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.

  1. I calculate the frequency response function by calculating the Fourier-Transformation $h=\mathcal F(g)$.
  2. 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)
  3. I do a inverse Fourier-Transformation $g_{inv}=\mathcal F^{-1}(h_{inv })$
  4. I plot $g$ (blue) and the calulated inverse $g_{inv}$ (red)

g and g_inv

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)];


hold on

1 Answer 1


Your result is correct, even if it seems a bit counterintuitive. Note that you don't compute the Fourier transform but the discrete Fourier transform (DFT). The system you want to invert has the impulse response


Obviously, the inverse system must advance the incoming signal by $19$ samples:


Clearly, that system is non-causal. Taking into account the implicit periodicity of the DFT (and IDFT) with length $N=100$, the signal $g_{inv}[n]$ cannot be distinguished from a signal $g_{k,inv}[n]=g_{inv}[n+kN]$, $k\in\mathbb{Z}$. Your result is just $g_{-1,inv}[n]$:



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