## 2 Concept of Time Frequency Analysis and Processing
### Choi-Williams Distribution  
Assume the signal to be analyzed is \(s(u)\), [PERSON] [12] introduced the core function \(\phi(\theta,\tau)\), and defined the general form of time frequency distribution of signal \(s(u)\) as  
\[P(t,\omega)=\frac{1}{4\pi^{2}}\iiint e^{-j\theta t-j\tau\omega+j\theta u}\phi( \theta,\tau)s^{*}(u-\frac{\tau}{2})s(u+\frac{\tau}{2})dud\tau d\theta \tag{1}\]where \(P(t,\omega)\) is the time frequency distribution of signal \(s(u)\), \(t\) and \(\omega\) are the instant time and instant frequency respectively. \(\theta\) and \(\tau\) are two parameters of the core function \(\phi\). Based on the general form of time frequency distribution, [PERSON] and [PERSON] argued that instead of devising procedures to eliminate the cross term from the Wigner distribution, one could devise a distribution of which the spurious values were minimal. Consider a signal which is made up of components \(s_{k}(t)\),  
\[s(t)=\sum_{k=1}^{N}s_{k}(t) \tag{2}\]  
Substituting (2) into (1), one obtains that  
\[P(t,\omega)=\sum_{k=1}^{N}P_{kk}(t,\omega)+\sum_{k,l=1,k\
eq l}^{N}P_{kl}(t,\omega) \tag{3}\]  
where  
\[P_{kl}(t,\omega)= \frac{1}{4\pi^{2}}\iiint e^{-\beta t-j\tau\omega+j\theta u}\phi( \theta,\tau)\cdot s_{k}^{*}(u-\frac{\tau}{2})s_{l}(u+\frac{\tau}{2})dud\tau d\theta \tag{4}\]  
[PERSON] and [PERSON] realized that by a judicious choice of the kernel, one could minimize the cross terms while retaining the desirable properties of the self terms. They found a particularly good choice for the kernel [32],  
\[\phi(\theta,\tau)=e^{-\theta^{2}\tau^{2}/\sigma} \tag{5}\]  
where \(\sigma\) is a constant. Substituting (5) into (1) and integrating over \(\theta\), one obtains  
\[P_{cw}(t,\omega)= \frac{1}{4\pi^{3/2}}\iint\frac{1}{\sqrt{\tau^{2}/\sigma}}e^{-[(u- t)^{2}/(4\tau^{2}/\sigma)]-j\tau\omega}s^{*}(u-\frac{\tau}{2})s(u+\frac{\tau}{2})dud\tau \tag{6}\]  
The ability to suppress the cross terms comes by means of controlling \(\sigma\). From (6), each value of the time-indexed autocorrelation function \(s^{*}\left(t-\tau/2\right)s(t+\tau/2)\) is obtained from a set of neighboring samples with selective weights \(e^{-[(u-t)^{2}/(4\tau^{2}/\sigma)]-j\tau\omega}\), so that \(s^{*}(u-\frac{\tau}{2})s(u+\frac{\tau}{2})\) has a large weight when \(u\) is close to \(t\), and a small weight when \(u\) is far away from \(t\). Furthermore, the size of the neighbor set is controlled by the variable \(\tau\), so that the size is increased for a large value of \(\tau\) and decreased for a small value of \(\tau\), and \(\sigma\) can be used to control the relative importance of \(\tau\). By designing the weight, [PERSON] and [PERSON] reduced the spurious terms effectively. Figure 1 shows the time frequency distributions of a 4-QAM modulated signal which is the addition of two separated signals with center frequencies of 11.52 MHz and 21.52 MHz, respectively. The four time frequency analysis methods are WVD [12], STFT [12], CWD [32], and SPWVD (Smoothed Pseudo Wigner-Ville Distribution, [33]). The simulation parameters are listed in Table 1. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Values** \\ \hline Pulse shaping filter & raised cosine roll-off filter \\ Raised cosine roll-off factor & 0.22 \\ Modulation type & 4-QAM \\ Symbol time period \(T_{b}\) & 0.26042 μs \\ Truncated length of Raised cosine pulse & 6\(T_{b}\) \\ Center frequency of the first signal & 11.52 MHz \\ Center frequency of the second signal & 21.52 MHz \\ Analysis window type & Hamming \\ Analysis window length & 4/3\(T_{b}\) \\ \(\sigma\) for Choi-Williams distribution & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Simulation Parameters. In principle, there should be only two signal components on the time frequency plane whose center frequencies are 11.52 MHz and 21.52 MHz with the bandwidths of 5 MHz, respectively. There should be no cross terms exist in other positions. Figure 1b shows that the linear time frequency method STFT does not produce cross terms.