From the results given in the Section 2, good performance is expected from the distributed ZF beamformer thanks to the added SNR and diversity gains. In this section, we discuss the effects of the residual CFO on those gains. In 4.1, we extend the system model given in Section 2 to the case where CFO is present. Then, the average SNR gain and diversity order are derived for the general case (*N*_{
t
}transmit antennas) in Sections 4.2 and 4.3.

### 4.1 System model with CFOs

The combination of the channel with the carrier frequency offset can be equivalently represented by the channel vector multiplied by the complex component {c}_{i}\left(t,{f}_{{\Delta}_{i}}\right)={e}^{{\varphi}_{i}\left(t,{{f}_{\Delta}}_{{}_{i}}\right)}={e}^{i2\pi {f}_{{\Delta}_{i}}t}, where *t* is the time index and {f}_{{\Delta}_{i}} denotes the CFO at the transmitter *i* with respect to the receiver's carrier frequency. Because of the CFO, the time coherency of the channel reduces, and the originally quasi-static channel now becomes time varying hence decreasing the performance of the beamforming scheme with static weights. As introduced earlier, we assume that the frequency offset is precompensated at the transmitters prior to transmission, i.e., only the residual CFOs {f}_{{\Delta}_{1}} and {f}_{{\Delta}_{2}} are left. We assume no initial phase offset between the transmitters. Equation (8), giving the instantaneous output SNR {\xi}_{1}\left(t,{f}_{{\Delta}_{1}},{f}_{{\Delta}_{2}}\right), can be written as follows

\begin{array}{c}\phantom{\rule{1em}{0ex}}\frac{1}{{\sigma}_{n}^{2}}{\left({c}_{1}\left(t,{f}_{{\Delta}_{1}}\right){h}_{11}^{H}{w}_{11}+{c}_{2}\left(t,{f}_{{\Delta}_{2}}\right){h}_{12}^{H}{w}_{21}\right)}^{2}\\ =\frac{1}{{\sigma}_{n}^{2}}\left({\left({h}_{11}^{H}{w}_{11}\right)}^{2}+{\left({h}_{12}^{H}{w}_{21}\right)}^{2}+{c}_{1}\left(t,{f}_{{\Delta}_{1}}\right){c}_{2}{\left(t,{f}_{{\Delta}_{2}}\right)}^{H}{h}_{11}^{H}{w}_{11}{\left({h}_{12}^{H}{w}_{21}\right)}^{H}\right.\\ \left(\right)close=")">\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}+{c}_{1}{\left(t,{f}_{{\Delta}_{1}}\right)}^{H}{c}_{2}\left(t,{f}_{{\Delta}_{2}}\right){\left({h}_{11}^{H}{w}_{11}\right)}^{H}{h}_{12}^{H}{w}_{21}\end{array}\n

(33)

We now average *ξ*_{1} over the channel realizations E\left[{\xi}_{1}\left(t,{f}_{{\Delta}_{1}},{f}_{{\Delta}_{2}}\right)\right] equals

\begin{array}{c}\frac{1}{{\sigma}_{n}^{2}}\left(E\left[{\left({h}_{11}^{H}{w}_{11}\right)}^{2}\right]+E{\left[{h}_{12}^{H}{w}_{21}\right]}^{2}+\left({c}_{1}\left(t,{f}_{{\Delta}_{1}}\right){c}_{2}{\left(t,{f}_{{\Delta}_{2}}\right)}^{H}\right.\right.\\ \left(\right)close=")">\phantom{\rule{1em}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\left(\right)close=")">+{c}_{1}{\left(t,{f}_{{\Delta}_{1}}\right)}^{H}{c}_{2}\left(t,{{f}_{\Delta}}_{{}_{2}}\right)E\left[{h}_{11}^{H}{w}_{11}\right]E\left[{h}_{12}^{H}{w}_{21}\right]\\ .\end{array}\n

(34)

