MIMO Systems and Channel Capacty Consder a MIMO system wth m Tx and n Rx antennas. x y = Hx ξ Tx H Rx The power constrant: the total Tx power s x = P t. Component-wse representaton of the system model, y m = g x ξ y = Gx ξ (13.1) j j j= 1 Lecture 13 15-Oct-15 1(31)
Theorem (Foschn, Telatar): Under sotropc sgnalng R = xx = ( P / n) I (e.g. no CSI at the Tx), the capacty of the channel n (13.1) s where H γ = r = P r t r x γ C = log det Im HH [ bt/s/hz] (13.) m P P G s the normalzed channel matrx, t 0 σ s the aggregate SNR at the Rx end. ( ) / P = P tr GG m s the total receved power, so that relable transmsson s possble at any rate R < C. t Proof: When x s a complex Gaussan vector, CN( 0, K x), ts entropy s where H K x s the correlaton matrx, ( ) = log ( πe) x m det x K (13.3) Lecture 13 15-Oct-15 (31)
Smlarly for the nose and the output vector: where Kξ = ξξ, K y = yy ELG513 Smart Antennas S.Loyka [ ] x = x = x j x j K xx K (13.4) n ( ) = ( π ) ( ) = ( π ) n H ξ log e det K ξ, H y log e det K y (13.5) Note: ths follows from the pdf of a complex Gaussan vector, and we have assumed that. 1 1 ρ ( x) = exp m x K x x π K x R R I I R I I R R I (13.6) x x = x x, x x = x x, x = x jx (13.7) where x R and x I are real and magnary parts of x. The mutual nformaton s Lecture 13 15-Oct-15 3(31)
( y x) ( x) ( y) ( x y) I, = H H H, = log K x K K xy y (13.8) where The capacty s x K K K xy = x y = y K yx K x xy ( ) y (13.9) C = max I, p( ) y x (13.10) x under the assumptons that the channel s fxed (statc). We further assume that the CSI s avalable at the Rx only (but not at the Tx). For Gaussan nose, the maxmum s acheved when x s zero-mean Gaussan. Under certan crcumstances (e.g. no Tx CSI), the covarance s a scaled dentty, subject to the power constrant trk x Pt. Hence, Pt K x = I m (13.11) m Lecture 13 15-Oct-15 4(31)
Here, we also assume that the nose s uncorrelated from one Rx antenna to another, and that the sgnal and nose are uncorrelated as well: K ξ = σ 0 I n, (13.1) * j x ξ = 0 (13.13) Lecture 13 15-Oct-15 5(31)
Under these assumptons, Usng the followng dentty, Pt K y = yy = GG σ0i n Pt Pt K xy = G, K yx = G m m n (13.14) A B 1 = A D - CA B C D (13.15) we fnd Fnally, 1 n xy = x y xy x yx = σ0 x K K K K K K K (13.16) C K x K y Pt = log = log det Im GG K xy σ0m (13.17) Lecture 13 15-Oct-15 6(31)
Let us normalze G as follows, where P r s the total receved power. Then, H = P P G (13.18) t r C γ = log det In HH [ bt/s/hz] (13.19) m where γ = P r σ 0 s the average SNR at the Rx. Ths gves the MIMO channel capacty n bt/s/hz. Ths s the celebrated Foschn-Telatar formula. Q.E.D. Lecture 13 15-Oct-15 7(31)
Important specal cases n parallel ndependent sub-channels, The capacty s H = I ( n = m) (13.0) C = mlog 1 γ m (13.1) Interpretaton: the ncomng bt stream s splt nto m ndependent sub-streams and transmtted ndependently over m channels. Snce the total Tx power s fxed to P t, perchannel SNR s γ m. Asymptotcally, Note that capacty growth lnearly wth γ much faster! γ m C (13.) ln Lecture 13 15-Oct-15 8(31)
Whle C(m) s monotoncally ncreasng wth m, the ncrease slows down for large n. Snce ncrease n n s related to huge ncrease n system complexty, there s some maxmum m, whch s approxmately m γ. In practce, one would keep m mmax. max Compare wth the case when all the bts are transmtted over one sub-channel only, Clearly, Another approach to ths problem s based on the cost model, C 1 = log( 1 γ ) (13.3) C C 1 for γ 1 (13.4) S = α m (13.5) where S = cost, α = cost/stream. Hence, the capacty/cost rato (capacty per unt cost) s C S 1 γ = log 1 α m (13.6) Ths s capacty per unt cost and t s always decreasng functon of n. However, the decrease s slow for moderate values of m, and t becomes large only when m ~ γ. Lecture 13 15-Oct-15 9(31)
