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Generalized Channel Capacity Theorem?

Started by Randy Yates January 12, 2006
We all know and love Shannon's original channel capacity theorem

  C = W * log_2 (1 + P/N).

However, this assumes a white noise spectral density. Is there a
formulation for channel capacity which takes into account a colored
noise spectrum?
-- 
%  Randy Yates                  % "Maybe one day I'll feel her cold embrace,
%% Fuquay-Varina, NC            %                    and kiss her interface, 
%%% 919-577-9882                %            til then, I'll leave her alone."
%%%% <yates@ieee.org>           %        'Yours Truly, 2095', *Time*, ELO   
http://home.earthlink.net/~yatescr

Randy Yates wrote:

> We all know and love Shannon's original channel capacity theorem > > C = W * log_2 (1 + P/N). > > However, this assumes a white noise spectral density. Is there a > formulation for channel capacity which takes into account a colored > noise spectrum?
Isn't it obvious that you should integrate the SNR over the bandwidth? Vladimir Vassilevsky DSP and Mixed Signal Design Consultant http://www.abvolt.com
Vladimir Vassilevsky <antispam_bogus@hotmail.com> writes:

> Randy Yates wrote: > >> We all know and love Shannon's original channel capacity theorem >> C = W * log_2 (1 + P/N). >> However, this assumes a white noise spectral density. Is there a >> formulation for channel capacity which takes into account a colored >> noise spectrum? > > > Isn't it obvious that you should integrate the SNR over the bandwidth?
Isn't it conceivable that the model that led to the simple channel capacity model breaks down under a non-white noise spectrum? In other words, no, it's not obvious to me. -- % Randy Yates % "Rollin' and riding and slippin' and %% Fuquay-Varina, NC % sliding, it's magic." %%% 919-577-9882 % %%%% <yates@ieee.org> % 'Living' Thing', *A New World Record*, ELO http://home.earthlink.net/~yatescr
in article ek3dj6yw.fsf@ieee.org, Randy Yates at yates@ieee.org wrote on
01/12/2006 19:40 (slight modification):

> We all know and love Shannon's original channel capacity theorem > > C = BW * log_2(1 + S/N) = BW * log_2((S+N)/N) > > However, this assumes a white noise spectral density. Is there a > formulation for channel capacity which takes into account a colored > noise spectrum?
BW C = integral{ log_2(1 + S(f)/N(f)) df } 0 BW = integral{ log_2( (S(f)+N(f))/N(f) ) df } 0 to prove it, think about a bunch of channels that are adjacent and disjoint in frequency with different S/N, add up the channel capacity of each channel, and then think Riemann integration. -- r b-j rbj@audioimagination.com "Imagination is more important than knowledge."
robert bristow-johnson <rbj@audioimagination.com> writes:

> in article ek3dj6yw.fsf@ieee.org, Randy Yates at yates@ieee.org wrote on > 01/12/2006 19:40 (slight modification): > >> We all know and love Shannon's original channel capacity theorem >> >> C = BW * log_2(1 + S/N) = BW * log_2((S+N)/N)
Hey Robert! That's not what I wrote! Please don't do that!
>> However, this assumes a white noise spectral density. Is there a >> formulation for channel capacity which takes into account a colored >> noise spectrum? > > BW > C = integral{ log_2(1 + S(f)/N(f)) df } > 0 > > > BW > = integral{ log_2( (S(f)+N(f))/N(f) ) df } > 0 > > > to prove it, think about a bunch of channels that are adjacent and disjoint > in frequency with different S/N, add up the channel capacity of each > channel, and then think Riemann integration.
Seems right. In that case you'd want to shape the signal spectrum to follow the noise spectrum, given a fixed signal power. -- % Randy Yates % "Remember the good old 1980's, when %% Fuquay-Varina, NC % things were so uncomplicated?" %%% 919-577-9882 % 'Ticket To The Moon' %%%% <yates@ieee.org> % *Time*, Electric Light Orchestra http://home.earthlink.net/~yatescr
in article slrsj2lp.fsf@ieee.org, Randy Yates at yates@ieee.org wrote on
01/12/2006 21:14:

> robert bristow-johnson <rbj@audioimagination.com> writes: > >> in article ek3dj6yw.fsf@ieee.org, Randy Yates at yates@ieee.org wrote on >> 01/12/2006 19:40 (slight modification): >> >>> We all know and love Shannon's original channel capacity theorem >>> >>> C = BW * log_2(1 + S/N) = BW * log_2((S+N)/N) > > Hey Robert! That's not what I wrote! Please don't do that!
sorry. i *did* say "slight modification". i did it because i wanted to use different symbols and wanted consistency in terms. it's sorta like having my cake and eating it, too. (so i ate yours.)
> Seems right. In that case you'd want to shape the signal spectrum to > follow the noise spectrum, given a fixed signal power.
i think that true, but dunno for sure. it's a sorta matched filter thingie. -- r b-j rbj@audioimagination.com "Imagination is more important than knowledge."
robert bristow-johnson <rbj@audioimagination.com> writes:
> [...] > in article slrsj2lp.fsf@ieee.org, Randy Yates at yates@ieee.org wrote on > 01/12/2006 21:14: >> Hey Robert! That's not what I wrote! Please don't do that! > > sorry. i *did* say "slight modification".
Oh. I didn't notice that. I don't have a problem with that. -- % Randy Yates % "Rollin' and riding and slippin' and %% Fuquay-Varina, NC % sliding, it's magic." %%% 919-577-9882 % %%%% <yates@ieee.org> % 'Living' Thing', *A New World Record*, ELO http://home.earthlink.net/~yatescr
in article 64ooj17g.fsf@ieee.org, Randy Yates at yates@ieee.org wrote on
01/12/2006 21:44:

> robert bristow-johnson <rbj@audioimagination.com> writes: >> [...] >> in article slrsj2lp.fsf@ieee.org, Randy Yates at yates@ieee.org wrote on >> 01/12/2006 21:14: >>> Hey Robert! That's not what I wrote! Please don't do that! >> >> sorry. i *did* say "slight modification". > > Oh. I didn't notice that. I don't have a problem with that.
it's a bad habit that i started about a year ago. it sorta creeps up on you, first you "correct" the quotee's spelling errors, and then it gets worse. i think i'll call up the doc for my next electro-shock therapy. that'll cure it. -- r b-j rbj@audioimagination.com "Imagination is more important than knowledge."

Randy Yates wrote:

> robert bristow-johnson <rbj@audioimagination.com> writes: >
>>to prove it, think about a bunch of channels that are adjacent and disjoint >>in frequency with different S/N, add up the channel capacity of each >>channel, and then think Riemann integration. > > > Seems right. In that case you'd want to shape the signal spectrum to > follow the noise spectrum, given a fixed signal power.
Yes, this does make sense. There is another complication with the Shannon formula: it is assumed that the channel is stationary and the noise is additive gaussian. You can apply this formula only if you can convert the real channel to the equivalent LTI AWGN. VLV
Randy Yates wrote:
> Vladimir Vassilevsky <antispam_bogus@hotmail.com> writes: > > >>Randy Yates wrote: >> >> >>>We all know and love Shannon's original channel capacity theorem >>> C = W * log_2 (1 + P/N). >>>However, this assumes a white noise spectral density. Is there a >>>formulation for channel capacity which takes into account a colored >>>noise spectrum? >> >> >>Isn't it obvious that you should integrate the SNR over the bandwidth? > > > Isn't it conceivable that the model that led to the simple channel > capacity model breaks down under a non-white noise spectrum? > > In other words, no, it's not obvious to me.
Randy, I think white is the worst case /if the transmission coding takes the noise spectrum into account/. For example, if most of the noise is crammed into one part of the channel, don't use that part. Otherwise, put more power where more is needed -- i.e. match the signal's spectrum to the noise's. I forget where I ran across that. It may have been something by Mischa Schwartz. Jerry -- Engineering is the art of making what you want from things you can get. &#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;