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[mathcad] Re: DSP method

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[mathcad] Re: DSP method
Author: David Burton    Posted: Wed, 11 Jan 2006 13:15:23 -0800
Do a correlation and the time difference is where the peak of the
correlation occurs.
David B

Mark Smith wrote:

> I have 2 digitised signals (A2D sampled) which are very similar in
> shape but differ by a fixed time difference, can anyone suggest
> methods which will allow me to calculate what the time difference is?
> I have the feeling that if i can somehow use all the data I should
> get a much more accurate value than say just comparing a zero crossing
> of the 2 traces which I think will be far more susceptable to noise in
> the signal.
>
> Thanks in advance
> mark
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[mathcad] RE: DSP method
Author: Pergande, Albert N    Posted: Wed, 11 Jan 2006 13:40:37 -0500
try correlating them with different index offsets. depending on the
data, you might only need a subset of the whole data stream.


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_____

From: Mark Smith
Sent: Wednesday, January 11, 2006 12:26 PM
To: Mathcad Discussion List
Subject: [mathcad] DSP method


I have 2 digitised signals (A2D sampled) which are very similar in shape
but differ by a fixed time difference, can anyone suggest methods which
will allow me to calculate what the time difference is?
I have the feeling that if i can somehow use all the data I should get
a much more accurate value than say just comparing a zero crossing of
the 2 traces which I think will be far more susceptable to noise in the
signal.

Thanks in advance
mark
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[mathcad] Re: DSP method
Author: Ted Diehl    Posted: Wed, 11 Jan 2006 14:05:36 -0500




Can you send me the two signals (ascii) files?
Email them to /> I would like to try a few things.

thanks

Ted Diehl





Mark Smith

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[mathcad] DSP method
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Please respond to

ptscience.com






I have 2 digitised signals (A2D sampled) which are very similar in shape
but differ by a fixed time difference, can anyone suggest methods which
will allow me to calculate what the time difference is?
I have the feeling that if i can somehow use all the data I should get a
much more accurate value than say just comparing a zero crossing of the 2
traces which I think will be far more susceptable to noise in the signal.

Thanks in advance
mark
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[mathcad] Re: DSP method
Author: Richard Jackson    Posted: Wed, 11 Jan 2006 14:22:52 -0500
Without seeing the data I have to guess a bit, but you could fit a spline
to one data set, so that it is represented by a function, f(x). Then least
squares fit f(x-delta) to the second data set, with delta as a parameter.
You will probably not be able to use the all the data though. Assuming
the two data sets are the same length, if you shift one relative to the
other parts of them will not overlap. You can use only the overlapping
sections.

A variation would be to also fit a spline to data set 2, to generate a
function g(x'). Then simultaneously fit g(x'-delta) to data set 1 and f(x-
delta) to data set 2.

Richard


On 11 Jan 2006 at 17:26, Mark Smith wrote:

>
> I have 2 digitised signals (A2D sampled) which are very similar in
> shape but differ by a fixed time difference, can anyone suggest
> methods which will allow me to calculate what the time difference is?
> I have thefeeling that if i can somehow use all the data I should get
> a much more accurate value than say just comparing a zero crossing of
> the 2 traces which I think will be far more susceptable to noise in
> the signal.
>
> Thanks in advance
> mark


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[mathcad] Re: DSP method
Author: Chris Whitford    Posted: Thu, 12 Jan 2006 09:14:00 +0000
You could take the cross-correlation and find the peak. To do this, Fourier
transform both waveforms, multiply one by the complex conjugate of the
other, then inverse transform. If the signals are periodic there will be
multiple peaks; use the one closest to zero.

Chris

At 17:26 11/01/2006 +0000, you wrote:
>I have 2 digitised signals (A2D sampled) which are very similar in shape
>but differ by a fixed time difference, can anyone suggest methods which
>will allow me to calculate what the time difference is?
> I have the feeling that if i can somehow use all the data I should get a
> much more accurate value than say just comparing a zero crossing of the 2
> traces which I think will be far more susceptable to noise in the signal.


