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T. Ratnamani et al.: Application of Time Frequency Analysis Methods in Inverse Scattering 79

Application of Time Frequency Analysis Methods in Inverse Scattering

T. Ratnamani1, Raj Kumar2 and O. Vijay Kumar1 1Naval Physical and Oceanographic Laboratory, Kochi- 682 021, India

2Defence Institute of Advanced Technology, Pune- 411 025, India [email protected]

Abstract: Ramp response technique is reported in literature for extraction of profiles of underwater objects. This paper presents the application of time frequency analysis tools like STFT and Wigner Distributions (WD’s), in extraction of profiles of underwater objects from reflections (echo). The profile extracted at different reflected angles are used in generation of acoustic image of the object. This method is developed for extraction of profiles, mainly from short/medium transmitted pulses, that are of interest in practical applications. This method has been verified for various pulse parameters of the incident wave, like pulse frequency, pulse length, type of modulation (CW, LFM, SFM), sampling frequency, as well as different size and shape of the scattering body. The results for the simulated echoes are presented here.

Keywords: Inverse Scattering, Time Frequency Analysis, Ramp Response Technique. 1. Introduction Inverse scattering techniques retrieve the shape and size information of an underwater object from available measurement or observation data, such as far-field backscattered frequency spectrum at certain aspects and frequencies. The forward scattering is found for given incident acoustic waves and the physical properties of the object. The exact analytic solutions [1, 2] can be derived and used for objects of simple geometries, such as sphere, infinite cylinder, and spheroid, while approximate methods are often used for objects of more general shapes.

The most popular method for inverse scattering is the ramp response technique. This is based on twice integrating the impulse response obtained from the underwater object. X.M.

Zhang et al. [2] developed a numerical method to obtain the ramp response for underwater objects in acoustic field, that requires limited data in frequency domain.

The concept of impulse response technique is first introduced by E.M. Kennaugh [8] and applied to electromagnetic scattering problems, based on the physical optics approximation.

The impulse response of an object was interpreted as the second derivative of the profile function. The profile function is defined as the cross sectional area along the direction of the incident wave. Consequently, by integrating the impulse response twice, it was found that the ramp response of the object is proportional to its profile function [4]. The ramp response technique has been utilized in electromagnetic inverse

80 Proceedings of SYMPOL-2011

scattering for profile reconstruction [1]. This technique has also been applied in acoustic wave filed for solving inverse problems, like underwater acoustical imaging and non-destructive evaluation methods.

The methods based on impulse response techniques have limited practical applications as the wide band returns from the underwater body are highly influenced by the media. The practical signals have limited bandwidth, generally very narrowband due to the limitations of the projectors/sensors. In addition, the pulse widths used is based on certain criteria and varies from short pulse to medium pulse. The ramp response technique cannot be applied for this type of signals. However, as the profile function is the cross sectional area along the direction of propagation, the reflected wave can be interpreted as sum of discretised waves returning from the scatterer. And can be considered as time series data, with time varying frequency content. Time series data have been traditionally analyzed in either the time or the frequency domains.

For signals with a time-varying frequency content, the combined time-frequency (TF) representations, based on the Cohen class of (generalized) Wigner Distributions (WD’s) offer a powerful analysis tool. Using them, it is possible to: 1) trace the time-evolution of the resonance features usually present in a standard sonar cross section (SCS), or in a radar cross section (RCS) and 2) extract target information that may be difficult to even notice in an ordinary SCS or RCS In the underwater applications. The time frequency techniques are generally used to get time and frequency localisation of the signal. This information is valuable in frequency estimation of signals, and detection and classification of targets.

Application of time frequency analysis for scattering/inverse scattering is not much reported in literature. Time-frequency analysis can be carried out, on the reflected signal, for the profile function extraction. The instantaneous amplitude correspond to the cross section of the

scatterer, that is The profile function is the time series of amplitude at discrete time intervals, can be obtained from summing the energy from all frequency bins corresponding to each time step. This technique is applied to extract profile function of two-dimensional objects. The results were verified from the ramp response technique for single cycle signal. The results also obtained are very close to the object shape simulated. The scatter was simulated by numerical method (Kirchhoff’s integration). This technique is verified on different pulse lengths and pulse parameters like CW, LFM couple of time frequency methods i.e., STFT and Winger Ville were used as time frequency analysis. The results were comparable for short pulses. But for pulses of longer duration, Winger Ville performs better.

The advantage of this method is that it can be applied on any type of signals and with any pulse length. The signals can narrow-band or broad-band based on the requirement. By appropriately choosing the pulse parameters and processing parameters this technique can be used on any pulse of practically used in underwater applications. 2-D objects of shapes given in Figure 1b have been selected for simulation and the echo from the object was simulated by numerical method. The simulated time series is used for profile extraction.

