On MIMO Fading Channels with Side Information at the Transmitter

Transcription

1 On MIMO Fading Channels with ide Information at the Transmitter EE850 Project, Ma 005 Tairan Wang ID: Abstract Transmission of information over a MIMO discrete-time flat fading channel with an average power constraint is considered. The scenario in which the transmitter alone has knowledge of the fading levels, as side information, is assumed. This knowledge ma be provided in either a causal or a non-causal manner. Upper and lower bounds are derived for each of the two cases. The tools developed are applied to the On/Off fading channel, and some useful strategies for transmitter adaption are discussed. I. INTRODUCTION Among the extensive research on fading channels, special attention has been given to the scenarios in which the fading coefficients are available to the communication sstem as side information. These scenarios include the case of channel side information (CI) available to both the receiver and the transmitter and the case of CI available to the receiver alone. In the non-coherent scenario, CI is not available to either the receiver or the transmitter, was considered. The scenario in which the fading levels are available onl to the transmitter was left mostl unconsidered. Aside from the scientific curiosit, this scenario gains practical use as well, for example, in OFDM-Discrete Multitone based sstems who have a-priori knowledge of all sub-carriers gains. Another motivation comes from the increase in computation resources at cellular base stations which ma use the available causal fading levels to design more sophisticated and powerful codes. uch is the case in Time Division Duplexing (TDD) based sstems where reciprocit facilitates channel measuring more accuratel at the transmitter, due to these increased processing capabilities. The receiver, for complexit reasons, avoids this operation. This project checks the results in paper [] carefull and then extend them to a MIMO counterpart. In [], problem of communicating through a flat fading AWGN channel with the fading coefficients available to the transmitter, as side information, was considered, in either a causal or a non-causal manner. Upper and lower bounds are derived for each of the two cases and Arimoto-Blahut like algorithms, to numericall compute capacit in each case, are presented when an average power constraint was imposed. The tools developed are applied to the On/Off fading channel, and to some restricted cases of a Raleigh fading channel. In the latter case the capacit per unit cost is examined and it is shown that transmitting at an arbitraril low E b /N 0 will sustain reliable communication at zero spectral efficienc, regardless of the causalit/non-causalit nature of the available side information, mimicking the case of full available CI at the transmitter and the receiver. II. CHANNEL MODEL There are M transmitter antennas and N receiver antennas. o, we consider the following MIMO channel model s s s M x z. = s s s M x z., () N s N s N s NM x M z N where x m C is the channel input from the mth transmitter antenna, n C is the channel output at the nth receiver antenna, s nm C is the flat fading coefficient from transmitter antenna m to receiver antenna n. The fading coefficients are independent (with respect to both n and m) and identicall distributed (i.i.d.). The additive noise at receiver antenna n is denoted z n C, and is independent (with respect to n), identicall distributed CN (0, ). An equivalent channel is = x z, ()

2 s s M where := [,..., N ] T, x := [x,..., x M ] T, z := [z,..., z N ] T, and := The fading s N s NM coefficients = (i) var with time i (discrete time index). We assume the noise processes are independent of the fading processes and of the channel inputs. We further assume a perfect knowledge of the fading coefficients (i) at the transmitter in either a causal manner (k) k i} or a non-causal manner (k) k }. Finall, it is assumed that the signalling is subject to the average power constraint E[ x ] P. (3) Gel fand and Pinsker [3] have found the capacit formula for a discrete memorless channel with random state known non-causall to the transmitter. Following the extensions made b Costa [5] for discrete-time channels with continuous alphabets and the introduction of constraints on the channel input X we have C nc = p(u ), F:U X, E[ F(U,) ] P I(U; Y ) I(U; )}, (4) where U is an auxiliar random variable, F is a deterministic function, and the joint distribution of the random variables, U, X and Y is given b p()p(u )p( x, ) if x = F(u, ), p(, u, x, ) = (5) 0 otherwise. It was shown b Cohen [4] that taking U to be independent of leads to a capacit formula equivalent to that given b hannon [] corresponding to the problem where the transmitter has causal CI. In the case we have C c = p(u), F:U X, E[ F(U,) ] P where the joint distribution of the random variables, U, X and Y is given b p()p(u)p( x, ) if x = F(u, ), p(, u, x, ) = 0 otherwise. I(U; Y ), (6) imilarl, the capacit formula form given b hannon (extended to continuous alphabets and including an average power constraint), C c = p(t), E[ T () ] P (7) I(T ; Y ), (8) can be shown to be a special case of the Gel fand-pinsker formula b taking the strategies probabilit distribution to be independent of the channel state, C nc = p(t ), E[ T () ] P I(T ; Y ) I(T ; )}, (9) where t T, the set of all possible mappings t : X which we will refer to as hannon strategies or simpl as strategies. We will use either form of each capacit formulae as suitable. III. LOWER BOUND The lower bound can be obtained b choosing an appropriate strateg. For the non-causal case, we examine I(U; Y ) I(U; ) = I(U;, Y ) I(U; Y ) (I(U;, Y ) I(U; Y )) = I(U; Y ) I(U; Y ). (0) The lower bound will be obtained b using the following choice of conditional probabilit distributions: } p L (x ) = arg I(X; Y ) p(x ) Ω provided it exists, () p L (u x, ) = δ(u gβ (x, )) if, Q β (u ) if, ()