+------------------+
+ Chris Whitford
+ Research Fellow, University of Leicester, Space Research Centre,
+ Physics and Astronomy Department, University Road, LEICESTER LE1 7RH, UK
+ Tel: +44 (0)116 252 3496, Fax: +44 (0)116 252 2464
+ email: http://www.star.le.ac.uk/
+ ------------------+


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[mathcad] Re: DSP method
Author: Eden Mei    Posted: Wed, 11 Jan 2006 10:38:31 -0800
Not sure what function is available, but the standard approach would be to run a correlation against the two streams and the peak value of the correlation function would be at the correct time offset

TTFN
----- Original Message -----
From: Mark Smith
To: Mathcad Discussion List
Sent: Wednesday, January 11, 2006 9:26 AM
Subject: [mathcad] DSP method


I have 2 digitised signals (A2D sampled) which are very similar in shape but differ by a fixed time difference, can anyone suggest methods which will allow me to calculate what the time difference is?
I have the feeling that if i can somehow use all the data I should get a much more accurate value than say just comparing a zero crossing of the 2 traces which I think will be far more susceptable to noise in the signal.

Thanks in advance
mark
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[mathcad] Re: DSP method
Author: Quang-Viet Nguyen    Posted: Wed, 11 Jan 2006 13:04:24 -0500
Hi Mark,

At 05:26 PM 1/11/2006 +0000, Mark Smith wrote:
>I have 2 digitised signals (A2D sampled) which are very similar in shape
>but differ by a fixed time difference, can anyone suggest methods which
>will allow me to calculate what the time difference is?
> I have the feeling that if i can somehow use all the data I should get a
> much more accurate value than say just comparing a zero crossing of the 2
> traces which I think will be far more susceptable to noise in the signal.
>
>Thanks in advance
>mark

You should check on the theory behind this, but I think there is a rather
fast way to do this with the built in FFT functions. You are trying to
calculate the cross-correlation function between the two signals, and
this can be found by doing the following:

Given two discrete time series signals f(t) and g(t),
1. take the FFT of each F(s) and G(s),
2. take the complex product of F(s) with the conjugate of G(s)
3. take the inverse FFT of the above product in (3) to get the
cross-correlation funtion R(t)
4. plot R(t) to see where the peaks are, and you can process R(t) to
automatically find the maxima.

The resulting function R(t) is the cross-correlation, and is in time units,
the peaks of this function correspond to maximum overlaps and correspond to
the difference in time you are looking for. If it is cyclic, you will get
multiple peaks.

I have done this to find the time delay between two digitized time-series
signals from transducers, microphones, etc. You also get the power
spectrum by plotting the FFT vs frequency.

I hope this helps.

Viet Nguyen

-------------------------------------------------------------------
Quang-Viet Nguyen, Ph.D.

NASA Glenn Research Center
21000 Brookpark Rd, MS 5-10
Cleveland, Ohio 44135

Voice: (216) 433-3574
Fax: (216) 433-5802
E-Mail: or />
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[mathcad] Re: DSP method
Author: Chris Whitford    Posted: Thu, 12 Jan 2006 14:14:30 +0000
Just a footnote to this: if you use the Fourier transform to get the
cross-correlation, you should pad each series with zeroes, at least as many
as the maximum shift you are looking for, because the FT "wraps around".

Chris

At 13:55 12/01/2006 +0000, you wrote:
>A huge thanks to all with their suggestion and at Ted's request attached are
>some generated data files with noise added which are the sort of thing I'll
>expect to get from the real world.
>regards
>mark


+------------------+
+ Chris Whitford
+ Research Fellow, University of Leicester, Space Research Centre,
+ Physics and Astronomy Department, University Road, LEICESTER LE1 7RH, UK
+ Tel: +44 (0)116 252 3496, Fax: +44 (0)116 252 2464
+ email: http://www.star.le.ac.uk/
+ ------------------+


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[mathcad] Re: DSP method
Author: Mark Smith    Posted: Mon, 16 Jan 2006 11:00:53 -0000
Hi again
Thanks to all who pointed me in the right direction of cross correlation.
I have a secondary question which hopefully someone might have an answer to.
I have done the cross correlation and plot its value against the shift to
find the maxima i.e. at what delay the 2 traces are most similar.
Is it valid for me to fit a polynomial to the points around the maximum &
solve for the maximum at a point intermediate to the sample points. In other
words can I resolve a time delay less than the period between samples or
must I increase the sample rate?

thanks again
regards
mark

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[mathcad] Re: DSP method
Author: Eden Mei    Posted: Mon, 16 Jan 2006 12:43:20 -0800
You could, but it's not obvious that you'll get much more mileage, since
those peaks tend to be broad.

