# Impulse Response from Frequency Response at Specific Values

I'm at a loss trying to answer the following question:

The frequency response $H(j\omega)$ of a length 4 FIR filter with real impulse response has the following specific values: $H(j\cdot0) = 2$, $H(j\cdot \pi/2 ) = 7 − 3j$, and $H(j\cdot\pi) = 0$. Determine the impulse response function $h(n)$.

I can't think of anything to do but apply the inverse DTFT but how would that work at specific values? Is there another method I'm forgetting?

## 1 Answer

If you look at the definition of $H(j\omega)$

$$H(j\omega)=\sum_{n=0}^{N-1}h(n)e^{-jn\omega}$$

then you can see that

$$H(0)=\sum_{n=0}^{N-1}h(n)=h(0)+h(1)+h(2)+h(3)$$

and

$$H(j\pi/2)=\sum_{n=0}^{N-1}h(n)e^{-jn\pi/2}=h(0)-jh(1)-h(2)+jh(3)$$

and

$$H(j\pi)=\sum_{n=0}^{N-1}h(n)e^{-jn\pi}=h(0)-h(1)+h(2)-h(3)$$

This gives you 4 equations for the 4 unknown coefficents: $$h(0)+h(1)+h(2)+h(3)=2\\ h(0)-h(2)=7\\ -h(1)+h(3)=-3\\ h(0)-h(1)+h(2)-h(3)=0$$

I guess you can take it from here.