# Generalized Channel Capacity Theorem?

Started by 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
```
```
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?

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
```
```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
```
```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
```
```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:
>
>
>>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.
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```