Hi Could someone help or give a hint on how to find the slowest possible rate at which a bandlimited signal x(t) can be sampled and still be able to find the area of that signal? How is it possible to find the area from the samples? Best Regard and thank you very much
Area of Bandlimited Signal
Started by ●October 10, 2004
Reply by ●October 10, 20042004-10-10
lathe_biosas@hotmail.com wrote:> Hi > > Could someone help or give a hint on how to find the slowest possible > rate at which a bandlimited signal x(t) can be sampled and still be > able to find the area of that signal? > > How is it possible to find the area from the samples? > > Best Regard and thank you very muchWhat do you mean by the area of a signal? A bandlimited signal can in theory be reconstructed from samples taken at a little more than twice its highest component frequency. In practice, more than just a smidgeon over is very helpful. Presumably, the reconstructed signal is sufficient for your need. If not, the original isn't either. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●October 11, 20042004-10-11
Jerry Avins <jya@ieee.org> wrote in message news:<PoOdnVJ36rjMSfTcRVn-vw@rcn.net>...> lathe_biosas@hotmail.com wrote: > > Hi > > > > Could someone help or give a hint on how to find the slowest possible > > rate at which a bandlimited signal x(t) can be sampled and still be > > able to find the area of that signal? > > > > How is it possible to find the area from the samples? > > > > Best Regard and thank you very much >see Reimann integration. at any sampling rate, you'll incurr an error in computing the area.
Reply by ●October 11, 20042004-10-11
<lathe_biosas@hotmail.com> wrote in message news:f3fb6877.0410100111.25c9d992@posting.google.com...> Hi > > Could someone help or give a hint on how to find the slowest possible > rate at which a bandlimited signal x(t) can be sampled and still be > able to find the area of that signal? > > How is it possible to find the area from the samples? > > Best Regard and thank you very muchHi Lathe - you should look for papers on Nyquist sampling and also sub-sampling then consider what is meant by band-limited. You will soon find out that this is really fundamental stuff that everyone (at least everyone I know - how sad is that?) is mad keen on but that people tend to over-simplify explanations - particularly if you are browsing the web ; in particular they will keep on saying that you have to sample at at least twice the highest frequency in the signal but they don't carefully define what they mean by the signal. I only have a very vague idea what you mean by finding the area and suspect that you should be looking for the average area in some unit of time. Best of luck - Mike
Reply by ●October 11, 20042004-10-11
lathe_biosas@hotmail.com wrote in message news:<f3fb6877.0410100111.25c9d992@posting.google.com>...> Hi > > Could someone help or give a hint on how to find the slowest possible > rate at which a bandlimited signal x(t) can be sampled and still be > able to find the area of that signal?Check out the Nyquist sampling criterion.> How is it possible to find the area from the samples?Eh... by Parseval's identity? Rune
Reply by ●October 11, 20042004-10-11
Jerry Avins <jya@ieee.org> wrote in message news:<PoOdnVJ36rjMSfTcRVn-vw@rcn.net>...> What do you mean by the area of a signal? A bandlimited signal can in > theory be reconstructed from samples taken at a little more than twice > its highest component frequency. In practice, more than just a smidgeon > over is very helpful. Presumably, the reconstructed signal is sufficient > for your need. If not, the original isn't either.Hi Jerry, Thanks for answering. By area of a signal it's meant: Area=\int_{-\infty}^{\infty} x(t)dt Regards
Reply by ●October 11, 20042004-10-11
"Mike Yarwood" <mpyarwood@btopenworld.com> wrote in message news:> I only have a very vague idea what you mean by finding the area and suspect > that you should be looking for the average area in some unit of time.Hi Mike Thanks for answering. The signal x(t) is a signal with Fourier transform X(f) such that X(f)=0 |f|>=W The signal x(t) is bandlimited to [-W,W] The area is defined as: Area=\int_{-infty}^{infty} x(t)dt What is the slowest possible rate at which we can sample x(t) and still be able to find the Area? How do we find the Area from the samples? Best Regards
Reply by ●October 11, 20042004-10-11
