Abstract - Fetal Phonocardiography (fPCG), digitized recording of fetal heart sound has proved very useful in the description and understanding of fetal heart sounds. It provides a visual display of the recorded waveform and allows computer aided signal processing techniques to characterize them. This paper presents an algorithm for estimation of Power Spectral Density (PSD) of the fPCG Signals. Fast Fourier Transformation (FFT) analysis is often used to transform such signals into the frequency domain but has certain technical limitations. In the presented work, the fetal heart sounds are recorded with the help of a highly sensitive Data Recording Module (DRM). These signals are de-noised using Wavelet Transform (WT) based noise suppression procedures. Averaged Periodogram based frequency domain analysis of these signals is performed in order to estimate PSD component of the fetal heart. The performance has been tested by Matlab simulation and the result shows that the proposed system provides narrow frequency peaks, permitting precise frequency identification and enhancing the ability to determine frequency changes at any time instance of the fPCG signal.
Keywords - Power spectral density, phonocardiography, wavelet de-noising, fetal heart sound.
I. INTRODUCTION
Historically, fetal heart sounds have been observed through auscultation of the fetal heart from the maternal abdominal surface, in which obstetricians listens to the heart through a stethoscope. A diagnosis of the fetal heart based on sounds heard through the stethoscope is subjective, varies from person to person and also depends on an obstetrician's experience. In order to remove subjectivity and to make fetal heart sound auscultation more quantitative a computerized and objective way would be most desirable. Fetal Phonocardiography (fPCG) is a recording of the acoustical waves produced by mechanical action of the fetal heart using modern digital signal processing techniques. In fPCG, fetal heart sound signals can be conveniently captured from the surface of mother's abdomen by placing a small and inexpensive acoustic sensor [1]. The main advantages of the fPCG technique are its passivity (non-invasiveness) and simplicity, which promises its potential for long term ambulatory home monitoring of the fetus [2].
Despite all these advantages of fPCG, this technique is not popular with the obstetricians because of its poor signal to noise ratio (SNR) at the time of recording [3]. The fPCG signals recorded from maternal abdominal surface are contaminated by various unwanted signals like maternal organ sounds, fetal movement effects and the ambient noise hence this technique requires robust signal processing to extract the fetal heart rate (FHR) signals [4]. Also, these signals exhibit marked changes with time and frequency, hence classified as non-stationary and non-deterministic signals that carry information about the anatomical and physiological state of the fetal heart. FFT analysis is generally used to transform such signals into frequency domain. This method has a few technical limitations such as: (i) use of deterministic algorithms that are valid only to periodic phenomena, (ii) necessity of windowing the data (iii) uncertainty in defining the relative powers of the various spectral components, and (iv) poor spectral resolution especially when short time frames are used [5].
In this work, Multi-resolution analysis technique based on WT is used for de-noising of the fPCG signals. Averaged Periodogram based frequency domain analysis of these signals is performed in order to estimate PSD component of the fetal heart. Matlab Simulink models are developed for de-noising and PSD estimation of the fPCG signals. The result is displayed with varying magnitude as a function of frequency, which is called power spectrum of the fPCG signals. It reflects the amplitude of the heart-rate fluctuations present at different frequencies. The generated information can be used for further diagnostic applications of the fetus.
The rest of the paper is structured as follows: A brief introduction to the wavelet transform and the PSD estimation is given in section 2. Section 3 presents the overall proposed procedure of wavelet de-noising and PSD estimation. Computer simulation methodologies and experimental results are discussed in section 4. In the end, conclusions are presented in section 5.
II. THEORETICAL BACKGROUND
Wavelet Transform:
In WT the time domain waveforms are mapped into a frequency-time domain while preserving both frequency and time information. The main idea of wavelet analysis is to measure the degree of similarity between the original waveform s(t) and the basic function of the WT also called the mother wavelet, through wavelet coefficients computation. The calculation process is performed on shifted version of the mother wavelet thus moving along the time, and on stretched or compressed version of the mother wavelet thus varying the frequency. The continuous wavelet transform (CWT) is defined as the convolution between the original signal s(t) and a wavelet .
------ (1)
Where s(t) is the input signal, a is the scaling factor, b is the translation parameter and Ψ(t) is the transforming function called mother wavelet. The mother wavelet is given by:
------ (2)
The DWT coefficients are usually sampled from the CWT on a dyadic grid, choosing parameters of translation b = n*2m and scale a = 2m. The mother wavelet in DWT is defined as:
------ (3)
DWT analyzes the signal by decomposing it into its coarse and detail information, which is accomplished by using successive high-pass and low-pass filtering operations, on the basis of the following equations:
------ (4)
------ (5)
Where and are the outputs of the high-pass and low-pass filters with impulse response h and g, respectively, after upsampling by 2 [6, 7].
2) Power Spectral Density:
Power Spectral Density Estimation is a conventional way of illustrating the signal characteristics in the frequency domain. It describes how the power of a signal is distributed with frequency. Estimation of power spectra is useful in a variety of applications, including the detection of signals buried in wide-band noise [8].
The power spectrum of a stationary random process s(t) is mathematically related to the correlation sequence by the discrete-time Fourier transform. In terms of normalized frequency, this is given by
------ (6)
This can be written as a function of physical frequency f (e.g., in hertz) by using the relation ω=2πf/fs, where fs is the sampling frequency.
------ (7)
The correlation sequence can be derived from the PSD by use of the inverse discrete-time Fourier transform:
The average power of the sequence s(t) over the entire Nyquist interval is represented by
------ (8)
The average power of a signal over a particular frequency band [ω1,ω2], 0 ≤ ω1 < ω2 ≤ π can be found by integrating the PSD over that band:
------ (9)
It can be seen from the above expression that Pxx(ω) represents the power content of a signal in an infinitesimal frequency band, hence it is called the power spectral density. The various methods of spectral estimation are categorized as follows:
Nonparametric Methods
Parametric Methods
Subspace Methods
Nonparametric methods are those in which the PSD is estimated directly from the signal itself. The simplest such method is the periodogram. An improved version of the periodogram is Welch's method. A more modern nonparametric technique is the multitaper method (MTM).
Parametric methods are those in which the PSD is estimated from a signal that is assumed to be output of a linear system driven by white noise. Examples are the Yule-Walker Autoregressive (AR) method and the Burg method. These methods estimate the PSD by first estimating the parameters (coefficients) of the linear system that hypothetically generates the signal. They tend to produce better results than classical nonparametric methods when the data length of the available signal is relatively short.
Subspace methods, also known as high-resolution methods or super-resolution methods, generate frequency component estimates for a signal based on an eigenanalysis or eigendecomposition of the correlation matrix. Examples are the multiple signal classification (MUSIC) method or the eigenvector (EV) method [9]. Do not use links to external files.
III. MATERIAL AND METHODS
The fetal heart sound signals are recorded from the maternal abdominal surface using an acoustic sensor. These recorded signals are saved in the personal computer for subsequent analysis. The further process of de-noising and PSD estimation of the recorded fPCG signal is summarized in following subsections:
Data Acquisition.
Wavelet De-noising.
Power Spectral Density Estimation.
A. Data Acquisition
The fetal heart sound is in the form of mechanical vibrations passing through tissue structures. These vibrations are relatively weak because of the physical distance and small size of the fetal heart valve [10]. In order to sense this weak heart sound from maternal abdomen, the transducer should be properly placed and its mechanical as well as electrical impedance should be matched with that of the system. To facilitate this, a highly sensitive and efficient Data Recording Module (DRM) is developed [11]. The block diagram of the data recording module is shown in Figure 1.
Fig.1. Block diagram of the Data Recording Module
The weak fPCG signals acquired from the microphone are pre-amplified, low-pass filtered and power amplified. These signals are then saved in a *.wav format of the personal computer for subsequent processing. An example of a recorded fPCG signal is shown in figure 2.
Fig.2. fPCG Signal
B. Wavelet De-noising
The WT is a two-dimensional timescale processing method for non-stationary signals with adequate scale values and shifting in time. It is capable of representing signals in different resolutions by dilating and compressing its basis functions. The main advantage of the WT is that it has a varying window size, being broad at low frequencies and narrow at high frequencies, thus leading to an optimal time-frequency resolution in all frequency ranges [12, 13].
The recorded fPCG signals are normalized to the maximum amplitude. This normalized signal is then decomposed into time-frequency representations using Discrete Wavelet Transform (DWT). The major advantage of the DWT is that it provides good time resolution. Because of its great time and frequency localization ability, the DWT can reveal the local characteristics of the input signal. The standard de-noising procedure affects the signal in both frequency and amplitude, and involves following steps:
Decomposition of the fPCG signal by applying DWT into N levels using band-pass filtering and decimation to obtain the approximation and detail coefficients. A suitable mother wavelet is selected for the decomposition of this signal.
Thresholding of these decomposition coefficients using appropriate de-noising algorithm.
Reconstruction of the fPCG signal from these thresholded detail and approximation coefficients using the inverse transform (IDWT).
In this work, the fPCG signal shown in figure 2 is decomposed to the 5 levels using fourth order Coiflets wavelet [14]. This wavelet family is characterized by its highest number of vanishing moments which is 2N. The general characteristics of this family are that it is compactly supported, orthogonal and their members are near to symmetry. The Rigorous SURE threshold algorithm with soft thresholding rule is used for the determination of threshold level [15].
C. Power Spectral Density Estimation
Estimating the power spectrum of a process is to simply find the Discrete-Time Fourier Transform of the samples of that process and take the magnitude squared of the result. This estimate is called the periodogram. An improved version of the periodogram is Welch's method. Nonparametric methods are those in which the PDS is estimated directly from the signal itself. In this work Welch's method is used for the PSD estimation of the fPCG signals. This method consists of dividing the time series data into segments, computing a modified periodogram of each segment, and then averaging the PSD estimates. A Hamming window is used to compute the modified periodogram of each segment. The averaging of modified periodograms tends to decrease the variance of the estimate relative to a single periodogram estimate of the entire data record. Although overlap between segments tends to introduce redundant information, this effect is diminished by the use of a nonrectangular window, which reduces the importance or weight given to the end samples of segments. The combined use of short data records and nonrectangular windows results in reduced resolution of the estimator. The parameters in Welch's method can be manipulated to obtain improved estimates relative to the periodogram, especially when the SNR is low. Here, the de-noised fetal heart sound signal is used as an input to the PSD estimation system. 512 samples of this signal are considered which are quite sufficient for computing power spectrum with good frequency resolution. The signal is detrended to remove slowly varying components [9]. The results are discussed in the following section.
IV. COMPUTER SIMULATION AND RESULTS
Wavelet de-noising and PSD estimation process of the fPCG signals are simulated using simulink toolbox of the matlab. The fPCG signals are recorded from the maternal abdominal surface using the DRM as discussed in the previous section. These signals are then de-noised with DWT based wavelet analysis. The simulink model for the implementation of wavelet de-noising algorithm is shown in figure 3.
Fig.3. Simulink Model of Wavelet De-noising Algorithm
The recorded fPCG signal is applied as an input from corresponding *.wav file, which is shown in figure 2. In this model, the Dyadic Analysis Filter Bank block decomposes the fPCG signal into a collection of subbands with smaller bandwidths and slower sample rates [16]. This block uses a series of highpass and lowpass FIR filters to repeatedly divide the input frequency range. The fPCG signal is decomposed to the 5 levels of fourth order Coiflets wavelet with Dyadic analysis filter bank. The rigorous SURE threshold algorithm with soft thresholding rule is adopted for the determination of threshold level. The Dyadic Synthesis Filter Bank block reconstructs the signal decomposed by the Dyadic Analysis Filter Bank block. This block takes in subbands of this signal, and uses them to reconstruct the signal by using a series of highpass and lowpass FIR filters. The reconstructed signal has a wider bandwidth and faster sample rate than the input subbands. The Vector Scope block is a comprehensive display tool similar to a digital oscilloscope. It is used to display time-domain response of the de-noised fPCG signal. The resultant waveform after the application of wavelet de-noising algorithm to a recorded fPCG signal (refer figure 2) is shown in figure 4.
Fig.4. Waveform of the De-noised fPCG Signal
The de-noised fPCG signal is again saved in *.wav file of the computer. This is then fed to the PSD estimation system. The simulink model for the implementation of PSD estimation is shown in figure 5. This system consists of a periodogram, buffers, db conversion, matrix transpose, selector and vector scope blocks [16]. The periodogram block computes a non-parametric estimate of the spectrum. This block averages the squared magnitude of the FFT computed over windowed sections of the input fPCG signal and normalizes the spectral average by the square of the sum of the window samples. The Buffer block redistributes the input samples to a new frame size, larger or smaller than the input frame size. Buffering to a larger frame size yields an output with a slower frame rate than the input. The dB Conversion block converts a linearly scaled power or amplitude input to dB. The Selector block generates as output selected elements of an input vector or matrix. The Vector Scope block is used to display frequency-domain response of the de-noised fPCG signal.
Fig.5. Simulink Model of PSD Estimation Algorithm
The experimental trials are carried out in a local hospital on 34 pregnant women with gestation age from 36th to 40th weeks. For each trail a 10 minutes data is captured and processed. Figure 2 shows an example of the fPCG signal. This signal is recorded on the abdominal surface of a woman with 38th week of pregnancy. Figure 4 is the de-noised form of this signal. It is observed that after applying the proposed wavelet de-noising technique, the SNR of the signal is improved effectively. Figure 6(a) & 6(b) is the PSD estimation and color spectrogram of the fPCG signal respectively.
Fig.6(a). PSD Estimation of the fPCG Signal
Fig.6(b). Color Spectrogram of the fPG Signal
The PSD estimate of figure 6(a) clearly displays fundamental harmonic which is dominant peak of the fPCG signal. The other harmonics are also displayed in the estimate. Figure 6(b) shows the color spectrogram of this signal. Spectrogram is color-based visualizations of the power spectrum of a sound signal, as the signal is swept through the time axis. The quantization of power concentration in complete frequency band provides useful tool in monitoring and diagnosing the fetal heart.
V. CONCLUSION
Monitoring the FHR variations has been considered as an indispensable procedure for fetal surveillance. Fetal Phonocardiography, with few inherited limitations, is found to be most suitable method for recording and processing of the fPCG signals. This technique is simple, non-invasive, inexpensive and can be used for long term measurement of FHR. In this work, the fPCG signals are acquired from the maternal abdominal surface using the DRM. These signals are then de-noised with the help of DWT based wavelet analysis. The fourth order Coiflets wavelet is used as mother wavelet. The level of threshold is determined by applying Rigorous SURE threshold algorithm with soft thresholding rule. The fPCG signals obtained from the wavelet de-noising are used as an input to the PSD estimation algorithm. Averaged Periodogram based frequency domain analysis of these signals is performed in order to estimate PSD component of the fetal heart. Matlab simulation models are developed for the implementation of wavelet de-noising and PSD estimation of the fPCG signal. The results are displayed in the form of spectral estimation of this signal along with corresponding color spectogram. These results obtained from the above mentioned wavelet de-noising and PSD estimation of fetal heart rate time series are capable of complementing the clinical examination results and leads to a better diagnosis of the fetus. Also, this will help to reinstate the fPCG technology for fetal monitoring which is found to be most suitable one for home care applications. The development of fPCG based prenatal diagnostic system for home monitoring application is the future object of the author.
ACKNOWLEDGMENT
The fetal heart sound recordings were done at district government women hospital and at Ratnaparkhi Nursing Home Gondia (M.S.). The authors of this paper would also like to thank Dr. Shrish Ratnaparkhi (Gynecologist) and Dr. (Mrs) Megha Ratnaparkhi (Obstetrician) for their kind support in carrying out observations with the help of developed prototype instrument. Pregnant women who volunteered to participate in clinical test are also appreciated for their kind gesture.
