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### Correlating GPS with a 1-bit A2D

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```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?
```
 0
Reply smcallis1 (7) 7/4/2012 4:37:04 AM

See related articles to this posting

```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
```
 0
Reply eric.jacobsen (2636) 7/4/2012 4:41:02 AM

```>
>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?
```
 0
Reply smcallis1 (7) 7/4/2012 4:46:20 AM

```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

--

r b-j                  rbj@audioimagination.com

"Imagination is more important than knowledge."

```
 0
Reply rbj (4090) 7/4/2012 4:53:20 AM

```"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

```
 0
Reply nospam (2805) 7/4/2012 6:41:21 AM

```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?

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
```
 0
Reply tim866 (481) 7/4/2012 5:40:02 PM

```On Wed, 04 Jul 2012 12:40:02 -0500, Tim Wescott

>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?
>

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
```
 0
Reply eric.jacobsen (2636) 7/4/2012 6:21:25 PM

```On Wed, 04 Jul 2012 18:21:25 +0000, Eric Jacobsen wrote:

> On Wed, 04 Jul 2012 12:40:02 -0500, Tim Wescott
>
>>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?
>>
>
> 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

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

(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
```
 0
Reply tim177 (4434) 7/4/2012 8:01:11 PM

```Guys

If both signals are 1 bit then you need to filter at least one of them befo=
re 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 =

It is actually possible to multiply 2 noise shaped 1bit streams and get a n=
oise shaped result if the noise shapers were coupled in a particular way, b=
ut that is topic for another day

Bob
```
 0

```On 7/4/12 2:21 PM, Eric Jacobsen wrote:
> On Wed, 04 Jul 2012 12:40:02 -0500, Tim Wescott
>>
>> 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 converter

Eric, 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
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."

```
 0
Reply rbj (4090) 7/5/2012 5:30:13 AM

```I think I mis-interpreted what the op meant by "1 bit ADC". To me this usua=
lly means a feedback sigma-delta converter but in this case it is more like=
ly just a comparator stuck at the end of an IF strip. I would also assume s=
ince this is a gps signal that you are looking for time of arrival differen=
ces by searching for the peak cross-correlation which can be done with a si=
mple XOR gate on the 2 1-bit signals followed by a filter or a average over=
a given number of samples.  So I guess even when the input to one of the c=
omparators has low SNR there will still be some residual correlation at the=
filtered output that can be detected. If the SNR is very bad then you migh=
t need to correlate for a long time to get a reliable result.=20

Bob
```
 0

```On Thu, 05 Jul 2012 05:00:54 -0700, Robert Adams wrote:

> I think I mis-interpreted what the op meant by "1 bit ADC". To me this
> usually means a feedback sigma-delta converter but in this case it is
> more likely just a comparator stuck at the end of an IF strip. I would
> also assume since this is a gps signal that you are looking for time of
> arrival differences by searching for the peak cross-correlation which
> can be done with a simple XOR gate on the 2 1-bit signals followed by a
> filter or a average over a given number of samples.  So I guess even
> when the input to one of the comparators has low SNR there will still be
> some residual correlation at the filtered output that can be detected.
> If the SNR is very bad then you might need to correlate for a long time
> to get a reliable result.
>
> Bob

You need to study up a bit on GPS.

early in the signal chain, followed by correlation and a _whole lot_ of
coding gain.

--
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
```
 0
Reply tim177 (4434) 7/5/2012 4:35:35 PM

```On Wed, 04 Jul 2012 14:44:10 -0700, Robert Adams wrote:

> 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.

And yet, the technique still works.

> 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

Who said anything about noise shaping?

