As stated at the beginning of this chapter, the impulse response of every causal, linear-phase, FIR filter is symmetric:
Assume that
is a zero-phase filter. Thus, every odd-length linear-phase filter can be expressed as a delay of some zero-phase filter,
By the shift theorem for z transforms (§6.3), the transfer function of a linear-phase filter is
and the frequency response is
which is a linear phase term times
For frequencies