linear phase and generalised linear phase filters

I know that for linear phase filters, all frequency components have equal delay times. That is, there is no distortion of signal due to the time delay of frequencies relative to one another.

However, for filters generalised linear phase, where the phase is of the form $$a\omega+b$$, and $$b$$ is not zero. Will the same be true for them? Thanks in advance!

• @hotpaw2: OK, so is there phase distortion? Feb 13 '14 at 1:32
• @hotpaw2, that doesn't sound right. The phase response $\phi=a\omega + b$ produces a delay of $d\phi / d\omega = a$, so a is nonzero for non-trivial causal filters. On the other hand, $b$ introduces a global phaseshift which requires a complex filter unless $b=n \pi$ for natural $n$. That is unless you allow a discontinuity of $b$ at $\omega=0$, which would not be obvious from what the OP stated. Feb 13 '14 at 15:17
• @Jazzmaniac: Can you provide an answer for my question above? Thanks! Feb 13 '14 at 15:41

Yes. The time delay of real-coefficient linear-phase N-point FIR filter is (N-1)/2 samples. The time delay of complex-coefficient generalized-linear-phase N-point FIR filter is also (N-1)/2 samples.

You can prove this to yourself. Design a narrowband linear-phase lowpass FIR filter, and plot its group delay. Then multiply that filter's coefficients by a complex exponential whose frequency is one fourth the sample rate, creating a complex-coefficient filter. Now plot the group delay of the complex filter. Compare the two group delay plots.

Yes the same will be true for generalized linear phase. The only difference is that there will be a processing delay, due to the filter being casual like hotpaw2 said. Normally for FIR filters this delays is half the number of coefficients of the filter.

• Will the waveform of the signal be retained? Can you show an ezample ? Feb 13 '14 at 5:08
• Absolutely not. Unless the magnitude of the frequency response is flat for all frequencies, which added to the linear phase makes a fancy delay line, which is pretty pointless. The whole point of filtering is modifying the waveform somehow, in order to obtain a desireable result out of it.
– bone
Feb 13 '14 at 5:12
• Ok, but for strictly linear phase filters, the waveform of the signal will be retained? Feb 13 '14 at 5:23
• No. Bone's answer is correct. Unless the filter is a pure delay (linear phase AND flat frequency response) the waveform will change. Feb 13 '14 at 9:55
• But there will be no phase distortion for both cases ? @Hilmar Feb 13 '14 at 11:50