Improving Frequency Resolution in Time Constrained Signals

For simplicity, suppose we are interested in detecting presence of either a
500Hz, 1kHz, and a 1.5kHz sinusoid in a signal that is being sampled at
100kHz. But lets say we only have 100 samples, or 1ms of data. This is
anywhere between 0.5 to 2 'cycles' of the sinusoids. So essentially, the
signal is well oversampled, but is not sampled long enough in time to
provide clean frequency resolution.

Question is: Are there particular techniques to improve frequency
resolution for oversampled, but time constrained, signals?

(One random thought considered was to decimate at different points in
time, say take every tenth sample starting at sample #1, then every tenth
starting at sample #2, to create 10 independent signals of same length.
Then concatenate to artificially lengthen the same signal. But I wasn't
sure if that would increase resolution since it doesn't add any new
data....)

Thanks!

On 30 Okt, 13:22, "tzoom84" <tszumow...@gmail.com> wrote:
> For simplicity, suppose we are interested in detecting presence of either a
> 500Hz, 1kHz, and a 1.5kHz sinusoid in a signal that is being sampled at
> 100kHz. But lets say we only have 100 samples, or 1ms of data. This is
> anywhere between 0.5 to 2 'cycles' of the sinusoids. So essentially, the
> signal is well oversampled, but is not sampled long enough in time to
> provide clean frequency resolution.
>
> Question is: Are there particular techniques to improve frequency
> resolution for oversampled, but time constrained, signals?

No, there aren't.

However, a far more interesting question is wheteher it is
possible to come up with a guesstimate about which of the
candidate sinusoidals is present, given data as stated.

Rune

tzoom84 wrote:

> For simplicity, suppose we are interested in detecting presence of either a
> 500Hz, 1kHz, and a 1.5kHz sinusoid in a signal that is being sampled at
> 100kHz. But lets say we only have 100 samples, or 1ms of data. This is
> anywhere between 0.5 to 2 'cycles' of the sinusoids. So essentially, the
> signal is well oversampled, but is not sampled long enough in time to
> provide clean frequency resolution.
> Question is: Are there particular techniques to improve frequency
> resolution for oversampled, but time constrained, signals?

The frequency resolution is the matter of SNR and the measurement time.
The limit is set by Cramer Rao bound.

Vladimir Vassilevsky
DSP and Mixed Signal Design Consultant
http://www.abvolt.com

>The frequency resolution is the matter of SNR and the measurement time.
>The limit is set by Cramer Rao bound.

Hmm, I had a feeling that I wouldn't be able to get past that
time/frequency resolution trade-off. 

tzoom84 wrote:
> For simplicity, suppose we are interested in detecting presence of either a
> 500Hz, 1kHz, and a 1.5kHz sinusoid in a signal that is being sampled at
> 100kHz. But lets say we only have 100 samples, or 1ms of data. This is
> anywhere between 0.5 to 2 'cycles' of the sinusoids. So essentially, the
> signal is well oversampled, but is not sampled long enough in time to
> provide clean frequency resolution.
> 
> Question is: Are there particular techniques to improve frequency
> resolution for oversampled, but time constrained, signals?
> 
> (One random thought considered was to decimate at different points in
> time, say take every tenth sample starting at sample #1, then every tenth
> starting at sample #2, to create 10 independent signals of same length.
> Then concatenate to artificially lengthen the same signal. But I wasn't
> sure if that would increase resolution since it doesn't add any new
> data....)
> 
> Thanks!

 From your description, it may be that all you want is to differentiate 
between which single sinuoid of 3 is present?
Actually you have stated two different problems:
1) detect the presence of ONE of the sinusoids.
2) get good frequency resolution - which may imply that you want to 
measure the frequency of one of the sinusoids.... is that right or not?

You have 1msec of data so presumably around 1kHz of frequency resolution.
A 1kHz bandwidth is all you need to differentiate between those sinusoids.
0Hz to 1kHz for the 500Hz signal
500Hz to 1.5kHz for the 1kHz signal
1kHz to 2kHz for the 1.5kHz signal.

So, you use a lowpass filter and a couple of bandpass filters.

