Limitation in both Time Frequency Domains ???

[rat_race]

hi everyone


i am wondering wether there exists signals limited both in time and frequency domains or not. actually i have been told by one of my professors that there is some. but i cant even imagine it ! could some one give me some examples : 3 at least ?

Regards

 

[Mansour_M]

Hi Dear rat_race!

I don't understand the word "Limitations". Please say more details.

Thanks!

 

[rat_race]

i mean any signal that has finite bandwidth in both frequency & time Domains.
imagine a simple Pulse with finite duration & its transform Sinc function, although it is finite in time but not in frequency. i should note that by finite , i mean exactly & strictly finite ! just like the pulse function in time domain. i do not mean effective fininte as may be defined arbitarily. any way this sounds impossible to me in mathematical terms, but the professor said that there exists some !!!

 

[Sal]

Hi

There are plenty of those signals, try looking at the Gaussian pulse exp(-t^2) and its n derivatives.

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Sal

 

[maharshi_qis]

ya..there r some signals limited both in time and freuency domain lik a simple pulse signal and also gate signal...some finite time domain signal may be finite even in freq domain also ..since both are invrese only..right..

 

[jayc]

None of the examples given are valid. A Gaussian is NOT time or band limited since the signal amplitude never reaches 0. And a pulse in time is a constant over all frequency and vice versa. I don't know what a "gate signal" is, but as far as I know, it is mathematically impossible to have a signal that is both time limited and band limited.

From Wikipedia https://en.wikipedia.org/wiki/Bandlimited:

A bandlimited signal cannot be also timelimited. More precisely, a function and its Fourier transform cannot both have finite support. This fact can be proved by using the sampling theorem.

Proof: Assume that a signal which has finite support in both domains exists, and sample it faster than the Nyquist frequency. This finite number of time-domain coefficients should define the entire signal. Equivalently, the entire spectrum of the bandlimited signal should be expressible in terms of the finite number of time-domain coefficients obtained from sampling the signal. Mathematically this is equivalent to requiring that a (trigonometric) polynomial can have infinitely many zeros in bounded intervals since the bandlimited signal must be zero on an interval beyond a critical frequency which has infinitely many points. However, it is well-known that polynomials do not have more zeros than their orders due to the fundamental theorem of algebra. This contradiction shows that our original assumption that a time-limited and bandlimited signal exists is incorrect.