3 3 using the following definitions, = I(X; Y ) 0, X p L (x )}, = I(X; Y ) = 0, X p L (x )}, β P/N, where g β is a chosen function which depends on the parameter β (the NR) and with the requirement that there is a one-to-one mapping between x and u for ever, and finall Q β is a chosen conditional probabilit distribution that depends on β as well. The idea behind taking Q β to be independent of x is that for those which it is defined for X has zero power. As for the causal case, the lower bounds will be derived from eq. (6) taking U to be a Gaussian random variable and X given = to be Gaussian as well. IV. UPPER BOUND It is well known that when complete side information is given to both the transmitter and the receiver the capacit of channel () is given b [8], p(x ) Ω I(X; Y ), (3) where Ω is the set of all conditional probabilit distributions p(x ) satisfing the constraint (3). Note that eq. (3) ma be regarded as a special case of the Gel fand-pinsker model when the state is added to the observation Y at the receiver s end. This, of course, is a trivial upper bound on the channel capacit when side information is available onl to the transmitter. A. Non-Causal CI In order to develop an upper bound on Gel fand-pinsker capacit, consider the capacit formulation given b eq. (9). Furthermore, for the time being we assume all the relevant alphabets are discrete. The associated average power constraint is seen to be, p()p(t ) t() P. (4),t Expanding the mutual information we see that I(T ; Y ) I(T ; ) = p()p(t )p( t, ) ln p(t ) p(t ),,t = } p(, t) q() p()p(t )p( t, ) ln ln q()p(t ) p(),,t =,,t p()p(t )p( t, ) ln,,t p()p(t )p( t, ) ln p(, t) q()p(t ) D(p() q()) p(, t) q()p(t ), for an probabilit distribution q() on Y, where equalit is achieved iff q() = p(). Finding the capacit can now be rewritten as the following optimization problem, C nc = max min p(, t) p()p(t )p( t, ) ln p(t ) q() q()p(t ), (5),,t with the constraints, t p(t ) =, p(t ) 0, t,,t p()p(t ) t() P. Lemma : The functional over which we optimize in (5) is concave in p(t ) and convex in q().

