SCIENCE Spectral analysis of short-time biomedical data using adaptive filters J. Braham Levy, B.Sc, M.Eng., and D.A. Linkens, B.Sc.(Eng.), M.Sc, Ph.D., C.Eng., M.I.E.E. Indexing terms: Biomedical engineering, Mathematical techniques, Spectral analysis, Adaptive filters Abstract: Conventional methods of spectral analysis are unable to track small but rapid variations in fre- quency. The LMS algorithm used as an adaptive line enhancer is found to track these signals, and provides insight into mechanical and physiological effects of the human body, when analysing electrical signals obtained from internally and externally placed electrodes. 1 Introduction Spectral analysis of biomedical signals is of considerable interest in the investigation and understanding of the workings of human physiology. The detection and resolution of frequencies, which may be time varying due to mechanical actions correlated to the electrical signals of the body, is a major motivation for this analysis. The analysis can be done by many methods, the stan- dard ones being that of Fourier transforms and associated techniques. These methods have a major drawback in that they require large quantities of data to produce significant results. As a result, any variation or perturbation in fre- quency can be smoothed out by the action of the algo- rithm; for example, the fast Fourier transform (FFT) will give a 'broad' spectral peak centred at the frequency of interest, particularly if using zero padding. As data from human patients cannot usually be obtained for long time series, analysis methods using shorter time series are sought. This has the double effect of being able to track any variation in frequency which occurs, and can be per- formed over a shorter time than that required for an FFT. Following earlier work in adaptive array systems [1] and geophysics [2], a whole set of algorithms has been used [3] to resolve these problems, the most attractive being that of autoregressive (AR) spectral analysis. These methods have not only been applied to biomedical signals, as mentioned above, but also a wealth of different algo- rithms have been developed producing significant results, full and detailed descriptions of which can be found in several recent publications, for example Reference 3. In this study an early developed algorithm, the least- mean squares (LMS) gradient search method [4], has been used to provide real-time analysis of biomedical data from both the human gastro-intestinal tract and processed ECG signals providing so-called heart rate variability (HRV) time series. 2 LMS algorithm Developed in the early 1960s by Widrow et al. [4], the advantage of this algorithm lies in its computational sim- plicity and ease of guaranteed convergence. Consider the prediction of a data point from solely past data: Paper 3O2OA(S9/E10), first received 21st June and in revised form 7th December 1983 Dr. Linkens is, and Mr. Levy was formerly, with the Department of Control Engi- neering, University of Sheffield, Mappin Street, Sheffield SI 3JD, England. Mr. Levy is now with Lucas Research Centre, Shirley, W. Midlands, England 164 x(k) = X a t (k)x(k - 1=1 (1) where x(m) is the data series, {a m {k)} is a coefficient weigh- ting vector m = 1, 2,..., P applied to {x(m)} and e(k) = x(k) - x(k) where e(k) is the prediction error. Consider the minimisation of e 2 (k) min [e 2 (k)] = min [{x(/c) - jc(/c)} 2 ] (2) (3) such that by performing this minimisation we will obtain the set of coefficients, a? I = 1, 2, ..., P, which are optimal in the mean-square sense. Thus, taking expected values: E[e 2 (k)li = E{lx(k) - x(/c)] 2 } and rewriting in vector notation, eqn. 1 becomes x(k) = X k A k = A k X k where x(/c - 1) x(k - 2) x(k - P) i.e. a vector of past data and A\ = [a y {k\ a 2 (k), ..., a p (k)'] the coefficient vector. Thus eqn. 4 now becomes By denning (4) (5) (6) (V) x(k)x(k - 1) \ (8) (9) (10) x(k)x(k-P)l which can be seen to be an autocorrelation vector, and jx(k - l)x(k - 1) x(k - 2)x(k - 2) \x(k-P)x(k-P) which is an autocorrelation matrix we can rewrite eqn. 8 as: E[e 2 (k)-] = £[x 2 (/c)] - 2P T k A k + AjR k A k (11) IEE PROCEEDINGS, Vol. 131, Pt. A, No. 3, MAY 1984