2. Ramp Response Technique When an incident pulse pt represented by a delta function in time (t) encounters an object in a homogeneous, isotropic Medium, the phenol- menon of wave diffraction or scattering occurs. The incident pulse is scattered in all directions. The waveform scattered in a direction opposite to the direction of incidence is called the backscattered impulse response. The ramp response is defined as a time-domain backscattered signal at a distant observation point in response to an incident ramp signal. The ramp signal is a function defined to be the second integral of the incident impulse function. In practice, the projector-hydrophone system is


used for the ramp response measurement. The projector generates an acoustic pulse and sends it out, while the hydrophone is used to receive the backscattered impulse response. Consequently, the backscattered impulse response is integrated twice with respect to time to yield ramp response. As we only need the backscattered signals, the projector and hydrophone should be set together:

PR(t) = ∫ t ∫ PI(t) ‘‘)dt’’ dt’ where, PR(t) is the profile function and the PI(t) impulse response of the object.

In the underwater acoustic far field, based on the physical optics approximation, the ramp response for rigid objects is proportional to the cross-sectional area of the object and can be expressed as below:

… (1)

Where, c represents the sound velocity in the water and A (ct/2) is the area of cross section of the object perpendicular to the line of sight. The function A (ct/2) is called a profile function, which is shown in Figure 2b. Here t denotes the propagation time of the acoustic wave. Zero propagation time is defined as when the wave crosses the centroid (geometric center) of the object. For the acoustically rigid bodies, Eq. (1) shows the linear relationship between the profile function and the ramp response of the object in the medium. By computing the ramp response of an object, the profile function can be obtained.

Fig. 1(a): An Incident Acoustic Wave

Illuminates an Object

3. Time-Frequency Analysis and Shape Extraction

On the other hand, it can be proven that the ramp response PR(t) and the normalized complex backscattering amplitude Γ(ω) are Fourier transform pairs in the acoustic far field. It is shown as below:

… (2)

Here the normalized complex backscattering amplitude G is defined as:

… (3)

where, S(ω) is the backscattered frequency spectrum in the acoustic far field. The symbol ‘‘*’’ denotes complex conjugate. Eqn. (2) is used to calculate the ramp response of the object. It should be noted that the normalized complex backscattering amplitude Γ is specified completely by scattering information of the object, and describes the frequency dependence of the scattering mechanisms, an inherent property of the object. The ramp response PR describes the shape, size, and orientation information of the object. Moreover, Eq. (2) can be further simplified as follows:

… (4) where Sr(ω) and Si(ω) are the real and imaginary parts of S(ω), respectively. In the acoustic far field, the relationship between the scattering field potential Ψs and the backscattered frequency spectrum S(ω) could be obtained as:


… (5)

Now, 1) to obtain the profile function (ramp response) for the band limited signal, the integral in the equation may be restricted to the signal bandwidth. 2) The integral, corresponding frequency components is obtained at discreet time instances. This method of profile function extraction is suitable for practical applications of band-limited signals.

However this requires back scattered signal to be represented/converted to time frequency representation. Profile function can be obtained by forming the integral of amplitude of all the frequency components for the corresponding time instances. However this may have some smearing/broadening effect on the profile function due to band limiting of the signal and can be compensated in image reconstruction.

Two Time-Frequency methods for this application that have been used and compared are STFT and Wigner –Ville distributions.

The Fourier Transform decomposes a signal into its constituent frequency components. Looking at the Fourier spectrum we can identify the frequencies. However, we cannot identify their temporal localization. Time Frequency Distri- butions (TFDs) map a one-dimensional signal into a two-dimensional function of time and frequency, and describe how the spectral content of the signal changes with time. From a strict mathematical point of view, we want to have a joint distribution, which will give us the fraction of the total energy of the signal at time t and frequency ω; i.e., we look for the function P(t, ω). The aim of time-frequency analysis is to come up with appropriate candidates for P(t, ω).

TFD are appropriate tools for non-stationary signal analysis, synthesis, and processing. Different types of time frequency distribution have been developed for that purpose. Two early forms of time-frequency analyses are: the

Short-Time Fourier Transform (STFT), used to generate the spectrogram (SP) [Koeing, Dunn and Lacy, 1946; Allen, and Rabiner, 1977], and the Wigner-Ville distribution (WVD) [Claasen and Mecklenbrauker, 1980]. Studies of the well known linear TFD Short-Time Fourier Transform (STFT) have been published by Nawad and Quatieri, [1988], and Allen [1977] among others. The time-frequency distribution ideally describes how the energy is distributed, and allows us to estimate the fraction of the total energy of the signal at time t and at frequency ω. The above statement states that the energy should be positive.

