It is observed from (3) that we wish to estimate the desired term

*S*(

*k*), from

*Y*(

*k*),

*k*=0,1,…,

*N*−1. In the framework of Wiener filter, the estimate of

*S*(

*k*), i.e.,

$\u015c\left(k\right)$, is determined by estimating a set of coefficients

${\left\{{\gamma}_{\mathrm{ki}}\right\}}_{i=0}^{N}$, in order to minimize the estimation mean square error (MSE) as

${\left\{{\widehat{\gamma}}_{\mathrm{ki},\phantom{\rule{0.3em}{0ex}}\text{opt}}\right\}}_{i=0}^{N}={\text{argmin}}_{{\left\{{\gamma}_{\mathrm{ki}}\right\}}_{i=0}^{N}}\mathbb{E}\left\{\left|S\right(k)-\u015c(k){|}^{2}\right\}$

(8)

$\phantom{\rule{0.3em}{0ex}}\mathrm{s.t.}\phantom{\rule{0.3em}{0ex}}\u015c\left(k\right)=\sum _{i=0}^{N-1}{\gamma}_{\mathrm{ki}}Y\left(i\right)+{\gamma}_{\mathrm{kN}},$

(9)

where

*γ*
_{
kN
} is a bias term that allows for nonzero means of

*S*(

*k*)and

*Y*(

*k*),

*k*=0,1,…,

*N*−1. Substituting (9) into (8) and setting the first derivative of the resulting (8) with respect to

*γ*
_{
kN
} to zero, we obtain the optimal estimate of

*γ*
_{
kN
}as

${\widehat{\gamma}}_{\mathrm{kN},\phantom{\rule{0.3em}{0ex}}\text{opt}}=\mathbb{E}\left\{S\right(k\left)\right\}-\sum _{i=0}^{N-1}{\gamma}_{\mathrm{ki}}\mathbb{E}\left\{Y\right(i\left)\right\}.$

(10)

Substituting (10) into (8), the MSE can be expressed as

$\begin{array}{ll}\hfill \mathbb{E}\left\{\left|S\right(k)-\u015c(k){|}^{2}\right\}=& {\Upsilon}_{k}^{H}{\Sigma}_{YY}{\Upsilon}_{k}-2\mathfrak{R}\left\{{\Upsilon}_{k}^{H}{\Sigma}_{YS\left(k\right)}\right\}\\ +{\Sigma}_{S\left(k\right)S\left(k\right)},\end{array}$

(11)

where

ϒ
_{
k
}=[

*γ*
_{
k 0},

*γ*
_{
k 1},…,

*γ*
_{
kN−1}]

^{
T
},

**Σ**
_{
Y
Y
} is the covariance matrix of

Y = [

*Y*(0),

*Y*(1),…,

*Y*(

*N*−1)]

^{
T
},

**Σ**
_{
Y S(k)} is the cross-covariance vector of

Y and

*S*(

*k*), Σ

_{
S(k)S(k)} is the variance of

*S*(

*k*). Setting the first derivative of (11) with respect to

ϒ
_{
k
} to zero, we obtain the optimal estimate of

ϒ
_{
k
}as

${\widehat{\Upsilon}}_{k,\mathrm{opt}}={\Sigma}_{YY}^{-1}{\Sigma}_{YS\left(k\right)}.$

(12)

Substituting (10) and (12) into (9), the optimal estimate of

*S*(

*k*) can thus be obtained as

${\u015c}_{\mathrm{opt}}\left(k\right)=\mathbb{E}\left\{S\right(k\left)\right\}+{\Sigma}_{YS\left(k\right)}^{H}{\Sigma}_{YY}^{-1}(Y-\mathbb{E}\{Y\left\}\right).$

(13)

Stacking all

*N* subcarriers, the optimal estimate of

S=[

*S*(0),

*S*(1),…,

*S*(

*N*−1)]

