High Resolution Cognitive Radio-Based Concatenated Spectrum Time-of-Arrival Estimation

Abstract

This paper proposes high-precision, time-of-arrival (TOA) estimation by concatenating non-contiguous and non-equal white spaces in the spectrum. White spaces are identified and allocated by the capabilities of cognitive radios (CR). A simple radio system that enables sequential concatenation of non-contiguous bands is introduced. The paper aims to design signal-processing techniques for the proposed radio that maintain a TOA performance comparable to that of utilizing a bandwidth equivalent to the addition of all white spaces. This work also investigates two potential high resolutions, TOA estimation architectures that are applicable to CR-based systems. The first approach estimates TOA over each sub-band and then optimally fuses the estimated TOAs. In the second technique, channels estimated over sub-bands are concatenated and a TOA-estimation method is applied to the whole available spectrum. The Cramer–Rao bound (CRB) of the TOA estimation for both approaches is computed. Minimum mean square estimation (MMSE) algorithm is applied to evaluate the performance of the proposed system. Simulations are concluded to study the performance in terms of the variance of error. Moreover, the complexity of the proposed concatenation method is evaluated.

Appendices

Appendix A: Linear Estimation Equations

Gauss–Markov theorem [14] is used to estimate TOAs from different bands. A linear model ${\user2 T} = {\user2 H\tau} + {\user2 E}$ is considered for the estimation, where ${\user2 T} = \left[ {\hat{\tau }_{1(1)} , \ldots ,\hat{\tau }_{1(K)} } \right]$, $\hat{\tau }_{1(k)}$ defined in (17), E represents error that is a zero-mean random variable with positive-definite covariance matrix R _E , and τ is an unknown constant vector and H is a full rank known mixing matrix. For this model, the minimum-variance unbiased linear estimator of τ observing T corresponds to:

$$ \hat{{\user2 \tau }} = \left( {{\user2 H}^{T} {\user2 R_{E}}^{ - 1} {\user2 H}} \right)^{ - 1} {\user2 H}^{T} {\user2 R_{E}}^{ - 1} {\user2 T} $$

(29)

In addition, the minimum mean square error (MMSE) of this estimator is:

$$ MMSE = \left( {{\user2 H}^{T} {\user2 R_{E}}^{ - 1} {\user2 H}} \right)^{ - 1} $$

(30)

Now, we use the system model of (17). Due to the independency of e _k , for $k \in \left\{ {1, \ldots ,K} \right\}$ in E, we have ${\user2 R}_{E} = diag\left( {\sigma_{1}^{2} , \ldots ,\sigma_{K}^{2} } \right)$, where $\sigma_{k}^{2}$ is the estimator error variance (e _k ). Accordingly, considering the estimator in (13), we have:

$$ \hat{\tau }_{1} = \left( {{\user2 H}^{T} {\user2 H}} \right)^{ - 1} {\user2 H}^{T} {\user2 T} $$

(31)

Here, ${\user2 T} = \left[ {\hat{\tau }_{1(1)} ,\, \ldots ,\,\hat{\tau }_{1(K)} } \right]^{T}$ and ${\user2 H} = \left[ {1,\, \ldots ,\,1} \right]^{T}$. Inserting these parameters into (31), we extract (18). Inserting R _E and H into (30), we observe that the variance of error of this estimator corresponds to:

$$ \sigma_{{\hat{\tau }_{1} }}^{2} = \frac{1}{{\mathop \sum \nolimits_{k = 1}^{K} \sigma_{k}^{ - 2} }} $$

(32)

Appendix B: Extraction the Cramer–Rao Band of TOA Estimation

Assume that the received signal (3) with bandwidth B is sampled at the Nyquest rate as $\bar{x} = \left[ {x_{1} , \ldots ,x_{N} } \right]$. It is easy to show that the elements of the Fisher information matrix (FIM) presented in (21) correspond to [23]:

$$ \begin{gathered} J_{mk} = \frac{{\alpha_{m} \alpha_{k}^{*} }}{{2B\sigma_{\upsilon }^{2} }}\mathop \sum \limits_{t = 1}^{N} \frac{{\partial s\left( {t - \tau_{m} } \right)}}{\partial t}\left( {\frac{{\partial s\left( {t - \tau_{k} } \right)}}{\partial t}} \right)^{*} + \frac{{\alpha_{k} \alpha_{m}^{*} }}{{2B\sigma_{\upsilon }^{2} }}\mathop \sum \limits_{t = 1}^{N} \frac{{\partial s\left( {t - \tau_{k} } \right)}}{\partial t}\left( {\frac{{\partial s\left( {t - \tau_{m} } \right)}}{\partial t}} \right)^{*} \hfill \\ \quad \quad \quad {\text{for }}m,k = 1, \ldots ,M \hfill \\ \end{gathered} $$