### 4.2 Average SNR gain with CFO

#### 4.2.1 Fixed CFO

Following the procedure for Equation (18) and from Equation (34), the average SNR is *E* E\left[{\xi}_{1}\left(t,{f}_{{\Delta}_{1}},{f}_{{\Delta}_{2}}\right)\right]

\begin{array}{c}=\frac{1}{{\sigma}_{n}^{2}}\left({P}_{1}E\left[\sum _{n=1}^{{N}_{t}-1}|{h}_{11}^{n}{|}^{2}\right]\right.+{P}_{2}E\left[\sum _{n=1}^{{N}_{t}-1}|{h}_{12}^{n}{|}^{2}\right]+{\left({c}_{1}\left(t,{f}_{{\Delta}_{1}}\right){c}_{2}\left(t,{f}_{{\Delta}_{2}}\right)\right.}^{H}\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\left(\right)close=")">\left(\right)close=")">+{c}_{1}{\left(t,{f}_{{\Delta}_{1}}\right)}^{H}{c}_{2}\left(t,{f}_{{\Delta}_{2}}\right)\sqrt{{P}_{1}{P}_{2}}E\left[\sqrt{\sum _{n=1}^{{N}_{t}-1}|{h}_{11}^{n}{|}^{2}}\right]E\left[\sqrt{\sum _{n=1}^{{N}_{t}-1}|{h}_{12}^{n}{|}^{2}}\right]\end{array}\n

(35)

where {c}_{1}\left(t,{f}_{{\Delta}_{1}}\right){c}_{2}{\left(t,{f}_{{\Delta}_{2}}\right)}^{H}+{c}_{1}{\left(t,{f}_{{\Delta}_{1}}\right)}^{H}{c}_{2}\left(t,{f}_{{\Delta}_{2}}\right) is equivalent to

{e}^{i2\pi f{\Delta}_{1}t}{e}^{-i2\pi {f}_{{\Delta}_{2}}t}+{e}^{-i2\pi {f}_{{\Delta}_{1}}t}{e}^{i2\pi {f}_{{\Delta}_{2}}t}=2cos\left(2\pi {f}_{\Delta}t\right),\phantom{\rule{1em}{0ex}}{f}_{\Delta}={f}_{{\Delta}_{1}}-{f}_{{\Delta}_{2}}.

(36)

The notation *f*_{
Δ
}refers to the relative CFO between two transmitters. This result shows that the average SNR is time-dependent and it varies over the transmission period *T*_{
p
}. We then compute the time-averaged result to obtain the exact average SNR

{E}_{T}\left[cos\left(2\pi {f}_{\Delta}t\right)\right]=\frac{1}{{T}_{p}}{\int}_{0}^{{T}_{p}}cos\left(2\pi {f}_{\Delta}t\right)\phantom{\rule{0.3em}{0ex}}\mathsf{\text{d}}t=\mathsf{\text{sinc}}\phantom{\rule{0.3em}{0ex}}\left(2\pi {f}_{\Delta}{T}_{p}\right)

(37)

From Equation (35), we obtain *E*[*ξ*_{1}(*T*_{
p
}, *f*_{
Δ
})]

\begin{array}{c}=\frac{1}{{\sigma}_{n}^{2}}\left({P}_{1}E\left[\sum _{n=1}^{{N}_{t}-1}|{h}_{11}^{n}{|}^{2}\right]\right.+{P}_{2}E\left[\sum _{n=1}^{{N}_{t}-1}|{h}_{12}^{n}{|}^{2}\right]\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\left(\right)close=")">+2\phantom{\rule{0.3em}{0ex}}\mathsf{\text{sinc}}\left(2\pi {f}_{\Delta}{T}_{p}\right)\sqrt{{P}_{1}{P}_{2}}E\left[\sqrt{\sum _{n=1}^{{N}_{t}-1}|{h}_{11}^{n}{|}^{2}}\right]E\left[\sqrt{\sum _{n=1}^{{N}_{t}-1}|{h}_{12}^{n}{|}^{2}}\right]& .\end{array}\n

(38)

As a result, based on the (19) and Equation (20) and following the procedure in Section 3.1, we can express the average SNR gain of the proposed scheme with residual CFO as *G*_{CFOfixed} equals to

10{log}_{10}\left(\left({P}_{1}+{P}_{2}\right)\left({N}_{t}-1\right)+2sinc\left(2\pi {f}_{\Delta}{T}_{p}\right)\sqrt{{P}_{1}{P}_{2}}{\left(\frac{\Gamma \left({N}_{t}-0.5\right)}{\left({N}_{t}-2\right)!}\right)}^{2}\right).

