From the frequency domain to the time domain

Equation (4) is a frequency-domain quadratic form that we minimized by varying a single parameter, a Fourier coefficient. Now we will look at the same problem in the time domain. We will see that the time domain offers flexibility with boundary conditions, constraints, and weighting functions. The notation will be that a filter f_t has input x_t and output y_t. In Fourier space this is Y=XF. There are two problems to look at, unknown filter F and unknown input X.
 

• Unknown filter
• Unknown input: deconvolution with a known filter