Multipath-channel estimation and application to ionospheric channels

Abstract

In this paper, the theory of linear time-variant systems is applied to multipath channels and, in particular, to ionospheric channels. Some useful formulas in time and frequency domains are given for both deterministic and random linear time-variant systems and multipath channels. The fundamental parameters of a multipath channel are the delay and the gain coefficient of each path, and the number of significant paths. We propose a methodology for estimating path-delays and path-coefficients and its application to the ionospheric channel estimation. Also, if path coefficients are stationary random processes, the power spectral density of the coefficients can be estimated from measured data of real (actual) channels. Finally, exhaustive computer simulations have been realized for testing the algorithms and a sample of the results is provided in the paper; also, estimation results for an actual narrowband HF ionospheric channel are provided, where up to four significant paths (rays) are detected with a Doppler spread lower than 0.2 Hz.

Appendices

Appendix 1

1.1 Frequency response of linear time-variant systems

The convolution integral Eq. 1 of a linear time-variant system can be expressed in equivalent ways as follows

$$ y(t) = \int\limits_{ - \infty }^{\infty } {h(\tau ;t)x(t - \tau )d\tau } = \int\limits_{ - \infty }^{\infty } {h(t - \tau ;t)x(\tau )d\tau } . $$

(27)

If $X(j\omega )$ is the Fourier transform of $x(t),$ we can write from Eq. 27

$$ y(t) = \int\limits_{ - \infty }^{\infty } {h(\tau ;t)\left[ {\frac{1}{2\pi }\int\limits_{ - \infty }^{\infty } {X(j\omega )e^{j\omega (t - \tau )} d\omega } } \right]d\tau } = \frac{1}{2\pi }\int\limits_{ - \infty }^{\infty } {X(j\omega )H(j\omega ;t)e^{j\omega t} d\omega } , $$

(28)

where $H(j\omega ;t)$ is the system function or the Fourier transform of $h(\tau ;t)$ with respect to the first variable $\tau$ (delay variable), i.e.

$$ H(j\omega ;t) = \int\limits_{ - \infty }^{\infty } {h(\tau ;t)e^{ - j\omega \tau } d\tau } $$

(29)

Formula Eq. 28 is important and has a similar interpretation in linear time-invariant systems, where $H(j\omega ;t)$ does not depend on time $t$. As an example, it is illustrative to consider the input signal $x(t) = e^{{j\omega_{0} t}} ,$ then from Eqs. 27 or 28, we have $y(t) = H(j\omega_{0} ;t) \cdot e^{{j\omega_{0} t}}$ that can be interpreted as an amplitude modulation of the input signal $e^{{j\omega_{0} t}}$ by the system function $H(j\omega_{0} ;t)$ or, in the frequency domain, the input spectral line $2\pi \delta (\omega - \omega_{0} )$ is spread out at the output according to the spectrum of $H(j\omega_{0} ;t)$.

Also, formula Eq. 28 can be expressed in the frequency domain, as follows

$$ Y(j\Upomega ) = \frac{1}{2\pi }\int\limits_{ - \infty }^{\infty } {X(j\omega )\tilde{H}(j\omega ;j(\Upomega - \omega ))d\omega } = \frac{1}{2\pi }\int\limits_{ - \infty }^{\infty } {X(j(\Upomega - \omega ))\tilde{H}(j(\Upomega - \omega );j\omega )d\omega } , $$

(30)

where $Y(j\Upomega )$ and $\tilde{H}(j\omega ;j\Upomega )$ are the Fourier transforms of $y(t)$ and $H(j\omega ;t),$ respectively, over time variable $t$. That is,

$$ \tilde{H}(j\omega ;j\Upomega ) = \int_{\, - \infty }^{\,\infty } {H(j\omega ;t)e^{ - j\Upomega \cdot t} dt} = \int_{\, - \infty }^{\,\infty } {\int_{\, - \infty }^{\,\infty } {h(\tau ;t)e^{ - j(\omega \tau + \Upomega t)} d\tau dt} } . $$

(31)

Remarks Compare formulas Eqs. 30 and 27 representing convolution integrals in the frequency domain and in the time domain, respectively. Also, note that if $h(\tau ;t) = h_{0} (\tau ),$ i.e. the system is time-invariant, then from Eq. 31 $\tilde{H}(j\omega ;j\Upomega ) = H_{0} (j\omega ) \cdot 2\pi \delta (\Upomega ),$ and from Eq. 30 we have $Y(j\Upomega ) = X(j\Upomega ) \cdot H_{0} (j\Upomega )$, being the well-known formula applied to linear time-invariant systems.

Finally, from Eqs. 2 and 31, we have

$$ \tilde{H}(j\omega ;j\Upomega ) = G(j\omega )\sum\limits_{k = 1}^{K} {\int\limits_{ - \infty }^{\infty } {a_{k} (t)e^{{ - j\omega \cdot \tau_{k} (t)}} \cdot e^{ - j\Upomega \cdot t} dt} } . $$

(32)

Assuming that delay $\tau_{k} (t)$ can be approximated as $\tau_{k} (t) \approx \tau_{k} + \nu_{k} \cdot t,$ then Eq. 32 could be expressed as

$$ \tilde{H}(j\omega ;j\Upomega ) \approx G(j\omega )\sum\limits_{k = 1}^{K} {e^{{ - j\omega \cdot \tau_{k} }} A_{k} (j(\Upomega + \omega \cdot \nu_{k} ))} , $$