A filter with a 1kHz bandwidth will likely have a length of around 1msec 
or so.  So, the attendant problem is that you will just get the filter 
"filled" with the data that you have available and only get out one or a 
very few "non-transient" data points.  Not great for a sinusoid.

All this suggests that you may have enough data if you handle it 
reasonably.  For example, what if you "baseband" the data using 500Hz, 
1kHz and 1.5kHz modulation?  I'm not sure this will work because of 
filtering requirements but it's worth some thought.  Something like 
multiply by sin and cos at each frequency then sum the squares.  Then 
lowpass each as best you can and the highest output "wins"......

The trick of course is differentiating signals that each fall in a 1kHz 
bandwidth with an "interfering" signal at the band edge.  And, the "band 
edges" aren't very sharply defined at that!  But, if the "interfering" 
signals aren't present then you could be OK.

Fred

On Oct 30, 8:22&#4294967295;am, "tzoom84" <tszumow...@gmail.com> wrote:
> For simplicity, suppose we are interested in detecting presence of either a
> 500Hz, 1kHz, and a 1.5kHz sinusoid in a signal that is being sampled at
> 100kHz. But lets say we only have 100 samples, or 1ms of data. This is
> anywhere between 0.5 to 2 'cycles' of the sinusoids. So essentially, the
> signal is well oversampled, but is not sampled long enough in time to
> provide clean frequency resolution.
>
> Question is: Are there particular techniques to improve frequency
> resolution for oversampled, but time constrained, signals?
>
> (One random thought considered was to decimate at different points in
> time, say take every tenth sample starting at sample #1, then every tenth
> starting at sample #2, to create 10 independent signals of same length.
> Then concatenate to artificially lengthen the same signal. But I wasn't
> sure if that would increase resolution since it doesn't add any new
> data....)
>
> Thanks!

With a sample rate = 100k and N = 100, you&#4294967295;ve got 100k/100 = 1k bin
spacing in the frequency domain (e.g.: DFT frequency index k = 0
corresponds to 0Hz, index k = 1 corresponds to 1kHz, index k = 2 is
2kHz, etc.).  So perhaps you&#4294967295;d do 3 DFTs, one at k = .5 (corresponds
to 500 Hz), one at k = 1 (corresponds to 1 kHz), and one at k = 1.5
(corresponds to 1.5 kHz).  Then compute magnitude squared of each and
find the largest.  Alternately, you could zero pad an FFT to 200
points and use outputs k = 1, 2 and 3 to represent f = 500 Hz, 1 kHz
and 1.5 kHz.  Keep in mind, of course, that zero padding doesn&#4294967295;t
really improve resolution - it provides an interpolation.  If you
start with a sampling interval that&#4294967295;s too short, you get a lousy
result.  Zero padding gives you more (interpolated) points of a lousy
result.

Of course, I&#4294967295;m presuming that only one tone is present at a time,
their amplitudes are the same, and you don&#4294967295;t have a transition between
the tones in your 100 point sampling interval.

There is a technique that will give you very good frequency
resolution, but it requires that you have at least a few bins&#4294967295; spacing
between your tonal inputs.  So you&#4294967295;re operating well below the margins
at which it will work.

Kevin McGee

On Oct 30, 5:22&#4294967295;am, "tzoom84" <tszumow...@gmail.com> wrote:
> For simplicity, suppose we are interested in detecting presence of either a
> 500Hz, 1kHz, and a 1.5kHz sinusoid in a signal that is being sampled at
> 100kHz. But lets say we only have 100 samples, or 1ms of data. This is
> anywhere between 0.5 to 2 'cycles' of the sinusoids. So essentially, the
> signal is well oversampled, but is not sampled long enough in time to
> provide clean frequency resolution.
>
> Question is: Are there particular techniques to improve frequency
> resolution for oversampled, but time constrained, signals?

If it's an either-or question, you don't need resolution, you just
need
to reject the 2 less probable alternatives.  1 kHz is periodic in the
data aperture, so will show a clean peak in a DFT.  If the clear 1 kHz
peak isn't there, then you can try rejecting the 1.5 kHz choice if the
(maybe low passed) waveform doesn't show enough inflection points
and zero crossings spaced about the appropriate distances apart.
What's left if you don't see any of the above is 0.5 kHz.

