The construction of the denoising or deblurring (with a filter to inverse) is not based on the knowledge of the desire signal itself. It is based only on on the knowledge of its Power Spectral Density (thus requiring also second order stationary condition).

y is the measured signal and o the desired signal to reconstruct, b the additive noise, independent from o and h the convolution filter (generally low-pass filter)

y = conv(o,h) + b

Following the orthogonality theorem, in the Fourier Domain, the Wiener inverse filter Gw is then defined by:

Gw(f) = H(f)PSD[o](f)/(|H(f)|^2PSD[o](f) + PSD[b](f))

Where:

- H(f) is the transfer function of h

- PSD[s](f) is the Power Spectral Density of the signal s

If b is independant identical gaussian noise of standard deviation \sigma_b, then PSD[b](f)= \sigma_b^2.

By doing an estimation of the PSD of y (with the Welsh method, for example), one can note that :

PSD[y](f) = |H(f)|^2PSD[o](f) + PSD[b](f)

Therefore, it is possible to have an estimation of the PSD of the desired signal without knowing it and then to construct with it the wiener inverse filter.

Dear Arun prakash

Wiener filter is used for time varying environment ( Mobile channel - here the person who is having mobile , is in motion[car, two wheeler, walking etc.,] and the other objects are also in motion[other vehicles in the road] ) . At that time channel impulse response in varying. To estimate the channel we have to transmit the training sequences [well known to both transmitter and receiver] periodically to the channel. That training sequence is called as desired response.

by

R.suresh Babu

The construction of the denoising or deblurring (with a filter to inverse) is not based on the knowledge of the desire signal itself. It is based only on on the knowledge of its Power Spectral Density (thus requiring also second order stationary condition).

y is the measured signal and o the desired signal to reconstruct, b the additive noise, independent from o and h the convolution filter (generally low-pass filter)

y = conv(o,h) + b

Following the orthogonality theorem, in the Fourier Domain, the Wiener inverse filter Gw is then defined by:

Gw(f) = H(f)PSD[o](f)/(|H(f)|^2PSD[o](f) + PSD[b](f))

Where:

- H(f) is the transfer function of h

- PSD[s](f) is the Power Spectral Density of the signal s

If b is independant identical gaussian noise of standard deviation \sigma_b, then PSD[b](f)= \sigma_b^2.

By doing an estimation of the PSD of y (with the Welsh method, for example), one can note that :

PSD[y](f) = |H(f)|^2PSD[o](f) + PSD[b](f)

Therefore, it is possible to have an estimation of the PSD of the desired signal without knowing it and then to construct with it the wiener inverse filter.

Though the reference or desired input is given to the filter, still in presence of noise and system non linearity, the adjustment of filter coefficients to reconstruct the signal is not an easy task. Here you should not forget the unavoidable presence of channel noise and other distortions where fixed filter will completely fails to track the signal. 

Dear Arun Prakash,

I really  understand your question because every time I have to teach "Wiener Theory" to my students I have the same kind of questions. In fact, as Philippe Blanc explained in his response, you should consider the two following steps:

1st question - Pure Theory (and here we assume  the desired response is apriori  known)  -  "What is the best linear filter w(n) that minimises the MSE of e(n)=d(n)-y(n) between a desired signal d(n) and the output y(n) of the filter given by y(n) = conv(x(n), w(n)) ?". Doing so, you will find the following optimal solution expressed either in the time domain

w(n)= inv(Rxx(n)) * Rdx[n) 

or in the frequency domain  W(f) = PSD[d,x](f) / PSD[x,x](f).

2nd question- In practice (we can't assume  the desired response is apriori  known) - "How can I implement this filter ?" . Now, the answer depends on the problem you are trying to solve. For example, if you are looking for a noise reduction filter then you have the input signal given by x(n) = s(n) + b(n) where s(n) is the target signal to be estimated, and  b(n) is a centered noise component that is uncorrelated to s(n). So we have :

PSD[x,x](f) = PSD[s,s](f) + PSD[n,n](f) .

Furthermore, you are now looking for a denoising filter. So the "desired" signal d(n) should be as closed as possible from the target signal, i.e. d(n) = s(n) and we have now :

PSD[d,x](f)= PSD[s,x](f)=PSD[s,s+b](f)=PSD[s,s](f) + PSD[s,b](f)=PSD[s,s](f)

We conclude that the denoising filter is given in the frequency domain by :

W(f) = PSD[d,x](f) / PSD[x,x](f)

=> W(f) =PSD[s,s](f) / (PSD[s,s](f) + PSD[n,n](f))

=>W(f) =  SNR(f) / (1+ SNR(f) )   where   SNR(f) = PSD[s,s](f)/ PSD[n,n](f)

So the problem is now to estimate the signal-to-noise ratio SNR(f) for each frequency bin  and you have plenty of solutions to do this !! But once your SNR estimator will be available, you will be able  to buid your optimal Wiener filter W(f) that will correspond to the theory analyzed in the 1st step.

I hope this will help.

Pascal.