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I have an exam in the following days and i have no clue of what is the response to this question in the exam:

given this signal:

$$g(t) = H\left(t+\frac{T}{3}\right) f(t) - H\left(t-\frac{2T}{3}\right)f(t)$$ $H(t)=$ Heaviside step function

$f(t) = \cos(\omega_1 t)$

Define a value of $T$ for which the signal $g(t)$ can be considered band limited.

This question doesn't make much sense to me. This is a windowed cosine function and it's bandwith depends on the value of $T$, it's always limited! Unless $T$ approches 0 and it can be seen as an impulse. Am I missing something?

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  • $\begingroup$ They're talking about band-limited, not time-limited. This is about the spectrum of $g(t)$. $\endgroup$ – Matt L. Nov 5 at 12:51
  • $\begingroup$ Sure, i know that it's about the fourier spectrum, my problem is that g(w) (the transform of g(t)) it's always band limited, being it's fourier transform a simple translated sinc, which is always limited $\endgroup$ – Dario Formi Nov 5 at 12:55
  • $\begingroup$ Why is a sinc function in the frequency domain band-limited? $\endgroup$ – Matt L. Nov 5 at 13:11
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    $\begingroup$ I don't think the question can be answered without a proper definition of "can be considered band limited". What's that supposed to mean ? $\endgroup$ – Hilmar Nov 5 at 13:19
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    $\begingroup$ My answer would be $T=0$, which makes $g(t)=0$, which is bandlimited. $\endgroup$ – MBaz Nov 5 at 16:07
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T=0 or T=infinity.

Any non-zero signal with finite length support in one domain has infinite support in the other domain.

cos(wt) is only theoretically band-limited if infinite in length.

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