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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?

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Consider using a minimum phase filter, which will have the least delay for a given magnitude response. The following response by MattL at the first post below details this further in using 'Leja Ordering' for converting large linear phase filter to a minimum phase filter using widely available algorithms (such as firls and firpm in MATLAB and firls and remez in Octave and Python scipy.signal). For filters of modest length it is fairly trivial to determine the zero locations for the linear phase filter and convert to a minimum phase filter by reflecting all the zeros to be inside the unit circle as detailed in the second post.

Minimum Phase - All Pass Decomposition For Large Linear Phase Filters

Minimum phase FIR method

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