Resolution of frequency measurement possible

Hi,

I am a newbie in DSP. I have got a requirement to measure the
frequency of a signal between 45KHz to 55 KHz with a resolution of
40Hz. The length of the signal which is available for measurement is
320 micro seconds. Is it possible to measure the frequency of this
signal with 40Hz resolution? If yes, how? With FFT, to get the
resolution of 40Hz the signal length should be atleast 25ms (...Am i
correct?). Does this represent a fundamental limit?

Thanks in advance,

Abins

On 27 Sep 2004 03:50:27 -0700, abinsj@gmail.com (Abins) wrote:

>Hi,
>
>I am a newbie in DSP. I have got a requirement to measure the
>frequency of a signal between 45KHz to 55 KHz with a resolution of
>40Hz. The length of the signal which is available for measurement is
>320 micro seconds. Is it possible to measure the frequency of this
>signal with 40Hz resolution? If yes, how? With FFT, to get the
>resolution of 40Hz the signal length should be atleast 25ms (...Am i
>correct?). Does this represent a fundamental limit?
>
>Thanks in advance,
>
>Abins

Hi,

  Yep, you're right.  To get 40 Hz resolution 
you need a time-domain sequence whose length 
is, ***at the very least***, 25 ms.

And yes it's a fundamental limit.  
The very smallest resolution you can achieve 
is the reciprocal of your time-domain 
signal's duration.  It's not only a good 
idea, it's the Law!

Good Luck,
[-Rick-]

Abins wrote:

> Hi,
> 
> I am a newbie in DSP. I have got a requirement to measure the
> frequency of a signal between 45KHz to 55 KHz with a resolution of
> 40Hz. The length of the signal which is available for measurement is
> 320 micro seconds. Is it possible to measure the frequency of this
> signal with 40Hz resolution? If yes, how? With FFT, to get the
> resolution of 40Hz the signal length should be atleast 25ms (...Am i
> correct?). Does this represent a fundamental limit?
> 
> Thanks in advance,
> 
> Abins

It depends on how pure the signal is. If you know -- with all the
conditions that Rune mentioned -- that the frequency is pure and that
there is very little noise, you can measure the time between zero
crossings. With the numbers you give, there will be at least 15 zero
crossings in your measurement interval, so by counting zero crossings,
you can tolerate more uncertainty in the measured time than if there
were much fewer. Noise or incoherent signal contamination will reduce
the accuracy of your measurement. You can work out an error budget for
this. When you do, it will become clear that higher the frequency (the
fine the time resolution) the more tolerable those error sources become.

Jerry
-- 
... they proceeded on the sound principle that the magnitude of a lie
always contains a certain factor of credibility, ... and that therefor
... they more easily fall victim to a big lie than to a little one ...
                                                                   A. H.
�����������������������������������������������������������������������

On 27 Sep 2004 03:50:27 -0700, abinsj@gmail.com (Abins) wrote:

>Hi,
>
>I am a newbie in DSP. I have got a requirement to measure the
>frequency of a signal between 45KHz to 55 KHz with a resolution of
>40Hz. The length of the signal which is available for measurement is
>320 micro seconds. Is it possible to measure the frequency of this
>signal with 40Hz resolution? If yes, how? With FFT, to get the
>resolution of 40Hz the signal length should be atleast 25ms (...Am i
>correct?). Does this represent a fundamental limit?
>
>Thanks in advance,
>
>Abins

The other advice you've gotten here is correct, there are some
fundamental limits.   However, depending on the input conditions this
can be treated in such a way that you can probably get pretty good
accuracy, probably close to 40Hz with some reasonable reliability.  It
becomes a bit of a stochastic problem in most cases.

Peter Kootsookos has an excellent website that contains good info on a
number of different ways to solve the problem.  See:

http://www.itee.uq.edu.au/~kootsoop/freqalgs.htm

The Fourier Coefficient technique links have some working matlab code
in there that might be helpful.


Eric Jacobsen
Minister of Algorithms, Intel Corp.
My opinions may not be Intel's opinions.
http://www.ericjacobsen.org

abinsj@gmail.com (Abins) wrote...
> I am a newbie in DSP. I have got a requirement to measure the
> frequency of a signal between 45KHz to 55 KHz with a resolution of
> 40Hz. The length of the signal which is available for measurement is
> 320 micro seconds. Is it possible to measure the frequency of this
> signal with 40Hz resolution? If yes, how? With FFT, to get the
> resolution of 40Hz the signal length should be atleast 25ms (...Am i
> correct?). Does this represent a fundamental limit?

If you know on, say, physical grounds, that your signal contains only
one frequency (i.e. one sinusoid), or a small bounded number of
frequencies, with at most a small amount of noise, then it is possible
get better frequency resolution than a naive Fourier transform.

One approach would be to simply least-squares fit your signal to a
sinusoid, but this gets tricky if you have more than one frequency
because the fitting process gets trapped in local minima, etcetera. 
As an alternative, a nice technique that was developed by Mandelshtam
for NMR studies which I've found to work well is available at
http://ab-initio.mit.edu/harminv/ .

However, if by "frequency" you just mean that you have a general
periodic signal, then you need some other approach.  Again, if you
have some solid a priori information, such as your signal having
really only one period, then there are ways to circumvent the DFT
uncertainty.

