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Lecture 5: Antenna Diversity and MIMO Capacity Theoretical Foundations of Wireless Communications=1Textbook: D. Tse and P. Visw Lecture 5 Antenna Diversity, MIMO Capacity Lars Kildehøj CommTh/EES/KTH Lecture 5: Antenna Diversity and MIMO Capacity Diversity 1 Antenna Diversity Theoretical Foundations of Wireless Communications MIMO Capacity Lars Kildehøj CommTh/EES/KTH Friday, April 27, 2018 9:30-12:00, Kansliet plan 3 1Textbook: D. Tse and P. Viswanath, Fundamentals of Wireless Communication 1 / 17 Overview Lecture 1-4: Channel capacity Lecture 5 Antenna Diversity, MIMO Capacity • Gaussian channels Lars Kildehøj • Fading Gaussian channels CommTh/EES/KTH • Multiuser Gaussian channels Diversity • Multiuser diversity Antenna Diversity MIMO Capacity Lecture 5: Antenna diversity and MIMO capacity 1 Diversity 2 Antenna/Spatial Diversity Receive Diversity (SIMO) Transmit Diversity (MISO), Space-Time Coding 2 × 2 MIMO Example 3 MIMO Capacity 2 / 17 Diversity Lecture 5 Multiuser diversity (lecture 4) Antenna Diversity, MIMO Capacity • Transmissions over independent fading channels. Lars Kildehøj CommTh/EES/KTH • Sum capacity increases with the number of users. Diversity ! High probability that at least one user will have a strong channel. Antenna Diversity MIMO Capacity 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 / 17 Antenna/Spatial Diversity Motivation: For narrowband channels with large coherence time or delay constraints, time diversity and frequency diversity cannot be exploited! Lecture 5 Antenna Diversity, MIMO Capacity Lars Kildehøj CommTh/EES/KTH Diversity Antenna Diversity SIMO MISO (D. Tse and P. Viswanath, Fundamentals of Wireless Communications.) MIMO Antenna diversity MIMO Capacity • Multiple transmit/receive antennas with sufficiently large spacing: • Mobiles: rich scattering ! 1=2 ::: 1 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 / 17 Antenna/Spatial Diversity { Receive Diversity (SIMO) • Channel model: flat fading channel, 1 transmit antenna, L receive antennas: Lecture 5 Antenna Diversity, MIMO Capacity y[m] = h[m] · x[m] + w[m] Lars Kildehøj CommTh/EES/KTH yl [m] = hl [m] · x[m] + wl [m]; l = 1;:::; L with Diversity • additive noise wl [m] ∼ CN (0; N0), independent across antennas, Antenna Diversity • Rayleigh fading coefficients h [m]. SIMO l MISO MIMO • Optimal diversity combining: maximum-ratio combining (MRC) MIMO Capacity r[m] = h[m]∗ · y[m] = kh[m]k2 · x[m] + h∗[m]w[m] • Error probability for BPSK (conditioned on h) Pr(x[m] 6= sign(r[m])) = Q(p2khk2SNR) with the (instantaneous) SNR 1 γ = khk2SNR = khk2Efjxj2g=N = LSNR · khk2 0 L 1 2 ! Diversity gain due to L khk and power/array gain LSNR. ! 3 dB gain by doubling the number of antennas. 