Transcript
SCIENCE
Spectral analysis of short-time biomedicaldata using adaptive filters
J. Braham Levy, B.Sc, M.Eng., and D.A. Linkens, B.Sc.(Eng.), M.Sc, Ph.D.,C.Eng., M.I.E.E.
Abstract: Conventional methods of spectral analysis are unable to track small but rapid variations in fre-quency. The LMS algorithm used as an adaptive line enhancer is found to track these signals, and providesinsight into mechanical and physiological effects of the human body, when analysing electrical signals obtainedfrom internally and externally placed electrodes.
1 Introduction
Spectral analysis of biomedical signals is of considerableinterest in the investigation and understanding of theworkings of human physiology. The detection andresolution of frequencies, which may be time varying dueto mechanical actions correlated to the electrical signals ofthe body, is a major motivation for this analysis.
The analysis can be done by many methods, the stan-dard ones being that of Fourier transforms and associatedtechniques. These methods have a major drawback in thatthey require large quantities of data to produce significantresults. As a result, any variation or perturbation in fre-quency can be smoothed out by the action of the algo-rithm; for example, the fast Fourier transform (FFT) willgive a 'broad' spectral peak centred at the frequency ofinterest, particularly if using zero padding. As data fromhuman patients cannot usually be obtained for long timeseries, analysis methods using shorter time series aresought. This has the double effect of being able to trackany variation in frequency which occurs, and can be per-formed over a shorter time than that required for an FFT.
Following earlier work in adaptive array systems [1]and geophysics [2], a whole set of algorithms has beenused [3] to resolve these problems, the most attractivebeing that of autoregressive (AR) spectral analysis. Thesemethods have not only been applied to biomedical signals,as mentioned above, but also a wealth of different algo-rithms have been developed producing significant results,full and detailed descriptions of which can be found inseveral recent publications, for example Reference 3.
In this study an early developed algorithm, the least-mean squares (LMS) gradient search method [4], has beenused to provide real-time analysis of biomedical data fromboth the human gastro-intestinal tract and processed ECGsignals providing so-called heart rate variability (HRV)time series.
2 LMS algorithm
Developed in the early 1960s by Widrow et al. [4], theadvantage of this algorithm lies in its computational sim-plicity and ease of guaranteed convergence.
Consider the prediction of a data point from solely pastdata:
Paper 3O2OA(S9/E10), first received 21st June and in revised form 7th December1983Dr. Linkens is, and Mr. Levy was formerly, with the Department of Control Engi-neering, University of Sheffield, Mappin Street, Sheffield SI 3JD, England. Mr. Levyis now with Lucas Research Centre, Shirley, W. Midlands, England
164
x(k) = X at(k)x(k -1=1
(1)
where x(m) is the data series, {am{k)} is a coefficient weigh-ting vector m = 1, 2 , . . . , P applied to {x(m)} and
e(k) = x(k) - x(k)
where e(k) is the prediction error.Consider the minimisation of e2(k)
min [e2(k)] = min [{x(/c) - jc(/c)}2]
(2)
(3)
such that by performing this minimisation we will obtainthe set of coefficients, a? I = 1, 2, . . . , P, which are optimalin the mean-square sense. Thus, taking expected values:
E[e2(k)li = E{lx(k) - x(/c)]2}
and rewriting in vector notation, eqn. 1 becomes
x(k) = Xk Ak = Ak Xk
wherex(/c - 1)x(k - 2)
x(k - P)
i.e. a vector of past data and
A\ = [ay{k\ a2(k), ..., ap(k)']
the coefficient vector.Thus eqn. 4 now becomes
By denning
(4)
(5)
(6)
(V)
x(k)x(k - 1) \
(8)
(9)
(10)
x(k)x(k-P)l
which can be seen to be an autocorrelation vector, and
jx(k - l)x(k - 1)x(k - 2)x(k - 2)
\x(k-P)x(k-P)