Cordially,
Steven G. Johnson

In article <4157f85e.256197546@news.sf.sbcglobal.net>,
Rick Lyons <r.lyons@_BOGUS_ieee.org> wrote:
>On 27 Sep 2004 03:50:27 -0700, abinsj@gmail.com (Abins) wrote:
>>I am a newbie in DSP. I have got a requirement to measure the
>>frequency of a signal between 45KHz to 55 KHz with a resolution of
>>40Hz. The length of the signal which is available for measurement is
>>320 micro seconds. Is it possible to measure the frequency of this
>>signal with 40Hz resolution? If yes, how? With FFT, to get the
>>resolution of 40Hz the signal length should be atleast 25ms (...Am i
>>correct?). Does this represent a fundamental limit?
...
>  Yep, you're right.  To get 40 Hz resolution 
>you need a time-domain sequence whose length 
>is, ***at the very least***, 25 ms.
>
>And yes it's a fundamental limit.  
>The very smallest resolution you can achieve 
>is the reciprocal of your time-domain 
>signal's duration.  

Doesn't the fundamental limit depend on the signal-to-noise ratio?
Even using only the FFT results, one can use (sync, for instance)
interpolation between bins, given a steady signal and sufficiently
high S/N.


IMHO. YMMV.
-- 
Ron Nicholson   rhn AT nicholson DOT com   http://www.nicholson.com/rhn/ 
#include <canonical.disclaimer>        // only my own opinions, etc.

"Ronald H. Nicholson Jr." <rhn@mauve.rahul.net> wrote in message
news:cjamig$460$2@blue.rahul.net...
> >And yes it's a fundamental limit.
> >The very smallest resolution you can achieve
> >is the reciprocal of your time-domain
> >signal's duration.
>
> Doesn't the fundamental limit depend on the signal-to-noise ratio?
> Even using only the FFT results, one can use (sync, for instance)
> interpolation between bins, given a steady signal and sufficiently
> high S/N.
>
>

Hello Ron et. al.,,

Yes the S/N ratio is most important. If you had 3 consecutive noise free
infinite precision samples from a single sinusoid, you can determine the
freq, amplitude and phase exactly. We will assume that we are not bandpass
sampling or if we are we know which spectral repeat we are operating in. The
problems arise from noise and lack of knowing how many sinusoids there are.

Clay

rhn@mauve.rahul.net (Ronald H. Nicholson Jr.) wrote in message news:<cjamig$460$2@blue.rahul.net>...
> 
> Doesn't the fundamental limit depend on the signal-to-noise ratio?
> Even using only the FFT results, one can use (sync, for instance)
> interpolation between bins, given a steady signal and sufficiently
> high S/N.
> 
> 
Thanks for your response. The signal-to-noise ratio is around 60dB. 
The signal is sinusoidal in nature. Could you please give some
references to the interpolation techniques which can be applied to the
FFT result?

On Mon, 27 Sep 2004 11:31:00 GMT, r.lyons@_BOGUS_ieee.org (Rick Lyons)
wrote:

>On 27 Sep 2004 03:50:27 -0700, abinsj@gmail.com (Abins) wrote:
>
>>Hi,
>>
>>I am a newbie in DSP. I have got a requirement to measure the
>>frequency of a signal between 45KHz to 55 KHz with a resolution of
>>40Hz. The length of the signal which is available for measurement is
>>320 micro seconds. Is it possible to measure the frequency of this
>>signal with 40Hz resolution? If yes, how? With FFT, to get the
>>resolution of 40Hz the signal length should be atleast 25ms (...Am i
>>correct?). Does this represent a fundamental limit?
>>
>>Thanks in advance,
>>
>>Abins
>
>Hi,
>
>  Yep, you're right.  To get 40 Hz resolution 
>you need a time-domain sequence whose length 
>is, ***at the very least***, 25 ms.
>
>And yes it's a fundamental limit.  
>The very smallest resolution you can achieve 
>is the reciprocal of your time-domain 
>signal's duration.  It's not only a good 
>idea, it's the Law!
>
>Good Luck,
>[-Rick-]
>

   Yikes, I should have been more specific.  When I 
wrote "resolution", I should have used the phrase
"FFT bin spacing".  (The freq spacing between 
FFT result samples.)

I shouldn't have used the word "resolution" 
because it means different things to 
different people.

So, to achieve no more than 40 Hz FFT bin spacing 
you need a time-domain sequence whose length 
is, ***at the very least***, 25 ms.


See Ya',
[-Rick-]

abinsj@gmail.com (Abins) wrote in message news:<d33e378e.0409280209.6bd63695@posting.google.com>...
> rhn@mauve.rahul.net (Ronald H. Nicholson Jr.) wrote in message news:<cjamig$460$2@blue.rahul.net>...
> > 
> > Doesn't the fundamental limit depend on the signal-to-noise ratio?
> > Even using only the FFT results, one can use (sync, for instance)
> > interpolation between bins, given a steady signal and sufficiently
> > high S/N.
> > 
> > 
> Thanks for your response. The signal-to-noise ratio is around 60dB. 
> The signal is sinusoidal in nature. Could you please give some
> references to the interpolation techniques which can be applied to the
> FFT result?

With a 100 MHz clock, you should be able to locate a (say,
positive-going) zero-crossing to within 10 nanoseconds. If you measure
for 10 cycles out of the possible 15 that gall within your available
interval, that's one nanosecond per cycle at each end of the
measurement. You will then know your approximately 18 microsecond
period to within two nanoseconds. Noise at 60 dB down can hardly
extend that beyomg 5 nanoseconds. It seems that a straightforward
measurement can give you accuracy to spare.

Jerry