Determining the Maximum Power Through a Band-Limited, Fixed-Point Signal Path

Hi Folks,

Here's a problem I've been scratching my head on for awhile now. It may
be obvious and I'm not seeing it. 

Assuming a fixed-point signal path, how would you determine the maximum
power that can sent through that signal path when it is band-limited?

For the sake of notationaly simplicity, assume the signal is scaled
such that full-scale is +/- 1.

For example, let's say I have a 16-bit two's complement signal path,
and it's band-limited to 1/4 of Nyquist. What is the maximum signal
power I can transmit through such a path? 

If anyone has any suggestions on how to solve this problem, I would
appreciate hearing them.

Here's how I'm approaching it, but it doesn't "feel" right. I assume
the input signal consists of independent samples uniformly distributed
with the maximum "spread" that will fit, i.e., the pdf is 1/2 from -1
to +1.

Since the samples are uniform, they're "white" over the Nyquist
bandwidth. I take this as the maximum power through the channel.

I then pass that through an FIR that defines the band-limiting. 
I then reason that each tap b_n in the filter changes the signal power 
by b_n^2 ([garcia], p.147). Also since the samples are independent,
the output variance is as follows:

  \sigma^2_y = \sigma^2_x \sum_{n=0}^{N-1} b_n^2

(Assume zero-mean).

This doesn't make me happy because, depending on the coefficients,
one can actually get more power out than in, EVEN when the filter
has unity gain in the passband. Arg. 

The one thing I haven't accounted for is that they output samples
are "spread" due to the summing, and by the Central Limit Theorem
approach Gaussian. So in reality you'd have to back off the filter
MACs so that you don't overflow. 

Maybe this is the totally wrong approach anyway. Any help is
appreciated.
-- 
%  Randy Yates                  % "Bird, on the wing,
%% Fuquay-Varina, NC            %   goes floating by
%%% 919-577-9882                %   but there's a teardrop in his eye..."
%%%% <yates@ieee.org>           % 'One Summer Dream', *Face The Music*, ELO
http://home.earthlink.net/~yatescr

Randy Yates <yates@ieee.org> writes:

> Hi Folks,
>
> Here's a problem I've been scratching my head on for awhile now. It may
> be obvious and I'm not seeing it. 
>
> Assuming a fixed-point signal path, how would you determine the maximum
> power that can sent through that signal path when it is band-limited?
>
> For the sake of notationaly simplicity, assume the signal is scaled
> such that full-scale is +/- 1.
>
> For example, let's say I have a 16-bit two's complement signal path,
> and it's band-limited to 1/4 of Nyquist. What is the maximum signal
> power I can transmit through such a path? 
>
> If anyone has any suggestions on how to solve this problem, I would
> appreciate hearing them.
>
> Here's how I'm approaching it, but it doesn't "feel" right. I assume
> the input signal consists of independent samples uniformly distributed
> with the maximum "spread" that will fit, i.e., the pdf is 1/2 from -1
> to +1.
>
> Since the samples are uniform, they're "white" over the Nyquist
> bandwidth. I take this as the maximum power through the channel.
>
> I then pass that through an FIR that defines the band-limiting. 
> I then reason that each tap b_n in the filter changes the signal power 
> by b_n^2 ([garcia], p.147). Also since the samples are independent,
> the output variance is as follows:
>
>   \sigma^2_y = \sigma^2_x \sum_{n=0}^{N-1} b_n^2

([garcia], p.272).

>
> (Assume zero-mean).
>
> This doesn't make me happy because, depending on the coefficients,
> one can actually get more power out than in, EVEN when the filter
> has unity gain in the passband. Arg. 
>
> The one thing I haven't accounted for is that they output samples
> are "spread" due to the summing, and by the Central Limit Theorem
> approach Gaussian. So in reality you'd have to back off the filter
> MACs so that you don't overflow. 
>
> Maybe this is the totally wrong approach anyway. Any help is
> appreciated.

@book{garcia,
  title = "Probability and Random Processes for Electrical Engineering",
  author = "{Alberto~Leon-Garcia}",
  publisher = "Addison-Wesley",
  year = "1989"}

-- 
%  Randy Yates                  % "She's sweet on Wagner-I think she'd die for Beethoven.
%% Fuquay-Varina, NC            %  She love the way Puccini lays down a tune, and
%%% 919-577-9882                %  Verdi's always creepin' from her room." 
%%%% <yates@ieee.org>           % "Rockaria", *A New World Record*, ELO   
http://home.earthlink.net/~yatescr

hi Randy,

>
> Assuming a fixed-point signal path, how would you determine the maximum
> power that can sent through that signal path when it is band-limited?

the signal *path* is bandlimited, not the signal?

