# For Fourier Transform with $x(t)$, how to obtain result in terms of $X(f)$?

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!

• FourierTransform just can't do that, but we can write a "shell" for it. Strongly related: mathematica.stackexchange.com/a/71393/1871 – xzczd Oct 9 '18 at 16:02
• 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? – H42 Oct 9 '18 at 17:50
• 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. – xzczd Oct 10 '18 at 7:38