Forums

Nyquist For Chirps

Started by John Harkes December 15, 2012
So when sampling a signal Nyquist (assuming first Nyquist zone) says you get half
your clock rate.  I have always followed the rule that you get closer to 40% of your
clock rate.  My question is what about chirps or swept FM.

Lets assume you chirp from 100MHz to 500MHz then I would want to run my A/D at
1.25GHz to swallow this signal.  If I chirp from 100MHz to 500MHz in 100ns that is a
very aggressive sweep rate, but everything works OK.  If I change my sweep period
from 100ns to 10ns then my demodulator starts to break down and everything becomes
very distorted.  I can fix everything by increasing the A/D clock rate.  That makes
me think this is an issue where I am not meeting some sampling criteria for my sweep
rate even though I am able to sample a 500MHz signal with a 1.25GHz A/D.  This is
all done in Matlab so it is not a case of hardware that cannot keep up.  If there is
a rule of thumb for sampling rate and sweep rate I would love to know what it is. 
Any help would be greatly appreciated.

Thanks.
>So when sampling a signal Nyquist (assuming first Nyquist zone) says you
ge=
>t half your clock rate. I have always followed the rule that you get
close=
>r to 40% of your clock rate. My question is what about chirps or swept
FM.
> >Lets assume you chirp from 100MHz to 500MHz then I would want to run my
A/D=
> at 1.25GHz to swallow this signal. If I chirp from 100MHz to 500MHz in
10=
>0ns that is a very aggressive sweep rate, but everything works OK. If I
ch=
>ange my sweep period from 100ns to 10ns then my demodulator starts to
break=
> down and everything becomes very distorted. I can fix everything by
incre=
>asing the A/D clock rate. That makes me think this is an issue where I am
=
>not meeting some sampling criteria for my sweep rate even though I am able
=
>to sample a 500MHz signal with a 1.25GHz A/D. This is all done in Matlab
s=
>o it is not a case of hardware that cannot keep up. If there is a rule of
=
>thumb for sampling rate and sweep rate I would love to know what it is.
An=
>y help would be greatly appreciated. > >Thanks. >
sampling rule is dependent on the frequency content of your signal irrespective of how it is produced. By increasing sweep rate you could be producing what you are not expecting either a different frequency range or discontinuities. Kadhiem
theoretically, there is no bandlimit to continuous-time chirps.  so no 
Nyquist for them either.

in fact, there is no bandlimit to any time-limited signal.  but we make 
approximations.  the longer (in time) a signal is, the more it can be 
treated as bandlimited above whatever are the apparent sinusoids 
contained therein.

if you want to figure out what is happening to linearly-swept chirps, 
you need to define a window for it.  it turns out that mathematically, 
it's easiest to understand with a gaussian window.  11 years ago, i 
wrote a dumb paper (it was my only paper for an IEEE function) that 
expresses some of this concisely.  if you (or anyone) want a pdf of it, 
lemme know, i'll try to scrape it up and send it.


-- 

r b-j                  rbj@audioimagination.com

"Imagination is more important than knowledge."


> >theoretically, there is no bandlimit to continuous-time chirps. so no >Nyquist for them either. > >in fact, there is no bandlimit to any time-limited signal. but we make >approximations. the longer (in time) a signal is, the more it can be >treated as bandlimited above whatever are the apparent sinusoids >contained therein. > >if you want to figure out what is happening to linearly-swept chirps, >you need to define a window for it. it turns out that mathematically, >it's easiest to understand with a gaussian window. 11 years ago, i >wrote a dumb paper (it was my only paper for an IEEE function) that >expresses some of this concisely. if you (or anyone) want a pdf of it, >lemme know, i'll try to scrape it up and send it. > > > > >r b-j rbj@audioimagination.com > >"Imagination is more important than knowledge." >
For a mathematical imagination we have the notion of +/- infinity. For a practical knowledge I will have my bandlimited signal wrap up correctly from end to start otherwise a break is just like sample errors and the discontinuity could be sharp such that I get glitches in real time spectrum analyser though matlab fft will not show any problem. For the case of chirp signal it is better to start from some zeros and end with some zeros, that implies good wrap up. Kadhiem
On 12/15/12 1:29 PM, kaz wrote:
> > For a mathematical imagination we have the notion of +/- infinity. For a > practical knowledge I will have my bandlimited signal wrap up correctly > from end to start
what do you mean by "wrap up"?
> otherwise a break is just like sample errors and the > discontinuity could be sharp such that I get glitches in real time spectrum > analyser
that discontinuity can only be constructed with very high frequency components.
> though matlab fft will not show any problem. > > For the case of chirp signal it is better to start from some zeros and end > with some zeros, that implies good wrap up.
you have to define your terms. my suspicion is that "wrap up" is equivalent to applying a window of some sort. maybe it's a rectangular window. i can't tell. rectangular windows normally cause sharp discontinuities. but to get to the bottom of what is going on, you need to be explicit with what you are doing to the ideal chirp to "wrap it up". if you do that, we might be able to get a handle on it. -- r b-j rbj@audioimagination.com "Imagination is more important than knowledge."
>On 12/15/12 1:29 PM, kaz wrote: >> >> For a mathematical imagination we have the notion of +/- infinity. For
a
>> practical knowledge I will have my bandlimited signal wrap up correctly >> from end to start > >what do you mean by "wrap up"? > >> otherwise a break is just like sample errors and the >> discontinuity could be sharp such that I get glitches in real time
spectrum
>> analyser > >that discontinuity can only be constructed with very high frequency >components. > >> though matlab fft will not show any problem. >> >> For the case of chirp signal it is better to start from some zeros and
end
>> with some zeros, that implies good wrap up. > >you have to define your terms. > >my suspicion is that "wrap up" is equivalent to applying a window of >some sort. maybe it's a rectangular window. i can't tell. rectangular >windows normally cause sharp discontinuities. > >but to get to the bottom of what is going on, you need to be explicit >with what you are doing to the ideal chirp to "wrap it up". if you do >that, we might be able to get a handle on it. > > >-- > >r b-j rbj@audioimagination.com > >"Imagination is more important than knowledge." > > >
I constantly supply my fellow engineers with vectors from matlab models and so I am really talking about offline vectors as original post but also applies to any frame based approach. Vectors for multiple cdma,lte,umts are all constructed so that the vector end merges with its start when they are played continuously through the actual circuit. If my fellow engineer comes back to me without a smile I know he got spikes (I mean in the spectrum!) because I had made some error at wrapping up of vector. I have also generated chirps manually(I mean not matlab function). Some tools may produce very bad vectors when it comes to this vector end/start continuity. But a chirp can made to start gently from zeros low frequencies then high frequencies all at same amplitude then amplitude is smoothly lowered towards zeros at high frequency such that when you connect end to start it looks the most beautiful dsp signal I have ever seen, though not quite a proper wrap up as cdma/lte cases. Applying a window is an alternative as it lets the start/end get a bit smoother. Kadhiem
On Sat, 15 Dec 2012 04:30:47 -0800, John Harkes wrote:

