I know it's a common technique to use a limiter (1-bit ADC) to digitize GPS. I'm curious how it's possible to correlate a signal when it's 20dB or more below the noise floor and you don't have enough bits to give you the dynamic range to pick it up. That is, with a 1-bit ADC and a -20dB SNR, the GPS signals aren't loud enough to ever cause a bit to toggle on their own, why isn't the information lost? The signals from the GPS satellites are also very close in amplitude on the ground, if they were substantially different, would you need more bits to get enough dynamic range?
Correlating GPS with a 1-bit A2D
Started by ●July 4, 2012
Reply by ●July 4, 20122012-07-04
On Tue, 03 Jul 2012 23:37:04 -0500, "gct" <smcallis@n_o_s_p_a_m.gmail.com> wrote:>I know it's a common technique to use a limiter (1-bit ADC) to digitize >GPS. I'm curious how it's possible to correlate a signal when it's 20dB or >more below the noise floor and you don't have enough bits to give you the >dynamic range to pick it up. That is, with a 1-bit ADC and a -20dB SNR, >the GPS signals aren't loud enough to ever cause a bit to toggle on their >own, why isn't the information lost? > >The signals from the GPS satellites are also very close in amplitude on the >ground, if they were substantially different, would you need more bits to >get enough dynamic range?Don't confuse the number of bits in the ADC with the number of bits processed for signal reception. Consider that the conversion rate of the ADC can be traded for more bits with processing gain, e.g, sample at a high rate, filter, decimate, etc. Eric Jacobsen Anchor Hill Communications www.anchorhill.com
Reply by ●July 4, 20122012-07-04
> >Don't confuse the number of bits in the ADC with the number of bits >processed for signal reception. Consider that the conversion rate of >the ADC can be traded for more bits with processing gain, e.g, sample >at a high rate, filter, decimate, etc. > > > > >Eric Jacobsen >Anchor Hill Communications >www.anchorhill.com >Hi Eric, I've definitely seen things like oversampling and then decimating to trade bandwidth for bits, I guess I'm just not understanding how any signal energy is preserved at all when the signal power is much less than .5 bits. It seems like it should always get rounded to zero and contribute nothing. Can you consider it as riding on top of the noise? Like the noise loads the A2D and the signal provides a teensy-weensy statistical wiggle on top of that that is sufficient to correlate?
Reply by ●July 4, 20122012-07-04
On 7/4/12 12:46 AM, gct wrote:>> >> Don't confuse the number of bits in the ADC with the number of bits >> processed for signal reception. Consider that the conversion rate of >> the ADC can be traded for more bits with processing gain, e.g, sample >> at a high rate, filter, decimate, etc. > > I've definitely seen things like oversampling and then decimating > to trade bandwidth for bits, I guess I'm just not understanding how any > signal energy is preserved at all when the signal power is much less than > .5 bits. It seems like it should always get rounded to zero and contribute > nothing. Can you consider it as riding on top of the noise? Like the > noise loads the A2D and the signal provides a teensy-weensy statistical > wiggle on top of that that is sufficient to correlate?a 1-bit converter isn't choosing between levels 0 and 1. it's between -1 and +1, or -V and +V. so, with a threshold at 0, it toggles between -1 and +1. and because a 1-bit converter has a negative feedback path, even if your input is slightly about 0 volts (midway between -1 and +1), it will not always go to +1. nearly half of the time it goes to -1, so that the average will be about the voltage of your input. -- r b-j rbj@audioimagination.com "Imagination is more important than knowledge."
