Overlay Cognitive Radio Based on OFDM with Channel Estimation Issues

Abstract

Cognitive radio (CR) has been proposed as a technology to improve the spectrum efficiency by giving an opportunistic access of the licensed-user spectra to unlicensed users. We consider an overlay CR consisting of a primary macro-cell and cognitive small cells of cooperative secondary base stations (SBS). We suggest studying a CR where an orthogonal frequency division multiplexing is used for both the primary users (PU) and the secondary users (SU). In order to cancel the interferences, a precoding is required at the SBS. Therefore, we first derive the interferences expression due to SU at the PU receiver. Then, zero forcing beamforming (ZFBF) is considered to cancel the interferences. However, applying ZFBF depends on the channels between the SBS and the PU. A channel estimation is hence necessary. For this purpose, we propose to approximate the channel by an autoregressive process (AR) and to consider the channel estimation issue by using a training sequence. The received signals, also called the observations, are considered to be disturbed by an additive white measurement noise. In that case, the AR parameters and the channel can be jointly estimated from the received noisy signal by using a recursive approach. Nevertheless, the corresponding state space representation of the system is non-linear. Then, we propose to carry out a complementary study by compare non-linear Kalman filter based approaches.

Appendices

Appendices

Kalman filter (KF) has been used in a wide range of applications, from speech enhancement to time-varying autoregressive parameters tracking, from biomedical applications to mobile communications. More particularly, in this paper, Kalman filter can be used to estimate the channel (see "Appendix 1"). In the "Appendices 2 and 3", we recall how to address the estimation of the state vector by using the extended Kalman filter (EKF) or the second-order EKF (SOEKF), where Taylor series expansion is used. It should be noted that the state noise and the measurement noise are still assumed to be Gaussian. In "Appendix 4", the UKF was proposed as an alternative to the EKF to avoid the linearization step. Like the UKF, the Quadrature Kalman filter (QKF) and Cubature Kalman filter (CKF) are a sigma point Kalman filter (SPKF) (See "Appendix 5"). The main difference between QKF, CKF and UKF is the way to choose the sigma points. In "Appendices 6 and 7" deal with how to choose the sigma points. Indeed, in QKF the sigma points are chosen by using the Gauss-Hermite quadrature where CKF differs from QKF by the considered way of approximation.

Appendix 1: Kalman filter
- initialize the value of $\hat{{\mathbf {x}}}(0\vert 0)$ and ${\mathbf {P}}(0\vert 0)$.
The prediction step
- update the state vector:
$\hat{{\mathbf {x}}}(n\vert n-1)={\mathbf {F}}(n)\hat{{\mathbf {x}}}(n-1\vert n-1)$
- update the error covariance matrix:
${\mathbf {P}}(n\vert n-1)={\mathbf {F}}(n){\mathbf {P}}(n-1\vert n-1){\mathbf {F}}^H(n)+{\mathbf {G}}{\mathbf {Q}}{\mathbf {G}}^H$
The filtering step
- update the Kalman gain:
${\mathbf {K}}(n)= {\mathbf {P}}(n\vert n-1){\mathbf {H}}(n)^H \big ( {\mathbf {H}}(n){\mathbf {P}}(n\vert n-1){\mathbf {H}}(n)^H +{\mathbf {R}}\big )^{-1}$
- update the state vector:
$\hat{{\mathbf {x}}}(n\vert n)=\hat{{\mathbf {x}}}(n\vert n-1)+{\mathbf {K}}(n)\left( {\mathbf {y}}(n)-{\mathbf {H}}(n)\hat{{\mathbf {x}}}(n\vert n-1)\right)$
- finally, update the error covariance matrix:
${\mathbf {P}}(n\vert n)=\{{\mathbf {I}}_U-{\mathbf {K}}(n){\mathbf {H}}(n)\}{\mathbf {P}}(n|n-1)$