5 / 17 Antenna/Spatial Diversity { Transmit Diversity (MISO), Space-Time Coding Channel model Lecture 5 Antenna Diversity, Flat fading channel, L transmit antennas, 1 receive antenna: MIMO Capacity Lars Kildehøj T CommTh/EES/KTH y[m] = h [m] · x[m] + w[m]; with Diversity • additive noise w[m] ∼ CN (0; N0), Antenna Diversity • SIMO vector h[m] of Rayleigh fading coefficients hl [m]. MISO MIMO MIMO Capacity Alamouti scheme • Rate-1 space-time block code (STBC) for transmitting two data symbols u1; u2 over two symbol times with L = 2 transmit antennas. T ∗ ∗ T • Transmitted symbols: x[1] = [u1; u2] and x[2] = [−u2 ; u1 ] . • Channel observations at the receiver (with channel coefficients h1; h2): ∗ u1 −u2 [y[1]; y[2]] = [h1; h2] ∗ + [w[1]; w[2]]: u2 u1 6 / 17 Antenna/Spatial Diversity { Transmit Diversity (MISO), Space-Time Coding • Alternative formulation Lecture 5 y[1] h1 h2 u1 w[1] Antenna Diversity, ∗ = ∗ ∗ + ∗ MIMO Capacity y[2] h2 −h1 u2 w[2] Lars Kildehøj | {z } CommTh/EES/KTH =y Diversity h1 h2 w[1] = ∗ u1 + ∗ u2 + ∗ Antenna Diversity h2 −h1 w[2] SIMO | {z } | {z } MISO =v1 =v2 MIMO MIMO Capacity ! v1 and v2 are orthogonal; i.e., the AS spreads the information onto two dimensions of the received signal space. 2 • Matched-filter receiver : correlate with v1 and v2 H 2 ri = vi y = khk ui +w ~i ; for i = 1; 2; 2 with independentw ~i ∼ CN (0; khk N0). 2 • SNR (under power constraint Efkxk g = P0): khk2 P SNR = 0 ! diversity gain of 2! 2 N0 2 The textbook uses a projection on the orthonormal basis v1=kv1k; v2=kv2k. 7 / 17 Antenna/Spatial Diversity { Transmit Diversity (MISO), Space-Time Coding Determinant criterion for space-time code design • Model: codewords of a space-time code with L transmit antennas Lecture 5 Antenna Diversity, and N time slots: Xi ,(L × N) matrix. MIMO Capacity Lars Kildehøj 8 T CommTh/EES/KTH y = [ y[1];:::; y[N]]; T ∗ T < ∗ y = h Xi + w with h = [ h1;:::; hL ]; Diversity T : w = [ w1;:::; wL ]: Antenna Diversity SIMO MISO Example: Alamouti scheme: Repetition coding: MIMO ∗ MIMO Capacity u1 −u2 u 0 Xi = ∗ Xi = u2 u1 0 u • Pairwise error probability of confusing XA with XB given h r ∗ 2 ! kh (XA − XB )k Pr(XA ! XB jh) = Q 2N0 r ! SNR h∗(X − X )(X − X )∗h = Q A B A B 2 (Normalization: unit energy per symbol ! SNR = 1=N0) 8 / 17 Antenna/Spatial Diversity { Transmit Diversity (MISO), Space-Time Coding • Average pairwise error probability Lecture 5 Antenna Diversity, MIMO Capacity Pr(XA ! XB ) = EfPr(XA ! XB jh)g Lars Kildehøj CommTh/EES/KTH • Some useful facts... ∗ ∗ Diversity • (XA − XB )(XA − XB ) is Hermitian (i.e., Z = Z). • ∗ Antenna Diversity (XA − XB )(XA − XB ) can be diagonalized by an unitary transform, SIMO (X − X )(X − X )∗ = UΛU∗; MISO A B A B MIMO ∗ ∗ 2 2 where U is unitary (i.e., U U = UU = I) and Λ = diagfλ1; : : : ; λLg, MIMO Capacity with the singular values λl of XA − XB . • And we get (with h~ = U∗h) 8 0s 19 PL ~ 2 2 < SNR l=1 jhl j λl = Pr(XA ! XB ) = E Q @ A ; : 2 ; L Y 1 ≤ 1 + SNR λ2=4 l=1 l 9 / 17 Antenna/Spatial Diversity { Transmit Diversity (MISO), Space-Time Coding Lecture 5 Antenna Diversity, MIMO Capacity 2 Lars Kildehøj • If all λl > 0 (only possible if N ≥ L), we get CommTh/EES/KTH L L Diversity Y 1 4 Pr(XA ! XB ) ≤ ≤ Antenna Diversity 2 L QL 2 1 + SNR λl =4 SNR λ SIMO l=1 l=1 l MISO L