^{
T
} can be written in matrix form as

$\begin{array}{ll}\phantom{\rule{-12.0pt}{0ex}}{\widehat{S}}_{\text{opt}}& \phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}\left[\begin{array}{c}\mathbb{E}\left\{S\right(0\left)\right\}\\ \phantom{\rule{1em}{0ex}}\mathbb{E}\left\{S\right(1\left)\right\}\\ \vdots \\ \mathbb{E}\left\{S\right(N-1\left)\right\}\end{array}\right]+\left[\begin{array}{c}{\Sigma}_{YS\left(0\right)}^{H}\\ {\Sigma}_{YS\left(1\right)}^{H}\\ \vdots \\ {\Sigma}_{YS(N-1)}^{H}\end{array}\right]{\Sigma}_{YY}^{-1}(Y-\mathbb{E}\{Y\left\}\right)\phantom{\rule{2em}{0ex}}\\ \triangleq \mathbb{E}\left\{S\right\}+{\Sigma}_{SY}{\Sigma}_{YY}^{-1}(Y-\mathbb{E}\{Y\left\}\right),\phantom{\rule{2em}{0ex}}\end{array}$

(14)

where

$\begin{array}{rcl}{\Sigma}_{SY}& =& \mathbb{E}\left\{\right(S-\mathbb{E}\left\{S\right\}\left){(Y-\mathbb{E}\{Y\left\}\right)}^{H}\right\}\\ =& \mathbb{E}\left\{S{Y}^{H}\right\}-\mathbb{E}\left\{S\right\}\mathbb{E}\left\{{Y}^{H}\right\},\end{array}$

(15)

and

$\begin{array}{rcl}{\Sigma}_{YY}& =& \mathbb{E}\left\{(Y-\mathbb{E}\{Y\left\}\right){(Y-\mathbb{E}\{Y\left\}\right)}^{H}\right\}\\ =& \mathbb{E}\left\{Y{Y}^{H}\right\}-\mathbb{E}\left\{Y\right\}\mathbb{E}\left\{{Y}^{H}\right\}.\end{array}$

(16)

In (15),

$\mathbb{E}\left\{S{Y}^{H}\right\}$ is derived as

$\begin{array}{rcl}\mathbb{E}\left\{S{Y}^{H}\right\}& =& \mathbb{E}\left\{S{(S+I+W)}^{H}\right\}\\ =& {\Phi}_{SS}+{\Phi}_{SI},\end{array}$

(17)

and in (16),

$\mathbb{E}\left\{Y{Y}^{H}\right\}$ is derived as

$\begin{array}{rcl}\mathbb{E}\left\{Y{Y}^{H}\right\}& =& \mathbb{E}\left\{(S+I+W){(S+I+W)}^{H}\right\}\\ =& {\Phi}_{\mathrm{SS}}+{\Phi}_{\mathrm{II}}+{\Phi}_{SI}+{\Phi}_{SI}^{H}+{\Phi}_{\mathrm{WW}},\end{array}$

(18)

where we have assumed that the desired term S and the ICI I=[*I*(0),*I*(1),…,*I*(*N*−1)]^{
T
} are independent of the noise W=[*W*(0),*W*(1),…,*W*(*N*−1)]^{
T
}, and Φ
_{
P
Q
} represents the correlation function between P and Q.

We now derive each term required to compute

${\widehat{S}}_{\mathrm{opt}}$ in (14). From (3), the elements of

Φ
_{
SS
}in (17) and (18) can be obtained as

$\begin{array}{rcl}\mathbb{E}\left\{S\left({k}_{1}\right){S}^{\ast}\left({k}_{2}\right)\right\}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}& =& \phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\mathbb{E}\left\{\left(H\left({k}_{1}\right)X\left({k}_{1}\right)J\left(0\right)\right){\left(H\left({k}_{2}\right)X\left({k}_{2}\right)J\left(0\right)\right)}^{\ast}\right\}\\ =& \phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\mathbb{E}\left\{H\left({k}_{1}\right){H}^{\ast}\left({k}_{2}\right)\right\}\\ \phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\times \phantom{\rule{0.3em}{0ex}}\mathbb{E}\left\{X\left({k}_{1}\right){X}^{\ast}\left({k}_{2}\right)\right\}\mathbb{E}\left\{J\left(0\right){J}^{\ast}\left(0\right)\right\},\end{array}$

(19)

where the fact that the transmit signal, the wireless channel, and the phase noise are independent from each other has been used. Using (5), the correlation function

