Filters and such

This is a continuation of the "Delta spectrum" post, so we start from there. I've realized that if not done incrementally, to obtain a convolution of two signals in the time domain, would be more computationally consuming, than to do it in the frequency domain.

FFT is with complexity, against the complexity of time domain convolution. So I'm going to explore this a little bit.

We start with a signal defined as:

Here H is the frequency response of our system, N is additive noise, X is the real (original signal) and Y is the observed, convoluted signal with noise. We would like to find a function G such as:

Whereas is an estimate of the original signal, which should minimize the mean square error.

Without going into further details, the operation is assumed to be carried out in the frequency domain as follows:

Taking into consideration that G is the Wiener filter defined as:

Now going back and accounting for our original signal and our frequency response, which are correspondingly a delta function and a Gaussian, the last equation is simplified enormously. For the impulse response we have:

And for the original signal one can write:

Where j is the imaginary unit.

The noise could be estimated by doing lowpass filtering and converging to a state where the mean square deviation from the original signal would be minimal. The parameters alpha and tau can be automatically estimated from the signal peaks, by doing nonlinear Gaussian fit in the time domain. The whole point of doing such a complicated process would be to try finding merged energy peaks, low intensity peaks, or estimating Doppler shift in the spectrum.