grammed into an online computer. Typical responses are shown for pulse, step and low-frequency mixed sinusoidal signals. Introduction: The problem of ...
ONLINE DIGITAL FILTERING The advantages of the recursive digital filter as a real-time signal processor are stated, and, as an example, a fourth-order Chebyshev lowpass filter has been synthetised and programmed into an online computer. Typical responses are shown for pulse, step and low-frequency mixed sinusoidal signals.
Introduction: The problem of selectively treating components of sampled-data series in different frequency bands has been approached in two ways. The first is to perform a Fourier transform of the data, apply the appropriate weighting to the components and then to recover the signal by the inverse transformation. This method, however, has the disadvantage that all the data have to be collected and stored before treatment. The second method is to synthetise a digital filter in the form of a simple recursive relationship, and this has the advantages that little storage is necessary and a signal may be treated in real time. The purpose of this letter is to show how a recursive digital filter may be used in the signals laboratory to treat continuous signals. Recent attempts to perform direct digital filtering appear to have been restricted to the development of special hardware,1 but, as many laboratories now have online computing facilities, it is worthwhile to point out the advantages of using the general-purpose digital computer as an online filter. These are: (a) It is a simultaneous input/output process, the computer and its analogue-digital/digital-analogue convertors becoming effectively a 2-port network (b) Drift of characteristics is absent, apart from errors associated with the conversion process (c) Given that certain quantisation criteria are fulfilled,2 the transfer function is absolutely stable and repeatable (d) A change of filter characteristics may be exercised simply by inserting a few coefficients on a data tape (e) For given coefficients, the cutoff frequency is precisely defined as a fraction of the sampling frequency, and is thus, to some extent, externally disposable (/) Several filters may be incorporated in a single program to give effective parallel operation of a number of channels.
This method is particularly powerful at very low frequencies, where continuous filters are virtually impracticable. We will illustrate this with an example of a lowpass filter designed to eliminate frequencies exceeding 0 • 1 Hz. Lowpassfilter:The filter employed belongs to the Chebyshev class, and it was synthetised by the method previously described.3'4 The specifications are as follows: ratio of cutoff frequency to sampling frequency = 0 005 maximum passband ripple =10% transition ratio =0-7 Thus, if the sampling frequency is set at 20Hz, the required cutoff frequency of 0 • 1 Hz is achieved. The pulse transfer function G(z~l) of this lowpass digital filter is given by
where Q,(z-') = 1 00477145 - 1-99950878Z-1 + 0-99571977z-2 Q 2 ( z -i) = 1-01099745 - 1-99985779Z-1 + 098914476z-2 and K is a positive real normalising constant. The realisation of this type of filter is achieved by means of cascaded biquadratics, each of which may be represented by the simple recurrence relationship n_2
2un_i
obtained by inverting the corresponding ztransform form, where vn and un are, respectively, the nth elements of the output and input time series {vn} and {«„} of the biquadratic. Implementation of the filter: A special interface has been developed so that sampling occurs at a rate controlled by an oscillator in the laboratory, and the data are transferred to a remote general-purpose computer from the laboratory a.d. convertor in the form of 8 bit words, giving a maximum quantisation error of 0-4% of full dynamic range. The
time. , s. Fig. 1 Responses of filter a Response to step input applied at the time marked by the arrow b Response to a pulse at the time marked by the arrow
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program for the simple filter described above involves storing only six coefficients. The loop for one sample consists of approximately 50 orders, of which only seven are the relatively slow multiplication orders. In order to demonstrate
problems. For example, an important diagnostic technique in electroencephalography involves the low-frequency wave analysis of e.e.g. signals. This is currently achieved by operating a large number of active filters in parallel, a technique
Fig. 2 Outputs of computer
with inherent difficulties of stability and calibration. A better result could be achieved by writing several recursive bandpass filters into a single online program.
a Programmed as a straight-through device, the input being a sum of sine waves of frequencies 0-05 Hz and 0-2Hz b Output with filter program entered. The sampling frequency was set at 20 Hz, giving a cutoff frequency of 0-1 Hz
the operation of the filter, the output d.a. convertor was connected to an x-y plotter. The transient responses to step and pulse signals are shown in Figs. \a and b. It will be seen that these are exactly analogous to the responses of a continuous filter except for their stepwise progression imposed by the zero-order hold d.a. convertor. The position of the pulse and step are indicated by arrows, and the small characteristic delay can be seen. In order to demonstrate the effectiveness of this relatively simple filter, we show in Fig. 26 the response of the filter to a signal which is the sum of two sinusoids, one at twice the cutoff frequency and one at half the cutoff frequency. This may be compared with Fig. la, for which the same signal was passed through the system with the filter replaced by the allpass scheme, vn = un. Discussion: Methods of constructing recursive digital filters of the lowpass, highpass, bandpass and bandstop type by direct synthesis and by frequency transformations have been previously reported.3'4 The above example illustrates the power andflexibilityof suchfiltersfor the treatment of continuous bandlimited signals in real time. We are investigating the application of these methods to a number of outstanding ELECTRONICS LETTERS
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We are grateful to J. Cooper, D. Griffiths and G. Rhodes for their valuable assistance in developing the hardware which made this work possible, and also to V. Price for permission to use the computer online. A. BUTTLE A. G. CONSTANTINIDES J. E. BRIGNELL
1st May 1968
Department of Electrical & Electronic Engineering The City University St. John Street, London EC1, England
References 1 JACKSON, L. B., KAISER, J. F., and MCDONALD, H. s.: 'Implementation of
digital filters', IEEE International Convention Digest, section 7F, March 1968 2 SANDBERG, i. w.: 'Floating-point-roundoff accumulation in digitalfilter realizations', Bell Syst. Tech. J., 1967, 46,_pp. 1775-1791 3 CONSTANTINIDES, A. G.: 'Synthesis of Chebyshev digital filters', Electronics Letters, 1967, 3, pp. 124-126 4 CONSTANTINIDES, A. c : 'Frequency transformations for digital filters', ibid., 1968, 4, pp. 115-116 253