$\mathbb{E}\left\{H\left({k}_{1}\right){H}^{\ast}\left({k}_{2}\right)\right\}$ can be derived as

$\begin{array}{rcl}\mathbb{E}\left\{H\right({k}_{1}\left){H}^{\ast}\right({k}_{2}\left)\right\}& =& \mathbb{E}\left\{\left(\sum _{{l}_{1}=0}^{L-1}h\left({l}_{1}\right){e}^{-j2\Pi \frac{{l}_{1}{k}_{1}}{N}}\right)\right.\\ \phantom{\rule{2em}{0ex}}\times \left.{\left(\sum _{{l}_{2}=0}^{L-1}h\left({l}_{2}\right){e}^{-j2\Pi \frac{{l}_{2}{k}_{2}}{N}}\right)}^{\ast}\right\}\\ =& \sum _{{l}_{1}=0}^{L-1}\sum _{{l}_{2}=0}^{L-1}\mathbb{E}\left\{h\left({l}_{1}\right){h}^{\ast}\left({l}_{2}\right)\right\}{e}^{-j2\Pi \frac{\left({l}_{1}{k}_{1}-{l}_{2}{k}_{2}\right)}{N}}\\ =& \sum _{l=0}^{L-1}{\sigma}_{l}^{2}{e}^{-j2\mathrm{\Pi l}\frac{\left({k}_{1}-{k}_{2}\right)}{N}}.\end{array}$

(20)

Therefore, substituting (20) into (19) we obtain that if

*k*
_{1}≠

*k*
_{2},

$\begin{array}{rcl}\mathbb{E}\left\{S\left({k}_{1}\right){S}^{\ast}\left({k}_{2}\right)\right\}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}& =& {E}_{s}\mathbb{E}\left\{J\left(0\right){J}^{\ast}\left(0\right)\right\}\phantom{\rule{0.3em}{0ex}}\sum _{l=0}^{L-1}\phantom{\rule{0.3em}{0ex}}{\sigma}_{l}^{2}{e}^{-j2\mathrm{\Pi l}\frac{\left({k}_{1}-{k}_{2}\right)}{N}}\delta \left({k}_{1}\phantom{\rule{0.3em}{0ex}}-\phantom{\rule{0.3em}{0ex}}{k}_{2}\right)\\ =& 0,\end{array}$

(21)

and if

*k*
_{1}=

*k*
_{2},

$\mathbb{E}\left\{S\right({k}_{1}\left){S}^{\ast}\right({k}_{1}\left)\right\}={E}_{s}\mathbb{E}\left\{J\right(0\left){J}^{\ast}\right(0\left)\right\}.$

(22)

Using (3) the elements of

Φ
_{
II
}in (18) can be derived as

$\begin{array}{ll}\mathbb{E}\left\{I\left({k}_{1}\right){I}^{\ast}\left({k}_{2}\right)\right\}=& \mathbb{E}\left\{\left(\sum _{{r}_{1}=0,{r}_{1}\ne {k}_{1}}^{N-1}H\left({r}_{1}\right)X\left({r}_{1}\right)J\left({k}_{1}-{r}_{1}\right)\right)\right.\\ \left.\times {\left(\sum _{{r}_{2}=0,{r}_{2}\ne {k}_{2}}^{N-1}H\left({r}_{2}\right)X\left({r}_{2}\right)J\left({k}_{2}-{r}_{2}\right)\right)}^{\ast}\right\},\end{array}$

(23)

and therefore, if

*k*
_{1}≠

*k*
_{2},

$\phantom{\rule{-12.0pt}{0ex}}\mathbb{E}\left\{I\left({k}_{1}\right){I}^{\ast}\left({k}_{2}\right)\right\}={E}_{s}\sum _{r=0,r\ne {k}_{1},{k}_{2}}^{N-1}\mathbb{E}\left\{J\left({k}_{1}-r\right){J}^{\ast}\left({k}_{2}-r\right)\right\},$

(24)

and if

*k*
_{1}=

*k*
_{2},

$\phantom{\rule{-12.0pt}{0ex}}\mathbb{E}\left\{I\left({k}_{1}\right){I}^{\ast}\left({k}_{1}\right)\right\}={E}_{s}\sum _{r=0,r\ne {k}_{1}}^{N-1}\mathbb{E}\left\{J\left({k}_{1}-r\right){J}^{\ast}\left({k}_{1}-r\right)\right\}.$