4 4 Proof is similar to the counterpart in paper [] and is omitted here. Following the work in [7] we can reformulate our maximization problem using the lagrange dual technique. Forming a partial lagrangian for (5) we have, L = p(, t) p()p(t )p( t, ) ln q()p(t ) ( λ ) p(t ) λ P p()p(t ) t(), (6),t, t,t where we introduce the lagrange multipliers λ R, λ 0 and notice that max p(t ) min q() min λ,λ L, such that p(t ) 0 and q() is a valid probabilit distribution, has the same optimal value as (5). Continuing to follow the ideas in [7] (note the exchange of the min and the max) we have the upper bound on problem (5) given b C nc min max q(),λ,λ p(t ) [ p()p(t ) p( t, ) ln,t ] p(, t) q()p(t ) λ p() λ t() λ λ P, (7) where p(t ) 0 and q() is a valid probabilit distribution. To find the solving p(t ) of the inner maximization, for a given q(), λ and λ, we differentiate L with respect to p(t ), L = p()p( t, ) ln p( t, )p(t )p( ) λ p()λ t() p(t ) q()p(t ) p()p(t )p( t, ) p(t ) p( )p(t )p( t, p( t, )p() ), p( )p(t )p( t, ) } = p() p( t, ) ln p( t, )p(t )p( ) λ q()p(t ) p() λ t() and we arrive to the following conditions on the solution to the maximization problem (7) (appling the Karush- Kuhn-Tucker (KKT) conditions) p( t, ) ln p( t, )p(t )p( ) λ q()p(t ) p() λ t() = 0, if p(t ) > 0, p( t, ) ln p( t, )p(t )p( ) λ q()p(t ) p() λ t() 0, if p(t ) = 0. (8) With the above conditions we can see that for the maximizing p(t ) in (7) we have min max L = min λ λ P. (9) q(),λ,λ p(t ) q(),λ,λ To conclude, we now have a dual problem to the problem (7) (the solution of which is an upper bound on problem (5)): C nc (P ) min λ λ P, (0) q(),λ,λ where the set over which we minimize consists of valid probabilit distributions q( ) on Y, real numbers λ and λ 0 for which a function p(t ) 0 exists such that, p( t, ) ln p( t, )p(t )p( ) λ q()p(t ) p() λ t() 0,, t, () and for an, t such that p(t ) 0 equalit in eq. () must hold. Choosing an feasible λ, λ and q() in eq. (0) such that () holds true will give us an upper bound on the Gel fand-pinsker capacit with an average power constraint.

5 5 B. Causal CI Again starting with the assumption that all relevant alphabets are discrete and examining eq. (6) we have the following, Lemma : For ever probabilit distribution p( ) on U, deterministic function F(u, ) and probabilit distribution q( ) on the channel output Y where Proof : we know that, I(U; Y ) = D(w F ( u) q( )) =,u,x, I(U; Y ) u p(u)d(w F ( u) q( )), () p( F(u, ), )p() ln p( F(u, ), )p( ). q() p( x, )p(x u, )p(u)p() ln,x p( x, )p(x u, )p(u)p( ) p(u),u,x p( x, )p(x u, )p(u )p( ) p( F(u, ), )p( ),u p( F(u, ), )p(u )p( ), =,u, p( F(u, ), )p(u)p() ln now for ever p(u), F(u, ) and q() we have, p(u) p( F(u, ), )p() ln u, therefore, = p(u) p( F(u, ), )p() ln u, = p( F(u, ), )p(u)p() ln,u = D( p() q()) 0, I(U; Y ) u p(u) p( F(u, ), )p() ln p( F(u, ), )p( ) q() I(U; Y ),u p( F(u, ), )p( )p(u ) q(),u p( F(u, ), )p( )p(u ) q() p( F(u, ), )p( ) q() = p(u)d(w F ( u) q( )). u Note we write w F ( ) to emphasize the dependence of w on F. The extension of the upper bound () to continuous alphabets and to constrained input ma be done to obtain: I(U; Y ) D(W F ( u) Q( ))dp (u) and the upper bound: C c (P ) = inf P (u),f γ 0 inf γ 0 P (u),f inf = inf γ 0 P (u),f γ 0 F u D(W F ( u) Q( )), (3) u ( I(U; Y ) γ P ( I(U; Y ) γ P u ( D(W F ( u) Q( )) γ P D(W F ( u) Q( )) γ )} F(u, ) dp ()dp (u) )} F(u, ) dp ()dp (u) ( P )} F(u, ) dp () )} F(u, ) dp (). (4)

6 6 V. APPLICATION AND DICUION To give an application example, we consider parallel fading channel, which is a degraded case of MIMO channel. In parallel fading channel, the number of transmitter antennas is the same as the number of receiver antennas, i.e., M = N = K; and the channel = diag(s,..., s K ) is a diagonal matrix. More important, b considering parallel fading channel, we can set the channel in real field R instead of complex field C, because after using an strateg and before transmitting, we can alwas add an extra phase to the transmitted signal to cancel the phase of the corresponding channel coefficient while satisfing the average power constraint (at least making s,..., s K to be real). Then, at the receiver side, it can alwas separate the real part and imaginar part of the received signals and process them respectivel. And the two parts have the same statistics, so analze one part is enough to understand the whole stor. o the following analsis will be in the real field R. We will take K = in the following example. A. On/Off Fading Channel In this example we shall consider a channel with binar fading, that is: P r (s = ) = P r (s = 0) = α, P r (s = ) = P r (s = 0) = α. To find the lower bound on the non-causal CI capacit we start b solving eq. () which gives (note the average power constraint (3) holds): δ(x )δ(x ) s = 0, s = 0 πp exp( x P )δ(x ) s =, s = 0 f L (x ) = (5) πp exp( x P )δ(x ) s = 0, s = πp 3 exp( x x P 3 ) s =, s = where P = P = P α α and P 3 = P α α b waterfilling over both space and time. For eq. () we shall take g β (x, ) = g β (x, s )g β (x, s ), g β (x, s = ) = x, g β (x, s = ) = x, and Q β (u ) = Q β (u s )Q β (u s ), Q β (u s = 0) = N (0, Nψ ), Q β (u s = 0) = N (0, Nψ ). Thus, we have f L (u x, ) = f L (u x, s )f L (u x, s ) (6) δ(u x ) s = f L (u x, s ) = exp( u πnψ Nψ ) s = 0 (7) δ(u x ) s = f L (u x, s ) = exp( u πnψ Nψ ) s = 0 (8) where ψ and ψ depend on P/N and will be determined through numerical optimization. Using eq. (5) (8) and after some work we get: p(u = diag(0, 0)) = p(u = diag(, 0)) = p(u = diag(0, )) = p(u = diag(, )) = exp( u πnψ Nψ ) exp( u πnψ Nψ ) πp/(α α ) exp( u P/(α α ) ) exp( u πnψ Nψ ) πp/(α α ) exp( u P/(α α ) ) exp( u πnψ Nψ ) πp/(α α ) exp( u P/(α α ) ) exp( u ), (9) P/(α α )

7 7 p( u, = diag(0, 0)) = p( u, = diag(, 0)) = p( u, = diag(0, )) = p( u, = diag(0, 0)) = exp( πn N ) exp( πn N ) exp( ( u ) ) exp( πn N πn N ) exp( πn N ) exp( ( u ) ) πn N exp( ( u ) ) exp( ( u ) ). (30) πn N πn N Using p( = diag(0, 0)) = ( α )( α ), p( = diag(, 0)) = α ( α ), p( = diag(0, )) = ( α )α, p( = diag(, )) = α α, and eq. (9) (30) in eq. (0), finall we reach the lower bound eq. (3) (no space for closed form). C nc I(U; Y ) I(U; ) = p()p(u )p( u, ) ln p( )p(u )p( u, ) p( )p(u )p( u, )du ddu. (3) p(u ) The lower bound on the causal CI capacit is obtained b taking the following probabilit distribution p(u) = f L (u) = f L (u )f L (u ) f L (u ) = πp/(α α ) exp( u P/(α α ) ) f L (u ) = πp/(α α ) exp( u P/(α α ) ) and the deterministic function F(u, ) = u and from eq. (6) we have the bound eq. (3) (no space for closed form). C c I(U; Y ) = p()p(u)p( u, ) ln p( )p( u, ) p( )p(u )p( u, ddu. (3) )du To develop the upper bound on the causal CI capacit we start b writing eq. (4) for this case, C c inf p(, F, u) [p(, F, u)] ln d d γ 0 F u q() γ(p ( α )( α ) F(u, s = 0, s = 0) ( α )α F(u, s = 0, s = ) } α ( α ) F(u, s =, s = 0) α α F(u, s =, s = ) ), (33) where F(u, ) = [F(u, s ) F(u, s )] T, p(, F, u) = α πn exp( N ) α πn exp( N ) α exp( πn N ) α exp( ( F(u, s = )) ) πn N α exp( πn N ) α exp( ( F(u, s = )) ) πn N α exp( ( F(u, s = )) α ) exp( ( F(u, s = )) ). (34) πn N πn N

