# Sub-Nyquist Sampling

Started by 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, Could you elaborate on this a little? It sounds too good to be true. I would have said it's impossible to distinguish the 8 aliased frequencies in your example, before I read the paragraph about phase differences, etc. What "delayed and undelayed signals" are you referring to? Can you provide a reference? I thought I understood sub-Nyquist sampling to mean: sampling a band-pass signal with part or all of its passband above the Nyquist frequency (Fs/2), with the requirement that each frequency in the analog pass-band maps uniquely to a frequency in the sampled base-band. If any two frequencies in the analog pass-band both alias to the same base-band frequency, you've lost the information to discriminate between them unambiguously. Right? And, of course, the total bandwidth has to be less Fs/2. I guess you could have multiple analog pass-bands that all map to mutually-exclusive regions of the base-band, but your example does not fit that model. Mark > -----Original Message----- > From: Roush, Craig Ryan (UMKC-Student) [mailto:] > Sent: Wed, September 03, 2003 10:30 AM > To: > Subject: [matlab] Sub-Nyquist Sampling > 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 > > ------------------------ Yahoo! Groups Sponsor > ---------------------~--> > Buy Ink Cartridges or Refill Kits for Your HP, Epson, Canon or Lexmark > Printer at Myinks.com. Free s/h on orders $50 or more to the > US & Canada. http://www.c1tracking.com/l.asp?cidU11 > http://us.click.yahoo.com/l.m7sD/LIdGAA/qnsNAA/wHYolB/TM > -------------------------- > -------~-> > > _____________________________________ > Note: If you do a simple "reply" with your email client, only > the author of this message will receive your answer. You > need to do a "reply all" if you want your answer to be > distributed to the entire group. > > _____________________________________ > About this discussion group: > > To Join: > > To Post: > > To Leave: > > Archives: http://www.yahoogroups.com/group/matlab > > More DSP-Related Groups: http://www.dsprelated.com/groups.php3 > > ">http://docs.yahoo.com/info/terms/  Mark- If you get a private reply from Craig about the reference URL, can you post? It's an interesting topic. Jeff Brower system engineer Signalogic "Egler, Mark" wrote: > > Craig, > > Could you elaborate on this a little? It sounds too good to be true. I would > have said it's impossible to distinguish the 8 aliased frequencies in your > example, before I read the paragraph about phase differences, etc. What > "delayed and undelayed signals" are you referring to? Can you provide a > reference? > > I thought I understood sub-Nyquist sampling to mean: sampling a band-pass > signal with part or all of its passband above the Nyquist frequency (Fs/2), > with the requirement that each frequency in the analog pass-band maps > uniquely to a frequency in the sampled base-band. If any two frequencies in > the analog pass-band both alias to the same base-band frequency, you've lost > the information to discriminate between them unambiguously. Right? And, of > course, the total bandwidth has to be less Fs/2. > > I guess you could have multiple analog pass-bands that all map to > mutually-exclusive regions of the base-band, but your example does not fit > that model. > > Mark > > > -----Original Message----- > > From: Roush, Craig Ryan (UMKC-Student) [mailto:] > > Sent: Wed, September 03, 2003 10:30 AM > > To: > > Subject: [matlab] Sub-Nyquist Sampling > > > > > > 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 > > > > > > > > ------------------------ Yahoo! Groups Sponsor > > ---------------------~--> > > Buy Ink Cartridges or Refill Kits for Your HP, Epson, Canon or Lexmark > > Printer at Myinks.com. Free s/h on orders$50 or more to the > > US & Canada. http://www.c1tracking.com/l.asp?cidU11 > > http://us.click.yahoo.com/l.m7sD/LIdGAA/qnsNAA/wHYolB/TM > > -------------------------- > > -------~-> > > > > _____________________________________ > > Note: If you do a simple "reply" with your email client, only > > the author of this message will receive your answer. You > > need to do a "reply all" if you want your answer to be > > distributed to the entire group. > > > > _____________________________________ > > About this discussion group: > > > > To Join: > > > > To Post: > > > > To Leave: > > > > Archives: http://www.yahoogroups.com/group/matlab > > > > More DSP-Related Groups: http://www.dsprelated.com/groups.php3 > > > > ">http://docs.yahoo.com/info/terms/ > > > > _____________________________________ > Note: If you do a simple "reply" with your email client, only the author of this message will receive your answer. You need to do a "reply all" if you want your answer to be distributed to the entire group. > > _____________________________________ > About this discussion group: > > To Join: > > To Post: > > To Leave: > > Archives: http://www.yahoogroups.com/group/matlab > > More DSP-Related Groups: http://www.dsprelated.com/groups.php3 > > ">http://docs.yahoo.com/info/terms/
 Mark, This analysis is coming from "Digital Techniques for Wideband Receivers" by James Tsui 1995 ISBN 0-89006-808-9 To give you a bit more background: Consider a instantaneous frequency measurement receiver (IFM). The main portion of this receiver is a correlator, the input to the correlator is a delayed and undelayed copy of the input signal. Thus you end up w/ E = \sin(2 \pi f \tau ) and F = \cos(2 \pi f \tau) \tau is the delay time and f is frequency. the freq can be obtained by \theta = \tan^-1 ( \frac{E}{F} ) = 2 \pi f \tau, however there is the following restriction \theta < 2 \pi, if \theta > 2\pi, there becomes this "ambiguity" problem. The maximum unambiguous bandwidth obtained is \Delta B = \frac{2 \pi }{2 \pi \tau} = \frac{1}{\tau}. This only relation only limits the bandwidth , but not the center frequency, thus, the center freq can be any value, ie ( if \tau = .5 ns, the unambiguous bandwidth is 2 GHz, the freq range can be either 0 to 2 GHz or 2 to 4 GHz, or any other values as long as the bandwidth is 2 GHz. Now for the sub-Nyquist approach the input signal is divided into 2 paths, and undelayed and one and a delayed one. After digitization, the digitized outputs can be processed through an FFT operation. Let X_{ru}(k) and X_{iu}(k) represent the real and imaginary components of the undelayed case and let X_{rd}(k) and X_{id}(k) represent the delayed case. The amplitude is obvious found from X_u (k) = [X_{ru}(k)^2 and X_{iu}(k)^2 ]^0.5. The delayed path has the same amplitude components, X_u(k_m) represents the maximum amplitude of the frequency component from the undelayed path, thus, X_u(k_m) can represent the input frequency. The phase difference between the delay and undelayed path can be written as : \theta = \theta_d - \theta_u = 2\pi f \tau where \theta_d = \tan^-1(\frac{X_{id}(k_m)}{X_{rd}(k_m)} ) \theta_u = \tan^-1(\frac{X_{iu}(k_m)}{X_{ru}(k_m)} ) From the phase difference \theta, the frequency of the input signal can be obtained because \tau is known. As long as the input frequency are sufficiently separated, the input freqs can be identified by observing those frequency bins whose magnitude exceed a threshold. The Tsui leads into the aforementioned example hope that helps you understand, I pretty much copied straight from chapter 5 of the book for you, the notation is in LaTeX in case you have trouble reading it. -Craig