An accurate analysis of the nonlinear power amplifier effects on SC-FDMA signals

Abstract

Single carrier-frequency division multiple access (SC-FDMA) is a multiple access technique in broadband wireless networks which has been adapted by 3GPP for uplink transmission in 4G mobile communications. In this paper, the nonlinear effect of practical power amplifier (PA) is studied on the power allocated SC-FDMA signals. The interference power on the estimated symbols of all users are derived by two approaches based on the polynomial model of nonlinear PA and allocated power of subcarriers. In the first approach, an accurate analysis is followed.An approximation of the accurate result is presented in the second approach to provide a closed form formula. A simulation study is conducted to verify the analytical outcomes. The simulation and the exact analytical results are significantly matched. Conversely, the approximate relations are extremely suitable for allocating power in system design due to their closed form nature where they provide acceptable accuracy for practical applications.

Appendices

Appendix 1: Deriving $\sigma _{{{k}},{{n}_{1}},{{n}_{2}},r'}^{k',r}$

For accurate calculation, it is necessary to sperate the summation on $q_1$ and $O_1$ in (16) when these parameters are equal to k. For this, according to (16), the $F_1$, $F_2$ and $F_3$ functions are defined as

$$\begin{aligned}&\sigma _{{{k}},{{n}_{1}},{{n}_{2}},r'}^{k',r}\nonumber \\ {}&= \frac{1}{{{(UN)}^{4}}{{N}^{2}}}\sqrt{{{p}_{{{n}_{1}}-Nr'}}} \sqrt{{{p}_{{{n}_{2}}-Nr'}}}\sum \limits _{s'=0}^{N-1} {\sum \limits _{s=0}^{N-1}{\frac{1}{\sqrt{{{p}_{s,r}}{{p}_{s',r}}}}}} \nonumber \\&E\left[ |{{a}_{k}}{{|}^{2}}\left( \sum \limits _{{{q}_{1}}=0}^{N-1}{|{{a}_{{{q}_{1}}}} {{|}^{2}}{{F}_{1}}\left( {{q}_{1}},{{n}_{1}}-Nr',s\right) } \right. \right. \nonumber \\&\sum \limits _{{{O}_{1}}=0}^{N-1}{|{{a}_{{{O}_{1}}}}{{|}^{2}} {{F}_{2}}\left( {{O}_{1}},{{n}_{2}}-Nr',s'\right) }+ \sum \limits _{{{q}_{1}}=0}^{N-1} {\sum \limits _{{{O}_{1}}=0}^{N-1}} \nonumber \\&\left. {|{{a}_{{{q}_{1}}}}{{|}^{2}}|{{a}_{{{O}_{1}}}}{{|}^{2}}{{F}_{3}} \left( {{q}_{1}},{{O}_{1}},{n}_{1}{-}Nr',{n}_{2}{-}Nr',s,s'\right) }\right) \nonumber \\&\left. {{e}^{\frac{j2\pi }{N}(-k({{n}_{1}}-Nr')-sk'+k({{n}_{2}}-Nr')-s'k')}}\right] \end{aligned}$$

(22)

where $p_{i,j}=0$ for $i<0$ and $i>N-1$ and

$$\begin{aligned}&{{F}_{1}}({{q}_{1}},{{n}_{1}}-Nr',s)\nonumber \\&=\sum \limits _{s1=r'N}^{{r}'N+N-1}{\sqrt{{{p}_{{{s}_{1}}-r'N,r'}}} \sqrt{{{p}_{{{s}_{1}}+{{n}_{1}}-s-r'N,r'}}}{{e}^{-\frac{j2\pi }{N}{{q}_{1}} (s-{{n}_{1}})}}}\nonumber \\&{{F}_{2}}\left( {{O}_{1}},{{n}_{2}}-Nr',s'\right) =\nonumber \\&\sum \limits _{{{l}_{1}}={r}'N}^{{r}'N+N-1}{\sqrt{{{p}_{{{l}_{1}}-r'N,r'}}} \sqrt{{{p}_{{{l}_{1}}-{{n}_{2}}+s'-r'N,r'}}}{{e}^{-\frac{j2\pi }{N}{{O}_{1}} ({{n}_{2}}-s')}}}\nonumber \\&{{F}_{3}}\left( {{q}_{1}},{{O}_{1}},{{n}_{1}}-Nr',{{n}_{2}}-Nr',s,s'\right) =\sum \limits _{{{s}_{1}}=r'N}^{{r}'N+N-1}{{}} \nonumber \\&\sum \limits _{{{l}_{1}}={r}'N}^{{r}'N+N-1}{\sqrt{{{p}_{{{s}_{1}}-r'N,r'}}} \sqrt{{{p}_{{{l}_{1}}-r'N,r'}}}\sqrt{{{p}_{{{s}_{1}}+{{n}_{1}}-s-r'N,r'}}}} \nonumber \\&\sqrt{{{p}_{{{l}_{1}}-{{n}_{2}}+s'-r'N,r'}}}{{e}^{-\frac{j2\pi }{N}{{q}_{1}}({{s}_{1}}-{{l}_{1}}+{{n}_{2}}-s'-r'N)}}\nonumber \\&{{e}^{\frac{j2\pi }{N}{{O}_{1}}(s1+{{n}_{1}}-s-{{l}_{1}}-r'N)}} \end{aligned}$$

