1.If signal in time domain is infinite ,it must be finite in frequency domain, vice versa. 2.If signal in time domain is infinite, 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. cheers
signal in time-frequency domain
Started by ●January 18, 2010
Reply by ●January 18, 20102010-01-18
>1.If signal in time domain is infinite ,it must be finite in frequency >domain, vice versa. > >2.If signal in time domain is infinite, 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. > >cheers >1.If signal in time domain is infinite ,it must be finite in frequency domain, vice versa. 2.If signal in time domain is infinite, it must be finite 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. cheers
Reply by ●January 18, 20102010-01-18
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
Reply by ●January 18, 20102010-01-18
On Jan 18, 2:36�am, "hyeewang" <hyeew...@yahoo.com.cn> wrote:> 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. > > cheersYou have to be more precise in your conjectures. 1. Your use of "infinite" and "finite" imply values. However, I think you mean extent. 2. You don't make a distinction between continuous time and discrete time. 3. You don't make a distinction between periodic and nonperiodic. Consider the function exp(-t^2), -inf < t < inf. it is infinite in extent and has an infinite bandwidth. Hope this helps. Greg
Reply by ●January 18, 20102010-01-18
"hyeewang wrote:> 1.If signal in time domain is infinite ,it must be finite in frequency > domain, vice versa.As Greg already pointed out, this statement (assuming by "is (in) finite" you mean "has (in)finite extent") is not correct.> 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?There is a nice discussion for continuous-time signals by Slepian on this topic: http://www.sm.luth.se/csee/courses/sms/022/2001/artiklar/Sle76e.pdf For discrete-time signals you can apply an argument very similar to what Robert Adams described in his interpolation article [1]. Regards, Andor [1] R. W. Adams: Nonuniform Sampling of Audio Signals, J. Audio Eng. Soc., Vol. 40,No. 11, p. 886-894, November 1992.
Reply by ●January 18, 20102010-01-18
On Mon, 18 Jan 2010 01:34:13 -0600, hyeewang wrote:>>1.If signal in time domain is infinite ,it must be finite in frequency >>domain, vice versa. >> >>2.If signal in time domain is infinite, 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. >> >>cheers >> >> > 1.If signal in time domain is infinite ,it must be finite in frequency > domain, vice versa. > 2.If signal in time domain is infinite, it must be finite 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. cheersEasy. Take the Fourier transform of x(t) = e^(-at), which is infinite in time. Note that it is (give or take a few sign changes) 1/(jw + a), which is infinite in frequency. Oops. -- www.wescottdesign.com
Reply by ●January 18, 20102010-01-18
On Mon, 18 Jan 2010 01:26:07 -0600, hyeewang wrote:> 1.If signal in time domain is infinite ,it must be finite in frequency > domain, vice versa. > > 2.If signal in time domain is infinite, 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. > > cheersTry: 1: If a signal is of finite extent in the time domain, then it must be of infinite extent in the frequency domain. 2: If a signal is of finite extent in the frequency domain, etc. See if that lasts a bit longer in the "a single case disproven disproves the whole" mill. -- www.wescottdesign.com
Reply by ●January 19, 20102010-01-19
>On Jan 18, 2:36=A0am, "hyeewang" <hyeew...@yahoo.com.cn> wrote: >> 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 > >You have to be more precise in your conjectures. > >1. Your use of "infinite" and "finite" imply values. However, > I think you mean extent. >2. You don't make a distinction between continuous time > and discrete time. >3. You don't make a distinction between periodic and > nonperiodic. > >Consider the function exp(-t^2), -inf < t < inf. it is infinite >in extent and has an infinite bandwidth. > >Hope this helps. > >Greg >thank you! Grey. 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.
Reply by ●January 19, 20102010-01-19
>>On Jan 18, 2:36=A0am, "hyeewang" <hyeew...@yahoo.com.cn> wrote: >>> 1.If signal in time domain is infinite ,it must be finite infrequency>>> domain, vice versa. >>> >>> 2.If signal in time domain is finite, it must be infinite infrequency>>> 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 oftheory.>>> >>> thank you in advance. >>> >>> cheers >> >>You have to be more precise in your conjectures. >> >>1. Your use of "infinite" and "finite" imply values. However, >> I think you mean extent. >>2. You don't make a distinction between continuous time >> and discrete time. >>3. You don't make a distinction between periodic and >> nonperiodic. >> >>Consider the function exp(-t^2), -inf < t < inf. it is infinite >>in extent and has an infinite bandwidth. >> >>Hope this helps. >> >>Greg >> > >thank you! Grey. >Yes. I mean "extent",not "value". > >1.If the extent of signal in time domain is infinite ,it must be finitein>frequency >domain, vice versa. >2.If the extent of signal in time domain is infinite, it must be finitein>frequency >domain,vice versa. >3. No matter periodic or non-periodic and continuous or discretetime,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. > >The property that you mention derives directly from the Time-Scaling property of the Fourier Transform: time comprension/expansion of a signal results in its spectral expansion/comprension. This implies that the bandwidth of a signal is inversely proportional to the time signal duratuion. This is also a direct consequence of the Heisenberg uncertainty principle (see http://www.ams.org/featurecolumn/archive/uncertainty.html). Hope that this helps
Reply by ●January 19, 20102010-01-19
On Jan 19, 12:05�am, "hyeewang" <hyeew...@yahoo.com.cn> wrote:> > 1.If the extent of signal in time domain is infinite ,it must be finite in > frequency > domain,......> 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 statement for which you are requiring a proof "if the extent of signal in time domain is infinite, it must be finite in frequency domain" is false, as Tim Westcott's example shows:>Take the Fourier transform of x(t) = e^(-at), which is infinite in >time. Note that it is (give or take a few sign changes) 1/(jw + a), >which is infinite in frequency....but maybe this is one of those examples that you reject? I wish you all the best of luck in finding a proof of your statement. --Dilip Sarwate
