Parallel Additive Gaussian Channels Let us assume that we have N parallel one-dimensional channels disturbed by noise sources with variances σ 2,,σ 2 N. N 0,σ 2 x x N N 0,σ 2 N y y N Energy Constraint: The total input energy is constrained on average to E the total average energy per channel use: N x 2 n = N E n = E The capacity of these parallel channels is achieved by the following input energy distribution: σn 2 E n = µ; σn 2 <µ E n = 0; σn 2 µ where µ is the Lagrange multiplier chosen such that n E n = E. Then the capacity of this set of parallel channels is given by the following Theorem: C = N 2 E n σn 2 Source: [], Section 7.5, pp. 343 ff. = N 2 µ σ 2 n
EE 7950: Statistical Communication Theory 2 Waterfilling Theorem: Proof: Let x =[x,,x N ] and y =[y,,y N ] and consider Ix; y N 2 Ix n ; y n 2 E n σn 2 } {{ } fe independent x n 2 Gaussian distributed x n Since equality can be achieved in both inequalities above the next step is to find the maximizing energy distribution E = [E,,E N ]. We make use of Theorem and identify the following sufficient and necessary conditions: fe λ E n λ 2σn 2 E n σn 2 E n 2λ = µ Q.E.D. This theorem is called the Waterfilling Theorem. Its functioning can be visualized in the following figure, where the power levels are in black: µ Power level:differnce µ σ 2 n level: σ 2 N Channels through N
EE 7950: Statistical Communication Theory 3 Correlated Parallel Channels MIMO Correlated channels arise from e.g., multiple antenna channels using N t transmit antennas and N r receive antennas: h x x 2 y y 2 h ij x Nt y Nr h Nt N r This channel is a multiple-input multiple-output MIMO channel described by the matrix equation: y = Hx n The transmitted signals x n are complex signals, as are the channel gains h ij and the received signals y n. The noise is complex additive Gaussian noise with variance N 0 that is N 0 /2 in each dimension. The path gains h ij are complex gain coefficients modeling a random phase shift and a channel gain. Often these are modeled as Rayleigh random variables modeling a scattering-rich or mobile radio transmission environment. MIMO Rayleigh Channel: The h ij are modeled as i.i.d. or correlated complex Gaussian random variables with variance /2 in each dimension.
EE 7950: Statistical Communication Theory 4 Channel Decomposition The correlated MIMO channel can be decomposed via the singular value decomposition SVD: H = UDV if r<t where U and V are unitary matrices, i.e., UU = I, and VV = I. The matrix D contains the singular values of H, which are the square roots of the eignevalues of HH and H H.IfH is square, the singular values are identical to the square of the eigenvalues of H. The channel equation can now be written in an equivalent form: y = Hx n = UDV x n U y = ỹ = D x ñ If N t >N r only the first N r signals of x will be received. If N r > N t the N r N t bottom channels will carry no signal. This leads to parallel Gaussian channels ỹ n = d n x n ñ n d N 0,σ 2 x d N 0,σ 2 N m = minnt,n r ỹ x m ỹ m
EE 7950: Statistical Communication Theory 5 MIMO Capacity The multiplicative factors d n can be eliminated by multiplying the received signal y with D. This leads back exactly to the parallel channel problem, and the capacity of the MIMO channel is determined by the waterfilling theorem: C = N d2 ne n 2σ 2 n = N d2 nµ 2σn 2 Note: The channels are complex, and hence there is no factor /2 and the variance is 2σn. 2 This capacity is achieved with the waterfilling power allocation: 2σ 2 n d 2 n E n = µ; σ 2 n <µ E n = 0; σ 2 n µ Optimal System: These considerations lead to the following optimal signalling strategy:. Perform the SVD of the channel H U, V, D. 2. Multiply the input signal vector x with V. This is matrix processing. 3. Multiply the output signal y with the matrix processor U. 4. Use each channel with signal-to-noise ratio d 2 ne n /2σ 2 n independently. x V x y U ỹ Matrix Processor MIMO Channel Matrix Processor
EE 7950: Statistical Communication Theory 6 Symmetric MIMO Capacity Drawback: the channel H needs to be known at both the transmitter and the receiver so the SVD can be computed. Fact: Channel knowledge is not typically available at the transmitter, and the only choice we have is to distribute the energy uniformly over all component channels. This leads to the Symmetric Capacity: C = N d2 ne = N d2 ne Noting that the d 2 n are the eigenvalues of HH, the above formula can be written in terms of matrix eigenvalues, using the fact detm = λm, and deti M = λm: C = N d2 ne = det I Nr = det I Nr E HH E H H Discussion: The capacity of a MIMO channel is goverend by the singular values of H, or in the symmetrical case by its eigenvalues. The eigenvalues determine the channels gains of the independent parallel channels. Channel H needs to be known at the receiver Channel Estimation We will study the behavior of channel eigenvalues later in this course.
EE 7950: Statistical Communication Theory 7 MIMO Fading Channels Assumed that the channel H is known at the receiver Ix;y, H = Ix; H Ix; y H = Ix; y H = E H [Ix; y H = H 0 ] We need to average the mutual information over all channel realizations. Ix; y H is maximized if x is circularly symmetric complex Gaussian with covariance Q, and Ix;y, H = E H [ det I r E ] Ix;y, H = E H [ det I r HQH E HUDU H The spectral decomposition of Q = UDU produces an equivalent channel H = HU with the same statistics, hence the maximizing Q is diagonal. Furthermore, concavity of the function det shows that Q = I, hence the maximizing Q is a multiple of the identity ] Capacity of the MIMO Rayleigh Channel: C = E H [ det I r By the law of large numbers: HH E ] HH N t N t I r and C = r E 2σ 2
EE 7950: Statistical Communication Theory 8 Evaluation of the Capacity Formula Following Telatar [2] define the random matrix W = { HH if N r <N t H H if N r N t W is an m m; m = minr, t non-negative definite matrix with real, non-negative eigenvalues µ n = d 2 n The capacity can be written in terms of these eigenvalues: C = E {µn } m E µ n for r = t symmetric channels, let {λ n } be the eigenvalues of H, then µ n = λ 2 n The eigenvalues of W follow a Wishart distribution: pµ,,µ m =α m e µ i µ n m pµ i,,µ m =α m i= i= e µ i µ n m i i det i<j µ i µ j 2 µ. µ m. µ m µ m m ; n = maxn r,n t C therefore depends only on the distribution of a single eigenvalue: C = me {µ} [ µe ] = mpµ µe dµ 0 This integration evaluates to C = 0 µe m k! k= k n m! [Ln m k µ] 2 µ n m e µ dµ L n m k x = dk k! ex dx e x x n mk isthelaguerre polynomial k of order k
EE 7950: Statistical Communication Theory 9 Large Systems As the number of antennas N t, N r, the number of eigenvalues Nµ, and the capacity formula C = E {µi } m i= µe m µem df 0 µ µ where F µ µ isthecumulative distribution CDF of the eigenvalues of W, which becomes continuous as m, n For random matrices like W, a general result states df µ µ dµ 2π µ n m τ, and µ ± = τ ± 2. µ µ µ ; for µ [µ,µ ] 0 otherwise In the limit, the capacity of the Rayleigh MIMO is given by C m = d µem µ 2π d µ µ µ dµ References [] R.G. Gallager, Information Theory and Reliable Communication, John Wiley & Sons, Inc., New-York, 968. [2] I.E. Telatar, Capacity of mulit-antenna Gaussian channels, Eur. Trans. Telecom., Vol. 0, pp. 585 595, Nov. 999.