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I am new to DSP and filter design. It is easy to design an FIR filter with linear phase by making the coefficient sequence symmetric. However, why make the phase linear?

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We aim to make the phase linear because distortion usually happens with non linear phased filters . If you look at phase in the time domain, it is time shift. If we have non linear phase, some parts of the signal are shifter more or less than the others thus causing distortion and an unreliable output. We mostly care about phase in Image processing because all the information is in the phase of the image. You can explore it more by seeing it. Filter using a butter-worth filter and a linear phase fir. You will see that the linear phase fir does a more adequate job at filtering.

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Because linear phase of a filter results in constant group delay according to the relation between group delay $\tau$ and phase response $\phi(\omega)$ of the filter: $$ \tau = - \frac{ d\phi(\omega) }{ d\omega} $$

But then you will ask: why then constant group delay? And the short answer will be that constant group delay will preserve the relative positions (in time or space) of different frequency (narrow band) components of the signal to each other.

Many applications, such as image processing, may require such a fidelity on the processed signals. But not all, and to some degree, non-constant group delay such as provided by IIR filters are well tolerated in applications like audio...

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The other way of putting it is that a non linear phase introduces dispersion.

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