Lecture 7: MIMO Architectures Theoretical Foundations of Wireless Communications 1. Overview. CommTh/EES/KTH

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1 : Theoretical Foudatios of Wireless Commuicatios 1 Thursday, May 19, :30-15:30, Coferece Room SIP 1 Textbook: D. Tse ad P. Viswaath, Fudametals of Wireless Commuicatio 1 / 1 Overview Lecture 6: MIMO Chael Modelig Aalysis of LOS chaels w/o reflectios. Stochastic models for MIMO fadig chaels. : 2 / 1

2 V-BLAST Optimal trasmitter for CSI-T 2 Time-ivariat chael model: y[m] = Hx[m] + w[m]. Use SVD decompositio (i.e., H = UΛV ad precode with V. V-BLAST Architecture (D. Tse ad P. Viswaath, Fudametals of Wireless Commuicatios. Geeralized structure with a uitary precodig matrix Q. Multiplexig coordiate system. data streams with capacity-achievig Gaussia codes with rates R k ad powers P k. Total rate R = k=1 R k ad power costrait k=1 P k = P. Joit decodig at the receiver. Examples: Q = V, waterfillig power allocatio; Q = I t, idepedet data streams. 2 CSI-T: chael state iformatio at the trasmitter. 3 / 1 V-BLAST Upper boud ohe highest reliable rate of commuicatio R < log det (I r + 1N0 HK xh bits/s/hz with the covariace matrix K x of the trasmitted sigal x K x = Q diag{p 1,..., P t } Q. Sphere packig A received vector (N time symbols lies with high probability i a ellipsoid of volume proportioal to det (N 0 I r + HK x H N. Noise spheres of radius N 0 ad volume proportioal to N r N 0. Maximum umber of codewords that ca be packed det (N 0 I r + HK x H N N r N. 0 The upper boud is achievable with V-BLAST! Example/special case: MISO system with Q = I t achieves ( t k=1 log 1 + h k 2 P k which is the upper boud from above! N 0 4 / 1

3 3 Capacity with CSI-R Fast fadig MIMO chael model: y[m] = H[m]x[m] + w[m]. Statioary, ergodic radom fadig process {H[m]}, E{ h 2 ij } = 1. Coheret commuicatios; i.e., the receiver tracks the fadig process {H[m]} exactly. Achievable rate i a give chael state H (with V-BLAST log det (I r + 1N0 HK xh. Codig over may coherece-time itervals (i.e., average over chael realizatios E H [log det (I r + 1N0 HK xh. Choose K x as a fuctio of the chael statistics such that [ C = max E log det (I r + 1N0 HK xh. K x :Tr[K x ] P K x depeds ohe statioary distributio of {H[m]}. 3 CSI-R: chael state iformatio at the receiver. 5 / 1 Capacity with CSI-R Special case 1: Oly few domiat paths which are time-ivariat. H ca be viewed as beig determiistic. Multiplex the streams ihe eige-directios of H H ad use waterfillig. Special case 2: May paths of approximately equal eergy. Use the agular represetatio: H a = U r HU t; etries of H a are geerated by differet physical paths (statistically idepedet. Assumptio: o domiat path; i.e., elemets h a ij have zero mea. Due to idepedece, trasmit i directios of Q = U t (i.e., sed iformatio i each trasmit agular widow: K x = U tλu t where Λ is a diagoal matrix with etries correspodig to the power allocatio. 6 / 1

4 Capacity with CSI-R Optimal power allocatio by solvig (with H = U r H a U t log det I r + 1 U r H a ΛH a U r N 0 C = max E Λ:Tr[Λ] P = max Λ:Tr[Λ] P E [ log det (I r + 1N0 H a ΛH a (Follows by usig det(aba 1 = det(a det(b det(a 1 = det(b. Special case: If h a ij CN (0, 1, i.i.d., equal power allocatio is optimal such that K x = (P/I t, ad the capacity is C = E = mi log det E log I r + SNR 1 + SNR λ 2 i HH, (with the radom, ordered sigular values λ 1 λ 2... λ mi. 7 / 1 Performace Gais: High SNR Regime By Jese s iequality we kow that mi mi log ( log 1 + SNR ( 1 + SNR λ 2 i [ ] 1 mi λ 2 i, mi with equality if sigular values are equal. High capacity if H is sufficietly radom ad well coditioed. Capacity at high SNR for i.i.d. Rayleigh chael C mi log SNR mi + E[log λ 2 i ] t (D. Tse ad P. Viswaath, Fudametals of Wireless Commuicatios. Asymptotic slope of a ( system:. Approximately imes the capacity of a (1 1 system. MISO: oly power gai. 8 / 1

5 Performace Gais: Low SNR Regime With the approximatio we get C = (D. Tse ad P. Viswaath, Fudametals of Wireless Commuicatios. log 2 (1 + x x log 2 e, mi mi = SNR = SNR E E log 1 + SNR λ 2 i SNR [ ] E λ 2 i log 2 e t E [Tr[HH ]] log 2 e [ ] h ij 2 log 2 e i,j = r SNR log 2 e bits/s/hz. Oly power gai a power gai of r over a (1 1 system sice, without CSI-T, o trasmit beamformig is possible! 9 / 1 Performace Gais: Large Atea Regime Cosider a ( system; capacity for i.i.d. Rayleigh fadig [ ( ] C (SNR = E log 1 + SNR λ2 i λ i / are the sigular values of H/. Iterestig result: the empirical distributio of the sigular values of H/ coverges to a determiistic distributio for almost all realizatios of H. Distributio of the squared sigular values: { f 1 1 (x = 1, 0 x 4, π x 4 0, else, quarter circle law. 10 / 1

6 Performace Gais: Large Atea Regime Covergece of the empirical distributio of H/ : (D. Tse ad P. Viswaath, Fudametals of Wireless Commuicatios. 11 / 1 Performace Gais: Large Atea Regime It follows that 1 ( 4 log 1 + SNR λ2 i log (1 + SNRx f (xdx 0 }{{} =c (SNR Sigificace of c (SNR C (SNR lim = c (SNR Large- approximatio Example: C (SNR c (SNR. 12 / 1 (D. Tse ad P. Viswaath, Fudametals of Wireless Commuicatios.

