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. > > SteveI 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.

# uniform scalar quantizer

Started by ●September 11, 2008

Reply by ●September 16, 20082008-09-16

Reply by ●September 16, 20082008-09-16

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

Reply by ●September 16, 20082008-09-16

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.

Reply by ●September 16, 20082008-09-16

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