I have a general question about sampling theory. Most of this comes from college coursework, and our friend Google. If my highest frequency of interest, F(in) was 500 Hz, and my frequency of sample, F(s), was 8 kHz then I meet the Nyquist criterion. However, I still need to have an anti-aliasing filter (low- pass, reconstruction filter, etc.) to properly attenuate any frequency components/ images above F(s)/2 assuming baseband sampling (Nyquist Zone 1). If my frequency of sample is 8 kHz, then my images should be centered around 8 kHz: F(s) +/- F(in). So my images would be 7.5 kHz and 8.5 kHz well outside F(s)/2. Images will also appear around every multiple of F(s) as well. But the images from every Nyquist Zone will still fold back into my baseband, correct? So even though I am 'over sampling' I still need an anti-aliasing filter. Many delta-sigma converters have high base sampling rates; e.g., 192 kHz, and can oversample 128*F(s), 256*F(s), 384*F(s), etc. Even here, the concept is the same. What I have always understood is oversampling spreads out the quantanization noise (Noise shaping?), and relaxes the anti-aliasing filter requirements. Regardless, a anti- aliasing filter is needed because images will fold back without one. Do I have the basics down?
An image in mixing schemes refers to unwanted frequencies being mixed (sampled) into your wanted frequencies and seeming to be part of your wanted frequencies even if they weren't before.. In your reference to
7.5 and 8.5, you seem to be referring to the wanted sidebands and not the image.
You're coming in with a signal bandwidth of 500 Hz. Your intention is that there will be no energy above that frequency, but you cannot guarantee it because there maybe interference and noise present. If you do not take steps to remove that interference and noise, then it will also be sampled. Now you will have a noisy sampled signal.
But But But......a noisy signal is bad enough, but there will only be images present if the noise extends above 4kHZ for which you will need your anti-aliasing filter.
Well, from what I read in ADI's Data Conversion Handbook (chap. 2) what you call 'wanted sidebands' are what they are refering to images around every multiple of F(s). Let me go read it again for clarity, maybe I missed something in their explanation.
I would ask Phil to clarify, but he does not have patience for 'stupid questions' and frankly I do not have patience for assholes who forget to take their medication. I wish Google Groups had an option to kill- file.
Unless you know ahead of time that the signal has no content up there -- anti-alias filtering is only necessary if there's something that will alias down to baseband.
Well, kinda, and kinda not. Depending on how you model sampling there is either no frequencies above Fs/2 after sampling, or the spectrum repeats itself every Fs. In the first case those "images" aren't there; in the second case they just replicate baseband.
See my first comment.
Oversampling by itself doesn't spread out quantization noise. Oversampling plus a sigma-delta modulator does. See
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for a simplified explanation.
Oversampling _does_ relax the requirements on the anti-alias filter, because the difference between the frequencies you must block out and the frequencies you want to keep gets bigger. If you are going to oversample and decimate digitally, however, you still need anti-aliasing filters in the digital domain -- these are usually called "decimation filters", but the function is the same.
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Tim Wescott
Wescott Design Services
http://www.wescottdesign.com
Do you need to implement control loops in software?
"Applied Control Theory for Embedded Systems" gives you just what it says.
See details at http://www.wescottdesign.com/actfes/actfes.html
It helps at this point if you have a background in telecoms; even a Ham Radio licence would help you. This is one of the difficulties of DSP; you really need a grounding in quite a number of areas of electronics.
Go back to the old medium wave stations running AM (Amplitude Modulation)
You get the carrier which is the main frequency. Now, even though AM suggests that the carrier will change in amplitude, it does not. If the DJ were to whistle into the mike at 1kHz, you'll get two sidebands, at frequencies of +/- 1kHz in addition to the main carrier. The reason for this is that AM is really a multiplication of the form (1+sin (modulation)) * sin (carrier).
If the DJ speaks into the mike....or.....applies your 500 Hz bandlimited signal, you'll get one sideband which is a copy of your original spectrum but shifted up from 0 Hz to the carrier frequency. The other sideband is an inverted copy of the original spectrum but decreasing downwards from the carrier.
So, AM is the multiplication of your signal by the carrier frequency.
-----ooooo-----
Come now to sampling. Instead of a carrier frequency of just one sine wave, you have a series of very short pulses. But, this comes down to (by Fourier Analysis; another bit of electronics you need under your belt) to a lot of carriers, each a multiple of the frequency of the basic sampling rate. Each one of these carriers becomes Amplitude Modulated by your original signal. (This time, sampling is seen as multiplication in the first instance and not as modulation) and you get the picture that you described of multiple carriers each with sidebands above and below each carrier.
This, I suspect is the picture that you describe as "images". Now, although this is a correct use of English, in that each sideband is an image of your original baseband signal, it is not what is meant by "imaging" or "image frequencies" when modulating one signal by another.
