I am looking for a lowpass FIR filter that is causal (all samples used are current or past) and exhibits very low passband ripple and has passband phase very close to zero, which necessitates a filter to be a nonlinear-phase filter. Meanwhile, for transition band and stopband, I want frequency response of the filter to remain below 1.
I tried hard to look at some design mechanisms or principles in use but so far came up empty. Maximally flat nonlinear-phase FIR filters seem to exhibit excessive transition band overshoots beyond magnitude of 1.
So what design mechanism should I use?