8 8 It is evident from the equation above that the optimal choice for F(u i, s i = 0) is 0, i =,. Taking q() = α πn exp( N ) α πn exp( N ) α exp( πn N ) α π(n P/(α α )) exp( (N P/(α α )) ) α exp( πn N ) α π(n P/(α α )) exp( (N P/(α α )) ) α α π(n P/(α α )) exp( (N P/(α α )) ) exp( ), (35) (N P/(α α )) and making the assignments γ = γp and δ = (α α )/P F(u, s = ), δ = (α α )/P F(u, s = ) in eq. (33) we get eq. (36), where we let δ, δ R and the change of variables allows us to exchange the over F and u with the over δ and δ. C c inf γ ( α δ α δ ) γ 0 α α δ δ α πn exp( N ) α πn exp( N ) ln p(,, δ, δ ) d d q(, ) α exp( πn N ) α exp( πn N ) ln p(,, δ, δ ) q(, ) d d α exp( πn N ) α exp( πn N ) ln p(,, δ, δ ) q(, d d ) α exp( πn N ) α exp( πn N ) ln p(,, δ }, δ ) q(, ) d d, (36) where = δ P N(α α ) and = δ P N(α α ). Looking at the limit as δ and δ of eq. (36) (similar to []), we get a necessar condition for the remum in (36) to exist: γ α α ( N(α α )/P ). (37) Taking γ in (36) to be the right side of eq. (37) we have the upper bound for this case, eq. (38) which will be evaluated numericall. α ( δ C c ) α ( δ ) ( N(α α )/P ) δ δ α πn exp( N ) α πn exp( The upper bound for non-causal case considered here is N ) ln p(,, δ, δ ) d d q(, ) α exp( πn N ) α exp( πn N ) ln p(,, δ, δ ) q(, ) d d α exp( πn N ) α exp( πn N ) ln p(,, δ, δ ) q(, d d ) α exp( πn N ) α exp( πn N ) ln p(,, δ }, δ ) q(, ) d d. (38) C nc C T RI = α α ln( P ), (39) N(α α ) where C T RI stands for the case where both the transmitter and the receiver have CI. This is a trivial upper bound on the capacities when CI is available to the transmitter alone, but it is tighter than the upper bound derived in ubsection IV-A.

9 9 Fig. and Fig., in the following pages, displa the bounds developed above, C lb c, C ub c, C lb nc (with numerical optimization performed when needed), where lb c stands for the lower bound on causal CI capacit, etc, and the capacit C T RI for several interesting values of α and α. Looking at the figures we note the following: ) As α and α grow larger the bounds become tighter. ) There is a clear advantage in knowing the side information in a non-causal manner over causal onl (at mid and high NR levels the upper bound on the causal CI capacit lies beneath the lower bound on the non-causal CI capacit). 3) For ver large NR the lower bound on the non-causal CI capacit becomes tight to the capacit in the full informed case. 0.7 α=0..5 α= R [nat/smbol] R [nat/smbol].5 0. C TRI C TRI 0. C ub c C lb nc 0.5 C ub c C lb nc C lb c P/N [db] C lb c P/N [db] Fig.. Bounds on capacit of a parallel binar fading channel with CI available at the transmitter, P r(s = ) = P r(s = ) = 0.. Fig.. Bounds on capacit of a parallel binar fading channel with CI available at the transmitter, P r(s = ) = P r(s = ) = 0.5. VI. OME UEFUL TRATEGIE The most difficult part during the process of finding an optimal strateg is to prove this strateg is optimal. For additive channel in Costa [5], optimal strateg as dirt paper coding can achieve the upper bound, i.e., capacit when complete side information is given to both the transmitter and the receiver. But for fading channel, this upper bound seems to be not tight in general, or sa there ma be a strict gap between the upper bound and the actual capacit. o, finding a useful (tight) upper bound will be helpful to find the optimal strateg in general. Unfortunatel, in paper [], the upper bound is not tight, especiall for non-causal CI. o we did not consider it in ection V. A. Channel Inversion A suboptimal but simple transmitter adaption scheme is channel inversion [9], i.e., it inverts the channel fading. The channel then appears to the encoder and the decoder as a time-invariant AWGN channel. Now, let γ be the instantaneous NR, and P (γ) be the instantaneous transmit power. We have the constraint γ P (γ)p(γ)dγ P. Then the power adaptation for channel inversion is given b P (γ)/p = σ/γ, where σ equals the constant received NR which can be maintained under the transmit power constraint. The constant σ thus satisfies σ/γp(γ) =, so σ = /E[/γ]. The fading channel capacit with channel inversion is just the capacit of an AWGN channel the NR σ: C(P ) = B log[ σ] = B log[ ]. (40) E[/γ] Channel inversion can exhibit a large capacit penalt in extreme fading environments. For example, in Raleigh fading E[/γ] is infinite, and thus the capacit with channel inversion is zero (so is the case for on/off channel).

