Volume 3 Signal Processing Reference Manual

Transcription

1 Contents Volume 3 Signal Processing Reference Manual Contents 1 Sampling analogue signals 1.1 Introduction Selecting a sampling speed References Digital filters 2.1 Introduction Simple digital filters Practical digital filters References Fourier analysis 3.1 Introduction Fourier series and the Fourier transform Reference Power spectral density 4.1 Introduction Computer calculation of power spectral density Window functions Calculation of signal properties from the PSD References Frequency response functions 5.1 Introduction Calculation of frequency response function Gain Phase change Coherence Time-at-level & probability density 6.1 Introduction Time-at-level analysis Probability density Computer calculation Interpretation of results Amplitude analysis 7.1 Introduction Rainflow cycle counting Level crossing analysis Peak and valley counting Range counting Summary The data chain 8.1 Introduction Data recording - analogue or digital Short term recording or long term analysis Analogue to digital conversion Sequential sample and hold Vol. 3 Contents Issue: 1 Date: Volume 3 Contents 1

2 Contents 8.6 Simultaneous sample and hold Reference Signal statistics Volume 3 Contents 2 Vol. 3 Contents Issue: 1 Date:

3 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 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). Vol. 3 Section 1 Issue: 1 Date: Volume 3 1-1

4 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, but the amplitude cannot. For the sine wave of a frequency greater than 125 Hz, two possible frequencies may be reconstructed. This suggests that for a sample rate of 250 Hz, the maximum frequency of sine wave that can be reconstructed is 125 Hz, i.e. half the sample rate. Further, sine waves of frequencies greater than half the sample rate will be reconstructed as sine waves of frequencies less than half the sample rate. In other words a 'folding' of the apparent frequency occurs, whereby frequencies greater than half the sample rate fold back to appear as frequencies less than half the sample rate. Indeed, a sine wave with a frequency equal to the sample rate (in this case, a sine wave of 250 Hz) would be sampled as shown. Volume Vol. 3 Section 1 Issue: 1 Date:

5 Sampling analogue signals Figure Hz signal sampled at 250 Hz The result is zero frequency and an unknown amplitude, so a signal frequency equal to the sample rate folds back to give an apparent frequency of zero Hz. On the frequency diagram, therefore, there is an axis of symmetry at half the sample rate. Sine waves of higher frequencies have their apparent frequency folded about this axis, to appear as a lower frequency. This phenomena is known as aliasing. It leads to one of the most important theorems of sampling, which is that if the highest frequency present in a signal is f 1, then at least 2 x f 1 samples per second must be taken in order to define this frequency. As has been shown above, the highest frequency present in the signal is not the same thing as the highest frequency of interest in the signal. The frequency which corresponds to half the sampling frequency is called the Nyquist frequency. It represents the maximum frequency which can be interpreted by a frequency domain analysis. Clearly, if it is proposed to analyse the signal in order to define the frequencies present, then a sample rate could be selected which is at least twice the maximum frequency present in the signal. This would avoid the problem of aliasing, and is therefore one possible criteria on which to select the sample rate. Other forms of analysis may require a higher sample rate. A sine wave sampled at four times its frequency could produce different sets of samples. Two possible sets of samples are shown in Figure 1.6. Figure 1.6 Possible samples from a signal sampled at four times the signal frequency In the first instance, the samples have occurred (by chance) on the peaks and valleys in the signal. In the second case, the peaks and valleys have not been accurately defined. The sample rate, of four times the frequency of the signal, is twice the Nyquist frequency, but is not adequate to define the amplitudes of the peaks and valleys. Clearly the conventional recommendation of the Nyquist frequency as a basis for selecting a sample rate is quite inadequate for analyses such as fatigue analysis, which require an accurate definition of the amplitudes of the peaks and valleys in the signal. It can be shown (ref 2) that the error in defining the peaks in a sine wave is given by P k = 2 sin f 2f s Vol. 3 Section 1 Issue: 1 Date: Volume 3 1-3

6 Sampling analogue signals where P k f f s = percent error on peaks/100 = frequency of the sine wave = sample frequency This equation is plotted below (Figure 1.7). It shows that at four points per cycle, the error in peak resolution can be up to 30% Figure 1.7 Error in sampling a sinewave In Ref 3, a narrow band Gaussian signal was used to investigate the error in fatigue life prediction which results from undersampling a signal. Taking a sample rate 100 times the signal frequency as giving the 'correct' answer, and normalising all other calculated lives by this value gave the results shown in Figure 1.8 for local stress-strain analysis and for analysis of welded joints using BS5400 fatigue life data. It can be seen that sampling at 10 times the signal frequency gave calculated lives 1.3 to 1.5 times the true value. A sampling rate of four times the signal frequency produces calculated lives which are two or three times the true value, and a sample rate based on the Nyquist frequency (two points per cycle) gives an error which is too large to be shown on the graphs. Relative life Narrow band Broad band Points/cycle Figure 1.8 Effect of sampling frequency on fatigue life estimation In practice, errors would be expected to be less than those shown, because most signals contain a mixture of frequencies, and the fatigue damage tends to be produced by large amplitude lower frequency components in the signal. However, a sample rate of ten times the maximum signal frequency generally gives a reasonable compromise between quantity of data and accuracy of analysis. Volume Vol. 3 Section 1 Issue: 1 Date:

7 Sampling analogue signals 1.3 References 1.1 Caxton C Foster REAL TIME PROGRAMMING - NEGLECTED TOPICS Addison Wesley Donaldson, K. FIELD DATA CLASSIFICATION AND ANALYSIS TECHNIQUES S.A.E. Paper Morton, K., Musiol, C., Draper, J. LOCAL STRESS-STRAIN ANALYSIS AS A PRACTICAL ENGINEERING TOOL Proc SEECO 1983, City University, London Vol. 3 Section 1 Issue: 1 Date: Volume 3 1-5

8 Sampling analogue signals Volume Vol. 3 Section 1 Issue: 1 Date:

9 Digital filters 2 Digital filters 2.1 Introduction A filter modifies an input signal to produce an output signal. Filters may be used for many purposes - to remove high frequency noise, to remove long term drift, or to shape computer-generated signals for mechanical testing. LOW PASS filters are designed to pass low frequencies, but eliminate higher frequencies. HIGH PASS filters are designed to pass high frequencies, but eliminate low frequencies. BAND PASS filters are designed to pass a certain bands of frequencies, but eliminate others. This introduction to digital filters will concentrate on the principles of low pass filters, although the information is generally applicable to high pass and band pass filters. For a low pass filter, the low frequencies will be passed with their amplitudes largely unchanged, whilst the higher frequencies will have their amplitudes reduced. The extent to which the filter modifies the amplitudes is expressed as the gain of the filter, and the gain will have different values at different frequencies. An ideal low pass filter will pass low frequencies without any modification to their amplitudes (gain = 1), and the higher frequencies will have their amplitudes reduced to zero. The gain diagram of an ideal filter, plotted as a function of frequency, would be as shown below. Figure 2.1 Gain diagram for an ideal filter The gain is the ratio of the output and input amplitudes at any specified frequency. The frequency at which the gain changes from unity to zero is the cut-off frequency. In practice it is difficult to obtain such a sharp step transition between the pass band and the stop band, and a typical filter will have a gain diagram as shown in Figure 2.1. Vol. 3 Section 2 Issue: 1 Date: Volume 3 2-1

10 Digital filters Figure 2.2 Typical gain diagram It is less obvious just what frequency represents a cutoff in this case, so a definition of the cutoff frequency is required. By definition, the cutoff frequency is the frequency at which the gain falls to a value of This number is not quite as arbitrary as it looks. In the generation of electrical power, power is proportional to (current) 2, so a gain of represents a reduction in power of (0.707) 2 = 0.5. The cutoff frequency is therefore the frequency at which the power is reduced to one-half of its value. Filter descriptions may specify both a cutoff frequency and a roll-off rate. The roll-off rate is usually defined as decibels/octave, or db/octave. One octave represents a doubling of frequency, and 3 db/octave gives a reduction in amplitude of over each octave, 6 db/octave gives a reduction of 0.5 in amplitude, etc. (See Appendix 2 for the Decibel scale). Most filters also effect the phase relationships within a signal. Consider a signal which is constructed by superimposing four sinewaves of different amplitudes and frequencies, as follows. Figure 2.3 Time-shifting of a series of sinewaves If this signal is shifted in time, so that the start point is at the line AA, without changing the shape of the waveform, then each of the component sine waves will have been shifted through a different angle. The highest frequency component has been shifted through 90 o, the next highest frequency through 45 o, and so on. In other words, the phase relationship between the component sine waves will have been changed. Plotting the phase change as a function of the frequency of each sine wave produces a phase diagram as shown. Volume Vol. 3 Section 2 Issue: 1 Date:

11 Digital filters Figure 2.4 Linear phase diagram for time-shifted signals It is clear that a linear phase change with frequency maintains the shape of the original signal. For mechanical engineering applications, and in particular fatigue analysis or testing, it is desirable that those frequencies which we wish to retain (the frequencies in the pass band) should be retained with the shape of the waveform unaltered. A linear phase diagram is therefore a desirable characteristic of a filter. Phase diagrams are usually plotted between limits of ±180 o, with a positive slope representing a phase delay. A linear phase diagram for the time-shifted signals would then appear as shown in Figure 2.5. Figure 2.5 Linear phase diagram for time-shifted signals, plotted between ±180 o The phase change which can be produced by filtering is illustrated below. A sinewave with added noise has been low-pass filtered, and the phase lag produced on the filtered signal can be clearly seen. Figure 2.6 Phase change produced by filtering Vol. 3 Section 2 Issue: 1 Date: Volume 3 2-3

