# How can one infer the input signal $x[n]$ from the output $y[n]$ of an LTI system with known Gain and Phase Response

I have the gain response of an amplifier and its phase response curves, in an appropriate frequency range. I also have a set of output (from the system) discrete data $$y[n]$$. How would one go about inferring the sequence $$x[n]$$ in the input? I can safely assume that the system is a so-called minimum phase system; so it is both causal and stable.

• This is a straight-forward problem because you know the full transfer function (magnitude and phase) and it's invertable if your LTI system is minimum phase. You will take your transfer function, compute the reciprocal of it (the inverse system) and then inverse Fourier transform that. Then you know what the inverse system is. Input $y[n]$ to this inverse system and out will come $x[n]$. Jan 2 at 22:15

I can safely assume that the system is a so-called minimum phase system

That is good because a the inverse of a causal minimum phase system is also causal and minimum phase.

For an LTI system input and output are related as

$$y[n] = x[n]*h[n]$$

where $$h[n]$$ is the impulse response of the system and $$*$$ the convolution operator. In the frequency domain this simply becomes

$$Y[k] = X[k] \cdot H[k]$$

where $$X[k]$$ is the discrete Fourier Transform (DFT) of $$x[n]$$ etc.

To recover the input from the output we need to inverse the system, i.e. $$X[k] = \frac{Y[k]}{H[k]} = Y[k] \cdot G[k]$$

So the order of operation here

1. Calculate the inverse transfer function $$G[k] = 1/H[k]$$
2. Calculate the inverse impulse response, $$g[n]$$ by doing an inverse DFT
3. Convolve the output with that impulse response $$\tilde{x}[n] = y[n]*g[n]$$

As with most "simple" things, the devil is often in the details but that's a good starting point.