10 0 A truncated inversion polic is also considered in [9], but this polic is not applicable for on/off channel, as it requires the receiver to know the threshold, which is equivalent to know the whole channel in this case. B. Log-DPC We find a useful strateg for scalar discrete-time flat fading channel model i = s i x i z i, (4) where x i 0 (the reason will be shown later) is the channel input, i R is the channel output, s i 0 are i.i.d. equivalent real random variables describing the fading coefficients after phase cancellation and z i are the i.i.d. Gaussian noise samples with variance N. We assume the noise processes are independent of the fading processes and of the channel inputs. We further assume a perfect knowledge of the fading coefficients s i at the transmitter in a non-causal manner s k k }. Finall, it is assumed that the signalling is subject to the average power constraint E[X i ] P. (4) Notice when P/N, we can alwas take logarithm to both sides of eq. (4) to get a channel with additive interference: ln i = ln s i ln x i. (43) Following the wa of dirt paper coding (e.g. modulo scheme [6] [0]), we can use this simple strateg: uppose U is the desired signal (between and ), X = ln x i is the transmitted signal, and = ln s i is the additive interference (known at TX through s i, but not at RX). Take modulo [, ] operation to get X = [U ] [,], and what is transmitted here is x i = exp(x) (this is the reason for x i 0). Receiver takes logarithm to the received signal Y = ln i, then performs modulo [, ] operation to get Y = [Y ] [,] = [X ] = [(U ) ] = [U]. o we can claim that for scalar discrete-time flat fading channel, as NR goes to infinit, C nc will approach the capacit when CI is known at both transmitter and receiver. This can be shown somehow in paper [], where at high NR, even the lower bound of the non-causal case will be close to the TRI bound. VII. CONCLUION In this project, we have tried to extend the results in paper [] to MIMO case and discussed some useful strategies. During this work, we realized that the complexit for the MIMO counterpart increases exponentiall with respect to the number of independent channels. The parallel fading channel was taken for simplicit. In m opinion, this work will not attract people s ees until some simple but meaningful strategies found to be able to easil implement in industr. REFERENCE [] Dan Goldsmith, hlomo hamai (hitz), and Yossef teinberg, On Fading Channels with ide Information at the Transmitter, submitted to IEEE Transactions on Information Theor, Nov [] C. E. hannon, Channels with side information at the transmitter, IBM Journal of Research and Development, vol., pp , Oct [3]. I. Gel fand and M.. Pinsker, Coding For Channel With Random Parameters, Problems of Control and Information Theor, vol. 9, no., pp. 9-3, 980. [4] A.. Cohen, Communication with ide Information, Graduate eminar 6.96, MIT, Cambridge, MA, pring 00 ( spring 00/schedule.html). [5] Max H. M. Costa, Writing on Dirt Paper, IEEE Transactions on Information Theor, vol. 9, no. 3, pp , Ma 983. [6] U. Erez and R. Zamir, Noise prediction for channels with side information at the transmitter, IEEE Transactions on Information Theor, vol. 46, no. 4, pp , Jul 000. [7] riram Vishwanath and Andrea Goldsmith, A Dualit Theor for Channel Capacit, Proc. 4st Annual Allerton Conference on Communications, Control and Computing, Allerton, IL, Oct. 00. [8] T. Cover and M. Chiang, Dualit Between Channel Capacit and Rate Distortion With Two-ided tate Information, IEEE Transactions on Information Theor, vol. 48, no. 6, pp , June 00. [9] Andrea Goldsmith and Pravin P. Varaia, Capacit of Fading Channels with Channel ide Information, IEEE Transactions on Information Theor, vol. 43, no. 6, pp , Nov [0] Multiuser Information Theor, pp. 6, Lecture Notes 9-0.