(23)

The final value of $\sigma _{{{k}},{{n}_{1}},{{n}_{2}},r'}^{k',r}$ by considering different cases in which $q_1$ and $O_1$ are equal to k or one of them is equal to k, are represented in (17).

Appendix 2: Deriving $C_{{{k}_{1}},{{k}_{2}},{{n}_{1}},{{n}_{2}},r'}^{k',r}$

In this subsection, $C_{{{k}_{1}},{{k}_{2}},{{n}_{1}},{{n}_{2}},r'}^{k',r}$ is calculated according to (19). Thus, each case in which Eq. (19) is not zero, is studied individually which is denoted by $A_i,i=1,\ldots ,4$.

In the first case $(q_2=k_1,q_1=k_2,O_1=O_2)$, for considering the exact value of $\gamma _4$, new functions $F_4$ and $F_5$ are defined as

$$\begin{aligned}&{{A}_{1}}=\frac{\sqrt{{{p}_{{{n}_{1}}-Nr',r'}}} \sqrt{{{p}_{{{n}_{2}}-Nr',r'}}}}{{{(UN)}^{4}}{{N}^{2}}} E\left[ \sum \limits _{s'=0}^{N-1}{\sum \limits _{s=0}^{N-1}}{\frac{|{{a}_{{{k}_{1}}}}{{|}^{2}} |a_{{{k}_{2}}}^{{}}{{|}^{2}}}{\sqrt{{{p}_{s,r}}{{p}_{s',r}}}}}\right. \nonumber \\&\sum \limits _{{{s}_{1}}={r}'N}^{{r}'N+N-1}{\sqrt{{{p}_{{{s}_{1}}-r'N,r'}}} \sqrt{{{p}_{{{s}_{1}}+{{n}_{1}}-s-r'N,r'}}}}{{e}^{-\frac{j2\pi }{N}({{k}_{2}} {{s}_{1}}{-}{{k}_{1}}({{s}_{1}}{+}{{n}_{1}}{-}s))}} \nonumber \\&\sum \limits _{{{l}_{1}}={r}'N}^{{r}'N+N-1}{\sum \limits _{{{O}_{1}}=0}^{N-1} {\sqrt{{p_{{{l}_{1}}-r'N,r'}}}\sqrt{{p_{{{l}_{1}}-{{n}_{2}}+s'-r'N,r'}}}}} |{{a}_{{{O}_{1}}}}{{|}^{2}}\nonumber \\&\left. {{e}^{-\frac{j2\pi }{N}{{O}_{1}}\left( {{n}_{2}}-s'\right) }}{{e}^ {\frac{j2\pi }{N}\left( -{{k}_{1}}\left( {{n}_{1}}-Nr'\right) -sk'+{{k}_{2}} \left( {{n}_{2}}-Nr'\right) -s'k'\right) }}\right] \nonumber \\&=\frac{\sqrt{{{p}_{{{n}_{1}}-Nr',r'}}}\sqrt{{{p}_{{{n}_{2}}-Nr',r'}}}}{{{(UN)}^{4}} {{N}^{2}}}E\left[ \sum \limits _{s'=0}^{N-1}{\sum \limits _{s=0}^{N-1} {\frac{|{{a}_{{{k}_{1}}}} {{|}^{2}}|a_{{{k}_{2}}}^{{}}{{|}^{2}}}{\sqrt{{{p}_{s,r}}{{p}_{s',r}}}}}} \right. \nonumber \\&|{{a}_{{{O}_{1}}}}{{|}^{2}}{{F}_{4}}({{k}_{1}},{{k}_{2}},{{n}_{1}},s){{F}_{5}} \left( {{O}_{1}},{{n}_{2}},s'\right) \nonumber \\&\left. {{e}^{\frac{j2\pi }{N}\left( -{{k}_{1}}({{n}_{1}}-Nr'\right) -sk'+{{k}_{2}} \left( {{n}_{2}}-Nr'\right) -s'k')}}\right] \end{aligned}$$

