Sampling the WVD; the analytic signal Because of the quadratic nature of the WVD, its sampling has to be done with care. Let us write it as follows: $$W_x(t,\nu)=2\int_{-\infty}^{+\infty} x(t+\tau)\ x^*(t-\tau)\ e^{-j4\pi \nu \tau}\ d\tau$$ If we sample $x$ with a period $T_e$, write $x[n]=x(nT_e)$, and evaluate the WVD at the sampling points $nT_e$ in time, we obtain a discrete-time continuous-frequency expression of it: $$W_x[n,\nu)=2\ T_e \sum_k x[n+k]\ x^*[n-k]\ e^{-j4\pi \nu k}.$$ As this expression is periodic in frequency with period $\frac{1}{2\ T_e}$ (contrary to period $\frac{1}{T_e}$ obtained for the Fourier transform of a signal sampled at the Nyquist rate), the discrete version of the WVD may be affected by a spectral aliasing, in particular if the signal $x$ is real-valued and sampled at the Nyquist rate. Two alternatives to this problem can be found. The first one consists in oversampling the signal by a factor of at least 2, and the second one in using the analytic signal. Indeed, as its bandwidth is half the one of the real signal, the aliasing will not take place in the useful spectral domain $[0,1/2]$ of this signal. This second solution presents another advantage: since the spectral domain is divided by two, the number of components in the time-frequency plane is also divided by two. Consequently, the number of interference terms decreases significantly. To illustrate this phenomenon, we consider the WVD of the real part of a signal composed of two atoms (see fig. 4.6): >> sig=atoms(128,[32,0.15,20,1;96,0.32,20,1]); >> tfrwv(real(sig)); Figure 4.6: WVD of a real signal composed of 2 gaussian atoms : when the analytic signal is not considered, spectral aliasing and additional interferences appear in the time-frequency plane We can see that four signal terms are present instead of two, due to the spectral aliasing. Besides, because of the components located at negative frequencies (between -1/2 and 0), additional interference terms are present. If we now consider the WVD of the same signal, but in its complex analytic form (see fig. 4.7), >> tfrwv(sig); Figure 4.7: WVD of the previous signal, but in its analytic form the aliasing effect has disappeared, as well as the terms corresponding to interferences between negative- and positive- frequency components.
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