>If you try to take the Fourier transform integral of a exp(j*omega_0*t), it will not >converge in the sense, how an improper integral's convergence is usually understood. >You will need to employ something like Cauchy principal value or Cesaro convergence >to make it converge to zero at omega!=omega_0. At omega=omega_0 the integral >diverges no matter in which sense you take it. So, strictly speaking, Fourier transform >of a sine doesn't exist.
"Usually understood" by who? If you are interested in mathematical rigor, you define the Fourier transform in terms of distributions, and it has no trouble at all handling sinusoids. This is the Schwarz stuff that Sampo mentioned. Another option is to use non-standard analysis and work in terms of hyperreal numbers. Then you can have your Dirac delta functions and your rigor! But both of these are overkill for day-to-day engineering practice. E On Mon, Jun 8, 2015 at 1:50 AM, Vadim Zavalishin < vadim.zavalis...@native-instruments.de> wrote: > If you try to take the Fourier transform integral of a exp(j*omega_0*t), > it will not converge in the sense, how an improper integral's convergence > is usually understood. You will need to employ something like Cauchy > principal value or Cesaro convergence to make it converge to zero at > omega!=omega_0. At omega=omega_0 the integral diverges no matter in which > sense you take it. So, strictly speaking, Fourier transform of a sine > doesn't exist. > > An equivalent look to this from the inverse transform's side is that the > spectrum of the sine is a Dirac delta function, which is not a function in > the normal sense. > > So, none of the statements of the Fourier transform theory (including the > sampling theorem, which assumes the existence of the Fourier transform), > taken rigorously, seem to apply to the sinusoidal signals. > > Regards, > Vadim > > > On 08-Jun-15 10:35, Victor Lazzarini wrote: > >> Not sure I understand this sentence. As far as I know the FT is defined >> as an integral between -inf and +inf, so I am not quite >> sure how it cannot capture infinite-lenght sinusoidal signals. Maybe you >> meant something else? (I am not being difficult, just >> trying to understand what you are trying to say). >> ======================== >> Dr Victor Lazzarini >> Dean of Arts, Celtic Studies and Philosophy, >> Maynooth University, >> Maynooth, Co Kildare, Ireland >> Tel: 00 353 7086936 >> Fax: 00 353 1 7086952 >> >> On 8 Jun 2015, at 08:19, vadim.zavalishin < >>> vadim.zavalis...@native-instruments.de> wrote: >>> >>> It might seem that such signals are unimportant, however even the >>> infinite sinusoidal signals, including DC, cannot be treated by the >>> sampling theorem, since the Dirac delta (which is considered as their >>> Fourier transform) is not a function in a normal sense and strictly >>> speaking Fourier transform doesn't exist for these signals. >>> >> >> -- >> dupswapdrop -- the music-dsp mailing list and website: >> subscription info, FAQ, source code archive, list archive, book reviews, >> dsp links >> http://music.columbia.edu/cmc/music-dsp >> http://music.columbia.edu/mailman/listinfo/music-dsp >> >> > > -- > Vadim Zavalishin > Reaktor Application Architect | R&D > Native Instruments GmbH > +49-30-611035-0 > > www.native-instruments.com > -- > dupswapdrop -- the music-dsp mailing list and website: > subscription info, FAQ, source code archive, list archive, book reviews, > dsp links > http://music.columbia.edu/cmc/music-dsp > http://music.columbia.edu/mailman/listinfo/music-dsp > -- dupswapdrop -- the music-dsp mailing list and website: subscription info, FAQ, source code archive, list archive, book reviews, dsp links http://music.columbia.edu/cmc/music-dsp http://music.columbia.edu/mailman/listinfo/music-dsp