12 Digital filters 2.2 Simple digital filters Three point smoothing filter Figure 2.6 shows a sinewave with added high frequency noise. A simple method of reducing the noise content of this signal would be to smooth the signal digitally. A simple smoothing program replaces each data point by the average of several adjacent data points. For example, a 3-point smoothing filter may form each output data point by averaging three input data points. In the notation used for these notes, y(t) is the output signal x(t) is the input signal y k, y k+1,x k, x k+1 etc are individual data points in the signals A three point smoothing filter could be expressed as y k = 1 3 (x k-1 + x k + x k+1 )...(2.1) The gain and phase diagrams for this filter can be obtained by filtering computer-generated white noise signals. Figure 2.7 Gain and phase diagrams for a three-point smoothing filter The three point smoothing filter could be expressed as y k = b 1 x k-1 + b 2 x k + b 3 x k+1...(2.2) The three constants b n, which before each had a value of 1, can now have any value, although for a 3 low pass filter the sum of the constants must equal unity if zero Hz is to be passed unaltered. The three point smoothing filter is in fact only one of a large class of filters which all produce the output signal by combining a number of input data points with different values of constants before each data point. It is a non-recursive filter, because it forms the output signal only from the input signal Recursive low pass filter Consider now a filter defined by the equation y k = b.x k + a.y k-1...(2.3) In this case the output signal y(t) is formed by mixing the input signal x(t) with previous values of the output signal. Volume Vol. 3 Section 2 Issue: 1 Date:

13 Digital filters To show that this is still a filter, the equation can be expanded, as follows. The value y k-1 will be given by y k-1 = b.x k-1 + a.y k-2 so that y k = b.x k + a.b.x k-1 + a 2.y k-2 Continuing the expansion, y k-2 = b.x k-2 + a.y k-3 so that y k = b.x k + ab.x k-1 + a 2 bx k-2 + a3.y k-3...(2.4) If this process of substitution is continued, the contribution of the term a n.y k-n will become negligible and the expression becomes y k = b.x k + ab.x k-1 + a 2 bx k-2 + a3 bx k (2.5) The output value y k is now made up only of input values x k, and the expression is similar to that of the three-point smoothing filter given in equation (2.1). If values are defined for a and b, (say) a = 7 8, b = 1 8, the filter has the characteristics shown : Figure 2.8 Gain and phase diagrams for the simple recursive filter The expression y k = b.x k + a.y k-1 is a recursive filter, because it mixes input and output to produce the new output. The process of expansion transformed the recursive filter into a non-recursive filter, with a slight approximation introduced by ignoring the final term in equation (2.4). In general, recursive filters can be expressed as non-recursive filters, although the non-recursive form is an approximation. Comparison of equations (2.3) and (2.4) shows that the recursive filter requires two multiplications and one addition Vol. 3 Section 2 Issue: 1 Date: Volume 3 2-5

14 Digital filters to produce each data point. Its expansion, even in its truncated form, requires seven multiplications and two additions to produce each data point. In computer operation, the non-recursive filter will be much slower. It is generally true that it is more efficient to achieve a required filter characteristic by using recursive filters. However, because recursive filters feed previous output back into the filter, they may require careful design if they are to be stable for all signals. It can be seen that these simple filters have gain and phase diagrams which are far from ideal - the gain diagram does not have a flat pass band and the phase diagram is not linear. To understand the design of more effective filters, we first need to define some filter characteristics. This will be done by reference to simple analogue filters Design of a simple recursive low pass filter The basic notation for filters was derived for analogue filters. Simple analogue filters consist of a resistor and a capacitor in series, and so have the general name of RC filters. Figure 2.9 Simple analogue filter The response of an RC filter to a step input takes the form shown in Figure 2.10 Input Output Figure 2.10 Response of a simple analogue filter to a step input This shows that if the input is held constant for some time, the output will reach the value of the input - so a very low frequency is passed with its amplitude unaltered. However, if the input is held for only a short interval of time, the output has insufficient time to reach the input value, so the amplitude of the higher frequency is reduced. The circuit is therefore acting as a low pass filter. In physical terms, at low frequencies the capacitor has time to fully charge and discharge, so the output voltage V 2 across the capacitor has time to equal the input voltage V 1. At higher frequencies the capacitor cannot fully charge and discharge in the time available, and so the ratio ( V2 / V1 ) falls towards zero as the frequency is increased. High frequencies are therefore rejected. If the capacitor is larger, it will require more time to 'fill'. Similarly, if the resistor has a high value, it will cause the capacitor to 'fill' more slowly. The maximum frequency that the filter could pass will therefore be inversely proportional to the size of the capacitor and the size of the resistor. Using the definition of the cut-off frequency as being the frequency at which the gain ( V2 / V1 ) falls to a value of 0.707, then the cutoff frequency for the RC filter is f cutoff = 1 2πRC... (2.6) Volume Vol. 3 Section 2 Issue: 1 Date:

15 Digital filters The term RC in equation (2.6) is called the time constant for the filter. If the input voltage is suddenly reduced to zero, the time constant is the time taken for the output voltage to fall to 1/e of its original value, where e = It was shown in Section 2.1 that a simple recursive low pass filter can be produced using an equation such as equation (2.3) y k = b.x k + a.y k-1 where x k is the input signal y k-1 is the output signal. and a and b are constants. A low pass filter has a gain of unity at a frequency of zero Hz. For zero Hz, the input is held constant, and so the output must also be constant and equal in value to the input. In equation (2.3) this will be true if a + b = 1... (2.7) The time constant for the filter can be calculated if the value of the input is suddenly reduced to zero. Figure 2.11 Step input at k = 0 Consider a filter defined by equation (2.3) with constants a = 7 8 b = 1 8 If the value of the input signal is held constant at 1, then reduced to zero when (say) k = 1, then a table can be constructed as follows Vol. 3 Section 2 Issue: 1 Date: Volume 3 2-7

