Sparse spectrum fitting algorithm using signal covariance matrix reconstruction and weighted sparse constraint

Abstract

Although sparse spectrum fitting algorithm has good multi-target resolution, the performance is decreased under the influence of strong interference. In response to this problem, a sparse spectrum fitting algorithm using signal covariance matrix reconstruction and weighted sparse constraint is proposed. The algorithm uses iterative adaptive approach (IAA) to estimate spatial spectrum and divide signal region. The signal region is integrated to reconstruct the signal covariance matrix. On the other hand, the $l_{1}$ norm constraint is weighted in different angles by the inverse spatial spectrum estimated by IAA. Finally, the spatial spectrum of the signal region is fitted to detect and resolve the weak targets. Simulation and theoretical analysis show that the proposed algorithm has outform performance under the influence of strong interference.

Appendix: The convergence proof for the iterative algorithm in Table 1

The negative log-likelihood function of $\left\{ {{\varvec{x}}(k)} \right\}_{k = 1}^{k = K}$ is

$$ {\text{In}}\left| {\varvec{R}} \right| + {\text{tr}}({\varvec{R}}^{ - 1} {\hat{\varvec{R}}}) $$

(A1)

where ${\hat{\varvec{R}}} = \frac{1}{K}\sum\limits_{k = 1}^{k = K} {{\varvec{x}}(k){\varvec{x}}^{{\text{H}}} } (k)$ is sample covariance matrix, ${\text{tr}}()$ is the trace of a matrix. Using matrix inversion lemma together with ${\varvec{Q}}(\theta_{l} ) = {\varvec{R}} - P(\theta_{l} ){\varvec{a}}(\theta_{l} ){\varvec{a}}^{{\text{H}}} (\theta_{l} )$, (A1) is equivalent to (A2)

$$ f(P(\theta_{l} )) = {\text{In}}\left( {1 + P(\theta_{l} ){\varvec{a}}^{{\text{H}}} (\theta_{l} ){\varvec{Q}}^{ - 1} (\theta_{l} ){\varvec{a}}(\theta_{l} )} \right) - \frac{{P(\theta_{l} ){\varvec{a}}^{{\text{H}}} (\theta_{l} ){\varvec{Q}}^{ - 1} (\theta_{l} ){\hat{\varvec{R}}\varvec{Q}}^{ - 1} (\theta_{l} ){\varvec{a}}(\theta_{l} )}}{{1 + P(\theta_{l} ){\varvec{a}}^{{\text{H}}} (\theta_{l} ){\varvec{Q}}^{ - 1} (\theta_{l} ){\varvec{a}}(\theta_{l} )}} $$

(A2)

Setting the first derivative of (A2) with respect to $P(\theta_{l} )$, to zero, gives

$$ \tilde{P}(\theta_{l} ) = \frac{{{\varvec{a}}^{{\text{H}}} (\theta_{l} ){\varvec{Q}}^{ - 1} (\theta_{l} )({\hat{\varvec{R}}} - {\varvec{Q}}(\theta_{l} )){\varvec{Q}}^{ - 1} (\theta_{l} ){\varvec{a}}(\theta_{l} )}}{{\left( {{\varvec{a}}^{H} (\theta_{l} ){\varvec{Q}}^{ - 1} (\theta_{l} ){\varvec{a}}(\theta_{l} )} \right)^{2} }} $$

(A3)

The second derivative of (A2), with respect to $P(\theta_{l} )$,is

$$ f^{^{\prime\prime}} (P(\theta_{l} )) = \frac{{\left[ {{\varvec{a}}^{{\text{H}}} (\theta_{l} ){\varvec{Q}}^{ - 1} (\theta_{l} ){\varvec{a}}(\theta_{l} )} \right]^{2} }}{{\left[ {1 + P(\theta_{l} ){\varvec{a}}^{{\text{H}}} (\theta_{l} ){\varvec{Q}}^{ - 1} (\theta_{l} ){\varvec{a}}(\theta_{l} )} \right]^{2} }} > 0 $$

(A4)

This means that $\tilde{P}(\theta_{l} )$ is the unique minimizer of $f(P(\theta_{l} ))$.Since $\tilde{P}(\theta_{l} )$ represents power, it showed been nonnegative. By using the matrix inversion lemma and the relationship between ${\varvec{Q}}(\theta_{l} )$ and ${\varvec{R}}$, (A3) become

$$ \tilde{P}(\theta_{l} ) = \max (0,P(\theta_{l} ) + \frac{{{\varvec{a}}^{{\text{H}}} (\theta_{l} ){\varvec{Q}}^{ - 1} (\theta_{l} )({\hat{\varvec{R}}} - {\varvec{R}}){\varvec{R}}^{ - 1} {\varvec{a}}(\theta_{l} )}}{{\left( {{\varvec{a}}^{{\text{H}}} (\theta_{l} ){\varvec{R}}^{ - 1} {\varvec{a}}(\theta_{l} )} \right)^{2} }}) $$

(A5)

The (A5) will be locally convergent if ${\varvec{R}}$ is recalculated by each updated $P(\theta_{l} )$ because of the cyclical maximization of the likelihood function (Yardibi et al., 2010), (A5) can rewritten as (assuming $\tilde{P}(\theta_{l} ) \ge 0$)

$$ P(\theta_{l} ) = P(\theta_{l} ) - \frac{1}{{{\varvec{a}}^{{\text{H}}} (\theta_{l} ){\varvec{R}}^{ - 1} {\varvec{a}}(\theta_{l} )}} + \frac{{{\varvec{a}}^{{\text{H}}} (\theta_{l} ){\varvec{R}}^{ - 1} {\hat{\varvec{R}}\varvec{R}}^{ - 1} {\varvec{a}}(\theta_{l} )}}{{\left( {{\varvec{a}}^{{\text{H}}} (\theta_{l} ){\varvec{R}}^{ - 1} {\varvec{a}}(\theta_{l} )} \right)^{2} }} $$

(A6)

Since the output of capon beamformer is $P(\theta_{l} ) \approx \frac{1}{{{\varvec{a}}^{{\text{H}}} (\theta_{l} ){\varvec{R}}^{ - 1} {\varvec{a}}(\theta_{l} )}}$, the (A6) become

$$ \tilde{P}(\theta_{l} ) = \frac{{{\varvec{a}}^{{\text{H}}} (\theta_{l} ){\varvec{R}}^{{ - 1}} {\hat{\varvec{R}}\varvec{R}}^{ - 1} {\varvec{a}}(\theta_{l} )}}{{\left( {{\varvec{a}}^{{\text{H}}} (\theta_{l} ){\varvec{R}}^{ - 1} {\varvec{a}}(\theta_{l} )} \right)^{2} }} \ge 0 $$

(A7)

that is step (4) (5) in Table 1, since (A7) is obtained from (A5), which is locally convergent, thus (A7) is locally convergent.