Appendix 2: Extended Kalman filter
- initialize the value of $\hat{{\mathbf {x}}}(0\vert 0)$ and ${\mathbf {P}}(0\vert 0)$.
- calculate the Jacobian matrix $\nabla _{{\mathbf {x}}} {\mathbf {f}}_{n}\vert _{\hat{{\mathbf {x}}}(n-1 \vert n-1)}$.
- update the a priori state vector:
$\hat{{\mathbf {x}}}(n\vert n-1)={\mathbf {f}}_{n}(\hat{{\mathbf {x}}}(n-1\vert n-1))$
- update the a priori error covariance matrix:
${\mathbf {P}}(n\vert n-1)= \nabla _{{\mathbf {x}}}{\mathbf {f}}_n|_{\mathbf {{\hat{x}}}(n-1|n-1)}{\mathbf {P}}(n-1|n-1) \nabla _{{\mathbf {x}}}{\mathbf {f}}_n^H|_{\mathbf {{\hat{x}}}(n-1|n-1)}+{\mathbf {G}}{\mathbf {Q}}{\mathbf {G}}^H$
- calculate the Jacobian matrix $\nabla _{{\mathbf {x}}} {\mathbf {h}}_{n}\vert _{\hat{{\mathbf {x}}}(n \vert n-1)}$.
- update the Kalman gain ${\mathbf {K}}(n)$:
${\mathbf {K}}(n)= {\mathbf {P}}(n\vert n-1)\nabla _{{\mathbf {x}}} {\mathbf {h}}_{n}^H\vert _{\hat{{\mathbf {x}}}(n \vert n-1)} \big ( \nabla _{{\mathbf {x}}} {\mathbf {h}}_{n}\vert _{\hat{{\mathbf {x}}}(n \vert n-1)}{\mathbf {P}}(n\vert n-1)\nabla _{{\mathbf {x}}} {\mathbf {h}}_{n}^H\vert _{\hat{{\mathbf {x}}}(n \vert n-1)} +{\mathbf {R}}\big )^{-1}$
- deduce the a posteriori state vector:
$\hat{{\mathbf {x}}}(n\vert n)=\hat{{\mathbf {x}}}(n|n-1)+{\mathbf {K}}(n)\{{\mathbf {y}}(n)- {\mathbf {h}}_n(\mathbf {{\hat{x}}}(n|n-1))\}$
- finally, update the error covariance error matrix:
${\mathbf {P}}(n\vert n)=\{{\mathbf {I}}_U-{\mathbf {K}}(n)\nabla _{{\mathbf {x}}}{\mathbf {h}}_n|_{\mathbf {{\hat{x}}}(n|n-1)}(n)\}{\mathbf {P}}(n|n-1)$

Appendix 3: Second-order extended Kalman filter
- calculate the Jacobian matrix $\nabla _{{\mathbf {x}}} {\mathbf {f}}_{n}\vert _{\hat{{\mathbf {x}}}(n-1 \vert n-1)}$ and $\overline{{\mathbf {f}}}^{mean}_{n-1}$ using:
$\begin{aligned}{{\overline{{\text {f}}}}}^{ \tiny mean }_{n}&={\mathbb {E}}\{ \mathbf {{\overline{f}}}_{n} |{\mathbf {y}}(0),\ldots , {\mathbf {y}}(n-1) \}=\sum ^U_{u=1} {\varTheta }_u^{\text {f}}{{\overline{{\text {f}}}}}^{ \tiny mean }_{n,u}\end{aligned}$
with ${\varTheta }_u^{\text {f}}$ is a $U \times 1$ vector with zeros everywhere except for the uth element
which is equal to 1. see [24]
- update the a priori state vector:
$\hat{{\mathbf {x}}}(n|n-1)= {\mathbf {f}}_{n}(\hat{{\mathbf {x}}}(n-1\vert n-1))+\frac{1}{2}\overline{{\mathbf {f}}}^{mean}_{n-1}$
- update the a priori error covariance matrix:
${\mathbf {P}}(n|n-1)= \nabla _{{\mathbf {x}}}{\mathbf {f}}_n|_{\mathbf {{\hat{x}}}(n-1|n-1)}{\mathbf {P}}(n-1|n-1) \nabla _{{\mathbf {x}}}{\mathbf {f}}_n^H|_{\mathbf {{\hat{x}}}(n-1|n-1)}+{\mathbf {G}}{\mathbf {Q}}{\mathbf {G}}^H$
- calculate the Jacobian matrix $\nabla _{{\mathbf {x}}} {\mathbf {h}}_{n}\vert _{\hat{{\mathbf {x}}}(n \vert n-1)}$ and $\mathbf {\overline{\overline{{h}}}}_{n}$ using (??).
- update the Kalman gain ${\mathbf {K}}(n)$ as follows:
${\mathbf {K}}(n)= {\mathbf {P}}(n\vert n-1)\nabla _{{\mathbf {x}}} {\mathbf {h}}_{n}^H\vert _{\hat{{\mathbf {x}}}(n \vert n-1)} \big ( \nabla _{{\mathbf {x}}} {\mathbf {h}}_{n}\vert _{\hat{{\mathbf {x}}}(n \vert n-1)}{\mathbf {P}}(n\vert n-1)\nabla _{{\mathbf {x}}} {\mathbf {h}}_{n}^H\vert _{\hat{{\mathbf {x}}}(n \vert n-1)} +{\mathbf {R}}\big )^{-1}$
- deduce the a posteriori state vector:
$\hat{{\mathbf {x}}}(n|n)=\hat{{\mathbf {x}}}(n|n-1)+{\mathbf {K}}(n)\{{\mathbf {y}}(n)- {\mathbf {h}}_n(\mathbf {{\hat{x}}}(n|n-1)) - \frac{1}{2} \mathbf {\overline{\overline{{h}}}}_n\}$
- finally, update the error covariance error matrix:
${\mathbf {P}}(n\vert n)=\{{\mathbf {I}}_U-{\mathbf {K}}(n)\nabla _{{\mathbf {x}}}{\mathbf {h}}_n|_{\mathbf {{\hat{x}}}(n|n-1)}(n)\}{\mathbf {P}}(n|n-1)$

Appendix 4: Unscented Kalman filter
- calculate the sigma points:
${{\mathscr {X}}}(n-1|n-1)= \Big [ \hat{{\mathbf {x}}}(n-1 \vert n-1), \hat{{\mathbf {x}}}(n-1 \vert n-1)\pm \big (\sqrt{(N_d+\lambda ){\mathbf {P}}(n-1 \vert n-1)} \big ) \Big ]$
- calculate the prediction of the system:
${{\mathscr {X}}}(n-1|n-1)= {\mathbf {f}}_{n} ({{\mathscr {X}}}(n-1|n-1))$
- update the a priori mean is:
$\hat{{\mathbf {x}}}^-(n|n-1)=\sum _{{\mathfrak {m}}=0}^{2U } w_{\mathfrak {m}}^{(c0)}{{\mathscr {X}}}_{{\mathfrak {m}}}(n|n-1)$
- update the a priori error covariance matrix:
$\begin{aligned}{\mathbf {P}}(n|n-1)&=\sum _{{\mathfrak {m}}=0}^{2U} w_{\mathfrak {m}}^{(c1)}\left[ {{\mathscr {X}}}_{{\mathfrak {m}}}(n|n-1)-\hat{{{\mathbf {x}}}}^-(n|n-1)\right] ^\star +{\mathbf {G}}{\mathbf {Q}}{\mathbf {G}}^H\end{aligned}$
- calculate $\hat{{\mathbf {y}}}^-(n|n-1)$:
$\hat{{\mathbf {y}}}^-(n|n-1)=\sum _{{m}=0}^{2U } w_{m}^{c0}{{\mathscr {Y}}}_{{m}}(n|n-1)$
where
$\begin{aligned}{{\mathscr {X}}}(n|n-1)={\mathbf {f}}_n\left( {{\mathscr {X}}}(n|n-1)\right) \end{aligned}$
- update the Kalman gain:
${\mathbf {K}}(n)= \lbrace {\mathbf {P}}^{xy}\rbrace \lbrace {\mathbf {P}}^{yy}\rbrace ^{-1}$
where ${\mathbf {P}}^{yy}$and ${\mathbf {P}}^{xy}$ are defined as:
$\begin{aligned}{\mathbf {P}}^{{{\mathbf {y}}}{{\mathbf {y}}}}(n)&=\sum _{{m}=0}^{2U} w_{m}^{c1}\left[ {{\mathscr {Y}}}_{{m}}(n|n-1)-\hat{{\mathbf {y}}}^-(n|n-1)\right] ^\star + {\mathbf {R}}\end{aligned}$ and
$\begin{aligned}{\mathbf {P}}^{{\mathbf {x}}{{\mathbf {y}}}}(n)=\sum _{{m}=0}^{2U} w_{\mathfrak {m}}^{c1}\left[ {{\mathscr {X}}}_{{m}}(n|n-1)-\hat{{\mathbf {x}}}^-(n)\right] \times \left[ {{\mathscr {X}}}_{{m}}(n|n-1)-\hat{{\mathbf {y}}}^-(n|n-1)\right] \end{aligned}$
- update the state vector:
$\hat{{\mathbf {x}}}(n|n)=\hat{{\mathbf {x}}}(n|n-1)+{\mathbf {K}}(n)\{{\mathbf {y}}(n)-\hat{{\mathbf {y}}}^-(n|n-1)\}$
- update the error covariance matrix:
${\mathbf {P}}(n|n)={\mathbf {P}}(n|n-1)-{\mathbf {K}}(n){\mathbf {P}}^{{{{\mathbf {y}}}}{{\mathbf {y}}}}(n){\mathbf {K}}^H(n)$

