Let $h_1[n]$ and $h_2[n]$ be impulse responses of type I FIR linear-phase systems with lengths $M$ and $N$. What can be said about the phase response of the following filters?

  1. $h_3[n] = h_1[n] + h_2[n]$

  2. $h_3[n] = h_1[n]h_2[n]$

  3. $h_3[n] = h_1[n] \star h_2[n]$

My try: We can easily show that $$H_1(e^{j\omega}) = e^{-j\omega M /2}(\sum_{k=0}^{M/2}a[k]\cos(k\omega)) , \ a[0] = h_1[M/2], a[k] = 2h_1[(M/2)-k] \ \\ k=1,\dots ,M/2$$ and similarly $$H_2(e^{j\omega}) = e^{-j\omega N /2}(\sum_{k=0}^{N/2}b[k]\cos(k\omega)) , \ b[0] = h_2[N/2], b[k] = 2h_2[(N/2)-k] \\ k=1,\dots ,N/2$$ So we can write $$H_1(e^{j\omega}) = e^{-j\omega M /2}A(\omega) \\ H_2(e^{j\omega}) = e^{-j\omega N /2}B(\omega)$$ where $A(\omega)$ and $B(\omega)$ are real-valued periodic functions. Since in the third case we have $$H_3(e^{j\omega}) = H_1(e^{j\omega})H_2(e^{j\omega}) = e^{-j\omega(M/2 + N/2)}A(\omega)B(\omega)$$ the phase is linear in this case but I don't know how to solve the other cases. Certainly the phase is linear in the first case if $M=N$ but what about $M\not = N$ and also the second case?

Edit: In the first case for $M\not = N$, we can choose $a[k]$ and $b[k]$ such that the $h_3[n]$ be of type I or II FIR system. For example let $$a[0] = b[0] = 1, a[k] = b[k] = 0 \ \ \ k \not = 0 \\ H_3(e^{j\omega}) = e^{-j\omega M/2} + e^{-j\omega N/2} = e^{-j\omega (N+M)/4}(e^{j\omega (M-N)/4} + e^{-j\omega (M-N)/4}) = \\ 2e^{-j\omega (N+M)/4}\cos((M-N)\omega/4)$$ If $M=6$ and $N=2$, we get type I system and if $M=4$ and $N=2$ the system is of type II. Certainly there are examples such that this factorization isn't possible. So in this case, what's the required condition for $h_1[n]$ and $h_2[n]$ to $h_3[n]$ be of linear phase?

For the second case, the analysis in the frequency domain doesn't seem possible. When $M=N$, it's easy to show $h_3[n]$ is of type I because of the symmetry around $n = M/2 = N/2$. I couldn't find an example such that $h_3[n] = h_1[n]h_2[n]$ be (anti)symmetric when $M \not = N$.


1 Answer 1



Think of the required symmetry of the impulse response of a type-I linear-phase FIR filter. Can this symmetry condition be met in cases $1$ and $2$ if $M\neq N$? What about $M=N$?

You've correctly answered case $3$. The delays just add up, and the phase of the combined system is still linear.

  • $\begingroup$ Thanks. Please see the edited question. $\endgroup$
    – S.H.W
    Nov 13, 2021 at 7:42

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.