Next consder dversty combnng, (one Tx and n Rx s). The capacty s [ ] T H = h1 h hn, m = 1 (13.7) C = log det I HH = log 1 γ (13.8) n γ h m = 1 Clearly, ths s the MRC. It proves once agan that the MRC s optmum (ths tme, from capacty (nformaton theoretc) vewpont). Selecton combnng can be represented as, ( ) { ( )} { } C = max log 1 γ h = log 1 γ max h (13.9) Hence, capacty-wse selecton combnng s the same as power-wse one. Tx dversty combnng, for fxed P t, s not the same as Rx combnng. Q: prove t! However, when per-branch Tx power s fxed, t s the same as Rx combnng. Q: prove t! Lecture 13 15-Oct-15 10(31)
SVD Decomposton and Channel Capacty Introduce a (nstantaneous) channel covarance matrx: Egenvalue decomposton of W s W = H H (13.30) W = QΛ Q (13.31) where Q s a mxm untary matrx of egenvectors, and dagonal matrx of egenvalues. Consder the SVD of H: [ ] Λ = dag λ λ λ 1 m H = UΣV (13.3) where U and V are untary matrces (of left and rght sngular vectors of H ), and Σ s dagonal (mxn) matrx of (non-negatve) sngular values. Note that W = H H = VΣ ΣV (13.33) Lecture 13 15-Oct-15 11(31)
Hence, Q = V, Λ = Σ Σ (13.33).e, egenvalues of W are squared sngular values of H, λ ( W) = σ ( H ). Usng the equatons above, the capacty can be presented as C m γ = log 1 λ m (13.34) = 1 Hence, W = σ λ = λ ( ) ( H ) are the channel egenmode power gans. Q: prove t! Ths s alternatve representaton of the Foschn-Telatar formula. Lecture 13 15-Oct-15 1(31)
channel Note: capacty doesn t change under transformaton Q: prove t! H H Lecture 13 15-Oct-15 13(31)
Example 1 Consder the all-1 channel H = 1, fnd ts capacty. Soluton: where Hence, j 1 1 1 1 T H = UΣV U = [ ], = mn, = [ 1 1 1] n 1 1 Σ V (13.35) m λ = mn, λ,, λ = 0 (13.36) ( n) m C = log 1 γ (13.37).e, t ncreases only logarthmcally wth n > ths s an example of a correlated (rankdefcent) channel. P Note : the effect of m s hdden n γ snce γ = r, where P σ r s the total Rx power, 0 P = mp. r r1 Lecture 13 15-Oct-15 14(31)
Example Consder a multpath channel of the form M 1 H = wv g (13.38) mn where v = Tx array manfold vector, w = Rx array manfold vector, and g - channel gans, all for th multpath component. Assume that = 1 j j j j (orthonormal). By nspecton, we conclude that w w = nδ, v v = mδ (13.39) M γ = 1 m λ = g, C= log 1 λ, f M < mn m, n ( ) (13.40) Hence the number of multpath components lmts MIMO capacty! Lecture 13 15-Oct-15 15(31)
In general case, the number of non-zero egenvalues s mn ( m, n, M ) γ M 0 = mn ( m, n, M ) C= log 1 λ m (13.41) Assumng that all the egenvalues are equal, = 1 γ C= mn ( m, n, M ) log 1 λ m Number of degrees of freedom: a factor n front of log s called a number of degrees of freedom, or capacty slope. Q.: What s the meanng of t? (13.4) Lecture 13 15-Oct-15 16(31)
Example 3 MISO channel (Tx dversty), n = 1, H = [ 1 1... 1] : The capacty s γ C = log 1 m = log 1 γ m ( ) (13.43). e., the same, as SISO wth the same total Tx power (the effect of coherent combnng s already ncluded n ρ, whch s the rato of total Rx power, n one Rx from all Tx s, to the nose power). Compare t wth SIMO (Rx dversty), m = 1, H = [ 1 1 1] T, Q.: explan the dfference! ( n) ( ) C = log 1 γ log 1 γ f n 1 (13.44) Lecture 13 15-Oct-15 17(31)