In order to achieve fine simultaneous time-frequency resolution in a non-stationary time series, we must deal with the uncertainty principle [Williams, Brown and Hero, 1991]. The uncertainty principle restricts us from achieving arbitrarily fine resolution simultaneously in both the time and the frequency domain. The condition to satisfy in the uncertainty principle is given by the inequality 4πΔω tΔ ≥ 1, where the selection of Δt (time resolution) and Δω (frequency resolution) parameters is not arbitrary. A trade-off between them should be considered in order to reach the desired “good” resolution. Next, an overview and a brief description of the advantages/limitations of the Spectrogram (SP), the Wigner-Ville (WVD), are presented.

3.1 Short-Time Fourier Transform (STFT) In order to introduce time-dependency in the Fourier transform, a simple and intuitive solution consists in pre-windowing the signal x(u) around a particular time t, calculating its Fourier transform, and doing that for each time instant t. The resulting transform, called the short-time Fourier transform (STFT, or short-time spectrum), is

… (6) Where h(t) is a short time analysis window localized around t = 0 and v = 0.


The STFT Fx(t, v, h) can be considered as the result of passing the signal x(u) through a band-pass filter whose frequency response is H¤( ξ− v), and is therefore deduced from a mother filter H(ξ) by a translation of v. So the STFT is similar to a bank of band-pass filters with constant bandwidth. The spectrogram is better able to resolve temporal evolution of frequency content, but has a trade-off in time resolution versus frequency resolution in accordance with the uncertainty principle. To overcome this restriction, several classes of Time-Frequency representations have been proposed, including wavelet methods and quadratic time-frequency distributions such as the Wigner-Ville distribution.

3.2 Wigner-Ville Distribution For a signal, s(t), with analytic associate x(t), the A time-frequency energy distribution, which is particularly interesting, is The Wigner-Ville distribution (WVD) defined as:

… (7) Or equivalently as:

… (8) The Wigner-Ville distribution Ws(t, ω) of a time series signal s(t) is defined as:

… (9) The time series function s(t) in Eqn. (9) can be either real or complex. The only type of complex signal considered in this paper is the analytic signal, which is defined as (s(t) + iH[s(t)]) where H denotes the Hilbert transform [e.g., Cohen, 1994]. The analytic signal has the same spectrum (or, to be precise, the spectrum multiplied by two) as that of the original real signal at positive frequencies but keeps the spectrum at negative frequencies zero.

A significant characteristic of WVD, which essentially correlates the signal with a time- and frequency-translated version of itself, is that it does not contain a windowing function as those in the Fourier and wavelet frameworks. This unique feature frees WVD from the

smearing effect due to the windowing function, and, as a result, the WVD provides the representation that has the highest possible resolution in the time-frequency plane. For example, if a wave packet is non-zero only for t € (t1, t2), it can be shown that the corresponding Ws have to be 0 for t outside this exact time interval. There is, however, a well-known disadvantage of using WVD. When there are multiple components in a signal, the WVD can become difficult to interpret with its cross terms.

A two-dimensional (2-D) low-pass filter can suppress the cross-term interference in the time frequency plane. Both these time frequency methods were applied obtain time frequency distributions of the signal. Integrating across the amplitude of the frequency band on each time instance the profile function is obtained.

4. Results and Discussions The 2D of scattering underwater object is shown in Figure 1b. The forward scatter of object has been simulated by numerical method (Kirchhoff’s integration) for single cycle, multiple cycles and for different frequencies. The profile function was obtained by using ramp response technique [2]. The shape of results obtained using profile function [2] is very close to the object shape simulated by numerical method. The several shapes were used in simulations for different pulse parameters, size of the objects, as well as different sizes of the spatial grid and sampling frequencies.

The parameters used in simulation are i.e., frequency; 25 KHz, transmitter angle θ1 = 950 and receiver angle θ2 = 950. The other parameters as taken like No. of cycles = 1; No. of finite elements; 500 × 50, Element type: rectangular, size of the body: 10 × 1 meters and sound velocity = 1500 meters/sec.

In Figure 1a, rectangular body is defined as the basic shape and another rectangle/triangle is either added or subtracted to obtain the shape. The forward scatter of object has been simulated by numerical method for different cycles and for different frequencies. The


parameters like placement of the alteration (P.S), width of the shape (w), and alteration type (a+ or a–) are described along with the result. Figure 2 to Figure 5 indicates the time series generated and profile function extracted for mono static case. Figure 2a denotes time series from simulation for P.S = 200, w = 10, type = a– and No. of cycles = 3. Figure 2b and Figure 2c denotes the corresponding profile functions extracted from time series generated in Figure 2a. By ramp response technique (2b), and Wigner Vile technique (2c).

