uniform scalar quantizer

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

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