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 σd 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 () 2 0.2 0.1
0 25 20 15 10 25 20 15 πσ 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 σπσ22πσ k=1 XAS− 2 N − F 1 2 = e σ 2 πσ XAS− 2 − F 2 =⋅const e σ
(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 =−1 aaaHH + 2 Ia nσ 37 stochastic output error minimization
the expected output error becomes:
−1 =−HH +σ 2 Jmin 1 aaa()n Ia (−−11 )()() =−HH +222 H + H + 1 aaa()()()nnnσσσ Iaaa aa ()()()−11− =−1 aaaHH +2 I aaa H +2 aaa H + 2 ()nnnσσ −−11 =−1 aaaHH+ 222 I aa H + Iaa Ha + nnnσσσ −1 σ =+1− aaaHHa 2 n σ
→ σ 2 2 →→σ 2 Jmin 1 i f n a Jmin 0 if n 0 38 colored noise
what if noise vector is not spatially white?
H assume noise has a known covariance E[nn ]=Rn≠σ2I
-1/2 ‘prewhiten’ the data with a square root factor Rn
=+ ⇒ −−1/ 2=+ 1/ 2 − 1/ 2 xAsnkkk RxRAsRn nknknk =+ xAsnkkk
==H −−1/ 2 1/ 2 = note that RnnRRRInkknnnE
2 so that the noise nk is white, with variance σ =1 39 colored noise
covariance matrices have a square root:
==HH RXXXUVX Σ
= HH RUVVUX ΣΣ 2 H = UUΣ
1/ 2 = H −−1/ 2= 1 H RUUX Σ RUUX Σ
check that 1/ 2 1/ 2 = 1/ 2− 1/ 2 = RRXX R X RRXX I 40 colored noise
we use the known results on the new variables: = −1/ 2 = −1/ 2 = −1/ 2 ARn A xRxknk nRnknk
the ZF equalizer becomes ==†1HH− yAxAAAxkk() k = HH−−−−−1/2 1/2 1 HH 1/2 1/2 ((ARn ) Rnnnk A ) AR ( ) R x ==HH−−11 − 1 H ()ARnnkk A AR x Wx ⇒ = −−−111H WRAARAnn()
the Wiener receiver is the same as before, since Rn is not used in the derivation check: ==−−−−−11/21/211/21/21 = − WRARxx()nnn RR R AR n RA x ⇒ ==HH−1/ 2 = H YWXWRn XWX ⇒ = −1/ 2 = −1 WARn W Rx 41 the matched filter
consider a single signal in white noise, =+H =σ 2 xakkks, nE nn kk I the ZF beamformer is given by − wa==()†1HH aaa ( ) =γ a
note: a scalar multiplication does not change the output SNR
w=a is known as a matched filter, classical beamformer and Maximum Ratio Combining (MRC) 42 the matched filter
with non-white noise, =+H = xakkks, nE nnR kk n
we have seen that the zero forcing beam former in non- white noise equals: = −−−111H WRAARAnn()
hence, in case of a single signal we get: ==−−−−1111H γ wRaaRann() Ra n
thus, the matched filter in non-white noise is = −1 wRan 43 the matched filter
similarly the Wiener filter, in white noise,
==−−121H +σ wRax () aa Ia σσσ−− =+(aaHHH21 I )()() a a a + 2 a a + 21 σσσ−− =+()()()aaHHH21 I aa a + 2 a a a + 21 σσσ−− =+()()()aaHHH21 I aaa + 2 Iaaa + 21 −− =+()()()aaHHH21 Iσσσ aa + 2 I a a a + 21 =+H 21− σ aa() a γ = a
it is also equal to a multiple of the matched filter! 44 the matched filter
Wiener Filter in colored noise, (using whitening)
= −1/ 2 aRn a ==−−11H + wRaxn() aaR a =+−−1/ 2H 1 RaaIan () =+−−1/ 2H 1 Raaan (1) = γ −1 Ran
this is again equal to a multiple of the matched filter for colored noise 45 the matched filter
the colored noise case is relevant also for the following reason: with more than one signal, we can write the model as =+=+′′ + xkkkk Asna1s () Asn kk this is of the form
=+ =H +σ 2 xakkks, n RAAn '' I
where the “noise” is colored due to the contribution of the interfering sources. 46 matched filter
in summary, • the matched filter is optimal with only one signal • equal to both zero-forcing and Wiener filter (with one signal there is no interference) • with colored noise, they also coincide • noise color can be used to capture interference of other signals 47 summary
•criteria: model matching vs. output error minimization • solutions: zero-forcing, Wiener filter • zero forcing optimizes: model matching, interference • Wiener filter optimizes: output error, SNR • matched filter: for a single signal both coincide and the beam former equal the response vector •both deterministic and stochastic analysis • colored noise