: Diversity-Multiplexing Tradeoff Theoretical Foundations of Wireless Communications 1 Rayleigh Friday, May 25, 2018 09:00-11:30, Kansliet 1 Textbook: D. Tse and P. Viswanath, Fundamentals of Wireless Communication 1 / 19
Rayleigh Overview Lecture 7+8: MIMO Architectures Transmitter architectures: V-BLAST and D-BLAST. Receiver architectures Linear decorrelator Linear MMSE Successive cancellation (decorrelator and MMSE) Capacity and outage probability : Diversity-Multiplexing Tradeoff (Ch. 9.1) 1 2 PAM and QAM 3 4 5 6 Rayleigh Geometric Interpretation 2 / 19
: Diversity-Multiplexing Tradeoff () Rayleigh Diversity gain d Important performance measure for slow fading channels. (Fixed rate R; highest achievable rate depends on channel realization and is a RV; outage probability.) Example: i.i.d. MIMO, d = n t n r. Outage probability: p out(r) 1/SNR d Multiplexing gain r Fast fading MIMO channels: multiplexing capability. (Coding over many channel realizations; average capacity.) Example: i.i.d. MIMO, C = n min log SNR + K, with n min = min(n r, n t), i.e., r = n min. 3 / 19
: Diversity-Multiplexing Tradeoff () Rayleigh To maximize the diversity gain, the rate R has to be fixed. If we want to communicate at a rate R = r log SNR (i.e., a fraction of the fast fading capacity), which diversity gain can we get? Diversity-Multiplexing Tradeoff () A diversity gain d (r) is achieved at multiplexing gain r if and or more precisely R = r log SNR p out SNR d (r), log p out(r log SNR) lim = d (r). SNR log SNR The curve d (r) is the diversity-multiplexing tradeoff of the slow fading channel. for (space-time) coding schemes: replace p out by p e (and d by d). 4 / 19
PAM and QAM PAM and QAM Rayleigh Scalar channel model: y[m] = hx[m] + w[m], with Additive white Gaussian noise, i.i.d., w CN (0, 1); Fading coefficient h CN (0, 1); Power constraint SNR. PAM error probability ( p e 1 2 for high SNR, where 1 D 2 min 4 + D 2 min ) 1 22R Dmin 2 SNR, We assumed that the constellation ranges from SNR to + SNR; The minimum distance is given by D min SNR/2 R. With R = r log SNR, we get p e 1 SNR 1 2r. Diversity-multiplexing tradeoff for PAM: d pam(r) = 1 2r, r [0, 1/2]. 5 / 19
PAM and QAM PAM and QAM QAM: 2 R/2 constellation points per real/imaginary dimension. Minimum distance and QAM error probability (high SNR) SNR D min and p 2 R/2 e 2R SNR. Diversity-multiplexing tradeoff for QAM: d qam(r) = 1 r, r [0, 1]. Rayleigh 6 / 19
PAM and QAM PAM and QAM Analysis, Case 1: d max := d(0) Classical diversity gain for a scheme with fixed rate. Describes how fast the error probability p e can be decreased with SNR for a fixed rate R = const. Example: Increasing the SNR by 6 db for fixed rate. Rayleigh p e decreases due to increasing D min with increasing SNR. 7 / 19
PAM and QAM PAM and QAM Analysis, Case 2: r max = arg d(r) = 0 Number of degrees of freedom. Describes how fast the rate R can be increased with SNR for a fixed error probability p e = const. Example: Increasing the SNR by 6 db. Rayleigh For p e = const (i.e., D min = const), the gain in SNR can be utilized for increasing the number of bits. 8 / 19
PAM and QAM Rayleigh Outage probability and high-snr approximation, { } p out = Pr log(1 + h 2 SNR) < r log SNR = { } Pr h 2 < SNRr 1 SNR 1 SNR. 1 r (Last step: for Rayleigh fading and small ɛ, Pr{ h 2 < ɛ} ɛ.) Uncoded QAM is optimal. Comment Assume p e = a p out, with a constant a, with 0 log(a) log SNR for high SNR. Then lim SNR log p e log SNR = lim SNR [ ] log a log pout log p out + = lim log SNR log SNR SNR log SNR. That a scheme is optimal does not mean that it achieves the outage probability. But it has the same exponential decay. 9 / 19
Rayleigh Model: L parallel channels, y l [m] = h l x l [m] + w l [m], with Additive noise w l CN (0, 1); Transmit power constraint SNR per sub-channel. Maximum diversity gain of d (0) = L. Assume target rate R = r log SNR bit/s/hz per sub-channel. Outage probability, { L } p out = Pr log(1 + h l 2 SNR) < Lr log SNR. l=1 Outage occurs if each of the sub-channels is in outage, accordingly ( { }) L p out Pr log(1 + h l 2 1 SNR) < r log SNR SNR. L(1 r) Optimal, d (r) = L(1 r), r [0, 1]. 10 / 19
