that the H n are correlated across n ) terms in (4). It turns out, howe ver , that one can find lower and upper bounds on I( SNR ) which are exponentially tight (and, hence, preserve the DM-tradeoff behavior) and analytically tractable. The next section formalizes this idea. B. J ensen channel and Jensen outage event W e start by noting that applying Jensen’ s inequality yields I( SNR ) = 1 N N − 1 X n =0 log det  I M R + SNR M T H n H H n  ≤ log det  I m + SNR M T N HH H  := J( SNR ) (7) where the “Jensen channel” is defined as H = ( [ H 0 H 1 . . . H N − 1 ] , if M R ≤ M T , [ H H 0 H H 1 . . . H H N − 1 ] , if M R > M T . In the following, we say that a Jensen outage event occurs if the Jensen channel H is in outage w .r .t. the rate R ( SNR ) = r log SNR , i.e., if J( SNR ) < R ( SNR ) . The corresponding out- age probability will be denoted as P J ( r , SNR ) and clearly sat- isfies P J ( r , SNR ) ≤ P O ( r , SNR ) . The operational significance of a Jensen outage will be established at the end of this section. W e shall first focus on characterizing the Jensen outage event analytically . Using (2), it is readily seen that H = H w ( R 1 / 2 H ⊗ I M ) , where H w is an i.i.d. C N (0 , 1) matrix with the same dimensions as H . Noting that H w U ∼ H w for U unitary and using the eigendecomposition R H ⊗ I M = U ( Λ ⊗ I M ) U H , where Λ = diag { λ 1 ( R H ) , λ 2 ( R H ) , . . . , λ ρ ( R H ) , 0 , . . . , 0 } , it follows that J( SNR ) = log det  I m + SNR M T N H w ( R H ⊗ I M ) H H w  ∼ log det  I m + SNR M T N H w ( Λ ⊗ I M ) H H w  . Next, observe that the following positive semidefinite ordering holds λ 1 ( R H ) diag { I ρ M , 0 }  Λ ⊗ I M  λ ρ ( R H ) diag { I ρ M , 0 } . (8) Since f ( A ) = log det( I + A ) is increasing over the cone of positiv e semidefinite matrices [12], we get the following bounds on the Jensen outage probability P  log det  I m + λ ρ ( R H ) SNR M T N H w H H w  < r log SNR  ≤ P J ( r , SNR ) ≤ P  log det  I m + λ 1 ( R H ) SNR M T N H w H H w  < r log SNR  (9) where H w = H w ([1 : m] , [1 : ρ M]) . T aking the e xponential limit (in SNR) in (9), it follows readily that P J ( r , SNR ) . = P  log det  I m + SNR H w H H w  < r log SNR  . (10) For later use, we define α := [ α 1 α 2 . . . α m ] with the singularity lev els α k = − log λ k ( H w H H w ) log SNR , k = 1 , 2 , . . . , m (11) and note that P J ( r , SNR ) . = P ( J ( r )) , where J ( r ) = ( α ∈ R m + : α 1 ≥ α 2 ≥ . . . ≥ α m , m X k =1 [1 − α k ] + < r ) . It is now natural to define the Jensen outage curve as d J ( r ) = − lim SNR →∞ log P J ( r , SNR ) log SNR . Based on (10), we can conclude that d J ( r ) is nothing but the DM-tradeoff curve of an effecti ve MIMO channel with ρ M transmit and m receiv e antennas. W e can therefore directly apply the results in [1] to infer that the Jensen outage curve is the piecewise linear function connecting the points ( r, d J ( r )) for r = 0 , 1 , . . . , m , with d J ( r ) = ( ρ M − r )(m − r ) . (12) Since, as already noted, P J ( r , SNR ) ≤ P O ( r , SNR ) , we obtain d C ( r ) ≤ d ? ( r ) ≤ d O ( r ) ≤ d J ( r ) , r ∈ [0 , m] , (13) for any family of codes C r . The optimal DM-tradeoff curve d ? ( r ) will be established in the next section by sho wing that codes satisfying d C ( r ) = d J ( r ) do exist and hence d ? ( r ) = d J ( r ) . I V . J E N S E N - O P T I M A L C O D E D E S I G N C R I T E R I O N The goal of this section is to derive a sufficient condition for a family of codes to achiev e d J ( r ) , and hence, by virtue of (13), to be DM-tradeoff optimal. A. Code design criterion Theor em 1: Consider a family of codes C r with block length N ≥ ρ M T that operates over the channel (1). If, for any codebook C r ( SNR ) ∈ C r and any two code words X , X 0 ∈ C r ( SNR ) , the codeword difference matrix E = X − X 0 is such that rank  R H  E H E  = ρ M T (14) then the error probability (for ML decoding) satisfies P e ( r , SNR ) . = SNR − d J ( r ) . Pr oof: W e start by deriving an upper bound on the av erage (w .r .t. the random channel) pairwise error probability (PEP). Assuming that X was transmitted, the probability of the ML decoder mistakenly deciding in fav or of codeword X 0 can be upper-bounded in terms of the codeword difference vectors e n = x n − x 0 n ( n = 0 , 1 , . . . , N − 1 ) as P ( X → X 0 ) ≤ E H ( exp − SNR 4M T N − 1 X n =0 || H n e n || 2 !) = E H  exp  − SNR 4M T T r  H w ΥH H w   where Υ = ( R 1 / 2 H ⊗ I M T ) diag  e n e H n  N − 1 n =0 ( R 1 / 2 H ⊗ I M T ) and H w denotes an M R × M T N i.i.d. C N (0 , 1) ma- trix. Straightforward manipulations reveal that rank( Υ ) = rank  R H  E H E  so that the assumption (14) implies rank( Υ ) = ρ M T . W ith the eigendecomposition Υ = UΛU H , we hav e T r  H w ΥH H w  ∼ T r  H w ΛH H w  , and hence P ( X → X 0 ) ≤ E H  exp  − SNR 4M T T r  H w ΛH H w   . Setting H w = H w ([1:M R ] , [1: ρ M T ]) and denoting the small- est nonzero eigen value of Υ as λ , we note that T r  H w ΛH H w  ≥ λ T r  H w H H w  (15) and thus P ( X → X 0 ) ≤ E H w  exp  − λ SNR 4M T T r  H w H H w   . (16) Next, note that T r  H w H H w  = T r  H w H H w  = m X k =1 λ k ( H w H H w ) = m X k =1 SNR − α k (17) where (17) follo ws from (11). W e can now write the PEP upper-bound in (16) in terms of the singularity lev els α k ( k = 1 , 2 , . . . , m ) characterizing the Jensen outage e vent: P ( X → X 0 ) ≤ E α ( exp − λ 4M T m X k =1 SNR 1 − α k !) . (18) Next, consider a realization of the random vector α and let S = { k : α k ≤ 1 } . W e hav e m X k =1 SNR 1 − α k ≥ X k ∈S SNR 1 − α k (i) ≥ |S | SNR 1 |S | P k ∈S (1 − α k ) (ii) = |S | SNR 1 |S | P m k =1 [1 − α k ] + (19) where (i) follows from the arithmetic-geometric mean inequal- ity and (ii) follows from the definition of S . Using (19) in (18), we get P ( X → X 0 ) ≤ E α  exp  − λ |S | 4M T SNR 1 |S | P m k =1 [1 − α k ] +  . (20) The dependence of the PEP upper bound (20) on the singular- ity lev els characterizing the Jensen outage e vent suggests to split up the ov erall error probability according to P e ( r , SNR ) = P (error , α ∈ J ( r )) + P (error , α / ∈ J ( r )) = P ( α ∈ J ( r )) P (error | α ∈ J ( r )) + P ( α / ∈ J ( r )) P (error | α / ∈ J ( r )) ≤ P ( α ∈ J ( r )) + P ( α / ∈ J ( r )) P (error | α / ∈ J ( r )) . (21) For any α / ∈ J ( r ) , we ha ve P m k =1 [1 − α k ] + ≥ r and |S | ≥ 1 , which upon noting that |C r ( SNR ) | = SNR N r , yields the following union bound based on the PEP in (20) P (error | α / ∈ J ( r )) ≤ SNR N r exp  − λ 4M T SNR r/ m  where we used |S | ≤ m . Hence, for any r > 0 , P (error | α / ∈ J ( r )) decays exponentially in SNR and we hav e P (error , α / ∈ J ( r )) = P ( α / ∈ J ( r )) | {z } ≤ 1 P (error | α / ∈ J ( r )) ≤ SNR N r exp  − λ 4M T SNR r/ m  . (22) Consequently , noting that P ( α ∈ J ( r )) . = P J ( r , SNR ) and using (22) in (21), we obtain P e ( r , SNR ) ˙ ≤ P J ( r , SNR ) . Since P J ( r , SNR ) ≤ P O ( r , SNR ) , it follows trivially that P J ( r , SNR ) ˙ ≤ P O ( r , SNR ) . In addition, for a specific family of codes C r , we hav e P O ( r , SNR ) ≤ P e ( r , SNR ) and hence P O ( r , SNR ) ˙ ≤ P e ( r , SNR ) . Putting the pieces together , we finally obtain P O ( r , SNR ) ˙ ≤ P e ( r , SNR ) ˙ ≤ P J ( r , SNR ) ˙ ≤ P O ( r , SNR ) which implies P e ( r , SNR ) . = P J ( r , SNR ) and hence (by definition of d J ( r ) ) P e ( r , SNR ) . = SNR − d J ( r ) . As a direct consequence of Theorem 1, a family of codes that