0
$\begingroup$

I have a function $ x(t) $ and I denoted its Fourier transformation as $ X(f) $. I want to get the Fourier transformation of $ x(t)\mathrm e^{2\mathrm i \pi f_0 t} $, and I know the result is $ X(f-f_0) $.

I tried to obtain this result by using Mathematica, but didn't obtain X[f-f0] as output:-

X[f] := FullSimplify@FourierTransform[x[t], t, f, FourierParameters -> {0, -2 Pi}]
FullSimplify@FourierTransform[x[t]*Exp[2 I \[Pi] f0 t], t, f, FourierParameters -> {0, -2 Pi}]

How can I obtain X[f-f0] as output?

Thanks!

$\endgroup$
  • 1
    $\begingroup$ FourierTransform just can't do that, but we can write a "shell" for it. Strongly related: mathematica.stackexchange.com/a/71393/1871 $\endgroup$ – xzczd Oct 9 '18 at 16:02
  • $\begingroup$ Thanks for your help. The shell helps a lot! I applied convolution properties etc. Just 1 issue, how can I obtain A[w-w0] instead of FourierTransform[a[t], t, w - w0, FourierParameters -> {0, -2 \[Pi]}] as output? $\endgroup$ – H42 Oct 9 '18 at 17:50
  • $\begingroup$ Try something like this: ft[x[t] Exp[2 I Pi f0 t], t, s] /. HoldPattern@FourierTransform[f_[x_], _, s_] :> Symbol[ToUpperCase@ToString@f][s]. BTW, personally I recommend not to rewrite a as A i.e. ft[x[t] Exp[2 I Pi f0 t], t, s] /. HoldPattern@FourierTransform[f_[x_], _, s_] :> f[s], just keep in mind now x actually represents $X$, this usually makes programming easier. $\endgroup$ – xzczd Oct 10 '18 at 7:38

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.