(39)

We can observe that the average SNR gain degrades, following a sinc function with the parameters *f*_{
Δ
}and *T*_{
p
}. As a result, for a long enough *T*_{
p
}, the sinc function produces a zero, i.e., no SNR gain is obtained.

#### 4.2.2 Uniform CFO

Assuming an uniformly distributed residual frequency offset, we obtain the average SNR by taking the expectation of the sinc function over the random *f*_{
Δ
}, i.e.,

E\left[\mathsf{\text{sinc}}\left(2\pi {f}_{\Delta}{T}_{p}\right)\right]={\int}_{-fc}^{fc}\mathsf{\text{sinc}}\left(2\pi x{T}_{p}\right)p\left(x\right)\mathsf{\text{d}}x

(40)

where *f*_{
c
}denotes the maximal frequency offset and *x* is a random variable uniformly distributed over the interval [-*f*_{
c
}*, f*_{
c
}]. Equation (40) is hence equivalent to

E\left[\mathsf{\text{sinc}}\left(2\pi {f}_{\Delta}{T}_{p}\right)\right]={\int}_{-2\pi {T}_{p}fc}^{2\pi {T}_{p}fc}\mathsf{\text{sinc}}\left(x\right)p\left(x\right)\mathsf{\text{d}}x=\frac{1}{2\pi {T}_{p}{f}_{c}}si\left(2\pi {T}_{p}{f}_{c}\right)

(41)

where *Si* (*x*) denotes the sine integral. As a result, the SNR gain with an uniformly distributed CFO in the interval [-*f*_{
c
}, *f*_{
c
}] (*G*_{CFOunif}) can be expressed as

10{log}_{10}\left(\left({P}_{1}+{P}_{2}\right)\left({N}_{t}-1\right)+\frac{si\left(2\pi {T}_{p}{f}_{c}\right)}{\pi {T}_{p}{f}_{c}}\sqrt{{P}_{1}{P}_{2}}{\left(\frac{T\left({N}_{t}-0.5\right)}{\left({N}_{t}-2\right)!}\right)}^{2}\right)

(42)

### 4.3 Diversity order

#### 4.3.1 Fixed CFO

As expressed in Section 3.2, the average error probability *P*_{
e
}is required to compute the diversity order *d*. However, when a residual CFO is present, the error probability *P*_{
e
}becomes time- and CFO-dependent. For *P*_{1} = *P*_{2} = 1, for a given transmit duration and a residual CFO (*f*_{
Δ
}), the error probability *P*_{
e
}(*t, f*_{
Δ
}) and a BPSK modulation, we have

{P}_{e}\left(t,{f}_{\Delta}\right)={\int}_{0}^{\infty}Q\left(\sqrt{2{\xi}_{1}\left(t,{f}_{\Delta}\right)}\right){{p}_{\xi}}_{{}_{1}}\left({\xi}_{1}\left(t,{f}_{\Delta}\right)\right)d{\xi}_{1}\left(t,{f}_{\Delta}\right),\phantom{\rule{1em}{0ex}}0\le t\le {T}_{p}

(43)

where *ξ*_{1}(*t, f*_{
Δ
}) denotes the equivalent signal at the time index *T*_{
p
}= *t* and, from Equation (33), can be expressed as

{\xi}_{1}\left(t,{f}_{\Delta}\right)=\frac{1}{{\sigma}_{n}^{2}}\left({\left({h}_{11}^{H}{w}_{11}\right)}^{2}+{\left({h}_{12}^{H}{w}_{21}\right)}^{2}+2cos\left(2\pi {f}_{\Delta}t\right){h}_{11}^{H}{w}_{11}{h}_{12}^{H}{w}_{21}\right).

