Chapter V – Discrete Sampling and Analysis of Time Varying Signals
CHAPTERV
Discrete Sampling and Analysis of Time Varying Signals
After acquiring a signal using a data acquisition card (A/D convertor), the signal is not analog (continuous) anymore. Digitalizing the signal means that only discrete values of the signal are known. The question is for a specific signal what is the minimum points required to correctly represent the original signal.
V.1. Sampling rate theorem
Definition:The sampling rate is the rate at which measurements are made.
An inappropriate choice of the sampling rate may lead to a wrong representation of the actual signal.
Figure 5.1.A 10 Hz sine wave.
Figure 5.2.A 10 Hz sine wave sampled at rate of 5 Hz (This is the case if the sampling rate is an integer fraction of the base frequency (fm): fm; fm/2; fm/3, … ).
Figure 5.3.A 10 Hz sine wave sampled at a rate of 11 Hz (the resulting 1 Hz signal corresponds to 11 Hz – 10 Hz).
Figure 5.4.A 10 Hz sine wave sampled at a rate of 18 Hz (the resulting apparent frequency is 8 Hz).
Definition: Aliases are artifacts of sampling process leading to a false frequency representation of the input signal.
Figure 5.5.A 10 Hz sine wave sampled at a rate of 20.1 Hz.
Sample rate theorem (Nyquist-Shannon theorem)In order to reconstruct a signal correctly: “Sampling rate () must be greater than twice the highest frequency component () of the original signal”
The problem with a simple application of the sampling rate theorem is that the representation of the original signal will not be unique. For example: the signal above (10 Hz sine wave) can be generated with a 30.1 Hz. This problem can be avoided by filtering at frequencies higher than 2.
- Folding diagram
A folding diagram can be used to determine the lowest alias frequency knowing the sampled frequency (), the sampling frequency () and the folding frequency ().
Figure 5.6.Folding diagram.
ExampleCompute the lowest alias frequency for =100 Hz and =70 Hz.
V.2. Spectral analysis of time varying signals
Spectral analysis means determining all the frequencies that exist in a specific signal (usually a complex signal).
Why do you need a spectral analysis?Before starting the experiments / After getting the results
- To determine the sampling rate. / - Post-processing data to get the frequencies that characterize the system.
- To determine the frequency response of a transducer.
Spectral analysis using Fourier-series analysis
Where is the angular frequency () corresponding to the fundamental or harmonic frequency.
And ;
T represents the period and will give the average value.
Figure 5.7. (left) Amplitude of harmonics for a sawtooth waveform; (right) Reconstruction of the signal using 3 harmonics.
One apparent problem of spectral analysis using Fourier series is that it is only applicable to period functions. This can be however avoided by duplicating the function.
ExampleFind the Fourier series of the function:
for
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Instrumentation and Measurements \ LK\ 2009