a single degree of freedom oscillator using a covariance-invariant digital filter. It is shown that this technique will produce an accurate response spectrum from.
BulleUnofthe SeismologicalSocmtyofAmerma,Vol 71,No 4, pp 1361-1367,August1981
COVARIANCE-INVARIANT DIGITAL F I L T E R I N G - - A B E T T E R DIGITAL PROCESSING T E C H N I Q U E FOR G R O U N D MOTION S T U D I E S BY J. E. EHRENBERG AND E. N. HERNANDEZ ABSTRACT This paper describes a new recursive digital algorithm for computing the response spectrum from acceleration data. The algorithm digitally implements a single degree of freedom oscillator using a covariance-invariant digital filter. It is shown that this technique will produce an accurate response spectrum from input data that have been sampled at a rate that is as low as three times the oscillator natural frequency, fn. The more common response spectrum digital filter obtained using a bilinear z transformation technique requires a sampling rate of ten times fn. The fact that the algorithm can be used for data with a low sampling rate makes it particularly well suited for real time, multi-frequency calculation of response spectra with a microprocessor system.
INTRODUCTION The response spectrum has proven to be a useful technique for measuring the effects of ground motion on structures. The response spectrum was originally obtained using either electronic or mechanical analog single degree of freedom oscillators. These analog systems were expensive and had questionable accuracy. In order to get around some of the problems inherent in analog response spectrum measurement systems, digital techniques are often used to determine the response spectra. There are two basic techniques that can be used to digitally implement the single degree of freedom oscillator. One approach is to directly solve the differential equation for the oscillator using various numerical integration techniques (Beaudot and Wolfson, 1970). The second approach is to design the digital processing algorithm to provide a good approximation to the frequency response of the analog single degree of freedom oscillator. The frequency domain design of a digital system is usually accomplished by first writing the Laplace domain representation of the system. Various techniques can then be used to transform the Laplace or s-plane system transfer function into a discrete or z-plane representation for the system. One popular method for going from the Laplace to the z domain is to use the bilinear z transformation. This approach was used by Stagner and Hart (1970) to digitally implement the single degree of freedom oscillator used in the calculation of the response spectra. Stagner and Hart compared the bflinear z digital system with the analog oscillator and found good agreement when the digital sampling frequency was 20 times the natural frequency of the oscillator. This sampling rate is a factor of ten greater than the minimum sampling rate specified by the Nyquist criterion. In many seismic applications one is interested in simultaneously calculating the response spectra at a number of different oscillator center frequencies for more than one motion sensor. In these cases it is extremely difficult, if not impossible, to do all the digital computation in real time when the signals are being sampled at a high rate compared to the Nyquist frequency. The problem with the bilinear z realization of the single degree of freedom oscillator is that the agreement between the analog and digital responses is poor when the sampling rate is low. This paper describes the use of a technique called covariance-invariant digital 1361
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J. E. EHRENBERG AND E. N. HERNANDEZ
filtering for digitally implementing the single degree of freedom oscillator needed to calculate the response spectra. It is shown that the digital filter obtained using this method requires a much lower sampling rate than filters obtained using the bilinear z transformation. DIGITAL FILTER D E S I G N
