# Linearity with Difference Period in Fourier Transform

I know that a system is linear if it satisfies

$$\mathscr{F}\{ a\,x(t)+b\,y(t) \} = a\,X(\omega)+b\,Y(\omega)$$

for Fourier transform, $$X(\omega)\triangleq\mathscr{F}\{x(t)\}$$

But what if $$x(t)$$ and $$y(t)$$ don't have the same period?

Can we use same linearity condition in Fourier transform?

• Hi MertEge. You have an answer? Are you waiting for another one to show up? You can leave a reply by upvoting if found useful, or accepting if it gave you te asnwer. Or ask for further clarification if posisble. – Fat32 Dec 3 '18 at 15:47

## 1 Answer

Continuous time Fourier transform does not require the signals to be periodic, and therefore the linearity assumption holds. CTFT is a linear operator.

Furthermore, if you mean CT Fourier series, which require that both signals to be periodic by (a common period) $$T_0$$, that's also linear.

But if input signals have no common periods, then you cannot apply the linearity test for CTFS.