(33)

where $A_{k} (j\Upomega )$ is the Fourier transform of $a_{k} (t),$ and from Eqs. 33 and 30

$$ \begin{aligned} Y(j\Upomega ) = & \sum\limits_{k = 1}^{K} {\frac{1}{2\pi }\int\limits_{ - \infty }^{\infty } {A_{k} (j(\Upomega - (1 - \nu_{k} )\omega ))G(j\omega )X(j\omega )e^{{ - j\omega \cdot \tau_{k} }} d\omega } } \\ = & \frac{1}{{2\pi (1 - \nu_{k} )}}\sum\limits_{k = 1}^{K} {A_{k} (j\Upomega ) * \left[ {G({{j\Upomega } \mathord{\left/ {\vphantom {{j\Upomega } {(1 - \nu_{k} )}}} \right. \kern-\nulldelimiterspace} {(1 - \nu_{k} )}})X({{j\Upomega } \mathord{\left/ {\vphantom {{j\Upomega } {(1 - \nu_{k} )}}} \right. \kern-\nulldelimiterspace} {(1 - \nu_{k} )}})e^{{ - {{j\Upomega \cdot \tau_{k} } \mathord{\left/ {\vphantom {{j\Upomega \cdot \tau_{k} } {(1 - \nu_{k} )}}} \right. \kern-\nulldelimiterspace} {(1 - \nu_{k} )}}}} } \right]} , \\ \end{aligned} $$

(34)

where symbol * means convolution operation.

Remarks Expression Eq. 34 has the following interpretation: for each path k, the path input-signal spectrum $G(j\Upomega )X(j\Upomega )$ undergoes a linear phase change due to the mean delay $\tau_{k} ,$ i.e. $G(j\Upomega )X(j\Upomega )e^{{ - j\Upomega \tau_{k} }}$; then, it undergoes a frequency expansion by a factor $1 - \nu_{k}$; hence, the result is convolved by $A_{k} (j\Upomega ),$ resulting in a larger spectral spread of the input-signal spectrum. Finally, the output-signal spectrum is the sum of the output-signal spectra corresponding to all paths.□

Appendix 2

2.1 Output power spectrum of linear time-variant systems

In the case of considering stationary random processes, we will develop the corresponding formulas for the power spectral density or the power spectrum of the output signal.

If $h(\tau ;t)$ and $x(t)$ are independent random processes and $x(t)$ is stationary, we can define the autocorrelation function $r_{y} (t,\Updelta t)$ of $y(t)$ as follows

$$ r_{y} (t,\Updelta t) = E\left\{ {y^{*} (t) \cdot y(t + \Updelta t)} \right\} = \int\limits_{ - \infty }^{\infty } {\int\limits_{ - \infty }^{\infty } {E\left\{ {h^{*} (u;t)h(v;t + \Updelta t)} \right\} \cdot r_{x} (\Updelta t + u - v)dudv} } , $$

(35)

where $y^{*} (t)$ is the complex conjugate of $y(t),\,r_{x} (\Updelta t) = E\left\{ {x^{*} (t) \cdot x(t + \Updelta t)} \right\}$ is the autocorrelation function of the stationary input $x(t)$ and $E\left\{ z \right\}$ is the statistical expectation of $z$. Identifying $\tau = u$ and $\Updelta \tau = v - u,$ we can write

$$ r_{y} (t,\Updelta t) = \int\limits_{ - \infty }^{\infty } {\int\limits_{ - \infty }^{\infty } {r_{h} (\tau ,\Updelta \tau ;t,\Updelta t) \cdot r_{x} (\Updelta t - \Updelta \tau )d\tau d(\Updelta \tau )} } , $$

(36)

where $r_{h} (\tau ,\Updelta \tau ;t,\Updelta t) = E\left\{ {h^{*} (\tau ;t)h(\tau + \Updelta \tau ;t + \Updelta t)} \right\}$ is the autocorrelation function of the impulse response $h(\tau ;t)$.

Finally, supposing $h(\tau ;t)$ is stationary in time t, then $y(t)$ is also stationary, and Eq. 36 can be written as

$$ r_{y} (\Updelta t) = \int\limits_{ - \infty }^{\infty } {r_{h} (\Updelta \tau ;\Updelta t) \cdot r_{x} (\Updelta t - \Updelta \tau )d(\Updelta \tau )} , $$

(37)

where

$$ r_{h} (\Updelta \tau ;\Updelta t) = \int\limits_{ - \infty }^{\infty } {r_{h} (\tau ,\Updelta \tau ;t,\Updelta t)d\tau } = \int\limits_{ - \infty }^{\infty } {E\left\{ {h^{*} (\tau ;t)h(\tau + \Updelta \tau ;t + \Updelta t)} \right\}d\tau }. $$

(38)

Compare equation Eq. 37 referred to autocorrelation functions of signals with Eq. 1 referred to signals.

In the frequency domain, if $S_{y} (j\Upomega )$ is the power spectral density of $y(t),$ i.e. the Fourier transform of $r_{y} (\Updelta t),$ and $S_{x} (j\Upomega )$ is the power spectral density of $x(t),$ then Eq. 37 is transformed into

$$ S_{y} (j\Upomega ) = \frac{1}{2\pi }\int\limits_{ - \infty }^{\infty } {S_{x} (j\omega )S_{h} (j\omega ; j(\Upomega - \omega ))d\omega } , $$

(39)

where

$$ S_{h} (j\omega ; j\Upomega ) = \int\limits_{ - \infty }^{\infty } {\int\limits_{ - \infty }^{\infty } {r_{h} (\Updelta \tau ;\Updelta t)e^{ - j(\omega \Updelta \tau + \Upomega \Updelta t)} d(\Updelta \tau )d(\Updelta t)} } . $$

(40)

Note the similarity between Eqs. 39 and 30, so a similar interpretation could be done here.