As stated at the beginning of this chapter, the impulse response of every causal, linear-phase, FIR filter is symmetric:
Assume that is odd. Then the filter
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 which is real. Since can go negative, the phase response is
For frequencies at which is nonnegative, the phase delay and group delay of a linear-phase filter are simply half its length: