### 4.1 Estimation algorithm for *w*_{h,k}

The *w*_{h,k} filter does not depend on interference but only on the desired signal propagation channel. Its estimation is then directly linked to the estimation of the propagation channel *h*_{x,k}.

In this work, we consider that propagation channels, i.e., useful and interference channels, can be modeled as *L* taps finite impulse response filters and with *L* < *K*. We introduce then these impulse responses, corresponding to the *m* th antenna, through the following vectors

{a}_{x}\left(m\right)=\left(\begin{array}{c}\hfill {a}_{x,0}\left(m\right)\hfill \\ \hfill \vdots \hfill \\ \hfill {{a}_{x,}}_{L-1}\left(m\right)\hfill \end{array}\right)

(37)

Thanks to this hypothesis we can introduce the *h*_{
x
}(*m*) vector that represents the frequency response of the propagation channel between the useful user and the *m* th antenna

{h}_{x}\left(m\right)=\left(\begin{array}{c}\hfill {h}_{x,0}\left(m\right)\hfill \\ \hfill \vdots \hfill \\ \hfill {{h}_{x,}}_{K-1}\left(m\right)\hfill \end{array}\right)

(38)

The two *a*_{
x
}(*m*) and *h*_{
x
}(*m*) vectors are linked by the following equation

{h}_{x}\left(m\right)={F}_{K,L}{a}_{x}\left(m\right)

(39)

where *F*_{K,L} is the *K* × *L* truncated Fourier rectangular matrix defined as follows

\begin{array}{ll}{F}_{K,L}\left(k,l\right)& =\frac{1}{\sqrt{K}}exp\left(-j\frac{2\mathit{\pi kl}}{K}\right),0\le k\\ <\phantom{\rule{0.25em}{0ex}}K\phantom{\rule{0.25em}{0ex}}\mathrm{and}\phantom{\rule{0.25em}{0ex}}0\le l\phantom{\rule{0.25em}{0ex}}<L\end{array}

(40)

In the context of a real transmission, a first estimation {\tilde{\mathit{h}}}_{x}\left(m\right) of *h*_{
x
}(*m*) can be proceeded through the classical least square as given in the following equation

{\tilde{h}}_{x}\left(m\right)=\left(\begin{array}{c}\hfill {y}_{0}\left(m\right)/{x}_{0}\hfill \\ \hfill \vdots \hfill \\ \hfill {y}_{K-1}\left(m\right)/{x}_{K-1}\hfill \end{array}\right)

(41)

As components of {\tilde{\mathit{h}}}_{x}\left(m\right) have not been averaged over a great number of observations, they are highly imprecise and they depend on the additive noise over the *y*_{
k
}(*m*) received samples. A smoothing operation [15], leading to a new estimated {\widehat{h}}_{x}\left(m\right) vector, can be proposed through the following equation

{\widehat{h}}_{x}\left(m\right)={F}_{K,L}{F}_{K,L}^{H}{\widehat{h}}_{x}\left(m\right)

(42)

This estimation algorithm is known as the indirect estimation [16, 17], and some papers [18] propose to enhance it through the introduction of a noise power estimation and an adaptive weight, able to take this estimation into consideration.

The main drawback of this algorithm comes from Equation 41 that involves knowledge of all transmitted symbols {*x*_{
k
}}_{k ∈ [0, K − 1]}. In a real transmission, only pilot symbols are known. If we consider that we have *K′* < *K* comb pilots {*x*_{0′}, *x*_{1′}, …, *x*_{K′}}, then Equation 41 becomes

\tilde{h}{\prime}_{x}\left(m\right)=\left(\begin{array}{c}\hfill {y}_{0},\left(m\right)/{x}_{0},\hfill \\ \hfill \vdots \hfill \\ \hfill {y}_{K\prime -1}\left(m\right)/{x}_{K\prime -1}\hfill \end{array}\right)

(43)

where \tilde{\mathit{h}}{\prime}_{x}\left(m\right) is a (*K*′ × 1) vector representing the first estimation of the propagation channel frequency responses on the pilot locations.

The first estimation for the propagation channel impulse response can then be obtained through

{\tilde{a}}_{x}\left(m\right)={F}_{K\prime ,L}^{H}\tilde{h}{\prime}_{x}\left(m\right)

(44)

Then, an interpolation step is required. It is performed by the following equation

{\widehat{h}}_{x}\left(m\right)={F}_{K\prime ,L}^{H}\tilde{h}{\prime}_{x}\left(m\right)

(45)

Equation 45 is devoted to the *m* th antenna, and it can be generalized to all antennas. If we consider now the *k* th component {\widehat{h}}_{x,k}\left(m\right) of all these {\left\{{\widehat{\mathit{h}}}_{x}\left(m\right)\right\}}_{m\in \left[0,M-1\right]} vectors, we can introduce the {\widehat{\mathit{h}}}_{x,k} vector defined as follows

{\widehat{h}}_{x,k}=\left(\begin{array}{c}\hfill {\widehat{h}}_{x,k}\left(0\right)\hfill \\ \hfill \vdots \hfill \\ \hfill {\widehat{h}}_{x,k}\left(M-1\right)\hfill \end{array}\right)

(46)

The estimation {\widehat{\mathit{w}}}_{h,k} of the *w*_{h,k} vector is then directly given by

{\widehat{w}}_{h,k}=\frac{{\widehat{h}}_{x,k}}{{\widehat{h}}_{x,k}^{H}{\widehat{h}}_{x,k}}

(47)

### 4.2 Estimation algorithm for *w*_{z,k}

The *w*_{z,k} vector is jointly dependent on interference, and propagation channels are devoted to interference sources cancellation. It is well known that a *M* antenna spatial filter is able to cancel *U* = *M* − 1 interferers. In our particular case, where we choose *M* = 2, we have then to cope with a unique interferer. In the sequel of this section, the *u* index that represents the interferer index will be omitted in equations.

We consider the transmission of a unique OFDM symbol, the eigendecomposition of *R*_{zz,k} is given by

{R}_{\mathit{zz},k}={U}_{k}\left(\begin{array}{cc}\hfill \phantom{\rule{0.25em}{0ex}}{\mu}_{0,k}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right){U}_{k}^{H}

(48)

And the eigendecomposition of *C*_{zz,k} is given as follows

{C}_{\mathit{zz},k}={U}_{k}\left(\begin{array}{c}\hfill 0\hfill \\ \hfill 0\hfill \end{array}\begin{array}{c}\hfill 0\hfill \\ \hfill \phantom{\rule{0.25em}{0ex}}{\mu}_{0,k}\hfill \end{array}\right){U}_{k}^{H}={\mu}_{0,k}{u}_{2,k}{u}_{2,k}^{H}

(49)

where *μ*_{2,k} is the second column vector of *U*_{
k
} orthogonal to the interference vector \left(\begin{array}{c}\hfill {h}_{z,k}\left(0\right)\hfill \\ \hfill {h}_{z,k}\left(1\right)\hfill \end{array}\right)*.* Therefore, the following scalar vector is null {\mathit{u}}_{2,k}^{T}\left(\begin{array}{c}\hfill {h}_{z,k}\left(0\right)\hfill \\ \hfill {h}_{z,k}\left(1\right)\hfill \end{array}\right)=0. Therefore, {\mathit{u}}_{2,k}=\alpha \left(\begin{array}{c}\hfill {h}_{z,k}\left(1\right)\hfill \\ \hfill -{h}_{z,k}\left(0\right)\hfill \end{array}\right)=0, where *α* is a complex scalar. In the sequel, we set *α* = 1.

The filter *w*_{z,k} will then be rewritten as

{w}_{z,k}=\frac{{\mu}_{0,k}{u}_{2,k}{u}_{2,k}^{H}{h}_{x,k}^{*}}{{\mu}_{0,k}{h}_{x,k}^{T}{u}_{2,k}{u}_{2,k}^{H}{h}_{x,k}^{*}}=\frac{{u}_{2,k}}{{h}_{x,k}^{T}{u}_{2,k}}

(50)

Finally, the solution is given by

{w}_{z,k}=\frac{{w}_{z,k,n}}{{w}_{z,k,d}}

(51)

with

{w}_{z,k,n}=\left(\begin{array}{c}\hfill {h}_{z,k}\left(1\right)\hfill \\ \hfill -{h}_{z,k}\left(0\right)\hfill \end{array}\right)

(52)

and

{w}_{z,k,d}={h}_{z,k}\left(1\right){h}_{x,k}\left(0\right)-{h}_{z,k}\left(0\right){h}_{x,k}\left(1\right)

(53)

The positive scalar *ρ*_{
k
} in the general formula of the optimal spatial filter given in Equation 34 becomes

{\rho}_{k}={\mu}_{0,k}\frac{{u}_{2,k}^{H}{{h}_{x,k}}^{2}}{{{h}_{x,k}}^{2}}

(54)

Therefore, *ρ*_{
k
} is the intercorrelation factor between the desired and the interference channel vectors. From the theorem stated above and Equation 50, we emit the following proposition.

**Proposition:** *In the case of two*-*antenna SIMO transmission disturbed by an interferer*, *the optimum combiner* *w*_{
k
}*is a weighted combination of the interference cancellation filter* *w*_{z,k}*and the maximum ratio combining filter* *w*_{h,k}.

The estimation of *w*_{z,k,n} and *w*_{z,k,d} is a complex task that involves the knowledge of the desired and interferer propagation channels. Nevertheless, the expression of *w*_{z,k,n} and *w*_{z,k,d} given by Equations 52 and 53, respectively, gives opportunities to project these components on a reduced Fourier basis.

#### 4.2.1 *w*_{
z,k,n
}estimation

On the first hand, we can notice that the components of *w*_{z,k,n} are simply those of the frequency response of the interferer propagation channels, corresponding to the *k* th subcarrier. Therefore, the components of this filter can then easily be expressed on a reduced Fourier basis. For that purpose, we introduce the (*K* × 1), *h*_{
z
}(*m*) vector that represents the frequency response of the propagation channel between the interferer and the *m* th antenna as follows

{h}_{z}\left(m\right)=\left(\begin{array}{c}\hfill {h}_{z,0}\left(m\right)\hfill \\ \hfill \vdots \hfill \\ \hfill {h}_{z,K-1}\left(m\right)\hfill \end{array}\right)

(55)

As in the previous section, the *h*_{
z
}(*m*) vector is linked to the *L* taps impulse response *a*_{
z
}(*m*) through the following equation

{h}_{z}\left(m\right)={F}_{K,L}{a}_{z}\left(m\right)

(56)

where the (*L* × 1) *a*_{
z
}(*m*) vector is defined as follows

{a}_{z}\left(m\right)=\left(\begin{array}{c}\hfill {a}_{z,0}\left(m\right)\hfill \\ \hfill \vdots \hfill \\ \hfill {a}_{z,L-1}\left(m\right)\hfill \end{array}\right)

(57)

Therefore, *w*_{z,n} can be expressed as

{w}_{z,n}={F}_{K,L}{v}_{z,n}

(58)

#### 4.2.2 *w*_{
z,k,d
}estimation

On the other hand, we can notice that the *w*_{z,k,d} is a scalar obtained by the product of two frequency response terms. It can then be viewed as the Fourier transform of the convolution of two impulses responses of propagation channels, and it can then be linked to a virtual 2 *L* taps impulse response.

If we introduce the (2 *L* × 1) *v*_{z,d} vector representing this virtual impulse response

{v}_{z,d}=\left(\begin{array}{c}\hfill {v}_{z,0,d}\hfill \\ \hfill \vdots \hfill \\ \hfill {v}_{z,2L-1,d}\hfill \end{array}\right)

(59)

Then we can introduce the *w*_{z,d} vector defined as

{w}_{z,d}=\left(\begin{array}{c}\hfill {w}_{z,0,d}\hfill \\ \hfill \vdots \hfill \\ \hfill {w}_{z,K-1,d}\hfill \end{array}\right)

(60)

with

{w}_{z,d}={F}_{K,2L}{v}_{z,d}

(61)

#### 4.2.3 Replica spatial filter structure

The decomposition of *w*_{z,k} in a numerator part and a denominator part as given by Equation 51 leads to propose a new spatial filter structure having two weights, represented by *w*_{z,k,n} acting over the received signal and a weight, represented by *w*_{z,k, d}*,* acting over the useful signal (Figure 4).

We can then introduce the error *e*_{
k
} defined by

{e}_{k}={w}_{z,k,d}{x}_{k}-\left[\begin{array}{cc}\hfill {y}_{k}\left(0\right)\hfill & \hfill {y}_{k}\left(1\right)\hfill \end{array}\right]{w}_{z,k,n}

(62)

At this stage, knowing that the *w*_{z,k} filter has to cancel the interference, we can propose to identify its two components through an error square minimization criterion

{w}_{z,k}=arg\underset{{w}_{z,k,n},{w}_{z,k,d}}{min}{\left|{e}_{k}\right|}^{2}

(63)

We have then to insert a constraint *ψ*(*w*_{z,k,n}, *w*_{z,k,d} ) in order to avoid the trivial solution: (*w*_{z,k,n} = 0, *w*_{z,k,d} = 0). Moreover, the minimization has to be done over all frequencies. It is then necessary to propose a global criterion. For that purpose, we introduce the *X* transmitted diagonal data matrix, where each element *x*_{
k
} corresponds to the desired symbol transmitted over the *k* th subcarrier

X=\mathrm{diag}\left\{{x}_{0},{x}_{1},\dots ,{x}_{K-1}\right\}

(64)

We introduce also the bi-diagonal matrix *Y* of the received signal over the two antennas

Y=\left(\begin{array}{cc}\hfill \begin{array}{ccc}\hfill {y}_{0}\left(0\right)\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \ddots \hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill {y}_{K-1}\left(0\right)\hfill \end{array}\hfill & \hfill \begin{array}{ccc}\hfill {y}_{0}\left(1\right)\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \ddots \hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill {y}_{K-1}\left(1\right)\hfill \end{array}\hfill \end{array}\right)

(65)

The \mathit{e}=\left(\begin{array}{c}\hfill {e}_{0}\hfill \\ \hfill \vdots \hfill \\ \hfill {e}_{K-1}\hfill \end{array}\right) vector representing errors over all subcarriers is then given by

e=X{w}_{z,d}-Y{w}_{z,n}

(66)

The filter weights are then given by the following minimization

\underset{{w}_{z,d},{w}_{z,n}}{{w}_{z}=arg\phantom{\rule{0.25em}{0ex}}min}\left({e}^{2}-\mathit{\mu \psi}\left({w}_{z,n},{w}_{z,d}\right)\right)

(67)

where *μ* is a Lagrange multiplier.

In [12, 19] and in a similar context, a constraint called maximum signal to interference plus noise constraint (MSINRC) is proposed. It is defined as

\psi \left({w}_{z,n},{w}_{z,d}\right)\phantom{\rule{0.25em}{0ex}}={\Vert X{w}_{z,d}\Vert}^{2}-1

(68)

Without loss of generality, we can consider that all transmitted symbols *x*_{
k
} are normalized: |*x*_{
k
}|^{2} = 1, we have then *X*^{H}*X* = *I*. The constraint presented in Equation 68 is then equivalent to ‖*w*_{z,d}‖^{2} = 1.

By nulling the partial derivative of ‖*e*‖^{2} − *μψ*(*w*_{z,n}, *w*_{z,d}) with respect to *w*_{
z,n
} and *w*_{z,d}[20], we obtain the following system of equations

\left\{\begin{array}{c}\hfill {X}^{H}Y{w}_{z,n}-{w}_{z,d}=\mu {w}_{z,d}\hfill \\ \hfill {w}_{z,n}={\left({Y}^{H}Y\right)}^{-1}{Y}^{H}X{w}_{z,d}\hfill \end{array}\right.

(69)

Merging the two equations of Equation 68, we arrive to

{X}^{H}\left(Y{\left({Y}^{H}Y\right)}^{-1}{Y}^{H}-I\right)X{w}_{z,d}=\mu {w}_{z,d}

(70)

It appears then that *w*_{z,d} is the eigenvector of the *X*^{H}(*Y*(*Y*^{H}*Y*)^{−1}*Y*^{H} − *I*)*X* matrix corresponding to the *μ* eigenvalue.

Left multiplying the two sides of Equation 70 by {\mathit{w}}_{z,d}^{H}, we obtain

{w}_{z,d}^{H}{X}^{H}\left(Y{\left({Y}^{H}Y\right)}^{-1}{Y}^{H}-I\right)X{w}_{z,d}=\mu {w}_{z,d}^{H}{w}_{z,d}

(71)

This last equation leads to

\frac{1}{\mu}=\frac{{w}_{z,d}^{H}{w}_{z,d}}{{w}_{z,d}^{H}\phantom{\rule{0.25em}{0ex}}{X}^{H}\left(Y{\left({Y}^{H}Y\right)}^{-1}{Y}^{H}-I\right)X{w}_{z,d}}

(72)

In the right side of Equation 72, we recognize the SINR formula; we can then conclude that

\frac{1}{\mu}=\mathrm{SINR}

Finally, as we have to maximize the SINR at the output of the filter, *w*_{z,d} has to be the generalized eigenvector which corresponds to the minimal eigenvalue *μ*. Using Equation 61, Equation 72 becomes

{X}^{H}\left(Y{\left({Y}^{H}Y\right)}^{-1}{Y}^{H}-I\right)X{F}_{K,2L}{v}_{z,d}=\mu {F}_{K,2L}{v}_{z,d}

(73)

Left multiplying the two sides of this equation by {\mathit{F}}_{K,2L}^{H} yields to

{F}_{K,2L}^{H}\phantom{\rule{0.25em}{0ex}}{X}^{H}\left(Y{\left({Y}^{H}Y\right)}^{-1}{Y}^{H}-I\right)X{F}_{K,2L}{v}_{z,d}=\mu {v}_{z,d}

(74)

It appears that from Equation 74, the virtual impulse response *V*_{z, d} is the (2 *L* × 1) eigenvector of the matrix {\mathit{F}}_{K,2L}^{H}\phantom{\rule{0.25em}{0ex}}{\mathit{X}}^{H}\left(\mathit{Y}{\left({\mathit{Y}}^{H}\mathit{Y}\right)}^{-1}{\mathit{Y}}^{H}-\mathit{I}\right)\mathit{X}{\mathit{F}}_{K,2L} corresponding to the minimal eigenvalue *μ*. This result is known and used by many authors. An enhanced maximum signal to interference plus noise constraint (EMSINRC) is proposed in [13]. It is based on the exploitation of the set *V*_{z,d} of all the eigenvectors of the previous matrix defined as

{V}_{z,d}=\left[\begin{array}{ccc}\hfill {v}_{z,d}^{0}\hfill & \hfill \dots \hfill & \hfill {v}_{z,d}^{2L-1}\hfill \end{array}\right]

(75)

The EMSINRC algorithm introduces a linear combination of elements of *V*_{
z,d
}, in order to propose a composite virtual impulse response {v}_{z,d}^{c} defined as follows

{v}_{z,d}^{c}={\displaystyle \sum _{i=0}^{2L-1}}{\eta}_{i}\frac{1}{{\mu}_{i}}{v}_{z,d}^{i}

(76)

where *μ*_{
i
} is the eigenvalue corresponding to the eigenvector {\mathit{v}}_{z,d}^{i}. The complex term *η*_{
i
} is defined such that {\eta}_{i}=arg{max}_{{\eta}_{i}}{\Vert {\mathit{v}}_{z,d}^{c}\Vert}^{2} and |*η*_{
i
}| = 1. This complex scalar is acting as a phase term that aligns all eigenvectors.

Concerning the vector *v*_{z,n}, it is obtained by merging Equation 58 into the second equation of the system (69). Hence, *v*_{z,n} is given as follows

{v}_{z,n}={F}_{\mathit{KK},\mathit{LL}}^{H}{\left({Y}^{H}Y\right)}^{-1}{Y}^{H}X{F}_{\mathit{KK},\mathit{LL}}{v}_{z,d}

(77)

*v*_{z,n} can then directly be obtained once *v*_{z,d} is determined.

Finally, having *w*_{z, d} and *w*_{z, n} form *v*_{z, d} and *v*_{z, n} by performing Equations 61 and 58, respectively, the interferer cancellation filter {\mathit{w}}_{z,k}=\frac{{\mathit{w}}_{z,k,n}}{{\mathit{w}}_{z,k,d}} for all subcarriers is obtained.

At this stage, all elements have been established and the optimal filter, given by Equation 34, can be estimated. The combination of Equation 34 is based on *ρ*_{
k
} that can be obtained directly from Equation 54 noticing that the interferer power *μ*_{0} and {\sigma}_{n}^{2} can be obtained through a noise plus interferer power estimation.