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I'm implementing a Python script that calculates the response of a system given an input. The system is $y(t) = x(t-2)$, i.e. it delays the signal by $2$, and the signal is $x(t) = \sin(3t)u(t)$.

As I'm learning the discrete Fourier transform (and it's inverse), I'm doing all the calculation in frequency domain. The procedure is very simple:

  1. Sample $x(t)$ between $[-10, 10]$ where the interval between each sample is $0.01$
  2. Calculate the DFT of $x(t)$ sampled (the result is $X(j\omega)$)
  3. Multiply the $X(j\omega)$ by $e^{-j2\omega}$ to shift the signal (the result is $Y(j\omega)$)
  4. Calculate the inverse DFT to obtain $y(t)$ sampled

In the end, the result is plotted and it's straightforward to check its correctness for $t > 0$. However, for $t < 0$, the output has some kind of strange behavior, as you can see below:

input/output

zoom at the beginning

Am I missing something? Can someone please explain me what causes this?

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That's not strange – that's expected!

The DFT is a circular operation – by shifting to the right, your signal "wrapped" around.

What you need to do is called fast convolution based on the DFT/FFT. The Wikipedia has a nice article on the Overlap-Add and the Overlap-Save methods, so I'd recommend starting with these, and then going for the literature that suits your understanding best.

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