MVDR and MPDR. Bhaskar D Rao University of California, San Diego

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1 MVDR ad MPDR Bhaskar D Rao Uiversity of Califoria, Sa Diego brao@ucsd.edu

2 Referece Books. Optimum Array Processig, H. L. Va Trees 2. Stoica, P., & Moses, R. L. (2005). Spectral aalysis of sigals (Vol. ). Upper Saddle River, NJ: Pearso Pretice Hall.

3 Narrow-bad Sigals Assumptios D x(ω c, ) = V(ω c, k s )F s [] + V(ω c, k l )F l [] + Z[] l= F s [], F l [], l =,.., D, ad Z[] are zero mea E( F s [] 2 ) = P s, ad E( F l [] 2 = P l, l =,..., D, ad E(Z[]Z [ ]) = σ 2 z I All the sigals/sources are ucorrelated with each other ad over time: E(F l []F m[p]) = P l δ[l m]δ[ p] ad E(F l []F s [p]) = 0 The sources are ucorrelated with the oise: E(Z[]F l [m]) = 0

4 Iterferece plus Noise sigal Covariace D I[] = V l F l [] + Z[], ; where V l = V(ω c, k l ) l= Properties of I[] I[] is zero mea The covariace of I[] is give by D S = E(I[]I H []) = P l V l Vl H l= + σ 2 z I

5 MVDR Beamformer Distortioless costrait o beamformer W : W H V s = Implicatio: W H x[] = W H V s F s [] + W H I[] () = F s [] }{{} +q[], where q[] = W H I[] (2) distortioless costrait Miimum Variace objective: Choose W to miimize MVDR BF desig E( q[] 2 ) = W H S W mi W W H S W subject to W H V s =.

6 MVDR beamformer MVDR BF desig Solutio: W mvdr = mi W W H S W subject to W H V s =. S Vs H S V s V s Derivatio: Ca be Obtaied usig Lagrage multipliers. Sice we are dealig with complex weights, eed to use Wirtiger calculus. See textbook for details

7 Iterpretatio W mvdr = S Vs H S V s V s tries to miimize E( q[] 2 ) = E( W H I[] 2 ) = W H S W, where S = D l= P lv l V H l + σ 2 z I It will try to place ulls at agular locatios cosistet with the iterferece plae waves if σ 2 z is small. If the umber of ateas is greater tha or equal to D, i.e. N D, the MVDR BF ca ull out all the (D ) iterferers. If N < D, the MVDR BF will attempt to cotrol the depth of the ulls to miimize iterferece. If σ 2 z is large compared to the power i the iterferig plae waves, the S σ 2 z I ad hece W mvdr V s

8 MVDR by SINR maximizatio W H V s 2 P s W H V s 2 W mvdr = arg max W W H = arg max S W W W H S W Note that the solutio is ot uique but is uique to scale. The scale ca be chose to satisfy the distortioless costrait. Sice S is positive defiite it admits a factorizatio S = LL H, where L is a N N matrix, a square root of S The square root is ot uique but ivertible because S is positive defiite. The W H S W = W H LL H W = W H W, where W = L H W or W = L H W The optimizatio over W ca be replaced by a optimizatio over W W o = arg max W W H L V s 2 W H W By the Cauchy-Schwarz iequality, W o L V s or W o = βl V s The W mvdr = L H Wo = βl H L V s = βs V s The distortioal less costrait Wmvdr H V s = leads to β = Hece W mvdr = S Vs H S V s V s Vs H S V s

9 Challeges with MVDR W mvdr = Vs H S S V s The mai challege is estimatig S? This requires coordiatio ad may ot always be possible. This leads to MPDR, miimum power distortioless respose beamformer. Most books refer to MPDR as MVDR. V s

10 MPDR, miimum power distortioless respose beamformer MPDR very similar to MVDR with respect to the costrait. Distortioless costrait o beamformer W : W H V s = Implicatio: W H x[] = W H V s F s [] + W H I[] (3) = F s [] }{{} +q[], where q[] = W H I[] (4) distortioless costrait Miimum Power objective: Choose W to miimize E( W H x[] 2 ), the power at the output of the beamformer E( W H x[] 2 ) = W H S x W, where S x = P s V s V H s + S MPDR BF desig mi W W H S x W subject to W H V s =.

11 MPDR beamformer MPDR BF desig Solutio: W mpdr = mi W W H S x W subject to W H V s =. S Vs H S x V s x V s Derivatio: Same as MVDR with S x replacig S Beefit: S x is easier to determie makig it computatioally attractive S x L x[]x H [] L = Same S x is eeded if you chage your mid o directio of iterest. Ca deal with multiple sigals of iterest with cosiderable ease.

12 Relatioship betwee MPDR ad MVDR For ucorrelated sources W mpdr = W mvdr Proof is based o the Matrix Iversio Lemma Note that (A + BCD) = A A B(DA B + C ) DA S x = P s V s V H s + S = S + V s P s V H s MPDR eeds iverse of S x. Usig the matrix iversio lemma with A = S, B = V s, C = P s, ad D = V H s, we have S x = S S V s (Vs H S V s + ) Vs H S P s

13 W mpdr = W mvdr S x = S S V s (Vs H S V s + ) Vs H S P s S x V s = S V s S V s (Vs H S V s + ) Vs H S P s = S V s S Vs H S V s V s Vs H S V s + P s = Note that Hece V H s S P s V s V s + S V s = βs V s where β = P s V H s S x W mpdr = Vs H S S x V s = x V s V s = βv H s S V s βvs H S βs V s V H s S P s V s + P s V s = Vs H S S V s = W mvdr V s

14 Spatial Power Spectrum usig MPDR Beamsteerig ad measurig power at the output of BF, i.e. E( (W d V(ψ T )) H x[] 2 ). FFT based processig for ULA MPDR based spatial power spectrum estimatio: Measure power at the output of the MPDR BF give by W mpdr = S Vs H S x V s x V s E( Wmpdr H x[] 2 ) = Wmpdr H S x W mpdr ( ) ( = Vs H S Vs H S x S x S x V s x V s Vs H S x ( = V H Vs H S s S x S x S ) x V s x V s Vs H S x V s = Vs H S x V s V s )