(24)

where

$$\begin{aligned}&{{F}_{4}}({{k}_{1}},{{k}_{2}},{{n}_{1}},s)= \sum \limits _{{{s}_{1}}={r}'N}^{{r}'N+N-1}{\sqrt{{{p}_{{{s}_{1}}-r'N,r'}}}}\nonumber \\&\sqrt{{{p}_{{{s}_{1}}+{{n}_{1}}-s-r'N,r'}}}{{e}^{-\frac{j2\pi }{N}({{k}_{2}}{{s}_{1}}-{{k}_{1}}({{s}_{1}}+{{n}_{1}}-s))}} \nonumber \\&{{F}_{5}}\left( {{O}_{1}},{{n}_{2}},s'\right) \nonumber \\&=\sum \limits _{{{l}_{1}}={r}'N}^{{r}'N+N-1}{\sqrt{{{p}_{{{l}_{1}}-r'N,r'}}}\sqrt{{{p}_{{{l}_{1}}-{{n}_{2}}+s'-r'N,r'}}}{{e}^{-\frac{j2\pi }{N}{{O}_{1}}\left( {{n}_{2}}-s'\right) }}} \end{aligned}$$

(25)

The cases where $O_1$ is equal to $k_1$ or $k_2$ should be septated to consider the $\gamma _4$. Thus,

$$\begin{aligned}&{{A}_{1}}=\frac{1}{{{(UN)}^{4}}{{N}^{2}}}\sqrt{{{p}_{{{n}_{1}}-Nr',r'}}} \sqrt{{{p}_{{{n}_{2}}-Nr',r'}}} \nonumber \\&\sum \limits _{s'=0}^{N-1}{\sum \limits _{s=0}^{N-1}{\frac{1}{\sqrt{{{p}_{s,r}}{{p}_{s',r}}}}}} \left( {{\gamma }_{2}}{{\gamma } _{4}}{{F}_{4}}({{k}_{1}},{{k}_{2}},{{n}_{1}},s){{F}_{5}} \left( {{k}_{1}},{{n}_{2}},s'\right) \right. \nonumber \\&+\,{{\gamma }_{2}}{{\gamma }_{4}}{{F}_{4}}({{k}_{1}},{{k}_{2}},{{n}_{1}} ,s){{F}_{5}}\left( {{k}_{2}},{{n}_{2}},s'\right) \nonumber \\&\left. +\,\gamma _{2}^{3} {\mathop{\mathop\sum\limits _{{{O}_{1}}=0}}\limits_{{{O}_{1}}\ne {{k}_{1}},{{k}_{2}}}}^{N-1}{{{F}_{4}}({{k}_{1}},{{k}_{2}},{{n}_{1}},s) {{F}_{5}}\left( {{O}_{1}},{{n}_{2}},s'\right) }\right) \nonumber \\&\left. {{e}^{\frac{j2\pi }{N}(-{{k}_{1}}\left( {{n}_{1}}-Nr'\right) -sk' +{{k}_{2}}\left( {{n}_{2}}-Nr'\right) -s'k')}}\right] \end{aligned}$$

(26)

Similarly, in the second one $(O_2=k_1,O_1=k_2,q_1=q_2)$, to sperate the cases in which $q_1$ is equal to $k_1$ or $k_2$, the function $F_6$ is defined as

