# Do antisymmetric FIR filters preserve the shape of their input waveforms?

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?

• Do you mean "antisymmetric FIR response"? Windows I know of are symmetric. – Juancho Feb 13 '14 at 11:49
• Hmm yeah I think so @Juancho – meta_warrior Feb 13 '14 at 11:55
• Can any filter preserve the shape of every possible input waveform? – 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 $\displaystyle\sum h[n] = 0$.