You might do better using extrapolation from the sides of the distribution,
e.g., extend the slopes of each side and use the intersection. That's
basically the techique used in star trackers.

TTFN

----- Original Message -----
From: "Mark Smith" /> To: "Mathcad Discussion List" /> Sent: Monday, January 16, 2006 3:00 AM
Subject: [mathcad] Re: DSP method


> Hi again
> Thanks to all who pointed me in the right direction of cross correlation.
> I have a secondary question which hopefully someone might have an answer
to.
> I have done the cross correlation and plot its value against the shift to
> find the maxima i.e. at what delay the 2 traces are most similar.
> Is it valid for me to fit a polynomial to the points around the maximum &
> solve for the maximum at a point intermediate to the sample points. In
other
> words can I resolve a time delay less than the period between samples or
> must I increase the sample rate?
>
> thanks again
> regards
> mark
>
> ---
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[mathcad] Re: DSP method
Author: Quang-Viet Nguyen    Posted: Tue, 17 Jan 2006 01:10:11 -0500
Hi Mark,

At 11:00 AM 1/16/2006 +0000, Mark Smith wrote:
>Hi again
>Thanks to all who pointed me in the right direction of cross correlation.
>I have a secondary question which hopefully someone might have an answer to.
>I have done the cross correlation and plot its value against the shift to
>find the maxima i.e. at what delay the 2 traces are most similar.
>Is it valid for me to fit a polynomial to the points around the maximum &
>solve for the maximum at a point intermediate to the sample points. In other
>words can I resolve a time delay less than the period between samples or
>must I increase the sample rate?

Since the data is sampled at discrete time-steps your time resolution is
Nyquist limited. So for example, if the data is sampled at 2 kHz, you
cannot resolve anything finer than 1 kHz - or 1 ms precision. So
increasing the sample rate to 2x greater than the highest frequency you
anticipate that you will need to resolve will work. Be careful to use the
correct low-pass filters to prevent aliasing the data.

Good Luck.

Viet Nguyen

-------------------------------------------------------------------
Quang-Viet Nguyen, Ph.D.

NASA Glenn Research Center
21000 Brookpark Rd, MS 5-10
Cleveland, Ohio 44135

Voice: (216) 433-3574
Fax: (216) 433-5802
E-Mail: or />
-------------------------------------------------------------------



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[mathcad] Re: DSP method
Author: David Burton    Posted: Mon, 16 Jan 2006 14:51:07 -0800
You can time shift a sampled sequence by arbitrary partial sample
intervals by multiplying it with a unity amplitude phase
ramp in the frequency domain. Use complex math to do this. If you
have a sharp transient in your signals that you can
use as an alignemt reference this can be done quite accurately.

See USPTO #5423325 for a discussing of aligning signals with finer
temporal resolution than the sample rate using this method.

Or you could increase the sample rate by upsampling the signals with
proper filtering.

David B

Mark Smith wrote:

>Hi again
>Thanks to all who pointed me in the right direction of cross correlation.
>I have a secondary question which hopefully someone might have an answer to.
>I have done the cross correlation and plot its value against the shift to
>find the maxima i.e. at what delay the 2 traces are most similar.
>Is it valid for me to fit a polynomial to the points around the maximum &
>solve for the maximum at a point intermediate to the sample points. In other
>words can I resolve a time delay less than the period between samples or
>must I increase the sample rate?
>
>thanks again
>regards
>mark


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[mathcad] Re: DSP method
Author: Chris Whitford    Posted: Tue, 17 Jan 2006 10:09:30 +0000
The Nyquist frequency doesn't limit the accuracy of the phase measurement.
This is determined more by the total length of the data, and how sharp the
correlation is. You can interpolate to find the peak of the correlation,
but this is only one sample from a distribution. If the same experiment
were repeated multiple times, one would expect to find a scatter on this
position. Depending on the quality of the data this scatter could more or
less than the time step.

Chris

At 01:10 17/01/2006 -0500, you wrote:
>Since the data is sampled at discrete time-steps your time resolution is
>Nyquist limited. So for example, if the data is sampled at 2 kHz, you
>cannot resolve anything finer than 1 kHz - or 1 ms precision. So
>increasing the sample rate to 2x greater than the highest frequency you
>anticipate that you will need to resolve will work. Be careful to use the
>correct low-pass filters to prevent aliasing the data.


