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Power distribution across aliased signals

Started by starman3 March 26, 2008
I am having trouble understanding the effect of aliasing/imaging on the
power of an output signal.
Imagine I am using a DAC to generate a single frequency sine wave with
frequency f well below fs/2 and with power P. 
Assume there is no reconstruction filter. So this frequency will image
repeatedly across the frequency domain at n.fs-f and n.fs+f.
I do not understand how the original power, P, is distributed across these
images? If the first image (the original intended signal) still has total
power P then there must be additional power coming from somewhere to
create the replica images so where does this power come from? 
On the other hand, if the original power is somehow spread across the
images then how does adding a reconstruction filter ever cause the right
amplitude to occur in the intended first image?
Additionally if the number of repeated images is infinite, all with the
same amplitude then this implies an infinite power source. So I can't help
assuming that there must be some decay in the image amplitude with
increasing frequency. But this never seems to be apparent from the texts
where the images are always shown in diagrams as exact replicas.

Can somebody help me to understand this?

Thanks


starman3 wrote:
> I am having trouble understanding the effect of aliasing/imaging on the > power of an output signal. > Imagine I am using a DAC to generate a single frequency sine wave with > frequency f well below fs/2 and with power P. > Assume there is no reconstruction filter. So this frequency will image > repeatedly across the frequency domain at n.fs-f and n.fs+f.
First of all, there is a reconstruction filter, just an inadequate one. In the usual case, the output of the DAC is maintained until the next sample's value replaces it; that's called a zero-order hold. It hreatly strengthens the baseband signal and weakens higher images.
> I do not understand how the original power, P, is distributed across these > images? If the first image (the original intended signal) still has total > power P then there must be additional power coming from somewhere to > create the replica images so where does this power come from?
In reality, there is no power at all. It's all just voltage.
> On the other hand, if the original power is somehow spread across the > images then how does adding a reconstruction filter ever cause the right > amplitude to occur in the intended first image?
Nothing is used up when numbers are reproduced.
> Additionally if the number of repeated images is infinite, all with the > same amplitude then this implies an infinite power source. So I can't help > assuming that there must be some decay in the image amplitude with > increasing frequency. But this never seems to be apparent from the texts > where the images are always shown in diagrams as exact replicas.
The images are shown _before_ the DAC. Only if the DAC outputs were impulses, not steps, then would the amplitudes of the images be equal. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
starman3 wrote:

> I am having trouble understanding the effect of aliasing/imaging on the > power of an output signal. > Imagine I am using a DAC to generate a single frequency sine wave with > frequency f well below fs/2 and with power P. > Assume there is no reconstruction filter. So this frequency will image > repeatedly across the frequency domain at n.fs-f and n.fs+f. > I do not understand how the original power, P, is distributed across these > images? If the first image (the original intended signal) still has total > power P then there must be additional power coming from somewhere to > create the replica images so where does this power come from?
It comes from the DAC. Assume that the DAC can output a 1V P-P sine wave. It can also output a 1V P-P square wave containing about sqrt(2) times as much RMS voltage, or twice as much power. (Assuming the transistors can stand it.)
> On the other hand, if the original power is somehow spread across the > images then how does adding a reconstruction filter ever cause the right > amplitude to occur in the intended first image?
> Additionally if the number of repeated images is infinite, all with the > same amplitude then this implies an infinite power source.
If you meant to output a sine and accidentally output a square wave instead, the harmonics go as (1/n) in amplitude, or (1/n)**2 in power. Fortunately sum (n=1 to infinity) (1/n)**2 converges. If you are closer to a sine, the harmonics will decrease even faster. -- glen
On Mar 26, 11:40 am, "starman3" <apl...@bluebottle.com> wrote:
> I am having trouble understanding the effect of aliasing/imaging on the > power of an output signal. > Imagine I am using a DAC to generate a single frequency sine wave with > frequency f well below fs/2 and with power P. > Assume there is no reconstruction filter. So this frequency will image > repeatedly across the frequency domain at n.fs-f and n.fs+f. > I do not understand how the original power, P, is distributed across these > images? If the first image (the original intended signal) still has total > power P then there must be additional power coming from somewhere to > create the replica images so where does this power come from? > On the other hand, if the original power is somehow spread across the > images then how does adding a reconstruction filter ever cause the right > amplitude to occur in the intended first image? > Additionally if the number of repeated images is infinite, all with the > same amplitude then this implies an infinite power source. So I can't help > assuming that there must be some decay in the image amplitude with > increasing frequency. But this never seems to be apparent from the texts > where the images are always shown in diagrams as exact replicas. > > Can somebody help me to understand this?
If you had a Dirac delta DAC, it would output around 1e44 Volts times your samples, with a pulse width of about 1e-44 Seconds (actually less, but there is no way to measure anything smaller than Planck dimension). 1e44 Volts through any finite impedance would likely have enough energy to create as many images as you could find up to and beyond any frequency you could detect, without decay. My guess is that your DAC actually uses wider pulses and lower voltages with a finite maximum slew rate, which means perhaps you just didn't notice the reconstruction filter implied by those limitations. IMHO. YMMV. -- rhn A.T nicholson d.0.t C-o-M http://www.nicholson.com/rhn/dsp.html
>starman3 wrote: > >> I am having trouble understanding the effect of aliasing/imaging on
the
>> power of an output signal. >> Imagine I am using a DAC to generate a single frequency sine wave with >> frequency f well below fs/2 and with power P. >> Assume there is no reconstruction filter. So this frequency will image >> repeatedly across the frequency domain at n.fs-f and n.fs+f. >> I do not understand how the original power, P, is distributed across
these
>> images? If the first image (the original intended signal) still has
total
>> power P then there must be additional power coming from somewhere to >> create the replica images so where does this power come from? > >It comes from the DAC. Assume that the DAC can output a 1V P-P >sine wave. It can also output a 1V P-P square wave containing >about sqrt(2) times as much RMS voltage, or twice as much power. >(Assuming the transistors can stand it.) > >> On the other hand, if the original power is somehow spread across the >> images then how does adding a reconstruction filter ever cause the
right
>> amplitude to occur in the intended first image? > >> Additionally if the number of repeated images is infinite, all with
the
>> same amplitude then this implies an infinite power source. > >If you meant to output a sine and accidentally output a square wave >instead, the harmonics go as (1/n) in amplitude, or (1/n)**2 in power. >Fortunately sum (n=1 to infinity) (1/n)**2 converges. If you are >closer to a sine, the harmonics will decrease even faster. > > >-- glen > > >> create the replica images so where does this power come from? > >It comes from the DAC. Assume that the DAC can output a 1V P-P >sine wave. It can also output a 1V P-P square wave containing >about sqrt(2) times as much RMS voltage, or twice as much power. >(Assuming the transistors can stand it.) > >> On the other hand, if the original power is somehow spread across the >> images then how does adding a reconstruction filter ever cause the
right
>> amplitude to occur in the intended first image? > >> Additionally if the number of repeated images is infinite, all with
the
>> same amplitude then this implies an infinite power source. > >If you meant to output a sine and accidentally output a square wave >instead, the harmonics go as (1/n) in amplitude, or (1/n)**2 in power. >Fortunately sum (n=1 to infinity) (1/n)**2 converges. If you are >closer to a sine, the harmonics will decrease even faster. > > >-- glen > >
ok thanks guys, it is becoming clearer to me. But I do have a few more questions. I now see where the power comes from and I see that I am actually transmitting a stepped sine wave (due to the nature of the DAC, as jerry posted). I don't quite follow how this makes an imperfect reconstruction filter. And if so what is it that determines the frequency characteristics of that imperfect filter? Would I be right to say that my stepped sine wave is theoretically equal to a sequence of square wave pulses, each of width 1/fs, modulated in amplitude with a sampled sine function? In a real case where the fundamental frequency of my sine wave is an exact submultiple of fs it should be possible to quantify exactly how the images will decay in amplitude. Can you point me in the right direction of the maths for this? And finally, (this might seem like an obvious question but I just want to clear it up) does an analogue reconstruction filter really just attenuate higher frequencies in the frequency domain or does it actually do something more complex like folding the higher images back onto the original? My intuition says it does the former but I just want to check I am not missing something. Thanks for your help --- paul
Hi Paul,
Here is some reading material that answers these questions.
Regards,
Steve

