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Technical Discussions
Pim van Andel
Pim van Andel
02:58 May-19-1998
sample rate for peak amplitude measurement

A low ADC sample rate gives a poor determination of peak amplitude
when we reconstruct the signal in the time domain by linear interpolation
between the successive sample points. Can we get a better estimate for
peak amplitude by applying a better reconstruction of the signal
(e.g. a 'spline like' technique in the time domain or zero padding in the
frequency domain)?.



 
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Dr. Ulrich Mletzko
R & D, Retired
Germany, Joined Nov 1998, 89

Dr. Ulrich Mletzko

R & D, Retired
Germany,
Joined Nov 1998
89
09:37 May-20-1998
Re: sample rate for peak amplitude measurement
: A low ADC sample rate gives a poor determination of peak amplitude
: when we reconstruct the signal in the time domain by linear interpolation
: between the successive sample points. Can we get a better estimate for
: peak amplitude by applying a better reconstruction of the signal
: (e.g. a 'spline like' technique in the time domain or zero padding in the
: frequency domain)?.

If you are using a low pass filter and if you are fullfilling the Shannon
sampling theorem, i.e. sampling frequency larger than two times the cut off
frequency of the filter, then you should be able to reconstruct the signal
completely.

But this can't be done by linear connection of the succesive sampling points!
You have to use the full set of the sampled data and make a Fourier transformation
into the frequency domain. Now use the obtained frequency data set (which in fact
are the amplitudes of all the sinus and cosinus functions of the spectrum,
respectively the real and imaginary parts) to calculate the time domain signal
by superposition of all the values for one (!) time point on the time axis of your
domain window. To reconstruct the full signal with high accuracy within the time
window you have to repeat the superposition calculation using a time resolution
much better than the sampling distance. This will give you the complete reconstruction
of your signal.

If you want to avoid that calculations, then you should use "oversampling". After low pass
filtering use a sampling frequency much higher than that of the Shannon theorem. For the
highest frequency in the spectrum (i.e. the cut off frequency of the filter) you will
have a "sampling error" of the amplitude value, which is depending on the oversampling rate.
Now you can use linear connection of the succesive sampling points. The obtained accuracy is
depending on your budget, as ADC PC cards of high sampling rate might be expensive.

Best regards

Uli Mletzko
NDT Group
State Materials TestingInstitute
University of Stuttgart
Germany





 
 Reply 
 
R Freemantle
R Freemantle
00:23 May-20-1998
Re: sample rate for peak amplitude measurement
: A low ADC sample rate gives a poor determination of peak amplitude
: when we reconstruct the signal in the time domain by linear interpolation
: between the successive sample points. Can we get a better estimate for
: peak amplitude by applying a better reconstruction of the signal
: (e.g. a 'spline like' technique in the time domain or zero padding in the
: frequency domain)?.

I have used zero padding of the complex frequency arrays to improve time domain
resolution for peak amplitude detection. I am pretty sure I have seen some
proofs that verify this method. I guess it is no different to zero
padding time domain data to improve frequency domain resolution.
Provided the frequency content of your data set is within the Nyquist limit
you should be okay. The technique is reasonably quick provided you have
a fast computer and a good FFT algorithm.

Hope this helps,

Richard.


=================================================
| Dr. R.J. Freemantle |
| |
| Research Fellow |
| UDSP Laboratory |
| Department of Physics |
| Keele University |
| Staffs, ST5 5BG, UK |
|-----------------------------------|
| Tel +44 (0)1782 584306 |
| Fax +44 (0)1782 711093 |
| Email r.j.freemantle@elec.keele.ac.uk |
| http://udsplab.elec.keele.ac.uk |
=================================================



 
 Reply 
 
Linas Svilainis
R & D,
Kaunas University of Technology, Lithuania, Joined Nov 1998, 68

Linas Svilainis

R & D,
Kaunas University of Technology,
Lithuania,
Joined Nov 1998
68
02:51 May-20-1998
Re: sample rate for peak amplitude measurement

