## Appendix A  
Simultaneous Equations (8) and (13), \(C_{z}(\widetilde{nT}_{c},\tau)\) can be derived as [21]:  
\[ \begin{array}{lll}C_{z}(\widetilde{nT}_{c},\tau)&=&\langle[z(nT_{c},\tau)- \langle z(nT_{c},\tau)\rangle][z(nT_{c}+\widetilde{nT}_{c},\tau)-\langle z(nT _{c}+\widetilde{nT}_{c},\tau)\rangle]\rangle\\ &=&\langle\langle[y(nT_{c},\tau)y^{*}(nT_{c},\tau)-\langle y(nT_{c},\tau)y^{* }(nT_{c},\tau)\rangle]\rangle\\ &\times[y(nT+\widetilde{nT}_{c},\tau)y^{*}(nT_{c}+\widetilde{nT}_{c},\tau)- \langle y(nT_{c}+\widetilde{nT}_{c},\tau)y^{*}(nT_{c}+\widetilde{nT}_{c},\tau )\rangle]\rangle\\ &=&\langle y(nT_{c},\tau)y^{*}(nT_{c},\tau)y(nT+\widetilde{nT}_{c},\tau)y^{* }(nT_{c}+\widetilde{nT}_{c},\tau)\rangle\\ &-\langle y(nT_{c},\tau)y^{*}(nT_{c},\tau)\rangle\langle y(nT+\widetilde{nT} _{c},\tau)y^{*}(nT_{c}+\widetilde{nT}_{c},\tau)\rangle\end{array}  \tag{13}\]In general, the complex waveforms follow the circular complex Gaussian statistics. The correlation function of the one-shot power waveforms can be simplified by using the complex Gaussian moment theorem [37]:  
\[ \begin{array}{l}\langle y(nT_{c},\tau)y^{*}(nT_{c},\tau)y(nT+\vec{n}T_{c},\tau )y^{*}(nT_{c}+\vec{n}T_{c},\tau)\rangle\\ =\langle y(nT_{c},\tau)y^{*}(nT_{c},\tau)\rangle\langle y(nT+\vec{n}T_{c},\tau )y^{*}(nT_{c}+\vec{n}T_{c},\tau)\rangle\\ +\langle y(nT_{c},\tau)y^{*}(nT_{c}+\vec{n}T_{c},\tau)\rangle\langle y^{*}(nT_ {c},\tau)y(nT+\vec{n}T_{c},\tau)\rangle\end{array}  \tag{10}\]  
Substituting Equation (10) into Equation (10), we can get:  
\[ \begin{array}{lcl}C_{z}(\vec{n}T_{c},\tau)&=&\langle y(nT_{c},\tau)y^{*}(nT _{c}+\vec{n}T_{c},\tau)\rangle\langle y^{*}(nT_{c},\tau)y(nT+\vec{n}T_{c},\tau )\rangle\\ &=&\left|C_{y}(\vec{n}T_{c},\tau)\right|^{2}\\ &&\\ C_{z}(0,\tau)&=&\left|C_{y}(0,\tau)\right|^{2}\\ &&\\ &=&\langle y(nT_{c},\tau)y^{*}(nT_{c},\tau)\rangle\langle y^{*}(nT_{c},\tau) y(nT_{c},\tau)\rangle\\ &=&\langle y(nT_{c},\tau)y^{*}(nT_{c},\tau)\rangle\langle y^{*}(nT_{c},\tau)y( nT_{c},\tau)\rangle\\ &=&\langle z(nT_{c},\tau)\rangle\langle z(nT_{c},\tau)\rangle\end{array}  \tag{11}\]  
According to Equations (10) and (14), the variance of the power waveforms after incoherent average can be expressed as [21]:  
\[ \begin{array}{lcl}\sigma_{Z}^{2}(\tau)&=&C_{Z}(0,\tau)\\ &=&\langle[Z(t,\tau)-\langle Z(t,\tau)\rangle][Z(t,\tau)-\langle Z(t,\tau) \rangle]\rangle\\ &=&\langle Z(t,\tau)Z(t,\tau)\rangle-\langle Z(t,\tau)\rangle\langle Z(t,\tau)\rangle \\ &=&\langle Z(t,\tau)Z(t,\tau)\rangle-\langle z(t,\tau)\rangle\langle z(t,\tau) \rangle\end{array}  \tag{12}\]  
Substitute Equation (10) into the first term of Equation (12), we can get:  
\[ \begin{array}{lcl}\langle Z(t,\tau)Z(t,\tau)\rangle&=&\frac{1}{N_{\rm I}^{2} }\left(\sum\limits_{i=0}^{N_{\rm I}-1}z(iT_{c},\tau)\sum\limits_{j=0}^{N_{\rm I }-1}z(jT_{c},\tau)\right)\\ &=&\frac{1}{N_{\rm I}^{2}}\sum\limits_{i=0}^{N_{\rm I}-1}\sum\limits_{j=0}^{N_ {\rm I}-1}\left\langle z(iT_{c},\tau)z(jT_{c},\tau)\right\rangle\end{array}  \tag{13}\]  
According to Equation (10), \(\langle z(iT_{c},\tau)z(jT_{c},\tau)\rangle\) can be represented as:  
\[ \begin{array}{lcl}\langle z(iT_{c},\tau)z(jT_{c},\tau)\rangle&=&C_{z}((i-j)T _{c},\tau)+\langle z(nT_{c},\tau)\rangle\langle z(nT_{c},\tau)\rangle\\ \\ \sigma_{Z}^{2}(\tau)&=&C_{Z}(0,\tau)\\ &=&\frac{1}{N_{\rm I}^{2}}\sum\limits_{i=0}^{N_{\rm I}-1}\sum\limits_{j=0}^{N_ {\rm I}-1}C_{z}((i-j)T_{c},\tau)\\ &=&\sum\limits_{k=-(N_{\rm I}-1)}^{N_{\rm I}-1}\frac{N_{\rm I}-k}{N_{\rm I}^{2 }}C_{z}(kT_{c},\tau)\\ &=&\frac{1}{N_{\rm I}}\sum\limits_{\vec{n}=-(N_{\rm I}-1)}^{N_{\rm I}-1}C_{z}( \vec{n}T_{c},\tau)\end{array}  \tag{14}\]  
Simultaneous Equations (11), (12) and (14), we can get:  
\[ \begin{array}{lcl}\frac{1}{N_{\rm eff}}&=&\frac{\sigma_{Z}^{2}(\tau)}{Z^{2}( \tau)}\\ &=&\frac{\sigma_{Z}^{2}(\tau)}{\langle z(nT_{c},\tau)\rangle\langle z(nT_{c}, \tau)\rangle}\\ &&\frac{N_{\rm I}^{-1}}{N_{\rm I}^{2}}\\ &=&\frac{n-(N_{\rm I}-1)}{N_{\rm I}}\left|C_{y}(nT_{c},\tau)\right|^{2}\\ \end{array}