# Zero-phase-shift filters must be symmetric about the origin

In reading Rafael Gonzalez Digital Image Processing, second edition, section 4.10.2, the author writes:

Notch filters are the most useful of the selective filters. A notch filter rejects (or passes) frequencies in a predefined neighborhood about the center of the frequency rectangle. Zero-phase-shift filters must be symmetric about the origin, so a notch with center at $$(u_o, v_o)$$ must have a notch at location $$-(u_o, v_o)$$.

Two questions:

1) Why does a zero phase shift filter have to be symmetric about the origin (with real coefficients I am assuming)?

2) What does the author mean here when he says a "notch"

Any insights appreciated.

• "Notch" filters are very narrow band-stop filters. Generally used to filter out impulsive noise in frequency domain. Jun 5 '20 at 10:19

Consider the Inverse Fourier Transform as the impulse response of the filter. In the case of an ideal brick-wall filter with zero-phase, the impulse response would be a Sinc function centered about zero in time (given by the Inverse Fourier Transform).

This is non-causal (time domain response is greater than 0 for $$t<0$$), but in order to be zero-phase as shown, the time domain response must be complex conjugate symmetric. It need not be real; in order for the frequency response to be real, the time response real components must be even and the imaginary components must be odd (complex conjugate symmetric).

But to visualize the above plot further and what "zero-phase" means, a delay in time is a linear phase in frequency, thus if we shift the impulse response to the right (as done prior to truncation for a causal solution), the magnitude response is unchanged but the phase in no longer "zero-phase". If we continued to see the next plot I have in this series, which shows the impulse response truncated by the dashed window in this plot below, we would then see that the magnitude response of the filter would also be distorted, but that is beyond the question being asked here.

I summarize the properties of odd and even functions as well as other universal Fourier Transform properties in the figure below. The time and frequency domains are interchangeable for these properties; for example, if a function is periodic in one domain, it is discrete in the other. If a function has only real components in one domain (zero-phase), then all real components in the other domain must be an even function about 0, and all imaginary components must be odd about zero (inverse sign). This results in complex conjugate symmetry.

The relationship of the Hilbert Transform mentioned for the causal and anticausal functions applies to minimum phase systems.

What is interesting is how these tie together and why I choose to describe the above relationships in terms of even and odd functions rather than simply saying the other domain will be complex conjugate symmetric. This way we can see how a causal function in time (or a single sided function in frequency) is the sum of even and odd functions, and therefore MUST be complex in the other domain given that we need both real and imaginary components to create such even and odd functions.

Regarding the OP’s second question: a notch filter usually implies a very narrow band of rejection to remove certain frequency components. For more details see this post here: Transfer function of second order notch filter

• Thank you for the answer! When you say this is non-causal, (time domain response is greater than 0 for t<0), I am not sure what you mean. From what I understand the sync function has both positive and negative components extending to both negative and positive infinity. Jun 5 '20 at 4:01
• Yes you are correct and anything that has non-zero values for negative time is called non causal. Since the impulse response is the filter output in response to an impulse at the input at time t=0, an output that occurs before the input is applied is “non-causal”. We can do this in simulation with post processing but it cannot be realized as an actual filter. Jun 5 '20 at 12:21
• So to be causal (and realizable) we must delay the impulse response in time to achieve the same magnitude bs frequency that would be associated with a real (zero-phase) frequency response. Further, we then need to truncate that impulse response for t<0. The delay adds linear phase (and makes the real frequency response complex) and the truncation adds distortion. Jun 5 '20 at 12:26