# Mathematical terms for “bandlimited” and “timelimited”?

A signal $$x(t)$$ is time limited if there exists a $$T>0$$ such that $$x(t)=0\qquad,\qquad |t|>T$$so we can define band limited (in frequency) signals where their FT is zero for $$\omega>\omega_0$$ and for some $$\omega_0$$. Also a signal limited on one domain (time or frequency) cannot be limited in the other domain either, but there are signals neither limited in time nor in frequency for example consider $$x(t)=e^{-|t|}$$ with corresponding FT $$X(\omega)=\dfrac{2}{1+\omega^2}$$. Neither $$x(t)$$ nor $$X(\omega)$$ are limited in time and frequency respectively. To show that, let $$x_T(t)$$ be a time limited version of signal $$x(t)$$ in $$|t| therefore$$x_T(t)=x(t)\Pi(\dfrac{t}{2T})$$by taking FT we have$$X_T(\omega)=X(\omega)*2Tsinc(\dfrac{T\omega}{\pi})=$$regardless of $$X(\omega)$$ being band limited or not, $$X_T(\omega)$$ is never band limited because of the convolution of $$X(\omega)$$ with $$sinc$$ function and this is what we wanted to show. The same argument can be used in dual case for band limited signals.
• I don't understand the point of the computations following "To show that...". Are you trying to prove that a time-limited signal cannot be band-limited? If so, then I fail to understand why "$X_T$ is never band-limited because of its convolution with $\text{sinc}$". – Giuseppe Negro Jan 14 at 18:16
• Yes that's right. A signal can't be limited in both time and frequency range. By "$X_T$ is never band-limited because of its convolution with sinc" I mean since sinc is spread throughout the frequency range, then even if $X(\omega)$ is limited, the convolution takes it to presence in higher frequencies too..... – Mostafa Ayaz Jan 14 at 18:22