Hello from a newbie to this forum, and to DSPs in general. I have a suspicion that the proper digital signal processor, running appropriate code, could pull an extremely low level signal from an extremely high background noise, but in that I'm unfamiliar with DSP technology, I need a little help to verify my assumption. I'm looking to detect a simple, but unknown, low frequency sine wave from an almost infinitely noisy background. The signal is expected to be a single frequency in the 30 to 300Hz range, +/- a fraction of a Hz, and will most likely contain several out of phase sine waves of the same frequency. Sampling time can be very long - up to several hours if necessary. Unambiguous detection of any one of the phases would be considered a success. What might be the worst possible signal to noise ratio one could expect to pull the sine wave from? Any mathematical proofs would be appreciated. This project, if technically feasible, will have a 'no limit budget', so any outlandish hardware, and/or programming effort, would most likely be acceptable. Thank you for any help and/or advice you may be able to offer.
DSP and SNR
Started by ●May 23, 2012
Reply by ●May 23, 20122012-05-23
"JRW" <jr@n_o_s_p_a_m.jrwhipple.com> writes:> Hello from a newbie to this forum, and to DSPs in general. > > I have a suspicion that the proper digital signal processor, running > appropriate code, could pull an extremely low level signal from an > extremely high background noise, but in that I'm unfamiliar with DSP > technology, I need a little help to verify my assumption. > > I'm looking to detect a simple, but unknown, low frequency sine wave from > an almost infinitely noisy background. The signal is expected to be a > single frequency in the 30 to 300Hz range, +/- a fraction of a Hz, and will > most likely contain several out of phase sine waves of the same > frequency.The sum of several sine waves at the same frequency but different phases can be represented by a single sine wave at the same frequency and an amplitde and phase that depends on the original sine waves.> Sampling time can be very long - up to several hours if necessary. > Unambiguous detection of any one of the phases would be considered a > success.Mathmematically, you never have completely unambiguous detection in the presence of noise. The probability of a missed detection may be so low that it's of no practical consequence, however.> What might be the worst possible signal to noise ratio one could expect to > pull the sine wave from? Any mathematical proofs would be > appreciated.You're generally correct that the longer the observation period, the lower the SNR the detection can deal with, given a maximum probability of missed detection. -- Randy Yates Digital Signal Labs http://www.digitalsignallabs.com
Reply by ●May 23, 20122012-05-23
JRW <jr@n_o_s_p_a_m.jrwhipple.com> wrote: (snip)> I'm looking to detect a simple, but unknown, low frequency sine wave from > an almost infinitely noisy background. The signal is expected to be a > single frequency in the 30 to 300Hz range, +/- a fraction of a Hz, and will > most likely contain several out of phase sine waves of the same frequency. > Sampling time can be very long - up to several hours if necessary. > Unambiguous detection of any one of the phases would be considered a > success.Do you mean a known frequency in that range, or any frequency in the range? At a single (infinitely narrow) frequency, a sine can have only one phase. Over a narrow, but not infinitely narrow range, (or if your clocks drift, same thing), it can look like a sum of sines of different frequencies and phases. If the signal is above the background, the S/N improves as the time (or number of samples) increases as the square root of the time. At or below background, it can be done if the phase and frequency are known. That is, if you have a frequency and phase reference. (This is known as synchronous, or coherent, detection.) (You can also detect the phase shift relative to the reference frequency.) (snip) -- glen
Reply by ●May 23, 20122012-05-23
On Wed, 23 May 2012 08:19:08 -0500, "JRW" <jr@n_o_s_p_a_m.jrwhipple.com> wrote:>Hello from a newbie to this forum, and to DSPs in general. > >I have a suspicion that the proper digital signal processor, running >appropriate code, could pull an extremely low level signal from an >extremely high background noise, but in that I'm unfamiliar with DSP >technology, I need a little help to verify my assumption. > >I'm looking to detect a simple, but unknown, low frequency sine wave from >an almost infinitely noisy background. The signal is expected to be a >single frequency in the 30 to 300Hz range, +/- a fraction of a Hz, and will >most likely contain several out of phase sine waves of the same frequency. >Sampling time can be very long - up to several hours if necessary. >Unambiguous detection of any one of the phases would be considered a >success.As others have mentioned, if the tones are really the same frequency, as opposed to nearly the same frequency, then a sum of several waves will be a single wave with some deterministic amplitude and phase.