The parameters used in simulation are i.e., frequency; 25 KHz, transmitter angle θ1 = 950 and receiver angle θ2 = 950. The other parameters

as taken like No. of cycles = 1; No. of finite elements; 500 × 50, Element type: rectangular, size of the body: 10 × 1 meters and sound velocity = 1500 meters/sec.

Figure 3a shows the time series simulated for P.S = 200, w = 10, type = a+ and No. of cycles = 3. And Figure 3b, and Figure 3d denotes the corresponding profile functions extracted from time series generated in Figure 3a by ramp response technique (3b), and Wigner Ville technique (3d) and In Figure 4a exhibits the time series from simulation for P.S = 1000, w = 20 type = a– and at P.S = 3000, w = 20 type = a– and No. of cycles = 20. The size of the body is 4000.

Fig. 1(b): 2D Shapes used for Simulation of Scatter from under water Objects


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Monostatic profile function extracted from Figure 4a by ramp response technique (4b), by STFT technique (4c), and Wigner Ville technique (4d) is shown in Figure 4b, Figure 4c, and Figure 4d. It is seen that with very narrow pulse, the time series generated clearly indicates the modulation of the signal by the shape of the object and the profile extracted is very close to the object shape selected for simulation of echo. The effect of pulse width is seen as super position of multiple returns and reflected in the amplitude of the time series shown in Figure 4a. However in the profile extracted by applying Wigner Ville is as per the object profile. However in the profile extracted by STFT, the features are elongated. This is due to limited localization of the frequency in STFT.

The simple integration method however is not efficient at all for the wide pulse. From the result it is clear that application of Wigner Ville, the profile extracted indicates the shape of the object. Therefore this method can be effectively used in underwater imaging. For the Figure 2a to 5a the “x-axis” corresponds to time (10 μs/sample), given as sample number and “y-axis” corresponds to amplitude (mille volts) of the signal at the receiver. For the Figures 2b to 5b the “x-axis” corresponds to length of the object, given as sample number and “y-axis” corresponds to height of the object.

In Figure 5a exhibits the time series from simulation for P.S = 1000, w = 20 type = a– and at P.S = 3000, w = 20 type = a– and No. of cycles = 20 for a bi-static reception, with incidence at 95 and receiver at 40. The size of the body is 4000. The profile function extracted from Figure 5a by ramp response technique (5b), by STFT technique (5c), and Wigner ville technique (5d) is shown in Figure 5b, Figure 5c, and Figure 5d.

From results, it is clear that the computational technique can be used on any shape of the object. However, the profile extraction function most suited on impulse incidence may not suitable for wide pulses. However for all practical applications the pulse does have finite pulse width. From the above results, it is also observed that shorter pulse gives the better definition than wide pulse.

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5. Conclusion For practical applications the ramp response method may not suitable for profile extraction. Methods based on time frequency distribution/ analysis presented here presented for extraction of profile function are more suitable for most practical applications. Out of the methods presented the Wigner Ville is better than the other methods, due to its inherent auto correlation property. This method is also better suited for noise added signals i.e., in practical environment. The method can be extended to resolve closely located features by applying frequency modulated signal for transmission and resolving the back scatter by appropriately shifting results in time


frequency distribution prior to integration. Hence this method is suitable for practical applications of inverse scattering of underwater objects. Another application of time frequency results may be for classification. The information obtained in the tine frequency plane may be considered for extraction of features set by appropriate method for classification.

Acknowledgments The authors gratefully acknowledge the support of Director, NPOL. The authors also acknowledge the support of Vice Chancellor, DIAT (DU), Pune for the support and encouragement.

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[3] Li, Wel, Liu, G.R., size Zhang, X.M., “Size Identification of Underwater Objects from Back Scattering Signals of Arbitrary Looking Angles”, Journal of Computational Acoustics, Vol. 12, No. 3 (2004).

[4] Li, Wel, Liu, G.R. and Varadan, V.K., “Acoustical Imaging of Underwater Objects Using the Bi-static Ramp Response Signals”, Smart Materials and Structure, Vol. 13, 2004.

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[7] Huerta-Lopez, Carlos I., Shin, Yongjune, Powers, Edward J. and Roesset, Jose M., “Time-Frequency Analysis of Earthquake Records”, 12WCEE, 2000.

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