Repetition coding: transmit identical QAM symbols over the parallel channels: d rep(r) = L(1 Lr), with r [0, 1/L]. Repetition coding achieves full diversity d (0) = L. Rate is reduced by a factor 1/L. Comparison Rayleigh 11 / 19
Model: n t transmit antennas, Rayleigh y[m] = h x[m] + w[m], with Additive noise w CN (0, 1); Overall transmit power constraint SNR. Maximum diversity gain of d (0) = n t. Assume target rate R = r log SNR bit/s/hz. Outage probability, p out = Pr { ( ) } log 1 + h 2 SNR < r log SNR. n t h 2 is a χ 2 random variable with 2n t degrees of freedom, and we have Pr{ h 2 < ɛ} ɛ nt such that Optimal. 1 p out =. nt (1 r) SNR d (r) = n t(1 r), r [0, 1]. 12 / 19
Alamouti converts a MISO channel into a scalar channel with the same outage behavior. Alamouti plus QAM is optimal. Repetition coding: transmitting the same symbol over the two antennas, one at a time; for n t = 2, d rep(r) = n t(1 2r). Rayleigh 13 / 19
Rayleigh Comparison of four schemes (uncoded QAM) Repetition coding, Alamouti, and V-BLAST with nulling (decorrelator). The schemes convert the channel into scalar channels. can be obtained as in the scalar case. V-BLAST with maximum likelihood (ML) decoding Pairwise error probability (PEP) between two codewords (averaged over the channel realizations and with average transmit energy normalize to one) Pr{x A x B } 16 SNR 2 x A x B 4 (see as well (3.92)). Worst-case PEP is of the order (X A and X B differ only in one dimension by the minimum distance of the QAM constellation) 16 2 R SNR 2 = 16 SNR (2 r). (with D AB = SNR x A x B, D min SNR/2 R/4, R = r log SNR) : d(r) = 2 r. Optimal : piecewise linear joining the points (0, 4), (1, 1), and (2, 0). 14 / 19
Comparison of the four schemes Rayleigh 15 / 19
Rayleigh Rayleigh Geometric Interpretation model: (n t n r ) MIMO channel with i.i.d. Rayleigh fading. Outage probability (for R = r log SNR), p mimo out (r log SNR) = min K x :Tr[K x ] SNR Pr{log det(int + HK x H ) < r log SNR}. Transmit strategy K x depends on the SNR. Assumption: K x = SNR/n ti nt. (Suboptimal but leads to the same decay rate as the optimal K x). { ( pout(r iid log SNR) = Pr log det I nt + SNR ) } HH < r log SNR. n t for the MIMO channel with i.i.d. Rayleigh fading can be shown to be a piecewise linear curve joining the points (k, (n t k)(n r k)), for k = 0,..., n min. 16 / 19
Rayleigh Rayleigh Geometric Interpretation Left: optimal for the MIMO channel with i.i.d. Rayleigh fading. Right: adding one transmit and receive antenna moves the entire curve by 1 (and not only the extreme point r max). Optimal can be achieved by space-time codes with length l = n t + n r 1. 17 / 19
Rayleigh Geometric Interpretation Rayleigh Geometric Interpretation Outage probability { ( p out(r log SNR) = Pr log det { nmin I nt + SNR ( = Pr log 1 + SNR ) λ 2 i n t i=1 with Random singular values λ i of the matrix H; n min eigenmodes with effective SNR, SNRλ 2 i /nt. Modes are... active if SNRλ 2 i /nt is in the order of SNR; inactive if SNRλ 2 i /nt is in the order of 1 or smaller; n t ) } HH < r log SNR < r log SNR Case 1, r 0: Outage happens if all modes are inacative (i.e., SNRλ 2 i /n t 1); happens if H is close to the zero matrix. }. Outage probability is in the order 1/SNR nr nt. 18 / 19
Rayleigh Geometric Interpretation Case 2, r is a positive integer. For high SNR, it can be shown that a typical outage event is characterized by r modes being fully effective and the remaining modes being fully ineffective. Outage happens if H is close to the space V r of all rank-r matrices. Rayleigh Geometric Interpretation A rank-r matrix H is described by rn t + (n r r)r parameters: rn t parameters to specify n r linearly independent row vectors. (n r r)r parameters to specify the remaining row vectors as linear combinations. V r is rn t + (n r r)r dimensional, and the orthogonal space is n tn r (rn t + (n r r)r) = (n t r)(n r r) dimensional, which is precisely the SNR exponent of the outage probability. The outage probability is proportional to the probability that (n t r)(n r r) parameters collapse (i.e., h i,j 2 < 1/SNR). 19 / 19