16 Digital filters k x (input) y (old output) y (new output) From equation (2.3), and using the values of b = 1 8, a = 7, the rest of the table can be completed. 8 k x (input) y (old output) y (new output) input reduced to zero This table is plotted in Figure 2.12 Volume Vol. 3 Section 2 Issue: 1 Date:

17 Digital filters Figure 2.12 Response of filter to step input at k = 0 By the 8th sample, the value has fallen to about 1 / e ( = ). In general, if the time between samples is T seconds, then it takes 8T seconds for the value of the output to fall to Using the notation of equation (2.3), it takes T b seconds, and this is the value of the time constant for the filter. It will be seen that the term T for a digital filter is analogous to the b term RC for an analogue filter, and it may be deduced that the cutoff frequency, which for an analogue filter is f cutoff = 1 2πRC will for a digital filter be given by f cutoff = 1 2π T b or f cutoff = b 2πT...(2.6) Notice that the cutoff frequency for the digital filter is not an absolute value, but is a function of sample rate (T is the time between samples). The constants a and b in equation (2.3) have to be recalculated for the filter each time it is used for signals of different sample rates. For this reason, filter characteristics are often expressed in terms of the Nyquist frequency, rather than absolute frequency. Example. For a signal sampled at 1000 Hz, a filter with a cutoff frequency of 100 Hz would require b = f cutoff 2πT and as T is the time between samples, i.e. 1/1000 seconds, b = π 1000 Vol. 3 Section 2 Issue: 1 Date: Volume 3 2-9

18 Digital filters so b = and from equation (2.3) a = = The impulse response for analogue filters was defined earlier. The impulse response of a digital filter can be obtained by holding the input values to zero, except when k = 0, when the input value is 1. For the simple filter, (say) a = 1 2, and b = 1 2, then y k = 0.5 x k y k-1 The impulse response can be calculated in a table k x y If the impulse value was a number other than 1, then the values in the table could be scaled accordingly. As any digital signal can be considered as a sequence of pulses at successive values of k, then the response of the filter to a signal can be calculated from the impulse response table shown above. For a triangular input signal : k = x = etc the response can be calculated by superimposition as Volume Vol. 3 Section 2 Issue: 1 Date:

19 Digital filters k x y 0 0 = = = = = = = = = 0.88 A B C D E F G where column A is the response of a unit impulse column B is the unit response multiplied by 2 to give the response to an impulse of 2, delayed by one sample column C is the unit response multiplied by 3 to give the response to an impulse of 3, delayed by one additional sample, etc. This result is plotted below (Figure 13), and shows that a time delay, or phase shift, has occurred between the input and output signals. Figure 13 Comparison of calculated input and output signals The above example also shows why output filters are needed on digital-to-analogue converters. Without filters, the digital signal is simply a series of pulses, rather than a continuous signal. Closing this section on simple recursive low pass filters, it has been shown above that the recursive filter could be expanded into a non-recursive filter which had an infinite number of terms in the expansion. For this reason, recursive filters are known as infinite impulse response (IIR) filters. Nonrecursive filters, which have (by definition) a finite number of terms in the series, are called finite impulse response (FIR) filters Recursive high pass filter The recursive low pass filter averaged the new input with the old output, so that any sudden changes (high frequencies) were smoothed out, and only long term trends (low frequencies) are retained. For a Vol. 3 Section 2 Issue: 1 Date: Volume

20 Digital filters high pass filter, only the sudden changes themselves are required, so the filter must operate by discarding long term trends. The formula y k = b.x k - a.y k-1...(2.7) subtracts the old (long term) output a.y k-1 from the input signal, and represents the equation of a high pass filter. The gain of such a filter is gain = b (1-a) and as the gain for a high-pass filter must be unity at the high frequencies, a + b = 1 as for the low pass filter Recursive band pass filter Band pass filters can be constructed by adding together a low pass and a high pass filter. The two cutoff frequencies can be calculated in the same way as for the individual filters. For the low pass filter y k = b 1.x k + a 1.y k-1... from (2.3) The output from this filter can be passed through a high pass filter z k = b 2.y k - a 2.z k-1... from (2.7) Adding the terms z k = b 1 b 2.x k + (a 1 - a 2 ) z k-1 + (a 1 a 2 ) z k-2...(2.8) to give the equation for a band pass filter More complex recursive filters The band pass filter was constructed by taking the output from one filter and passing it through another filter. This process can be used to construct more complex filters, low pass and high pass as well as band pass. The output from a low pass filter y k = b.x k + a.y k-1... from (2.3) can be used as input to a second low pass filter z k = b.y k + a.z k-1 Adding the terms gives a single expression z k = 2a.z k-1 + a 2.z k-2 + b 2.x k... (2.9) Volume Vol. 3 Section 2 Issue: 1 Date:

21 Digital filters Methods such as these, and much more complex ones, can be used to construct digital filters with characteristics to suit particular requirements. 2.3 Practical digital filters An ideal low-pass filter would have the frequency response shown in Figure 6, with all the frequencies below the cut-off frequency passed completely unaltered, and all frequencies above the cut-off frequency completely rejected. None of the simple filters described in the previous sections come close to meeting these criteria. The design of practical filters involves a compromise between the sharpness of the cutoff, and the smoothness of the frequency response. Three types of filter are commonly encountered in engineering data acquisition - Butterworth, Chebyshev and Elliptical filters. Butterworth filters have a smooth frequency response in both the pass band and the stop band, and their effect on the phase relationships in the signal is reasonably linear over much of the pass band range. Typical gain and phase diagrams are shown in Figure 2.14 for a low pass Butterworth filter with a roll-off of -18dB per octave Figure 2.14 Gain and phase diagrams for a butterworth low pass filter (See Section 2, page 3, for description of roll-off rate). The Butterworth filter is a recursive filter, defined by a set of polynomial equations. These equations may be expanded to produce a non-recursive filter expression. For a low-pass filter, two design criteria must be specified - the cut-off frequency, and the roll-off rate. A first-order Butterworth filter has a single term polynomial, and a roll-off rate equal to -6dB/octave. A second-order filter has a - 12dB/octave roll-off, a third-order filter -18dB/octave, and so on. In the terminology used in filter design, a first-order Butterworth filter may be referred to as a 2-pole filter, a second order filter as a 4- pole filter, etc. As the number of poles approaches infinity, the gain and phase diagrams approach the ideal filter characteristics. Chebyshev filters produce a sharper corner frequency, but at the expense of ripple in either the pass band (type 1 Chebyshev filters) or in the stop band (type 2). The ripple in the pass band allows the filter to have a gain very close to unit a zero Hz and close to the cut-off frequency. Three design criteria are required - the cut-off frequency and roll-off rate, plus the amount of ripple to be permitted. Elliptical filters produce ripple in both the pass and stop bands, so four design criteria are required. They produce a linear phase diagram. Figure 2.15 shows a comparison of the gain diagrams for the three types of filter. Vol. 3 Section 2 Issue: 1 Date: Volume

22 Digital filters 2.4 References Figure 2.15 Comparison of butterworth, chebyshev and elliptical filters For the description of recursive filters, this section has drawn on the following book Caxton C Foster REAL TIME PROGRAMMING - NEGLECTED TOPICS Addison-Wesley 'Joy of Science' paperback 1981 ISBN X This is an excellent book for non-programmers as well as programmers, covering many hard-to-find topics in data acquisition and test control in a very readable way. Volume Vol. 3 Section 2 Issue: 1 Date:

23 Fourier analysis 3 Fourier analysis 3.1 Introduction Section 1 referred to the analysis of the frequency content of signals and introduced the concept of plotting the amplitudes present in a signal against the frequencies at which those amplitudes occurred. Consider a sine wave with an amplitude of 1, and a frequency of 1 Hz. Figure 3.1 A 1 Hz sine wave of amplitude 1 It can be represented by the equation where a is the amplitude of the sine wave. y 1 = a sin ω 1 t A second sine wave, of amplitude 1 3, and frequency 3 Hz y 2 = b sin ω 2 t can be added to the first sine wave, to give Figure 3.2 A 3 Hz sinewave of amplitude 1 3 y = a sin ω 1 t + bsin ω 2 t Figure 3.3 Summation of the two sinewaves A third sine wave, of amplitude 1, and frequency 5 Hz, when added, gives 5 Vol. 3 Section 3 Issue: 1 Date: Volume 3 3-1

24 Fourier analysis After a total of ten or so such sine have been superimposed, the resulting signal is a fair representation of a square wave. Figure 3.4 Summation of ten sinewaves It seems a reasonable argument that if a square wave can be produced by adding together a number of sine waves, then almost any signal can be produced from sine waves of various amplitudes, frequencies, and phase relationships. This is broadly true, providing the signal obeys certain rules. These rules will be defined fully later on, but in general terms the signal must repeat over a certain time interval, and be reasonably constant in its properties. Engineers have extended this argument by the convenient assumption that a length of signal that doesn't repeat at all represents one occurrence (or repeat) of a signal. As in section 1, the amplitudes could be plotted at their respective frequencies. Figure 3.5 Amplitude vs frequency for a synthesized square wave The process of extracting from a signal the amplitudes and frequencies present is the basis of frequency domain analysis. In essence it is the reverse of the procedure to construct the square wave - it shows what frequencies and amplitudes were used in its construction. 3.2 Fourier series and the Fourier transform To calculate the amplitudes and frequencies present in a signal, a band pass filter could be used to filter out all but a narrow band of frequencies, and the procedure repeated for a succession of pass bands to cover the frequency range of interest. This was the original method used, but its disadvantage was its susceptibility to the characteristics of the filters. Modern methods are based on Fourier analysis, and in particular the Fast Fourier Transform. Volume Vol. 3 Section 3 Issue: 1 Date:

25 Fourier analysis The square wave was derived using a series of sine waves y(t) = a 1 sin ω 1 t + a 2 sin ω 2 t + a 3 sin ω 3 t (3.1) In general, the series is y(t) = a 0 + a 1 cos 2πt T + b 1 sin 2πt T + a 2 + b 2 cos 4πt T sin 4πt T + a 3 + b 3 cos 6πt T +... sin 6πt T (3.2) Note that T is the period of the signal, i.e. the length of time between repeats of the signal. In strict terms Fourier series only apply to signals which repeat after a fixed interval of time. The series can be expressed in more compact form as k= y(t) = a 0 + k=1 a k cos 2πkt T + b k 2πkt sin T...(3.3) where a 0 is the mean of the signal, and by definition T 2 a 0 = 1 T - T 2 y(t).dt T 2 a k = 2 T - T 2 y(t).cos 2πkt T.dt, k 1 T 2 b k = 2 T - T 2 y(t).sin 2πkt T.dt, k 1...(3.4) From the square wave example, each term in the series represents a discrete frequency, and equation (3.4) shows that the frequencies are spaced at intervals of 2π/T. Vol. 3 Section 3 Issue: 1 Date: Volume 3 3-3