(44)

Similar to Section 3.2, we express the average BER as

{P}_{e}\left(t,{f}_{\Delta}\right)=\frac{1}{\pi}{\int}_{0}^{\frac{\pi}{2}}{\int}_{0}^{\infty}exp\left(-\frac{2{\xi}_{1}\left(t,{f}_{\Delta}\right)}{2{sin}^{2}\varphi}\right){p}_{{\xi}_{1}}\left({\xi}_{1}\left(t,{f}_{\Delta}\right)\right)d{\xi}_{1}\left(t,{f}_{\Delta}\right)d\varphi .

(45)

Using the characteristic function (CHF) method to evaluate the PDF *p*(*γ*_{1}) requires to obtain {\Psi}_{{\gamma}_{1}}\left(jv\right). Where *Ψ*_{
x
}(*jv*) is the CHF of the random variable *x*

{\Psi}_{x}\left(jv\right)=E\left[{e}^{jvx}\right]={\int}_{-\infty}^{\infty}{e}^{jvx}p\left(x\right)\mathsf{\text{d}}x.

(46)

Assuming that the transmitters have a same power of 1, i.e., *P*_{1} = *P*_{2} = 1, from Equations (8), (17) and (46), the equivalent CHF using the CHF method to evaluate the PDF *p*(*ξ*_{1}(*t*,*f*_{
Δ
})) requires to compute the {\Psi}_{{\xi}_{1}}\left(jv,t,{f}_{\Delta}\right). From Equations (44) and (35), the equivalent CHF {\Psi}_{{\xi}_{1}}\left(jv,t,{f}_{\Delta}\right) can be expressed as

\begin{array}{ll}\hfill E\left[{e}^{jv{\xi}_{1}}\right]& =E\left[{e}^{jv}\left({\left({h}_{11}^{H}{w}_{11}\right)}^{2}+{\left({h}_{12}^{H}{w}_{21}\right)}^{2}+2cos\left(2\pi {f}_{\Delta}t\right){h}_{11}^{H}{w}_{11}{h}_{12}^{H}{w}_{21}\right)\right]\phantom{\rule{2em}{0ex}}\\ =E\left[{e}^{jv}\sum _{n=1}^{{N}_{t}-1}\left|{h}_{n}{|}^{2}{e}^{jv}{\sum}_{n=1}^{{N}_{t}-1}\right|{h}_{n}{|}^{2}{e}^{jv2cos\left(2\pi {f}_{\Delta}t\right)}\sqrt{{\sum}_{n=1}^{{N}_{t}-1}|{h}_{11}^{n}{|}^{2}}\sqrt{{\sum}_{n=1}^{{N}_{t}-1}|{h}_{12}^{n}{|}^{2}}\right].\phantom{\rule{2em}{0ex}}\end{array}

(47)

However, similar to the results in Section 3.2, the third term in Equation (47) is not independent from the two others and the expectation operator cannot be separated, obtaining the equivalent PDF (and hence *P*_{
e
}) in a general closed form is difficult.

In addition, the average BER must be integrated over the time index t for a given transmission duration (*T*_{
p
}). Therefore, to compute the diversity order of the considered cooperative scheme, we approximate numerically the error probability Pe for a varying number of antennas. We then obtain the average probability of error by integrating the different *P*_{
e
}, i.e., for a given residual CFO (*f*_{
Δ
}), over the transmission duration

{P}_{e|{T}_{p}}=\frac{1}{{T}_{p}}{\int}_{0}^{{T}_{p}}{P}_{e}\left(t\right)\mathsf{\text{d}}t.

(48)

We then use these numerical approximations of the error probability Pe to compute the diversity order of the considered cooperative scheme when CFO is present. The Section 5.2 presents the resulting diversity order.

#### 4.3.2 Uniform CFO

We study the effects of a random and uniformly distributed CFO on the diversity order. In such a case, the diversity order is obtained by numerically approximating the PDF of the equivalent channel for a random variable *Δ*_{
f
}uniformly distributed over the interval [-*f*_{
c
}, *f*_{
c
}]. Section 5 presents the results from the diversity order with an uniformly distributed CFO.