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)?.

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

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 | =================================================

02:51 May-20-1998 Linas Svilainis R & D, Kaunas University of Technology, Lithuania, Joined Nov 1998 ^{67}

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

00:35 Jun-16-1998 Tom Nelligan Engineering, retired, USA, Joined Nov 1998 ^{390}

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.