Hi, sure I can provide more detail. a can be any factor, but regarding the implementation I used a=1.

As to what I have tried, I wanted to offrain from using too much detail because it will streamline the discussion into a certain direction, eg using fftw, which implements the discrete fourrier transform and I am not sure about the relationship between discrete fourrier transform and the actual fourrier transform.

I tried the following:

I tried using julia for the computation and doing some plots

```
x = -5:dx:5
p = fftfreq(length(x),2pi/dx)
scatter(p, real.(fft(step.(x))))
scatter!(p, imag.(fft(step.(x))))
```

however the real and imaginary parts dont really seem to match the expected result at all.

However the absolute value only seems to be off by a constant

Also when looking at the formula, the discrete fourrier transform is given by

X_k=\sum_{n=0}^{N-1} x_n \cdot e^{-\frac{i 2 \pi}{N} k n}

while the fourrier transform is given by

\hat{f}(\xi)=\int_{-\infty}^{\infty} f(x) e^{-i 2 \pi \xi x} d x

I am however unsure how they are related.

I do see some relationship between the two, in particular if I would to approximate the integral by a Riemann sum

\int_{-\infty}^{\infty} f(x) \, dx \approx \frac{1}{N}\sum_{N}f(x[N])

this reletionship would suggest divideing by N to obtain the correct fourrier transform but that doesâ€™t lead to a correct result eather, also it doesnâ€™t help with the feact that the real and complex components donâ€™t match.

I also looked at [this post][2] from maths stackexchange, but it did not really prove to be useful, because the answers dont answer my question.