--
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
```
 0
Reply tim177 (4434) 7/5/2012 4:37:03 PM

```On Thu, 05 Jul 2012 11:35:35 -0500, Tim Wescott <tim@seemywebsite.com>
wrote:

>On Thu, 05 Jul 2012 05:00:54 -0700, Robert Adams wrote:
>
>> I think I mis-interpreted what the op meant by "1 bit ADC". To me this
>> usually means a feedback sigma-delta converter but in this case it is
>> more likely just a comparator stuck at the end of an IF strip. I would
>> also assume since this is a gps signal that you are looking for time of
>> arrival differences by searching for the peak cross-correlation which
>> can be done with a simple XOR gate on the 2 1-bit signals followed by a
>> filter or a average over a given number of samples.  So I guess even
>> when the input to one of the comparators has low SNR there will still be
>> some residual correlation at the filtered output that can be detected.
>> If the SNR is very bad then you might need to correlate for a long time
>> to get a reliable result.
>>
>> Bob
>
>You need to study up a bit on GPS.
>
>early in the signal chain, followed by correlation and a _whole lot_ of
>coding gain.

That's the whole idea right there.  The processing gain provides the
dynamic range for signal separation.

Eric Jacobsen
Anchor Hill Communications
www.anchorhill.com
```
 0
Reply eric.jacobsen (2636) 7/5/2012 5:33:33 PM

```You're right, not my area. I just get excited when I see the term "1bit a/d" and assume its the kind I actually know something about.
```
 0

```On Thu, 05 Jul 2012 01:30:13 -0400, robert bristow-johnson
<rbj@audioimagination.com> wrote:

>On 7/4/12 2:21 PM, Eric Jacobsen wrote:
>> On Wed, 04 Jul 2012 12:40:02 -0500, Tim Wescott
>>>
>>> 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)?

Ideally there aren't rails, just a threshold above which the output is
1 and below which the output is 0, or vice-versa.   It can be thought
of as the sign bit of a two's complement converter.

>>>  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
>
>Eric, 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.

In an rf system the dynamic range at the converter location will be
known by the system design, e.g., an AGC controlled +/- 1vpp or
whatever.  So the quantization step is constrained to a known range
and one can think of it that if it were beyond that range then a
second bit would be needed.  This bounds the problem in a useful way.

>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).

It doesn't have to be optimum to be useful or relevant.

>> 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
>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.

As I mention above there's a viewpoint that can nicely bound the
problem to lend itself to useful analysis.

>
>> 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.

There are lots of ways to get processing gain.

Eric Jacobsen
Anchor Hill Communications
www.anchorhill.com
```
 0
Reply eric.jacobsen (2636) 7/5/2012 7:59:42 PM

```On 7/5/12 3:59 PM, Eric Jacobsen wrote:
> On Thu, 05 Jul 2012 01:30:13 -0400, robert bristow-johnson
> <rbj@audioimagination.com>  wrote:
>
>> On 7/4/12 2:21 PM, Eric Jacobsen wrote:
>>> On Wed, 04 Jul 2012 12:40:02 -0500, Tim Wescott
>>>>
>>>> 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)?
>
> Ideally there aren't rails, just a threshold above which the output is
> 1 and below which the output is 0, or vice-versa.   It can be thought
> of as the sign bit of a two's complement converter.
>

yup.  i would say "just a threshold above which the output is 1 and
below which the output is [-1] or vice-versa." not 0.  sorta like (-1)^bit.

anyway, the output rails here are +1 and -1.  even with dither added, if
your input level gets high enough (or negative enough), the output is
pinned against either +1 or -1, and those levels of the input indicate
where the input rails are.  and you can't increase the dither amplitude
indefinitely to widen out the space between the rails because eventually
the dither noise will be little else than noise and your signal is
drowned out.

>>>>   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
>>
>> Eric, 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.
>
> In an rf system the dynamic range at the converter location will be
> known by the system design, e.g., an AGC controlled +/- 1vpp or
> whatever.

sure.

>  So the quantization step is constrained to a known range
> and one can think of it that if it were beyond that range then a
> second bit would be needed.

with 1-bit, there really *isn't* a quantization step (from the POV of
the input, in the "staircase function" this would be the horizontal
length of the step).  if the input is + you get +1 at the output, and if
the input is - you get -1.  but, until you add that second bit, there is
no quantization step for the input which is essentially why 1-bit flash
A/D is qualitatively different than a 2-bit or 3-bit or more-bit A/D.

