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Topic: signal in time-frequency domain (Read 6074 times) previous topic - next topic
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signal in time-frequency domain

1.If signal in time domain is infinite ,it must be finite in frequency domain, vice versa.

2.If signal in time domain is finite, it must be infinite in frequency domain,vice versa.

How to prove them? Which articles/textbooks implemented that?

Do not just give a example to deduct them,I require a proof of theory.

thank you in advance.

cheers

signal in time-frequency domain

Reply #2 –
Quote from: lvqcl on
Quote from: hyeewang on
1.If signal in time domain is infinite ,it must be finite in frequency domain, vice versa.

IIRC no.

Quote from: hyeewang on
How to prove them? Which articles/textbooks implemented that?

Do not just give a example to deduct them,I require a proof of theory.

  ?


The Foiurier transform is used in signal processing, usually digital signal processing, only to approximate when the limits of integration are very large but in reality still finite. They are large enough that people get away with the definite integral that can be calculated exactly that has infinite limits of integration. Usually, such integration is done in the complex plane with a semicircle, the straight part on the real axis and the arc from the left side or lower limit on the real axis to the right side which is the upper limit on the real axis. The integral becomae a line integral around a closed semicircular path and the Cauchy Goursant (I am not certain of the spelling of his name) Theorem uses residues inside the semicircle and the radius of the semicircle approaches infinity as the integral converges to a finite limit.

Then this finite limit which is the exact value of the integral is very close to the value of an integral with large enough values of its limits that it can be used in the time domain.
Dr Barney

signal in time-frequency domain

Reply #3 –
Yes. I mean "extent",not "value".

1.If the extent of signal in time domain is infinite ,it must be finite in frequency
domain, vice versa.
2.If the extent of signal in time domain is infinite, it must be finite in frequency
domain,vice versa.
3. No matter periodic or non-periodic and continuous or discrete time,the two statements above must keep.

How to prove them? Which articles/textbooks implemented that?
Do not just give a example to deduct them,I require a proof of theory.

signal in time-frequency domain

Reply #4 –
Are you talking about continuous-time or discrete-time domain?
There are some ideal analogue signals like step functions or Dirac delta function which have infinite amount of spectral components no matter if the signal is finite or infinite in the time domain. I think one example could be a 1Hz square wave extending from zero to +- infinity (infinite in both the time and the frequency domain which violates your rule no. 1).

signal in time-frequency domain

Reply #5 –
Quote from: hyeewang on
1.If signal in time domain is infinite ,it must be finite in frequency domain, vice versa.

This is not correct. Take a step function with an infinite support. Its spectrum is infinite too.