26 Fourier analysis Figure 3.6 Spacing of fourier coefficients at intervals of 2π T If the time T over which the signal repeats becomes larger, the spacing between the discrete frequencies becomes smaller. Real signals, which do not repeat, have an infinitely large value of T, and so the individual coefficients merge into a continuous series, called a Fourier Transform The derivation of the equations for the Fourier Transform is as follows. The x-axes in Figure 3.7 represent the series 2πk/T. Let ω k = 2πk T and let the interval, 2π T, be ω. Then equation (3.3) can be written as k= y(t) = k=1 2 T 2 T -T 2 y(t) cos 2πkt T 2πkt.dt cos T k= + k=1 2 T 2 T -T 2 y(t) sin 2πkt T 2πkt.dt sin T or, substituting ω for 2πk/T, and ω for 2π/T Volume Vol. 3 Section 3 Issue: 1 Date:

27 Fourier analysis k= y(t) = k=1 ω T 2 π -T 2 y(t) cos ( ω k t ).dt cos ( ω k t ) k= y(t) = k=1 ω T 2 π -T 2 y(t) sin ( ω k t ).dt sin ( ω k t ) As the time over which the signal repeats becomes longer, T, ω dω, and the summations become integrals from ω = 0 to ω =, so that y(t) = ω= dω π y(t) cos( ) - ω=0 ωt.dt cos ( ωt ) + ω= dω π y(t) sin( ) - ω=0 ωt.dt sin ( ωt ) In order to simplify this expression, let two new terms be introduced : A(ω) = 1 2π - y(t) cos( ωt ).dt B(ω) = 1 2π - y(t) sin( ωt ).dt...(3.5) then y(t) = 2 A(ω) cos ωt.dω B(ω) sin ωt.dω...(3.6) Equation (3.6) can be thought of as the definition of the components of the signal y(t), consisting of two 'amplitude' terms, A(ω) and B(ω), which have different values at different frequencies. A(ω) and B(ω) are the Fourier Transforms of y(t), as they represent the components of the signal. Equation (3.6) is the Inverse Fourier Transform of y(t), as it represents the method of reconstructing the signal from its components. Vol. 3 Section 3 Issue: 1 Date: Volume 3 3-5

28 Fourier analysis It is usual to introduce a i notation into these equations, where i = -1 If, by definition e iθ = cos θ + i sin θ then, also by definition, Y(ω) = A(ω) - i B(ω). Note that these terms are definitions only. A(ω) and B(ω) are defined in equation 3.5, so that or more concisely as Y(ω) = 1 2π - y(t) (cos ωt - i sin(ωt)).dt Y(ω) = 1 2π - y(t) e -iωt.dt Y(ω) is the Fourier Transform of y(t) in this new notation. It represents the conversion of the signal from the time domain into the frequency domain. It can be shown that y(t) = - Y(ω) e -iωt.dω is the Inverse Fourier Transform, which represents the reconstruction of the signal from the frequency description. 3.3 Reference Newland, D E AN INTRODUCTION TO RANDOM VIBRATION AND SPECTRAL ANALYSIS Longman (2nd ed) Volume Vol. 3 Section 3 Issue: 1 Date:

29 Power spectral density 4 Power spectral density 4.1 Introduction The power spectral density diagram, or PSD, contains a description of the amplitudes and frequencies present in a signal. In Section 3 it was shown that complex signals could be constructed by adding together sine and cosine waves of various amplitudes, frequencies and phase relationships. Figure 4.1, below, shows a signal constructed from a series of sinewaves. Figure 4.1 Signal constructed by adding sinewaves Power spectral density analysis is essentially the reverse of this signal construction process - it determines the amplitudes and frequencies present in a signal. The rms amplitudes of the sinewaves shown in Figure 4.1 may be plotted at the appropriate frequency. Figure 4.2 RMS amplitudes plotted vs frequency More complex signals contain a large number of different amplitudes and frequencies so the graph in Figure 4.2 would be a continuous function. For these signals a series of filters could be used to determine the amplitudes present between any two frequencies, and the rms amplitudes plotted in the centre of each frequency band of the filter. In fact, is it usual to plot the (amplitude) 2, as in electrical power generation power is proportional to (current) 2. Also, in order to remove the effect of the width of the filter, the power is divided by the width of the filter to produce a density diagram. The power spectral density diagram is therefore a display of (rms amplitude) 2 frequency versus frequency Vol. 3 Section 4 Issue: 1 Date: Volume 3 4-1

