Comparison of the parameterization functions

To illustrate the differences between some of the presented distributions, we represent their weighting (parameterization) function in the ambiguity plane, along with the result obtained by applying them on a two-component signal embedded in white gaussian noise: the signal is the sum of two linear FM signals, the first one with a frequency going from 0.05 to 0.15, and the second one from 0.2 to 0.5. The signal to noise ratio is 10dB.

On the left-hand side of the figures 4.16 and 4.17, the parameterization functions are represented in a schematic way by the bold contour lines (the weighting functions are mainly non-zeros inside these lines), superimposed to the ambiguity function of the signal. The AF-signal terms are in the middle of the ambiguity plane, whereas the AF-interference terms are distant from the center. On the right-hand side, the corresponding time-frequency distributions are represented.

**Figure 4.16:** Two chirps embedded in a 10 dB white gaussian noise analyzed by different quadratic distributions. On the left-hand side, the parameterization function is represented by a bold contour line, superimposed to the ambiguity function of the signal. The AF-signal terms are in the middle of the ambiguity plane, whereas the AF-interference terms are distant from the center. On the right-hand side, the corresponding time-frequency distribution is represented
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**Figure 4.17:** Two chirps embedded in a 10 dB white gaussian noise analyzed by different quadratic distributions (concluding)
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From these plots, we can conclude that the ambiguity plane is very enlightening with regard to interference reduction in the case of multicomponent signals. On this example, we notice that the smoothed-pseudo-WVD is a particularly convenient and versatile candidate. This is due to the fact that we can adapt independently the time-width and frequency-width of its weighting function. But in the general case, it is interesting to have several distributions at our disposal since each one is well adapted to a certain type of signal. Besides, for a given signal, as a result of the different interference geometries, these distributions offer complementary descriptions of this signal.

Eric Chassande-Mottin 2005-10-26