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Basic Sub-Nyquist Sampling

Started by Craig September 3, 2003
I am having some difficulty understanding the sub-nyquist sampling
theory.

For example: 
if you have an ADC that can only operate at 250MHz, then the max
bandwidth is 125MHz.  However, if I wanted 1000MHz and I want to use
the same ADC, the bandwidth would "fold" into 125MHz output band.  So
I could potentially have 8 solutions if I injected in a 40MHz signal. 
40, 210, 290, 460, 540, 710, 790, and 960MHz.

What I do not follow is how we can determine which zone the signal is
in.  From my reading it appears that you take the phase difference
between the delayed and undelayed signal input, and since you know the
what the delay is, you can extract the true frequency.  The delayed
and undelayed phase can be determined straight from the arctan of the
ratio of I and Q data.  And you can determine the max unambiguous
bandwidth by 1/tau (where tau is the delay).

I am just not clear why this works, and how to prove that this works,
I realize there are some fine points about this, but I want to
understand the basic concept then move forward with the remaining
concepts.

Thanks,
-Craig
Craig wrote:
> > I am having some difficulty understanding the sub-nyquist sampling > theory. > > For example: > if you have an ADC that can only operate at 250MHz, then the max > bandwidth is 125MHz. However, if I wanted 1000MHz and I want to use > the same ADC, the bandwidth would "fold" into 125MHz output band. So > I could potentially have 8 solutions if I injected in a 40MHz signal. > 40, 210, 290, 460, 540, 710, 790, and 960MHz. > > What I do not follow is how we can determine which zone the signal is > in. From my reading it appears that you take the phase difference > between the delayed and undelayed signal input, and since you know the > what the delay is, you can extract the true frequency. The delayed > and undelayed phase can be determined straight from the arctan of the > ratio of I and Q data. And you can determine the max unambiguous > bandwidth by 1/tau (where tau is the delay). > > I am just not clear why this works, and how to prove that this works, > I realize there are some fine points about this, but I want to > understand the basic concept then move forward with the remaining > concepts. > > Thanks, > -Craig
The Nyquist sampling rate is twice the bandwidth of the signal. The sampler must be preceded by an analog filter that removes all components outside the band. When the band of interest extends down to DC, a low-pass filter suffices. Your 250 MHz ADC can -- with reservations touched on later -- sample any band 125 MHz wide PROVIDED THAT THERE ARE NO SIGNAL COMPONENTS OUTSIDE THAT BAND. With that done, the ambiguity you discovered goes away. Although bandpass sampling works, the jitter requirement on the samples and the timing of the sample aperture are as stringent as on a full-rate sampler. Not all slow ADCs can meet these specs. Moreover, not all combinations of sample rate and band-edge location are possible, even if the Nyquist criterion is met. This is explained and diagrammed in detail in Richard G. Lyons' book, "Understanding Digital Signal Processing." Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Hi Craig,

crrea2@umkc.edu (Craig) wrote in message news:<82396605.0309030627.2da324b4@posting.google.com>...
> I am having some difficulty understanding the sub-nyquist sampling > theory. > > For example: > if you have an ADC that can only operate at 250MHz, then the max > bandwidth is 125MHz. However, if I wanted 1000MHz and I want to use > the same ADC, the bandwidth would "fold" into 125MHz output band. So > I could potentially have 8 solutions if I injected in a 40MHz signal. > 40, 210, 290, 460, 540, 710, 790, and 960MHz. > > What I do not follow is how we can determine which zone the signal is > in. From my reading it appears that you take the phase difference > between the delayed and undelayed signal input, and since you know the > what the delay is, you can extract the true frequency. The delayed > and undelayed phase can be determined straight from the arctan of the > ratio of I and Q data. And you can determine the max unambiguous > bandwidth by 1/tau (where tau is the delay). > > I am just not clear why this works, and how to prove that this works, > I realize there are some fine points about this, but I want to > understand the basic concept then move forward with the remaining > concepts.
I'm not sure what you are trying to do here. Your first paragraph looks like you are real sampling. The second paragraph looks like you are complex sampling. If you are real sampling you don't have the I&Q data to play with. If you are complex sampling, your unambiguous bandwidth is 250MHz, not 125MHz, so you only have 4 points of ambiguity. If you are complex sampling, the I&Q will be at quadrature within one of the spans of ambiguity, and usually not within the others. If the stimulus is a single pure tone, the actual phase relationship between your I&Q data can be measured to determine the real frequency of the stimulus. The details may depend on how your I&Q signals were derived. In real systems, signals are seldom pure and simple, especially across such a large frequency range. If the signal has any complexity, trying to resolve ambiguities in this way is normally impossible. However, if your stimulus is continuous, you can switch sampling rates a few times, and shift the ambiguities around. By correlating across the ambiguous data sets you have captured, you can determine the real frequencies in arbitrarily complex stimulii. Note that the tolerable sampling jitter in an ADC used in this way is the same as the tolerable jitter would be in a straight 2G sample/s ADC - i.e. bloody hard to achieve! Regards, Steve