Reply by ●July 4, 20122012-07-04
"gct" <smcallis@n_o_s_p_a_m.gmail.com> wrote in message news:Zo2dnZZ8cr_tVW7SnZ2dnUVZ_gudnZ2d@giganews.com...>I know it's a common technique to use a limiter (1-bit ADC) to digitize > GPS. I'm curious how it's possible to correlate a signal when it's 20dB > or > more below the noise floor and you don't have enough bits to give you the > dynamic range to pick it up. That is, with a 1-bit ADC and a -20dB SNR, > the GPS signals aren't loud enough to ever cause a bit to toggle on their > own, why isn't the information lost? > > The signals from the GPS satellites are also very close in amplitude on > the > ground, if they were substantially different, would you need more bits to > get enough dynamic range?All information left after hard limiting is in the phase, hence 3dB loss in SNR compared to ideal linear processing. This loss is traded for great simplification of the processing. Strong narrowband interference does represent a problem to such receivers, however that is not very important for many applications. VLV
Reply by ●July 4, 20122012-07-04
On Tue, 03 Jul 2012 23:37:04 -0500, gct wrote:> I know it's a common technique to use a limiter (1-bit ADC) to digitize > GPS. I'm curious how it's possible to correlate a signal when it's 20dB > or more below the noise floor and you don't have enough bits to give you > the dynamic range to pick it up. That is, with a 1-bit ADC and a -20dB > SNR, the GPS signals aren't loud enough to ever cause a bit to toggle on > their own, why isn't the information lost? > > The signals from the GPS satellites are also very close in amplitude on > the ground, if they were substantially different, would you need more > bits to get enough dynamic range?No one actually answered the question you're asking. Wow. The bandwidth is so wide that -- except for the strong interference that Vladimir was talking about -- in absolute terms all of the desired signals are swamped by noise. The noise would seem to be a bad thing -- except it is what makes the limiter end up acting, on average, in a more or less linear fashion with respect to the desired signals. Basically what happens in the time domain is that the limiter output is flipping back and forth madly in response to noise. The presence of a desired signal at the limiter affects the statistics of the limiter output in a way that pretty much means that -- as long as the limiter input is broad-band enough -- the signal shows up on the limiter output, a bit extra noisy and scaled by the total amount of noise on the limiter input, but otherwise unmolested. -- Tim Wescott Control system and signal processing consulting www.wescottdesign.com
Reply by ●July 4, 20122012-07-04
On Wed, 04 Jul 2012 12:40:02 -0500, Tim Wescott <tim@seemywebsite.please> wrote:>On Tue, 03 Jul 2012 23:37:04 -0500, gct wrote: > >> I know it's a common technique to use a limiter (1-bit ADC) to digitize >> GPS. I'm curious how it's possible to correlate a signal when it's 20dB >> or more below the noise floor and you don't have enough bits to give you >> the dynamic range to pick it up. That is, with a 1-bit ADC and a -20dB >> SNR, the GPS signals aren't loud enough to ever cause a bit to toggle on >> their own, why isn't the information lost? >> >> The signals from the GPS satellites are also very close in amplitude on >> the ground, if they were substantially different, would you need more >> bits to get enough dynamic range? > >No one actually answered the question you're asking. Wow.I think we did, just not completely.>The bandwidth is so wide that -- except for the strong interference that >Vladimir was talking about -- in absolute terms all of the desired >signals are swamped by noise. The noise would seem to be a bad thing -- >except it is what makes the limiter end up acting, on average, in a more >or less linear fashion with respect to the desired signals. > >Basically what happens in the time domain is that the limiter output is >flipping back and forth madly in response to noise. The presence of a >desired signal at the limiter affects the statistics of the limiter >output in a way that pretty much means that -- as long as the limiter >input is broad-band enough -- the signal shows up on the limiter output, >a bit extra noisy and scaled by the total amount of noise on the limiter >input, but otherwise unmolested.I think you're essentially describing dithering, which works the same whether it's a 1-bit converter or a twenty-bit converter when the signal is near the LSB (which it has to be with a 1-bit converter, I suspect). The idea is the same, trade sampling bandwidth for bits via processing gain. Eric Jacobsen Anchor Hill Communications www.anchorhill.com
Reply by ●July 4, 20122012-07-04
On Wed, 04 Jul 2012 18:21:25 +0000, Eric Jacobsen wrote:> On Wed, 04 Jul 2012 12:40:02 -0500, Tim Wescott > <tim@seemywebsite.please> wrote: > >>On Tue, 03 Jul 2012 23:37:04 -0500, gct wrote: >> >>> I know it's a common technique to use a limiter (1-bit ADC) to >>> digitize GPS. I'm curious how it's possible to correlate a signal >>> when it's 20dB or more below the noise floor and you don't have enough >>> bits to give you the dynamic range to pick it up. That is, with a >>> 1-bit ADC and a -20dB SNR, the GPS signals aren't loud enough to ever >>> cause a bit to toggle on their own, why isn't the information lost? >>> >>> The signals from the GPS satellites are also very close in amplitude >>> on the ground, if they were substantially different, would you need >>> more bits to get enough dynamic range? >> >>No one actually answered the question you're asking. Wow. > > I think we did, just not completely. > >>The bandwidth is so wide that -- except for the strong interference that >>Vladimir was talking about -- in absolute terms all of the desired >>signals are swamped by noise. The noise would seem to