which is an autocorrelation matrix we can rewrite eqn. 8as:
E[e2(k)-] = £[x2(/c)] - 2PTkAk + AjRk Ak (11)
IEE PROCEEDINGS, Vol. 131, Pt. A, No. 3, MAY 1984
which can be seen to be a quadratic function in the coeffi-cient vector Ak. Taking the gradient of eqn. 11 with respecttoAk:
= -2Pk + 2R.A, (12)
and by setting eqn. 11 to zero, we obtain the optimum(Weiner) coefficient vector:
A* — P - 1 Pk — k k
(13)
The LMS algorithm uses the method of steepest descent sothat the coefficient vector Ak is updated by a change pro-portional to the negative gradient — Vfc:
Ak+1=Ak-nVk (14)
An expression for the gradient can be shown to be:
Vf c= -2e(k)Xk (15)
and so the LMS algorithm is given by
(16)
where pi is a step size and determines the convergence ofthe algorithm. It has been shown [1] that convergence isguaranteed by setting
> (17)
where kmax is the largest eigenvalue of the Toeplitz matrixRk. It has further been shown [5] that
> 0
where
= 4 Ir
(18)
(19)
i.e. the signal power.The predictor can now be viewed as an all-pole adapt-
ive filter taking data (x(m)} and producing a white noise
output (e(m)}. The output will be white if the prediction isessentially correct.
The transfer function of this filter is, thus, an estimate ofthe power spectral density (PSD) of the signal:
««**>-TTrfep (20)
A simpler estimate, which omits absolute gain knowledgebut preserves frequency information, is given by [5]:
Q(a>, k) =1
1 "(21a)
(216)
The denominator of eqn. 216 can be evaluated simply by azero padded FFT yielding a correct result (no biasing) asall the information required is contained within the datalength of the FFT; the AR adaptive filter coefficients canbe viewed as forming an impulse response. This is then theadaptive line enhancer (ALE) [1].
3 Methods
It can be seen that the LMS algorithm has considerablecomputational simplicity and is thus suitable for imple-mentation on a real-time microprocessor system. To thisend, a Texas Instrument FS990/4 development system,based around a TMS9900 16-bit microprocessor andincorporating 28K words of memory, analogue to digitaland digital to analogue convertors (Analog Devices RTS-124X series) was employed. As a programming languageTexas Instruments Microprocessor Pascal (TIMPP) waschosen, the reasons for this choice being three fold:
(a) very high speed of operation is not essential with theparticular biomedical signals being used, sample periodsbeing as low as 1 s. Hence, a high-level language can beadvantageously used for the obvious reason that the devel-opment time to produce a working system is less
system start up andglobal data initilisation
ALE processinitialisation
yes
perform ALE
algorithm
yes
LMS processinitialisation
user I/O
initialisation
yes
signal ALE andLMS to stop
Fig. 1 Flow chart for concurrent Pascal adaptive filtering programs
All three processes appear to run simultaneously
IEE PROCEEDINGS, Vol. 131, Pt. A, No. 3, MAY 1984 165
(b) TIMPP has its own real-time operating systemwhich allows 'concurrent' programming. This allows theLMS algorithm to proceed for every sample: the ALEoperation, i.e. FFT on the zero padded coefficients pro-duced by the LMS algorithm and display of such, at adifferent slower rate; and a user interaction process for thecontrol of the overall system, to appear to be all happeningsimultaneously, i.e. concurrently, (see Fig. 1)
(c) the authors preference for the Pascal programminglanguage [6].
Digestive tract data were gathered from internal elec-trodes stitched to the serosal (outside) wall of the stomachand duodenum during human operations, and also fromAg/AgCl electrodes (as used in heart beat monitoring) onthe surface of the abdomen.
The HRV data were obtained from ECG signals inhumans, which were processed to an analogue form to givea 'beat-to-beat' variability signal. The resulting variabilitywas partially synchronised to an external thermal stimulusapplied to the patient at a set frequency [7].
