# Combination of type I FIR linear-phase systems

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 $$M \not = N$$.

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.