Definition and properties A function of particular interest, especially in the field of radar signal processing, is the narrow-band ambiguity function (noted AF), defined as $$A_x(\xi,\tau)=\int_{-\infty}^{+\infty} x(s+\tau/2)\ x^*(s-\tau/2)\ e^{-j2\pi \xi s}\ ds.$$ This function, also known as the (symmetric) Sussman ambiguity function, is a measure of the time-frequency correlation of a signal $x$, i.e. the degree of similarity between $x$ and its translated versions in the time-frequency plane. Unlike the variables '$t$' and '$\nu$' which are "absolute" time and frequency coordinates, the variables '$\tau$' and '$\xi$' are "relative" coordinates (respectively called delay and doppler). The AF is generally complex-valued, and satisfies the Hermitian even symmetry: $$A_x(\xi,\tau) = A_x^*(-\xi,-\tau).$$ An important relation exists between the narrow-band ambiguity function and the WVD, which says that the ambiguity function is the two-dimensional Fourier transform of the WVD: $$A_x(\xi,\tau)=\int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} W_x(t,\nu)\ e^{j2\pi(\nu \tau-\xi t)}\ dt\ d\nu.$$ Thus, the AF is the dual of the WVD in the sense of the Fourier transform. Consequently, for the AF, a dual property corresponds to nearly all the properties of the WVD. Among these properties, we will restrict ourselves to only three of them, which are important for the following: Marginal properties The temporal and spectral auto-correlations are the cuts of the AF along the $\tau$-axis and $\xi$-axis respectively: $$r_x(\tau)=A_x(0,\tau) \mbox{ and } R_x(\xi)=A_x(\xi,0).$$ The energy of $x$ is the value of the AF at the origin of the $(\xi,\tau)$-plane, which corresponds to its maximum value: $$\vert A_x(\xi,\tau)\vert\ \leq\ A_x(0,0)\ =\ E_x,\ \forall \xi, \tau.$$ TF-shift invariance Shifting a signal in the time-frequency plane leaves its AF invariant apart from a phase factor (modulation): $$y(t) = x(t-t_0)\ e^{j2\pi \nu_0 t}\ \Rightarrow A_y(\xi,\tau) = A_x(\xi,\tau)\ e^{j2\pi(\nu_0 \tau-t_0 \xi)}$$ Interference geometry In the case of a multi-component signal, the elements of the AF corresponding to the signal components (denoted as the AF-signal terms) are mainly located around the origin, whereas the elements corresponding to interferences between the signal components (AF-interference terms) appear at a distance from the origin which is proportional to the time-frequency distance between the involved components. This can be noticed on a simple example: * Example: The M-file ambifunb.m of the TF Toolbox implements the narrow-band ambiguity function. We apply it on a signal composed of two linear FM signals with gaussian amplitudes: >> N=64; sig1=fmlin(N,0.2,0.5).*amgauss(N); >> sig2=fmlin(N,0.3,0).*amgauss(N); >> sig=[sig1;sig2]; Let us first have a look at the WVD (see fig. 4.12): >> tfrwv(sig); Figure 4.12: WVD of 2 chirps with gaussian amplitudes and different slopes We have two distinct signal terms, and some interferences oscillating in the middle. If we look at the ambiguity function of this signal (see fig. 4.13), >> ambifunb(sig); Figure 4.13: Narrow-band ambiguity function of the previous signal : the AF-signal terms are located around the origin, whereas the AF-interference terms are located away from the origin we have around the origin (in the middle of the image) the AF-signal terms, whereas the AF-interference terms are located away from the origin. Thus, applying a 2-D low pass filtering around the origin on the ambiguity function, and returning to the WVD by 2-D Fourier transform will attenuate the interference terms. Actually, this 2-D filtering is operated, in the general expression of the Cohen's class, by the parameterization function $f$, as we discuss it now.
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