>On Dec 13, 7:42 am, Jerry Avins <j...@ieee.org> wrote: >> Anatol wrote: >> >> "Anatol" <uana...@yahoo.com> writes: >> >> >>> Hello, >> >>> Could someone explain me please how the sampling theorem >> >>> formula for narrow band signals is obtained. >> >>> In the literature and on the web one can find a good >> >>> explanation of the sampling theorem of band limited signals, >> >>> fs >=3D 2fmax. >> >>> The explanation of the formula for narrow band signals >> >>> fs >=3D 2fmax/k >> >>> is not so intuitive and not very clear. >> >>> Could you give me a link to the sampling theorem >> >>> for narrow band signals or some hints about how >> >>> this formula was obtained or how it can be justified. >> >>> Thank you, >> >>> Anatol >> >> Hi Anatol, >> >> >> If you can get to a library and find the book Signals and Systems >> >> [signalsandsystems], you will find a good model of the samplingprocess=> >> >> there that can be used to understand any type of sampling, whether >> >> bandpass, narrowband, or otherwise. In order to understand it youwill>> >> first need to know about the following >> >> >> 1. convolution >> >> 2. the Dirac delta function and its sifting property >> >> 3. the Fourier transform and some of its properties >> >> >> Good luck. >> >> >> --Randy >> >> >> @BOOK{signalsandsystems, >> >> title =3D "{Signals and Systems}", >> >> author =3D "{Alan~V.~Oppenheim, Alan~S.~Willsky, withIan~T.~Young}",>> >> publisher =3D "Prentice Hall", >> >> year =3D "1983"} >> >> >> -- >> >> % Randy Yates % "Bird, on the wing, >> >> %% Fuquay-Varina, NC % goes floating by >> >> %%% 919-577-9882 % but there's a teardrop in his >> > eye..." >> >> %%%% <ya...@ieee.org> % 'One Summer Dream', *Face TheMusic*,=> >> > ELO >> >>http://www.digitalsignallabs.com >> >> > Thank you Randy, >> >> > Yes, the convolution product of the Fourier transform >> > of the original signal with the shifted Dirac function >> > explaines very well the formula Fs > 2Fmax, but is does >> > not explain for me the formula Fs > 2Fmax/K. >> > I believe there must be a short and clear idea >> > encoded in this formula. The reasoning about the >> > greatest integer K that is smaller then Fmax/Fband >> > is not sufficient for me. >> >> Try it without any math at all. The sampling process creates images. >> Baseband signals have images that begin above Fs/2. Any part of that >> image spectrum that extends below Fs/2 is an alias. When the initial >> spectrum is such that the images created by sampling don't overlap, >> there is no alias. Plot the locations and extents of the images in the >> sampling paradigm that puzzles you and you will understand. >> >> Jerry >> -- >> Engineering is the art of making what you want from things you canget.>>=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=>=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF= >=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF > >You are thinking of this... > >Bounds on the sampling frequency fs > >2fh/k <=3Dfs<=3D 2fl/(k-1) for k=3D1,2...N > >Here k can range from 1 to N with k=3DN yielding the smalest value >fs=3D2fh/N and k=3D1 corresponding to teh Nyquist rate fs>=3D2fh. > >fl and fh here being the lowest and highest frequencies of the signal >you are sampling. > > > >HardyHi Hardy, Can you tell me please what is N in these relations ? Is it bounded ? Anatol