+------------------+
+ Chris Whitford
+ Research Fellow, University of Leicester, Space Research Centre,
+ Physics and Astronomy Department, University Road, LEICESTER LE1 7RH, UK
+ Tel: +44 (0)116 252 3496, Fax: +44 (0)116 252 2464
+ email: http://www.star.le.ac.uk/
+ ------------------+


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[mathcad] Re: DSP method
Author: Mark Smith    Posted: Tue, 17 Jan 2006 10:51:18 -0000
Hi Chris

You wrote... "The Nyquist frequency doesn't limit the accuracy of the phase
measurement.
This is determined more by the total length of the data, and how sharp the
correlation is. You can interpolate to find the peak of the correlation,
but this is only one sample from a distribution. If the same experiment
were repeated multiple times, one would expect to find a scatter on this
position. Depending on the quality of the data this scatter could more or
less than the time step."

Thanks for this it is as I had a feeling that this was the case but is there
a way to determine what the theoretical accuracy is from the data. I want to
only have to sample at a rate sufficient for the accuracy I need given the
level of noise present, slow A2D's are cheeper than fast ones.
I can improve accuracy by averaging many sets of data to reduce scatter but
again I'd like to keep this to a minimum.

cheers
mark

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[mathcad] Re: DSP method
Author: Michael Inggs    Posted: Tue, 17 Jan 2006 12:55:04 +0200
Hi

Chris is quite right about the length determining the resolution
possible. Pushing up the sample rate just makes more Fourier space
points available, and results in essentially interpolation of the data
anyway.

There is a whole literature on Superresolution techniques, which
essentially let you extend the time duration of your samples by fitting
a model to the data, and in that way achieving the long sample length
and more sample points. Look for Prony's method, Autoregression, MUSIC
algorithm, and so on. These might be much more appropriate to your need
for higher correlation resolution. Remember, all of these are a trade
off against noise: high noise means bad model...

Regards

Mike

Chris Whitford wrote:
> The Nyquist frequency doesn't limit the accuracy of the phase

--
Michael Inggs: Dept. Electrical Engineering, University of Cape Town
Private Bag, Rondebosch 7701, South Africa
Tel: +27 21 650 2799 Fax: +27 21 650 3465
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[mathcad] Re: DSP method
Author: Oakley, Philip SELEX UK    Posted: Tue, 17 Jan 2006 11:19:28 +0000


The potential accuracy can only be determined from your knowledge of the signal (and its digitisation).
Basically you need to know that it is band-limited in some way such that there is a particular part of the band that has no confounding alias components. For a regularly sampled data stream, any frequency f(w.t) is confused with f([s+/-w].t) where s=sampling frequency (which may include jitter). [this is for Real data, rather than I-Q sampled Complex data where only the + out of the +/- applies]

As long as you can identify a suitable spot where you no aliases then you can get quite fine sub-resolution.
The sub resolution depends on the number of sample point. This is obvious when n=1 sample [all you get is 'before' or 'after' for time resolution!!]

Philip

-----Original Message-----
From: Mark Smith /> Sent: 17 January 2006 10:51
To: Mathcad Discussion List
Cc: /> Subject: [mathcad] Re: DSP method


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Hi Chris

You wrote... "The Nyquist frequency doesn't limit the accuracy of the phase
measurement.
This is determined more by the total length of the data, and how sharp the
correlation is. You can interpolate to find the peak of the correlation,
but this is only one sample from a distribution. If the same experiment
were repeated multiple times, one would expect to find a scatter on this
position. Depending on the quality of the data this scatter could more or
less than the time step."

Thanks for this it is as I had a feeling that this was the case but is there
a way to determine what the theoretical accuracy is from the data. I want to
only have to sample at a rate sufficient for the accuracy I need given the
level of noise present, slow A2D's are cheeper than fast ones.
I can improve accuracy by averaging many sets of data to reduce scatter but
again I'd like to keep this to a minimum.

cheers
mark

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[mathcad] Re: DSP method
Author: Quang-Viet Nguyen    Posted: Tue, 17 Jan 2006 09:40:34 -0500
At 10:09 AM 1/17/2006 +0000, you (Chris Whitford) wrote:
>The Nyquist frequency doesn't limit the accuracy of the phase measurement.
>This is determined more by the total length of the data, and how sharp the
>correlation is. You can interpolate to find the peak of the correlation,
>but this is only one sample from a distribution. If the same experiment
>were repeated multiple times, one would expect to find a scatter on this
>position. Depending on the quality of the data this scatter could more or
>less than the time step.