: A low ADC sample rate gives ...
Not just that-if it's chosen too low, frequency
components aliasing occurs. This might give you additional
artefacts in the signal. Nyquist criterion defines the
frequency should be used. The sampling frequency should
be twice the maximal frequency STILL PRESENT in the signal
spectra. Here we need to settle the deal what is called
the CUT-OFF FREQUENCY. Usualy it is used in notation
of Chebyshev filter response, where the attenuation is
NOT BELOW THE RIPPLE (usually even less than -3dB,please correct me if you have another
information). Even if we are talking about the
"cut-off frequency", as the frequency where we have the sufficient
attenuation, we have to decide which level it is and, in
addition to that, SIGNAL FREQUENCY SPECTRA should be
taken on account here. Wider discussion on that issue
you can find at http://www.ndt.net/article/0598/linas_eq/linas_eq.htm
Here we present what's happening with the signal when
conventional way of sampling frequency choice is used.
Fig http://www.ndt.net/article/0598/linas_eq/fig5.gif
presents the signal spectra in dB from center frequency
The low pass filter with 5MHz@-3dB(-6dB/oct) was applied here
so the signal was sampled using 10MHz frequency.
Fig http://www.ndt.net/article/0598/linas_eq/fig8.gif
is presenting the error signal in percent from original
signal(oversampled) peak amplitude.
Because signals in reality will always have some contents
left behind the anti-aliasing filter response, there will
always be some aliasing. The sufficient filter attenuation
in stop-band+sufficient sampling frequency term means
that we are satisfied with errors introduced.
We are suggesting to use the comparable errors concept
in order to decide about the required sampling frequency.
The A/D converter word length/resolution is used to decide
about the sufficient sampling frequency. Refer to the
same figure for levels to determine such frequency for
8,10 and 12 bit A/D converter. Note, that signal was
1024 times averaged in order to reduce electrical noise influence.

:...a poor determination of peak amplitude
: when we reconstruct the signal in the time domain by linear interpolation
: between the successive sample points. Can we get a better estimate for
: peak amplitude by applying a better reconstruction of the signal
: (e.g. a 'spline like' technique in the time domain or zero padding in the
: frequency domain)?.
If Nyquist criterion is satisfied at agreed level
of errors, then we have ALL THE INFORMATION NECESSARY
for SIGNAL REASTORATIION at agreed level of errors
(or peak amplitude reconstruction). The zero padding
is ideal restoration procedure, but again, one has to
beware of circular convolution introduced errors.
Slightly degraded results can be obtained using time-limited
version (FIR) of sinc function convolution (which in fact is
the time domain equivalent for zero pading-the proof of
relevance of such interpolation is in sampling theorem).
But if time-limited interpolation is applied, influence of
circular convolution in time domain is greatly reduced
and there is no more need for FFT for faster interpolation-
filtering with Lagrange 4,6 points interpolation or cubic
spline interpolation are almost of the same choice.
Modified cubic spline is a bit smoother because of
less ringing in stop-band. In case of finite time response
filter (cubic spline/Lagrange) it's even faster the FFT,
because it has much lower complexity.
One note - we are NOT CREATING ADDITIONAL INFORMATION
with interpolation - once Nyquist limit is satisfied,
all the information is here - we're just extracting what
we need.

More information on amplitude quantisation errors - on
http://www.ndt.net/article/wsho0597/linas/linas.htm#2

Good luck,

Linas



 
 Reply 
 
Tom Nelligan
Engineering,
retired, USA, Joined Nov 1998, 390

Tom Nelligan

Engineering,
retired,
USA,
Joined Nov 1998
390
00:35 Jun-16-1998
Re: sample rate for peak amplitude measurement

: A low ADC sample rate gives a poor determination of peak amplitude
: when we reconstruct the signal in the time domain by linear interpolation
: between the successive sample points. Can we get a better estimate for
: peak amplitude by applying a better reconstruction of the signal
: (e.g. a 'spline like' technique in the time domain or zero padding in the
: frequency domain)?.

One of my colleagues has asked me to post the following excerpt from ASTM specification E 1065-96, Standard Guide for Evaluating Characteristics of Ultrasonic Search Units, with the thought that the general guideline given there might be of interest to you.

NOTE A4.2 For accurate measurement of the time response of a digitized rf waveform, an 8-bit digitizer is needed. A sufficient number of samples per cycle should be taken that a curve through the sampled values provides a smooth waveform that resembles the original analog waveform. For reliable measurement of peak or low level waveforms, a minimum samples of 36 samples per cycle is recommended. An 8-bit digitizer is inherently limited to displaying 48 dB of dynamic range and only half of this range is useable for evaluating an rf waveform. To evaluate low level signals may require increasing the gain of the amplifier. Averaging a number of waveforms increases the reliability. Specific requirements may be established between the supplier and user.


 
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