IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 37, NO. 4. APRIL 1990 417

Communications

An Adaptive “Wiener” Filter for Estimating the Time-Derivative of the Left Ventricular

Pressure Signal

S. PERIYALWAR, A. E. MARBLE, S . T. NUGENT, A N D D. N. SWINGLER

Abstract-This communication presents the development of an op- timal digital differentiating filter based on the Wiener theory for esti- mating the peak derivative of the left ventricular pressure (LVP) sig- nal. The magnitude coherence function is used to estimate the signal and noise spectra. The peak derivative obtained by this method is found to be within 2% of the results obtained by a seven-point second-order data fit and a DFT technique.

INTRODUCTION Accurate estimation of the rate of ventricular pressure change is

critical when determining the contractility index of the myocardial muscle. The left ventricular pressure waveform transduced from humans is approximately periodic with a fundamental frequency of approximately 1.1 Hz, and a range of 0.8-2 Hz. It has been shown, using Fourier analysis that, for either normal or diseased heart muscle, the left ventricular pressure waveform generated by these hearts contains no more than twelve harmonics which are signifi- cant in representing the LVP waveform to within 99% of the orig- inal waveform [I ] . Hence, to obtain a good estimate of dP/dt , the primary requirements of the derivative algorithm are linear gain and phase response over the range of frequencies between zero and the frequency corresponding to the 12th harmonic of the funda- mental frequency. Flicker noise, white noise, and line interference are added to the left ventricular pressure signal during transduction and amplification; these undesirable artifacts have to be dealt with during the differentiation process. Several approaches to develop differentiators can be found in the literature [2]-[5] . Our main in- terest in this communication is the presentation of a digital differ- entiator, based on Wiener theory, which: 1) adapts to the quasi- periodic nature of the signal, and 2) adaptively deals with the noise present, such that the resultant signal-to-noise ratio of the deriva- tive is optimized.

ADAPTIVE DIGITAL DIFFERENTIATOR Adaptive filters are data-driven filters in which the transfer func-

tion is adapted to manipulate with minimum degradation, the de- sired components of the signal, and to attenuate the interfering sig- nals. In our case, the input signal characteristics are not known a priori and are to be estimated.

For our purpose the observed signal y ( t ) may be modeled as an originating sequence x ( t ) , which is corrupted by noise n ( t ) . Op-

Manuscript received August 16, 1988; revised June 23, 1989. S . Periyalwar and A. E. Marble are with Department of Electrical En-

gineering, Technical University of Nova Scotia, Halifax, Nova Scotia, Canada B3J 2x4.

S . T. Nugent is with the Department of Electrical Engineering, Dalhou- sie University, Halifax, Nova Scotia, Canada B3J 2x4.

D. N. Swingler is with the Department of Electrical Engineering, Saint Mary’s University, Halifax, Nova Scotia, Canada 835 2x4.

IEEE Log Number 8933600.

timal signal estimation in the presence of additive noise was at- tempted initially by Wiener on assuming that the signal and noise are stochastic, stationary, and uncorrelated processes with known power density spectra. The “Wiener” differentiator in the fre- quency domain can be estimated as

whereAfi(w) is the estimated filter transfer function and l k ‘ (w ) l 2 and 1 N ( o ) 1’ the estimated power density spectra of the signal and noise. Note if l ] i r ( ~ ) ) ~ = 0 then H ( w ) = j w as expected. Wiener is put between quotation marks since the estimated derivative is, in general, not optimal in the least square error sense. This is due to the fact that here H ( U ) is an estimate itself.

In order to implement the “Wiener” differentiator the magni- tude coherence spectrum is used to separate the signal and noise spectra [7]. The coherence function between two wide sense sta- tionary random processes y I ( t ) and yz( t ) is estimated as [8]

where Yl, ( U ) and Y2, ( U ) are the fast Fourier transform outputs of the nth data segments of y l ( t ) and y 2 ( t ) (the length of each data segment corresponds to a full period of the LVP signal), N is the number of data segments and “*” denotes complex conjugate. The yI( t ) and y2( t ) signals required to estimate the coherence function are obtained from the periodic observed signal, using the model shown in Fig. 1. If one assumes uncorrelated noise and coherent signal between y l ( t ) and y 2 ( t ) , then the power in y 2 ( t ) due to co- herent signal and power due to uncorrelated noise at frequency w is expressed as [9]

