On the transition width of finite impulse-response digital filters

Several properties of finite-duration impulse-response (FIR) digital filters designed to have the maximum possible number of ripples are discussed and illustrated with examples. Such filters have been called extraripple filters. Among the properties of such filters are as follows. 1) Extraripple low-pass filters with fixed passband ripple δ
1
and stopband ripple δ
2
achieve the local minimum of transition width in the class of linear phase filters with fixed impulse-response duration of N samples. 2) For the case δ
1
= δ
2
the minimum transition width is roughly independent of F
p
, the passband cutoff frequency. 3) For the case δ
2
< δ
1
, the minimum transition width decreases with increasing bandwidth. Several figures are included to show the relation between the transition width and bandwidth for low-pass filters.