Anyone know of a good derivation of the ~1.96 dB implimentation loss when using a 1-bit hard limiting A2D for GPS tracking ( I assume this applies to any DSSS tracking ). I have found very few, it seems it may be empericaly found mostly and is 10*log10(2/pi) = -1.96dB _____________________________ Posted through www.DSPRelated.com
1 bit A2D GPS loss
Started by ●March 21, 2014
Reply by ●March 21, 20142014-03-21
On Fri, 21 Mar 2014 10:54:48 -0500, jacobfenton wrote:> Anyone know of a good derivation of the ~1.96 dB implimentation loss > when using a 1-bit hard limiting A2D for GPS tracking ( I assume this > applies to any DSSS tracking ). I have found very few, it seems it may > be empericaly found mostly and is 10*log10(2/pi) = -1.96dB > > _____________________________ > Posted through www.DSPRelated.comIt's a nonlinear problem, so it may be that empirical solutions are the only ones that can be found. Why don't you try the following: Model your noise as bandlimited Gaussian with zero mean. Keep both the bandwidth and the spectral density as symbolic throughout until the end. Now add in some signal of known amplitude and significantly lower bandwidth than the noise bandwidth. Now calculate the probability that the resulting signal will be a positive number. You should now have enough information to model a 1-bit "ADC" that turns the incoming signal into a +1 if it is positive and a -1 if it is negative. You should be able to compute the signal power and the noise power at the output of your "ADC". At this point you should have SNR in and SNR out -- then you can compare the two and see if it's 1.96dB. I wouldn't be at all surprised if the actual loss number depends on the bandwidth and noise level: in that case then the 1.96dB loss figure _is_ empirical, and it's probably what you get when everything else is tweaked for optimal system performance. -- Tim Wescott Wescott Design Services http://www.wescottdesign.com
Reply by ●March 21, 20142014-03-21
There is mention of 1.96 dB loss in Gardner re to the use of limiting in a PLL. IIRC, the 1.96 dB value is the worst case when the SNR is worse than 0 dB. Mark
Reply by ●March 26, 20142014-03-26
On Friday, March 21, 2014 11:54:48 AM UTC-4, jacobfenton wrote:> Anyone know of a good derivation of the ~1.96 dB implimentation loss when > > using a 1-bit hard limiting A2D for GPS tracking ( I assume this applies to > > any DSSS tracking ). I have found very few, it seems it may be empericaly > > found mostly and is 10*log10(2/pi) = -1.96dB > > > > _____________________________ > > Posted through www.DSPRelated.comTo the OP and Tim: the 1.96 dB loss asymptote at 1-bit ADC is well-known and the derivation of that number is analytical. It comes from information-theoretic methods. Or maybe I'm not understanding what you mean by "empirical". Here are applications to GPS as you asked for. https://www.mitre.org/sites/default/files/pdf/09_3995.pdf http://mediatum.ub.tum.de/doc/1160962/1160962.pdf http://mediatum.ub.tum.de/doc/1163025/1163025.pdf
Reply by ●March 26, 20142014-03-26
On Wed, 26 Mar 2014 11:17:37 -0700, julius wrote:> On Friday, March 21, 2014 11:54:48 AM UTC-4, jacobfenton wrote: >> Anyone know of a good derivation of the ~1.96 dB implimentation loss >> when >> >> using a 1-bit hard limiting A2D for GPS tracking ( I assume this >> applies to >> >> any DSSS tracking ). I have found very few, it seems it may be >> empericaly >> >> found mostly and is 10*log10(2/pi) = -1.96dB >> >> >> >> _____________________________ >> >> Posted through www.DSPRelated.com > > To the OP and Tim: the 1.96 dB loss asymptote at 1-bit ADC is well-known > and the derivation of that number is analytical. It comes from > information-theoretic methods. > > Or maybe I'm not understanding what you mean by "empirical". > > Here are applications to GPS as you asked for. > https://www.mitre.org/sites/default/files/pdf/09_3995.pdf > > http://mediatum.ub.tum.de/doc/1160962/1160962.pdf > http://mediatum.ub.tum.de/doc/1163025/1163025.pdfWell, by empirical _I_ certainly meant "found from real-world experimentation". It sounds like it ain't empirical. -- Tim Wescott Control system and signal processing consulting www.wescottdesign.com