4 Results
The results are shown graphically in Figs. 2-14. Thesegraphs are two-dimensional representations of three-dimensional data such that frequency is traversed alongthe horizontal axis, while time, in seconds, (correspondingto iteration number) is along the vertical axis. The tri-angles represent significant amplitude peaks in the ALE.The graphs were obtained from either a Tektronix 4010terminal or from a Hewlett-Packard 7225A plotter.
Figs. 2 and 3 show results obtained from human duo-denal data. In Fig. 2 a clear frequency component at 0.2Hz is seen, as is one at 0.4 Hz. 0.2 Hz is known to be theelectrical 'slow-wave' frequency of the duodenum, andsuperimposed on this can be seen the perturbations, due tomechanical actions, occurring over 2-3 cycles of data. Thetracking of this signal is particularly good due to its nearsinusoidal waveform and high signal/noise ratio. Thetracked 0.4 Hz component is the second harmonic of thedueodenum 0.2 Hz signal. In Fig. 3 clear tracking at 0.2 Hzis seen, again with very large perturbations in frequency atseveral points lasting for 2-3 cycles of data. No harmonic
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Fig. 2 ALE tracking on serosal duodenal recording for subject HNinth-order filterSample time = 1.0 sa = 0.1651Adaptation time = 54 iterations
is detected here, but a subharmonic component (0.1 Hz) isapparent, as is a signal of 0.3 Hz likely to be due to respir-ation of the patient. The 0.1 Hz component is possibly a
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Fig. 3 ALE tracking on serosal duodenal recording for subject D
Tenth-order filterSample time = 1.0 sa =0.1653Adaptation time = 60 iterations
beat frequency between the duodenum and respiratory fre-quencies.
Results from external electrodes on the abdomen areshown in Figs. 4-7. A sample of the input data to the algo-rithm from a surface recording is shown in Fig. 4 with an
5x10
500
5x108r
1 2 3 . A 5frequency x10 Hz
Fig. 4 Time-series data and FFT for surface recording for subject P
FFT performed on this data length (512 points = 8 min 32s of data.) It can be seen from the FFT that a clear fre-quency component of 0.05 Hz is detected, as are minorpeaks, due to the fluctuations in this frequency, adjacent tothis peak. Only a small second-harmonic component isshown on the FFT due to the scaling of the graph. Fig. 5shows the ALE performed on the same data. Clearly, afrequency of 0.05 Hz is detected known to be the frequencyof the human slow wave, as are harmonics at 0.1, 0.15 and0.2 Hz. Some breakthrough of respiration is seen later inthe graph at 0.3 Hz, as is a very low-frequency componentprobably due to thermal effects, which is also seen on theFFT. Perturbations in the 0.05 Hz signal lasting 2-3 cyclesare clearly seen on the ALE, but are not visible on theFFT.
Figs. 6 and 7 show the ALE performed on the similardata using different lengths of adaptive filter. Again, the0.05 Hz signal is clearly detected along with harmonics, asare the perturbations due to the previously mentioned
166 IEE PROCEEDINGS, Vol. 131, Pt. A, No. 3, MAY 1984
mechanical activity. The large number of extraneous peaksat the beginning of the plot shows the adaptive algorithm
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A
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Fig. 5 ALE tracking for surface recording from subject P25th-order modelSample time = 1.0 sa = 0.1661Adaptation time = 150 iterations
show clearly the basic 0.1 Hz HRV signal modulated bythe stimulus frequency. The failure of Fig. 13 to track
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Fig. 7 ALE tracking data for Fig. 6 with Wth-order filter
19th-order filter.Sample time = 1.0 sa = 0.1659Adaptation time =114 iterations
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Fig. 6 ALE tracking for surface recording from subject H using 25 th-order filter25th-order filterSample time = 1.0 sa = 0.1661Adaptation = 150 iterations
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Fig. 8 ALE tracking for surface recording from subject D using 25th-order filter25th-order filterSample time = 1.0 sa = 0.1661Adaptation = 150 iterations
'tuning' due to a fairly low signal/noise ratio. Figs. 8 and 9show similar results on equivalent data from a differentpatient for two different filter lengths.
Fig. 10 shows how the ALE can go wrong! Fig. 11shows a sample waveform of the recording used for thesegraphs, which was produced from internal electrodesstitched to the serosal surface of the gut. The signal is quiteclearly rich in harmonic content, as can be seen on theassociated FFT. The ALE in attempting to detect thesefrequencies produces spurious frequency componentsgiving a line-splitting effect around 0.1 Hz.
Figs. 12-14 show ALE graphs performed on heart ratevariability data at external thermal stimulus frequencies of0.05 Hz, 0.07 and 0.1 Hz, respectively. Figs. 12 and 14
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clearly is due to the frequency of the stimulus not beingharmonically related to the basic 0.1 Hz rhythm.Occasional breakthrough of respiration at 0.3 Hz is seen inall these graphs. These results qualitatively agree withthose produced by Kitney [7] using running lowpass fil-tered event series (LPFES) spectra.
5 Conclusions
The ALE is particularly good at detecting sinusoids inwhite noise and to some extent coloured noise. When thenoise is harmonically rich, namely in Fig. 11, where thesignal harmonics can be regarded as noise, the ALE breaks
167
down as a tracker of the major, predominant frequency.Other effects of noise can be seen in start-up transients (for
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Fig. 9 ALE tracking data from Fig. 8 with 19th-order filter19th-order filterSample time = 1.0 sa = 0.1659Adaptation time = 114 iterations
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Fig. 10 Incorrect tracking ofserosal gastric recording from subject H225th-order filterSample time = 1.0 sa = 0.1661Adaptation = 150 iterations
example Figs. 6 and 7) during the adaptive tuning period.Both of these can be viewed in the following way: initially,the all-pole filter will attempt to place its poles equallyaround the unit circle in the z-plane. As tuning continuesand the signal is detected, those poles due to the noisemigrate away from the unit circle and thus become lessand less significant. With harmonically rich signals thisdoes not happen as there is essentially no one predominantfrequency.
The choice of model order (P) and the convergence ratedetermined by n (the search step size) affects the resultsobtained. Empirical rules [8] seem to be borne out bypractical examples, although recently an attempt has beenmade to quantify this for the lattice filter ALE [9, 10].These results, although apparently good for pure sinusoids
plus white noise, do not work for the biomedical signalsunder investigation here when applied to the LMS/ALEcombination. The quantification of choice of model order
10x10'
-10 100 200 300 400 500
3x10^
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Fig. 11 Time-series data and FFT for serosal gastric recording fromsubject H2Nasal gastric tube: data and FFT
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Fig. 13 ALE tracking as for Fig. 12, but stimulus frequency of 0.07 Hz20th-order modelSample time = 1.0 sa = 0.3300Adaptation time = 60 iterations
168 IEE PROCEEDINGS, Vol. 131, Pt. A, No. 3, MAY 1984
(P) still has some way to go, and, for the present, empiricalrules seem adequate to produce good results.
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Fig. 14 ALE tracking as for Fig. 13, but stimulus frequency of 0.1 Hz20th-order modelSample time = 1.0 sa = 0.3300Adaptation time = 60 iterations
As for fi, which determines the convergence of the algo-rithm, this can be decided using a modification of theinequality (eqn. 18):
(22)
(23)
™ rx(O)P
0 <a < 2
where n is adapted continuously online dependent on a (auser entered parameter) and rx(0). The signal power can beestimated online by several methods, one such being:
a2(k) = o\k -
a1 = rx(0)
- V)o\k) (24)
The authors, however, prefer to use a sliding windowmethod which estimates o2(k) using the past P number ofsamples by eqn. 19. This can be viewed as a fixed-lengthmoving filter, whereas eqn. 24 is essentially a lowpass filter.Both methods will track variations in signal powerstrength, essential in real-time signal processing. Themethod used is one of personal choice.
The time to adaption is governed by fi and, hence, inour case a, and it has been shown that the adaptive time,xa, is given approximately by [5]:
- 1In (1 -
(25)
where lavg is the average eigenvalue in the matrix R:
This is equivalent to
- 1T_ —
In (1 - a/P)where 0 < a < 2 (26)
This online computation of n given a greatly aids therobustness of the LMS/ALE algorithm by always guar-anteeing the convergence within a set time, as can be seenfrom the results.
The successful demonstration of the LMS/ALE algo-rithm in real time given by the results above leads to apossible clinical application in postoperative care of
IEE PROCEEDINGS, Vol. 131, Pt. A, No. 3, MAY 1984
patients who have undergone abdominal surgery [11]. Aportable system incorporating graphical display would aidgreatly in the detection of migrating myoelectrical com-plexes (MMC) occurring in the stomach, using surfaceand/or serosal electrodes. This could indicate a return tonormal electrical patterns of behaviour after surgical pro-cedures.
The use of the LMS/ALE algorithm on HRV signalsshould also aid in the study of thermoregulatory and otherinfluences on heart rate variability where work may bedone online instead of the present offline work [7]. Usehas also been made of spectral analysis for the study ofReynauld's disease [12], a particularly disabling disease ofthe circulatory system of humans. Online analysis by theLMS/ALE should be able to detect this problem easily andquickly, and also aid in monitoring its progress undertreatment. Further use of the LMS algorithm has beenmade in obtaining parameter estimates for control systemsin, for example, adaptive or optimal control schemes [13].The inherent simplicity and robustness of the algorithmshould produce control algorithms which could easily beadapted on a continuous basis.
6 Acknowledgments
The authors would like to thank the following for theirassistance and support in this study: the Departments ofMedical Physics and Surgery, University of Sheffield, par-ticularly Dr. R.H. Smallwood, Dr. C.J. Stoddard and Pro-fessor A. Johnson for digestive tract data; Dr. R. Kitney,Imperial College, London and Dr. O. Rompleman, DelftUniversity, Netherlands for the HRV data; the Depart-ment of Control Engineering, University of Sheffield for itsfacilities; and the UK SERC for its financial support forone of the authors (J.B. Levy).
7 References
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2 BURG, J.P.: 'Maximum entropy spectral analysis'. Proc. 37thMeeting of Society of Explorations Geophysicists, 1967, pp. 34-41
3 KAY, S., and MARPLE, S.L.: 'Spectrum analysis—a modern per-spective', Proc. IEEE, 1981,69, pp. 1380-1419
5 GRIFFITH, L.J.: 'The rapid measurement of digital instantaneousfrequency', IEEE Trans., 1975, ASSP-23, pp. 207-222
6 JENSEN, K., and WIRTH, N.: 'Pascal user manual and report'(Springer-Verlag, 1975, 2nd edn.)
7 KITNEY, R.I., and ROMPLEMAN, O. (Eds.): 'The study of heart-rate variability (Oxford University Press, 1980)
8 LINKENS, D.A.: 'Empirical rules for the selection of parameters forauto regressive spectral analysis of biomedical rhythms', SignalProcess., 1979, 1, pp. 243-258
9 REDDY, V.U., EGARDT, B., and KAILATH, T.: 'Optimized lattice-form adaptive line enhancer for a sinusoidal signal in broad-bandnoise', IEEE Trans., 1981, ASSP-29, pp. 702-709
11 LEVY, J.B., LINKENS, D.A., RIMMER, S.R., JOHNSON, A., andSTODDART, C.J.: 'Adaptive tracking of human slow-wave frequencychanges using surface and serosal signals'. Proc. 1st European Sym-posium on Gastrointestinal Mobility, Bologna, Italy, September 1982
12 DE TRAFFORD, J.C., LAFFERTY, K., KITNEY, R.I., COTTON,L.T., and ROBERTS, V.C.: 'Modelling of the human vasomotorcontrol system and its application to the investigation of arterialdisease', IEEE Proc. A, 1982,129, pp. 646-650
13 BITMEAD, R.R.: 'Convergence in distribution of LMS-type adaptiveparameter estimates', IEEE Trans., 1983, AC-28, pp. 54-60
