# In filter design, why isn't it possible to have a frequency response with phase 0?

A linear phase filter will cause a delay in the output signal of the filter.

If I'm not mistaken, a filter with phase equal 0 would cause no delays in the output signal.

Can someone explain me why isn't it possibly to generate a 0 phase response filter? I mean, theoretically...

I'm studying for a test and I can't find the answer in textbooks.

## 2 Answers

A zero-phase frequency response is real-valued, and, by a property of the Fourier transform, the corresponding impulse response is symmetric. Since it is symmetric (i.e. it is non-zero for $t<0$) it can't be causal, and consequently, it can't be implemented (without adding delay, which defeats the purpose).

• Zero phase noncausal filters can be implemented, if future values of signals are available such as the case in image processing, where the whole image data (including the current and future index values) is already stored and available for computation. In this case causal does not refer strictly to time n of h[n] but just space index n1,n2 of impulse response h[n1,n2] . – Fat32 Jan 17 '15 at 20:07
• As a matter of fact those few lines of code will convince you to see it: I=imread('cameraman.tif'); b=fir1(14,0.35'); b2=b'*b; b2=b2/sum(b2(:)); I2=filter2(b2,I); figure,imshow(I); figure,imshow(I2/255); – Fat32 Jan 17 '15 at 20:11
• @BulentS.: When talking about spatial coordinates instead of time, the concept of causality makes no sense. "If future values ... are available" is exactly what causality is about, because they aren't. Pixels of an image are not "in the future", just because they are further away from some point of reference. – Matt L. Jan 17 '15 at 21:02
• @BulentS.: Esteban mentions the term "delay" in the first sentence of his question. That's why I concluded that he is talking about time-domain filters. And regardless of the terms Lim may use in his book, his filters are causal according to the standard definition of causality. – Matt L. Jan 18 '15 at 6:40
• I'm sorry. I didn't see these comments ! I did refer to the question of causality when I posted the question, but both of your answers gave me useful insight. – Esteban Filardi Feb 18 '15 at 23:09

Do you mean a zero phase for one single frequency only? Think of the classical bandpass circuit (passive or active), which has zero phase shift at the center frequency.

More than that, there are two typical delay categories: Phase delay and group delay. And the group delay is determined NOT by the actual phase value but by the SLOPE of the phase function (even if the phase is zero).