I know that we can design a linear phase FIR filter by choosing filter coefficients symmetric or antisymmetric, what is the intuitive idea behind that?


The reason is Euler's formula, from which you get




If you have symmetric or anti-symmetric coefficients, the corresponding frequency response can always be decomposed in purely real-valued cosine terms $(1)$ or purely imaginary sine components $(2)$, plus a linear phase term if the filter is not centered around $n=0$.

Take as a simple example a filter of length $N=5$ with filter coefficients

$$\mathbf{h}=\big[1, 2, 3, 2, 1]\tag{3}$$

starting at index $n=0$. The corresponding frequency response is


Since the last term in brackets is purely real-valued, $H(e^{j\omega})$ has a linear phase response $\phi(\omega)=-2\omega$.

For even-length filters you get an additional delay of half a sample, and for anti-symmetric filters there's an additional phase shift of $\pi/2$ due to $j$ on the left side of $(2)$.

Also take a look at this answer discussing the four types of linear phase FIR filters.

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  • $\begingroup$ Thank you very much. $\endgroup$ – Ece Su Ildiz Jun 10 at 10:55

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