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Gauss-Markov Theorem and Wiener Filtering

Module by: Clayton Scott, Robert Nowak. E-mail the authors

Let xx and yy be jointly Gaussian distributed: ( x y )∼𝒩( m x m y )( R xx R xy R yx R yy ) x y m x m y R xx R xy R yx R yy Then the conditional distribution of y y given x x is y | x ∼𝒩 m y + R yx ⁢ R xx -1⁢(x− m x )Q y | x m y R yx R xx x m x Q where Q= R yy − R yx ⁢ R xx -1⁢ R xy Q R yy R yx R xx R xy

We know that the conditional mean y ^= m y + R yx ⁢ R xx -1⁢(x− m x ) y m y R yx R xx x m x is the best estimate of yy give xx in a (mean) squared error sense.

Example 1

x=y+W x y W where y∼𝒩0 R yy y 0 R yy and W∼𝒩0 R WW W 0 R WW and yy and WW are independent. ( x y )∼𝒩( 0 0 )( R yy + R WW R yy R yy R yy ) x y 0 0 R yy R WW R yy R yy R yy y | x ∼𝒩 R yx ⁢ R xx -1⁢x R yy − R yx ⁢ R xx -1⁢ R xy y | x R yx R xx x R yy R yx R xx R xy y ^= R yy ⁢ R yy + R WW -1⁢x=H⁢x y R yy R yy R WW x H x where H H is the Wiener filter. Minimum MSE estimator of yy given xx. R yx = R yy R yx R yy and R xx = R yy + R WW R xx R yy R WW .

Direct Optimization

x=y+W x y W y ^=G⁢x y G x H=argminGE(y−G⁢x)T⁢(y−G⁢x) H G y G x y G x where E(y−G⁢x)T⁢(y−G⁢x) y G x y G x is the MSE.

MS=E(y−G⁢x)T⁢(y−G⁢x)=Etr⁢(y−G⁢x)⁢(y−G⁢x)T=tr⁢E(y−G⁢x)⁢(y−G⁢x)T MS y G x y G x tr y G x y G x tr y G x y G x
(1)
Minimizing MSE is equivalent to minimizing
ε2=E(y−G⁢x)⁢(y−G⁢x)T= R yy −G⁢ R xy − R yx ⁢GT+G⁢ R xx ⁢GT ε 2 y G x y G x R yy G R xy R yx G G R xx G
(2)
Taking the derivative with repsect to GG ∂ε2∂ G =2⁢⁢ R yx +2⁢G⁢ R xx =0 G ε 2 -2 R yx 2 G R xx 0 This implies
H= R yx ⁢ R xx -1= R yy ⁢ R yy + R WW -1 H R yx R xx R yy R yy R WW
(3)

Figure 1: y= y ^+(y− y ^) y y y y , where y− y ^ y y is the error. The error and the estimate are statistically orthogonal to each other.
Geometrical Interpretation
Geometrical Interpretation (geoint.png)

Orthogonality Condition

The optimal Wiener filter H= R yy ⁢ R yy + R WW -1 H R yy R yy R WW satisfies the following condition E(y− y ^)⁢ y ^T=0 y y y 0

E(y− y ^)⁢ y ^T=E(y⁢xT⁢HT−H⁢x⁢xT⁢HT)= R yx ⁢HT−H⁢ R xx ⁢HT= R yy ⁢HT−H⁢( R yy + R WW )⁢HT= R yy ⁢ R yy + R WW -1⁢ R yy − R yy ⁢ R yy + R WW -1⁢( R yy + R WW )⁢ R yy + R WW -1⁢ R yy =0 y y y y x H H x x H R yx H H R xx H R yy H H R yy R WW H R yy R yy R WW R yy R yy R yy R WW R yy R WW R yy R WW R yy 0
(4)

The Classical Wiener Filter

Figure 2: y⁢t y t is a stationary random signal, w⁢t w t is stationary random noise, and the filter is H H
Figure 2 (block1.png)
We want to find H⁢ω H ω that minimizes the MSE
ε2=E y⁢t ^−y⁢t2= R ŷ ŷ ⁢0−2× R y ŷ ⁢0+ R yy ⁢0 ε 2 y t y t 2 R ŷ ŷ 0 2 R y ŷ 0 R yy 0
(5)
where R yy ⁢τ=Ey⁢t⁢y⁢t+τ R yy τ y t y t τ . We can express the MSE in the frequency domain by noting that R yy ⁢0=12⁢π⁢∫−∞∞ S yy ⁢ωd ω R yy 0 1 2 ω S yy ω where S yy ⁢ω S yy ω is the power spectrum of y⁢t y t .

