Subspace-Based Algorithms for Blind ML Frequency and Transition Time Estimation in Frequency Hopping Systems

Abstract

Frequency hopping spread spectrum (FHSS) is a technology for combating narrow band interference. Two parameters required for estimation in FHSS are transition time and hopping frequency. In this paper, blind subspace-based schemes with a maximum likelihood (ML) criterion for estimating frequency and transition time without using reference signals are proposed. The selection of the related parameters is discussed. Subspace-based algorithms are applied with the help of the proposed block selection scheme. The performance is improved with a block selection algorithm to overcome the unbalanced processing block problems in various algorithms. The proposed method significantly reduces computational complexity compared with a greedy search ML-based algorithm. The performance is shown to outperform an existing iterative ML-based algorithm with a comparable complexity.

Appendices

Appendix 1: Derivation of Multiroots in ( 10 ) and ( 11 )

In this appendix, we demonstrate the multiroot problem of (10) and (11) by the following development. As these two equations are similar, (10) is used in the derivation. The same development can be applied to (11).

$\text{Im} ({\mathbf{X}}_{1}^{H} {\mathbf{D}}_{1} {\mathbf{S}}_{1} {\mathbf{S}}_{1}^{H} {\mathbf{X}}_{1} )$ is expanded. After some manipulations, we have

$$\begin{aligned} & \text{Im} \left( {{\mathbf{X}}_{1}^{H} {\mathbf{D}}_{1} {\mathbf{S}}_{1} {\mathbf{S}}_{1}^{H} {\mathbf{X}}_{1} } \right) \\ & \quad = \text{Im} \left[ {r_{K} (1)e^{{j\omega_{1} T_{s} }} + 2r_{K} (2)e^{{j2\omega_{1} T_{s} }} + \cdots + (K - 1)r_{K} (K - 1)e^{{j(K - 1)\omega_{1} T_{s} }} } \right] \\ \end{aligned}$$

(27)

where

$$r_{K} (m) = \left\{ {\begin{array}{*{20}l} {\sum\limits_{i = 1}^{K - m} {x(i)x^{*} (i + m),} } \hfill &\quad {1 \le m < K} \hfill \\ 0 \hfill &\quad {m \ge K} \hfill \\ \end{array} } \right.$$

(28)

Consider the situation without noise. By substituting $m = 1, 2, \ldots ,K - 1$ into (28), we obtain

$$\begin{aligned} & r_{K} (1) = \sum\limits_{i = 1}^{K - 1} {x(i)x^{*} (i + 1) = (K - 1)\left| {a_{1} } \right|^{2} e^{{ - j\omega T_{s} }} ;} \\ & r_{K} (2) = \sum\limits_{i = 1}^{K - 2} {x(i)x^{*} (i + 2) = (K - 2)\left| {a_{1} } \right|^{2} e^{{ - j2\omega T_{s} }} ;\quad \ldots ;} \\ & r_{K} (K - 1) = \sum\limits_{i = 1}^{1} {x(i)x^{*} (i + K - 1) = (1)\left| {a_{1} } \right|^{2} e^{{ - j(K - 1)\omega T_{s} }} } \\ \end{aligned}$$

(29)

By substituting (28) into (26), we obtain

$$\begin{aligned} &\text{Im} \left( {{\mathbf{X}}_{1}^{H} {\mathbf{D}}_{1} {\mathbf{S}}_{1} {\mathbf{S}}_{1}^{H} {\mathbf{X}}_{1} } \right) \hfill \\ &\quad = \text{Im} \left[ {r_{K} (1)e^{{j\omega_{1} T_{s} }} + 2r_{K} (2)e^{{j2\omega_{1} T_{s} }} + \cdots + (K - 1)r_{K} (K - 1)e^{{j(K - 1)\omega_{1} T_{s} }} } \right] \hfill \\ &\quad \cong \left| {a_{1} } \right|^{2} \text{Im} \left[ {(K - 1)e^{{ - j\omega T_{s} }} e^{{j\omega_{1} T_{s} }} + 2(K - 2)e^{{ - j2\omega T_{s} }} e^{{j2\omega_{1} T_{s} }} } \right. \hfill \\ &\left. {\quad \quad + \cdots + (K - 1)(1)e^{{ - j(K - 1)\omega T_{s} }} e^{{j(K - 1)\omega_{1} T_{s} }} } \right] \hfill \\ &\quad = \left| {a_{1} } \right|^{2} \text{Im} \left[ {(K - 1)e^{{j\left( {\omega_{1} - \omega } \right)T_{s} }} + 2(K - 2)e^{{j2\left( {\omega_{1} - \omega } \right)T_{s} }} + \cdots + (K - 1)(1)e^{{j(K - 1)\left( {\omega_{1} - \omega } \right)T_{s} }} } \right] \hfill \\ &\quad = \left| {a_{1} } \right|^{2} \left( {(K - 1)\sin \left( {\bar{\omega }_{1} - \bar{\omega }} \right) + 2(K - 2)\sin \left( {2\left( {\bar{\omega }_{1} - \bar{\omega }} \right)} \right)} \right. \hfill \\ &\quad \quad \left. { + \cdots + (K - 1)(1)\sin \left( {\left( {K - 1} \right)\left( {\bar{\omega }_{1} - \bar{\omega }} \right)} \right)} \right) = 0 \hfill \\ \end{aligned}$$

