For antisymmetric FIR windows, the phase response is of the form $a\omega+b$, where $b$ is $\pm\frac{\pi}{2}$, so it is not strictly linear. Does the waveform of the signal gets retained?

  • $\begingroup$ Do you mean "antisymmetric FIR response"? Windows I know of are symmetric. $\endgroup$ – Juancho Feb 13 '14 at 11:49
  • $\begingroup$ Hmm yeah I think so @Juancho $\endgroup$ – meta_warrior Feb 13 '14 at 11:55
  • $\begingroup$ Can any filter preserve the shape of every possible input waveform? $\endgroup$ – Dilip Sarwate Feb 13 '14 at 18:14

Such phase response is called Generalized linear phase, where you are allowed the $\pm \pi/2$ constant phase, and zero crossings (which add $\pi$ jumps in phase at one or more frequencies).

Filters with an antisymmetric impulse response do in fact have the $\pm \pi/2$ constant phase.

Simplest case: $h[n] = \delta[n] - \delta[n-1]$, with frequency response $H(e^{j\theta}) = 2j e^{-j \theta /2} \sin( \theta/2 )$.

In practical terms, these filters are types of differentiators: they all have a zero at $z=1$ (0 frequency) due to the fact that $\displaystyle\sum h[n] = 0$.

It is clear that these filters do not retain the waveform of the signal.

Normally you say that linear phase filters "retain the waveform" in case of low-pass filters, where you want to filter out noise but retain the shape of a pulse. More importantly, you want a good eye-pattern when decoding, and a non-linear-phase filter will result in asymmetric step responses and a poor eye pattern.


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