Reply by ●March 26, 20142014-03-26
Tim Wescott <tim@seemywebsite.please> wrote:> On Wed, 26 Mar 2014 11:17:37 -0700, julius wrote:(snip, someone wrote)>> To the OP and Tim: the 1.96 dB loss asymptote at 1-bit ADC is well-known >> and the derivation of that number is analytical. It comes from >> information-theoretic methods.>> Or maybe I'm not understanding what you mean by "empirical".(snip)> Well, by empirical _I_ certainly meant "found from real-world > experimentation". It sounds like it ain't empirical.I am not sure that they are exclusive. Consider the Rayleigh Criterion for resolution in optics. (Signal processing in cylindrical coordinates.) In deciding whether two points are too close to separate in a diffraction limited system, there is a constant, 1.220, that comes from the first zero of a Bessel function, and pi. The analytical value, according to the wikipedia Angular Resolution page, came after the empirical value, and was appropriately close to the empirical value. With deconvolution techniques, one can do a little better, and in a less perfect system, worse. While it seems to come from diffraction theory, it is, about as much, empirical. Just because it fits an analytical model, and conforms to information theory, doesn't mean that it isn't empirical. With different signal processing methods, seems to me likely that you get different values. (As for resolution and deconvolution, what is possible depends much on the S/N ratio. The early Hubble pictures were of bright objects where the deconvolution worked well. The PSF was known extremely well for the unrepaired Hubble telescope.) -- glen
Reply by ●March 27, 20142014-03-27
Gardner 2nd edition around page 127 discusses loss due to limiting in a PLL. THe number there is -1.05 dB. -1.59 dB is associated with Shannons theorem, not sure how that does or doesn't apply here. Mark
Reply by ●March 28, 20142014-03-28
makolber@yahoo.com writes:> Gardner 2nd edition around page 127 discusses loss due to limiting in a PLL. > > THe number there is -1.05 dB. > > -1.59 dB is associated with Shannons theorem, not sure how that does > or doesn't apply here.You can see this for a derivation of that number: www.digitalsignallabs.com/shannonlimit.pdf -- Randy Yates Digital Signal Labs http://www.digitalsignallabs.com
Reply by ●March 28, 20142014-03-28
On Friday, March 28, 2014 9:00:52 AM UTC-4, Randy Yates wrote:> makolber@yahoo.com writes: > > > > > Gardner 2nd edition around page 127 discusses loss due to limiting in a PLL. > > > > > > THe number there is -1.05 dB. > > > > > > -1.59 dB is associated with Shannons theorem, not sure how that does > > > or doesn't apply here. > > > > You can see this for a derivation of that number: > > > > www.digitalsignallabs.com/shannonlimit.pdf > > > > -- > > Randy Yates > > Digital Signal Labs > > http://www.digitalsignallabs.comRandy's paper doesn't explicitly state the result in db but -1.59 db = 10*log(ln(2)) Clay
Reply by ●March 28, 20142014-03-28
clay@claysturner.com writes:> On Friday, March 28, 2014 9:00:52 AM UTC-4, Randy Yates wrote: >> makolber@yahoo.com writes: >> >> >> >> > Gardner 2nd edition around page 127 discusses loss due to limiting in a PLL. >> >> > >> >> > THe number there is -1.05 dB. >> >> > >> >> > -1.59 dB is associated with Shannons theorem, not sure how that does >> >> > or doesn't apply here. >> >> >> >> You can see this for a derivation of that number: >> >> >> >> www.digitalsignallabs.com/shannonlimit.pdf >> >> >> >> -- >> >> Randy Yates >> >> Digital Signal Labs >> >> http://www.digitalsignallabs.com > > Randy's paper doesn't explicitly state the result in db but -1.59 db = 10*log(ln(2)) > > ClayClay et al., It looks like my derivation was wrong. I removed the paper until I can correct it. -- Randy Yates Digital Signal Labs http://www.digitalsignallabs.com