(3)

The confidence in the estimation of the coherence function in- creases with N and the signal-to-noise ratio [9]. In the case of the LVP signal the signal-to-noise ratio in the bandwidth of the signal is reasonably high and increasing N greater than 16 does not sub- stantially change the estimated coherence function. On the other hand, the respiration of the subject, which has a duration between 7 and 15 cardiac cycles, affects the LVP signal. It is observed that with large values of N the drift in fundamental frequency is much greater and the condition of stationarity may no longer hold. Also, larger values of N greatly increase the computation time. Consid- ering these compromising aspects we have used 16 data segments in Y I ( ~ ) and ~ 2 ( t ) .

SOME NONADAPTIVE DIFFERENTIATORS In examining the performance of our “Wiener” differentiator,

it is useful to compare the derivative obtained by this method to that obtained by using other numerical analysis methods. The al-

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418 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING. VOL. 31 . NO. 4, APRIL 1990

Periodic x ( t ) y ( t ) * ~ , Signal D t l n r ( LVP 1

Y2 ( t )

Fig. 1 . Model used to obtain y , ( t ) and yz( t ) from the observed signal.

gorithms used to make a comparison are taken from [6] and listed below :

Piecewise linear approximation (PLA), expressed as

( 5 )

where x ( t , ) is a discrete value of a continuous function ~ ( t ) and h is the time period between samples.

Three-point Lagrange derivative approximation (TPL), ex- pressed as

(6) x(t,+1) - x ( t , - , )

2h x ’ ( t , ) =

Five-point Lagrange derivative approximation (FPL), expressed as

x( t r -2) - 8 x ( t , - , ) + 8 X ( f , + l ) - 4 t , + 2 ) , ( 7 ) 12h x’(t ,) =

0 0 0 1 0 2 0 3 0 4 0 5 NORMALIZED FREQUENCY

Fig. 2 . Magnitude response of digital filters.

where x ( t ) is the signal content, n ( t ) is the noise content, and N is the number of data points in one period of the signal. The signal- to-{ noise + distortion} ratio of the outputs from the different methods is estimated by the following relationship

where n‘ ( t ) is the theoretical derivative and y’ ( t ) the derivative from the differentiator.

r‘(t,) = 10h . (8) The filters using the numencal analysis techniques and the DFT

Five-point second-order data fit (FPS), expressed as

-2X(f,-*) - X ( t , - l ) + x ( t , + , ) + 2x(t,+*)

Seven-point second-order data fit (SPS), expressed as

(9) - 3 ~ ( t , - ~ ) - 2x(t,-2) - 4 t , - 1 ) + x ( ~ , - I ) + 2 4 [ , + ~ 2 ) + 3n(tI+3)

28h x’(t ,) =

Seven-point third-order data fit (SPT), expressed as

The discrete Fourier transform technique (DFT) expressed as

Y ( t ) = D F T - I [jnoo x ~ ( n w ~ ) ] n = 0, 1 , . . . , 12 (11) where wo is the fundamental frequency of the signal. The DFT technique involves the frequency domain differentiation of the first twelve harmonics in every period of observed signal. The contri- bution of all higher harmonics, which contain mostly noise, is set to zero.

The magnitude response of the digital filters are shown in Fig. 2, for normalized sampling frequency.

ASSESSMENT OF ALGORITHMS ON SIMULATED TEST SIGNALS A test signals is generated by summing twelve weighted cosine

terms as indicated

15 ~ ( t ) = C COS (nod) t = 0, t,, 24, . . . , ( N - l ) t s

n = I n

where a,, is the fundamental frequency of the signal, N is the num- ber of sample points in one period of the signal, and t, is the time between samples to get N points per period. The energy contribu- tion of the 12th harmonic to the test signals is about 0.43 % .

With Gaussian white noise added to the input signal, the signal- to-noise ratio is estimated as

-i [ X ( f ) l Z

t = l [.(t,l’

(13) t = I

SNRlN = 10 log,,, N

technique can operate on just one period of data. In the case of the “Wiener” technique, each of the yl( t ) and y2( t ) signals comprised of 16 periods of the test signal to which noise is added. In this case, the average signal-to-noise ratios are calculated.

Noiseless computer generated signals with N = 64 and 512 points per period are first used to obtain the signal-to-distortion ratio of the derivative obtained by the various techniques. The per- centage distortion of the peak derivative in the absence of noise is calculated. These results are shown in Table I.

Next a noisy version of the test signal generated with 1 Hz fun- damental frequency is used. The signal-to-noise ratio of the input signal is varied from 0 dB to about 50 dB by varying the amount of zero mean random Gaussian white noise added and the corre- sponding signal-to-noise ratios of the derivatives obtained from the different methods are estimated. To make the comparison realistic the input signal and noise is band limited to 30 Hz by low-pass filtering.

From plots (Fig. 3) of input signal-to-noise ratio versus output signal-to-{ noise + distortion} ratio for the various conditions of the theoretical signal and from other experiments not detailed here, it is clear that: 1) the performance of the “Wiener” derivative filter is superior to the numerical analysis techniques, and is at least comparable if not superior to the DFT technique (when the cutoff frequency is accurately known); 2) the performance of all the filters improves with the increase in the number of sample points used, and reduction in the contribution of higher harmonics to the signal; 3) the improvement in performance is more pronounced at higher signal-to-noise ratios than at lower signal-to-noise ratios; and 4)

IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 37, NO. 4, APRIL 1990 419

TABLE I PERFORMANCE OF THE FILTERS ON A TEST SIGNAL IN THE ABSENCE OF NOISE

Signal-to-Deviation % Distortion at Peak Ration in dB d P / d t

N = 512 Method N = 64 N = 512 N = 64

PLA TPL FPL FPS SPS SPT DFT “Wiener”

9.10 19.30 33.10 10.00 5.50

19.10 > 200 > 200

26.90 55.00

104.10 44.40 38.10 88.30 > 200 > 200

14.50 6.10 0.90

18.60 33.00 4.50 0.00 0.00

0.01 0.08 0.00 0.26 0.56 0.00 0.00 0.00

1 00 15 00 31 00 47 00 -20 00 -

-1.00 15.00 31 00 47.00

INPUT SNR IN DB

- PU - OFT (IO harmonics) _ _ _ - TPL .- - - - _- FPL

FPS ~ - - r “Wiener’ SPS SPT

- DFT (1 2 harmonics) - OFT (I 4 harmonics) __- _ _ _ _ _ -

Fig. 3. Sample plots of the effect of noise on the digital filters

changes in fundamental frequency do not change the performance of the filters.

In the numerical analysis techniques, the improvement in per- formance with increase in the number of sample points per period of the signal is attributed to the reduction in ts, the sampling pe- riod. Considering the “Wiener” technique, the improvement in performance can be explained by the fact that better estimation of the coherence function is possible with a large number of samples, which in turn results in more accurate estimation of the signal and noise spectra [9]. This results in a better estimate of the optimum “Wiener” filter transfer function and hence a better estimate of the signal derivative.

ASSESSMENT OF DERIVATIVE ALGORITHMS ON HUMAN LVP SIGNALS

The left ventricular pressure signals, from ten patients who had undergone cardiac catheterization, were recorded on an HP 3964A FM tape recorder from a USCI 8 French catheter connected to a P23 dB Statham pressure transducer which applied the pressure sig- nal to an electronics for medicine ( E for M ) amplifying and re- cording system. The LVP signal of these ten patients was played back on the tape recorder and digitized at a sampling frequency of 1000 Hz. The fundamental frequency of each patient’s signal is estimated by detecting the rising and falling edges of each pulse. Knowing the total number of samples per period of the signal, it is possible to select I28 approximately equidistant samples. They,( t ) and y z ( t ) signals required by our “Wiener” technique, are then obtained, the corresponding points separated by approximately one time period.

The choice of 128 samples per period is due to the fact that it is

Eight periods of the LVP signal before processing m I

& 160 001 I

f, = 0.92 Hz

-

2 PLA TPL FPL FPS SPS SPT DFT“Wiener”

Derivative of the LVP signal obtained by the different techniques 7

x

m 3 220.00 L L 0

0.00 z

n n

3 Fig. 4. Analysis of LVP signal of patient 1

desirable to have at least eight to ten data points on the leading edge of the pressure curve. A larger number of points, on the other hand, would result in larger CPU time which is again a constraint. A typical LVP signal and its derivative obtained using the different digital techniques discussed is shown in Fig. 4 , for one of the sub- jects of this study.

The magnitude coherence function is estimated as described ear- lier and used to determine the signal-to-noise ratio before and after differentiation using the various algorithms. For the ten patients studied, the average peak derivative obtained using PLA, TPL, and FPL, is correspondingly 26.7, 17.3, and 19.1% higher, than the value obtained using the DFT. Among the data fitting techniques SPS seems to give better results compared to FPS and SPT tech- niques. Not only are the signal-to-noise ratios higher, but also the peak LVP derivatives are lower. It is observed that on an average, the mean peak derivative obtained by using FPS and SPT are 8.8 and 17% higher than that obtained by the DFT for the ten patients studied. The mean peak derivative obtained using SPS, DFT, and the “Wiener” technique are on an average within 2 % of each other. This seems to suggest that the best numerical analysis technique would be the SPS.

It is observed that the standard deviation of the peak derivatives obtained by the DFT technique are the lowest. This suggests that for the ten patients studied the derivatives obtained using the DFT technique are the least effected by the noise present in the LVP signal. It is seen from the magnitude spectra of the patient data, that most of the signal content is in the first few harmonics. This is why the DFT method gives better results as long as the noise in the first twelve harmonics is low. The problem with this method is that even the signal content above the 12th harmonic gets rejected.

The ‘‘Wiener” technique in the frequency domain is analogous to averaging in the time domain. The coherence coefficients used in the “Wiener” technique can be considered to be the average normalized frequency spectrum. Hence the use of this averaging process may result in the removal of some of the signal content in addition to the removal of noise. This effect is not noticeable in the case of signals that are highly corrupted by noise but can be seen in signals with high signal-to-noise ratios. In the case of the data with the lowest signal-to-noise ratio in the study group, the “Wie- ner” technique performs much better than all the other techniques. This is because the “Wiener” technique, unlike the other tech- niques, is capable of handling noise in the frequency band of in- terest.

CONCLUSION From the clinical results of this study it is concluded that when

the signal-to-noise ratio of the LVP signal is low, better derivatives can be obtained using the “Wiener” technique. Of all the tech-

420 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING. VOL. 31. NO. 4. APRIL 1990

niques studied the “Wiener” filter is the only one that tries to ex- tract the noise present in the frequency band of interest. Unlike the DFT technique this method does not reject signal content in the higher harmonics. The “Wiener” filter is truly adaptive in that it not only adapts its sampling rate to sample exactly 128 sample points per period of the signal, but also has an adaptive transfer function that changes to give an optimum estimate of the signal derivative in the mean square error sense. While quantitative mea- sures of the performance of the different techniques are difficult to produce in practice, since their performance depends on the signal- to-noise ratio and the amount of overlap between the power density spectra of the signal and the noise, it would be reasonable to sug- gest, that an accurate differentiator of the left ventricular pressure waveform, which deals optimally with noise both inside and out of the pass band, can be based on the adaptive “Wiener” filter out- lined here.

REFERENCES

[I] A. E. Marble, J. W. Ashe, D. H. K . Tsang, D. Belliveau, and D. N. Swingler, “Assessment of algorithms used to compute the fast Fourier transform of the left ventricular pressure on a microcomputer,” Med. Biol. Eng. Cornput., vol. 23, pp. 190-194, 1985.

[2] J. Spriet and J . Bens, “Optimal design and comparison of wide-band digital on-line differentiators,” IEEE Trans. Acousf., Speech, Signal Processing, vol. ASSP-27, pp. 46-52, 1979.

[3] A. Antoniou, “Design of digital differentiator satisfying prescribed specifications,” IEE Proc., vol. 127-E, pp. 24-30, 1980.

[4] S . Usui and I. Amidror, “Digital low-pass differentiation for biologi- cal signal processing,” IEEE Trans. Biomed. Eng., vol. 29, pp. 686- 693, 1982.

[5] T. Gasser, W. Kohler, C. Jennen-Steinmetz, and L. Sroka, “The anal- ysis of noisy signals by nonparameteric smoothing and differentia- tion,” IEEE Trans. Biomed. Eng., vol. BME-33, pp. 1129-1133, 1986.

[6] A. E. Marble, C. M. McIntyre, R. Hastings-James, and C. W. Hor, “A comparison of algorithms used in computing the derivative of the left ventricular pressure,” IEEE Trans. Biorned. Eng. , vol. BME-28, pp. 524-529, 1981.

[7] C. Charayaphan, “An assessment of signal-to-noise ratio of physio- logical signal using magnitude coherence spectrum,” in Proc. CMBEC,

[SI G. M. Jenkins and D. G. Watts, Spectral Analysis and Its Applica- tions.

[9] E. H. Scannell, Jr. and G. Clifford Carter, “Confidence bounds for magnitude-squared coherance estimates.” IEEE Trans. Acousf., Speech, Signal Processing, vol. ASSP-26, 1978.

vol. 13, 1987, pp. 203-204.

San Francisco, CA: Holden-Day, 1969.

Low-Pass Diff erentiators for Biological Signals with Known Spectra: Application to ECG Signal

Processing

PABLO LAGUNA, NITISH V. THAKOR, PERE CAMINAL, A N D

RAIMON JANE

Abstract-Digital low-pass filtering and differentiation (LPD) are useful in real-time processing of many biomedical signals. A general

Manuscript received October 27, 1988; revised May 23, 1989. This work was supported by the C.I.R.I.T. Generalitat de Catalunya, Spain, and Grants NS24282 and HL01509 from the National Institute of Health, and Grant 1240184 from CAICYT, Spain.

P. Laguna, P. Caminal, and R. Jane are with Institut de Cibernetica, Universitat Politecnica de Catalunya-CSIC, Barcelona, Spain.

N. V. Thakor is with the Department of Biomedical Engineering, The Johns Hopkins University, School o f Medicine, Baltimore, MD 21218.

IEEE Log Number 8933601.

method is presented for determining the coefficients of a differentiator that maximizes the signal-to-noise ratio or minimizes the error between actual and ideal LPD filters, when signal and noise spectra are known. Several examples of digital filters suitable for QRS complex and P-T wave processing in ECG are presented.

I. INTRODUCTION Biological signals usually have a band-limited spectrum. Signal

recordings in practice are corrupted by noise from biological and environmental sources. For example, ECG signal recordings may be corrupted by broad-band muscle noise. Low-pass filters that re- ject noise frequencies higher than the cutoff frequency of the signal are desirable. Differentiation of biological signal is a useful signal processing tool to extract information about rapid transients in the signal. Low-pass filtering and differentiation are usually imple- mented in two cascaded stages of digital filters [l] , [2]. Some stud- ies have been made to combine these into a single linear filter low- pass differentiator (LPD) [31-[5].

The current methods do not take into consideration signal and noise spectra. Often signal and noise spectra are known a priori, from studies of a large number of experimental recordings [6]. LPD can be optimized from the knowledge of these spectra. Another important consideration is the specific application of our LPD; we may be interested in maximizing the signal-to-noise ratio (SNR) or in preserving the differentiated signal shape while noise content is reduced. We present finite impulse response (FIR) implementa- tions of LPD designed to meet these objectives. We present theo- retical criteria for determining the best LPD coefficients and give the values obtained for: 1) QRS detection in noise, 2) P , T waves detection in noise, 3) P , T waves enhancement with respect to QRS.

11. THEORY We employ the notation of [3] where the ideal LPD frequency

response is given by (Fig. 1)

where an , (0 < a i I ) , denotes the upper limit of the differen- tiation band and j is the complex imaginary unit. For the sake of simplicity, we assume the sampling interval to be equal to one unit

~~

of time ( T = 1 ). Let fr be the sampling I

frequency of the LPD, so that

f: L a = 2 - .

The ideal filter in its most general form is m

te and f: be the cutoff

(2 )

j n w ) . (3a)

Alternately, a different base exp ( j ( n - 1 / 2 ) w ) leads to m

H ( a , 0) = c CAuexp ( J ( n - 1/2)w) (3b)

where CAT and CA,, are the coefficients of the ideal filter, which have an implicit dependence of parameter a.

The approximate FIR filter obtained with N finite number of coef- ficients can be expressed as

n = -m

N

F, (w) = j C c,,~ sin ( n u ) (4a) n = I

N

~ , ( w ) = j C c”,, sin ( ( n - 1 / 2 ) w ) n = l

where Cn>, Cna are the coefficients of the filter approximation.

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