Phase of the Ideal Lowpass Filter

Note that the specification of the ideal lowpass filter in Chapter 1 was actually only a specification of $G(\omega)$. Often we wish to modify only the spectrum energy at certain frequencies to make them more or less audible. In these cases, a particular $G(\omega)$ is desired, and we do not care about $\Theta(\omega)$ as long as it does not make any audible difference. One might initially guess that we should set the desired phase response to zero, but this turns out to be an impossible specification for causal filters, as we will see below. The choice that's most natural for recursive filters is minimum phase, so we will need to look at that case. Another popular choice is linear phase, but this turns out to be impossible for recursive filters as well. (It is an option for nonrecursive filters--i.e., FIR filters having no feedback.) We will look at these cases in the following sections.