(25)

Similarly, the elements of

Φ
_{
SI
}in (17) and (18) can be derived as

$\begin{array}{ll}\phantom{\rule{-15.0pt}{0ex}}\mathbb{E}\left\{S\left({k}_{1}\right){I}^{\ast}\left({k}_{2}\right)\right\}=& \mathbb{E}\left\{\left(H\right({k}_{1}\left)X\right({k}_{1}\left)J\right(0\left)\right)\right.\\ \left.\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\times {\left(\sum _{r=0,r\ne {k}_{2}}^{N-1}H\left(r\right)X\left(r\right)J({k}_{2}-r)\right)}^{\ast}\right\},\end{array}$

(26)

and therefore, if

*k*
_{1}≠

*k*
_{2}
$\mathbb{E}\left\{S\left({k}_{1}\right){I}^{\ast}\left({k}_{2}\right)\right\}={E}_{s}\mathbb{E}\left\{J\left(0\right){J}^{\ast}\left({k}_{2}-{k}_{1}\right)\right\},$

(27)

and if

*k*
_{1}=

*k*
_{2}
$\mathbb{E}\left\{S\left({k}_{1}\right){I}^{\ast}\left({k}_{1}\right)\right\}=0.$

(28)

It is noted from (19)–(28) that the correlation function $\mathbb{E}\left\{J\left({k}_{1}\right){J}^{\ast}\left({k}_{2}\right)\right\}$ plays a pivotal role in the computation of Φ
_{
SS
}, Φ
_{
II
}, and Φ
_{
SI
}.

Using (4), the correlation function

$\mathbb{E}\left\{J\left({k}_{1}\right){J}^{\ast}\left({k}_{2}\right)\right\}$ can be derived as

$\begin{array}{rcl}\mathbb{E}\left\{J\left({k}_{1}\right){J}^{\ast}\left({k}_{2}\right)\right\}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}& =& \phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\frac{1}{{N}^{2}}\mathbb{E}\left\{\left(\sum _{{n}_{1}=0}^{N-1}{e}^{\mathrm{j\varphi}\left({n}_{1}\right)}{e}^{-j2\Pi \frac{{n}_{1}{k}_{1}}{N}}\right)\right.\\ \phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}\times \left.{\left(\sum _{{n}_{2}=0}^{N-1}{e}^{\mathrm{j\varphi}\left({n}_{2}\right)}{e}^{-j2\Pi \frac{{n}_{2}{k}_{2}}{N}}\right)}^{\ast}\right\}\\ =& \phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\frac{1}{{N}^{2}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\sum _{{n}_{1}=0}^{N\phantom{\rule{0.3em}{0ex}}-1}\sum _{{n}_{2}=0}^{N\phantom{\rule{0.3em}{0ex}}-1}\phantom{\rule{0.3em}{0ex}}\mathbb{E}\phantom{\rule{0.3em}{0ex}}\left\{{e}^{j\left(\varphi \left({n}_{1}\right)-\varphi \left({n}_{2}\right)\right)}\right\}{e}^{\phantom{\rule{0.3em}{0ex}}-j2\Pi \frac{\left({n}_{1}{k}_{1}-{n}_{2}{k}_{2}\right)}{N}}.\end{array}$

(29)

### Nonstationary phase noise

Substituting (7) into (29), we obtain

$\begin{array}{rcl}\mathbb{E}\left\{J\left({k}_{1}\right){J}^{\ast}\left({k}_{2}\right)\right\}& =& \frac{1}{{N}^{2}}\sum _{{n}_{1}=0}^{N-1}\sum _{{n}_{2}=0}^{N-1}\\ \mathbb{E}\left\{{e}^{j\left(\sum _{{m}_{1}=0}^{{n}_{1}}v\left({m}_{1}\right)-\sum _{{m}_{2}=0}^{{n}_{2}}v\left({m}_{2}\right)\right)}\right\}\\ \times {e}^{-j2\Pi \frac{\left({n}_{1}{k}_{1}-{n}_{2}{k}_{2}\right)}{N}}\\ =& \frac{1}{{N}^{2}}\sum _{{n}_{1}=0}^{N-1}\sum _{{n}_{2}=0}^{N-1}\\ \mathbb{E}\left\{{e}^{j\text{sgn}\left({n}_{1}-{n}_{2}\right)\left(\sum _{m=0}^{|{n}_{1}-{n}_{2}|-1}v\left(m\right)\right)}\right\}\\ \times {e}^{-j2\Pi \frac{\left({n}_{1}{k}_{1}-{n}_{2}{k}_{2}\right)}{N}},\end{array}$