> So when sampling a signal Nyquist (assuming first Nyquist zone) says you > get half your clock rate. I have always followed the rule that you get > closer to 40% of your clock rate. My question is what about chirps or > swept FM.
Nyquist didn't say that: http://www.wescottdesign.com/articles/Sampling/sampling.pdf
> Lets assume you chirp from 100MHz to 500MHz then I would want to run my > A/D at 1.25GHz to swallow this signal. If I chirp from 100MHz to 500MHz > in 100ns that is a very aggressive sweep rate, but everything works OK. > If I change my sweep period from 100ns to 10ns then my demodulator > starts to break down and everything becomes very distorted. I can fix > everything by increasing the A/D clock rate. That makes me think this > is an issue where I am not meeting some sampling criteria for my sweep > rate even though I am able to sample a 500MHz signal with a 1.25GHz A/D. > This is all done in Matlab so it is not a case of hardware that cannot > keep up. If there is a rule of thumb for sampling rate and sweep rate I > would love to know what it is. Any help would be greatly appreciated.
You are mistaking the period of the wave within your chirp for the spectral content of your chirp. They are different. I'm fairly sure that a chirp with a nice linear frequency ramp has a closed-form Fourier transform. I'm sure that if it does, it involves Bessel functions. The older I get (and the better computers get at numerical calculations), the more I translate "has Bessel functions" to mean, in a practical sense "run away, now". Make up your chirp with its fast ramp in Matlab, take its FFT, and plot it. Keep increasing the sampling rate until you're sure that the spectrum of the chirp is being accurately represented. What this is will vary by what your problem is, but when you get to the point where the energy around f/2 is less than 1% of the tallest peak, then you're probably there. Now make up a slow chirp, and do the same FFT, and plot _that_. Plot them one on top of the other. Now ponder that spectrum, and mentally pick out sampling rates, drawing a dotted line at your f/2 point, and visualizing how the spectrum will fold over that line for each signal. It should become obvious why the fast chirp gives you more grief, and why faster sampling helps. -- Tim Wescott Control system and signal processing consulting www.wescottdesign.com
robert bristow-johnson <rbj@audioimagination.com> writes:

> 11 years ago, i wrote a dumb paper (it was my only paper for an IEEE > function) that expresses some of this concisely. if you (or anyone) > want a pdf of it, lemme know, i'll try to scrape it up and send it.
A "dumb rb-j paper" is an oxymoron. I would like to see it, Robert. -- Randy Yates Digital Signal Labs http://www.digitalsignallabs.com
<snip>
>Make up your chirp with its fast ramp in Matlab, take its FFT, and plot >it. Keep increasing the sampling rate until you're sure that the >spectrum of the chirp is being accurately represented. What this is will >vary by what your problem is, but when you get to the point where the >energy around f/2 is less than 1% of the tallest peak, then you're >probably there. > >Now make up a slow chirp, and do the same FFT, and plot _that_. Plot >them one on top of the other. > >Now ponder that spectrum, and mentally pick out sampling rates, drawing a >dotted line at your f/2 point, and visualizing how the spectrum will fold >over that line for each signal. It should become obvious why the fast >chirp gives you more grief, and why faster sampling helps.
Well if your phase modulation can not be represented correctly within Fs/2 then your signal will be distorted even if the carrier wave is within Fs/2? Sampling theory is not about the instantaneous frequency right? This was a question. Mark DeArman
On Sat, 15 Dec 2012 23:51:54 -0800, Mac Decman
<dearman.mark@gmail.com> wrote:

<snip>

>Well if your phase modulation can not be represented correctly within >Fs/2 then your signal will be distorted even if the carrier wave is >within Fs/2? Sampling theory is not about the instantaneous frequency >right? This was a question. > >Mark DeArman
Just a P.S: I use a worksheet like the following to do IF and phase calculations for pulsed sweep measurements. The one in the picture is just an example showing Log vs. variable Sinh vector sweep. Building them symbolic makes it easy to extract all the essential parameters required for sampling them. http://foxy.cascadeacoustic.com/worksheet.jpg Mark DeArman