$$\begin{aligned}&{{A}_{2}}=\frac{\sqrt{{{p}_{{{n}_{1}}-Nr',r'}}} \sqrt{{{p}_{{{n}_{2}}-Nr',r'}}} }{{{(UN)}^{4}}{{N}^{2}}} E\left[ \sum \limits _{s'=0}^{N-1}{\sum \limits _{s=0}^{N-1}} {\frac{|{{a}_{{{k}_{1}}}}{{|}^{2}}|a_{{{k}_{2}}}^{{}}{{|}^{2}}}{\sqrt{{{p}_{s,r}}{{p}_{s',r}}}}}\right. \nonumber \\&\sum \limits _{{{l}_{1}}={r}'N}^{{r}'N+N-1}{\sqrt{{{p}_{{{l}_{1}}-r'N,r'}}} \sqrt{{{p}_{{{l}_{1}}-{{n}_{2}}+s'-r'N,r'}}}} {{e}^{-\frac{j2\pi }{N} \left( {{k}_{2}}{{l}_{1}}-{{k}_{1}}\left( {{l}_{1}}-{{n}_{2}}+s'\right) \right) }} \nonumber \\&\sum \limits _{{{s}_{1}}={r}'N}^{{r}'N+N-1}{\sum \limits _{{{q}_{1}}=0}^{N-1} {\sqrt{{{p}_{{{s}_{1}}-r'N,r'}}}\sqrt{{{p}_{{{s}_{1}}+{{n}_{1}}-s-r'N,r'}}}}} |{{a}_{{{q}_{1}}}}{{|}^{2}} \nonumber \\&\left. {{e}^{-\frac{j2\pi }{N}{{q}_{1}}(s-{{n}_{1}})}}{{e}^{\frac{j2\pi }{N} \left( -{{k}_{1}}\left( {{n}_{1}}-Nr'\right) -sk'+{{k}_{2}}\left( {{n}_{2}}-Nr'\right) -s'k'\right) }}\right] \nonumber \\&=\frac{\sqrt{{{p}_{{{n}_{1}}-Nr',r'}}}\sqrt{{{p}_{{{n}_{2}}-Nr',r'}}}}{{{(UN)}^{4}} {{N}^{2}}}E\left[ \sum \limits _{s'=0}^{N-1}{\sum \limits _{s=0}^{N-1} {\frac{|{{a}_{{{k}_{1}}}} {{|}^{2}}|a_{{{k}_{2}}}^{{}}{{|}^{2}}}{\sqrt{{{p}_{s,r}}{{p}_{s',r}}}}}} \right. \nonumber \\&\sum \limits _{{{q}_{1}}=0}^{N-1}{|{{a}_{{{q}_{1}}}}{{|}^{2}}{{F}_{6}}\left( {{k}_{1}}, {{k}_{2}},{{n}_{2}}-Nr',s'\right) {{F}_{7}}\left( {{q}_{1}},{{n}_{1}}-Nr',s\right) } \nonumber \\&\left. {{e}^{\frac{j2\pi }{N}\left( -{{k}_{1}}\left( {{n}_{1}}-Nr'\right) -sk'+{{k}_{2}}\left( {{n}_{2}}-Nr'\right) -s'k'\right) }}\right] \end{aligned}$$

(27)

where

$$\begin{aligned}&{{F}_{6}}({{k}_{1}},{{k}_{2}},{{n}_{2}}-Nr',s') \nonumber \\&= \sum \limits _{{{l}_{1}}={r}'N}^{{r}'N+N-1}{\sqrt{{{p}_{{{l}_{1}}-r'N,r'}}}\sqrt{{{p}_{{{l}_{1}}-{{n}_{2}}+s'-r'N,r'}}}}{{e}^{-\frac{j2\pi }{N}({{k}_{2}}{{l}_{1}}-{{k}_{1}}({{l}_{1}}-{{n}_{2}}+s'))}} \nonumber \\&{{F}_{7}}({{q}_{1}},{{n}_{1}}-Nr',s) \nonumber \\&=\sum \limits _{{{s}_{1}}={r}'N}^{{r}'N+N-1}{\sqrt{{{p}_{{{s}_{1}}-r'N,r'}}}\sqrt{{{p}_{{{s}_{1}}+{{n}_{1}}-s-r'N,r'}}}{{e}^{-\frac{j2\pi }{N}{{q}_{1}}(s-{{n}_{1}})}}} \end{aligned}$$

(28)