be a bad thing -- >>except it is what makes the limiter end up acting, on average, in a more >>or less linear fashion with respect to the desired signals. >> >>Basically what happens in the time domain is that the limiter output is >>flipping back and forth madly in response to noise. The presence of a >>desired signal at the limiter affects the statistics of the limiter >>output in a way that pretty much means that -- as long as the limiter >>input is broad-band enough -- the signal shows up on the limiter output, >>a bit extra noisy and scaled by the total amount of noise on the limiter >>input, but otherwise unmolested. > > I think you're essentially describing dithering, which works the same > whether it's a 1-bit converter or a twenty-bit converter when the signal > is near the LSB (which it has to be with a 1-bit converter, I suspect). > > The idea is the same, trade sampling bandwidth for bits via processing > gain.The only reason I wouldn't just call it "dithering" is because no dither signal is being intentionally introduced -- one is just going by whatever energy impinges on the antenna and whatever noise is generated in one's RF front end. If the word 'dither' came into the conversation I would admit that it is, indeed, dither of a sort -- but to my mind 'dither' conjures up a signal that is intentionally introduced, by a circuit block that was specifically designed to generate it. That's certainly not the case with ADCs. I've used this intrinsic dither extensively with high-speed ADCs, because such devices almost universally have enough noise that the last, and often the last few, bits are affected. On the down side this means that any one reading isn't going to match the expected number of bits, but on the bright side it means that -- to some extent -- the ADC nonlinearities are averaged out at the same time that a mechanism for extending resolution is handed to you for free. (And yes, y'all answered the guy's question in an academic sort of way, but I felt that you left out the part that would turn the light on for him). -- My liberal friends think I'm a conservative kook. My conservative friends think I'm a liberal kook. Why am I not happy that they have found common ground? Tim Wescott, Communications, Control, Circuits & Software http://www.wescottdesign.com
Reply by ●July 4, 20122012-07-04
Guys If both signals are 1 bit then you need to filter at least one of them before you multiply. Otherwise the high frequency components fold down and your result will be meaningless. Remember that a noise shaped 1 bit stream has very high dynamic range over a specific frequency range. Otherwise none of your iPods would work. It is actually possible to multiply 2 noise shaped 1bit streams and get a noise shaped result if the noise shapers were coupled in a particular way, but that is topic for another day Bob
Reply by ●July 5, 20122012-07-05
On 7/4/12 2:21 PM, Eric Jacobsen wrote:> On Wed, 04 Jul 2012 12:40:02 -0500, Tim Wescott > <tim@seemywebsite.please> wrote: >> >> Basically what happens in the time domain is that the limiter output is >> flipping back and forth madly in response to noise.wouldn't the 1-bit output flip back-and-forth madly for a DC input around zero (or whatever is halfway between the rails)?>> The presence of a >> desired signal at the limiter affects the statistics of the limiter >> output in a way that pretty much means that -- as long as the limiter >> input is broad-band enough -- the signal shows up on the limiter output, >> a bit extra noisy and scaled by the total amount of noise on the limiter >> input, but otherwise unmolested. > > I think you're essentially describing dithering, which works the same > whether it's a 1-bit converter or a twenty-bit converterEric, i don't think that is correct. a 1-bit converter is qualitatively different than any other positive number of bits. it's because the 1-bit converter does not really have a step size. it has rails, but there is no staircase function that has an ostensible slope and step-size to it to define the quantizer gain parameter and uniform p.d.f. additive noise parameter (you know, that (delta^2)/12 thing). if you have a multi-bit (flash) converter, you have a staircase function and you can conceptually "lay a plank" on the staircase and the slope of that plank is the gain of the quantizer. but try do use that to determine the gain of a 1-bit comparator? it's just one step and there is no unique slope for your plank. the size of that step is more comparable to what the rails values are than a quantizer step-size. and it is not clear what would be the correct or optimal dither strength for a 1-bit converter whereas we know that for a multi-bit converter the optimal dither is triangular of width of 2 LSBs and this turns the variance of the total quantization error from (delta^2)/12 to 3*(delta^2)/12, a 4.77 dB increase in noise power (and the payoff is that the mean and variance of the quantization error is completely decoupled from the actual value of the quantity being quantized).> when the signal is near the LSB (which it has to be with a 1-bit> converter, I suspect). no, Eric. i don't think that is the case. there is no LSB with a 1-bit converter. i'll bet Bob Adams will remember long ago (when i was living in Maine, ca. 1990) i drove down to Wilmington MA and talked with him about this "gain of the comparator" issue (you need to figure it out probabilistically) and later he pointed me to a 1987 paper from John Paulos (who was at Crystal semi, i think) that had a result that agreed with mine.> The idea is the same, trade sampling bandwidth for bits via processing > gain.but you get only one additional bit for each time you trade away 2 octaves of bandwidth. unless there's feedback like with sigma-delta. -- r b-j rbj@audioimagination.com "Imagination is more important than knowledge."