lathe_biosas@hotmail.com wrote:> Jerry Avins <jya@ieee.org> wrote in message news:<PoOdnVJ36rjMSfTcRVn-vw@rcn.net>... > > >>What do you mean by the area of a signal? A bandlimited signal can in >>theory be reconstructed from samples taken at a little more than twice >>its highest component frequency. In practice, more than just a smidgeon >>over is very helpful. Presumably, the reconstructed signal is sufficient >>for your need. If not, the original isn't either. > > > Hi Jerry, > > Thanks for answering. > > By area of a signal it's meant: > > Area=\int_{-\infty}^{\infty} x(t)dt > > RegardsI won't do you homework for you, but you can have these hints: The minimum sample rate for finding the area is is the minimum sample rate to reproduce the signal. You probably don't actually want to to the integration. You probably do want the equivalent of Area=\int_{-\infty}^{\infty} |x(t)|dt. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●October 11, 20042004-10-11
Hello nameless, Since you mention you have a bandlimited signal, then we know that the signal must be sampled at least twice as fast as its highest frequency. You may wish to look up the Nyquist Theorem for more details on this. Now assuming uniform sampling with a sampling period of T. The WKS sampling theory says in order to reconstruct the continuous signal from the samples, we interpolate using a sinc() function. given our sampled signal X[n] = ...,x0,x1,x2,x3,.... thus X(t) = sum over all n X[n]* sinc(pi*(t-nT)/T) where sinc(x) == sin(x)/x Now the integral of X(t) can be expanded as a sum of integrals int (X(t)) dt = sum over all n X[n]* int( sinc(pi*(t-nT)/T)) dt int (sinc(pi*(t-nT)/T)) dt is resolved by x = pi*(t-nT)/T and dx = (pi/T)*(t-nT)dt So our component integral is just int (sinc(x) dx) = (T/pi)*pi = T (The above intergal is not too hard to do, if you know the trick - but it is a standard one in integral tables) Since each integral in the sum has a value of T The total integral is simply T * sum of all X[n] or shown another way Area == T* ( ...+X[0]+X[1]+X[2]+X[3]+X[4]+... ) IHTH, Clay S. Turner p.s. General Comments about this approach, While what I've given you about is the theoretical answer, it is not too practical in that the convergence is very slow. The slow convergence stems from using an interpolant that allows for frequencies right up one half of the sampling rate. If you were to oversample your signal, then you could use a different interpolant with better convergence properties. For example a raised cosine would be better. The same type of mathematical process would be carried out as with the sinc. The difference in an error analysis results from the length of the tails of the interpolant. One thing to remember is with sampled systems, one is usually concerned with sinc() like interpolants. If you look in a standard book on numerical analysis, you will see the classical formulae on numerical integration. These are based on polynomial interpolants. So that is the difference between what you are doing here and what they are doing. The question arises "What kinds of signals do you need to integrate and how accurately do you need the results?" You can get samples from a polynomial process and use a classical numerical integration formula. <lathe_biosas@hotmail.com> wrote in message news:f3fb6877.0410100111.25c9d992@posting.google.com...> Hi > > Could someone help or give a hint on how to find the slowest possible > rate at which a bandlimited signal x(t) can be sampled and still be > able to find the area of that signal? > > How is it possible to find the area from the samples? > > Best Regard and thank you very much
Reply by ●October 11, 20042004-10-11
<lathe_biosas@hotmail.com> wrote in message news:f3fb6877.0410111118.366bfd1@posting.google.com...> "Mike Yarwood" <mpyarwood@btopenworld.com> wrote in message news: > > > I only have a very vague idea what you mean by finding the area andsuspect> > that you should be looking for the average area in some unit of time. > > Hi Mike > > Thanks for answering. > > The signal x(t) is a signal with Fourier transform X(f) such that > X(f)=0 |f|>=W > The signal x(t) is bandlimited to [-W,W] > The area is defined as: > > Area=\int_{-infty}^{infty} x(t)dt > > What is the slowest possible rate at which we can sample x(t) and > still be able to find the Area? > How do we find the Area from the samples? > > Best RegardsHi Lathe , Several other people have responded to "What is the slowest possible rate at which we can sample x(t) and still be able to find the Area?" directly and given good hints as to what to look for, so I'll leave that alone. You can't find the area of your analytic signal by looking at the samples unless you make the assumption that it is also time limited and that you have all of the non-zero sample values. If you don't want to make this assumption you could take the samples that you do have and work out what the average area in some appropriate unit of time is: again methods for reconstructing the signal have been given by several people so you have the information you need. Best of Luck - Mike
