# Sampling above the Nyquist rate but

1. Jan 30, 2008

### brendan_foo

http://s188.photobucket.com/albums/z65/liznbren/stuff/?action=view&current=envelope.jpg

If I sample at just fractionally above the Nyquist rate, and I obtain a sequence as is shown in the linked image, is it reasonable to expect perfect interpolation (say with a Sinc function ideally) as it is taken over infinite time.

I am using a real-time DSP chip and I get a modulated output, as as evident in the image linked.

Any comments on this sort of sequence would be great. It has a discrete frequency of about 0.9 pi.

Thanks...
Brendan

2. Jan 30, 2008

### rbj

the answer is: theoretically yes. that sinusoid shown in your referred figure is (barely) sufficiently sampled. any time a continuous-time signal is sufficiently sampled (that is sampled at a rate that exceeds twice the highest frequency component of the signal), the original continuous-time signal can be reconstructed from the samples by use of the sinc( ) function as

$$x(t) = \sum_{n=-\infty}^{+\infty} x[n] \ \mathrm{sinc}\left( \frac{t-nT}{T} \right)$$

where

$$\mathrm{sinc}(u) \equiv \frac{\sin(\pi u)}{\pi u}$$

and

$$F_s = \frac{1}{T}$$ is your sampling frequency.

now the problem you have with your real-time DSP chip is that no matter how fast it is (yet finite in speed), you can't add up an infinite number of terms. so your sinc() function will get truncated somehow. simply cutting it off is applying the rectangular window which is pretty bad, so you will likely use a different window to make the reach of your sinc() function finite. in audio, i have been pretty much satisified with a reach of +/- 16 samples from the interpolated point that you're calculating. but i know there are sample rate conversion chips that have a reach of +/- 32 samples (a summation of 64 terms).

this is a good forum and i wouldn't mind seeing more DSP discussion here, but the real action is at the comp.dsp USENET group. that's where i would take it.

3. Jan 30, 2008

### brendan_foo

Thanks! I guessed this was so, I tried using MATLAB to create such a convolution of sinc( ) functions and as the length got arbitrarily large the results were convergent to the theory.

The practicalities of actual implemented DSP are very interesting indeed it seems.

All of this is for Direct-Digital-Synthesis by means of a wave-lookup table. It serves me well to look at it as the dual of sampling an actual sinusoid and proceeding by such means.

Thanks for the reference to the USENET group!!!

-Brendan

4. Dec 25, 2008

### Petre

5. Dec 25, 2008

### Staff: Mentor

Re: Sampling above the Nyquist rate...but/ sine signal?

At first I was worried that this post was spam, but after skimming the first paper, it seems perfectly appropriate for this thread. Thanks, Petre, and welcome to the PF.

Quiz Question for brendan -- What practical reason comes into play that would drive you in a practical system to not sample just barely above the Nyquist frequency? Very important consideration....