Optimum signal synthesis for time-scale estimation

In signal analysis, the joint estimation of the time-scale parameters which can affect a known signal (Doppler effect or scale effect, delay. . . ) may be a problem of interest. An important result has shown that, even if the quality of the time delay estimation is classically given by the inverse spread of the signal spectral density, the quality of the scale estimation only depends on the inverse of the signal spread in Mellin space. This spread has a direct interpretation in the time-frequency plane and can be precisely estimated when duration, bandwidth and relative bandwidth of the signal are known. We propose here to develop two methods of optimum signal synthesis which minimize the variance of the estimates given by the Cramer-Rao lower bounds. The first method is based on the stationary phase principle, applied on frequency and Mellin spaces, which allows to construct signals with given autocorrelation functions in scale and time spaces. The second method is devoted to the construction of a frequency phase law depending on the Mellin variable with the spreads in frequency and Mellin spaces related to the expected scale and time-delay resolutions. 1. FORMULATION OF THE PROBLEM We are dealing with the problem of the joint estimation of the timescale parameters of a known signal embedded in gaussian white noise. This is, for example, the case encountered in broad-band radar or sonar theory, when looking for parameters such as the velocity (related to scale parameter) or the position (related to timeshift parameter) of a target. In this case, the questions we are trying to answer are : which is the best signal to use for minimizing the variances of the estimates ? Can we develop synthesis methods which allow to construct such a signal ? The answer to the first question has already been developed in [6] and is briefly recalled here in order to develop the synthesis. Let z(t) be the transmitted and analytic signal. Its Fourier transform Z(f) has therefore no negative frequency. The general transformation x(t) of the signal z(t) can be expressed as : x(t, θ0) = A0Tθ0z(t) e iφ0 + b(t) (1) where Tθ0 is a time-scale action of the affine group which transforms the signal z(t) with a set θ0 = (a0, b0) of unknown parameters (time scale a0 and time shift b0). The parameter A0 is the amplitude, φ0 a phase change and b(t) a zero-mean white gaussian noise with σ variance. When the probability density of the Presented at IEEE-ICASSP’98, May 12-15, 1998, Seattle, Washington, USA parameters A0 and φ0 is unknown, the Maximum Likelihood ratio Λ to maximize, according to the Maximum Likelihood estimation theory, is given by the square modulus of the broad-band crossambiguity function :