>What might be the worst possible signal to noise ratio one could expect to >pull the sine wave from? Any mathematical proofs would be appreciated.Also as previously mentioned, there is a tradeoff between reliability of detection and SNR. As SNR goes down the reliability of detection goes down. Processing gain due to increased observation time can be exploited, but there is a limit to how well one can do. You might try searching around for tone or frequency detection articles that talk about the "threshold effect".>This project, if technically feasible, will have a 'no limit budget', so >any outlandish hardware, and/or programming effort, would most likely be >acceptable. > >Thank you for any help and/or advice you may be able to offer.Eric Jacobsen Anchor Hill Communications www.anchorhill.com
Reply by ●May 23, 20122012-05-23
eric.jacobsen@ieee.org (Eric Jacobsen) writes:> [...]> Processing gain due to increased observation time can be exploited, > but there is a limit to how well one can do. You might try searching > around for tone or frequency detection articles that talk about the > "threshold effect".Eric, I had heard of the "threshold effect" in relation to FM detection, but not in the more general sense of "detection" as in "detection and estimation." I did a quick Google, and also searched and [kayII] and [vantrees] and did not find any such general definition. I also do not recall any such threshold in the general case. Is there another name for the effect you mention? --Randy @BOOK{kayII, title = "{Fundamentals of Statistical Signal Processing, Volume II: Detection Theory}", author = "Steven~M.~Kay", publisher = "Prentice Hall", year = "1998"} @book{vantrees, title = "Detection, Estimation, and Modulation Theory, Part I", author = "Harry L. Van Trees", publisher = "Wiley", year = "2001"}> >>This project, if technically feasible, will have a 'no limit budget', so >>any outlandish hardware, and/or programming effort, would most likely be >>acceptable. >> >>Thank you for any help and/or advice you may be able to offer. > > > Eric Jacobsen > Anchor Hill Communications > www.anchorhill.com-- Randy Yates Digital Signal Labs http://www.digitalsignallabs.com
Reply by ●May 23, 20122012-05-23
Randy Yates <yates@digitalsignallabs.com> writes:> eric.jacobsen@ieee.org (Eric Jacobsen) writes: >> [...] > >> Processing gain due to increased observation time can be exploited, >> but there is a limit to how well one can do. You might try searching >> around for tone or frequency detection articles that talk about the >> "threshold effect". > > Eric, > > I had heard of the "threshold effect" in relation to FM detection, but > not in the more general sense of "detection" as in "detection and > estimation." I did a quick Google, and also searched and [kayII] and > [vantrees] and did not find any such general definition. > > I also do not recall any such threshold in the general case. > > Is there another name for the effect you mention? > > --Randy > > > @BOOK{kayII, > title = "{Fundamentals of Statistical Signal Processing, Volume II: Detection Theory}", > author = "Steven~M.~Kay", > publisher = "Prentice Hall", > year = "1998"} > @book{vantrees, > title = "Detection, Estimation, and Modulation Theory, Part I", > author = "Harry L. Van Trees", > publisher = "Wiley", > year = "2001"}Oh, well duh. I guess you are implying an architecture (since we're only searching for a single, unknown sinusoid) of an FM discriminator (e.g., d arctan(z) / dt) followed by an averager. True, we are attempting a frequency detection, not a "pulse" detection. -- Randy Yates Digital Signal Labs http://www.digitalsignallabs.com
Reply by ●May 23, 20122012-05-23
On Wed, 23 May 2012 12:15:44 -0400, Randy Yates <yates@digitalsignallabs.com> wrote:>Randy Yates <yates@digitalsignallabs.com> writes: > >> eric.jacobsen@ieee.org (Eric Jacobsen) writes: >>> [...] >> >>> Processing gain due to increased observation time can be exploited, >>> but there is a limit to how well one can do. You might try searching >>> around for tone or frequency detection articles that talk about the >>> "threshold effect". >> >> Eric, >> >> I had heard of the "threshold effect" in relation to FM detection, but >> not in the more general sense of "detection" as in "detection and >> estimation." I did a quick Google, and also searched and [kayII] and >> [vantrees] and did not find any such general definition. >> >> I also do not recall any such threshold in the general case. >> >> Is there another name for the effect you mention? >> >> --Randy >> >> >> @BOOK{kayII, >> title = "{Fundamentals of Statistical Signal Processing, Volume II: Detection Theory}", >> author = "Steven~M.