Recall:

R yy ⁢τ R yy τ and S yy ⁢ω S yy ω are FT pairs: S yy ⁢ω=∫−∞∞ R yy ⁢τ⁢e−(i⁢ω⁢τ)d τ S yy ω τ R yy τ ω τ R yy ⁢τ=12⁢π⁢∫−∞∞ S yy ⁢ω⁢ei⁢ω⁢τd ω R yy τ 1 2 ω S yy ω ω τ

Random Signal Response of Linear Systems

Figure 3: x⁢t x t is a stationary random process and H H is a linear time-invariant system.
Figure 3 (block2.png)
y⁢t ^=∫−∞∞h⁢u⁢x⁢t−ud u y t u h u x t u
E y⁢t ^=∫−∞∞h⁢u⁢Ex⁢t−ud u = m x ⁢∫−∞∞h⁢ud u y t u h u x t u m x u h u
(6)
since Ex⁢t−u x t u is a constant. In particular, if x⁢t x t is zero-mean then so is y⁢t ^ y t .

Autocorrelation of Output Process

R ŷ ŷ ⁢τ=E y⁢t ^⁢ y⁢t+τ ^=E∫h⁢s⁢x⁢t−sd s ⁢∫h⁢u⁢x⁢t+τ−ud u =∫∫Ex⁢t−s⁢x⁢t+τ−u⁢h⁢s⁢h⁢ud s d u =∫∫ R xx ⁢τ+s−u⁢h⁢s⁢h⁢ud s d u = R xx ⁢τ*h⁢−τ*h⁢τ R ŷ ŷ τ y t y t τ s h s x t s u h u x t τ u u s x t s x t τ u h s h u u s R xx τ s u h s h u R xx τ h τ h τ
(7)

Power Spectrum

S ŷ ŷ ⁢ω= S xx ⁢ω⁢H⁢ω¯⁢H⁢ω= S xx ⁢ω⁢∥H⁢ω∥2 S ŷ ŷ ω S xx ω H ω H ω S xx ω H ω 2
(8)

Cross-Correlation and Cross-Spectrum

R ŷ x ⁢τ=E y⁢t ^⁢x⁢t+τ=E∫h⁢s⁢x⁢t−sd s ⁢x⁢t+τ=∫h⁢s⁢ R xx ⁢τ+sd s = R xx ⁢τ*h⁢−τ R ŷ x τ y t x t τ s h s x t s x t τ s h s R xx τ s R xx τ h τ
(9)
This implies S ŷ x ⁢ω= S xx ⁢ω⁢H⁢ω¯ S ŷ x ω S xx ω H ω R ŷ ŷ ⁢0=12⁢π⁢∫−∞∞∥H⁢ω∥2⁢ S xx ⁢ωd ω R ŷ ŷ 0 1 2 ω H ω 2 S xx ω R y ŷ ⁢0=12⁢π⁢∫−∞∞H⁢ω¯⁢ S yx ⁢ωd ω R y ŷ 0 1 2 ω H ω S yx ω S xx ⁢ω= S yy ⁢ω+ S ww ⁢ω S xx ω S yy ω S ww ω S yx ⁢ω= S yy ⁢ω S yx ω S yy ω since y⁢t y t and w⁢t w t are independent. Thus, the expression for the MSE becomes ε2=12⁢π⁢∫−∞∞∥H⁢ω∥2⁢( S yy ⁢ω+ S ww ⁢ω)−2⁢H⁢ω¯⁢ S yy ⁢ω+ S yy ⁢ωd ω ε 2 1 2 ω H ω 2 S yy ω S ww ω 2 H ω S yy ω S yy ω ε2 ε 2 is minimized by minimizing the integrand each frequency ωω. This implies H⁢ω⁢( S yy ⁢ω+ S ww ⁢ω)= S yy ⁢ω H ω S yy ω S ww ω S yy ω H⁢ω= S yy ⁢ω S yy ⁢ω+ S ww ⁢ω H ω S yy ω S yy ω S ww ω

Example 2

Figure 4
Figure 4 (syy.png)
Figure 5
Figure 5 (sww.png)
Figure 6
Figure 6 (h.png)

Comparison

Signal Vector (discrete-time)

H= R yy ⁢ R yy + R ww -1 H R yy R yy R ww Here H H is a matrix, R yy R yy is the signal "power", and R ww R ww is the noise "power".

Classical (continuous-time)

H⁢ω= S yy ⁢ω S yy ⁢ω+ S ww ⁢ω= S yy ⁢ω⁢ S yy ⁢ω+ S ww ⁢ω-1 H ω S yy ω S yy ω S ww ω S yy ω S yy ω S ww ω
(10)
Which means that the Wiener Filter is defined as the ratio of signal power to the sum of signal power and noise power.

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