## Appendix A
\] (A-12)  
The first term in Equation A-12 is the cross-correlation of noise free interference signal \(i_{NF}(t)=e^{-at^{2}}\,\cos(t)\) and its delayed version \(i_{NF}(t-t_{d})=e^{-a(t-t_{d})^{2}}\cos(t-t_{d})\), which is calculated as follows:  
\[i_{NF}(t)\otimes i_{NF}(t-t_{d})=i_{NF}(t)*i_{NF}(t-t_{d})=\left(e^{-at^{2}} \cos t\right)*\left(e^{-a(t-t_{d})^{2}}\,\cos(t-t_{d})\right)\] (A-13)Well-known Fourier transform relationships useful for the calculation of Equation A-13 are:  
\[A(t)*B(t)\Leftrightarrow A(w)B(w)\] (A-14) \[A(t)B(t)\Leftrightarrow\frac{1}{2\pi}\,A(w)*B(w)\] (A-15) \[\frac{1}{|\alpha|}\,A\!\left(\frac{t}{\alpha}\right) \Leftrightarrow A(\alpha w)\] (A-16)  
where:  
\[A(t)\Leftrightarrow A(w)\text{ and }B(t)\Leftrightarrow B(w)\] (A-17)  
and \(w\) is the frequency domain variable. If we only consider the envelope of \(i_{NF}(t)=e^{-at^{2}}\cos t\), i.e. \(i_{NF,\,env}(t)=e^{-at^{2}}\) and use the Fourier transform relation:  
\[e^{-at^{2}}\Leftrightarrow\sqrt{\frac{\pi}{a}}e^{-\frac{w^{2}}{4a}}\] (A-18)  
then Fourier transform of Equation A-13 becomes:  
\[i_{NF,\,env}(t)*i_{NF,\,env}(t-t_{d})=\left(e^{-at^{2}}\right)*\left(e^{-a(t-t_ {d})^{2}}\right)\Leftrightarrow(\sqrt{\frac{\pi}{a}}e^{-\frac{w^{2}}{4a}}\sqrt {\frac{\pi}{a}}e^{-\frac{w^{2}}{4a}})\,e^{-jnt_{d}}=(\sqrt{\frac{\pi}{a}} \sqrt{\frac{\pi}{a}}e^{-\frac{(\sqrt{2}w)^{2}}{4a}})e^{-jnt_{d}}\] (A-19)  
Then \(i_{NF,\,env}(t)*i_{NF,\,env}(t-t_{d})\) is the inverse Fourier transform of \((\sqrt{\frac{\pi}{a}}\sqrt{\frac{\pi}{a}}e^{-\frac{(\sqrt{2}w)^{2}}{4a}})e^{- jnt_{d}}\), which is given as:  
\[i_{NF,\,env}(t)*i_{NF,\,env}(t-t_{d})=\sqrt{\frac{\pi}{a}}\,\frac{1}{\sqrt{2} }\,e^{-a(\frac{t-t_{d}}{\sqrt{2}})^{2}}\] (A-20)  
Before cross-correlation, from Equation A-10 and Equation A-11, peak value of \(i_{S}(t)\) is 1 and \(i_{S}(t)\) reduces down to \(e^{-1}\) of its peak value at:  
\[t=\pm\frac{1}{\sqrt{a}}\,(\text{for Equation A-10 })\text{ and }\,t=t_{d}\pm\frac{1}{\sqrt{a}}\,(\text{for Equation A-11})\] (A-21)  
After cross-correlation, from Equation A-20, \(e^{-1}\) of \(i_{S}\otimes i_{S}(t)\) happens at:  
\[t=t_{d}\pm\frac{\sqrt{2}}{\sqrt{a}}\] (A-22)  
And the coherence length of the cross-correlation(effective coherence length) has increased by a factor of \(\sqrt{2}\). In other words, \(L_{C,\,eff}=\)\(\sqrt{2}\)\(L_{C}\)  
The second part of appendix A shows that 14 dB _SNR_ improvement is obtained at zero order fringe peak after cross-correlation. Peak value (zero-order fringe peak) of cross-correlation \(i(t)\) happens when \(t=t_{d}\) and its value \(i_{S}*i_{R}(t=t_{d})\) is calculated by substituting \(t=t_{d}\) in Equation A-12 as follows:  
\[ \begin{split}& i_{S}*i_{R}(t_{d})=\int\limits_{-\infty}^{+ \infty}((e^{-a\lambda^{2}}\,\cos\lambda\ )\ (e^{-a(-\lambda)^{2}}\,\cos(-\lambda))\,d\lambda\ +\int\limits_{-\infty}^{+\infty}(e^{-a\lambda^{2}}\,\cos \lambda\ )\ (w_{S}(t_{d}-\lambda))\,d\lambda\\ &+\int\limits_{-\infty}^{+\infty}(w_{R}(\lambda))\ (e^{-a(- \lambda)^{2}}\,\cos(-\lambda))\,d\lambda\ +\int\limits_{-\infty}^{+\infty}w_{R}(\lambda)\ w_{S}(t_{d}-\lambda))\,d\lambda \end{split} \] (A-23)The first term of Equation A-23 is the signal component which is the energy of noise free interference signal \(i_{NF}(t)=e^{-at^{2}}\cos(t)\) and the other three terms of Equation A-23 are noise terms. Then the variance of three noise terms are given as:  
\[(2\int\limits_{-\infty}^{+\infty}(e^{-a\lambda^{2}}\cos\lambda\ )^{2}\ \ d\lambda)\times\sigma_{N}^{2}+(\int\limits_{-\infty}^{+\infty}w_{R}( \lambda)\ ^{2}\ d\lambda\ )\times\sigma_{N}^{2}\] (A-24)  
Then _SNR_ at the peak of cross-correlation \(i\left(t\right)\) is given as the ratio between magnitude of peak signal to the standard deviation of noise as shown in Equation A-25.