> For the sake of notationaly simplicity, assume the signal is scaled
> such that full-scale is +/- 1.
>
> For example, let's say I have a 16-bit two's complement signal path,
> and it's band-limited to 1/4 of Nyquist. What is the maximum signal
> power I can transmit through such a path?

i still think that the maximum signal power is still the same, (+/-
1)^2 or 1.  maybe it's (+/- 1)^2 / 4 or 1/4.

if your (received) signal is bandlimited to 1/4 Nyquist one out of
every 4 samples is needed to adequately represent the signal and the
other 3 are are actually functions of the 1/4 of the samples that are
independently defined.  so, in the frequency domain, with no channel
bandlimiting, the maximum power samples are all +/- 1 and you can
define them (using maximum length sequences) so that the spectrum is
full and flat all the way to Nyquist.  then filter that with a
brickwall to Nyquist/4 and your power has to go down to 1/4.  the same
original analog signal that was toggling +/- 1 could be filtered and
sampled at 1/4 of the rate (but now we don't know if the filtered
signal remains between +/- 1).

dunno, Randy.  it's a good question.

r b-j

Randy Yates <yates@ieee.org> writes:

> Since the samples are uniform, they're "white" over the Nyquist
> bandwidth. 

Doh! I meant to say this:

  Since the samples are independent, they're "white" over the Nyquist
  bandwidth. 
-- 
%  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://home.earthlink.net/~yatescr

On Jun 25, 4:52 pm, Randy Yates <y...@ieee.org> wrote:
> Hi Folks,
>
> Here's a problem I've been scratching my head on for awhile now. It may
> be obvious and I'm not seeing it.
>
> Assuming a fixed-point signal path, how would you determine the maximum
> power that can sent through that signal path when it is band-limited?
>
> For the sake of notationaly simplicity, assume the signal is scaled
> such that full-scale is +/- 1.
>
> For example, let's say I have a 16-bit two's complement signal path,
> and it's band-limited to 1/4 of Nyquist. What is the maximum signal
> power I can transmit through such a path?
>
> If anyone has any suggestions on how to solve this problem, I would
> appreciate hearing them.
>

A constant value of 1.0 or -1.0 would put through the maximum
power and be band limited to a tiny fraction of the Nyquist frequency.

> Here's how I'm approaching it, but it doesn't "feel" right. I assume
> the input signal consists of independent samples uniformly distributed
> with the maximum "spread" that will fit, i.e., the pdf is 1/2 from -1
> to +1.
>
> Since the samples are uniform, they're "white" over the Nyquist
> bandwidth. I take this as the maximum power through the channel.
>

What do whiteness or uniformity have to do with maximum power?
Is this a requirement due to something left out of the problem
statement?


Dale B. Dalrymple
http://dbdimages.com

On Jun 25, 9:02 pm, dbd <d...@ieee.org> wrote:
> On Jun 25, 4:52 pm, Randy Yates <y...@ieee.org> wrote:
>
>
>
> > Hi Folks,
>
> > Here's a problem I've been scratching my head on for awhile now. It may
> > be obvious and I'm not seeing it.
>
> > Assuming a fixed-point signal path, how would you determine the maximum
> > power that can sent through that signal path when it is band-limited?
>
> > For the sake of notationaly simplicity, assume the signal is scaled
> > such that full-scale is +/- 1.
>
> > For example, let's say I have a 16-bit two's complement signal path,
> > and it's band-limited to 1/4 of Nyquist. What is the maximum signal
> > power I can transmit through such a path?
>
> > If anyone has any suggestions on how to solve this problem, I would
> > appreciate hearing them.
>
> A constant value of 1.0 or -1.0 would put through the maximum
> power and be band limited to a tiny fraction of the Nyquist frequency.

If the signal path were not bandlimited, then any signal
consisting of only 1.0 and -1.0 would put through the maximum
power.  So even if the signal path were AC-only, take a square
wave with the lowest frequency near a maxima in the path response,
and filter out only the portions above the required bandlimit
(then gain the abs(max) back up to 1.0 if necessary).  What's
left is pretty close to the DC power max as per the above.


IMHO. YMMV.
--
rhn A.T nicholson d.0.t C-o-M

dbd <dbd@ieee.org> writes:

> On Jun 25, 4:52 pm, Randy Yates <y...@ieee.org> wrote:
>> Hi Folks,
>>
>> Here's a problem I've been scratching my head on for awhile now. It may
>> be obvious and I'm not seeing it.
>>
>> Assuming a fixed-point signal path, how would you determine the maximum
>> power that can sent through that signal path when it is band-limited?
>>
>> For the sake of notationaly simplicity, assume the signal is scaled
>> such that full-scale is +/- 1.
>>
>> For example, let's say I have a 16-bit two's complement signal path,
>> and it's band-limited to 1/4 of Nyquist. What is the maximum signal
>> power I can transmit through such a path?
>>
>> If anyone has any suggestions on how to solve this problem, I would
>> appreciate hearing them.
>>
>
> A constant value of 1.0 or -1.0 would put through the maximum
> power and be band limited to a tiny fraction of the Nyquist frequency.

