Sampling question

[slacker]

Hi all,

I found (through Google) this excellently written article when looking for information regarding the Hilbert transform: www.complextoreal.com/tcomplex.htm

Having a signal processing background, a lot of the article already makes sense to me. However the application example specified at the end left me confused. I was hoping to get some clarification and/or extra explanation.

We do all this because of something Nyquist said. He said that in order to properly reconstruct a signal, any signal, baseband or passband, needs to be sampled at least two times its highest spectral frequency. That requires that we sample at frequency of 200.

But we just showed that if we take a modulated signal and go through all this math and create an analytic signal (which by the way does not require any knowledge of the original signal) we can separate the information signal the baseband signal s(t)) from the carrier. We do this by dividing the analytic signal by the carrier. Now all we have left is the baseband signal. All processing can be done at a sampling frequency which is 6 (two times the maximum frequency of 3) instead of 200.

This sounds very similar to the basis on which oscilloscopes work (which sample high frequency signals with a low sampling frequency). However I understand that application because there's a tunable band pass filter applied before sampling in this application. With this tunable band pass filter, the sampled signal is still aliased, however because we know the relationship the alias has to our real signal it is recoverable.

In this communications example, I do not understand how sampling the signal at 6 can possibly work when the frequency is 100. Won't the alias images overlap >15 fold completely destroying the data? Where and how does this analytic signal get around this problem?

Thanks,
Tom

[slacker]

Having given it more thought, I think I understand better how this works now. Because the side sided bandwidth is less than half of the sampling rate of 6, the many alias images never actually overlap - similar to the principle of the oscilloscope.

So the Hilbert transform and analytic signal is just a formal way of describing how to recover the base band information from the carrier.

Any comments on this understanding I've developed would be appreciated.

Thanks.

[slacker]

To better understand this, I've started experimenting in Matlab:

ws = 10;
fs = ws/(2*pi);
dt = 1/fs;
t = 0:dt:10*(2*pi);
N = length(t);
n = 0:N-1;
s1 = (4*cos(2*t)-6*sin(3*t));
s2 = (4*cos(2*t)-6*sin(3*t)).*cos(100*t);
S = fft(s2);
w = 2*pi*n/(N*dt);
mag = 2*abs(S)/N;

figure;
subplot(211);
plot(t,s2);
grid on;
axis([0 10 -10 10]);
subplot(212);
plot(w,mag);
grid on;
axis([0 ws 0 7]);

As I thought, there is no alias distortion due to sampling at a ws = 10 rad/sec. However where does the Hilbert transform and the analytic signal fit into all of this to recover the original signal?

(image of graph produced by above Matlab code snippet removed)

[tmcdavid]

Slacker Tom:

I found this site by exactly the same sequence as you. Please reference my subject thread Quadrature Demodulation, last post Feb. 12 '07.

Your conclusion about the sampling is exactly correct. You can undersample the carrier as long as you oversample the baseband frequency range. But one caveat is the one that you touched on that dictates that the sampling rate is still high enough that the sidebands of the desired modulation do not come too close to the sidebands of the aliased images either above or below the actual carrier.

A related caveat results from the fact that all the noise in the image and alias areas will fold into the desired band, unless an anti-aliasing filter precedes the sampling. So it is requisite in the system we are discussing that an analog bandpass filter precede the sampling to "pick out" the single aliased spectrum band actually containing the energy of interest.

Now this analog filter throws a condition back to the choice of the sampling rate. Not only must the modulation images not overlap, they also must not intrude into the pass-band or transition bands of the analog anti-alias filter. This last has caused me to have to oversample the baseband at a higher rate that I ordinarily would, due to lack of a (reasonably priced) narrow-band analog filter.

Another constraint is that the sampling can not alias the carrier down to DC, which would cause the two sidebands to alias on to one-another. This is actually part of the first caveat, not separate condition. The resulting sample stream is still centered around a carrier, just a much lower frequency carrier.

Finally, what to do with the signals after the sampling: For the analytic signal you want an inline (I) and quadrature (Q) pair. The basic theory is to apply a Hilbert transform to the sampled signal (I) to get the quadrature companion. However, since the Hilbert Xfm is band-pass by nature, this results in the I and Q channels having a somewhat different amplitude response. I needed to keep the transform operator short because the higher-than-I-wish sampling rate was running me short of processing power. This would definitely impact the signal spectrum. My solution was to generate a bandpass zero phase desired spectrum having a flat bandpass and raised-cosine transitions (to moderate filter ringing) and make two copies. Then I shift all the spectrum by +pi/4 for one copy and by -pi/4 for the other copy. These two filter models are then transformed to the time domain and windowed in the time domain to keep the operators short. For a check, I transform back to frequency domain to see if I can tolerate the unavoidable sidelobes.

The sampled version of the analog signal is then passed through the two filters separately to get the I and Q versions of the analytic signal. The rest of the demodulation can proceed along one of several lines, most of them classic.

For example, each pair of output samples at a given sample instant are the I and Q components of the analytic signal, so the phase at that instant is the arctangent of their ratio. Now the signal is represented as a single stream of phase samples. Since frequency is the derivative of phase the stream of first differences is the frequency. It will extend above and below the aliased carrier frequency, so the phase step constant of the ideal carrier can be subtracted from the stream of first differences with the result being the baseband frequency deviation.

If one is interested in phase demodulation, don't bother with the sequential differences, just subtract the phase ramp that represents the aliased carrier and the result is the baseband phase deviation.

Hope that has added to your vision of the process.

Regards,

Terry

[slacker]

Wow. I didn't expect to see a reply here

Thanks a lot Terry. You have confirmed my thoughts and have certainly added to my understanding.

I was a little naive to think before that my signals processing background should be sufficient for me to understand communications. How wrong I was. Although the two areas seem to share much of the mathematics they are very different in principle, as I now know.

Thanks again!

[kcknaik]

what is main use of sampling

[kcknaik]

what is alias and anti-aliasing