There are a variety of techniques that can be used to synthesize digital filters that approximate a continuous time system. Each synthesis procedure will result in a different digital filter with its own particular set of characteristics. None of the digital filters have all the same properties as the continuous time system, and some trade-offs must be made in choosing the digital filter design procedure. One of the most common digital filter synthesis techniques uses the bilinear z technique that maps the Laplace transfer function for the system into a discrete time (z domain) transfer function such that the positive i~o axis in the s plane becomes the upper half of the unit circle in the z plane. In other words, the bilinear z transformation compresses the continuous system frequency range of (0, ~) into a discrete time frequency of (0, f~/2) where f~ is the sampling rate for the discrete system. Other digital filter synthesis techniques do not have this characteristic, and consequently part of the continuous system frequency response above fs/2 is folded back into the frequency interval from 0 to f~/2. Unfortunately, the price that is paid for the frequency compression property of the bilinear z technique is a rather severe distortion of the discrete system frequency response. Another synthesis procedure is to design the digital filter such that the discrete time and continuous systems have the same response to a particular input, such as a unit step. The synthesis procedures that match the continuous and discrete system responses for a step input and an impulsive input are called the step invariant and impulse invariant techniques. These filters often provide a closer match between the continuous and discrete frequency response than the bilinear z filter. The covariance-invariant digital filtering technique is similar to the impulse and step invariant design techniques in that it matches the responses of the continuous and digital system. However, the covariance-invariant filter matches the response for a random input rather than for a deterministic input. In particular, the technique produces a digital filter that, when excited by discrete time white noise, has an output time series with a covariance function that matches the covariance of the output of the continuous time system when it is excited by white noise. The result is a very close match between the continuous and digital system magnitude-squared frequency responses. The theory and synthesis procedure for covariance-invariant digital filters has been discussed in a paper by Perl and Scharf (1977). The differential equation for the relative displacement, y(t), for a single degree of freedom oscillator subject to ground motion is y ( t ) + 2ficonfc(t) + ~on2y(t) = - - a ( t )
(1)
where ~n is the undamped natural frequency of the oscillator, fi is the critical damping ratio, and a(t) is the ground acceleration. The Laplace domain transfer function for the oscillator is Hc(s)
-
Y(s) - A(s)
-
-1 s 2 + 2fi~ns + ~n 2
(2)
COVARIANCE-INVARIANT
DIGITAL
1363
FILTERING
where A (s) and Y ( s ) are the Laplace transforms of a(t) and y(t). The transfer function, Hc(s), can be expanded in partial fractions to yield Oil
Hc(s) - - -
Oi2
~
8 "~ $1
(3)
S + 82
where Oil = i/2OZn 41 -- ~2 Oi2 ~
--Oil
S l = fi(~)n - - g~-On 82 = S l *
,/1
-
= fl~On "~- i ~ .
f12
4 1 - f12.
It is shown in the paper by Perl and Scharf that the second order covarianceinvariant digital filter has discrete time transfer function, H ( z ) , that satisfies H(z)H(z-l)
N(z) = 2 I-[ (z - e-~JT)(z -1 -- e -~T)
(4)
.l~l
where N ( z ) is a second degree polynomial given by 2
2
N ( z ) = ~ alH~(sj)(1 - e -2~T) [[ (z J=l
-
e-skT)(z-1
--
e-SkT).
(5)
k=l k~.,'
The numerator polynomial for the oscillator transfer function in equation (4) can be factored after a considerable amount of algebra to yield N(z) = -kl(z
- rD(z -x - r l ) / r l
(6)
where k~ = (e-AT[--2B cos(BT) + 2A sin(BT)] + e-~AT[2B cos(BT) + 2A s i n ( B T ) ] } / 8 A B ( A A = fl,,~ B = ~On~/1 __fi2
and where -ko 1 r~ = ~ + -~ ~/iko/kl) 2 - 4 OF
-ko 1 ]-1 rl -~ [ - ~ 1 + -2 ~((ko/kl)2 - 4
2 + B 2)
1364
J.
:E. E H R E N B E R G
AND
E.
N.
HERNANDEZ
where ko = { - 4 A e -2AT s i n ( 2 B T ) + 2B(1 - e - c a r ) } / 8 A B ( A
2 + BZ).
E x c e p t for a scale factor, b o t h values of rl p r o v i d e t h e s a m e m a g n i t u d e r e s p o n s e as a f u n c t i o n of f r e q u e n c y . T h e filter will h a v e m i n i m u m phase, if t h e value of rl w i t h m a g n i t u d e g r e a t e r t h a n one is used. T h e z d o m a i n t r a n s f e r f u n c t i o n is ~ / ~ - ( k j r l ) ( z - rl) H ( z ) = z2 _ [ e _ A T 2 c o s ( B T ) ] z + e_2A T. I
1.0-
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/1\
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5
FIG. 1. Magnitude of the frequency response for the continuous time system, covarianee-invanant, and bihnear z digital filters for f#f, = 6 and B = 0.05. T h e c o r r e s p o n d i n g difference e q u a t i o n for t h e digital filter is y(n) = bly(n-
1) + b 2 y ( n - 2 ) + c [ x ( n -
where a
=
--rl
bl = 2e -AT c o s ( B T ) b2
=
--e -2AT
1) + a x ( n - 2)]
(8)
1365
COVARIANCE-INVARIANT DIGITAL FILTERING