(30)

where sgn(

*n*) represents the sign operation, i.e.,

$\text{sgn}\left(n\right)=\left\{\begin{array}{c}1,\phantom{\rule{1em}{0ex}}n>0\\ 0,\phantom{\rule{1em}{0ex}}n\le 0\end{array}\right.$

(31)

Since

*v*(

*n*) is an i.i.d zero-mean Gaussian variable with variance

${\sigma}_{v}^{2}$,

${\left\{\text{sgn}\left({n}_{1}-{n}_{2}\right)v\left(m\right)\right\}}_{m=0}^{|{n}_{1}-{n}_{2}|-1}$ are also i.i.d. Gaussian variable with zero mean and variance given by

$\begin{array}{rcl}{\Sigma}_{\left(\text{sgn}\left({n}_{1}-{n}_{2}\right)v\left(m\right)\right)\left(\text{sgn}({n}_{1}-{n}_{2})v\left(m\right)\right)}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}& =& \phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\mathbb{E}\left\{{\left(\text{sgn}\left({n}_{1}\phantom{\rule{0.3em}{0ex}}-\phantom{\rule{0.3em}{0ex}}{n}_{2}\right)\right)}^{2}{\left(v\right(m\left)\right)}^{2}\right\}\\ =& \phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}{\sigma}_{v}^{2}.\end{array}$

(32)

Therefore, the variance of

$\text{sgn}\left({n}_{1}\phantom{\rule{0.3em}{0ex}}-\phantom{\rule{0.3em}{0ex}}{n}_{2}\right)\left(\sum _{m\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}0}^{|{n}_{1}\phantom{\rule{0.3em}{0ex}}-\phantom{\rule{0.3em}{0ex}}{n}_{2}|\phantom{\rule{0.3em}{0ex}}-\phantom{\rule{0.3em}{0ex}}1}v\left(m\right)\right)$ can be derived as

$\begin{array}{l}{\Sigma}_{\left(\text{sgn}\left({n}_{1}-{n}_{2}\right)\left(\sum _{m=0}^{|{n}_{1}-{n}_{2}|-1}v\left(m\right)\right)\right)\left(\text{sgn}\left({n}_{1}-{n}_{2}\right)\left(\sum _{m=0}^{|{n}_{1}-{n}_{2}|-1}v\left(m\right)\right)\right)}\\ =|{n}_{1}-{n}_{2}|{\sigma}_{v}^{2}.\end{array}$

(33)

Notice that for a Gaussian variable

*α* with mean

*μ* and variance

*ψ*
^{2}, its characteristic function is given by [

23]

$\mathbb{E}\left\{{e}^{\mathrm{jt\alpha}}\right\}={e}^{\mathrm{j\mu t}-\frac{{\psi}^{2}{t}^{2}}{2}}.$

(34)

Substituting (33) into (34) and letting

*μ* = 0 and

*t* = 1, we obtain

$\mathbb{E}\left\{{e}^{j\text{sgn}\left({n}_{1}-{n}_{2}\right)\left(\sum _{m=0}^{|{n}_{1}-{n}_{2}|-1}v\left(m\right)\right)}\right\}={e}^{-\frac{|{n}_{1}-{n}_{2}|{\sigma}_{v}^{2}}{2}}.$

(35)

Substituting (35) into (30), the correlation function $\mathbb{E}\left\{J\left({k}_{1}\right){J}^{\ast}\left({k}_{2}\right)\right\}$ for the nonstationary phase noise case can be obtained.