MIMO 1 4 = L · ∗ MIMO Capacity SNR det[(XA − XB )(XA − XB ) ] ! Diversity gain of L is achieved. ! Coding gain is determined by the determinant ∗ det[(XA − XB )(XA − XB ) ] (determinant criterion): 10 / 17 Antenna/Spatial Diversity { 2 × 2 MIMO Example Channel Model Lecture 5 Antenna Diversity, • 2 transmit antennas, 2 receive antennas: MIMO Capacity Lars Kildehøj y1 h11 h12 x1 w1 CommTh/EES/KTH = · + y2 h21 h22 x2 w2 Diversity | {z } | {z } | {z } | {z } y H x w Antenna Diversity SIMO MISO • Rayleigh distributed channel gains hij from transmit antenna j to MIMO receive antenna i. MIMO Capacity • Additive white complex Gaussian noise wi ∼ CN (0; N0). ! 4 independently faded signal paths, maximum diversity gain of 4. H11 H12 H11 H12 H H = 21 H21 H22 H22 11 / 17 Antenna/Spatial Diversity { 2 × 2 MIMO Example Degrees of freedom • Number of dimensions of the received signal space. Lecture 5 • MISO: one degree of freedom for every symbol time. Antenna Diversity, MIMO Capacity ! Repetition coding (L = 2): 1 dimension over 2 time slots. Lars Kildehøj ! Alamouti scheme (L = 2): 2 dimension over 2 time slots. CommTh/EES/KTH • SIMO: one degree of freedom for every symbol time. Diversity ! Only one vector is used to transmit the data, Antenna Diversity y = hx + w: SIMO MISO • MIMO MIMO: potentially two degrees of freedom for every symbol time. MIMO Capacity ! Two degrees of freedom if h1 and h2 are linearly independent. y = h1x1 + h2x2 + w: (D. Tse and P. Viswanath, Fundamentals of Wireless Communications.) 12 / 17 Antenna/Spatial Diversity { 2 × 2 MIMO Example Spatial multiplexing Lecture 5 • Motivation: Neither repetition coding nor the Alamouti scheme Antenna Diversity, MIMO Capacity utilize all degrees of freedom of the channel. Lars Kildehøj CommTh/EES/KTH • Spatial multiplexing (V-BLAST) utilizes all degrees of freedom. ! Transmit independent uncoded symbols over the different Diversity antennas and the different symbol times. Antenna Diversity SIMO • Pairwise error probability for transmit vectors x1, x2 MISO MIMO 1 2 16 MIMO Capacity Pr(x1 ! x2) ≤ ≤ 2 2 4 1 + SNR kx1 − x2k =4 SNR kx1 − x2k ! 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): y~ = H−1y 13 / 17 MIMO Capacity • MIMO channel model with n transmit and n receive antennas: Lecture 5 t r Antenna Diversity, MIMO Capacity y = Hx + w; with w ∼ CN (0; N0I): Lars Kildehøj CommTh/EES/KTH • x 2 Cnt , y 2 Cnr , and H 2 Cnr ×nt . Diversity • Channel matrix H is known at the transmitter and receiver. 2 Antenna Diversity • Power constraint Efkxk g = P. MIMO Capacity • Singular value decomposition (SVD): H = UΛV∗, where • U 2 Cnr ×nr and V 2 Cnt ×nt are unitary matrices; nr ×nt • Λ 2 R is a matrix with diagonal elements λ1; : : : ; λnmin and off-diagonal elements equal to zero; • λ1; : : : ; λnmin , with nmin = minfnr ; nt g are the ordered singular values of the matrix H; • λ2; : : : ; λ2 are the eigenvalues of HH∗ and H∗H. 1 nmin nmin • P ∗ Alternative formulation: H = λi ui vi .
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