>  This bounds the problem in a useful way.

i'm trying to understand what the bound you mean is and what is useful.

>
>> 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).
>
> It doesn't have to be optimum to be useful or relevant.

that's true.  but i remember back in the 90s when this was discussed a
lot in the AES because Crystal Semiconductor and Analog Devices (i.e.
Bob Adams and crew) were beating each other up a little about the
first-generation 1-bit audio converters that were all the rage back
then, that the amount of dither and the p.d.f. of the dither (the
triangular p.d.f. no longer was important) became a much discussed issue.

>>> 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
>> 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.
>
> As I mention above there's a viewpoint that can nicely bound the
> problem to lend itself to useful analysis.

i wouldn't mind understanding that viewpoint better.  admittedly, i am
audio-centric, but the issues should apply to other uses as well.

>>> 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.
>
> There are lots of ways to get processing gain.

you mean S/N on the data?

bandwidth-reduction, sample rate increase, increasing acquisition time,
additional A/D converters (with independent dither), getting more bits
in the A/D.  can't think of much else.

1-bit converters for audio).  but, as far as i can tell, the issues
transcend audio vs. RF vs. whatever application space.

--

r b-j                  rbj@audioimagination.com

"Imagination is more important than knowledge."

```
 0
Reply rbj (4090) 7/5/2012 9:29:38 PM

```On Thu, 05 Jul 2012 17:29:38 -0400, robert bristow-johnson
<rbj@audioimagination.com> wrote:

>On 7/5/12 3:59 PM, Eric Jacobsen wrote:
>> On Thu, 05 Jul 2012 01:30:13 -0400, robert bristow-johnson
>> <rbj@audioimagination.com>  wrote:
>>
>>> On 7/4/12 2:21 PM, Eric Jacobsen wrote:
>>>> On Wed, 04 Jul 2012 12:40:02 -0500, Tim Wescott
>>>>>
>>>>> 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)?
>>
>> Ideally there aren't rails, just a threshold above which the output is
>> 1 and below which the output is 0, or vice-versa.   It can be thought
>> of as the sign bit of a two's complement converter.
>>
>
>yup.  i would say "just a threshold above which the output is 1 and
>below which the output is [-1] or vice-versa." not 0.  sorta like (-1)^bit.
>
>anyway, the output rails here are +1 and -1.  even with dither added, if
>your input level gets high enough (or negative enough), the output is
>pinned against either +1 or -1, and those levels of the input indicate
>where the input rails are.  and you can't increase the dither amplitude
>indefinitely to widen out the space between the rails because eventually
>the dither noise will be little else than noise and your signal is
>drowned out.
>
>>>>>   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
>>>
>>> Eric, 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.
>>
>> In an rf system the dynamic range at the converter location will be
>> known by the system design, e.g., an AGC controlled +/- 1vpp or
>> whatever.
>
>sure.
>
>>  So the quantization step is constrained to a known range
>> and one can think of it that if it were beyond that range then a
>> second bit would be needed.
>
>with 1-bit, there really *isn't* a quantization step (from the POV of
>the input, in the "staircase function" this would be the horizontal
>length of the step).  if the input is + you get +1 at the output, and if
>the input is - you get -1.  but, until you add that second bit, there is
>no quantization step for the input which is essentially why 1-bit flash
>A/D is qualitatively different than a 2-bit or 3-bit or more-bit A/D.
>
>>  This bounds the problem in a useful way.
>
>i'm trying to understand what the bound you mean is and what is useful.

In the relevant receiver architectures the benefits of using a hard
limiter depend on a few things, but generally the idea is that one can
get rid of expenses like an AGC and ADC.   However, the output of the
limiter, with certain important but practical assumptions, is
indistinguishable from a multiple-bit ADC with an AGC circuit that
keeps the quantization limited to one LSB.  It is then pretty easy to
analyze with more mundane and linear analysis.

>>
>>> 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).
>>
>> It doesn't have to be optimum to be useful or relevant.
>
>that's true.  but i remember back in the 90s when this was discussed a
>lot in the AES because Crystal Semiconductor and Analog Devices (i.e.
>Bob Adams and crew) were beating each other up a little about the
>first-generation 1-bit audio converters that were all the rage back
>then, that the amount of dither and the p.d.f. of the dither (the
>triangular p.d.f. no longer was important) became a much discussed issue.
>
>>>> 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
>>> 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.
>>
>> As I mention above there's a viewpoint that can nicely bound the
>> problem to lend itself to useful analysis.
>
>i wouldn't mind understanding that viewpoint better.  admittedly, i am
>audio-centric, but the issues should apply to other uses as well.
>
>>>> 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.
>>
>> There are lots of ways to get processing gain.
>
>you mean S/N on the data?
>
>bandwidth-reduction, sample rate increase, increasing acquisition time,
>additional A/D converters (with independent dither), getting more bits
>in the A/D.  can't think of much else.

Correlation, for one.   The signal is spread-spectrum and the
correlator looking for the relevant spreading code has high gain.
This is what makes the system practical with a limiter as the front
end.

>1-bit converters for audio).  but, as far as i can tell, the issues
>transcend audio vs. RF vs. whatever application space.

The signal processing concepts are general, as they usually are, but
knowing how it all works together can be tricky, as usual.

Eric Jacobsen
Anchor Hill Communications
www.anchorhill.com
```
 0
Reply eric.jacobsen (2636) 7/5/2012 9:46:36 PM

```"gct" <smcallis@n_o_s_p_a_m.gmail.com> writes:

> 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?

Because any quantizer, with the proper amount and type of dither at its
input, is indistinguishable from an analog signal with wideband, benign
noise, and with such signals the noise in the system is sufficient to
act as dither.

So the question then becomes, "How can I detect a signal with such a low
--
Randy Yates
Digital Signal Labs
http://www.digitalsignallabs.com
```
 0
Reply yates9428 (616) 7/7/2012 8:49:40 AM

```robert bristow-johnson <rbj@audioimagination.com> writes:
> [...]
> On 7/5/12 3:59 PM, Eric Jacobsen wrote:
>>  So the quantization step is constrained to a known range
>> and one can think of it that if it were beyond that range then a
>> second bit would be needed.
>
> with 1-bit, there really *isn't* a quantization step (from the POV of
> the input, in the "staircase function" this would be the horizontal
> length of the step).  if the input is + you get +1 at the output, and
> if the input is - you get -1.  but, until you add that second bit,
> there is no quantization step for the input which is essentially why
> 1-bit flash A/D is qualitatively different than a 2-bit or 3-bit or
> more-bit A/D.

In a general bipolar N-bit converter with rails at +/- A, the
quantization step is A / 2 ^ (N - 1). So for a 1-bit converter, why
isn't the quantization step A? That is, A is the analog voltage that
the +1 in digital corresponds to.

But you say (rightly so) that there IS no A. It's not like the multi-bit
converter which "saturates" when the input magnitude is greater than A.

OK, so here's my conjecture: the dither noise (which necessarily
corresponds to an analog quantization step) you choose to use sets A.

> [...]
> that's true.  but i remember back in the 90s when this was discussed a
> lot in the AES because Crystal Semiconductor and Analog Devices (i.e.
> Bob Adams and crew) were beating each other up a little about the
> first-generation 1-bit audio converters that were all the rage back
> then, that the amount of dither and the p.d.f. of the dither (the
> triangular p.d.f. no longer was important) became a much discussed
> issue.

As far as I can tell, a one-bit A/D still requires the triangular PDF.
It's just that the amount depends on your desired rail.
--
Randy Yates
Digital Signal Labs
http://www.digitalsignallabs.com
```
 0
Reply yates9428 (616) 7/7/2012 9:04:35 AM

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