http://www.dspguide.com/ch3/3.htm
On Mar 27, 1:16 am, "starman3" <apl...@bluebottle.com> wrote:
> >starman3 wrote: > > >> I am having trouble understanding the effect of aliasing/imaging on > the > >> power of an output signal. > >> Imagine I am using a DAC to generate a single frequency sine wave with > >> frequency f well below fs/2 and with power P. > >> Assume there is no reconstruction filter. So this frequency will image > >> repeatedly across the frequency domain at n.fs-f and n.fs+f. > >> I do not understand how the original power, P, is distributed across > these > >> images? If the first image (the original intended signal) still has > total > >> power P then there must be additional power coming from somewhere to > >> create the replica images so where does this power come from? > > >It comes from the DAC. Assume that the DAC can output a 1V P-P > >sine wave. It can also output a 1V P-P square wave containing > >about sqrt(2) times as much RMS voltage, or twice as much power. > >(Assuming the transistors can stand it.) > > >> On the other hand, if the original power is somehow spread across the > >> images then how does adding a reconstruction filter ever cause the > right > >> amplitude to occur in the intended first image? > > >> Additionally if the number of repeated images is infinite, all with > the > >> same amplitude then this implies an infinite power source. > > >If you meant to output a sine and accidentally output a square wave > >instead, the harmonics go as (1/n) in amplitude, or (1/n)**2 in power. > >Fortunately sum (n=1 to infinity) (1/n)**2 converges. If you are > >closer to a sine, the harmonics will decrease even faster. > > >-- glen > > >> create the replica images so where does this power come from? > > >It comes from the DAC. Assume that the DAC can output a 1V P-P > >sine wave. It can also output a 1V P-P square wave containing > >about sqrt(2) times as much RMS voltage, or twice as much power. > >(Assuming the transistors can stand it.) > > >> On the other hand, if the original power is somehow spread across the > >> images then how does adding a reconstruction filter ever cause the > right > >> amplitude to occur in the intended first image? > > >> Additionally if the number of repeated images is infinite, all with > the > >> same amplitude then this implies an infinite power source. > > >If you meant to output a sine and accidentally output a square wave > >instead, the harmonics go as (1/n) in amplitude, or (1/n)**2 in power. > >Fortunately sum (n=1 to infinity) (1/n)**2 converges. If you are > >closer to a sine, the harmonics will decrease even faster. > > >-- glen > > ok thanks guys, it is becoming clearer to me. But I do have a few more > questions. I now see where the power comes from and I see that I am > actually transmitting a stepped sine wave (due to the nature of the DAC, > as jerry posted). I don't quite follow how this makes an imperfect > reconstruction filter. And if so what is it that determines the frequency > characteristics of that imperfect filter?
The edge rate (as well as the higher derivatives) is a primary contributor. Take each rectangular step and turn it into a nicely rounding "Sinc-like" pulse with the same area, and the result will approach some bandlimit, depending on the steepness (inverse of main lobe width) of the Sinc.
> And finally, (this might seem like an obvious question but I just want to > clear it up) does an analogue reconstruction filter really just attenuate > higher frequencies in the frequency domain or does it actually do > something more complex like folding the higher images back onto the > original? My intuition says it does the former but I just want to check I > am not missing something.
Would an RC lowpass filter warm up when presented with a high power signal well above its passband, or would it find some way to send that power to the speaker box? IMHO. YMMV. -- rhn A.T nicholson d.0.t C-o-M
Ron N. wrote:

   ...

> Would an RC lowpass filter warm up when presented with > a high power signal well above its passband, or would it > find some way to send that power to the speaker box?
Good question! Now replace that filter with an LC. What then? 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;
Jerry Avins wrote:

> Ron N. wrote:
>> Would an RC lowpass filter warm up when presented with >> a high power signal well above its passband, or would it >> find some way to send that power to the speaker box?
Ideal R and C, or more realistic. Real C turn inductive at high enough frequencies.
> Good question! Now replace that filter with an LC. What then?
Better use a superconducting L just to be sure. Real capacitors are closer to ideal than real inductors. -- glen