Capacty of an Ergodc Raylegh-fadng Channel We assume that Rx knows the channel (H), but Tx doesn t. Start wth the general MIMO capacty expresson for a gven (fxed) H : γ C = log det I HH (13.45) m Snce H s random for a fadng channel, C s random as well. For any gven realzaton of H, C can be evaluated, but t vares from realzaton to realzaton,.e. random varable. Defne the mean (ergodc) capacty, γ C = C = log det I HH (13.46) H m Telatar gves a detaled formulaton, based on the mutual nformaton,(, ) proves that the maxmum s acheved when x are..d. Gaussan. H ( ) I x y H, and Note that the mean capacty makes practcal sense for ergodc channels only,.e., when the expectaton over realzatons s the same as expectatons over tme. Lecture 13 15-Oct-15 18(31)
Interpretaton of ergodc capacty: t can be proved (based on nformaton theory) that there exsts a sngle code that acheves the capacty n (13.46). Alternatvely, an adaptve system can be bult that acheves the nstantaneous capacty n (13.45) and ts average capacty s as n (13.46). For a Raylegh channel, h j are..d. complex Gaussans wth unt varance, CN 0,1. Telatar descrbes n detals the evaluaton of capacty n ths case hj ( ) (analytcal approach). Consder some specal cases. Lecture 13 15-Oct-15 19(31)
Example 1 n s fxed, and m. In ths case, 1 1 m * = hj hkj n m m j= 1 HH I (13.47) n probablty due to the central lmt theorem. Hence, the capacty s C = nlog ( 1 γ ) (13.48).e., the same as the capacty of n parallel ndependent channels, AWGN, no fadng. Ths s the effect of Tx dversty. Consder n = 1, m then, C = log( 1 γ ) (13.49).e, an nfnte-order dversty transforms the Raylegh-fadng channel nto a fxed (nonfadng) AWGN channel. Example : m s fxed and n. Homework. References: [1-3, 5, 6]. Lecture 13 15-Oct-15 0(31)
Summary MIMO capacty. Basc concepts (entropy, mutual nformaton) for random vectors. Canoncal form of the MIMO capacty. Large n lmt. Comparson to conventonal systems. Capacty of dversty combnng systems. The mpact of multpath, Tx and Rx antenna number. MIMO capacty of a Raylegh channel. The mpact of fadng. Lecture 13 15-Oct-15 1(31)
References 1. J.R. Barry, E.A. Lee, D.G. Messerschmtt, Dgtal Communcaton, 003 (Thrd Edton). Ch. 10, 11.. D. Tse, P. Vswanath, Fundamentals of Wreless Communcatons, Cambrdge, 005. 3. P.P. Vadyanathan et al, Sgnal Processng and Optmzaton for Transcever Systems, Cambrdge Unversty Press, 010; Ch. 6,, App. B, C. 4. D.W. Blss, S. Govndasamy, Adaptve Wreless Communcatons: MIMO Channels and Networks, Cambrdge Unversty Press, 013. 5. A. Paulraj, R. Nabar, D. Gore, Introducton to Space-Tme Wreless Communcatons, Cambrdge Unversty Press; 003. 6. E.G. Larson, P. Stoca, Space-Tme Block Codng for Wreless Communcatons, Cambrdge Unversty Press, 003. 7. I.E. Telatar, "Capacty of Mult-Antenna Gaussan Channels," AT&T Bell Lab. Internal Tech. Memo., June 1995 (European Trans. Telecom., v.10, N.6, Dec.1999). 8. Foschn, G.J., Gans M.J.: On Lmts of Wreless Communcatons n a Fadng Envronment when Usng Multple Antennas, Wreless Personal Communcatons, vol. 6, No. 3, pp. 311-335, March 1998. 9. Raylegh, G.G., Goff, J.M.: "Spato-Temporal Codng for Wreless Communcatons," IEEE Trans. Commun., v.46, N.3, pp. 357-366, 1998. Lecture 13 15-Oct-15 (31)
Appendx 1: SVD and optmzaton of MIMO Channel Capacty When the x T sgnal correlaton matrx { } P = E xx I, the MIMO capacty s 1 C = log 1 HPH (1) σ If CSI (channel state nformaton) s avalable at the T x, P can be chosen to maxmze C, subject to the total T x power constran: where P s the -th T x power. Consder the m n MIMO channel, Usng the SVD of H, m = 1 where U,V are n n and m m untary matrces, 0 P = tr( P ) PT () y = Hx ξ (3) H = UΣV (4) U U = V V = I, and Lecture 13 15-Oct-15 3(31)