### Stationary phase noise

Since

*ϕ*(

*n*) is modeled as a stationary Gaussian process with zero mean,

*ϕ*(

*n*
_{1})−

*ϕ*(

*n*
_{2})is also a stationary Gaussian process with zero mean and variance given by

$\begin{array}{rcl}{\Sigma}_{\left(\varphi \left({n}_{1}\right)-\varphi \left({n}_{2}\right)\right)\left(\varphi \left({n}_{1}\right)-\varphi \left({n}_{2}\right)\right)}& =& \mathbb{E}\left\{{\left(\left(\varphi \left({n}_{1}\right)-\varphi \left({n}_{2}\right)\right)\right)}^{2}\right\}\\ =& 2{\Phi}_{\varphi \left(0\right)\varphi \left(0\right)}-2{\Phi}_{\varphi \left({n}_{1}-{n}_{2}\right)\varphi \left({n}_{1}-{n}_{2}\right)}.\end{array}$

(36)

Therefore, substituting (36) into (34) and letting

*μ* = 0 and

*t* = 1, we obtain

$\mathbb{E}\left\{{e}^{j\left(\varphi \left({n}_{1}\right)-\varphi \left({n}_{2}\right)\right)}\right\}={e}^{{\Phi}_{\varphi \left({n}_{1}-{n}_{2}\right)\varphi \left({n}_{1}-{n}_{2}\right)}-{\Phi}_{\varphi \left(0\right)\varphi \left(0\right)}}.$

(37)

Similar to the nonstationary one, substituting (37) into (29), the correlation function $\mathbb{E}\left\{J\left({k}_{1}\right){J}^{\ast}\left({k}_{2}\right)\right\}$ for the stationary phase noise case can be obtained.

Finally, considering that

$\begin{array}{rcl}\mathbb{E}\left\{S\right(k\left)\right\}& =& \mathbb{E}\left\{H\right(k\left)\right\}\mathbb{E}\left\{X\right(k\left)\right\}\mathbb{E}\left\{J\right(0\left)\right\}\\ =& 0,\end{array}$

(38)

and

$\begin{array}{rcl}\mathbb{E}\left\{Y\right(k\left)\right\}& =& \mathbb{E}\left\{H\right(k\left)\right\}\mathbb{E}\left\{X\right(k\left)\right\}\mathbb{E}\left\{J\right(0\left)\right\}\\ +\sum _{r=0,r\ne k}^{N-1}\mathbb{E}\left\{H\right(r\left)\right\}\mathbb{E}\left\{X\right(r\left)\right\}\mathbb{E}\left\{J\right(k-r\left)\right\}+\mathbb{E}\left\{W\right(k\left)\right\}\\ =& 0,\end{array}$

(39)

we have for

$\mathbb{E}\left\{S\right\}$ in (15) as

$\mathbb{E}\left\{S\right\}={0}_{N},$

(40)

and for

$\mathbb{E}\left\{Y\right\}$ in (15) and (16) as

$\mathbb{E}\left\{Y\right\}={0}_{N}.$

(41)

Since the FFT does not change the noise distribution,

Φ
_{
WW
} in (18) is given by

${\Phi}_{\mathrm{WW}}={\sigma}^{2}{I}_{N}.$

(42)

In summary, substituting the derived results (21) and (22), (24) and (25), (27) and (28), (40)–(42) into (14), the optimal estimate ${\widehat{S}}_{\text{opt}}$ can be obtained. Notice that for the nonstationary phase noise case, (30) and (35) should be adopted; for the stationary phase noise case, (29) and (37) should be adopted.

It is noted that without changing the structure of conventional OFDM systems, the proposed Wiener filter preprocessing algorithm is based on the statistics of phase noise (which can be obtained from measurements or data sheets) and performs directly on the received signal Y(see Figure 1 for illustration). Subsequently, the algorithms of previous researches on phase noise cancelation (e.g., [12–17]) can be performed based on the preprocessed received signal ${\widehat{S}}_{\text{opt}}$ instead of on the received signal Y. We will show in the next section that by utilizing the correlation inherently exists in phase noise process, the proposed Wiener filter preprocessing algorithm can effectively mitigate the ICI which results from phase noise and lower the error floor, and therefore significantly improve the performances of previous researches phase noise cancelation, especially in severe phase noise case.