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Mathematics for Signal Processing (Signal Processing for Communication) mathematics for signal processing (signal processing for communication) 2007 spatial processing / beam forming reader: sections 3.1-3.3 Marc Geilen PT 9.35 [email protected] http://www.es.ele.tue.nl/education/5ME00/ Eindhoven University of Technology 2 overview • optimal beam formers – deterministic approach – stochastic approach • colored noise • the matched filter 3 data model assume we receive d (narrow band) signals on an antenna array: d =+=+ xanAsnkiikkkk∑ s , i =1 objective: construct a receiver weight vector w such that = H ykkwx is an estimate of one of the sources, or all sources: = H yWxkk 4 deterministic approach =⇔= noiseless case xAskk XAS dxN MxNMxd objective: find W such that WXH = S minimize WXH − S we consider two scenarios: • A is known • S is known 5 deterministic approach scenario 1 use direction finding methods to determine how many signals there are and what their response vectors are A is known: = … Aa[]1 ad (α2) (α1) (t) (α) 6 deterministic approach scenario 2 the sending signal makes use of a known training sequence, agreed in the protocol S is known, goal: select W alternatively, this could be the case via decision feedback Im(sk) 1/ 2 Re(s ) −1/ 2 1/ 2 k −1/ 2 7 deterministic approach =⇔= noiseless case xAskk XAS objective: find W such that WXH = S with A known − XAS=⇔ AXS††1 =,() A = AAAHH hence, we set WAH = † all interference is cancelled (if M ≥ d): H = WA Id 8 deterministic approach =⇔= noiseless case xAskk XAS objective: find W such that WXH = S with S known − WSXSXXXHHH==†1††(), AXSW == () H (after training, W is used to estimate the unknown S) again, all interference is cancelled (M ≥ d, N ≥ d): WAH = I 9 deterministic approach noisy case XASN=+ two possible optimization criteria model matching: adapting the model of A or S to minimize residual: XAS− 2 F output error minimization 2 WXH − S F 10 deterministic approach noisy case XASN=+ model matching: adapting the model of A (or S) to minimize residual: 2 XAS− F A Aˆ 11 deterministic approach noisy case XASN=+ output error minimization 2 WXH − S F n k s x y k A k W k e k - 12 model matching with A known: zero-forcing solution 2 ˆˆSXASsuch that − is minimal F ˆSAX= ††⇒ WH = A WHA=I: all interference is cancelled (hence zero-forcing) (under what conditions?) the ZF beamformer maximizes the output Signal-to-Interference Ratio (SIR) 13 example model matching the ZF beamformer satisfies WHA=I let w1 be the first column of W, the beamformer of the first signal. HH= ⇒ ……= WA I w12[][] a ad 0 0 ⊥ { … } waa12,,d thus, w1 projects out all other sources, except source 1. but what about noise? d ==HHH + yt11()wx () t∑ wa 1ii st () wn 1 () t i=1 =+H st11()wn () t the effect on the noise is not considered! 