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.
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