and dag[ ] Σ1 0 Σ = 0 0 (5) Σ 1 = σ1, σ,.. σ k are non-zero sngular values of H. Usng (4) and (3), y = UΣV x ξ, yɶ = Σxɶ ξ ɶ (6) y = σ x ξ, = 1,.. k (7) where ɶ, ɶ y = U y x = V x, ξ ɶ = U ξ. Note that multplcaton by a untary matrx does not change statstcs of a random vector and, hence, does not affect the mutual nformaton and the capacty. Hence, the channel n (7) has the same capacty as the orgnal channel (3). But the channel n (7) s as of k ndependent sub-channels, wth per-sub-channel SNR: and ɶ P (8) γ = λ σ 0 λ = σ are the egenvalues of HH, and P = P. Its capacty s k p C = log 1 λ (9) = 1 σ0 Lecture 13 15-Oct-15 4(31)
Optmum P can be found usng water-fllng (WF) technque as follows: where [ x] σ 0 P = µ, = 1,,.. k λ 1 (10) k1 P = PT (11) = 1 = x f x>0 and 0 otherwse; k 1 s the number of actve egenmodes (.e. wth non-zero P ), and constant µ s found from (10). Note that (10) and (11) also gve (mplctly) k 1. Lecture 13 15-Oct-15 5(31)
Water-fllng technque can be formulated as teratve algorthm as follows [1]: 1) order egenvalues, set teraton ndex p=0 ) fnd µ as follows 1 k p 1 µ = pt σ 0 k p = 1 λ (1) 3) set P usng (10) wth k 1 = k p 4) f there s zero P, set p = p 1, elmnate λ and go to step 5) fnsh when all P (=1,,..k-p) are non-zero. Ths algorthm gves all non-zero P. All the other P are zeros (.e., those egenmodes are not used). Proof of the water-fllng technque : usng Lagrange multplers wth the followng goal functon, Lecture 13 15-Oct-15 6(31)
λp F = log(1 ) α P PT σ0 df dp (13) df = 0, = 0 (14) dα where α s a Lagrange multpler. From (14), one obtans (10) and (11). Fnally, optmum P s found usng (4) where,,..,0...0. D = dag p1 p p k 1 P = VDV (15) Lecture 13 15-Oct-15 7(31)
Effect of T x CSI on the Capacty [] Compare the MIMO channel capacty n cases: 1) no T x CSI (unnformed T x -UT) ) full T x CSI (nformed T x -IT) PT In case 1, the capacty s gven by (9) wth P = m k PT = 1 mσ0 C = log(1 λ ) (15) UT In case, the capacty s gven by (9) wth P gven by (10) C IT k1 λp = 1 0 = log(1 ) (18) σ σ P = µ λ 0 (18a) Lecture 13 15-Oct-15 8(31)
Consder the rato when PT 0 / σ.e. hgh SNR mode. k1 P = PT (18b) = 1 C IT β = (19) CUT Assumng that P T =const and σ0, t s clear from (18a) and (18b) that P =P T /m (assumng k=m,.e. full-rank channel), and CIT P 1 as C σ UT T 0 (0) Hence, optmum power allocaton does not provde advantage n hgh SNR mode parallel transmsson (spatal multplexng) wth equal powers s optmum. Lecture 13 15-Oct-15 9(31)
Consder the case of low SNR, one fnds that ndex. Hence, Smlarly, Hence, P max T 0 P / σ 0. Assume that T = P and all the other 0 C IT P σ0, then from (18)-(18b) =, where max s the largest egenmode λ P λ = log 1 C C IT C UT UT = 1 max T max σ0 σ0 m PT m σ0 = 1 λ log e P T log e mλmax mλmax ( HH ) = m tr( HH ) λ (1) () Important concluson: n low SNR case, the best strategy s to use the largest egenmode only ths s beamformng! Lecture 13 15-Oct-15 30(31)
In hgh SNR mode, the best strategy s to use spatal multplexng (parallel transmsson on all egenmodes). References: 1. A. Paulraj, R. Nabar, D. Gore, Introducton to Space-Tme Wreless Communcatons, Cambrdge Unversty Press, 003.. G. Larsson, P. Stoca, Space-Tme Block Codng for Wreless Communcatons, Cambrdge Unversty Press, 003. Lecture 13 15-Oct-15 31(31)