and c is a gain constant t h a t can be set to give the desired o u t p u t level. Figures 1 and 2 show the magnitude of the frequency response for the covarianceinvariant and the bilinear z realization of the single degree of freedom oscillator. T h e ratio of sampling frequency, fs, to oscillator natural frequency was 6 for Figure 1 and 3 for Figure 2. B o t h Filters were normalized to have unity magnitude at the natural frequency, f~. T h e bilinear z design was optimized by the usual prewarping procedure (Stanley, 1975). T h e results clearly show the poor agreement between the continuous Filter response and the bilinear z response. This is especially true for the
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FREQUENCY
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NORMALIZED FREQUENCY,f/fn FIG. 2. Magmtude of the frequency response for the continuous time system, covanance-mvaraant, and bflinear z digatal filters for fJfn = 3 and fl = 0.05. lower sampling rate f~ = 3f~. On the other hand, the frequency response of the covariance-invariant filter closely matches the continuous filter response at all frequencies. T h e r e is a slight difference in the response near the folding frequency, fs/2, due to folding back of the continuous filter frequency response above fs/2. T h e phase response of the m i n i m u m phase covariant-invariant and bilinear z realizations of the filter is shown in Figure 3 for f~ = 5fn. T h e phase response of the bilinear z design more closely matches the continuous system phase response. This can be expected since the covariant-invariant design procedure is only based on the m a g n i t u d e response. T h e response spectra of a ground motion signal primarily depends on the distribution of energy as a function of frequency. T h e magnitude
1366
J. E. E I - I R E N B E R G
AND
E. N . I - I E R N A N D E Z
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ANALOG TRANSFER FUNCTION -----BILINEAR Z DESIGN
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NORMALIZED FREQUENCY, f/fn FIG. 3 P h a s e r e s p o n s e for t h e c o n t i n u o u s t u n e s y s t e m , c o v a r i a n c e - i n v a n a n t , a n d b i h n e a r z d i g i t a l f i l t e r s for fs/fn = 5 a n d fi = 0.05.
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fs/fn Fro. 4. P e r c e n t of e r r o r m t h e m a g m t u d e of t h e r e s p o n s e s p e c t r u m for a s t e p i n p u t for t h e c o v a n a n c e m v a r i a n t a n d b i l i n e a r z f i l t e r for fl = 0.02.
COVARIANCE-INVARIANT DIGITAL FILTERING
1367
response of the response spectra filter is therefore more important than its phase response. The superiority of the covariance-invariant design is further demonstrated by comparing the outputs of the two digital realizations of the response spectra filter for a known input. The errors in magnitude of the response spectra as a function of the ratio of sampling frequency to natural frequency for a step input for the covariance-invariant filter and bilinear z filter are shown in Figure 4. Both filters were normalized to have unity response at the natural frequency and had a critical damping ratio, fl, equal to 0.02. The errors in the response spectra for a step input for other values of fl have been investigated, and results similar to those in Figure 4 were obtained. CONCLUSION
In this paper we have discussed the use of the covariance-invariant digital filtering technique for the calculation of response spectra. It was shown that in terms of the agreement between the magnitude of the continuous and digital filter frequency response and in terms of the magnitude of the response spectra error, the covarianceinvariant filter is significantly better than the results obtained with the commonly used bilinear z digital filtering method. The significance of this is that, for a certain specified maximum response spectra error, the covariance-invariant digital filter will require a much smaller sampling rate than that required by the bilinear z filter. By taking advantage of this reduction in computation rate, it is now possible to replace large, costly computers with small, inexpensive microprocessors or minicomputers for real time calculation of the response spectra. REFERENCES Beaudet, P. R. and S. J. Wolfson (1970}. Digital filter for response spectra, Bull, Sezsm Soc Am. 60, 1001-1012 Perl, G. and L. L. Scharf (1977) Covariance-invariant chgltal filtering, IEEE Trans Acoust. Speech S~gnal Processing ASSP-25, 143-151. Stagner, J. R. and G. C. Hart (1970). Application of the bLhnear z transform method to ground motion studms, Bull. Se~sm. Soc. Am 60, 809-817. Stanley, W D. {1975). D~g~talSignal Processing, Reston Publishing Co., Inc., Reston, Virginia DEPARTMENT OF ELECTRICAL ENGINEERING AND APPLIED PHYSICS LABORATORY UNIVERSITY OF WASHINGTON SEATTLE,WASHINGTON 98195 (J E.E.)
PRESIDENT TERRA TECHNOLOGY CORPORATION 3860 148TH AVENUE N E. REDMOND, WASHINGTON 98052 (E.N H.)
