- Email: [email protected]
Comparison of FFT and adaptive ARMA methods in transcranial Doppler signals recorded from the cerebral vessels
Computers in Biology and Medicine 32 (2002) 445 – 453 www.elsevier.com/locate/compbiomed
Comparison of FFT and adaptive ARMA methods in transcranial Doppler signals recorded from the cerebral vessels ˙ Inan G+ulera; ∗ , F-rat Hardala/ca , Memduh Kaymazb a
Department of Electronics and Computer Education, Biomedical Engineering Group, Faculty of Technical Education, Gazi University, 06500 Teknikokullar, Ankara, Turkey b Department of Neurosurgery, Faculty of Medicine, Gazi University, 06500 Bes%evler, Ankara, Turkey Received 9 November 2001; received in revised form 6 June 2002
Abstract In this work, transcranial Doppler signals recorded from the temporal region of the brain on 35 patients were transferred to a personal computer by using a 16-bit sound card. Fast Fourier transform and adaptive auto regressive-moving average (A-ARMA) methods were applied to transcranial Doppler frequencies obtained from the middle cerebral artery in the temporal region. Spectral analyses were obtained to compare both methods for medical diagnoses. The sonograms obtained using A-ARMA method give better results for spectral resolution than the FFT method. The sonograms of A-ARMA method o
1. Introduction Transcranial Doppler study of the adult intracerebral circulation has increased enormously in the last 14 years. Transcranial Doppler has been used to evaluate intracranial stenoses, cerebral arteriovenous malformations, cerebral vasospasm, and cerebral hemodynamics in general [1]. When the bifurcation of a cerebral vessel has a structural defect or aneurysm, it is required to use more sophisticated signal processing methods. ∗
Corresponding author. Tel.: +90-312-212-3976; fax: +90-312-212-0059. ˙ G+uler). E-mail address: [email protected] (I.
0010-4825/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 1 0 - 4 8 2 5 ( 0 2 ) 0 0 0 3 6 - 7
I˙. G+uler et al. / Computers in Biology and Medicine 32 (2002) 445 – 453
446
It is easy to make simple measurements of major blood vessels, but intra-brain blood =ow measurements require sophisticated imaging methods. Utilizing the Doppler principle in medicine varies slightly from classical Doppler method in that the targets do not spontaneously emit a radiation. It is therefore necessary to transmit a signal into the body to observe the changes in frequency that occur when it is re=ected or scattered from the targets. It can be shown that under these conditions there is a shift in the ultrasound frequency given by fd = ft − fr = 2ft v cos =c;
(1)
where ft and fr are the transmitted and received ultrasound frequencies, respectively, v the velocity of the target, c the velocity of sound in the medium, and the angle with ultrasound beam and the direction of motion of the target. Since the Doppler frequency is not a single frequency, the suitable signal processing method should be chosen. The suitable signal analysis method should meet the following conditions [2–9]: the spectral envelope should be clearly obtained, there should be no noise frequency components in the sonogram, the resolution of sonograms should be high, and Nnally sonograms should provide some information about blood speed and pressure for the clinicians. The qualities of these sonograms have great importance for clinicians to make an accurate diagnosis. In this work, Transcranial Doppler signals recorded from the middle cerebral arteries of the temporal region of 35 patients were transferred to a personal computer (PC) by using a 16-bit sound card. FFT and A-ARMA methods were applied to the recorded signals in order to obtain their sonograms. The sonograms were compared with magnetic resonance imaging (MRI) records for veriNcation of the proposed work.
2. Materials and method 2.1. Hardware The hardware of the system consists of four units as shown in Fig. 1. There is a 2 MHz ultrasound transducer, analog Doppler unit (Multi Doppler Transducer XX, DWL Gmb, Uberlingen, Germany), analog=digital interface board (Sound Blaster Pro-16), and PIII 600 MHz microprocessor PC with printer. The Doppler unit is also equipped with imaging software that makes it possible to focus the sample volume at a desired location in the temporal region. The analog Doppler unit can work in both continuous and pulse wave modes. The sample volume sizes can be adjusted according to vessel diameter for recording di
Analog Doppler Unit
Analog/Digital Interface Board
Fig. 1. Block diagram of measurement system.
Computer &Printer
I˙. G+uler et al. / Computers in Biology and Medicine 32 (2002) 445 – 453
447
2.2. Spectral analysis of Doppler signal Spectral analysis methods are widely used to investigate the information from Doppler blood =ow signals. Spectral analysis methods examine the Transcranial Doppler signal against time. Transcranial Doppler signals obtained from the middle cerebral artery are sampled and grouped in suitable frames. The most commonly used frame lengths are 64; 128 and 256. After the framing process, a power spectral density P(f) of each frame can be calculated using FFT and A-ARMA methods. These are combined to construct three-dimensional sonograms. The horizontal axis represents time (t) and the vertical axis represents frequency (f). The gray scale of the diagrams shows the power of the frequency component of the graph P(f). As the gray scale turns into black, it means that the power of related frequency component is increasing, otherwise it is decreasing [6]. The frequency content of the signal will determine the sampling rate to be used, since the maximum frequency analyzed is half of the sampling frequency. Since the maximum frequency is not known a priori and depends upon the vessel on which the measurements are performed, it is useful to implement the system such that it is capable of operating at various sampling frequencies. The available sampling frequencies in the system are 2:56; 5:12, and 10:24 kHz while the expected transcranial Doppler frequencies change between 0:5 and 5:0 kHz even in stenoses. In general, the Doppler signal is nonstationary. The signal may be assumed to be stationary for 10 ms or greater time periods if the =ow is laminar and the velocity is not very high. However, this assumption is not valid for high-velocity turbulent =ows such as vessel narrowing where the Doppler spectrum changes very rapidly. In this case, the frame length should be shortened to validate the above assumption. On the other hand, very short frame lengths may yield statistically poor spectral resolution. Therefore, selection of frame length is an important factor in Doppler spectral analysis. The frame length used in this work is 128. 2.3. Fast Fourier transform method In order to take the FFT of Nnite transcranial Doppler signal, an integer power of 2, it must be framed such as 64, 128, 256. Windowing is applied to the frequency spectrum of the frame. Windowing prevents the nonexisting frequency components from appearing in the spectrum. In addition, zero padding is applied to the same signal after the windowing process. This creates a certain overhead on the process, although it increases readability of the spectrum. Discrete Fourier transform of a discrete time periodic signal is deNned as the following: N −1 Xk = x(n)e(−jkn(2=N )) ; (2) n=0
where Xk is expressed as discrete Fourier coeQcients, N is the frame size and x(n) is the input signal on time domain. To obtain the frequency spectrum of this signal, logarithmic values of the squares of absolute values of Xk are found as shown below P(k) = 10 log |Xk |2 :
(3)
However, the FFT method becomes insuQcient for recording transcranial blood =ow signal which is generally nonstationary. In order to assume the transcranial blood =ow signal is stationary, the frame size of the window should be shortened. FFT and other classical spectral analysis methods give the
I˙. G+uler et al. / Computers in Biology and Medicine 32 (2002) 445 – 453
448
best result when the frame size is larger. When the middle cerebral artery has an aneurysm, there is turbulent =ow. In this case the performance of FFT method decreases, spectrum is broadening, and frequency resolution is poor. These drawbacks force the use of the A-ARMA method for transcranial Doppler signal processing [6 –8,10]. 2.4. Adaptive ARMA method The parametric methods are based on modeling the data sequence x(n) as the output of a linear system characterized by a rational system function of the form [10] q −k B(z) k=0 ck z H (z) = = ; (4) p A(z) 1 + k=1 ak z −k where ak (k = 1; 2; : : : ; p) and ck ; (k = 1; 2; : : : ; q) are called ARMA parameters. The random process x(n) generated by the pole-zero model in Eq. (4) is called an ARMA process of order (p; q) and it is usually denoted as ARMA (p; q). If q = 0 and b0 = 1, the resulting system model has a system function H (z) = 1=A(z) and its output x(n) is called an autoregressive (AR) process of order p. This is denoted as AR(p). The third possible model is obtained by setting A(z) = 1, so that H (z) = B(z). Its output x(n) is called a moving average (MA) process of order q and denoted as MA(q). Of these three linear models, the AR model is by far the most widely used. The reasons are twofold. First, the AR model is suitable for representing spectra with narrow peaks. Second, the AR model results in very simple linear equations for the AR parameters. On the other hand, the MA model requires many more coeQcients to represent a narrow spectrum. Consequently, it is rarely used by itself as a model for spectrum estimation. By combining poles and zeroes, the ARMA model provides a more eQcient representation, from the viewpoint of the number of model parameters, of the spectrum of a random process [10]. The adaptive ARMA spectral analysis is a parametric method. In this method, it is assumed that a signal sj (j is time instant) can be generated by exciting an ARMA process using a random time series, nj . A time series sj can be modeled as an ARMA process as the following: p q sJ = − ak sj − k + ck nj − k + n j ; k=1
(5)
k=1
where nj is a white noise process. The ARMA parameters ak and ck are calculated by Burg and Levinson methods. Once the parameters of the ARMA model are identiNed, the power spectrum of the signal, sj , can be calculated from these ARMA parameters as the following [7]: 2 |1 + qk=1 ck exp(−i!k)|2 ; (6) Ps (w) = |1 + pk=1 ak exp(−i!k)|2 √ where 2 is the variance of the white noise, i = −1, and ! is the angular frequency. To model the transcranial Doppler signal, xj , one simply proceeds in the opposite direction and constructs a so-called A-ARMA Nlter, as shown in Fig. 2. In Fig. 2, z −1 stands for one sample delay. xj is the input signal at time j that is to be analyzed. The sets akj and ckj are, respectively,
I˙. G+uler et al. / Computers in Biology and Medicine 32 (2002) 445 – 453 Xj
+
449
ej
z-1 z-1
a1j
z-1 . . .
a2j +
c1j
a3j +
c2j z-1
z-1
apj +
...
c3j z-1
z-1
+
yj
cqj ...
z-1
Fig. 2. Adaptive ARMA Nlters detail structure.
the feedforward and feedback weights of the A-ARMA Nlter. In Fig. 2, yj is the estimate of the input signal by the ARMA Nlter and is expressed as the following [7,11–13]: p p yj = akj xj−k + ckj ej−k ; (7) k=1
k=1
where ej is the estimation error expressed as ej = xj − yj :
(8)
Eqs. (7) and (8) imply that yj is an ARMA estimate of xj signal if the error signal ej is a white noise. After the convergence of the ARMA Nlter weights, the power spectrum of the signal xj can be estimated. A least mean square (LMS) algorithm for the adaptation of the weights of the adaptive ARMA Nlter is described as the following: In this case, the output of adaptive ARMA Nlter yj can be written as yj = ATj Xj + CjT Ej
(9)
and the error signal as ej = xj − ATj Xj − CjT Ej
(10)
where Aj and Cj are feedforward and feedback weight vectors, respectively, and Xj and Ej are feedforward and feedback input vectors, respectively. These vectors are deNned as the following: ATj = [a1; j ; : : : ; ap; j ];
(11)
CjT = [c1; j ; : : : ; cq; j ];
(12)
XjT = [xj−1 ; : : : ; xj−p ];
(13)
EjT = [ej−1 ; : : : ; ej−q ]:
(14)
450
I˙. G+uler et al. / Computers in Biology and Medicine 32 (2002) 445 – 453
Applying the steepest descent method and approximating the gradient of the mean square error with respect to the Nlter weight vectors by the gradient of the squared error, we have @Ej2 Aj+1 = Aj − "a ; (15) @Aj @Ej2 : (16) Cj+1 = Cj − "c @Cj To have a simple adaptation algorithm, we assume that the feedback input vector Ej is not a function of the Nlter parameters. Under this assumption, the LMS algorithm can be easily derived and is expressed as the following: Aj+1 = Aj + 2"a ej Xj ;
(17)
Cj+1 = Cj + 2"c ej Ej ;
(18)
or ak; j+1 = ak; j + 2"a ej xj−k ;
k = 1; 2; : : : ; p;
(19)
ck; j+1 = ck; j + 2"c ej ej−k ;
k = 1; 2; : : : ; q;
(20)
where step-sizes "a and "c are small constants controlling the adaptation speed of the LMS algorithm. The algorithm states that the Nlter parameters at each successive time step, ak; j+1 and ck; j+1 are equal to their current values, akj and ckj , plus a modiNcation term. The number of Nlter parameters is equal to q + p. 3. Results and discussion It is easier to follow the frequency components of the Doppler signal by forming a sonogram. The envelope of the sonogram is the maximum frequency curve to investigate di
I˙. G+uler et al. / Computers in Biology and Medicine 32 (2002) 445 – 453
451
Table 1 Statistics of 35 patients Number Age
Sex
Resistive Index
Diagnosis
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35
Female Female Male Male Male Female Female Female Male Male Female Female Male Male Female Female Female Female Male Male Male Male Female Female Female Male Male Male Male Female Female Male Male Male Male
0.67 0.66 0.68 0.67 0.65 0.70 0.68 0.69 0.69 0.70 0.62 0.61 0.61 0.60 0.79 0.80 0.79 0.78 0.81 0.78 0.79 0.78 0.72 0.71 0.70 0.75 0.73 0.72 0.74 0.62 0.61 0.63 0.60 0.61 0.63
Aneurysm Aneurysm Aneurysm Aneurysm Aneurysm Hemorrhage Hemorrhage Hemorrhage Hemorrhage Hemorrhage In=ammation In=ammation In=ammation In=ammation Oedema Oedema Oedema Oedema Oedema Oedema Oedema Oedema Trauma Trauma Trauma Trauma Trauma Trauma Trauma Tumour Tumour Tumour Tumour Tumour Tumour
57 65 55 58 62 25 30 36 21 23 65 71 60 67 3 13 21 38 22 30 34 40 12 22 30 28 35 38 41 47 56 48 52 53 55
In order to calculate the cerebral pressure before and after the surgery operation accurately, the peak and valley of systole and diastole curves should be clearly imaged on the sonograms. Transcranial Doppler results and resistive index values of 35 patients are given in Table 1. These data have been measured at the State Hospital in Ankara. As shown in Table 1, 5 of 35 patients have an aneurysm, 7 patients have had brain trauma, 6 patients have a tumour, 4 patients have in=ammation, 8 patients have oedema and 5 patients have hemorrhage. Normal resistive index of adults is less than 60%. If the patient has an aneurysm, (s)he is likely to be hypertensive. This causes high blood velocity in middle cerebral arteries in the temporal region. For this reason, the di
452
I˙. G+uler et al. / Computers in Biology and Medicine 32 (2002) 445 – 453
Fig. 3. Doppler sonogram recorded from a 58 yr old patient with an aneurysm: (a) FFT method, (b) Adaptive ARMA method.
Fig. 4. Doppler sonogram recorded from a 58 yr old patient with an aneurysm: (a) FFT method (three-dimensional), (b) Adaptive ARMA method (three-dimensional).
have higher values than controls with no di
I˙. G+uler et al. / Computers in Biology and Medicine 32 (2002) 445 – 453
453
sonogram shown in Fig. 4b provides a better spectral resolution in the case of instantaneous frequency than the FFT sonogram shown in Fig. 4a. 4. Conclusion The main purpose of this work is to increase the spectral resolution of transcranial Doppler signals when used for cerebral vessels. This is very useful when blood velocity in cerebral arteries is turbulent. The A-ARMA method is more useful than FFT for processing of blood =ow signals in cerebral arteries. References [1] D.H. Evans, W.N. McDicken, R. Skidmore, J.P. Woodcock, Doppler Ultrasound: Physics, Instrumentation and Clinical Applications, Wiley, Chichester, 1989. [2] N.F. G+uler, M.K. K-ym-k, I˙ . G+uler, Comparison of FFT and AR-based sonogram outputs of 20 MHz pulsed Doppler data in real time, Comp. Biol. Med. 25 (1995) 383–391. [3] F.S. Schlindwein, D.H. Evans, A real time autoregressive spectrum analyzer for Doppler ultrasound signals, Ultrasound Med. Biol. 15 (1989) 263–272. ˙ G+uler, F. Hardala/c, S. M+uld+ur, Determination of aorta failure with the application of FFT, AR and wavelet methods [4] I. to Doppler technique, Comp. Biol. Med. 31 (2001) 229–238. [5] M.K. Steven, Modern Spectral Estimation, Signal Processing Series, Prentice-Hall, Englewood Cli
Comparison of FFT and adaptive ARMA methods in transcranial Doppler signals recorded from the cerebral vessels ˙ Inan G+ulera; ∗ , F-rat Hardala/ca , Memduh Kaymazb a
Department of Electronics and Computer Education, Biomedical Engineering Group, Faculty of Technical Education, Gazi University, 06500 Teknikokullar, Ankara, Turkey b Department of Neurosurgery, Faculty of Medicine, Gazi University, 06500 Bes%evler, Ankara, Turkey Received 9 November 2001; received in revised form 6 June 2002
Abstract In this work, transcranial Doppler signals recorded from the temporal region of the brain on 35 patients were transferred to a personal computer by using a 16-bit sound card. Fast Fourier transform and adaptive auto regressive-moving average (A-ARMA) methods were applied to transcranial Doppler frequencies obtained from the middle cerebral artery in the temporal region. Spectral analyses were obtained to compare both methods for medical diagnoses. The sonograms obtained using A-ARMA method give better results for spectral resolution than the FFT method. The sonograms of A-ARMA method o
1. Introduction Transcranial Doppler study of the adult intracerebral circulation has increased enormously in the last 14 years. Transcranial Doppler has been used to evaluate intracranial stenoses, cerebral arteriovenous malformations, cerebral vasospasm, and cerebral hemodynamics in general [1]. When the bifurcation of a cerebral vessel has a structural defect or aneurysm, it is required to use more sophisticated signal processing methods. ∗
Corresponding author. Tel.: +90-312-212-3976; fax: +90-312-212-0059. ˙ G+uler). E-mail address: [email protected] (I.
0010-4825/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 1 0 - 4 8 2 5 ( 0 2 ) 0 0 0 3 6 - 7
I˙. G+uler et al. / Computers in Biology and Medicine 32 (2002) 445 – 453
446
It is easy to make simple measurements of major blood vessels, but intra-brain blood =ow measurements require sophisticated imaging methods. Utilizing the Doppler principle in medicine varies slightly from classical Doppler method in that the targets do not spontaneously emit a radiation. It is therefore necessary to transmit a signal into the body to observe the changes in frequency that occur when it is re=ected or scattered from the targets. It can be shown that under these conditions there is a shift in the ultrasound frequency given by fd = ft − fr = 2ft v cos =c;
(1)
where ft and fr are the transmitted and received ultrasound frequencies, respectively, v the velocity of the target, c the velocity of sound in the medium, and the angle with ultrasound beam and the direction of motion of the target. Since the Doppler frequency is not a single frequency, the suitable signal processing method should be chosen. The suitable signal analysis method should meet the following conditions [2–9]: the spectral envelope should be clearly obtained, there should be no noise frequency components in the sonogram, the resolution of sonograms should be high, and Nnally sonograms should provide some information about blood speed and pressure for the clinicians. The qualities of these sonograms have great importance for clinicians to make an accurate diagnosis. In this work, Transcranial Doppler signals recorded from the middle cerebral arteries of the temporal region of 35 patients were transferred to a personal computer (PC) by using a 16-bit sound card. FFT and A-ARMA methods were applied to the recorded signals in order to obtain their sonograms. The sonograms were compared with magnetic resonance imaging (MRI) records for veriNcation of the proposed work.
2. Materials and method 2.1. Hardware The hardware of the system consists of four units as shown in Fig. 1. There is a 2 MHz ultrasound transducer, analog Doppler unit (Multi Doppler Transducer XX, DWL Gmb, Uberlingen, Germany), analog=digital interface board (Sound Blaster Pro-16), and PIII 600 MHz microprocessor PC with printer. The Doppler unit is also equipped with imaging software that makes it possible to focus the sample volume at a desired location in the temporal region. The analog Doppler unit can work in both continuous and pulse wave modes. The sample volume sizes can be adjusted according to vessel diameter for recording di
Analog Doppler Unit
Analog/Digital Interface Board
Fig. 1. Block diagram of measurement system.
Computer &Printer
I˙. G+uler et al. / Computers in Biology and Medicine 32 (2002) 445 – 453
447
2.2. Spectral analysis of Doppler signal Spectral analysis methods are widely used to investigate the information from Doppler blood =ow signals. Spectral analysis methods examine the Transcranial Doppler signal against time. Transcranial Doppler signals obtained from the middle cerebral artery are sampled and grouped in suitable frames. The most commonly used frame lengths are 64; 128 and 256. After the framing process, a power spectral density P(f) of each frame can be calculated using FFT and A-ARMA methods. These are combined to construct three-dimensional sonograms. The horizontal axis represents time (t) and the vertical axis represents frequency (f). The gray scale of the diagrams shows the power of the frequency component of the graph P(f). As the gray scale turns into black, it means that the power of related frequency component is increasing, otherwise it is decreasing [6]. The frequency content of the signal will determine the sampling rate to be used, since the maximum frequency analyzed is half of the sampling frequency. Since the maximum frequency is not known a priori and depends upon the vessel on which the measurements are performed, it is useful to implement the system such that it is capable of operating at various sampling frequencies. The available sampling frequencies in the system are 2:56; 5:12, and 10:24 kHz while the expected transcranial Doppler frequencies change between 0:5 and 5:0 kHz even in stenoses. In general, the Doppler signal is nonstationary. The signal may be assumed to be stationary for 10 ms or greater time periods if the =ow is laminar and the velocity is not very high. However, this assumption is not valid for high-velocity turbulent =ows such as vessel narrowing where the Doppler spectrum changes very rapidly. In this case, the frame length should be shortened to validate the above assumption. On the other hand, very short frame lengths may yield statistically poor spectral resolution. Therefore, selection of frame length is an important factor in Doppler spectral analysis. The frame length used in this work is 128. 2.3. Fast Fourier transform method In order to take the FFT of Nnite transcranial Doppler signal, an integer power of 2, it must be framed such as 64, 128, 256. Windowing is applied to the frequency spectrum of the frame. Windowing prevents the nonexisting frequency components from appearing in the spectrum. In addition, zero padding is applied to the same signal after the windowing process. This creates a certain overhead on the process, although it increases readability of the spectrum. Discrete Fourier transform of a discrete time periodic signal is deNned as the following: N −1 Xk = x(n)e(−jkn(2=N )) ; (2) n=0
where Xk is expressed as discrete Fourier coeQcients, N is the frame size and x(n) is the input signal on time domain. To obtain the frequency spectrum of this signal, logarithmic values of the squares of absolute values of Xk are found as shown below P(k) = 10 log |Xk |2 :
(3)
However, the FFT method becomes insuQcient for recording transcranial blood =ow signal which is generally nonstationary. In order to assume the transcranial blood =ow signal is stationary, the frame size of the window should be shortened. FFT and other classical spectral analysis methods give the
I˙. G+uler et al. / Computers in Biology and Medicine 32 (2002) 445 – 453
448
best result when the frame size is larger. When the middle cerebral artery has an aneurysm, there is turbulent =ow. In this case the performance of FFT method decreases, spectrum is broadening, and frequency resolution is poor. These drawbacks force the use of the A-ARMA method for transcranial Doppler signal processing [6 –8,10]. 2.4. Adaptive ARMA method The parametric methods are based on modeling the data sequence x(n) as the output of a linear system characterized by a rational system function of the form [10] q −k B(z) k=0 ck z H (z) = = ; (4) p A(z) 1 + k=1 ak z −k where ak (k = 1; 2; : : : ; p) and ck ; (k = 1; 2; : : : ; q) are called ARMA parameters. The random process x(n) generated by the pole-zero model in Eq. (4) is called an ARMA process of order (p; q) and it is usually denoted as ARMA (p; q). If q = 0 and b0 = 1, the resulting system model has a system function H (z) = 1=A(z) and its output x(n) is called an autoregressive (AR) process of order p. This is denoted as AR(p). The third possible model is obtained by setting A(z) = 1, so that H (z) = B(z). Its output x(n) is called a moving average (MA) process of order q and denoted as MA(q). Of these three linear models, the AR model is by far the most widely used. The reasons are twofold. First, the AR model is suitable for representing spectra with narrow peaks. Second, the AR model results in very simple linear equations for the AR parameters. On the other hand, the MA model requires many more coeQcients to represent a narrow spectrum. Consequently, it is rarely used by itself as a model for spectrum estimation. By combining poles and zeroes, the ARMA model provides a more eQcient representation, from the viewpoint of the number of model parameters, of the spectrum of a random process [10]. The adaptive ARMA spectral analysis is a parametric method. In this method, it is assumed that a signal sj (j is time instant) can be generated by exciting an ARMA process using a random time series, nj . A time series sj can be modeled as an ARMA process as the following: p q sJ = − ak sj − k + ck nj − k + n j ; k=1
(5)
k=1
where nj is a white noise process. The ARMA parameters ak and ck are calculated by Burg and Levinson methods. Once the parameters of the ARMA model are identiNed, the power spectrum of the signal, sj , can be calculated from these ARMA parameters as the following [7]: 2 |1 + qk=1 ck exp(−i!k)|2 ; (6) Ps (w) = |1 + pk=1 ak exp(−i!k)|2 √ where 2 is the variance of the white noise, i = −1, and ! is the angular frequency. To model the transcranial Doppler signal, xj , one simply proceeds in the opposite direction and constructs a so-called A-ARMA Nlter, as shown in Fig. 2. In Fig. 2, z −1 stands for one sample delay. xj is the input signal at time j that is to be analyzed. The sets akj and ckj are, respectively,
I˙. G+uler et al. / Computers in Biology and Medicine 32 (2002) 445 – 453 Xj
+
449
ej
z-1 z-1
a1j
z-1 . . .
a2j +
c1j
a3j +
c2j z-1
z-1
apj +
...
c3j z-1
z-1
+
yj
cqj ...
z-1
Fig. 2. Adaptive ARMA Nlters detail structure.
the feedforward and feedback weights of the A-ARMA Nlter. In Fig. 2, yj is the estimate of the input signal by the ARMA Nlter and is expressed as the following [7,11–13]: p p yj = akj xj−k + ckj ej−k ; (7) k=1
k=1
where ej is the estimation error expressed as ej = xj − yj :
(8)
Eqs. (7) and (8) imply that yj is an ARMA estimate of xj signal if the error signal ej is a white noise. After the convergence of the ARMA Nlter weights, the power spectrum of the signal xj can be estimated. A least mean square (LMS) algorithm for the adaptation of the weights of the adaptive ARMA Nlter is described as the following: In this case, the output of adaptive ARMA Nlter yj can be written as yj = ATj Xj + CjT Ej
(9)
and the error signal as ej = xj − ATj Xj − CjT Ej
(10)
where Aj and Cj are feedforward and feedback weight vectors, respectively, and Xj and Ej are feedforward and feedback input vectors, respectively. These vectors are deNned as the following: ATj = [a1; j ; : : : ; ap; j ];
(11)
CjT = [c1; j ; : : : ; cq; j ];
(12)
XjT = [xj−1 ; : : : ; xj−p ];
(13)
EjT = [ej−1 ; : : : ; ej−q ]:
(14)
450
I˙. G+uler et al. / Computers in Biology and Medicine 32 (2002) 445 – 453
Applying the steepest descent method and approximating the gradient of the mean square error with respect to the Nlter weight vectors by the gradient of the squared error, we have @Ej2 Aj+1 = Aj − "a ; (15) @Aj @Ej2 : (16) Cj+1 = Cj − "c @Cj To have a simple adaptation algorithm, we assume that the feedback input vector Ej is not a function of the Nlter parameters. Under this assumption, the LMS algorithm can be easily derived and is expressed as the following: Aj+1 = Aj + 2"a ej Xj ;
(17)
Cj+1 = Cj + 2"c ej Ej ;
(18)
or ak; j+1 = ak; j + 2"a ej xj−k ;
k = 1; 2; : : : ; p;
(19)
ck; j+1 = ck; j + 2"c ej ej−k ;
k = 1; 2; : : : ; q;
(20)
where step-sizes "a and "c are small constants controlling the adaptation speed of the LMS algorithm. The algorithm states that the Nlter parameters at each successive time step, ak; j+1 and ck; j+1 are equal to their current values, akj and ckj , plus a modiNcation term. The number of Nlter parameters is equal to q + p. 3. Results and discussion It is easier to follow the frequency components of the Doppler signal by forming a sonogram. The envelope of the sonogram is the maximum frequency curve to investigate di
I˙. G+uler et al. / Computers in Biology and Medicine 32 (2002) 445 – 453
451
Table 1 Statistics of 35 patients Number Age
Sex
Resistive Index
Diagnosis
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35
Female Female Male Male Male Female Female Female Male Male Female Female Male Male Female Female Female Female Male Male Male Male Female Female Female Male Male Male Male Female Female Male Male Male Male
0.67 0.66 0.68 0.67 0.65 0.70 0.68 0.69 0.69 0.70 0.62 0.61 0.61 0.60 0.79 0.80 0.79 0.78 0.81 0.78 0.79 0.78 0.72 0.71 0.70 0.75 0.73 0.72 0.74 0.62 0.61 0.63 0.60 0.61 0.63
Aneurysm Aneurysm Aneurysm Aneurysm Aneurysm Hemorrhage Hemorrhage Hemorrhage Hemorrhage Hemorrhage In=ammation In=ammation In=ammation In=ammation Oedema Oedema Oedema Oedema Oedema Oedema Oedema Oedema Trauma Trauma Trauma Trauma Trauma Trauma Trauma Tumour Tumour Tumour Tumour Tumour Tumour
57 65 55 58 62 25 30 36 21 23 65 71 60 67 3 13 21 38 22 30 34 40 12 22 30 28 35 38 41 47 56 48 52 53 55
In order to calculate the cerebral pressure before and after the surgery operation accurately, the peak and valley of systole and diastole curves should be clearly imaged on the sonograms. Transcranial Doppler results and resistive index values of 35 patients are given in Table 1. These data have been measured at the State Hospital in Ankara. As shown in Table 1, 5 of 35 patients have an aneurysm, 7 patients have had brain trauma, 6 patients have a tumour, 4 patients have in=ammation, 8 patients have oedema and 5 patients have hemorrhage. Normal resistive index of adults is less than 60%. If the patient has an aneurysm, (s)he is likely to be hypertensive. This causes high blood velocity in middle cerebral arteries in the temporal region. For this reason, the di
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Fig. 3. Doppler sonogram recorded from a 58 yr old patient with an aneurysm: (a) FFT method, (b) Adaptive ARMA method.
Fig. 4. Doppler sonogram recorded from a 58 yr old patient with an aneurysm: (a) FFT method (three-dimensional), (b) Adaptive ARMA method (three-dimensional).
have higher values than controls with no di
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sonogram shown in Fig. 4b provides a better spectral resolution in the case of instantaneous frequency than the FFT sonogram shown in Fig. 4a. 4. Conclusion The main purpose of this work is to increase the spectral resolution of transcranial Doppler signals when used for cerebral vessels. This is very useful when blood velocity in cerebral arteries is turbulent. The A-ARMA method is more useful than FFT for processing of blood =ow signals in cerebral arteries. References [1] D.H. Evans, W.N. McDicken, R. Skidmore, J.P. Woodcock, Doppler Ultrasound: Physics, Instrumentation and Clinical Applications, Wiley, Chichester, 1989. [2] N.F. G+uler, M.K. K-ym-k, I˙ . G+uler, Comparison of FFT and AR-based sonogram outputs of 20 MHz pulsed Doppler data in real time, Comp. Biol. Med. 25 (1995) 383–391. [3] F.S. Schlindwein, D.H. Evans, A real time autoregressive spectrum analyzer for Doppler ultrasound signals, Ultrasound Med. Biol. 15 (1989) 263–272. ˙ G+uler, F. Hardala/c, S. M+uld+ur, Determination of aorta failure with the application of FFT, AR and wavelet methods [4] I. to Doppler technique, Comp. Biol. Med. 31 (2001) 229–238. [5] M.K. Steven, Modern Spectral Estimation, Signal Processing Series, Prentice-Hall, Englewood Cli