dbd wrote:> 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. >FWIW Wikipedia agrees with you: <http://en.wikipedia.org/wiki/Bandlimited>. Paul
Determining the Maximum Power Through a Band-Limited, Fixed-Point Signal Path
Started by ●June 25, 2007
Reply by ●June 26, 20072007-06-26
Reply by ●June 26, 20072007-06-26
Randy Yates 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. > > 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.We've established in another thread that you mean a band-pass system. I gather that you are asking how to determine, for a given system, the input signal and level that will give you the maximum output power? I.e. you are free to choose not only the input level, but the shape of the input signal as well? That's an interesting problem statement. I don't know the answer, but I'll bet you that a random input signal is the wrong one. If your system limits the output most strongly, and whether or not you have an implicit limitation that you don't want to hit any nonlinearities, your best bet would probably be an input signal that gives you an output signal that's as close to a square wave as possible. This probably means a periodic input at close to the lower limit of your band, with tailoring to square up the corners of the output as much as possible while avoiding any ringing. If you can hit nonlinearities, then just jam in a sine wave and measure your +1, -1 square wave at the output. -- Tim Wescott Wescott Design Services http://www.wescottdesign.com Do you need to implement control loops in software? "Applied Control Theory for Embedded Systems" gives you just what it says. See details at http://www.wescottdesign.com/actfes/actfes.html
Reply by ●June 26, 20072007-06-26
On Jun 26, 9:45 am, Tim Wescott <t...@seemywebsite.com> wrote: > Randy Yates wrote: >>... > We've established in another thread that you mean a band-pass system. > If 'band-pass system' is an update to the original post, could you provide a definition of 'band-pass system' or a reference to the defining thread, please? Dale B. Dalrymple http://dbdimages.com
Reply by ●June 26, 20072007-06-26
dbd wrote:> On Jun 26, 9:45 am, Tim Wescott <t...@seemywebsite.com> wrote: > > Randy Yates wrote: > >>... > > > We've established in another thread that you mean a band-pass > system. > > > > If 'band-pass system' is an update to the original post, could you > provide > a definition of 'band-pass system' or a reference to the defining > thread, please? >(a) pardon me, I meant "what you meant by 'band-pass system'" (b) Are you contributing, or trying to turn this thread into a troll? -- Tim Wescott Wescott Design Services http://www.wescottdesign.com Do you need to implement control loops in software? "Applied Control Theory for Embedded Systems" gives you just what it says. See details at http://www.wescottdesign.com/actfes/actfes.html
Reply by ●June 26, 20072007-06-26
On Tue, 26 Jun 2007 07:01:50 -0400, Randy Yates <yates@ieee.org> wrote:>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.Since for most people, and I'd argue most applications, "band limited" means low-pass, then I'd say that in general "band-limited" as a general term would be expected to include DC. I couldn't figure out why this was a difficult problem since the obvious answer is DC at plust or minus 1. Amending the conditions to exclude DC and instead make the system band-pass, which is apparently your intent, would have cleared that up from the beginning. And I agree with Tim Wescott's suggestion to try a few high-power inputs (maybe a sine wave?) and crank up the input power and see how the output power characteristics tend. One might also try that with white noise at the input, then crank up the input power and see what happens. If there's a notable inflection in the input/output power curve for either the sine or noise input cases then the peak of the output power may be, or be very close to, the maximum output power waveform. If nothing else it might point you in a productive direction. Eric Jacobsen Minister of Algorithms Abineau Communications http://www.ericjacobsen.org
Reply by ●June 26, 20072007-06-26
dbd <dbd@ieee.org> writes:> 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.Yes, in that example it was a lowpass system.> 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.After consulting a few texts, it does appear that the common usage of the term "bandlimited" implies a lowpass system or signal, and that is not what I had in mind. A "bandpass system" is not really what I intended either. What I mean is a system that has a constant frequency response of G(f) = 1 or G(f) = 0 in a finite number of intervals between 0 and Nyquist. In my actual, real-world problem, I am looking at the maximum energy from 0 to 50 Hz and from 15 kHz to 32 kHz. I apologize for the confusion. -- % 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
Reply by ●June 26, 20072007-06-26
Tim Wescott <tim@seemywebsite.com> writes:> [...] > We've established in another thread that you mean a band-pass system.Well, not really that either, but I hope if you see my definition the question is now clear.> I gather that you are asking how to determine, for a given system, the > input signal and level that will give you the maximum output power? > I.e. you are free to choose not only the input level, but the shape of > the input signal as well?Yes, exactly.> That's an interesting problem statement. I don't know the answer, but > I'll bet you that a random input signal is the wrong one.I think you're probably right.> If your system limits the output most strongly, and whether or not you > have an implicit limitation that you don't want to hit any > nonlinearities, your best bet would probably be an input signal that > gives you an output signal that's as close to a square wave as > possible. This probably means a periodic input at close to the lower > limit of your band, with tailoring to square up the corners of the > output as much as possible while avoiding any ringing. > > If you can hit nonlinearities, then just jam in a sine wave and > measure your +1, -1 square wave at the output.Good suggestion, Tim. I'll think about approaching it this way. -- % 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
Reply by ●June 26, 20072007-06-26
Randy Yates <yates@ieee.org> writes:> In my actual, real-world problem, I am looking at the maximum energy > from 0 to 50 Hz and from 15 kHz to 32 kHz.Doh! Arghhhh!!!! Then in my system DC is the answer! Crap! I was so focused on the intrigue of the general question that I didn't see the answer for my particular application is simple. I'll go hide under a rock now... -- % 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
Reply by ●June 28, 20072007-06-28
Randy Yates 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. > > 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.Can you get more power out than you put in is all the b_n are less than one? There's no contradiction if the b_n exceed one; that's amplification.> 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.In some worlds, the maximum power is determined by arcing the dielectric and melting the conductor, but once the input is normalized *and* amplification is disallowed, I think you're right. Another high-"power" waveform is a very slow max-amplitude square wave that is filtered to stay in band. Except for a transition region near what had been the discontinuities, it will be +/-1 most places, and you can't get more power than that under your constraints. Jerry -- Engineering is the art of making what you want from things you can get. ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
Reply by ●June 28, 20072007-06-28
dbd wrote: ...> 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.Not zero mean, though. You need to switch once in a while. ... Jerry -- Engineering is the art of making what you want from things you can get. ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