The separation for considering the exact value of $\gamma _4$ is as follows

$$\begin{aligned}&{{A}_{2}}=\frac{\sqrt{{{p}_{{{n}_{1}}-Nr',r'}}}\sqrt{{{p}_{{{n}_{2}}-Nr',r'}}}}{{{(UN)}^{4}}{{N}^{2}}}\sum \limits _{s'=0}^{N-1}{\sum \limits _{s=0}^{N-1}{\frac{1}{\sqrt{{{p}_{s,r}}{{p}_{s',r}}}}}} \nonumber \\&({{\gamma }_{2}}{{\gamma }_{4}}{{F}_{6}}({{k}_{1}},{{k}_{2}},{{n}_{2}}-Nr',s'){{F}_{7}}({{k}_{2}},{{n}_{1}}-Nr',s) \nonumber \\&+{{\gamma }_{2}}{{\gamma }_{4}}{{F}_{6}}({{k}_{1}},{{k}_{2}},{{n}_{2}}-Nr',s'){{F}_{7}}({{k}_{1}},{{n}_{1}}-Nr',s) \nonumber \\&+\gamma _{2}^{3}\mathop{\sum\limits_{{{q}_{1}}=0}}\limits_{{{q}_{1}}\ne {{k}_{1}},{{k}_{2}}}^{N-1}{{{F}_{6}}({{k}_{1}},{{k}_{2}},{{n}_{2}}-Nr',s'){{F}_{7}}({{q}_{1}},{{n}_{1}}-Nr',s)}) \nonumber \\&{{e}^{\frac{j2\pi }{N}(-{{k}_{1}}({{n}_{1}}-Nr')-sk'+{{k}_{2}}({{n}_{2}}-Nr')-s'k')}} \end{aligned}$$

(29)

In the third case ($q_2=k_1,O_1=k_2,q_1=O_2$), the cases where $q_1$ is equal to $k_1$ or $k_2$ should be separated for considering exact value of $\gamma _4$. But note that these cases are considered in $A_1$. Thus, these cases should be omitted from the calculation which leads to the definition of function $F_8$ as

$$\begin{aligned}&{{A}_{3}}=\frac{1}{{{(UN)}^{4}}{{N}^{2}}}\sqrt{{{p}_{{{n}_{1}}-Nr',r'}}}\sqrt{{{p}_{{{n}_{2}}-Nr',r'}}} \nonumber \\&E\left[ \sum \limits _{s'=0}^{N-1}{\sum \limits _{s=0}^{N-1}{\frac{|{{a}_{{{k}_{1}}}}{{|}^{2}}|a_{{{k}_{2}}}^{{}}{{|}^{2}}}{\sqrt{{{p}_{s,r}}{{p}_{s',r}}}}}} \sum \limits _{{{l}_{1}}={r}'N}^{{r}'N+N-1}{\sum \limits _{{{s}_{1}}={r}'N}^{{r}'N+N-1}{\mathop{\sum\limits_{{{q}_{1}}=0}}\limits_{{{q}_{1}}\ne {{k}_{1}},{{k}_{2}}}^{N-1}{{}}}}\right. \nonumber \\&\sqrt{{{p}_{{{l}_{1}}-{{n}_{2}}+s'-r'N,r'}}}\sqrt{{{p}_{{{l}_{1}}-r'N,r'}}}\sqrt{{{p}_{{{l}_{1}}-{{n}_{2}}+s'-r'N,r'}}} \nonumber \\&\sqrt{{{p}_{{{l}_{1}}-r'N,r'}}}{{e}^{-\frac{j2\pi }{N} \left( {{q}_{1}}{{s}_{1}}-{{k}_{1}}({{s}_{1}}+{{n}_{1}}-s)+{{k}_{2}}{{l}_{1}}-{{q}_{1}}({{l}_{1}}-{{n}_{2}}+s')\right) }} \nonumber \\&\left. {{e}^{\frac{j2\pi }{N}(-{{k}_{1}}({{n}_{1}}-Nr')-sk'+{{k}_{2}}({{n}_{2}}-Nr')-s'k')}}\right] \nonumber \\&= \frac{\sqrt{{{p}_{{{n}_{1}}-Nr',r'}}}\sqrt{{{p}_{{{n}_{2}}-Nr',r'}}}}{{{(UN)}^{4}}{{N}^{2}}}E\left[ \sum \limits _{s'=0}^{N-1}{\sum \limits _{s=0}^{N-1}} {\frac{|{{a}_{{{k}_{1}}}}{{|}^{2}}|a_{{{k}_{2}}}^{{}}{{|}^{2}}}{\sqrt{{{p}_{s,r}}{{p}_{s',r}}}}}\right. \nonumber \\& \mathop{\sum\limits_{{{q}_{1}}=0}}\limits_{{{q}_{1}}\ne {{k}_{1}},{{k}_{2}}}^{N-1}{|{{a}_{{{q}_{1}}}}{{|}^{2}}{{F}_{8}}(k_1,k_2,q_1,n_1{-}Nr',n_2{-}Nr',s,s')}\nonumber \\&\left. {{e}^{\frac{j2\pi }{N}(-{{k}_{1}}({{n}_{1}}-Nr')-sk'+{{k}_{2}}({{n}_{2}}-Nr')-s'k')}}\right] \end{aligned}$$

