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. 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 11, 20082008-09-11
On Sep 11, 12:25�pm, Thomas Arildsen <tha.es-aau...@spamgourmet.com> wrote:> 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. > > Thomas ArildsenThe starting point will have to be Lloyd's algorithm: # Stuart P. Lloyd. Least Squares Quantization in PCM. IEEE Transactions on Information Theory, vol. 28, no. 2, pp. 129-137, 1982. # J. Sabin and R. Gray. Global Convergence and Empirical Consistency of the Generalized Lloyd Algorithm. IEEE Transactions on Information Theory , vol. 32, no. 2, pp. 148-155, 1986. Gray and Gersho's "Vector Quantization and Signal Compression" book is a very good reference on this topic. Hope this helps, Julius
Reply by ●September 12, 20082008-09-12
On Thu, 11 Sep 2008 18:42:35 -0700, julius wrote:> On Sep 11, 12:25 pm, Thomas Arildsen <tha.es-aau...@spamgourmet.com> > wrote: >> 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. >> >> Thomas Arildsen > > The starting point will have to be Lloyd's algorithm: > > # Stuart P. Lloyd. Least Squares Quantization in PCM. IEEE Transactions > on Information Theory, vol. 28, no. 2, pp. 129-137, 1982. # J. Sabin and > R. Gray. Global Convergence and Empirical Consistency of the Generalized > Lloyd Algorithm. IEEE Transactions on Information Theory , vol. 32, no. > 2, pp. 148-155, 1986.I know Lloyd's algorithm - it's not a uniform quantizer.> Gray and Gersho's "Vector Quantization and Signal Compression" book is a > very good reference on this topic.I have this book. However I can't really figure out how to derive "given a Gaussian input of variance x, here's how to calculate delta" from anything in it. 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 12, 20082008-09-12
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. Vladimir Vassilevsky DSP and Mixed Signal Design Consultant http://www.abvolt.com
Reply by ●September 12, 20082008-09-12
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
Reply by ●September 15, 20082008-09-15
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.) > > SteveI 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.
Reply by ●September 15, 20082008-09-15
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
Reply by ●September 15, 20082008-09-15
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
Reply by ●September 15, 20082008-09-15
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
Reply by ●September 16, 20082008-09-16
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 -- All email to sender address is lost. My real adress is at es dot aau dot dk for user tha.