7 Performace Gais: Liear Scalig Why does the capacity scale liearly with the umber of ateas? Or what makes the differece? Special case 1: MISO chael with a large trasmit atea. For the ( 1 MISO chael (i.i.d. Rayleigh we get C 1(SNR = E log 1 + SNR h i 2. As, by the law of large umbers, 1 C 1 log(1 + SNR = C AWGN, sice lim h i 2 E[ h i 2 ]. Icreasig the umber of trasmit ateas reduces the fluctuatios ihe istataeous SNR. I cotrast, a (1 1 system suffers from Jese s loss, C 11 (SNR = E [ log ( 1 + SNR h 2 log ( 1 + SNR E[ h 2 ] = C AWGN. 13 / 1 Performace Gais: Liear Scalig Special case 2: SIMO chael with a large trasmit atea. For the (1 SIMO chael we get (with receive beamformig C 1(SNR = E log 1 + SNR h i 2. For large, C 1 log(snr = log( + log(snr. Ihe SIMO case, we have a power gai which icreases liearly with but o degree of freedom gai. (D. Tse ad P. Viswaath, Fudametals of Wireless Commuicatios. 14 / 1

8 Performace Gais: Liear Scalig Special case 3: AWGN chael with ifiite badwidth. With power costrait P ad AWGN with spectral desity N 0/2, we get ( C = lim W log 1 + P = P W N 0W N 0 Capacity is bouded due to a fixed receive power although the umber of degrees of freedom icreases. The MIMO case: capacity of a ( system icreases liearly i sice simultaeously: there is a liear icrease ihe total received power ad there is a liear icrease ihe degrees of freedom (due to the radomess ad well-coditioedess of H. Requiremet: keep the atea spacig fixed at half the wavelegth, ad icrease the aperture! 15 / 1 Full CSI So far: CSI-R full degree of freedom gai. Full CSI: CSI-R plus CSI-T additioal power gai. Trasceiver architecture SVD: H[m] = U[m]Λ[m]V[m]. Preprocessig with V[m]. Postprocessig with U[m]. Power allocatio with waterfillig. AWGN capacity achievig code o each stream. Capacity mi C E log 1 + SNR λ 2 i mi Power gai by the factor / mi due to CSI-T. With CSI-T, the trasmitted eergy is focused ohe o-zero eigemodes of the chael, whereas without CSI-T the eergy is trasmitted across all directios i C t. For large, the distributio of λ i / coverges agai agaist f ; i.e., o power allocatio over time oly over space (chael hardeig. 16 / 1

9 Slow Fadig MIMO Chael Chael model: y[m] = Hx[m] + w[m], with H fixed but radom. Full CSI-R while trasmitter kows oly chael statistics. Reliable commuicatio at rate R is possible as log as log det (I r + 1N0 HK xh > R subject to Tr[K x] P. Iformatioheory guaratees the existece of a chael-state idepedet codig scheme that achieves reliable commuicatios wheever this coditio is met. Uiversal code: works for all chaels that satisfy the above coditio. If the coditio is ot satisfied, we are i outage. Outage probability [ ] pout MIMO (R = mi Pr log det (I r + 1N0 HK xh < R. K x :Tr[K x ] P Choose the trasmit strategy (i.e., K x to miimize the outage probability. Solutio depeds o statistics of H. 17 / 1 Slow Fadig MIMO Chael Slow fadig versus fast fadig Fast fadig: maximize the log-term average rate of iformatio flow (i.e., the ergodic capacity, with C = max E H [f (K x, H, K x f (K x, H = log det (I r + 1N0 HK xh. Maximize expected value of f (K x, H. Slow fadig: miimize (for a sigle realizatio of the chael the probability that the iformatio rate falls below the target rate R, p out(r = mi Pr[f (K x, H < R]. K x Miimize the tail probability that f (K x, H is less thahe target rate. Special case: i.i.d. Rayleigh fadig chael. Optimal covariace matrix for fast fadig: K x = P/I t. Cojecture [119]: it is as well the optimal covariace for the i.i.d. Rayleigh slow fadig at high SNR. 18 / 1

Lecture 8: MIMO Architectures (II) Theoretical Foundations of Wireless Communications 1. Overview. Ragnar Thobaben CommTh/EES/KTH

Lecture 8: MIMO Architectures (II) Theoretical Foundations of Wireless Communications 1. Overview. Ragnar Thobaben CommTh/EES/KTH MIMO : MIMO Theoretical Foundations of Wireless Communications 1 Wednesday, May 25, 2016 09:15-12:00, SIP 1 Textbook: D. Tse and P. Viswanath, Fundamentals of Wireless Communication 1 / 20 Overview MIMO