OK, an example off the shelf (and be prepared for some silly errors as I'm thinking as I'm typing.
Let us suppose that your sampling at 8kHz, which is what you suggested, and that you've your baseband signal of 500 Hz Bandwidth, but that you've also got a signal at 7.8kHz, which is crucially above the Fs/2 that everyone talks about. When we multiply them all by the sampling signal, what do we get?
(I've deliberately chosen 7.8kHz to guarantee to be able to illustrate imaging into your wanted 500 Hz bandwidth)
Well first of all, the sampling operations is sin(modulation) * sin(carrier) and not (1+sin(modulation)) * sin (carrier) as per the medium wave station, so we only get our copies of the sidebands around each multiple of the sampling frequency. WE DO NOT GET THE UNCHANGING CARRIERS.
Right from the example, (0.5kHz +7.8kHz) * 8kHz will give us 200 Hz (which is the lower sideband from 7.8kHz), 7.5kHz (which is the lower sideband from 0.5kHz), 8.5kHz (which is the upper sideband from 0.5kHz) and 15.8kHz (which is the upper sideband from 7.8kHz)
Now, it seems at first sight as though there isn't a problem but so far we've only considered the first harmonic at 8kHz. We have to consider _ALL_ the harmonics including the "zeroth" harmonic of DC, because they're all there in the sampled signal.
Lets look at the DC or baseband. Originally we just wanted your 0.5kHz signal, but now as the result of the modulation at 8kHz, we've also got a signal at 200Hz resulting from the sampling of the 7.8kHz signal.
This 200Hz signal IS WHAT IS KNOWN AS AN IMAGE.You can't filter it out once you've got it because it's right in your bandwidth.
OK, now look at what happens at the next harmonic of the sampling frequency at 16 kHz......you can work out the wanted numbers from 16+/- 0.5 and 16+/- 7.8.
But.....the 8kHz modulation has created another image at 15.8 kHz, which appears as an image as a 200 Hz from (16 - 15.8) in what should be the lower sideband of the 16 kHz carrier.
And so it goes on for every subsequent harmonic of the original sampling frequency.
Right, I'm getting bored with this, it's very tedious! There's one final point to make and it is to introduce where the Nyquist criterion comes from.
Consider two of the carrier, say the 8kHz and 16kHz already discussed.
Provided no upper sideband of the 8kHz carrier is above 12 kHz, and provided no lower sideband of the 16kHz carrier is below 12 kHz, then there won't be any aliasing. ie, the bandwidth of the sidebands has to be less than the distance between two carriers, or, put another way, the maximum frequency in the baseband must not exceed half the sampling frequency.....as I said, you need to have a bit of a background in radio.
(I am a Radio Ham with the callsign G....4....S....D....W)
I'm not sure what you mean about the carrier not changing in amplitude. The energy in the spectrum at the carrier frequency falls as the energy in the sidebands rises. (Otherwise, we'd have a neat method of power generation!) This is a real change in amplitude: To duplicate this exact waveform as a summation of
3 sinusoids, the carrier component would need to have its amplitude reduced compared to the unmodulated case.
Those who want to get a good hands-on feel for AM (or FM, PWM, etc) can download my Daqarta software and use the free built-in signal generator. You can set up various carrier and modulator frequencies and depths, and view and measure the resultant waveforms and spectra (and hear the results on your sound card).
There is no need to purchase Daqarta for this... the signal generator (and everything else that doesn't use sound card input signals) will keep working after the trial period expires, and you are welcome to use it as long as you like. (PS: On some systems, users have reported no trial period at all. If that happens to you, let me know via the Contact page on the site and I will create a special trial key for you. The upcoming version, in a few weeks, should correct this problem.)
Best regards,
Bob Masta D A Q A R T A Data AcQuisition And Real-Time Analysis
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Scope, Spectrum, Spectrogram, Signal Generator Science with your sound card!
I wonder if Bob was talking about a balanced modulator which produces DSB suppressed carrier, and not about DSB with full carrier which is the stuff of AM on the medium wave?
Whoops! My apologies to all! I had forgotten that I needed to use a slightly modified form of the AM equation to prevent clipping of the output. (Digital systems are not forgiving in this respect.) I have been using the modified forn for so long now that I'd forgotten about this difference with the standard form. The two different equations are discussed at
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One nice feature of the modified version is that as you increase modulation depth above 100%, the output smoothly goes to pure multiplication at 200% depth.
Again, my apologies for misleading anyone.
Best regards,
Bob Masta D A Q A R T A Data AcQuisition And Real-Time Analysis
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Scope, Spectrum, Spectrogram, Signal Generator Science with your sound card!
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