SIGNAL TO NOISE RATIO OPTIMIZATION USING TIME-FREQUENCY AND AMBIDUITY FUNCTION MATCHED FILTERS *

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1 APPLIED PYSICS UNDERWATER SOUNDS SIGNAL TO NOISE RATIO OPTIMIZATION USING TIME-FREQUENCY AND AMBIDUITY FUNCTION MATCED FILTERS * A. DĂNIŞOR Ovidius Universit of Constanta Received December 21, 2004 This paper proposes a new method of signal to noise ratio maximization for better underwater targets detection. Studing the properties of the underwater noise we couldn t conclude anthing about his stationer, even in the restraint sense. Due this fact, the time-frequenc signal processing, signal affected b the underwater noise is justified. This paper proposes a new method of signal to noise ratio optimization, method based on a quadratic time-frequenc processing (Wigner-Ville transform) of the SONAR signals. This optimization is realized using a time-frequenc matched filter, similar with the classic matched filter. Using this quadratic matched filter we obtained a signal to noise ratio visible better than the signal to noise ratio in the time domain, using the classic matched filter. INTRODUCTION Time frequenc methods have long been used for detection in applications such as sonars. Time-frequenc methods are of interest because of nonstationar nature of these signals. The underwater noise is a nonstationar one and that s wh it is necessar to use quadratic detectors for better performances. Quadratic detectors theor has recentl been exploited to derive optimal quadratic time frequenc detectors. Unfortunatel, for an efficient implementation of the matched time frequenc filter presented in this paper it is necessar to use performant computers. TE QUADRATIC MATCED FILTER The time-frequenc signal representation is utilized for subaquatic targets detections and their parameters estimation, especiall when the medium is affected b a nonstationar noise like the underwater noise. This paper presents a time * Paper presented at the 5th International Balkan Workshop on Applied Phsics, 5 7 Jul 2004, Constanţa, Romania. Rom. Journ. Phs., Vol. 51, Nos. 1 2, P , Bucharest, 2006

2 102 A. Dănişor 2 frequenc filter that maximizes the signal to noise ratio, a time frequenc matched filter. The time frequenc transfer function of this filter reseambles to the matched filter impulse response of the linear detectors. It is known that the quadratic detectors have good frequenc selectivit, short time transient response and noise suppression. If we consider the signal received b a sonar sstem: t () = st () + nt () if the target is present and: if the target is absent, where: (t) is the received signal s(t) the reflected signal n(t) the noise the quadratic detector s response is: t () = nt () e= Q(, t τ) R (, t τ) dtdτ (1) where R (, t τ ) is the instantaneous autocorrelation function of the received signal: R t (, ) t τ 2 τ t τ = + (2) 2 and Qtτ (, ) ma be considered the impulse response of the nonstationar detector. The statistical autocorrelation function is, in fact, the statistical mean of the instantaneous autocorrelation function: τ τ = + = 2 2 * r E t t E R t E(*) denotes the statistical mean. Using the Wigner-Ville distribution s definition: j2π fτ WV(, t f ) R(, t τ) e d ( ( )) (3) = τ (4) the instantaneous autocorrelation function can be written as: 2 (, ) (, ) j π f τ τ R t = WV t f e df (5)

3 3 Signal to noise ratio optimization using time-frequenc 103 To have a similarit with the definition of instantaneous autocorrelation function, equation (2),we choose for the detector impulse response Qtτ (, ) an expression like: Qt (, ) h t τ τ τ =, t+ (6) 2 2 and using the Wel smbol of the impulse response of the nonstationar detector, defined as: τ τ j 2πντ L (, t ν) = h t+, t e d 2 2 τ (7) we can express the quadratic detector response b the equation: Qt (, ) L (, t ) e j 2 πντ τ = ν dν (8) Using the relations (5) and (8) the response (1) becomes: e L (, t ν) WV (, t f ) e j f dtdfdνdτ = (9) * 2 π( ν) τ Equation (9) shows the relation between the response o the quadratic detector and the Wel smbol associated with the detector. Using Fourier transform of the Dirac impulse relation (9) becomes more simple: e= L * (, t f) WV (, t f) dtdf (10) The essential problem is to find a suitable formula for Wel smbol of the timefrequenc filter L (t,f), formula which maximize the response of the quadratic detector. Like the impulse response formula of the classic matched filter we choose for the Wel smbol an expression similar with the smmetric Wigner-Ville distribution of the reflected signal. The smmetr between the Wel smbol and the Wigner-Ville distribution is in time domain as in frequenc domain: * * s L (, t f) = WV ( t, f) (11) It is known that the Wigner-Ville distribution of a real signal is real too, previous equation (11) can be wrote as: L * (, t f) = WV ( t, f) (12) s