And if there's no noise, you can just randomly pick 3 points and
try to solve the equation for a sinusoid with 3 unknowns.  Repeat
if necessary.

.

IMHO. YMMV.
--
rhn A.T nicholson d.0.t C-o-M
 http://www.nicholson.com/rhn/dsp.html

"tzoom84" <tszumowski@gmail.com> wrote in message 
news:OaOdnfUdh9nrR3fXnZ2dnUVZ_uqdnZ2d@giganews.com...
> For simplicity, suppose we are interested in detecting presence of either 
> a
> 500Hz, 1kHz, and a 1.5kHz sinusoid in a signal that is being sampled at
> 100kHz. But lets say we only have 100 samples, or 1ms of data. This is
> anywhere between 0.5 to 2 'cycles' of the sinusoids. So essentially, the
> signal is well oversampled, but is not sampled long enough in time to
> provide clean frequency resolution.
>
> Question is: Are there particular techniques to improve frequency
> resolution for oversampled, but time constrained, signals?
>
> (One random thought considered was to decimate at different points in
> time, say take every tenth sample starting at sample #1, then every tenth
> starting at sample #2, to create 10 independent signals of same length.
> Then concatenate to artificially lengthen the same signal. But I wasn't
> sure if that would increase resolution since it doesn't add any new
> data....)
> Thanks!

Can you store a history of samples in memory, like a buffer of 2048 samples 
perhaps?

On Oct 30, 10:32=A0am, Vladimir Vassilevsky <nos...@nowhere.com> wrote:
> tzoom84 wrote:
> > For simplicity, suppose we are interested in detecting presence of eith=
er a
> > 500Hz, 1kHz, and a 1.5kHz sinusoid in a signal that is being sampled at
> > 100kHz. But lets say we only have 100 samples, or 1ms of data. This is
> > anywhere between 0.5 to 2 'cycles' of the sinusoids. So essentially, th=
e
> > signal is well oversampled, but is not sampled long enough in time to
> > provide clean frequency resolution.
> > Question is: Are there particular techniques to improve frequency
> > resolution for oversampled, but time constrained, signals?
>
> The frequency resolution is the matter of SNR and the measurement time.
> The limit is set by Cramer Rao bound.
>

Actually, resolution has very little to do with SNR. Resolution is the
ability to separate 2 distinct signals. The resolution is mostly
determined by the length of the observation time.

Accuracy, on the other hand is very much SNR dependent. This is when
you are trying to estimate the frequency of 1 particular signal.

Cheers,
David

Dave wrote:

> On Oct 30, 10:32 am, Vladimir Vassilevsky <nos...@nowhere.com> wrote:
> 
>>tzoom84 wrote:
>>
>>>For simplicity, suppose we are interested in detecting presence of either a
>>>500Hz, 1kHz, and a 1.5kHz sinusoid in a signal that is being sampled at
>>>100kHz. But lets say we only have 100 samples, or 1ms of data. This is
>>>anywhere between 0.5 to 2 'cycles' of the sinusoids. So essentially, the
>>>signal is well oversampled, but is not sampled long enough in time to
>>>provide clean frequency resolution.
>>>Question is: Are there particular techniques to improve frequency
>>>resolution for oversampled, but time constrained, signals?
>>
>>The frequency resolution is the matter of SNR and the measurement time.
>>The limit is set by Cramer Rao bound.
>>
> 
> 
> Actually, resolution has very little to do with SNR. Resolution is the
> ability to separate 2 distinct signals. The resolution is mostly
> determined by the length of the observation time.

Then the resolution = Euclidean distance between the signals divided by 
noise, i.e. SNR.

> Accuracy, on the other hand is very much SNR dependent.

accuracy ~ log(SNR)

> This is when
> you are trying to estimate the frequency of 1 particular signal.

Engineers don't use words like "many, little, mostly, very, much".


Vladimir Vassilevsky
DSP and Mixed Signal Design Consultant
http://www.abvolt.com