## 3 Time Frequency Transforms
If the inverse Fourier transform of the instantaneous auto-correlation function \(R(t,\tau)\) is taken with respect to \(t\) instead of \(\tau\), the ambiguity function can be obtained as follows:  
\[AF(\tau,\uptheta)=\int_{-\infty}^{+\infty}R(t,\tau)e^{j\uptheta t}\text{d}t= \int_{-\infty}^{+\infty}y_{a}(t+\frac{\tau}{2})y_{a}^{*}(t-\frac{\tau}{2})e^ {j\uptheta t}\text{d}t \tag{17}\]  
The ambiguity function can be used to monitor the disturbing effect in the received GNSS signals. The WVD has a number of desirable properties that make it a good indicator of how the energy of the signal can be viewed as a function of time and frequency. First, the WVD of any signal is always real. Second, it satisfies the time marginal condition:  
\[\int_{-\infty}^{+\infty}WVD(t,\upomega)\text{d}\upomega=|y_{a}(t)|^{2} \tag{18}\]  
That is, by summing the TF distribution over all frequencies, the instantaneous energy of the signal at a particular time instant can be obtained. Similarly, the WVD also satisfies the frequency marginal condition:  
\[\int_{-\infty}^{+\infty}WVD(t,\upomega)\text{d}t=|Y_{a}(\upomega)|^{2} \tag{19}\]  
In this case, by summing the TF distribution over all time, the power spectrum of the signal at a particular frequency can be obtained. Although WVD has many good properties and provides nearly the best resolution among all of the TF techniques, its main drawback comes from undesirable cross-term interference. The WVD is said to be bilinear, because the analyzed signal enters twice in its calculation. Consider the signal \(y(t)=y_{1}(t)+y_{2}(t)\), where \(y(t)\), \(y_{1}(t)\) and \(y_{2}(t)\) are analytic. Expanding the instantaneous auto-correlation function of \(y(t)\), we can obtain:  
\[R_{y}(t,\uptau)=R_{y_{1}}(t,\uptau)+R_{y_{2}}(t,\uptau)+R_{y_{1}y_{2}}(t, \uptau)+R_{y_{2}y_{1}}(t,\uptau) \tag{20}\]  
where \(R_{y_{1}y_{2}}(t,\uptau)\) and \(R_{y_{2}y_{1}}(t,\uptau)\) are the instantaneous cross-correlation functions (e.g., \(R_{y_{1}y_{2}}(t,\uptau)=y_{1}(t+\frac{\uptau}{2})y_{2}^{*}(t-\frac{\uptau}{2})\)). Taking Fourier transforms of Equation (20) with respect to \(\uptau\), it is easy to know that:  
\[WVD_{y}(t,\upomega)=WVD_{y_{1}}(t,\upomega)+WVD_{y_{2}}(t,\upomega)+2\text{ Re}\{WVD_{y_{1}y_{2}}(t,\upomega)\} \tag{21}\]  
where \(WVD_{y_{1}}(t,\upomega)\) and \(WVD_{y_{2}}(t,\upomega)\) are the WVDs of \(y_{1}(t)\) and \(y_{2}(t)\), respectively, and the last term is the cross-WVD (XWVD) between \(y_{1}(t)\) and \(y_{2}(t)\), provided as:  
\[STFT(t,\upomega)=\frac{1}{2\uptau}e^{-j\upomega t}\int_{-\infty}^{+\infty}Y_{ a}(\upomega^{\prime})H(\upomega^{\prime}-\upomega)e^{j\upomega^{\prime}t} \text{d}\upomega^{\prime} \tag{13}\]0  
Thus, the WVD of the sum of two signals is not the sum of their corresponding WVDs, but also of their XWVDs. This means that the spectrum energy density of the sum of two signals does not reduce to the sum of the individual densities (unless the signals are spectrally disjoint). If \(y_{1}(t)\) and \(y_{2}(t)\) are mono-component signals, \(WVD_{y_{1}}(t,\upomega)\) and \(WVD_{y_{2}}(t,\upomega)\) are the auto-terms, while \(2\text{Re}\{WVD_{y_{1}y_{2}}(t,\upomega)\}\) is a cross-term. As a result, if a signal contains more than one component, in the TF plane, its WVD suffers from spurious features containing cross-terms that occur halfway between each pair of auto-terms. The magnitude of these oscillatory cross-terms can be twice as large as the auto-terms, and they do not possess any physical meaning. As an example, the time-domain signal, which contains four Gaussian components, is shown in \[STFT(t,\upomega)=\frac{1}{2\uptau}e^{-j\upomega t}\int_{-\infty}^{+\infty}Y_{ a}(\upomega^{\prime})H(\upomega^{\prime}-\upomega)e^{j\upomega^{\prime}t} \text{d}\upomega^{\prime} \tag{13}\]1 The WVD of this signal is provided in Figure 2a, and correspondingly, the contour of the computed WVD is presented in Figure 2b.