4 104 A. Dănişor 4 Based on these considerations, the output of the matched time-frequenc filter becomes: ot (, f) = WVs( (' t t), ( f' f)) WV(', t f') dtdf ' ' (13) B similarit with the classic matched filter, for a phsical realization it is necessar to introduce a dela time T, and a supplementar shift frequenc F, equivalent to the time dela, but on the frequenc axis. Based on these considerations, the output of the matched time-frequenc filter becomes: ot (, f) = WVs( T t' + tf, f' + f) WV( t', f') dtdf ' ' (14) The maximum response of the time-frequenc filter is obtained at the time t = T and for the frequenc f = F and its expression appears as: e = o ( t, f ) = WV ( t ', f ') WV ( t ', f ') dt ' df ' (15) t= T, f = F Relation (15) represents in fact the output of the quadratic detector. In the previous relation T represents in fact the duration of the hidroacoustic signal and F his frequenc band. To appreciate the efficienc of time frequenc matched filter we must compute the signal to noise ratio. Signal to noise ratio is defined as a rapport between the power of the peak signal and the medium power of the filter output if onl the noise is present, both measured at the output of the time frequenc matched filter. PS SNR = (16) P It is known that the maximum value at the output of the matched time frequenc filter is obtained at: s t = T f = F In consequence the maximum power of the signal at the output of the matched time frequenc filter is: n 2 Ps = e (17)

5 5 Signal to noise ratio optimization using time-frequenc 105 The medium power of the time frequenc matched filter output is: TF 1 P = ot (, f) dtdf (18) n () t = n() t T 0 0 where o(t,f) is the output of the time frequenc matched filter in absence of the reflected signal (target absent). We can obtain a ver interesting result using the ambiguit function. It is known that the ambiguit function of the received signal t () is defined as: j2πν t A(, τν) R(,) t τ e dt = (19) The ambiguit function represents the bidimensional Fourier transform of the Wigner-Ville distribution: j2 π( fτ νt) WV( f, t) A( τν, ) e d d = τ ν If the Wigner-Ville distribution is assimilated to the time representation of a signal, the ambiguit function can be assimilated to the frequenc domain representation of the same signal. That is wh the output of the time-frequenc matched filter (13) is similar to: 2 ( ) (, ) (, ) (, ) j πν ot f A t τ f s A e d d = ντ ντ ν τ (20) But it is known that the relation (13) defines an unrealisable filter and that is wh the output ot (, f) in the relation (12) is unrealisable too. For a realisable quadratic filter, which output is expressed in the equation (14), using the ambiguit function we obtain: 2 ( ( ) ( )) (, ) (, ) (, ) j πν t ot f A T τ f F As e d d = ντ ντ ν τ (21) The maximum value of the output filter in the relation (21) is: e= o(, t f) = A ( ντ, ) A ( ντ, ) dτdν t= T, f = F s

6 106 A. Dănişor 6 EXPERIMENTAL RESULTS To validate the concepts outlined in the preceding section an experimental stud on two kind of signals affected b a real subaquatic noise. We considered, first, a sinus sonar signal received and processed b the time frequenc matched filter. Its frequenc is 1.8 kz. The signal to noise ratio at the filter s input is unitar. The aspect of the time frequenc output filter is presented in Figure 1. In this case the signal to noise ratio at the filter s output, computed with (16) is: SNR = 53.85dB If the complexit of the sonar signal increase, the signal to noise ratio gains about 5dB. For a chirp hdroaccoustic signal, with a linear variation of frequenc between 1.5 and 1.8 [kz] the signal to noise ratio is: SNR = db The time frequenc matched filter response in this case is shown in Figure 2. In the absence of a reflected signal, onl the noise present, the filter s output is shown in Figure 3. To detect the presence of a submarine target it must sample the output quadratic detector onl when his output signal is maximum (at t=t and f=f) obtaining an eventuall maximum response defined b equation (15). If a digital sstem is used to sample the signal, the filter s response can be observed after the acquisition of ever new sample. The maximum response of the time frequenc matched filter, after the reception of ever sample is presented in Figure 4, if the signal is sinus signal and in Figure 5, in case of a chirp signal. Fig. 1 Time frequenc matched filter output. Sinus signal.

7 7 Signal to noise ratio optimization using time-frequenc 107 Fig. 2 Time frequenc matched filter output. Chirp signal. Fig. 3 Time frequenc matched filter output. Noise onl presence.

8 108 A. Dănişor 8 Fig. 4 Maximum response of the time frequenc matched filter. Sinus signal. Fig. 5 Maximum response of the time frequenc matched filter. Chirp signal.

9 9 Signal to noise ratio optimization using time-frequenc 109 REFERENCES 1. Bouvet, M., Traitements des signaux pour les sstems sonar- Ed. Masson, Paris Coates M., Time frequenc modeling - Universit of Cambridge, PhD., Dănişor A., The stud of the underwater noise Ovidius Universit. Annals of Phsics, Constanţa, Dănişor A, Optimizarea detecţiei ţintelor submarine Telecomunicaţii, Bucureşti, Maurice, A., Time frequenc analsis; a real time differential spectral methods for an application on sonar signals - J.Accostic Societ of America, Aug., Saeed, A., Optimal kernels for nonstationar spectral estimation - IEEE Transaction on information theor, Feb