## 3 Testing Wide-Sense Stationary Assumption Fulfilment  
A random process is wide-sense stationary if its mean value and its autocorrelation function are time invariant. The relation between the WSS bandpass process \(r(t)\), and its complex envelope \(y(t)\), has been described in [17] as:  
\[r(t)=\text{Re}[y(t)e^{j2\pi f_{0}t}]=\frac{1}{2}y(t)e^{j2\pi f_{0}t}+\frac{1}{ 2}y^{*}(t)e^{-j2\pi f_{0}t} \tag{2}\]  
The complex envelope \(y(t)\) must be a zero-mean process in order that \(E[r(t)]\) be independent of \(t\). Moreover, the autocorrelation function \(R_{rr}(t+\Delta t,t)=E[r(t+\Delta t)r(t)]\) of bandpass signal \(r(t)\) is related to its complex envelope \(y(t)\) in the following manner:  
\[R_{rr}(t+\Delta t,t)=\frac{1}{2}\text{Re}[R_{yy}(\Delta t)e^{\alpha(\Delta t) }]+\frac{1}{2}\text{Re}[R_{yy*}(\Delta t)e^{j\alpha(\Delta t)+j\beta(t)}] \tag{3}\]  
where \(\alpha(\Delta t)=2\pi f_{0}\Delta t\), \(\beta(t)=4\pi f_{0}t\), \(R_{yy}(\Delta t)=E[y(t+\Delta t)y^{*}(t)]\), and \(R_{yy^{*}}(\Delta t)=E[y(t+\Delta t)y(t)]\). If \(r(t)\) is WSS, then the \(t\)-dependent term in Equation (3) vanishes; i.e., \(R_{yy^{*}}(\Delta t)=0\). The process \(y(t)=I(t)+jQ(t)\) is a complex value; thus, \(R_{yy^{*}}(\Delta t)\) can be rewritten as:  
\[R_{yy^{*}}(\Delta t)=R_{II}(\Delta t)-R_{QQ}(\Delta t)+j[R_{IQ}(\Delta t)+R_{ QI}(\Delta t)] \tag{4}\]  
where \(R_{II}(\Delta t)=E[I(t+\Delta t)I(t)]\) and \(R_{QQ}(\Delta t)=E[Q(t+\Delta t)Q(t)]\) are the autocorrelation functions of \(I(t)\) and \(Q(t)\), and \(R_{IQ}(\Delta t)=E[I(t+\Delta t)Q(t)]\) and \(R_{QI}(\Delta t)=E[Q(t+\Delta t)I(t)]\) are their cross-correlation functions. Thus, the condition for wide-sense stationarity of bandpass signal \(r(t)\) requires that:  
\[R_{II}(\Delta t)=R_{QQ}(\Delta t)\quad\text{and}\quad R_{IQ}(\Delta t)=-R_{ QI}(\Delta t) \tag{5}\]  
Moreover, the complex envelope \(y(t)\) should be a zero-mean process. According to [6], the similarity of autocorrelation functions \(R_{II}(\Delta t,\tau)\) and \(R_{QQ}(\Delta t,\tau)\) can be expressed as the cross-correlation coefficient:  
\[c_{IQ}=\max_{\Delta t^{\prime}}[E[R_{II}(\Delta t+\Delta t^{\prime}),R_{QQ}( \Delta t)]] \tag{6}\]and similarity of cross-correlation functions \(R_{IQ}(\Delta t,\tau)\) and \(R_{QI}(\Delta t,\tau)\), can be described as coefficient:  
\[c_{IQQI}=\max_{\Delta t^{\prime}}[E[R_{IQ}(\Delta t+\Delta t^{\prime}),-R_{QI}( \Delta t)]] \tag{7}\]  
In [14] the method of testing, if the impulse response measured with the probe signal \(s(t)\) satisfies the assumption of WSS, it is presented. It is based on testing the similarity of the autocorrelation functions: \(R_{II}^{h}(\Delta t,\tau)=E[h_{I}(t+\Delta t,\tau)h_{I}(t,\tau)]\), \(R_{QQ}^{h}(\Delta t,\tau)=E[h_{Q}(t+\Delta t,\tau)h_{Q}(t,\tau)]\), and cross-correlation functions: \(R_{IQ}^{h}(\Delta t)=E[h_{I}(t+\Delta t,\tau)h_{Q}(t,\tau)]\), \(R_{QI}^{h}(\Delta t)=E[h_{Q}(t+\Delta t,\tau)h_{I}(t,\tau)]\), defined similarly as in Equations (6) and (7) for in-phase \(h_{I}(t,\tau)\) and quadrature \(h_{Q}(t,\tau)\) components of the impulse response \(h(t,\tau)\). The cross-correlation coefficients \(c_{IQ}^{h}\) and \(c_{IQQI}^{h}\) are calculated as functions of delay \(\tau\):  
\[c_{IQ}^{h}(\tau) =\max_{\Delta t^{\prime}}[E[R_{II}^{h}(\Delta t+\Delta t^{\prime},\tau),R_{QQ}^{h}(\Delta t,\tau)]] \tag{8}\] \[c_{IQQI}^{h}(\tau) =\max_{\Delta t^{\prime}}[E[R_{IQ}^{h}(\Delta t+\Delta t^{\prime },\tau),-R_{QI}^{h}(\Delta t,\tau)]] \tag{9}\]  
For each TVIR, mean values of these coefficients can be obtained as: \(\hat{c}_{IQ}^{h}=E[c_{IQ}^{h}(\tau)]\) and \(\hat{c}_{IQQI}^{h}=E[c_{IQQI}^{h}(\tau)]\). The calculation of these coefficients requires the calculation of the TVIR estimate first. As mentioned in Section 2, the DFT transform is usually used to implement the matched filtering algorithm.