This is analogous to the use of a phase-sampled o-scope at limited
bandwidth to get higher accuracy phase-measurements on periodic signals by
averaging the signal. But it doesn't replace an o-scope with a higher
speed digitizer to capture a non-periodic (or single-shot transient) signal
- which will be Nyquist limited. If the data is periodic then longer
sample lengths permit a type of averaging which improves phase
resolution. For a cross-correlation on non-periodic data streams, does
having a longer sampling length improve phase resolution?

-------------------------------------------------------------------
Quang-Viet Nguyen, Ph.D.

NASA Glenn Research Center
21000 Brookpark Rd, MS 5-10
Cleveland, Ohio 44135

Voice: (216) 433-3574
Fax: (216) 433-5802
E-Mail: or />
-------------------------------------------------------------------




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[mathcad] Re: DSP method
Author: Eden Mei    Posted: Tue, 17 Jan 2006 07:18:00 -0800
Your correlation function will only be sharp if tow things occur:
> High information "distance," e.g., off-peak deltas result in large decorrelations. You could do this by filtering out signals that don't help the correlation
> Sufficient resolution OR phasing of the FFT to decrease the spectral leakage into adjacent bins, i.e., a 50 Hz signal will correlate better if it's not spread out into 40, 45,50, 55, 60 Hz bins. Small changes our sample rate can alter the behavior.

TTFN,

----- Original Message -----
From: Mark Smith
To: Mathcad Discussion List
Cc:
Sent: Tuesday, January 17, 2006 2:51 AM
Subject: [mathcad] Re: DSP method


Hi Chris

You wrote... "The Nyquist frequency doesn't limit the accuracy of the phase
measurement.
This is determined more by the total length of the data, and how sharp the
correlation is. You can interpolate to find the peak of the correlation,
but this is only one sample from a distribution. If the same experiment
were repeated multiple times, one would expect to find a scatter on this
position. Depending on the quality of the data this scatter could more or
less than the time step."

Thanks for this it is as I had a feeling that this was the case but is there
a way to determine what the theoretical accuracy is from the data. I want to
only have to sample at a rate sufficient for the accuracy I need given the
level of noise present, slow A2D's are cheeper than fast ones.
I can improve accuracy by averaging many sets of data to reduce scatter but
again I'd like to keep this to a minimum.

cheers
mark

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[mathcad] Re: DSP method
Author: Oakley, Philip SELEX UK    Posted: Tue, 17 Jan 2006 16:18:54 +0000

Nguyen

Did you realy mean ;)
"non-periodic data streams, does having a longer sampling length improve phase resolution"

If it is non periodic it won't have a phase anyway, only correlation intervals.... clearly an area riddled with inconsistent terminology and expectations....

You can have alias free sampling, at least theoretically [the method uses the fact that there are two infinities (countable & uncountable) which follow the rule: all infinities are equal but some are more equal than others]

Philip



-----Original Message-----
From: Quang-Viet Nguyen /> Sent: 17 January 2006 14:41
To: Mathcad Discussion List
Subject: [mathcad] Re: DSP method


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At 10:09 AM 1/17/2006 +0000, you (Chris Whitford) wrote:
>The Nyquist frequency doesn't limit the accuracy of the phase measurement.
>This is determined more by the total length of the data, and how sharp the
>correlation is. You can interpolate to find the peak of the correlation,
>but this is only one sample from a distribution. If the same experiment
>were repeated multiple times, one would expect to find a scatter on this
>position. Depending on the quality of the data this scatter could more or
>less than the time step.

This is analogous to the use of a phase-sampled o-scope at limited
bandwidth to get higher accuracy phase-measurements on periodic signals by
averaging the signal. But it doesn't replace an o-scope with a higher
speed digitizer to capture a non-periodic (or single-shot transient) signal
- which will be Nyquist limited. If the data is periodic then longer
sample lengths permit a type of averaging which improves phase
resolution. For a cross-correlation on non-periodic data streams, does
having a longer sampling length improve phase resolution?

-------------------------------------------------------------------
Quang-Viet Nguyen, Ph.D.

NASA Glenn Research Center
21000 Brookpark Rd, MS 5-10
Cleveland, Ohio 44135

Voice: (216) 433-3574
Fax: (216) 433-5802
E-Mail: or />
-------------------------------------------------------------------




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