# 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?

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.