I want to calculate the inverse impulse response of a LTI system in the frequency domain. I generate a simple impulse response $g$. For this I generate a vector of 100 zeros. I set the value of the 20th sample to 0.5.
- I calculate the frequency response function by calculating the Fourier-Transformation $h=\mathcal F(g)$.
- I invert every single frequency bin in $h$ where $h_{inv}[k]=\frac{h[k]^*}{|h[k]|^2}$ (This is the formula for the inverse of a complex number)
- I do a inverse Fourier-Transformation $g_{inv}=\mathcal F^{-1}(h_{inv })$
- I plot $g$ (blue) and the calulated inverse $g_{inv}$ (red)
Question: Why is the impulse of $g_{inv}$ at 82 samples? Is $g_{inv}$ my inverse impulse response?
Matlab code:
g=[zeros(1,19) 0.5 zeros(1,80)];
h=fft(g);
h_inv=conj(h)./(abs(h).^2);
g_inv=ifft(h_inv);
figure
plot(g)
hold on
plot(g_inv)