~Kay", >> publisher = "Prentice Hall", >> year = "1998"} >> @book{vantrees, >> title = "Detection, Estimation, and Modulation Theory, Part I", >> author = "Harry L. Van Trees", >> publisher = "Wiley", >> year = "2001"} > >Oh, well duh. I guess you are implying an architecture (since we're only >searching for a single, unknown sinusoid) of an FM discriminator (e.g., >d arctan(z) / dt) followed by an averager. > >True, we are attempting a frequency detection, not a "pulse" detection. >-- >Randy Yates >Digital Signal Labs >http://www.digitalsignallabs.comAlso, an obvious method for detecting a single tone would be a long DFT/FFT in order to exploit the processing gain. Tone detection in a DFT is subject to the threshold effect. Peter K. has an old paper on the topic: B.G. Quinn and P. J. Kootsookos, "Threshold Behaviour of the Maximum Likelihood Estimator of Frequency," IEEE Trans. Signal Processing, Vol. 42, pp 3291-3294, November 1994. It shows up here in Fig. 2(a). pretty noticeably: http://www.ericjacobsen.org/FTinterp.pdf Eric Jacobsen Anchor Hill Communications www.anchorhill.com
Reply by ●May 23, 20122012-05-23
JRW, Other things that come to mind: - if you have a reference for the signal / phase of interest then you might have a chance to do what you want regarding phase. Otherwise there is no hope unless the frequencies are different as well. Others have already commented on this. A description of the physics involved always helps. - since you are just now delving into DSP then for very low SNR one might postulate this "problem": "If the SNR is very low yet the bandwidth is very low as well then a very long time sample will be necessary at least. This is a technique used in space communications by limiting the bandwidth. So, let's say that's going to work in principle. But, if the data is going to be sampled and digitized then there will be quantization noise added that is related to the A/D resolution / word length. Perhaps someone else here can readily comment on the effect of quantization noise in a very low SNR situation. I can't off the top of my head. Fred
Reply by ●May 23, 20122012-05-23
On Wed, 23 May 2012 10:47:41 -0700, Fred Marshall <fmarshallxremove_the_x@acm.org> wrote:>JRW, > >Other things that come to mind: > >- if you have a reference for the signal / phase of interest then you >might have a chance to do what you want regarding phase. Otherwise >there is no hope unless the frequencies are different as well. Others >have already commented on this. A description of the physics involved >always helps. > >- since you are just now delving into DSP then for very low SNR one >might postulate this "problem": > >"If the SNR is very low yet the bandwidth is very low as well then a >very long time sample will be necessary at least. This is a technique >used in space communications by limiting the bandwidth. So, let's say >that's going to work in principle. But, if the data is going to be >sampled and digitized then there will be quantization noise added that >is related to the A/D resolution / word length. > >Perhaps someone else here can readily comment on the effect of >quantization noise in a very low SNR situation. I can't off the top of >my head. > >FredThe noise will act as effective dithering and the quantization noise can be overcome with processing gain. Again, increasing the observation time allows more processing gain, but there's always a limit. Eric Jacobsen Anchor Hill Communications www.anchorhill.com
Reply by ●May 23, 20122012-05-23
eric.jacobsen@ieee.org (Eric Jacobsen) writes:> On Wed, 23 May 2012 10:47:41 -0700, Fred Marshall > <fmarshallxremove_the_x@acm.org> wrote: > >>JRW, >> >>Other things that come to mind: >> >>- if you have a reference for the signal / phase of interest then you >>might have a chance to do what you want regarding phase. Otherwise >>there is no hope unless the frequencies are different as well. Others >>have already commented on this. A description of the physics involved >>always helps. >> >>- since you are just now delving into DSP then for very low SNR one >>might postulate this "problem": >> >>"If the SNR is very low yet the bandwidth is very low as well then a >>very long time sample will be necessary at least. This is a technique >>used in space communications by limiting the bandwidth. So, let's say >>that's going to work in principle. But, if the data is going to be >>sampled and digitized then there will be quantization noise added that >>is related to the A/D resolution / word length. >> >>Perhaps someone else here can readily comment on the effect of >>quantization noise in a very low SNR situation. I can't off the top of >>my head. >> >>Fred > > The noise will act as effective dithering and the quantization noise > can be overcome with processing gain. Again, increasing the > observation time allows more processing gain, but there's always a > limit.I'm not sure narrow-band noise will dither correctly. At a minimum, you could ensure there is a wideband dither noise, and if the gain is set reasonably and the narrow-band analog SNR is low, then the degradation in SNR out of the ADC due to quantization noise is going to be small. -- Randy Yates Digital Signal Labs http://www.digitalsignallabs.com
