Sensor array signal tracking using a data-driven window approach
Introduction
In typical nonstationary array processing scenarios, the interval of data stationarity tends to vary with time, i.e., the received data may include both highly nonstationary and almost stationary blocks. Another typical situation occurs where some sources move rapidly within the window exploited, whereas the motion of the remaining part of sources is weak.
In such scenarios, the lag window length becomes one of the most important parameters. In the traditional fixed-window approach, the use of short windows is well known to increase the variance of direction finding techniques. With longer lag windows, the estimation variance can be lowered but the DOA estimates become biased and, therefore, are unable to track rapidly moving sources. As a result, the traditional fixed-window approach does not enable tracking multiple sources with severely different intervals of stationarity.
In this paper, we develop a new adaptive-window approach to DOA tracking. In our technique, multiple data-driven windows are used, i.e., a separate adaptive window is employed for each source. Our algorithm combines the developed adaptive multiwindow subspace tracker and the popular root-MUSIC technique [1], [12]. The adaptive-window selection procedure is based on the approximate minimization of the mean squared estimation error using the bias-to-variance tradeoff approach developed originally for another class of problems [7], [8], [9]. Comparisons with conventional fixed-window algorithms demonstrate a potential of the developed adaptive-window approach. A natural price for the improvements achieved is a higher computational cost. Also, our approach is restricted by scenarios with ‘well separated’ sources.
Section snippets
Signal model
Assume that a uniform linear array (ULA) of n sensors receives q (q<n) narrowband signals impinging from the unknown varying directions {θ1(t),θ2(t),…,θq(t)}. The output vector of the array at the discrete time t can be expressed aswhere the n×q time-varying direction matrixis composed of the source direction vectorsλ is the wavelength, d is the interelement spacing, (·)T stands for the transpose, and
Conventional fixed-window approach
In this section, we revisit the traditional fixed-window approach with the rectangular sliding window containing M independent data snapshots.2 Write the data matrix asThe lag window estimate of the array covariance matrixis given bywhere , is the identity matrix, σ2 is the sensor noise variance, and (·)H stands for the Hermitian
Adaptive-window approach
Let us make the following assumptions:(A1) The array is large () so that the sources are well separated in the sense of Rayleigh criterion [3]. (A2) The source powers are subject to much slower variations than their DOA's. (A3) The number of sources is known.
The first assumption is almost always true for large arrays. Although the high-resolution root-MUSIC algorithm will be exploited for DOA tracking, we stress that this algorithm is chosen because of other reasons than its high-resolution
Simulations
We have assumed a ULA of five omnidirectional sensors with the half-wavelength spacing. SNR=1.25 dB has been assumed for each source in a single sensor. The simplest two-window algorithm was implemented with the window lengths equal to 8 and 128 snapshots (i.e., ). In all figures given below, the true source trajectories are indicated by dashed lines.
In the first two examples, we simulated the single source scenario. Fig. 2(a)–(c) displays the estimated trajectories for the first example
Conclusions
In several practically important source tracking applications, the interval of source stationarity may vary with time, so that the array observations may contain both almost stationary and nonstationary data intervals. Even more complicated situations may occur where some sources move rapidly within the window, whereas the motion of the other sources is weak. In such scenarios, the traditional fixed-window approach may be nonoptimal because it may result in a significant degradation of the
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The work of the first author is supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada.