Thomas Arildsen  <tha.es-aau-dk@spamgourmet.com> wrote:

>On Tue, 16 Sep 2008 09:14:12 +0000, Steve Pope wrote:

>> Thomas Arildsen  <tha.es-aau-dk@spamgourmet.com> wrote:

>>>I am trying to select the step size, for a uniform scalar quantizer,
>>>that will minimize the mean squared error between the quantized and the
>>>original signal. The original signal is Gaussian of known variance (and
>>>mean - let's just say zero for convenience).

>> You're still leaving out a design input parameter -- perhaps the number
>> of bits in the quantizer.  If this is unconstrained, then the answer is
>> to make the step size very small.

>I didn't mean to leave that out, sorry. I also assume I know which number 
>of bits I want.

Thanks.   Thus you are trying to minimize the MS error
in the presence of both quantizing error and saturation, for
Gaussian input.

A closed form expression for this is a nice exercise, but in
practical terms it is both trivial and necessary to simulate, so in 
a practical sense the only reason to pursue an analytic solution is
curiosity.   The real-world signal will be bandlimited,
and the real-world ADC will exhibit suboptimalities which will
shift the optimum level.

Steve

On Tue, 16 Sep 2008 09:14:12 +0000, Steve Pope wrote:

> Thomas Arildsen  <tha.es-aau-dk@spamgourmet.com> wrote:
> 
cut...
> 
>>I am trying to select the step size, for a uniform scalar quantizer,
>>that will minimize the mean squared error between the quantized and the
>>original signal. The original signal is Gaussian of known variance (and
>>mean - let's just say zero for convenience).
> 
> You're still leaving out a design input parameter -- perhaps the number
> of bits in the quantizer.  If this is unconstrained, then the answer is
> to make the step size very small.
> 

I didn't mean to leave that out, sorry. I also assume I know which number 
of bits I want.

Thomas Arildsen
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Thomas Arildsen  <tha.es-aau-dk@spamgourmet.com> wrote:

>On Mon, 15 Sep 2008 16:49:29 +0000, Steve Pope wrote:

>> Thomas Arildsen  <tha.es-aau-dk@spamgourmet.com> wrote:

>>>I was thinking MSE-optimal. I was looking for a more formal
>>>specification and a reference to cite, but this seems to confirm my
>>>suspicion that this is hard to find. Well, thanks anyway.

>> I'm still not sure you have said what you are trying to optimize.

>I am trying to select the step size, for a uniform scalar quantizer, that 
>will minimize the mean squared error between the quantized and the 
>original signal. The original signal is Gaussian of known variance (and 
>mean - let's just say zero for convenience).

You're still leaving out a design input parameter -- perhaps the number
of bits in the quantizer.  If this is unconstrained, then the
answer is to make the step size very small.

>Well, I seem to have found what I was looking for in Bucklew & Gallagher, 
>"Some Properties of Uniform Step Size Quantizers", 1980 (the example in 
>the discussion of Property 8). Specific quantizers are also listed in 
>Jayant & Noll, "Digital Coding of Waveforms", Table 4.1, quoted from Max, 
>"Quantizing for Minimum Distortion", 1960.

Cool

Steve

On Mon, 15 Sep 2008 16:49:29 +0000, Steve Pope wrote:

> Thomas Arildsen  <tha.es-aau-dk@spamgourmet.com> wrote:
> 
>>On Fri, 12 Sep 2008 16:13:29 +0000, Steve Pope wrote:
> 
>>> It's not clear what optimum means in the above, but it could mean the
>>> RMS error from quantizing is on the same order as the RMS erros from
>>> saturation of the ADC.
> 
>>I was thinking MSE-optimal. I was looking for a more formal
>>specification and a reference to cite, but this seems to confirm my
>>suspicion that this is hard to find. Well, thanks anyway.
> 
> I'm still not sure you have said what you are trying to optimize.
> 
> Steve

I am trying to select the step size, for a uniform scalar quantizer, that 
will minimize the mean squared error between the quantized and the 
original signal. The original signal is Gaussian of known variance (and 
mean - let's just say zero for convenience).
Well, I seem to have found what I was looking for in Bucklew & Gallagher, 
"Some Properties of Uniform Step Size Quantizers", 1980 (the example in 
the discussion of Property 8). Specific quantizers are also listed in 
Jayant & Noll, "Digital Coding of Waveforms", Table 4.1, quoted from Max, 
"Quantizing for Minimum Distortion", 1960.

Thomas Arildsen
-- 
All email to sender address is lost.
My real adress is at es dot aau dot dk for user tha.

On Mon, 15 Sep 2008 13:02:43 -0400, Randy Yates wrote:

> Randy Yates <yates@ieee.org> writes:
> 
>> Thomas Arildsen <tha.es-aau-dk@spamgourmet.com> writes:
>>
>>> How does one design the optimum uniform scalar quantizer for a
>>> specific Gaussian input?
>>> All the literature I can find on quantization seems to deal with
>>> analyzing specific quantizers, quantization noise probability
>>> densities and power spectra etc., but I cannot seem to find any
>>> specifics on just the design - selecting delta. I have seen it derived
>>> under entropy constraints, but I would like to know the approach
>>> without entropy coding after the quantizer.
>>
>> I don't know if it has what you're looking for, but there is some
>> coverage of quantizers in the rate-distortion chapter of [cover].
>>
>> --Randy
>>
>> @book{cover,
>>   title = "Elements of Information Theory", author = "Thomas M. Cover
>>   and Joy A. Thomas", publisher = "John Wiley and Sons, Inc.", year =
>>   "1991"}
> 
> PS: There is a newer edition.

Thanks, I have the 2006 edition. I looked at it but couldn't find the 
specific details. I did however find what I was looking, see my other 
post.

Thomas Arildsen
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Randy Yates <yates@ieee.org> writes:

> Thomas Arildsen <tha.es-aau-dk@spamgourmet.com> writes:
>
>> How does one design the optimum uniform scalar quantizer for a specific 
>> Gaussian input?
>> All the literature I can find on quantization seems to deal with 
>> analyzing specific quantizers, quantization noise probability densities 
>> and power spectra etc., but I cannot seem to find any specifics on just 
>> the design - selecting delta. I have seen it derived under entropy 
>> constraints, but I would like to know the approach without entropy coding 
>> after the quantizer.
>
> I don't know if it has what you're looking for, but there is some
> coverage of quantizers in the rate-distortion chapter of [cover].
>
> --Randy
>
> @book{cover,
>   title = "Elements of Information Theory",
>   author = "Thomas M. Cover and Joy A. Thomas",
>   publisher = "John Wiley and Sons, Inc.",
>   year = "1991"}

PS: There is a newer edition.
-- 
%  Randy Yates                  % "My Shangri-la has gone away, fading like 
%% Fuquay-Varina, NC            %  the Beatles on 'Hey Jude'" 
%%% 919-577-9882                %  
%%%% <yates@ieee.org>           % 'Shangri-La', *A New World Record*, ELO
http://www.digitalsignallabs.com

Thomas Arildsen <tha.es-aau-dk@spamgourmet.com> writes:

> How does one design the optimum uniform scalar quantizer for a specific 
> Gaussian input?
> All the literature I can find on quantization seems to deal with 
> analyzing specific quantizers, quantization noise probability densities 
> and power spectra etc., but I cannot seem to find any specifics on just 
> the design - selecting delta. I have seen it derived under entropy 
> constraints, but I would like to know the approach without entropy coding 
> after the quantizer.

I don't know if it has what you're looking for, but there is some
coverage of quantizers in the rate-distortion chapter of [cover].

--Randy

@book{cover,
  title = "Elements of Information Theory",
  author = "Thomas M. Cover and Joy A. Thomas",
  publisher = "John Wiley and Sons, Inc.",
  year = "1991"}

-- 
%  Randy Yates                  % "Ticket to the moon, flight leaves here today 
%% Fuquay-Varina, NC            %  from Satellite 2"
%%% 919-577-9882                % 'Ticket To The Moon' 
%%%% <yates@ieee.org>           % *Time*, Electric Light Orchestra
http://www.digitalsignallabs.com

Thomas Arildsen  <tha.es-aau-dk@spamgourmet.com> wrote:

>On Fri, 12 Sep 2008 16:13:29 +0000, Steve Pope wrote:

>> It's not clear what optimum means in the above, but it could mean the
>> RMS error from quantizing is on the same order as the RMS erros from
>> saturation of the ADC.

>I was thinking MSE-optimal. I was looking for a more formal specification 
>and a reference to cite, but this seems to confirm my suspicion that this 
>is hard to find. Well, thanks anyway.

I'm still not sure you have said what you are trying to optimize.

Steve

On Fri, 12 Sep 2008 16:13:29 +0000, Steve Pope wrote:

> Vladimir Vassilevsky  <antispam_bogus@hotmail.com> wrote:
> 
>>Thomas Arildsen wrote:
> 
>>> How does one design the optimum uniform scalar quantizer for a
>>> specific Gaussian input?
> 
>>By the tedious consideration of the sum of the probable errors at every
>>quantization step.
> 
>>For the sufficiently high number of the quantization steps, the RMS of
>>gaussian should be about 1/3...1/4 of the full scale of the ADC.
> 
> It's not clear what optimum means in the above, but it could mean the
> RMS error from quantizing is on the same order as the RMS erros from
> saturation of the ADC.
> 
> I agree with your ballpark range.  ("Full scale" being 1/2 of the
> rail-to-rail ADC range.)
> 
> Steve

I was thinking MSE-optimal. I was looking for a more formal specification 
and a reference to cite, but this seems to confirm my suspicion that this 
is hard to find. Well, thanks anyway.

Thomas Arildsen

-- 
All email to sender address is lost.
My real adress is at es dot aau dot dk for user tha.

Vladimir Vassilevsky  <antispam_bogus@hotmail.com> wrote:

>Thomas Arildsen wrote:

>> How does one design the optimum uniform scalar quantizer for a specific 
>> Gaussian input?

>By the tedious consideration of the sum of the probable errors at every 
>quantization step.

>For the sufficiently high number of the quantization steps, the RMS of 
>gaussian should be about 1/3...1/4 of the full scale of the ADC.

It's not clear what optimum means in the above, but it
could mean the RMS error from quantizing is on the same order
as the RMS erros from saturation of the ADC.

I agree with your ballpark range.  ("Full scale" being 1/2 of
the rail-to-rail ADC range.)

Steve