Appendix 5: Quadrature Kalman filter and Cubature Kalman Filter
Prediction:
- compute the Cholesky factor of ${\mathbf {P}}(n-1|n-1)$:
${\mathbf {P}}(n-1|n-1)={\mathbf {S}}(n-1|n-1){\mathbf {S}}(n-1|n-1)^H$
- compute the sigma points and the weights according to "Appendix 6 (Gauss-Hermite
quadrature points for QKF) or according to "Appendix 7" (the cubature points for CKF).
- evaluate the sigma points:
${{\mathscr {X}}}_l(n-1|n-1)=\hat{{{\mathbf {x}}}}(n-1|n-1)+{\mathbf {S}}(n-1|n-1){\xi } _l$
- propagate the sigma points through the non-linear update function:
${{\mathscr {X}}}_l^*(n|n-1)={\mathbf {f}}_{n} ({{\mathscr {X}}}_l(n-1|n-1))$
- estimate the a priori state vector by combining the a posteriori sigma points:
$\hat{{{\mathbf {x}}}}(n|n-1)= \sum _l w_l{{\mathscr {X}}}_l^*(n-1|n-1)$
- estimate the a priori estimation error covariance matrix:
${\mathbf {P}}(n|n-1)= \sum _l w_l{{\mathscr {X}}}_l^*(n-1|n-1){{\mathscr {X}}}_l^*(n-1|n-1)^H$
$-\hat{{{\mathbf {x}}}}(n|n-1)\hat{{{\mathbf {x}}}}(n|n-1)^H +{\mathbf {G}}{\mathbf {Q}}{\mathbf {G}}^H$
Update
- compute the Cholesky factor of ${\mathbf {P}}(n|n-1)$:
${\mathbf {P}}(n|n-1)={\mathbf {S}}(n|n-1){\mathbf {S}}(n|n-1)^H$
- compute again the sigma points and the weights according to "Appendix 6" (Gauss-Hermite
quadrature points for QKF) or according to "Appendix 7" (the cubature points for CKF).
- evaluate the sigma points:
${{\mathscr {X}}}_l(n|n-1)=\hat{{{\mathbf {x}}}}(n|n-1)+{\mathbf {S}}(n|n-1){\xi } _l$
- propagate the sigma points through the non-linear observation function:
${{\mathscr {Y}}}(n|n-1)={\mathbf {h}}_{n} ({{\mathscr {X}}}_l(n|n-1)$
- calculate $\hat{{\mathbf {y}}}^-(n|n-1)$:
$\hat{{{\mathbf {y}}}}^-(n|n-1)= \sum _l w_l{{\mathscr {Y}}}(n|n-1)$
- update the Kalman gain:
${\mathbf {K}}(n)= \lbrace {\mathbf {P}}^{xy}\rbrace \lbrace {\mathbf {P}}^{yy}\rbrace ^{-1}$
where
${\mathbf {P}}^{xy}= \sum _l w_l{{\mathscr {X}}}_l(n|n-1){{\mathscr {Y}}}(n|n-1)^H-\hat{{{\mathbf {x}}}}(n|n-1)\hat{{{\mathbf {y}}}}^-(n|n-1)^H$
and
${\mathbf {P}}^{yy}= \sum _l w_l{{\mathscr {Y}}}(n|n-1){{\mathscr {Y}}}(n|n-1)^H-\hat{{{\mathbf {y}}}}^-(n|n-1)\hat{{{\mathbf {y}}}}^-(n|n-1)^H + {\mathbf {R}}$
- update the state vector:
$\hat{{\mathbf {x}}}(n|n)=\hat{{\mathbf {x}}}(n|n-1)+{\mathbf {K}}(n)\{{\mathbf {y}}(n)-\hat{{\mathbf {y}}}^-(n|n-1)\}$
- update the error covariance matrix:
${\mathbf {P}}(n|n)={\mathbf {P}}(n|n-1)-{\mathbf {K}}(n){\mathbf {P}}^{{{{\mathbf {y}}}}{{\mathbf {y}}}}(n){\mathbf {K}}^H(n)$