(30)

where

$$\begin{aligned}&{{F}_{8}}({{k}_{1}},{{k}_{2}},{{q}_{1}},{{n}_{1}}-Nr',{{n}_{2}}-Nr',s,s') \nonumber \\&= \sum \limits _{{{l}_{1}}={r}'N}^{{r}'N+N-1}{\sum \limits _{{{s}_{1}}={r}'N}^{{r}'N+N-1}{\sqrt{{{p}_{{{l}_{1}}-r'N,r'}}}}} \nonumber \\&\sqrt{{{p}_{{{l}_{1}}-{{n}_{2}}+s'-r'N,r'}}}\sqrt{{{p}_{{{l}_{1}}-r'N,r'}}}\sqrt{{{p}_{{{l}_{1}}-{{n}_{2}}+s'-r'N,r'}}} \nonumber \\&{{e}^{-\frac{j2\pi }{N}({{q}_{1}}{{s}_{1}}-{{k}_{1}}({{s}_{1}}+{{n}_{1}}-s)+{{k}_{2}}{{l}_{1}}-{{q}_{1}}({{l}_{1}}-{{n}_{2}}+s'))}} \end{aligned}$$

(31)

Thus, the final value of $A_3$ is as follows

$$\begin{aligned}&A_3= \frac{\sqrt{{{p}_{{{n}_{1}}-Nr',r'}}}\sqrt{{{p}_{{{n}_{2}}-Nr',r'}}}}{{{(UN)}^{4}}{{N}^{2}}}\sum \limits _{s'=0}^{N-1}{\sum \limits _{s=0}^{N-1}{\frac{\gamma _2^3}{\sqrt{{{p}_{s,r}}{{p}_{s',r}}}}}} \nonumber \\&\mathop{\sum\limits_{{{q}_{1}}=0}}\limits_{{{q}_{1}}\ne {{k}_{1}},{{k}_{2}}}^{N-1}{{{F}_{8}}({{k}_{1}},{{k}_{2}},{{q}_{1}},{{n}_{1}}-Nr',{{n}_{2}}-Nr',s,s')} \nonumber \\&{{e}^{\frac{j2\pi }{N}(-{{k}_{1}}({{n}_{1}}-Nr')-sk'+{{k}_{2}}({{n}_{2}}-Nr')-s'k')}} \end{aligned}$$

(32)

Again, in the forth case $O_2=k_1,q_1=k_2,O_1=q_2$, the cases where $q_1$ is equal to $k_1$ or $k_2$ are calculated in $A_2$. Thus, theses cases are omitted by defining $F_9$ as

