# Signal Processing

1. Sep 26, 2008

### Tac-Tics

One subject that I've always been interested in, but have never found the proper introduction to is signal processing. I thought PF might be an appropriate place to ask (tough, I'm not even sure which forum it would be most appropriate, but since I'm stronger in math than physics, I'm posting here).

This is sort of an open-ended inquery, but I'll list the kinds of problems I'd like to learn how to solve:

1) Given a continuous input signal, you can sample the signal at regular intervals. Given that your sampling rate is sufficiently high (something about the nyquist frequency), you can reconstruct the continuous signal from the samples. How is this done?

2) Given a recording of a violin or piano, how can you analyze what note is being played? I know the fourier series is used for when you know the period of your signal, but with sampled data coming from a microphone, the signal is not guaranteed periodic. I believe this has to do with the fourier transformation, but no source I've found has a clear definition of how it works or what it represents other than it is an extrapolation of the fourier series for aperiodic signals.

3) In music, the pitch of a note generally correlates with the frequency of a signal. If the frequency distribution of a signal is constant, how do you reconcile this with the fact that the pitches (and thus, frequencies) in the music vary with time?

4) I found a listing of some standard functions with their fourier transformations listed in a table. Many of them included a delta (which as far as I understand is being used as a Dirac delta function). The Dirac delta function, is of course, not really a function since it diverges to infinity, (but is still useful because it gives you the correct answer). Is it typical that the fourier transformations of signals diverge like this?

You can see I have some rough notions of the subject already, but I want to know good resources for fitting all the pieces together. Any help would be very much appreciated.

2. Sep 26, 2008

This thread should almost certainly be in the Electrical Engineering forum.

A lowpass filter. In a nutshell, the sampling process creates "copies" of the signal at different frequencies (multiples of the sampling rate, specifically). Provided the sampling rate is high enough, these copies will not overlap (in the frequency domain) with the original component. So, to recover the original component, you use a lowpass filter to eliminate all the copies.

There are lots of way to go about this. One method is to take the (short time) Fourier transform, and look for strong components. Another way is to use time-domain correlation methods to determine the pitch period. In all cases, there is a trade-off between the accuracy of the frequency estimation, and the time resolution of the system.

You need to use the short-time Fourier Transform. I.e., you break the signal apart into little chunks called "frames" and take the FFT of each one. If you use shorter frames, you get better time resolution, but worse frequency resolution. If you use longer frames, the frequency resolution improves, but the time resolution goes down. In the limit, as the frame becomes very long, you have no time resolution at all, and so don't know when the note in question was played.

Well, I'd say that it's common for people in signal processing not to bother making this rigorous. But there are a few common functions with Dirac delta divergences; the constant function, for example.

3. Sep 26, 2008

### Tac-Tics

Thanks for the quick response!

Yeah, that might have been a better place. I'll leave it to an admin to move it if it seems more appropriate there. Maybe I should have titled the thread fourier analysis or something instead =-)

I'll look over your answers and try to make sense of them. I'd also like to know of any good books or tutorials on the subject if you know any.

4. Sep 26, 2008

Well, the Wikipedia pages on sampling are a pretty decent introduction to that topic. Pitch tracking and joint time/frequency analysis are more advanced, and consequently I don't know of a good intro-level resource on those...

5. Dec 25, 2008

### Petre

Re: Signal Processing/Nyquist?

Dear All,
I think that the papers below will help you to clarify the subject of Nyquist (?) frequency, sampling and processing.

ET 4 CO 198.pmd
www.ieindia.org/pdf/88/88ET104.pdf

9 CP PE 8.pmd
www.ieindia.org/pdf/89/89CP109.pdf

Revaluation and replacement of basic terms in the sampling theory
www.pueron.org/pueron/nauchnakritika/Th_Re.pdf