Appendix 6: Calculation of Gauss-Hermite quadrature points and weights
- Set the number of hermite polynomial points m and thus $U^m$ points where
U is the size of the state vector.
- generate a symmetric tridiagonal matrix with zero diagonal elements ${\mathbf {J}}$ such that:
$J_{i,i+1}=\sqrt{\frac{i}{2}}$ where $i=1, \cdots , m$
- compute $\left\{ \lambda _i\right\} _{i=1, \cdots , m}$, which are the eigenvalues of ${\mathbf {J}}$
- set $\xi _l=\sqrt{2}\lambda _i$.
- set $w_i=(e_i)_1^2$ where $(e_i)_1$ is the first element of the $i^{\text {th}}$ normalized eigenvector of ${\mathbf {J}}$.
- extend the one-dimensional quadrature point set of m points in one dimension to
a lattice of $U^m$ cubature points in U dimensions by using the product rule:
$\sum _{i_1, i_2, \cdots i_U} w_{i_1}w_{i_2}\cdots w_{i_U} f(x_1^{i_1}, x_2^{i_2}, \cdots x_U^{i_U})$
- the weights for these Gauss-Hermite cubature points are calculated
by the product of the corresponding one-dimensional weights.
- by changing the variable ${\mathbf {x}}=\sqrt{2}\varvec{\varSigma }+ {\mu }$ we get Gauss-Hermite weighted
sum approximation for multidimensional Gaussian integral where ${\mu }$ is the mean
and $\varvec{\varSigma }$ is the covariance of Gaussian $(\varvec{\varSigma }=\sqrt{\varvec{\varSigma }}\sqrt{\varvec{\varSigma ^H}})$.
- Then, let $\int c({\mathbf {x}}) {\mathscr {N}} \left\{ {\mathbf {x}};{\mu };\varvec{\varSigma } \right\} d{\mathbf {x}}\approx \sum _{i_1, i_2, \cdots i_U} w_{i_1}w_{i_2}\cdots w_{i_U} f(\sqrt{\varvec{\varSigma }} {\xi } _{i_1, i_2, \cdots i_U}+{\mu })$.
- $w_{{i_1, i_2, \cdots i_U}}=\frac{1}{\pi ^{m/2}}w_{i_1}w_{i_2}\cdots w_{i_U}$
- ${\xi } _{i_1, i_2, \cdots i_U}=\sqrt{2} (x_1^{i_1}, x_2^{i_2}, \cdots x_U^{i_U})$.

Appendix 7: Calculation of cubature points and weights
- Set the number of cubature points, $m=2U$ where U is the size of the state vector.
-set ${\xi }_i=\sqrt{\frac{m}{2}}\left[ {\mathbf {1}}\right] _i$ where $[{\mathbf {1}}]$ is $i^{\text {th}}$ column of the matrix
$[{\mathbf {1}}]=\left\{ \begin{array}{cccccc}1&{} 0&{} \cdots &{} -1 &{} 0&{} \cdots \\ 0&{} 1 &{} \cdots &{} 0&{} -1&{} \cdots \\ \vdots &{} 0 &{} \cdots &{} \vdots &{} 0&{} \cdots \\ 0 &{} 0 &{} \cdots &{} 0 &{} 0&{} \cdots \end{array} \right\}$
- set the weight to $w_i=\frac{1}{m}$