Connecting the Vector Space and Classical Wiener Filters Suppose we observe x y w which are all N 1 vectors and where w 0 σ 2 I . Given x we wish to estimate y. Think of y as a signal in additive white noise w. x is a noisy observation of the signal. Taking a Bayesian approach, put a prior on the signal y: y 0 R yy which is independent of noise w. The minimum MSE (MMSE) estimator is y R yx R xx x Under the modeling assumptions above R yx y y w y y y w y y R yy since y w 0 and since y and w are zero-mean and independent. R xx x x y w y w y y y w w y w w R yy R ww since w w R ww . Hence y R yy R yy R ww x H opt x Where H opt is the Wiener filter. Recall the frequency domain case H opt f S yy f S yy f S ww f Now let's look at an actual problem scenario. Suppose that we know a priori that the signal y is smooth or lowpass. We can incorporate this prior knowledge by carefully choosing the prior covariance R yy . Recall the DFT k k 0 , … , N - 1 𝒴 k 1 N n 0 N Y k 2 k n N or in vector notation k k 0 , … , N - 1 𝒴 k y u k where u k 1 2 k N 2 2 k N … 2 N 1 k N N (H dehotes Hermitian transpose) u k u k u k u k 1 , k l k l u k u l u k u l 0 , i.e., k k 0 , … , N - 1 u k is an orthonormal basis. The vector u k spans the subspace corresponding to a frequency band centered at frequency f k 2 k N ("digital" frequency on 0 1 ). If we know that y is lowpass, then y u k 2 𝒴 k 2 should be relatively small (compared to y u 0 2 ) for high frequencies. Let σ k 2 y u k 2 A lowpass model implies σ 0 2 σ 1 2 … σ N 2 2 , assuming N even, and conjugate symmetry implies j j 1 , … , N 2 σ N - j 2 σ j 2 Furthermore, let's model the DFT coefficients as zero-mean and independent 𝒴 k 0 𝒴 k 𝒴 l σ k 2 l k 0 l k This completely specifies our prior y 0 R yy R yy U D U where D σ 0 2 0 … 0 0 σ 1 2 … 0 ⋮ ⋮ ⋱ ⋮ 0 0 … σ N - 1 2 and U u 0 u 1 … u N - 1 𝒴 U y is the DFT and y U 𝒴 is the inverse DFT. With this prior on y the Wiener filter is y U D U U D U σ 2 I x Since U is a unitary matrix U U I and therefore y U D U U D σ 2 I U x U D U U D σ 2 I U x U D D σ 2 I U x If A , B , C are all invertible, compatible matrices, then A B C C B A . U U , U U . Now take the DFT of both sides 𝒴 U y D D σ 2 I 𝒳 where 𝒳 U x and is the DFT of x. Both D and D σ 2 I are diagonal so 𝒴 k d k k d k k σ 2 𝒳 k σ k 2 σ k 2 σ 2 𝒳 k Hence the Wiener filter is a frequency (DFT) domain filter 𝒴 k H k 𝒳 k where 𝒳 k is the k th DFT coefficient of x and the filter response at digital frequency 2 k N is H k σ k 2 σ k 2 σ 2 Assuming σ 0 2 σ 1 2 … σ N 2 2 and j j 1 , … , N 2 σ N - j 2 σ j 2 . The filter's response is a digital lowpass filter!

A Digital Lowpass Filter!