satisfies (14) for all codew ord difference matrices in any codebook C r ( SNR ) ∈ C r realizes a DM-tradeoff curve d C ( r ) = d J ( r ) and hence, by (13) d J ( r ) ≤ d ? ( r ) ≤ d J ( r ) which implies d ? ( r ) = d J ( r ) . (23) The optimal DM-tradeoff curve for selecti ve-fading MIMO channels is therefore given by the DM-tradeoff curve of the associated Jensen channel. Put differently , Theorem 1 shows that, ev en though J ( r ) ⊆ O ( r ) by definition, we still hav e P ( J ( r )) . = P ( O ( r )) which essentially says that the “original” channel has the same high-SNR outage behavior as its associated Jensen channel. The code design criterion in Theorem 1 provides a sufficient condition for achieving the DM-tradeoff curve. Interestingly , the classical rank criterion [13]–[18], aimed at maximizing the div ersity gain for r = 0 , can be shown [19] to be equiv alent to the criterion in Theorem 1. W e emphasize, howe ver , that optimality w .r .t. the DM-tradeoff at multiplexing rate r requires that (14) is satisfied for all codew ord difference matrices in any codebook C r ( SNR ) ∈ C r , in particular also for SNR → ∞ . W e next state a sufficient condition for DM-tradeoff optimality which makes this aspect explicit and establishes a connection to the approximately universal code design criterion in [7]. Cor ollary 1: A family of codes C r of block length N ≥ ρ M T is DM-tradeoff optimal if there exists an  > 0 such that λ m ( SNR ) ˙ ≥ SNR − ( r −  ) (24) where λ ( SNR ) = min k =1 , 2 ,...,ρ M T E = X − X 0 , X , X 0 ∈C r ( SNR )  λ k ( R H  E H E ) > 0  . Pr oof: Using (24) in (22), we obtain P (error , α / ∈ J ( r )) ≤ SNR N r exp − SNR / m 4M T ! which, following the same logic as in the proof of Theorem 1, implies that P e ( r , SNR ) . = SNR − d J ( r ) . Note that the quantity λ m ( SNR ) is tri vially a lo wer bound on the product of the m smallest nonzero eigen values of any codew ord difference matrix in the codebook C r ( SNR ) . Consequently , in the case of non-selectiv e fading, where R H  E H E = E H E , any family of codes C r satisfying (24) will also be approximately universal in the sense of [7, Th. 3.1]. Moreov er, if λ ( SNR ) remains strictly positiv e as SNR → ∞ , C r fulfills the non-vanishing determinant criterion [2], [3] and will, by (22) and the same arguments as in the proof of Theorem 1, be DM-tradeoff optimal. B. Application to the frequency-selective case As an example, we shall next specialize our results to frequency-selecti ve fading MIMO channels, recovering the results reported previously in [8], [9]. For the sake of sim- plicity of exposition, we shall employ a cyclic signal model, as obtained in an OFDM system for example. The channel’ s transfer function is giv en by H ( e j 2 πθ ) = L − 1 X l =0 H ( l ) e − j 2 πlθ , 0 ≤ θ < 1 where the H ( l ) have i.i.d. C N  0 , σ 2 l  entries and satisfy E n v ec( H ( l )) vec( H ( l 0 )) H o = σ 2 l δ ( l − l 0 ) I M T M R . W ith H n = H ( e j 2 π n N ) , n = 0 , 1 . . . , N − 1 , the channel’ s cov ariance matrix follows as R H = F diag  σ 2 0 , σ 2 1 , . . . , σ 2 L − 1 , 0 , . . . , 0  F H where F is the N × N FFT matrix. Since rank( R H ) = L , inserting ρ = L into (12) and using (23) yields the optimal DM-tradeoff curve as the piecewise linear function connecting the points ( r , d ? ( r )) for r = 0 , 1 , . . . , m , with d ? ( r ) = ( L M − r )(m − r ) . (25) This is the optimal DM-tradeof f curve for frequency-selecti ve fading MIMO channels reported pre viously in [9]. Specializing (25) to the case M T = M R = 1 and noting that d ? ( r ) = ( L − r )(1 − r ) = L (1 − r ) for r = { 0 , 1 } , yields the results reported in [8]. 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