14 model matching zero-forcing solution WXHH=+ S() A† N the output noise depends on A+ and can be large, since = ⇒ †1H = − AUAAAΣΣ V() A U AA V A -1 this happens if ΣA is large, i.e. if A is ill-conditioned σ 1 1 σ d e.g. if directions are very close 15 model matching σ 2 11u σ 2 22u a2 a1 to discriminate a1 and a2, the ZF beamformer amplifies noise in the direction of u2 16 model matching with S known: 2 AXASˆˆ such that − is minimal F − AXSXSSSˆ ==†1HH() this does not specify the beamformer, but it is natural to set WAH = ˆ† 17 output error minimization objective: minimize the output error with S known 2 WWXSHHsuch that − is minimal F WSXH = † − note thatXXXX†1= HH() , so that 11−− WSXXXRRWRRHHH=(H11 )= ˆˆ , = ˆˆ -1 NN XS X X XS 1 RXXˆ = H : sample data covariance matrix X N 1 RXSˆ = H : sample correlation between the sources XS N and the received data 18 output error minimization H 2 objective: minimize the output error WX− S F with A known and assuming independent sources: observe that 11 SSHH→→ I,0 SN NN 11 1 RXSASSNSAˆ ==HHH +→ XS NN N =→ˆˆ-1− 1 WRRXXS RA X = H RxxX E[]is the true data covariance matrix. with finite samples, we take the estimate from XXH: = ˆ−1 WRAX 19 deterministic approach = ˆ−1 WRAX this is the Linear Minimum Mean Square Error (LMMSE) or Wiener receiver. • it maximizes the Signal-to-Interference-plus-Noise Ratio (SINR) at the output. • it does not cancel all interference: WAH ≠ I (because the loss in interference is compensated by the gain in removing noise) 20 Wiener filtering remember that for a convolution filter h(t) the optimal receiver in terms of output signal-to-noise ratio is given by: Hf()() Sf Gf*()= Hf()2 Sf ()+ Nf () spectral decomposition of array receiver H XASN=+ AUV= Σ •columns of V are vectors in ‘signal-space’ • that obtain the same attenuation (singular values of Σ) • their impact in ‘antenna-space’ are the columns of U 21 spectral decomposition H AUV= Σ u3 x2 σ u A σ 11 22u x1 x3 signal-space antenna-space 22 Wiener filtering spectral decomposition of array receiver H XASN=+ AUV= Σ define: XUXSVSNUN**===HHH * then X* =+UASNH () HH H =+UUΣ VS UN ** = ΣSN+ 23 Wiener filtering signal and noise are ‘spatially’ white in (transformed) signal-space and antenna-space respectively RnnUnUnUnnUUIUI=E**H==== E ( HHHHHH )( ) E σσ 2 2 n* nn RssVsVsVssVVIVI=E**HHHHHHH==== E ( )( ) E s* 24 Wiener filtering covariance of the transformed antenna samples: Rxxsnsn=E**HHHH=+ E ( ** )( ** += ) E ssnn **** + x* ΣΣ ΣΣ =+=+σσ222 ΣΣIInn Σ I a diagonal matrix! in this space, ‘virtual’ antenna elements are independent! Rxssnsssns=E**HHHH=+ E ( * * ) * = E ** + ** xs** ΣΣ **HH ** =+=+=ΣΣΣEEss ns 0 25 Wiener filtering thus, the Wiener receiver in the transformed space is WX***HH== S⇒ W * SX **† −1 ***H**HHH11 − 1 *-1 WSXXXRRWRR=,== ˆˆ** * ˆˆ * ** NNXS X X XS −1 *22→+σ WI()ΣΣn a diagonal matrix (!) with σ Hf()() Sf w * = k compare: Gf*( ) = kk, σσ22+ 2 kn Hf() Sf ()+ Nf () (in this space, beam forming is trivial) YVYVWX==****HHH = VWUX WUWV= * H 26 comparing receivers output error interference zero-forcing receiver (solid) Wiener receiver Wiener receiver (dashed) (what about ZF?) 27 comparing receivers ==H σσ… AUVΣΣ, diag(1 ,d ) • zero forcing = H WUWVZF 1 W diagonal with w ZF = ZF k, k σ k • Wiener receiver = H WUWWiener V σ W diagonal with wWiener = k Wiener k, k σσ22+ kn 28 stochastic approach assume a model with 1 source =+ =HH = + H xakkks, nys k wxwawn k ()() k k we make the following assumptions: 2 ==H E[sskkk ] 1,E[]n0 and define (spatial noise ‘color’): = H Rnnn :E[kk ] so that 2 =+HHHHH + E[|ys | ] E[(wakk wn )( wa s kk wn ) ] =+++HHHHHHHH2 E[wa |ssskkkkkkk | aw wa nw wn aw wnnw ] =HH +++ H H E[waaw 0 0 wnnwkk ] =+HH H ()()waaw wRwn 29 stochastic approach 2 =+HH H E[|y | ] waaw wRwn signal noise so the Signal to Noise Ratio (SNR) at the output of the Wiener receiver is: E[| (waH )s |2 ] waawHH SNR ()w ==k out H 2 H E[| (wnk ) | ] wRwn 30 stochastic approach two stochastic optimization criteria • maximum likelihood the likelihood of a set of data X is the probability (density) of obtaining that particular set of data, given the transmitted signal S. The Maximum Likelihood Estimate of S is the value of S that maximizes the probability of receiving that particular X. (compare deterministic model matching) • stochastic output error minimization the error at the output y of the receiver is a stochastic process, which depends on the weight vector w. Choose the weight vector w, which minimizes the variance of this process. (compare deterministic output error minimization) 31 stochastic model matching assume a model with d sources =+ =… ⇔ =+ xAsnkkk(1kN,,) XASN assume sk to be deterministic noise i.i.d. in time (temporally white), and spatially white 2 (Rn=σ I) and jointly complex Gaussian distributed 2 nk − 0.4 2 1 σ 2 σ ⇔= 0.3 n0Inkk~(,)CN pe () 2πσ 0.2 0.1 0 25 20 15 10 25 20 15 5 10 5 0 0 32 stochastic model matching nk=xk-Ask so the probability (density) to receive a certain vector xk, when sk has been transmitted is: − 2 − xAskk ==−=1 σ 2 pp(|)xskk ( n k x k As k ) e 2πσ − 2 ∑ xAs− 2 N −−xAskk N kk 1122 pe(|)XS==∏ σσ e 22πσ πσ k=1 XAS− 2 N − F 1 2 = e σ 2πσ XAS− 2 − F =⋅const e σ 2 (p(X|S) is the likelihood of receiving a certain data matrix X, for a given transmitted data matrix S) 33 stochastic model matching XAS− 2 − F pconste(|)XS=⋅σ 2 deterministic maximum likelihood technique: estimate S as the one which maximizes the likelihood of the actually received X ˆSXSsuch that p( | ) is maximal XAS- 2 - F ⇔ ˆS such that eσ 2 is maximal ⇔ ˆSXASsuch that -2 is minimal F for white Gaussian noise, Maximum Likelihood is equivalent to deterministic model matching (hence, the solutions are also the same) 34 stochastic derivation stochastic output error minimization minimize the Linear Minimum Mean Square Error cost: 2 Jw()=− E wxH s kk it can be worked out as follows: 2 Jw()=−=− EwxHHHH s E ()() wx s wx − s kk kk kk =−−+HH H[] H 2 wEEsEsEs xxwwkk x kk kk xw k =−HH[] − H +2 wRwx wEssEss akk kk a k w Es k 2 if sk is regarded stochastic with E[|sk| ]=1, then =HHH −−+ Jw()wRwx wa aw 1 35 stochastic output error minimization =HHH −−+ J()wwRwwaawx 1 differentiate with respect to w: let w=u-jv wit u and v real valued, then the gradient is ∂∂ JJ ∂∂ uv11 11 1 1 ∇=∇−∇=JJjJ»» − j wu22 v 2 2 ∂∂ JJ ∂∂ uvdd ∇=∇=∇HH H = with properties wwwwa a,0 aw wRwxx Rw ⇒ ∇= − wJ Rwx a the minimum J(w) is attained for ∇= ⇒ = −1 wJ 0wRax we thus obtain the Wiener receiver 36 stochastic output error minimization the minimum J(w) is attained for the Wiener receiver = −1 wRax the expected output error becomes: =HHH−−1111 −−+ − − Jmin aRxxx RR a aR x a aR x a 1 =−H −1 1 aRx a − =−HHσ 2 + 1 1 aaa()s Rn a −1 =−HH +σ 2 1 aaa()n Ia 37 stochastic output error minimization the expected output error becomes: −1 =−HH +σ 2 Jmin 1 aaa()n Ia −−11
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