(33)

If $s\left( t \right)$ does not change significantly during the time interval between samples, the sum in (33) can be approximated by the following integral [25, 26].

$$ J_{mk} = \frac{{\alpha_{m} \alpha_{k}^{*} }}{{\sigma_{\upsilon }^{2} }}\int\limits_{0}^{N} \frac{{\partial s\left( {t - \tau_{m} } \right)}}{\partial t}\left( {\frac{{\partial s\left( {t - \tau_{k} } \right)}}{\partial t}} \right)^{*} dt + \frac{{\alpha_{k} \alpha_{m}^{*} }}{{\pi N_{0} }}\int\limits_{0}^{N} \frac{{\partial s\left( {t - \tau_{k} } \right)}}{\partial t}\left( {\frac{{\partial s\left( {t - \tau_{m} } \right)}}{\partial t}} \right)^{*} dt $$

(34)

The FIM can be expressed as a scaled product of three simpler matrices:

$$ J = \left( {\sigma_{\upsilon }^{2} } \right)^{ - 1} ({\user2 A}{\user2 S}{\user2 A}^{*} + {\user2 A}^{*} {\user2 S}^{*} {\user2 A}) $$

(35)

where $A = diag\left( {\alpha_{1} , \ldots ,\alpha_{M} } \right)$ and the element of S are:

$$ S_{mk} = \int\limits_{0}^{N} \frac{{\partial s\left( {t - \tau_{m} } \right)}}{\partial t}\left( {\frac{{\partial s\left( {t - \tau_{k} } \right)}}{\partial t}} \right)^{*} dt + \int\limits_{0}^{N} \frac{{\partial s\left( {t - \tau_{k} } \right)}}{\partial t}\left( {\frac{{\partial s\left( {t - \tau_{m} } \right)}}{\partial t}} \right)^{*} dt $$

(36)

Assuming the observation interval is long enough to contain all multi-paths, we observe that the integral (36) is a function of $\left| {\tau_{1} - \tau_{2} } \right|$. In that case, S is a real symmetric Toplitz matrix [14].

Now, we find the inverse of matrix J. The m ^th diagonal element of J ⁻¹ is the bound for the variance of m ^th multi-path delay estimate:

$$ CRB\left( {\tau_{m} } \right) = \left( {J^{ - 1} } \right)_{mm} = \frac{{\sigma_{\upsilon }^{2} }}{{\left| {\alpha_{m} } \right|^{2} }}\left( {S^{ - 1} } \right)_{mm} $$

(37)

We observe that $\left( {S^{ - 1} } \right)_{mm}$ depends only on the signal waveform and the relative delays of echoes. Therefore, the CRB for estimation of m ^th multi-path delay depends on the power of the m ^th echo and relative delay of all echoes.

Next, we simplify (37) assuming M = 2, i.e., the availability of only two dominant multi-path, which models the effect of closely spaced multi-path in wireless channel [25, 26]. As result, we have:

$$ S^{ - 1} = \frac{1}{{\left[ {\mathop \smallint \nolimits \left| {\dot{s}} \right|^{2} \left( t \right)dt} \right]\left[ {1 - \rho^{2} \left( {\left| {\tau_{1} - \tau_{2} } \right|} \right)} \right]}} \times \left( {\begin{array}{*{20}c} 1 & { - \rho \left( {\left| {\tau_{1} - \tau_{2} } \right|} \right)} \\ { - \rho \left( {\left| {\tau_{1} - \tau_{2} } \right|} \right)} & 1 \\ \end{array} } \right) $$

(38)

Now, based on (37) and (38), after some simplification, we can find (21).