Contents Volume 3 – Signal Processing Reference Manual Contents 1 Sampling analogue signals 1.1 Introduction...................................................................................................................................................................1-1 1.2 Selecting a sampling speed .......................................................................................................................................1-1 1.3 References.....................................................................................................................................................................1-5 2 Digital filters 2.1 Introduction...................................................................................................................................................................2-1 2.2 Simple digital filters ....................................................................................................................................................2-4 2.3 Practical digital filters............................................................................................................................................... 2-13 2.4 References.................................................................................................................................................................. 2-14 3 Fourier analysis 3.1 Introduction...................................................................................................................................................................3-1 3.2 Fourier series and the Fourier transform .................................................................................................................3-2 3.3 Reference.......................................................................................................................................................................3-6 4 Power spectral density 4.1 Introduction...................................................................................................................................................................4-1 4.2 Computer calculation of power spectral density ...................................................................................................4-2 4.3 Window functions.........................................................................................................................................................4-3 4.4 Calculation of signal properties from the PSD.......................................................................................................4-8 4.5 References.................................................................................................................................................................. 4-10 5 Frequency response functions 5.1 Introduction...................................................................................................................................................................5-1 5.2 Calculation of frequency response function...........................................................................................................5-2 5.3 Gain.................................................................................................................................................................................5-2 5.4 Phase change ................................................................................................................................................................5-3 5.5 Coherence......................................................................................................................................................................5-4 6 Time-at-level & probability density 6.1 Introduction...................................................................................................................................................................6-1 6.2 Time-at-level analysis .................................................................................................................................................6-1 6.3 Probability density .......................................................................................................................................................6-4 6.4 Computer calculation ..................................................................................................................................................6-5 6.5 Interpretation of results...............................................................................................................................................6-5 7 Amplitude analysis 7.1 Introduction...................................................................................................................................................................7-1 7.2 Rainflow cycle counting.............................................................................................................................................7-1 7.3 Level crossing analysis...............................................................................................................................................7-9 7.4 Peak and valley counting........................................................................................................................................ 7-11 7.5 Range counting ......................................................................................................................................................... 7-12 7.6 Summary ...................................................................................................................................................................... 7-13 8 The data chain 8.1 Introduction...................................................................................................................................................................8-1 8.2 Data recording - analogue or digital ........................................................................................................................8-1 8.3 Short term recording or long term analysis ............................................................................................................8-3 8.4 Analogue to digital conversion ................................................................................................................................8-4 8.5 Sequential sample and hold.......................................................................................................................................8-5 ©safe technology limited Volume 3 Contents 1 Vol. 3 Contents Issue: 1 Date: 31.10.02 Contents 8.6 Simultaneous sample and hold .................................................................................................................................8-6 8.7 Reference.......................................................................................................................................................................8-7 9 Signal statistics Volume 3 Contents 2 ©safe technology limited Vol. 3 Contents Issue: 1 Date: 31.10.02 Sampling analogue signals 1 Sampling analogue signals 1.1 Introduction The world is an analogue world, which must be converted into digital form if it is to be processed by computer. The process of conversion is called sampling. The way in which signals are sampled has an effect on all the subsequent signal analysis and so it is appropriate to consider the subject of sampling before studying the analysis of the measured data. Sampling may be carried out using an analogue to digital converter (ADC). An ADC takes samples of analogue signals at specified times, and converts the samples into binary digits for analysis or storage. The way in which ADC's operate in described in Section 8. 1.2 Selecting a sampling speed Consider an ADC which is set to sample at 250 Hz, i.e. 250 samples per second. The diagram below (Figure 1.1) shows sine waves, with arbitrarily chosen frequencies of 55 Hz and 71 Hz, sampled by the ADC at 250 samples/second. It is reasonable to suppose that these sine waves could be reconstructed from the samples taken. Figure 1.1 Signals of 55Hz and 71Hz, sampled at 250Hz At the same sample frequency of 250 Hz, sine waves of 100 Hz and 150 Hz produce the same set of samples (Figure 1.2). Figure 1.2 Signals of frequency 100Hz and 150Hz sampled at 250Hz produce the same set of samples A reconstruction of a signal from these samples would give the signal with the lower frequency, and it would not be possible to reconstruct the higher frequency sine wave from this set of samples. Consider one further sine wave, with a frequency of 125 Hz, i.e. a frequency between the two sine waves shown above. If this is sampled at 250 Hz, a number of sets of samples are possible (Figure 1.3). ©safe technology limited Volume 3 1-1 Vol. 3 Section 1 Issue: 1 Date: 31.10.02 Sampling analogue signals Figure 1.3 Possible samples from a signal of 125Hz sampled at 250Hz In each case, it is possible to reconstruct a sine wave of the correct frequency, but a number of different amplitudes are possible. A graph of the frequency of the sine waves, and the amplitudes and frequencies that would be reconstructed from the samples taken by the ADC, could be plotted as shown below (Figure 1.4). Figure 1.4 Amplitude - frequency plot for the sinewaves It can be seen that when the sine wave has a frequency less than 125 Hz, both the amplitude and frequency can be reconstructed correctly. For a sine wave of 125 Hz the frequency can be reconstructed correctly,