: Antenna Diversity and Theoretical Foundations of Wireless Communications Wednesday, May 4, 206 9:00-2:00, Conference Room SIP Textbook: D. Tse and P. Viswanath, Fundamentals of Wireless Communication / Overview Lecture -4: Channel capacity Gaussian channels Fading Gaussian channels Multiuser Gaussian channels Multiuser diversity : Antenna diversity and MIMO capacity 2 /
Diversity Multiuser diversity (lecture 4) Transmissions over independent fading channels. Sum capacity increases with the number of users. High probability that at least one user will have a strong channel. Fading channels (point-to-point links) Use diversity to mitigate the effect of (deep) fading. Diversity: let symbols pass through multiple paths. Time diversity: interleaving and coding, repetition coding. Frequency diversity: for example OFDM. Antenna Diversity. 3 / Motivation: For narrowband channels with large coherence time or delay constraints, time diversity and frequency diversity cannot be exploited! (D. Tse and P. Viswanath, Fundamentals of Wireless Communications.) Antenna diversity Multiple transmit/receive antennas with sufficiently large spacing: Mobiles: rich scattering /2... carrier wavelength. Base stations on high towers: tens of carrier wavelength. Receive diversity: multiple receive antennas, single-input/multiple-output (SIMO) systems. Transmit diversity: multiple transmit antennas, multiple-input/single-output (MISO) systems. Multiple transmit and receive antennas, multiple-input/multiple-output (MIMO) systems. 4 /
Receive Diversity (SIMO) Channel model: flat fading channel, transmit antenna, L receive antennas: y[m = h[m x[m + w[m y l [m = h l [m x[m + w l [m, l =,..., L with additive noise w l [m CN (0, N 0 ), independent across antennas, Rayleigh fading coefficients h l [m. Optimal diversity combining: maximum-ratio combining (MRC) r[m = h[m y[m = h[m 2 x[m + h [mw[m Error probability for BPSK (conditioned on h) Pr(x[m sign(r[m)) = Q( 2 h 2 SNR) with the (instantaneous) SNR γ = h 2 SNR = h 2 E{ x 2 }/N 0 = LSNR L h 2 Diversity gain due to L h 2 and power/array gain LSNR. 3 db gain by doubling the number of antennas. 5 / Channel model Flat fading channel, L transmit antennas, receive antenna: y[m = h T [m x[m + w[m, with additive noise w[m CN (0, N 0), vector h[m of Rayleigh fading coefficients h l [m. Alamouti scheme Rate- space-time block code (STBC) for transmitting two data symbols u, u 2 over two symbol times with L = 2 transmit antennas. Transmitted symbols: x[ = [u, u 2 T and x[2 = [ u 2, u T. Channel observations at the receiver (with channel coefficients h, h 2): [ u u2 [y[, y[2 = [h, h 2 u 2 u + [w[, w[2. 6 /
Alternative formulation [ [ [ y[ h h 2 u y[2 = h2 h u 2 }{{} =y [ [ h h2 = u + h 2 }{{} =v h [ w[ + w[2 } {{ } =v 2 [ w[ u 2 + w[2 v and v 2 are orthogonal; i.e., the AS spreads the information onto two dimensions of the received signal space. Matched-filter receiver 2 : correlate with v and v 2 r i = v i H y = h 2 u i + w i, for i =, 2, with independent w i CN (0, h 2 N 0). SNR (under power constraint E{ x 2 } = P 0): SNR = h 2 2 P 0 N 0 diversity gain of 2! 2 The textbook uses a projection on the orthonormal basis v / v, v 2/ v 2. 7 / Determinant criterion for space-time code design Model: codewords of a space-time code with L transmit antennas and N time slots: X i, (L N) matrix. y T = [ y[,..., y[n, y T = h X i + w T with h = [ h,..., h L, w T = [ w,..., w L. Example: Alamouti scheme: [ u u2 X i = u 2 u Repetition coding: [ u 0 X i = 0 u Pairwise error probability of confusing X A with X B given h ( ) h (X A X B ) Pr(X A X B h) = Q 2 2N 0 ( ) SNR h (X A X B )(X A X B ) = Q h 2 (Normalization: unit energy per symbol SNR = /N 0) 8 /
