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I have read

"Signals that are bandlimited are not timelimited" and the reverse; "Signals that are timelimited are not bandlimited".

Q1: Is this because of the Fourier transform?

Q2: What are the mathematical terms for "bandlimited" and "timelimited"?

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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 $|t|<T$ 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.

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  • $\begingroup$ 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 $\text{sinc}$". $\endgroup$ – Giuseppe Negro Jan 14 at 18:16
  • $\begingroup$ 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..... $\endgroup$ – Mostafa Ayaz Jan 14 at 18:22

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