Short-time-series spectral analysis ofbiomedical data

D.A. Linkens, B.Sc.(Eng.), M.Sc, Ph.D., C.Eng., M.I.E.E.

Indexing terms: Biomedical engineering, Signal processingAbstract: In the paper a review is given of modern parametric spectral analysis methods based on difference-equation models, in contrast with classical spectral analysis based on Fourier transform approaches. Thealgorithms discussed are based on an autoregressive structure as this leads to simple, efficient estimators.To provide fast algorithms, recursion in model order is introduced, while a lattice-filter structure can alsoassist in improving the speed of certain algorithms. The problem of spectral leakage is considered, withemphasis being given to algorithms which produce fine frequency resolutions with small bias. Recursion intime, which produces a sequential estimation, is described, and gives rise to a range of algorithms rangingfrom Kalman-filter-type methods to simple LMS algorithms due to Widrow. A wide range of examples isgiven based on simulated data and biomedical cases. These illustrate the problems caused by initial phaseshift in a short time series, but also the considerable improvement in frequency resolution over the FFTapproach. The use of parametric spectral analysis in tracking fine frequency fluctuations is illustrated withgastrointestinal electrical activity data.

1 Introduction

The transformation from time-series recordings to frequency-domain information forms one of the most common forms ofsignal processing in the applied sciences. The advent of fast,cheap computing power together with fast algorithms hasmade spectral analysis very popular. In biomedical engineeringthe use of spectral analysis often gives information aboutunderlying causal processes from a knowledge of the frequencycomponents contained in a particular signal. The spectrummay thus give information which can be used for diagnosticpurposes or for elucidation of physiological dynamics.

A frequent problem associated with biomedical spectralanalysis is that spectral analysis usually assumes and requireslong stretches of time-series data having the statistical propertyof stationarity. Very few biomedical applications fulfil thecondition of stationarity, and in addition it is rare that verylong recordings can be obtained. Given conditions of non-stationarity the normal procedure is to analyse the data as asequence of short segments, and to investigate the time-varyingproperties of the spectrum, e.g. tracking the changes in fre-quency of one or more major components in the spectrum. Inboth this case and the situation where long records are notavailable, the requirement for short-time-series spectral analy-sis arises.

2 Parametric spectral analysis

Estimation of the power spectral density (PSD) of sampleddata containing deterministic and/or stochastic components isconventionally performed by the fast Fourier transform (FFT).Spectral analysis based on the Fourier transform can be tracedback to Schuster [1] who introduced the term 'periodogram',and attempted to fit a Fourier series to sun-spot data fordetermining hidden periodicities. Wiener [2] provided a theo-retical framework for the analysis of stochastic processesusing a Fourier transform approach. The autocorrelationapproach introduced by Wiener was given practical PSD usageby Blackman and Tukey [3]. This Blackman-Tukey approachwas the most popular technique until the introduction of theFFT algorithm, generally credited to Cooley and Tukey [4].FFT spectral estimation has dominated the scene for manyyears. There are however, a number of problems related tospectral analysis based on the periodogram approach.

The major limitation of FFT spectral analysis is that offrequency resolution, i.e. the ability to distinguish the spectral

Paper 2275A, received 13th October 1982Dr. linkens is with the Department of Control Engineering, Universityof Sheffield, Mappin Street, Sheffield Si 3JD, England

components of two or more signals. The frequency resolution,as defined by the separation between spectral components, isapproximately the inverse of the available data time length.This becomes a big problem when attempting to analyse shorttime series. The requirement to analyse short time recordingsarises either because of availability of data (e.g. received radarpulses), or because the data have time-varying spectra (e.g.many biomedical signals). A second problem with FFT analysisis the 'windowing' of the data that is inherent in the method.Windowing causes 'leakage' in the PSD estimates, which meansthat energy in the main lobe of a spectrum leaks into the side-lobes, obscuring and distorting other spectral components.Although careful selection of data-window tapering functionscan reduce sidelobe leakage, it always results in worse fre-quency resolution.

The above limitations of FFT spectral analysis have led tothe investigation of a number of alternative PSD estimatorsduring the past decade, particularly for the analysis of data withtime-varying spectral components. Some of these methods arerelated to early work by Prony [5] who attempted to fitsummed exponentials to time-series data based on the expan-sion of various gases. The Prony method has been adapted forthe case of real undamped sinusoids in noise, and is describedby Hildebrand [6]. The method does not require estimationof the autocorrelation lags, but necessitates the solution oftwo sets of simultaneous linear equations and polynomial rootsolving.

A method which gives the ultimate resolution betweensinusoidal signals buried in additive noise is that of 'HarmonicDecomposition' due to Pisarenko [7, 8] . It does however,require autocorrelation lag estimates, and also involves acomputationally complex eigenequation solution. Also, in realdata analysis it tends to give spurious spectral lines togetherwith frequency and power estimates which are biased. Afurther method is referred to as maximum likelihood (ML)spectral estimation, and was originated by Capon [9] forseismic array frequency-wave number analysis. In this approachthe PSD estimate is obtained by measuring the power out ofa set of narrowband filters [10]. The shape of these filters isgenerally different for each frequency, whereas in the periodo-gram approach the shape is fixed. The maximum likelihoodfilters effectively adapt to the process for which the PSD isbeing estimated. The ML method gives better resolution thanFFT analysis, but is inferior to the autoregressive methodswhich are emphasised in this Section. It is worth noting thatthe peaks in the ML spectrum are proportional to power,whereas the autoregressive approach gives power as the areaunder the spectrum.

The method which has found most widespread use as an

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alternative to FFT is that of autoregressive (AR) spectralanalysis. This will be illustrated with a number of biomedicalexamples. It should be noted that apart from improved reso-lution and reduced leakage, which make the AR method suit-able for frequency tracking of time-varying signals, it alsooffers large data-reduction facilities, because of its parametricbasis. Thus, potentially, a large number of data points can berepresented with a small number of filter weights.

This has been of considerable importance in speech synthesis,and could also be valuable in storage of large biomedical databases. It also offers the possibility of an objective assessmentof spectral content in a signal, unlike the visual pattern recog-nition approach of the FFT. This is possible because of theinherent smoothing involved in AR spectral analysis. Thiscontrasts with FFT analysis, which gives large spectral varianceas the Fourier transform of a zero-mean random process isitself a zero-mean random process. It should be noted that thelarge improvements in spectral resolution claimed for ARanalysis in the literature are only obtained under large signal-to-noise ratios (SNRs). For low SNR modern spectral methodsdo not give improved frequency resolution over classicalmethods.

Like other methods, the AR technique has quite a longhistory. Yule [11] and Walker [12] used AR models for fore-casting trends in economic time series and determining period-icities in sunspot data. Burg [13] introduced the maximumentropy (ME) method for the analysis of geophysical data, andParzen [14] formally proposed AR spectral estimation. Theequivalence between the ME and AR PSD estimators for one-dimensional stationary Gaussian processes was shown by vanden Bos [15]. Since that time AR spectral analysis has beenapplied in radar, sonar, imaging, radio astronomy,biomedicine,oceanography, ecological systems etc. At this point, mentionshould be made of a large review paper on modern spectralanalysis by Kay and Marple [16] and selected reprint papersavailable in Childers [17], which is one of the IEEE reprintseries. Both these publications contain extensive bibliographies.

The classical discrete Fourier transform (DFT) is given by

J V - l

Ym = exp (-)2nmn/N)

m = O , . . . . , N - 1 0)The quantity usually computed as the periodogram from theDFT is

(NT)2 m

\ N-l

= 1^ Z ^nexp(-/27rm«/AT)|2 (2)

It should be noted that eqn. 2 is not, however, scaled appro-priately as a PSD, as the peak is equal to power in the assumedperiodic signal, instead of area under the peak.

It is shown by Kay and Marple [16] that the periodogramPSD estimate is itself a least-squares fit of the data to a har-monic model comprising a Fourier series. In this case thefrequencies are preselected, whereas in Prony's method [5] notonly are powers estimated but also the number and frequenciesof the sinusoids are also identified. It should be noted that inthe periodogram approach noise is modelled by harmonicsinusoids, which clearly leads to a nonparsimonious modeldescription. The information contained in the data could, intheory, be accurately descibed by a model with a significantlylower number of coefficients, i.e. a parsimonious model. Amajor difference between parametric spectral analysis and

general systems identification is that only the output processfrom the model is available in spectral analysis.

Parametric spectral analysis is usually based on a so-calledARM A model, which is founded on the fact that many deter-ministic and stochastic signals can be represented in a rationaltransfer function format. Thus discrete data can often berepresented by a linear difference equation:

m =0

p

~~ X atn=0

(3)

where xk is the input sequence to the model and yk is theoutput from the model and which is the signal to be analysed.In transfer function form this is

(4)

where

anz

B(z~l) = I bmz.mm = 1

The A(z~l) polynomial defines the autoregressive (AR)terms, and the B(z~l) polynomial defines the moving average(MA) terms. The input driving sequence is usually assumed tobe white noise with zero mean and variance a2. Usually it isassumed that a0 = b0 = 1, as any filter gain (equivalent tosignal amplitude) can be incorporated in a2.

When A(z~l)= 1 an MA model is obtained, and whenB(z~1)=\ an AR (or all-pole) model results. The Wolddecomposition theorem [18] asserts that an ARMA or MAprocess can be represented as a unique AR model of possiblyinfinite order, and vice versa. Thus, although it may not beparsimonious, an ARMA model can be approximated by ahigher-order AR model. As the estimation of AR model para-meters is a linear algorithm, in contrast to that for an ARMAmodel, the majority of parametric spectral analysis is concen-trated on an AR model formulation.

The power spectrum relating to eqn. 4 is obtained byevaluating the expression around the unit circle, i.e. z"1 =exp (-j2nfT). For an AR model the PSD can thus bewritten as

= |//{exp(-/27r/T)}|2<I>u(/)

2>Bexp(-/2ir/>!r)l"

(5)

The PSD can therefore be determined solely from a knowledgeof the coefficients ax, a2,. . • , ap, and a2. The AR coefficientscan be found directly from a linear least-squares approach bysolving the so-called Yule-Walker normal equations, whichare given by

r(0)

r(p-l)

'(1) • r(p-\)

r(0)

(6)

664 IEEPROC, Vol. 129, Ft. A, No. 9, DECEMBER 1982

These equations can also be augmented to incorporate the a2

equation, giving

r(0)

r(0) .

r(P)

1

&:

ap

~o2

0

0

(7)

where r(i) are estimates of the autocorrelation terms of thesignal yk, usually obtained from a mean-lagged productsalgorithm given either by

AR parameter identification is closely related to the theoryof linear prediction, in which the prediction of yk based onthe previous p samples is given by

aiyh.i (12)

An important lattice filter interpretation of the fast Levinsonalgorithm is given by considering the prediction error for apth-order linear predictor defined as epk

ePk = yk + Z apiyk _,

using the relationship of eqn. 11 this can be written as

(13)

r(i) = t' (8)pi

Z (flp-i,i

or = ep-

KO = *yy(0 =2

vv i TyJ+iyj (9)

Use of eqn. 8 can produce bias effects if N is not large, and useof eqn. 9 can produce invalid autocorrelation functions whichcause solutions to blow-up. Later algorithms attempt to over-come these problems.

The correlation matrix in eqn. 6 is of the Toeplitz form(i.e. the elements along each diagonal are equal), and is positivedefinite. The Toeplitz form can be solved recursively withoutmatrix inversion using a method first formulated by Levinson[19] and improved by Wiggins and Robinson [20]. Theserecursive methods are much faster than matrix inversions ascomputing time is proportional to p2, instead of p 3 for stan-dard solution methods, where p is the number of model coef-ficients being determined. The Levinson algorithm calculatesrecursively the coefficient sets (an, a2), {a2l, a22, o\) etc.,where the leading suffix now defines the iteration number.The algorithm is initialised by setting

an = -r(l)/r(0) (10)

o\ =

with the following recursions I = 2,3,... ,p given by

i

o2 = ( l - k , , l a ) o ? - i (11)

As successively higher-order models are being estimated withthis algorithm, examination of the a2 terms can assist indetermining a suitable model order for the data being analysed.Thus, for a perfect AR process a2 will first reach its minimumvalue at the correct model order.

The coefficients (an, a22, • • • >aPP) are often called the

reflection coefficients and written as (Kt, K2,. .. , Kp). Acondition for the autocorrelation matrix to be valid and forthe poles of A(z~x) to lie on or outside the unit circle of thez"1 plane is that \K{\< 1 for /= 1 ,2 , . . . ,p.

p-\,k

where bpk is the backward prediction error given by

apiyk.p + i

(14)

(15)

The predictor coefficients for the backwards predictor arecomplex conjugates of those for the forward predictor, andhence one can write

bnb — >-l,k (16)

The relationships in eqns. 14 and 16 can be represented in thelattice filter structure shown in Fig. 1.

An alternative viewpoint of AR PSD estimation is its con-cept of autocorrelation extrapolation. In conventional periodo-gram analysis the autocorrelation beyond the known lags isassumed to be zero, which provides the well known phenom-enon of sidelobe leakage caused by such windowing. In ARmethods, however, the autocorrelation function is extrapolatedbeyond the known lags, and it is reponsible for the highresolution of AR spectral analysis. Clearly, the success of theanalysis will depend on the accuracy of the extrapolation, andthis is a function of the particular algorithm being used, as willbe illustrated in following examples.

Burg [13] introduced the concept that extrapolation shouldbe made so that the time series characterised by the extra-polated autocorrelation function has maximum entropy. Analternative viewpoint is that the PSD should be the one withthe flattest (whitest) spectrum corresponding to the knownautocorrelation function terms. This maximum entropy cri-terion maximises the randomness of the time series, thusproducing a minimum bias solution. Use of Lagrange multipliertechniques shows that for a Gaussian random process themaximum entropy solution is identical to the AR PSD alreadydescribed [15].

A number of different techniques exist for the estimationof the AR parameters, which can be split into the classes ofoff-line (i.e. batch) or on-line (i.e. sequential) algorithms. Note

pn

Fig. 1 Lattice formulation of prediction error filters

IEEPROC, Vol. 129, Pt. A, No. 9, DECEMBER 1982 665

that the sequential estimators can involve recursion both inmodel order and in time. For the off-line methods eitherforwards and backwards prediction is employed. For the firstclass consider the forward linear prediction error to be givenby

(17)i = 0

where ap0 = 1.In matrix form eqn. 17 can be written as

E Y

ep,N-l

dpi

*pp

(18)

The prediction error is given by

EP = (19)

where the summation range depends on the range employed inthe matrices of eqn. 18 in determination of the predictionerrors. Eqn. 18 can be written in normal equation format as

(YfYt)A =

0

(20)

To extend the range of epi in eqn. 18 from p,.. . ,N — 1 to0 , . . . ,N + p + 1 one has to assume that measurements arezero for k < 0 and k > N — 1. This implies a data windowingof the sequence, which will give rise to leakage and a corres-ponding degradation of frequency resolution. Using datamatrix Yx in eqn. 18 corresponds to the Yule-Walker equationsand produces a Toeplitz ( Y*f 7f) matrix with its fast Levinsonalgorithm, but having leakage associated with its data window.Using Y2 no windowing is assumed, but the (Y1? Yt) matrix isno longer Toeplitz, and the normal equations are often referredto as covariance equations in speech analysis [21]. Data matrixy3 produces a prewindowed structure, and YA gives a post-windowed structure. Each of the data matrices is Toeplitz,but only Yt gives a Toeplitz product matrix in the normalequation. Fast algorithms have, however, been developed foreach structure and details are given in Morf etal. \21\ for thecovariance case, and Friedland et al. [23] for the prewindowedcase.

A number of problems exist in algorithms referred to aboveusing forward linear prediction alone. For short time series theYx method (termed autocorrelation method) gives least

resolution, while the covariance method gives false peaks andperturbations from the correct frequency locations, togetherwith a greater sensitivity to noise. Spectral line splitting, i.e.the existence of two closely spaced spectral peaks where onlyone should occur, has been observed in all four methods (e.g.by Fougere et al. [24]) and is explained by Kay and Marple[25]. Some of these problems have been alleviated by the useof combined forwards and backwards prediction, first intro-duced by Burg [13], and distinct from the maximum entropyconcept.

The Burg algorithm estimates the AR parameters byminimising the error given by

J V - l N-\

Ep = 1 \epi\2+ I \bpi\ (21)

i=P i=P

subject to the constraint that the AR parameters satisfy theLevinson recursion

dpi = ap-i,i+app<*p- i.p-i (22)

for all orders from 1 to p. This constraint ensures a stable ARfilter (all poles outside the unit circle) and gives reflectioncoefficients with a magnitude less than unity. In this wayminimisation with respect only to ait is required, yielding thealgorithm

N-l

+\et-iti\'(23)

Recursive algorithms exist for these relationships, and furtherdetails can be found in Ulrych and Bishop [26]. Problemsstill exist, however, in the Burg algorithm of spectral line split-ting and biased frequency estimates for short data sequences.If ep in eqn. 21 is minimised with respect to all the api fori = 1 , . . . ,p then these problems are reduced. This approachremoves the Levinson constraint and was suggested indepen-dently by Ulrych and Clayton [27] and Nuttall [28]. In thisalgorithm a matrix equation is obtained and given by

(24)

where

app

(0,0) (0,p)

rp(p,0) . . . rp{p,p)

where

N-p-l

rP0,f) = I1 = 0

and

EP =P

7 = 0

(25)

(26)

666 IEEPROC, Vol. 129, Pt. A, No. 9, DECEMBER 1982

Rp can be expressed as sums and products of Toeplitz andHankel matrices, enabling a recursive fast algorithm to bedeveloped requiring 0(p2) operations [29]. The LS methodgives less bias and reduced variance in frequency estimation,and obviates spectral line splitting.

Sequential estimation of AR parameters has been attemptedvia a number of approaches. First, a Kalman filtering methodcan be used, giving the following recursion equations:

(27)

(28)

(29)

where

m + l

"1P - r YT Y I

Ym =

The Kalman approach requires 0(p3) operations for each newdata point, whereas a simpler approach using the LMS algorithmof adaptive filtering due to Widrow [30] requires only Q(p)operations. This method uses updating giving by

L m + 1 (30)

This is similar to eqn. 27 except that n is fixed, whereas theKalman gain is variable. Convergence is guaranteed for 0 < /i <1/Xmox, where Xmax is the maximum eigenvalue of Rxx.Choice of (i involves a compromise between rate of conver-

gence to Am and the amount of steady-state variance (calledmisadjustment).

A third approach uses lattice recursive relationships whichupdate the reflection coefficients with 0(p) operations, andinfrequently update the AR parameters using a Levinson algor-ithm with 0(p) operations. Equivalent formulations for theoff-line estimators previously described are given by Morfetal. [31,32].

A suitable model order is an important parameter whichmust be chosen carefully, as too low an order gives a highlysmoothed spectrum, whereas too high a model producesspurious peaks in the spectrum. The prediction error power is

0.468F,Hz

Fig. 2 Effect of initial phase on spectral analysis of 16 points ofsimulated data from 1.0 cos (0.4 T + <p) + 1.0 noise

F.Hz0.312 0.468

a LS forwards and backwards algorithmb Yule-Walker algorithmc FFT with zero padding to 128 points

IEEPROC, Vol. 129, Pt. A, No. 9, DECEMBER 1982667

not a satisfactory criterion to use, as it is a mathematicallydecreasing function of the model order p .

Several criteria have been suggested as objective bases forthe selection of model order, two of which are due to Akaike.In one of these [33] a final prediction error (FPE) is examinedfor a minimum and is given by

(31)

The second method [34] minimises an information theoreticfunction, giving the Akaike information criterion (AIC)

AICP = 2(p+l)/N (32)

Another method, proposed by Parzen [35], is called thecriterion autoregressive transfer (CAT) function, and is given by

(33)CATO = I-L-I-7T

where

Ep = {N/(N-j)}Ep

1.0

—0.8

3

02

0.0

10 15 20 25 30 35 40 45 50 55 60time,s

significant frequencies0.1515

0.00 0.05 0.10 0.15 0.20 Q25 030 0.35 O40c frequency,Hz(resolution 0.0152 Hz)

0.45

Bir'iA~izV

1.00

50

-5-10

or15

T3-20£-252-30f-35

-45-50-55-60

significant frequencies0.06360.11750.15870.2000

000 0O5 OX) 0.15 Q20 025 030 035 0.4O 0.45b frequency,Hz(resolution 0.6250E-03Hz)

-0.04

0 20 40 60 80 100 120 140 160 180 200 220 240e residuals plotted at 1.0000 intervals,s

1 . 0 0 - 0 . 7 4 Z ' 1 • 0 .62Z" 3 • 0.33Z"* + 0 . 0 5 Z " - 0.06Z""1 - 0 . 0 3 Z " 4 * 0 . 0 6 Z " ' • 0 . 0 1 Z " 9 - 0 . I 1 Z ' ' + 0 .03Z" 1 0 • 0 . 0 6 Z " " + O . ' 2Z~ 1 2 • O . ' I Z " 1 1 + 0 . 0 4 Z " " - 0 .05Z" 1

.5

.0

0.5

0.0

-0 .5

-1 .0

' - 1 . 5 - 1 . 0 - 0 . 5 0 . 0 0 . 5 i . O 1 . 5

Fig. 3 Spectral analysis of simulated amplitude-modulated data

a Data comprising 64 pointsb AR 20th-order spectral analysis via Yule-Walker algorithmc FFT spectral analysisd z-plane plot, AR polynomial coefficients and frequency componentse Autocorrelation function from AR model residuals

± 1 SD668

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y SEAL

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1 - /!/

0.0'25

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0.0'06

POLKS

i - a/

0.0619

0.0125

0.0007

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1.0000

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1.5569

REAL

O.SSSEOO

0.700F.00

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HARMONIC

Is1

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2 *

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0.312E00

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oor"0 9 Z ",,r.i

'SZ~"

0 6 Z n

IEEPROC, Vol. 129, Pt. A, No. 9, DECEMBER 1982

Although results using simulated data have been reasonablygood using the above criteria, when used on actual data thereis a tendency to under-estimate the model order (e.g. Jones[36]). For short data stretches none of the criteria work will(e.g. Ulrych and Clayton [27]) and for this condition Ulrychand Ooe [37] recommend a model order of N/3 to N/2. Themodel order depends on SNR and the closeness of spectralcomponents in the data. Linkens [38] gives empirical rulesfor model order selection based on the above conditions.Certainly, subjective assessment is required at present for theselection of the model order. Evidence exists that once a suit-able model order has been selected for an application, it islikely to remain suitable even when there are time-variationsin the characteristics of the data, as in biomedicine.

3 Examples of AR spectral analysis

In spectral analysis of biomedical signals the problem of time-varying characteristics is nearly always present. Thus, thetracking of frequency components is a difficult requirement,and yet may often have physiological significance. In mostbiological signals considerable noise is present, and spectralline splitting does not appear to be a problem. It mostly occursfor very high SNR, and in any case has not been discovered

10 15 20 25 30samples plotted at 0.1667 intervals

0

5

-10

-15

m-20

-25

-30

-35

0.0 0.2 0.4 0.6 0J8 1.0 12 1.4 1.6 1fl 20 Z2 23 2.6 2Bb frequency,resolution 0.0038

Q0 Q2 0.4 0.6 0.8 1.0 12 1.4 1.6 1.8 20 2.2 2.4 26 2.8c frequency, resolution 0.0462

Fig. 4 Spectral analysis of rat locomotor activity

a Data (200 points)b AR spectrum showing oestral cyclec FFT

using the forward-backward LS algorithm, which also gives thebest frequency estimation.

A major problem is that, for a process consisting of asinusoid plus noise, the peak location in the AR spectrumdepends critically on the initial phase of the sinusoid [40].It has been observed by Ulrych and Clayton [27] that the LSmethod gives the best performance with respect to this problem,although an alternative method based on an analytic signal[41] is also successful in reducing the phase dependency. Itshould be noted that the classic periodogram also gives a peakfrequency shift for short data records, particularly noticeableif zero padding is used. This is illustrated in Fig. 2 for 16 datapoints of a noise-contaminated sinusoid. The Figure showsclearly the reduced phase dependency and improved frequencyresolution of the LS method (Fig. 2a) over a Yule-Walkerapproach of the periodogram.

Ia

s x s 3 : 5 R c s i s s : ; ;

DATA

Fig. 5 Spectral analysis of 64 points of human digit blood flowfluctuation caused by thermal stimulation

0312F H z

0.624 0.936

Fig. 6 LS spectra for 16-point blocks of canine duodenal electricalactivity incremented ten points at a time, showing frequency variability

IEEPROC, Vol. 129, Pt. A, No. 9, DECEMBER 1982 669

The ability of the spectral analysis to detect low-frequencyrhythms is shown in Fig. 4 for data comprising locomotoractivity of a female rat (Fig. 4a). In Fig. 4b the major peak at

The ability of AR analysis to discriminate multiple peaksfrom short data records is illustrated from amplitude-modu-lated data in Fig. 3 a. An indication of successful spectralanalysis is obtained by examining the autocorrelation of theresiduals left after model fitting (Fig. 3e). Fig. 3 b showsthe AR spectrum with clear frequency resolution of the sidebands, in contrast with the FFT of Fig. 3c. Fig. 3d shows thez-plane pole locations together with a listing of significantfrequencies.

1.0 corresponds to the well known circadian activity. The peakat 2.0 is the second harmonic of the circadian component,while the peak at 0.25 represents activity caused by the oestralreproductive cycle of the rat. Neither of these latter compo-nents can be distinguished from background spectral variabilityin the FFT of Fig. 4c. The ability of the AR techniques toproduce a smoothed spectrum which emphasise importantcomponents is demonstrated in Fig. 5 for 64 points of a humandigit blood flow signal. In this case the component at 0.05 Hz,visible in Yule-Walker and Burg but invisible in the FFT, wascasued by an external thermal stimulus at that frequency. Theuse of the LS method for tracking frequency variations in real

025

ata o -e3o - °o „ -

0 a 50 time.min 100

Q25

,0 -

50 time min 100

Fig. 7 Frequency tracking of human gastrointestinal data showing internal and surface electrode spectral information against time with 2 minepochs

a Internal gastric b Surface

0.25

n r-, a ,-,

a Da ^

-....^.^a-

D = =

-

° ca a

a a

• D

0.25

time.min 30

CD

CD

a

•

•

•

DD

DD

•a

D

D

a

= •

D

D

D

D

DDD a n

OnDcDa

a

•

°

•

time.min 30

Fig. 8 Frequency tracking similar to Fig. 7 on another patient

a Internal gastric b Surface

670 IEEPROC, Vol. 129, Pt. A, No. 9, DECEMBER 1982

data is now illustrated in Fig. 6 where, significant, and appar-ently regular, variations occur in a canine duodenal signal.

AR methods have been applied to long recordings of gastro-intestinal data to obtain frequency tracking information. Inthese spectrogram displays the vertical axis represents frequencyand the horizontal axis represents time epochs. The significanceof particular spectral components is assessed by the closenessof the AR pole pair to the unit circle of the z-plane. Thesize of the rectangular indicator is directly related to theamplitude of the component. A significance threshold is usedto limit the number of components plotted. One applicationof this approach has been to the analysis of surface electrodeabdominal recordings used to indicate gastric electrical activity.The correspondence between internal electrode and noninvasivesurface electrode spectra is illustrated in Fig. 7. In this examplethe sample rate was 1 Hz, with 128 points epochs (i.e. 2mindata stretches), and the Burg algorithm was used with a20th-order model. The analysis was performed in real time.In this case there is clear similarity between the internal andexternal spectra. The absence of spectral components duringcertain epochs for the internal recording was caused by theBurg algorithm 'blowing up' due to a very small noise content.In the Figure the height of the rectangular boxes is directlyrelated to amplitude, and the width of the box is inverselyrelated to the bandwidth of the component (i.e. an indicationof stationarity of the signal). An example on another patientis shown in Fig. 8 where correspondence between internal andexternal spectra is confined to the fundamental component at0.05 Hz. In this case the surface recording spectral componentsare generally lower than for the internal electrode.

The AR frequency tracking profile for a human duodenalrecording is shown in Fig. 9. The small transient changes inboth fundamental and second-harmonic components areassociated with mechanical contractions which affect theamplitude, waveshape and frequency of the electrical activity.The transient frequency changes are 'smoothed' because of thelength of the epochs (2 min). In Fig. 10 1 min epochs havebeen used on another patient, and the frequency transientsassociated with mechanical muscular contractions are clearlyvisible, being less smoothed.

4 Conclusions

The so-called 'modern' methods of spectral analysis based onlinear prediction and autoregressive models offer advantages ofgreater frequency resolution and reduced spectral leakage overconventional FFT analysis. These advantages are, however,only realised after careful selection of sampling frequency,model order and type of algorithm. The greatest improvementsare obtained for good signal-to-noise ratios, although it shouldbe noted that some noise contamination is mandatory to avoidsome of the algorithms from blowing-up numerically. Evensome unfiltered gastrointestinal electrical signals have required

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IEEPROC, Vol. 129, Pt. A, No. 9, DECEMBER 1982 671

the artificial addition of noise to give numerical robustnesson occasions.

For short-time-series analysis, where frequencies are changingrapidly, the forwards and backwards LS algorithm proposedby Ulrych and Clayton seems to give good performance inmany applications. The algorithms described and illustratedsuffer from the disadvantage of being computationally complex,and limit their on-line capability to relatively low-frequencysignal processing. For higher-frequency processing the muchsimpler adaptive line enhancement approach based on theLMS adaptive filtering algorithm due to Widrow offers theadvantage of a much lower computational burden. It does,however, suffer from problems in numerical convergence, andmay require rather long tracking adaptation times to reducethe variance in the frequency estimates.

Finally, it should be emphasised that the methods describedin this paper represent one approach to the estimation offrequency components in time-varying rhythmic data. Anotherapproach uses a raster scan pattern with interactive selectionof parameters to obtain spectral information [42]. This methoduses graphics facilities on a microcomputer with a very simplealgorithm. The method is, however, not amenable to on-lineanalysis, and requires large amounts of data storage. Somequantification of frequency profiles is possible, but the methodis basically a visual approach, and is best suited to stronglyoscillatory signals.

5 References

1 SCHUSTER, A.: 'On the investigation of hidden periodicities withapplication to a supposed 26 day period of meteorological phenom-ena', Terrestrial Magnetism, 1898, 3, pp. 13-41

2 WIENER, N.: 'Generalised harmonic analysis', Acta Mathematica,1930, 55, pp. 117-258

3 BLACKMAN, R.B., and TUKEY, J.W.: 'The measurement of powerspectra from the point of view of communications engineering'(Dover, New York, 1959)

4 COOLEY, J.W., and TUKEY, J.W.: 'An algorithm for machinecalculation of complex Fourier series', Math. Comput., 1965,19, pp. 297-301

5 PRONY, G.R.B.: 'Essai experimental et analytique, etc.', /. deL'EcolePoly'technique, 1795, 1, pp. 24-76

6 HILDEBRAND, F.B.: 'Introduction to numerical analysis' (McGraw-Hill, New York, 1956)

7 PISARENKO, V.F.: 'On the estimation of spectra by means of non-linear functions of the covariance matrix', Geophys. J. R. Astron.Soc, 1972, 28, pp. 511-531

8 PISARENKO, V.F.: The retrieval of harmonics from a covariancefunction', ibid., 1973, 33, pp. 347-366

9 CAPON, J.: 'High-resolution frequency-wavenumber spectrumanalysis', Proc. IEEE, 1969, 57, pp. 1408-1418

10 LACROSS, R.T.: 'Data adaptive spectral analysis methods', Geo-physics, 1971, 36, pp. 661-675

11 YULE, G.U.: 'On a method of investigating periodicities in dis-turbed series, with special reference to Wolfer's sunspot numbers',Phil. Trans. R. Soc. London A, 1927, 226, pp. 267-298

12 WALKER, G.: 'On periodicity in series of related terms', ibid.,1931, 131, pp. 518-532

13 BURG, J.P.: 'Maximum entropy spectral analysis'. 37th Ann. Int.Meet. Soc. Explor. Geophys., Oklahoma City, Oct. 1967

14 PARZEN, E.: 'Statistical spectral analysis (single channel case) in1968'. Dept. Statistics, Stanford Univ., Tech. Rep. 11,1968

15 VAN DEN BOS, A.: 'Alternative interpretation of maximum entropyspectral analysis', IEEE Trans., 1971, IT-17, pp. 493-494

16 KAY, S.M., and MARPLE, S.L.: 'Spectrum analysis - a modernperspective', Proc. IEEE, 1981,69, pp. 1380-1419

17 CHILDERS, D.G. (Ed.): 'Modern spectral analysis' (IEEE Press,1978)

18 WHITMAN, E.C.: 'The spectral analysis of discrete time series interms of linear regressive models'. Naval Ord. Labs. Rep. NOLTR-70-109, White Oak MD, 1974

19 LEVINSON, N.: The Wiener RMS (root mean square) error criterionin filter design and prediction', / . Math. Phys., 1947, 25, pp.261-278

20 WIGGINS, P.A., and ROBINSON, E.A.: 'Recursive solution to themultichannel filtering problem, J. Geophys. Res., 1965, 70, pp.1885-1891

21 MAKHOUL, J.: 'Linear prediction: A tutorial review', Proc. IEEE,1975,63, pp. 561-580

22 MORF, M., DICKINSON, B., KAILATH, T., and VIEIRA, A.:'Efficient solution of covariance equations for linear prediction',IEEE Trans., 1977, ASSP-25, pp; 429-433

23 FRIEDLANDER, B., MORF, M., KAILATH, T., and LJUNG, L.:'New inversion formulas for matrices classified in terms of theirdistance from Toeplitz matrices', Linear Algebra & Appl., 1979,27, pp. 31-60

24 FOUGERE, P.F., ZAWALICK, EJ., and RADOSKI, H.R.: 'Spon-taneous line splitting in maximum entropy power spectrum analysis',Phys. Earth & Planet. Inter., 1976,12, pp. 201-207

25 KAY, S.M., and MARPLE, S.L.: 'Sources of remedies for spectralline splitting in autoregressive spectrum analysis'. Int. conf. acous-tics, speech and signal processing, 1979, pp. 151—154

26 ULRYCH, T.J., and BISHOP, T.N.: 'Maximum entropy spectralanalysis and autoregressive decomposition', Rev. Geophys. &Space Phys., 1975,13, pp. 183-200

27 ULRYCH, T.J., and CLAYTON, R.W.: Time series modelling andmaximum entropy', Phys. Earth & Planet. Inter., 1976, 12, pp.188-200

28 NUTALL, A.H.: 'Spectral analysis of a univariate process with baddata points, via maximum entropy, and linear predictive methods'.Naval Underwater Systems Centre, Tech. Rep. 5303, New LondonCT, 1976

29 MARPLE, S.L.: 'A new autoregressive spectrum analysis algorithm',IEEE Trans., 1980, ASSP-28, pp. 441-454

30 WIDROW, B.: 'Adaptive noise cancelling: Principles and applica-tions',Proc. IEEE, 1975, 63, pp. 1692-1716

31 MORF, M., KAILATH, T., and LJONG, L.: 'Fast algorithms forrecursive identification'. Proc. IEEE conf. on decision and control,Clearwater FL, 1976, pp. 916-921

32 MORF, M., VIEIRA, A., and LEE, D.T.: 'Ladder forms for identi-fication and speech processing'. Proc. IEEE conf. on decision andcontrol, New Orleans, 1977, pp. 1074-1078

33 AKAIKE, H.: 'Fitting autoregressive models for prediction', Ann.Inst. Statist. Math., 1969, 21, pp. 243-247

34 AKAIKE, H.: 'Use of an information theoretic quantity for statis-tical model identification', Proc. 5th Hawaii int. conf. on systemsciences, 1972, pp. 249-250

35 PARZEN, E.: 'Some recent advances in time series modelling',IEEE Trans., 191 A, AC-19, pp. 723-730

36 JONES, R.H.: 'Autoregression order selection', Geophysics, 1976,41, pp. 771-773

37 HAYKIN, S.S. (Ed.): 'Nonlinear methods of spectral analysis'(Springer-Verlag, New York, 1979)

38 LINKENS, D.A.: 'Empirical rules for the selection of parametersfor autoregressive spectral analysis of biomedical rhythms', SignalProcess., 1979,1, pp. 243-258

39 SMALLWOOD, R.H., LINKENS, D.A., KWOK, H.L., and STOD-DARD, C.J.: 'Use of autoregressive modelling techniques for theanalysis of colonic myoelectrical activity in man', Med. & Biol.Eng. & Comput., 1980, 18, pp. 591-600

40 CHEN, W.Y., and STEGEN, G.R.: 'Experiments with maximumentropy power spectra of sinusoids', / . Geophys. Res., 1974, 79,pp. 3019-3022

41 KAY, S.M.: 'Maximum entropy spectral estimation using theanalytical signal', IEEE Trans., 1978, ASSP-26, pp. 467-471

42 LINKENS, D.A., KITNEY, R.I., and ROMPELMAN, O.: 'A raster-scan method for observing physiological entrainment phenomena',Med. & Biol. Eng. & Comput., 1982, 20, pp. 483-488

Derek A. Linkens received a B.Sc. (Eng.)in electrical engineering from the ImperialCollege of Science & Technology, London,in 1960, and an M.Sc. in systems engineer-ing from the University of Surrey in1968. He was awarded a Ph.D. from theUniversity of Sheffield in 1977 for workon coupled nonlinear oscillators. From1960 he worked on underwater weapondevelopment with the Plessey Co., Ilford,and then on aircraft flight control with

Elliott Automation, Rochester. Since 1969 he has been withthe Control Engineering Department of the University ofSheffield, where he is presently a Reader. His research in-terests include modelling and analysis of gastrointestinalelectrical activity, estimation of lung parameters, identi-fication and control of muscle relaxation anaesthesia, anddesign of autopilots for surface ships.

672 IEEPROC, Vol. 129, Pt. A, No. 9, DECEMBER 1982