$$\begin{aligned}&A_4=\frac{\sqrt{{{p}_{{{n}_{1}}-Nr',r'}}}\sqrt{{{p}_{{{n}_{2}}-Nr',r'}}}}{{{(UN)}^{4}}{{N}^{2}}}E \left[ \sum \limits _{s'=0}^{N-1}{\sum \limits _{s=0}^{N-1}{\frac{|{{a}_{{{k}_{1}}}}{{|}^{2}}|a_{{{k}_{2}}}^{{}}{{|}^{2}}}{\sqrt{{{p}_{s,r}}{{p}_{s',r}}}}}}\right. \nonumber \\&\sum \limits _{{{l}_{1}}={r}'N}^{{r}'N+N-1}{\sum \limits _{{{s}_{1}}={r}'N}^{{r}'N+N-1}{\mathop{\sum\limits_{{{q}_{1}}=0}}\limits_{{{q}_{1}}\ne {{k}_{1}},{{k}_{2}}}^{N-1}{{}}}}\sqrt{{{p}_{{{l}_{1}}-r'N,r'}}} \nonumber \\&\sqrt{{{p}_{{{l}_{1}}-{{n}_{2}}+s'-r'N,r'}}} \sqrt{{{p}_{{{l}_{1}}-r'N,r'}}}\sqrt{{{p}_{{{l}_{1}}-{{n}_{2}}+s'-r'N,r'}}}\nonumber \\&{{e}^{-\frac{j2\pi }{N}({{k}_{2}}{{s}_{1}}-{{q}_{1}}({{s}_{1}}+{{n}_{1}}-s)+{{q}_{1}}{{l}_{1}}-{{k}_{1}}({{l}_{1}}-{{n}_{2}}+s'))}} \nonumber \\&\left. {{e}^{\frac{j2\pi }{N}(-{{k}_{1}}({{n}_{1}}-Nr')-sk'+{{k}_{2}}({{n}_{2}}-Nr')-s'k')}}\right] \nonumber \\&= \frac{\sqrt{{{p}_{{{n}_{1}}-Nr',r'}}}\sqrt{{{p}_{{{n}_{2}}-Nr',r'}}}}{{{(UN)}^{4}}{{N}^{2}}}E \left[ \sum \limits _{s'=0}^{N-1}{\sum \limits _{s=0}^{N-1}{\frac{|{{a}_{{{k}_{1}}}}{{|}^{2}}|a_{{{k}_{2}}}^{{}}{{|}^{2}}}{\sqrt{{{p}_{s,r}}{{p}_{s',r}}}}}}\right. \nonumber \\&\mathop{\sum\limits_{{{q}_{1}}=0}}\limits_{{{q}_{1}}\ne {{k}_{1}},{{k}_{2}}}^{N-1}{|{{a}_{{{q}_{1}}}}{{|}^{2}}{{F}_{9}}({{k}_{1}},{{k}_{2}},{{q}_{1}},{{n}_{1}}-Nr',{{n}_{2}}-Nr',s,s')} \nonumber \\&\left. {{e}^{\frac{j2\pi }{N}(-{{k}_{1}}({{n}_{1}}-Nr')-sk'+{{k}_{2}}({{n}_{2}}-Nr')-s'k')}}\right] \nonumber \\ \end{aligned}$$

(33)

where

$$\begin{aligned}&{{F}_{9}}({{k}_{1}},{{k}_{2}},{{q}_{1}},{{n}_{1}}-Nr',{{n}_{2}}-Nr',s,s') \nonumber \\&= \sum \limits _{{{l}_{1}}={r}'N}^{{r}'N+N-1}{\sum \limits _{{{s}_{1}}={r}'N}^{{r}'N+N-1}{\sqrt{{{p}_{{{l}_{1}}-r'N,r'}}}}} \nonumber \\&\sqrt{{{p}_{{{l}_{1}}-{{n}_{2}}+s'-r'N,r'}}}\sqrt{{{p}_{{{l}_{1}}-r'N,r'}}}\sqrt{{{p}_{{{l}_{1}}-{{n}_{2}}+s'-r'N,r'}}} \nonumber \\&{{e}^{-\frac{j2\pi }{N}({{k}_{2}}{{s}_{1}}-{{q}_{1}}({{s}_{1}}+{{n}_{1}}-s)+{{q}_{1}}{{l}_{1}}-{{k}_{1}}({{l}_{1}}-{{n}_{2}}+s'))}} \end{aligned}$$

(34)

Thus, the final value of $A_4$ is as follows

$$\begin{aligned}&{{A}_{4}}=\frac{\sqrt{{{p}_{{{n}_{1}}-Nr',r'}}}\sqrt{{{p}_{{{n}_{2}}-Nr',r'}}}}{{{(UN)}^{4}}{{N}^{2}}}\sum \limits _{s'=0}^{N-1}{\sum \limits _{s=0}^{N-1}{\frac{\gamma _2^3}{\sqrt{{{p}_{s,r}}{{p}_{s',r}}}}}} \nonumber \\&\mathop{\sum\limits_{{{q}_{1}}=0}}\limits_{{{q}_{1}}\ne {{k}_{1}},{{k}_{2}}}^{N-1}{{{F}_{9}}({{k}_{1}},{{k}_{2}},{{q}_{1}},{{n}_{1}}-Nr',{{n}_{2}}-Nr',s,s')} \nonumber \\&{{e}^{\frac{j2\pi }{N}(-{{k}_{1}}({{n}_{1}}-Nr')-sk'+{{k}_{2}}({{n}_{2}}-Nr')-s'k')}} \nonumber \\ \end{aligned}$$

(35)