Time domain versus frequency domain

Equation () 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. The time domain offers new 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.