Average pairwise error probability Pr(X A X B ) = E{Pr(X A X B h)} Some useful facts... (X A X B )(X A X B ) is Hermitian (i.e., Z = Z). (X A X B )(X A X B ) can be diagonalized by an unitary transform, (X A X B )(X A X B ) = UΛU, where U is unitary (i.e., U U = UU = I) and Λ = diag{λ 2,..., λ2 L }, with the singular values λ l of X A X B. And we get (with h = U h) Pr(X A X B ) = E Q SNR L l= h l 2 λ 2 l 2, L l= + SNR λ 2 l /4 9 / If all λ 2 l > 0 (only possible if N L), we get Pr(X A X B ) = L l= Diversity gain of L is achieved. + SNR λ 2 l /4 4 L SNR L L l= λ2 l SNR 4 L L det[(x A X B )(X A X B ) Coding gain is determined by the determinant det[(x A X B )(X A X B ) (determinant criterion). 0 /
2 2 MIMO Example Channel Model 2 transmit antennas, 2 receive antennas: [ [ y h h 2 = y 2 h 2 h 22 }{{}}{{} y H [ x x 2 }{{} x [ w + w 2 } {{ } w Rayleigh distributed channel gains h ij from transmit antenna j to receive antenna i. Additive white complex Gaussian noise w i CN (0, N 0). 4 independently faded signal paths, maximum diversity gain of 4. H H 22 H 2 H 2 H = H H 2 H 2 H 22 / 2 2 MIMO Example Degrees of freedom Number of dimensions of the received signal space. MISO: one degree of freedom for every symbol time. Repetition coding (L = 2): dimension over 2 time slots. Alamouti scheme (L = 2): 2 dimension over 2 time slots. SIMO: one degree of freedom for every symbol time. Only one vector is used to transmit the data, y = hx + w. MIMO: potentially two degrees of freedom for every symbol time. Two degrees of freedom if h and h 2 are linearly independent. y = h x + h 2x 2 + w. (D. Tse and P. Viswanath, Fundamentals of Wireless Communications.) 2 /
2 2 MIMO Example Spatial multiplexing Motivation: Neither repetition coding nor the Alamouti scheme utilize all degrees of freedom of the channel. Spatial multiplexing (V-BLAST) utilizes all degrees of freedom. Transmit independent uncoded symbols over the different antennas and the different symbol times. Pairwise error probability for transmit vectors x, x 2 [ Pr(x x 2) + SNR x x 2 2 /4 2 6 SNR 2 x x 2 4 Diversity gain of 2 (not 4) but higher coding gain as compared to the Alamouti scheme (see example in the book). Spatial multiplexing is more efficient in exploiting the degrees of freedom. Optimal detector, joint ML detection: complexity grows exponentially with the number of antennas. Linear detection, e.g., decorrelator (zero forcing): ỹ = H y 3 / MIMO channel model with n t transmit and n r receive antennas: y = Hx + w, with w CN (0, N 0I). x C nt, y C nr, and H C nr nt. Channel matrix H is known at the transmitter and receiver. Power constraint E{ x 2 } = P. Singular value decomposition (SVD): H = UΛV, where U C nr nr and V C nt nt are unitary matrices; Λ R nr nt is a matrix with diagonal elements λ,..., λ nmin and off-diagonal elements equal to zero; λ,..., λ nmin, with n min = min{n r, n t} are the ordered singular values of the matrix H; λ 2,..., λ2 n min are the eigenvalues of HH and H H. n min Alternative formulation: H = λ i u i vi. Sum of rank- matrices λ i u i vi. H has rank n min. 4 /
(D. Tse and P. Viswanath, Fundamentals of Wireless Communications.) SVD can be used to decompose the MIMO channel into n min parallel SISO channels. x = V x, ỹ = U y, ỹ = Λ x + w w = U w with w CN (0, N 0I nr ) and x 2 = x 2 ; i.e., the energy is preserved. MIMO capacity (with waterfilling) n min ( ) C = log + P i λ 2 i N 0 with P i = [ + µ N0 λ 2 i with µ chosen to satisfy the total power constraint P i = P. 5 / SVD architecture for MIMO communications (D. Tse and P. Viswanath, Fundamentals of Wireless Communications.) 6 /
Capacity at high SNR Uniform power allocation is asymptotically optimal; i.e., P i = P/k. C k ( ) log + Pλ2 i k log SNR + kn 0 k spatial degrees of freedom; if H has full rank k = n min. k ( ) λ 2 log i k With Jensen s inequality C k k k ( log + P ) λ 2 i k log kn 0 ( + P kn 0 ( k Maximum capacity in high SNR if all singular values are equal. k λ 2 i )) Condition number: max i λ i / min i λ i, H is well conditioned if CN. Capacity at low SNR Allocate power only to the strongest eigenmode C P (max λ 2 i ) log N 2 e 0 i 7 /