30 Power spectral density Figure 4.3 Area under the psd is the variance of the signal The area under the curve, between any two frequencies, gives an (rms amplitude) 2 term, and so power spectral density diagrams show not the 'power' at a given frequency, but the power between two chosen frequencies. As shown in Appendix 1, the average value of (rms amplitude) 2 in a signal is the variance, so the area under the PSD between any two frequencies is the variance of the signal between these frequencies. 4.2 Computer calculation of power spectral density In Section 3 it was shown that Fourier analysis could be used instead of band pass filters to obtain the frequencies and amplitudes present in a signal. The computer calculation of power spectra uses a Fourier analysis. The derivation in Section 3 of the Fourier Transform and Inverse Fourier Transform applies to continuous signals. Sampled signals are not continuous, but are represented by discrete samples. Discrete versions of the Fourier relationships, called Discrete Fourier Transforms (DFT's), are used in the analysis of sampled signals. From section 3, the Fourier transform Y(ω) of a signal y(t) is given by Y(ω) = 1 2π y(t) (cos ωt - i sin ωt).dt...(4.1) - and by definition, Y (ω) = A(ω) - i B(ω)....(4.2) The complex Fourier coefficients A(ω) and B(ω) are calculated from the measured signal. The procedure is to take a length of signal, form the complex Fourier coefficients, and take the complex conjugate of the coefficients to produce the (amplitude) 2 terms. (The complex conjugate of a pair of complex numbers is obtained by multiplying together the real parts of the numbers, and adding the product of the imaginary parts of the numbers.) The computer algorithm used to form the Fourier coefficients is usually a Fast Fourier Transform (FFT) algorithm (Ref 1). It works most efficiently when the length of the signal is a whole power of two data values, i.e. 2 N, where N is a whole number. Typical numbers of samples would be 64, 128, , 8192, and so on. The program takes a set of (say) 256 data value, forms the Fourier coefficients and their complex conjugates. As the next set of 256 data values will produce a slightly different set of coefficients, the program averages each set of coefficients with those already produced. The set of 256 averaged coefficients will produce a description of the amplitudes in the signal at 128 frequencies covering a range from zero to the Nyquist frequency (i.e. from zero to a maximum frequency equal to half the sampling frequency). For a signal sampled at 100 Hz, the maximum frequency contained in the power spectrum will be 50 Hz, and 128 discrete values would be displayed between zero and 50 Hz. Figure 4 shows the PSD of a computer-generated signal. Most of the area of Volume Vol. 3 Section 4 Issue: 1 Date:

31 Power spectral density the PSD is concentrated within a fairly narrow band of frequencies, so this signal is a narrow band signal. Figure 4.4 PSD of a narrow band random signal A signal which contains many frequencies has a PSD as shown in Figure 4.5, and is a broad band signal. 4.3 Window functions Figure 4.5 PSD of a broad band random signal The discontinuities at the end of each set of 2 n data points have an effect on the ability of the FFT algorithm to resolve frequencies. A PSD of a single sinewave formed as described above is shown in Figure 4.8. The vertical axis is plotted on a log scale to emphasise small amplitudes. It can be seen that the frequency of the sinewave is correctly determined, but the analysis shows a number of adjacent frequencies which are not really present in the signal. These spurious frequencies are side lobes, and the phenomenon is known as leakage. Vol. 3 Section 4 Issue: 1 Date: Volume 3 4-3

32 Power spectral density The magnitude of the side lobes can be reduced by a technique called spectral windowing. In time domain windowing, each section of the signal is multiplied by a window function before the Fourier coefficients are calculated. For example, a random signal passed through a triangular window function would look as shown in Figure 4.6. Figure 4.6 An example of a window function Window functions suppress the side lobes, but at the expense of making the peaks in the spectrum broader. Of the many window functions, two are of particular interest to mechanical engineers. The Hanning window is a cosine function and is shown in Figure 4.7. It is very successful at suppressing spurious side lobes, but produces a rather broad central lobe. It is built into many commercial signal analysers. Hanning window 10% cosine window Figure 4.7 Hanning and 10% cosine window functions Volume Vol. 3 Section 4 Issue: 1 Date:

33 Power spectral density A 10% cosine taper is used in some signal processing software for analysis of random signals. It produces a narrower central lobe and therefore is better able to resolve closely adjacent frequencies, but its side lobe suppression is less than the Hanning window. If an accurate measure of amplitude is required, the PSD must be corrected for the effect of the window function by scaling the vertical axis of the PSD. Typical factors are 1 / for the Hanning window, and 1 /0.875 for the 10% cosine. These will usually be built into analysis software. Figure 4.8 shows the effect of a 10% cosine taper on the PSD of a 10 Hz sine wave. Note that the vertical axis is a log axis, covering a range from 0.1 to When plotted on a linear vertical axis the effect is much less dramatic (Figure 4.9). No window 10% cosine taper Figure 4.8 Effect of windowing on the psd of a 10 Hz sine wave (log axes) No window 10% cosine taper Figure 4.9 Effect of windowing on the psd of a 10 Hz sine wave (linear axes) Figure 4.10 shows the effect of the 10% cosine taper on the PSD of a narrow band random signal. Vol. 3 Section 4 Issue: 1 Date: Volume 3 4-5

34 Power spectral density No window 10% cosine taper Figure 4.10 Effect of windowing on the PSD of a narrow band gaussian signal (log axes) Using computer software to calculate a PSD, the length of the data sections, 2 n, must be selected. A large value of 2 n means that the frequencies are closely spaced, so the peaks in the PSD are narrow. A smaller value of 2 n produces lower broader peaks with less ability to resolve closely adjacent frequencies. Figures 4.11 and 4.12 show the effect of the length of data section on the PSD of a constant amplitude sine wave, plotted as linear and log axes. 2 n = n = n = 128 Figure 4.11 effect of window length on the psd of a 10 Hz sine wave (linear axes) Volume Vol. 3 Section 4 Issue: 1 Date:

35 Power spectral density 2 n = n = n = 128 Figure 4.12 Effect of window length on the PSD of a 10 Hz sine wave (log axes) The length of the data sections determines the number of such sections used to obtain the average values of the ordinates of the PSD. This may be important for short lengths of signal. If the mean value of the estimate of any ordinate in the PSD is µ, and the standard deviation of this estimate is σ, then the ratio of σ / µ is related to the number of data sections, L. σ µ = 1L The variation in the estimates of any spectral ordinate form a χ 2 k distribution the ratio distribution, and for this σ µ = 2 K = 1 L so the number of degrees of freedom, k = 2L The confidence limits for the estimates of the ordinates in the PSD can be calculated from standard tables. For example, if a signal contains data points, and the sections are of length 2 n = 2048, then approximately 20 sections of data will be analysed. The number of degrees of freedom, k = 2L = 40, and from tables of the χ 2 k distribution, there is 90% confidence that the true value will lie between 0.7 and 1.5 times the value obtained in the analysis. Irrespective of the length of the data section of 2 n data points, the area under the PSD should be the same, as this represents the variance of the signal. Figure 4.13 shows the integral of the three PSD's illustrated in Figure It can be seen that the final value obtained for the integral is almost identical in each case. Vol. 3 Section 4 Issue: 1 Date: Volume 3 4-7

36 Power spectral density Figure 4.13 Integral of the psd for different window sizes Two final comments should be made before ending this section on the computer calculation of PSD's. It is usual for each section of the signal to overlap the preceding section. The 10% cosine taper would use a 10% overlap, the Hanning taper would use a 50% overlap, and so on. A signal with a non-zero mean value would produce a very large peak in the PSD at zero Hz. This may dominate the display of the PSD, and may hinder mathematical interpretation. The zero Hz component may be eliminated by subtracting the mean of the signal from each data point before analysis. This process is sometimes called normalisation. To summarise Sections : The maximum frequency that can be displayed in the PSD is equal to the Nyquist frequency of the signal, i.e. half the sampling rate used for the signal. The resolution of the PSD is determined by the number of data values used for each spectral estimate. For example, sections of 256 data values will produce 128 Fourier coefficients, and therefore 128 points on the PSD display. The fineness of peaks and the resolution of detail is determined by the length of these data sections. However, the statistical validity of the results is determined by the number of data sections used to provide the average spectral estimate. For short signals, therefore, there is a trade-off between spectral resolution and statistical accuracy, and for all signals, the computer analysis of any signal will be quicker if short data sections are used. A fine resolution will produce higher, sharper peaks in the display, but maintain more or less the same area under the peak. The units of the vertical axis are (signal units) 2 /Hz. This emphasises that the display is a density diagram, and the relevant information is obtained from the area under the diagram, not from the vertical-axis ordinates themselves. The original method of estimating PSD's using band pass filters emphasises this point. So also do displays at differing spectral resolutions. 4.4 Calculation of signal properties from the PSD A number of signal properties can be calculated by taking moments of area of the PSD about the zero Hz axis. Volume Vol. 3 Section 4 Issue: 1 Date:

37 Power spectral density m n = Ax n Figure 4.14 Definition of moments of the PSD Integrating the PSD provides the variance of the signal, and taking its square root gives the rms of the signal: 1 rms σ = (m ) 2 o The number of positive slope zero crossings per unit time is λ ο = m 2 m o 1 2 where m n denotes the n th moment. The number of peaks per unit time is µ = m 4 m The irregularity factor of a signal is defined as the number of positive slope zero crossings divided by the number of peaks. It is calculated from the PSD using the above equations. Section 3 briefly touched on the fact that for a Fourier analysis to be valid, the signal must obey certain statistical rules. A PSD program will produce a PSD whether the signal obeys these rules or not. It is then the responsibility of the user to ensure that the analysis is valid. For the PSD to be valid, the signal should be: RANDOM: the probability of the occurrence of a given value can be estimated, but the actual value cannot be predicted. STATIONARY: the statistical properties of the signal (the mean, standard deviation, etc) do not vary with absolute time, in other words different sections of the signal should produce the same estimates of mean, standard deviation, etc Vol. 3 Section 4 Issue: 1 Date: Volume 3 4-9

38 Power spectral density ERGODIC: the statistical estimates made within a section of a signal are the same as those made from the section as a whole, and, from the condition of stationarity, are the same as those made from any other section of the signal. Real measured signals start and end, so they are never strictly stationary or ergodic, so the signal should obey the rules within reasonable limits. 4.5 References Of the many books on signal processing, the following is very much oriented to mechanical engineers and practical applications. D Brook and RJ Wynne SIGNAL PROCESSING - PRINCIPLES AND APPLICATIONS Edward Arnold paperback 1988 Also recommended, as in Section 3, is Newland, D E AN INTRODUCTION TO RANDOM VIBRATION AND SPECTRAL ANALYSIS Longman (2nd ed) Volume Vol. 3 Section 4 Issue: 1 Date: