OPTIMAL SCALING VALUES FOR TIME-FREQUENCY DISTRIBUTIONS IN DOPPLER ULTRASOUND BLOOD FLOW MEASUREMENT

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1 OPTIMAL SCALING VALUES FOR TIME-FREQUENCY DISTRIBUTIONS IN DOPPLER ULTRASOUND BLOOD FLOW MEASUREMENT F. García Nocetti, J. Solano González, E. Rubio Acosta Departamento de Ingeniería de Sistemas Computacionales y Automatización, Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, Universidad Nacional Autónoma de México, Circuito Escolar S/N, Ciudad Universitaria, México D. F., 04510, México Contacting fabian.garcia@iimas.unam.mx Abstract: Time-frequency distributions (TFD) are an alternative for signal analysis associated to Doppler ultrasound blood flow measurement, since these do not suppose that the signal is stationary. TFD have a scaling factor with which an optimization problem to get spectral estimations can be proposed. The optimization problem can be solved analytically or experimentally considering the characteristics of the studied signals that correspond to three simulated Doppler ultrasound quasi-stationary signals represent a typical blood flow in the Carotid, Coronary and Femoral arteries. Modified-B, Choi Williams, Born Jordan and Bessel distributions are considered. In this work the optimal scaling factor values, the so-called optimal parameters have been determined experimentally for different conditions of SNR and window length to estimate pseudo instantaneous mean frequency () and RMS bandwidth (). Modified-B distribution produces the best and spectral estimations. Keywords: Time-Frequency Distributions, Signal Analysis, Doppler ultrasound blood flow. 1.- INTRODUCTION The analysis of a signal ( t) x using the Fourier transform magnitude reveals which frequency components are present but it does not locate them temporarily. If a signal is multiplied by a sampling window ( t) W, with support T < t < T, then the temporal location of the frequency components capability is artificially introduced. In this way, the S t, ω is obtained: spectrogram ( ) S ( t ω) W ( τ t) x( τ ) j, e ωτ dτ (1) Nevertheless, as the support of ( t) W diminishes, the temporal resolution increases but the frequency resolution diminishes. In the opposite case, as the support of W ( t) increases, the temporal resolution diminishes but the frequency resolution increases. Further, the use of the Fourier transform supposes that the signal is stationary. For the case of the signals associated to Doppler ultrasound blood flow measurement, this supposition is fulfilled as the support of ( t) W diminishes, sacrificing the frequency resolution. Time frequency distributions (TFD) (Cohen, 1989) are an alternative for signal analysis associated to Doppler ultrasound blood flow measurement, since they do not suppose that the signal is stationary. Further, the temporal location of the frequency components is done in an intrinsic way and they do not suffer from the commitment between the temporal and frequency resolutions explained previously. The Time Frequency Distributions have a scaling factor with which an optimization problem to get spectral estimations can be proposed. Two spectral estimations are considered: pseudo instantaneous mean frequency and RMS bandwidth. This optimization problem can be solved analytically (Boashash, and Sucic, 003) or experimentally considering characteristics of the studied signals. The considered signals are three simulated Doppler ultrasound quasi-stationary signals that represent a typical blood flow in the Carotid, Coronary and Femoral arteries. Modified-B, Choi Williams, Born Jordan and Bessel distributions are considered. Previous works have suggested optimal scaling factor values experimentally calculated (García, et. al, 00 b; Cardoso, et al., 1996).. TIME FREQUENCY DISTRIBUTIONS The time frequency distributions (TFD) of the Cohen class considered in this work are the Bessel, the Born Jordan, the Choi Williams and the Modified-B distributions. The discrete TFD of a complex signal x(n) of length LN-1, whose elements are numbered from 1-N to

2 N-1, when it is evaluated at discrete time n0, and optimized (Boashash, and Black, 1987) is: DTFD k N 1 W W f e τ 0 W 0 W 0 f 0,0 j N ( 0, ) 4Re ( τ) ( τ) ( 0, τ) () where f (, ) ( ) ( ) ( ) πkτ nτ is the generalized autocorrelation function, W(n) is a (Hanning) sampling window of length LN-1, and k is the discrete frequency taking integer values from 0 to N-1. for the discrete Bessel TFD (Guo, and Durand, 1994) is: f ( 0, τ ) min{ ατ, N 1 τ} 1 µ (3) 1 x( µ + τ ) x ( µ τ) µ max{ ατ, N + 1+ τ} πα τ ατ where α is a scaling factor taking the half of any f 0,0 x 0 x 0. natural value. Note that ( ) ( ) ( ) for the discrete Born Jordan TFD (Cohen, 1989) is: f ( 0, τ ) (4) min{ ατ, N 1 τ} 1 x( µ + τ ) x ( µ τ) µ max{ ατ, N + 1+ τ} 4ατ where α is a scaling factor taking the half of any f 0,0 x 0 x 0. natural value. Note that ( ) ( ) ( ) for the discrete Choi-Williams TFD (Choi, and Williams, 1989) is: N 1 τ µ N + 1+ τ f ( 0, τ ) µ 1 4τ σ e x 4πτ σ + x ( µ τ ) ( µ τ) (5) RMS error of a spectral estimation. The value of the optimal parameter depends on the spectral estimation that is realized. Also it depends on the length of the sampling window (L) and noise to signal ratio (SNR). In this work it is considered the spectral estimations of the pseudo instantaneous mean frequency () and RMS bandwidth (). In consequence, optimal parameters are calculated for each of them. In this work an experimentally determination of optimal parameters is accomplished. 4. DOPPLER ULTRASOUND SIGNAL SIMULATION In order to characterize the pseudo instantaneous mean frequency () and the RMS bandwidth () error estimations when the TFD are used, it has been proposed the utilization of three simulated Doppler ultrasound quasi-stationary signals that represent a typical blood flow in the Carotid, Coronary and Femoral arteries. Their characteristics are well documented (Evans, 000) (De Lazzari, et al., 006). Briefly, the signals duration is cardiac cycles at 61.6 ppm; they have a constant of Hz and their wave form are shown in figures 1a to 1c. The simulation procedure is accurate described in (Cardoso, et al., 1996). In this work, a sampling rate f o 1800Hz is considered. Note that the sampling rate must be four times the signal s maximum frequency when TFD are used. A white noise is added to the whole signal before starting the signal analysis procedure, according to typically prescribed signal noise ratios (SNR). In this work, SNR of -db and noiseless cases are considered (the minus sign will be omitted). where σ is a scaling factor taking any positive real f 0,0 x 0 x 0. value. Note that ( ) ( ) ( ) for the discrete Modified-B TFD (Hussain, and Boashash, 00) (Boashash, et al, 013) is: f ( 0, τ ) α (6) N 1 τ Γ( α ) 1 x 1 ( µ τ ) x α ( µ τ ) µ N + 1+ τ ( α ) + cosh ( µ ) Γ where α is a scaling factor taking any positive real value. 3.- OPTIMAL PARAMETER An optimal parameter is that time frequency distribution scaling factor value that minimizes the Fig 1a: Signal s pseudo instantaneous mean frequency () wave form of the simulated Doppler ultrasound quasi-stationary signal that represents a typical blood flow in the Carotid artery.

3 Finally, the pseudo instantaneous power distribution (PIPD) of this TFD is calculated. Its elements are also numbered in the discrete frequency domain from N/ to N/-1. The PIPD is defined as: PIPD ( 0, k) TFD ( 0, k) TFD( 0, k) 0 TFD( 0, k) 0 < 0 (7) In case of the calculation, the pseudo instantaneous mean frequency associated to the n th window signal is stated by: Fig 1b: Signal s pseudo instantaneous mean frequency () wave form of the simulated Doppler ultrasound quasi-stationary signal that represents a typical blood flow in the Coronary artery. Fig 1c: Signal s pseudo instantaneous mean frequency () wave form of the simulated Doppler ultrasound quasi-stationary signal that represents a typical blood flow in the Femoral artery. 5. SPECTRAL ESTIMATION The spectral estimation of both the and the is worked out as in (Cardoso, et al., 1996; Fan, and Evans, 1994). Their procedures have a common part. First, a signal piece of length L is taken from the n th to the (n+l-1) th elements of the whole signal, it will be called the n th signal window. In this work, L can be 17, 55, 511 and 103, and LN-1. The signal window s elements are numbered in the discrete time domain from 1 N to N-1. The quadrature signal s elements are also numbered in the discrete time domain from 1-N to N-1. Second, the TFD of this quadrature signal is calculated using equation () and (3), (4), (5) or (6) depending on the study case, considering prescribed scaling factors. The TFD s elements are numbered in the discrete frequency domain from N/ to N/-1. ( ) n N 1 k N N k 1 kn ( 0, ) f PIPD k PIPD ( 0, k ) (8) where f k is the real frequency associated to discrete frequency k. Observe that n can be considered as the whole signal s discrete time variable, running from 0 to T-L. Indeed, it represents the total amount of fully overlapped signal windows of length L in the whole signal (an overlapping of L-1 elements). That is, the (1) correspond to the 1 st signal window; the (), to the nd signal window; and so on. On the other hand, in case of the calculation, the RMS bandwidth associated to the n th window signal is stated by: ( ) n N 1 kn ( ( n) fk ) PIPD ( 0, k) N 1 kn PIPD ( 0, k) with the same considerations as in equation (8). 6. ERROR ESTIMATION (9) Typically, in any spectral estimation, the error has two independent components (Cardoso, et al., 1996). The first component represents the mean of the errors of the estimated values respect to the theoretic values. That error will be called the bias. The second component represents the standard deviation of those errors. Then, the root mean square (RMS) error is estimated according to: error RMS + bias std (10) In case of calculating the error estimation of the, it can be done with: ( ) ( ) ( ) error n n n estimated theoretic (11) 1 m m n 1 ( ) bias error n (1)

4 1 m std error n bias m n 1 ( ( ) ) (13) where m is the total amount of fully overlapped signal windows of length L in the whole signal of length T, in consequence, m T-L+1. Whereas, in case of calculating the error estimation of the, it can be done with: ( ) ( ) ( ) error n n n estimated theoretic (14) with the same considerations as in equations (11), (1) and (13). 7.- RESULTS Bessel, Instantaneous Frequency, L Fig a: estimation error as a function of scaling factor value using Bessel distribution. Optimal parameter for L511 without noise is α 3. The following graphs show the spectral estimation errors which are obtained for different values of scaling factors. Note that the graphs have a minimum point defining the optimum value of the scaling factor, that is, the optimal parameter. The analysis is done separately for and. Similar results are obtained when analyzing the Carotid, Femoral or Coronary blood flow simulated signals Bessel Distribution Bessel, RMS Bandwidth, L Figures a and b shows the results obtained using Bessel distribution for window length of L511 and without noise. Table 1 shows optimal parameters for different window lengths and noise levels for both, and L Noiseless SNR Noiseless SNR Table 1: Optimal parameters for Bessel distribution Born Jordan Distribution Figures 3a and 3b shows the results obtained using Born Jordan distribution for window length of L511 and without noise. Table shows optimal parameters for different window lengths and noise levels for both, and Fig b: estimation error as a function of scaling factor value using Bessel distribution. Optimal parameter for L511 without noise is α 3 L Noiseless SNR Noiseless SNR Table : Optimal parameters for Bessel distribution Born Jordan, Instantaneous Frequency, L Fig 3a: estimation error as a function of scaling factor value using Born Jordan distribution. Optimal parameter for L511 without noise is α.

5 Born Jordan, RMS Bandwidth, L51 Choi Williams, RMS Bandwidth, L Sigma Fig 3b: estimation error as a function of scaling factor value using Born Jordan distribution. Optimal parameter for L511 without noise is α 1. Fig 4b: estimation error as a function of scaling factor value using Choi Williams distribution. Optimal parameter for L511 without noise is σ Choi Williams Distribution Figures 4a and 4b shows the detailed results obtained using Choi Williams distribution for window length of L511 and without noise. Table 3 shows optimal parameters for different window lengths and noise levels for both, and L Noiseless SNR Noiseless SNR Table 3: Optimal parameters for Choi Williams distribution Modified-B Distribution Figures 5a and 5b shows the results obtained using Modified-B distribution for window length of L511 and with SNRdB. Table 4 shows optimal parameters for different window lengths and noise levels for both, and L Noiseless SNR Noiseless SNR Table 4: Optimal parameters for Modified-B distribution. Choi Williams, Instantaneous Frequency, L51 Modified B, Instantaneous Frequency, L Sigma Fig 4a: estimation error as a function of scaling factor value using Choi Williams distribution. Optimal parameter for L511 without noise is σ 0.3. Fig 5a: estimation error as a function of scaling factor value using Modified-B distribution. Optimal parameter for L511 with SNRdB is α

6 Modified B, RMS Bandwidth, L Fig 5b: estimation error as a function of scaling factor value using Modified-B distribution. The optimal parameter for L511 with SNR is α RESULTS ANALYSIS Tables 1, 3 and 4 show optimal parameters (optimal scaling factor values) experimentally obtained for Bessel, Born Jordan, Choi Williams and Modified-B respectively. Note that optimal parameters may depend on window length, SNR. Also, optimal parameters may be different for and spectral estimation. Figure 6 compares the error in the estimation for the different TFD considered, using optimal parameters, without noise, and considering different window lengths (L17, 55, 511 and 103). It is observed that the TFD that more accurately estimates the is the B-Modified, followed by Born Jordan and Choi Williams, estimating it nearly alike; finally Bessel distribution. 0.0, SNRdB, TI It is observed that the TFD that more accurately estimates the are the B-Modified and Choi Williams, estimating it nearly alike, followed by Born Jordan and, finally, Bessel distribution , SNRdB, TI0 L Fig 7.- in estimating the with optimal parameters. 9.- CONCLUSIONS In this work the optimal scaling factor values, the so called optimal parameters, have been determined experimentally for different conditions of SNR and window length to estimate and. The results are shown in tables 1 to 4. The considered signals were three simulated Doppler ultrasound quasi-stationary signals that represent a typical blood flow in the Carotid, Coronary and Femoral arteries. Bessel, Born Jordan, Choi Williams and Modified-B distributions were considered; tables 1, 3 and 4 show optimal parameters (optimal scaling factor values) experimentally obtained respectively. Note that optimal parameters may be different for estimating and. Also, the Modified-B distribution produces the best and spectral estimations. 103 Modified B Bessel Born Jordan Choi Williams ACKNOWLEDMENTS L Bessel Choi Williams Born Jordan Modified B The authors acknowledge project DGAPA-UNAM- PAPIIT (IN10113), project Consorciado CYTED (P6PIC095) by the financial support. Also we want to acknowledge to M. Fuentes, J. Contreras, S. Padilla and M. Vazquez for their technical support in the development of this work. Fig. 6.- in estimating the with optimal parameters. Figure 7 compares the error in the estimation for the different TFD considered, using optimal parameters, without noise, and considering different window lengths (L17, 55, 511 and 103). REFERENCES Boashash, B. and P. Black (1987). An Efficient Real- Time Implementation of the Wigner-Ville Distribution. IEEE Transactions on Acoustics, Speech, and Signal Processing. ASSP

7 Boashash B. and Sucic V. (003) Resolution Measure Criteria for the Objective Assessment of the Performance of Quadratic Time- Frequency Distributions. IEEE Transactions on Signal Processing. 51, Boashash B. Azemi G. and O Toole J.M. (013) Time-Frequency Processing of Nonstationary Signals. IEEE Signal Processing Magazine. November 013, Cardoso, J. G. Ruano and P. Fish (1996). Nonstationary Broadening Reduction in Pulsed Doppler Spectrum Measurements Using Time- Frequency Estimators. IEEE Transactions on Biomedical Engineering Choi, H. and W. Williams (1989). Improved Time- Frequency Representation of Multicomponent Signals Using Exponential Kernels. IEEE Transactions on Acoustics, Speech and Signal Processing Cohen, L. (1989). Time-Frequency Distributions -A Review. Proceedings of the IEEE De Lazzari C., et al. (006) Coronary Blood Flow: Comparison between in Vivo and Numerical Simulation Data. Computers in Cardiology Evans D., McDickn N. (000) Doppler Ultrasound. Physics, Instrumentation and Signal Processing. Ed. John Wiley & Sons, LTD. England. Second Edition. Fan, L. and D. Evans (1994). Extracting Instantaneous Mean Frequency Information from Doppler Signals Using the Wigner Distribution Function. Ultrasound in Med. & Biol García-Nocetti F., Solano J., Rubio E (00) Precision enhancement of Doppler Ultrasound spectral estimation by finding TFD optimal parameters. Forum Acusticum Sevilla. Special Issue of the Revista de Acústica. 33. September 00. Sevilla, Spain. García-Nocetti F., Solano J., Rubio E., Moreno, E. (00) High Performance Computing of Time Frequency Distributions for Doppler Ulltrasound Signal Analysis. Preprints of the 15 th Triennial World Congress of the IFAC. July 00, Barcelona Spain. Guo, Z., L. Durand and H. Lee (1994). The Time- Frequency Distributions of Nonstationary Signals Based on a Bessel Kernel. IEEE Transactions on Signal Processing Hussain Z. And Boashash B. (00). Adaptive Instantaneous Frequency Estimation of Multicomponent FM Signals Using Quadratic Time-Frequenc Distributions. IEEE Transactions on Signal Processing