Model fitting by least squares

From the frequency domain to the time domain

Where data fitting uses the notation $\bold m \rightarrow \bold d$, linear algebra and signal analysis often use the notation $\bold x \rightarrow \bold y$. Equation (4) is a frequency-domain quadratic form that we minimized by varying a single parameter, a Fourier coefficient. Now, we look at the same problem in the time domain. We see that the time domain offers flexibility with boundary conditions, constraints, and weighting functions. The notation is that a filter $f_t$ has input $x_t$ and output $y_t$. In Fourier space, it is expressed $Y=XF$. There are two applications to look at, unknown filter $F$ and unknown input $X$.