Of course that's true... in the special case that the bandwidth includes
DC. I was looking for a more general result.

>> Here's how I'm approaching it, but it doesn't "feel" right. I assume
>> the input signal consists of independent samples uniformly distributed
>> with the maximum "spread" that will fit, i.e., the pdf is 1/2 from -1
>> to +1.
>>
>> Since the samples are uniform, they're "white" over the Nyquist
>> bandwidth. I take this as the maximum power through the channel.
>>
>
> What do whiteness or uniformity have to do with maximum power?
> Is this a requirement due to something left out of the problem
> statement?

No. If my solution doesn't make sense, then ignore it and propose a
solution to the problem as stated.
-- 
%  Randy Yates                  % "She has an IQ of 1001, she has a jumpsuit
%% Fuquay-Varina, NC            %            on, and she's also a telephone."
%%% 919-577-9882                % 
%%%% <yates@ieee.org>           %        'Yours Truly, 2095', *Time*, ELO   
http://home.earthlink.net/~yatescr

On Jun 25, 11:38 pm, Randy Yates <y...@ieee.org> wrote:

> dbd <d...@ieee.org> writes:

>
> > A constant value of 1.0 or -1.0 would put through the maximum
> > power and be band limited to a tiny fraction of the Nyquist frequency.
>
> Of course that's true... in the special case that the bandwidth includes
> DC. I was looking for a more general result.
>
> >> ...

>
> No. If my solution doesn't make sense, then ignore it and propose a
> solution to the problem as stated.
> --

Well, DC -is- the maximum power solution, but given the original
problem
statement, that is the only thing special about it.

If you want a different form of solution, amend the problem statement.

This is Usenet, people at used to bait and switch.

Dale B. Dalrymple
http://dbdimages.com

dbd <dbd@ieee.org> writes:

> On Jun 25, 11:38 pm, Randy Yates <y...@ieee.org> wrote:
>
>> dbd <d...@ieee.org> writes:
>
>>
>> > A constant value of 1.0 or -1.0 would put through the maximum
>> > power and be band limited to a tiny fraction of the Nyquist frequency.
>>
>> Of course that's true... in the special case that the bandwidth includes
>> DC. I was looking for a more general result.
>>
>> >> ...
>
>>
>> No. If my solution doesn't make sense, then ignore it and propose a
>> solution to the problem as stated.
>> --
>
> Well, DC -is- the maximum power solution, but given the original
> problem
> statement, that is the only thing special about it.
>
> If you want a different form of solution, amend the problem statement.

I'm not sure where the misunderstanding is. I stated that I wanted
to know the maximum power through a bandlimited, fixed-point signal path.
That, in general, does not include DC. 

> This is Usenet, people at used to bait and switch.

I don't appreciate being falsely accused, Dale. Unless you can be more
specific, you seem to be the one that's confused.
-- 
%  Randy Yates                  % "She tells me that she likes me very much,
%% Fuquay-Varina, NC            %     but when I try to touch, she makes it
%%% 919-577-9882                %                            all too clear."
%%%% <yates@ieee.org>           %        'Yours Truly, 2095', *Time*, ELO  
http://home.earthlink.net/~yatescr

On Jun 26, 4:01 am, Randy Yates <y...@ieee.org> wrote:
> dbd <d...@ieee.org> writes:
> > On Jun 25, 11:38 pm, Randy Yates <y...@ieee.org> wrote:
> > If you want a different form of solution, amend the problem statement.
>

> I'm not sure where the misunderstanding is. I stated that I wanted
> to know the maximum power through a bandlimited, fixed-point signal path.
> That, in general, does not include DC.
>

'Bandlimited' is commonly used to mean 'having a zero psd above a
specific frequency'.

In the original post, the bandlimited example was '1/4 of Nyquist'.
This was taken by myself and others as an upper boundary. No
lower boundary was suggested.

This, 'in general', includes DC.

Randy, if you have a different definition of bandlimited, please
add it to the problem statement as I've asked.

> > This is Usenet, people at used to bait and switch.
>
> I don't appreciate being falsely accused, Dale. Unless you can be more
> specific, you seem to be the one that's confused.

Using an unstated alternate definition for a term with a common
usage is the most common form of Usenet bait and switch.
It is probably the single greatest contributer to high response
counts on Usenet threads.

Dale B. Dalrymple
htp://dbdimages.com