Can a causal filter without phase shifts exist?

Question

When I was studying dispersion of refraction index in semiconductors and dielectrics, my professor tried to explain that if a filter (like a dielectric absorbing some light frequencies, or an electric RC-filter) removes some frequencies, then the remaining ones must be phase shifted to compensate for those frequencies (which are infinitely spread in time as usual monochromatic signals) being subtracted from the whole signal, to preserve causality.

I intuitively understand what he was talking about, but what I'm not sure of is whether his argument is really justified - i.e. whether there can exist a non-trivial filter, which absorbs some frequencies and leaves remaining ones not shifted, but still preserving causality. I can't seem to construct one, but can't prove it doesn't exist as well.

So the question is: how can it be (dis)proved that a causal filter must shift phases of frequencies relative to each other?

Answer 1

Suppose that a linear filter has impulse response $h(t)$ and frequency response/transfer function $H(f) = \mathcal F [h(t)]$, where $H(f)$ has the property that $H(-f) = H^*(f)$ (conjugacy constraint).

Now, the response of this filter to complex exponential input $x(t) = e^{j2\pi f t}$ is $$y(t) = H(f)e^{j2\pi f t} = |H(f)|e^{j(2\pi f t + \angle H(f))}$$ and if we want this filter to cause no phase shift, it must be that $\angle H(f) = 0$ for all $f$.

How about if, instead of no phase shift, we are willing to allow a fixed constant phase shift for all frequencies? That is, $\angle H(f) = \theta$ for all $f$ is acceptable to us where $\theta$ need not be $0$? The extra latitude does not help very much, because $\angle H(-f) = -\angle H(f)$, and so $\angle H(f)$ cannot have fixed constant value for all $f$ unless that value is $0$.

We conclude that if a filter does not change the phase at all, then $H(f)$ is a real-valued function, and because of the conjugacy constraint, it is also an even function of $f$. But then its Fourier transform $h(t)$ is a an even function of time, and thus the filter cannot be causal (except in trivial cases): if its impulse response is nonzero for any particular $t > 0$, then it is also nonzero for $-t$ (where $-t < 0$).

Note that the filter need not be doing any frequency suppression, that is, we did not need the assumption that some frequencies are "removed" by the filter (as the OP's professor's filter does) to prove the claim that zero phase shift is not possible with a causal filter, frequency suppressor or not.

Answer 2

There are filters that cause a ,,linear'' phase shift, that is, constant delay. It is not possible to filter anything at all (causally) without causing any delay.

Answer 3

Phase shift is due to time delay i.e. Time taken by the signal to reach from input to the output of a system. Now if system is not causing any phase shift then it means time delay is zero. Now think of a system which is providing output at the same instant when input is applied. Will that be possible ? Of course not .if there's a system then it must be performing some kind of job on the signal that produces delay and finally phase shift

Answer 4

You can have a filter without phase shift. It's called an observer (predictor). It is no longer just a filter though but rather a mathematical model of how multiple sensor readings relate to each other. So you are able to predict the signal and thus have the best possible prediction of the real signal at the same instant that you take your measurements (no phase shift).