(30)

In (30), there are (K − 1) $\sin ( \cdot )$ terms. It can be viewed as multi-order harmonic sine waves $m(\bar{\omega }_{1} - \bar{\omega })$, $m = 2, \ldots , K - 1$ and thus there are multiple roots (up to K − 1 roots).

Based on the above result, it leads that the iteration algorithm (41) in [13] has a problem to converge to the correct estimate. Since it has the multi-root problem, $\hat{\bar{\omega }}_{1} (K)$ in (41) may or may not converge to the correct region (either infected by the initial value of $\hat{\bar{\omega }}_{1} (1)$ or by the noise). Figure 6 shows the example of SNR = 4 dB by using the iterative ML algorithm of [13]. The scatter plot of the estimation errors is displayed in 10,000 trial runs of f ₁ = 18,000 Hz, f ₂ = 29,000 Hz, M = 128, and K = 64. Observed from Fig. 6, the majority of the estimation errors are located around the region of −1567, 0, and 1567 Hz (which are relative to the zero points at 16,433, 18,000, and 19,567 Hz). Due to the above-mentioned problem, the estimation errors disperse in the frequency band.

Fig. 6

f₁ estimation error versus trial runs (4 dB)

Appendix 2: Derivation of $\hat{\omega }_{1} = \mathop {\arg }\limits_{{\omega_{1} }} \left\{ {\hbox{max} \left( {\left\| {{\hat{\mathbf{s}}}_{1}^{H} (\omega_{1} ){\mathbf{x}}_{1} } \right\|^{2} }/K \right)} \right\}$

By expanding (8), we have $\varphi_{1} (a_{1} ,\omega_{1} ,K) = \left\| {{\mathbf{x}}_{1} - a_{1} {\mathbf{s}}_{1} } \right\|^{2} = \left\| {x_{1} } \right\|^{2} - a_{1}^{H} {\mathbf{s}}_{1}^{H} {\mathbf{x}}_{1} - a_{1} {\mathbf{x}}_{1}^{H} {\mathbf{s}}_{1} + \left\| {a_{1} {\mathbf{s}}_{1} } \right\|^{2}$.

To minimize φ ₁(a ₁, ω ₁, K), by setting ∂φ ₁/∂a ₁ = 0, we have

$${\raise0.7ex\hbox{${\partial \varphi_{1} }$} \!\mathord{\left/ {\vphantom {{\partial \varphi_{1} } {\partial a_{1} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\partial a_{1} }$}} = - {\mathbf{x}}_{1}^{H} {\mathbf{s}}_{1} + Ka_{1}^{H} = 0 \Rightarrow \hat{a}_{1} = \frac{1}{K}{\mathbf{s}}_{1}^{H} {\mathbf{x}}_{1} .$$

(31)

Putting $\hat{a}_{1} = \frac{1}{K}{\text{s}}_{1}^{H} {\mathbf{x}}_{1}$ in φ ₁(a ₁, ω ₁, K), we have

$$\varphi_{1} (\omega_{1} ,K) = \left\| {\left( {{\mathbf{I}} - {\mathbf{s}}_{1} {\mathbf{s}}_{1}^{H} /K} \right){\mathbf{x}}_{1} } \right\|^{2} .$$

(32)

With $\hat{a}_{1}$ and $\hat{\omega }_{1} ,$ the objective function becomes

$$\varphi_{1} (K) \equiv \left\| {{\mathbf{x}}_{1} } \right\|^{2} - \left\| {{\hat{\mathbf{s}}}_{1}^{H} {\mathbf{x}}_{1} } \right\|^{2} /K.$$

(33)

For a given K, the minimization of φ ₁(a ₁, ω ₁, K) is equivalent to finding a frequency ω ₁ satisfying

$$\hat{\omega }_{1} = \mathop {\arg }\limits_{{\omega_{1} }} \left\{ {\hbox{max} \left( {\left\| {{\hat{\mathbf{s}}}_{1}^{H} {\mathbf{x}}_{1} } \right\|^{2} /K} \right)} \right\}$$

(34)