Detection of a Band-Limited Signal Using an Orthonormal, Fully-Decimated Filter-Bank

Abstract

In this paper, two methods are proposed for the detection of a band-limited signal in unknown variance white Gaussian noise. The complex amplitude and the frequency of the signal and the noise variance are assumed as unknown parameters. Using wavelet concepts, an orthonormal, fully-decimated filter-bank is employed to decompose the signal into its subband components. It is shown that, in this process, the noise is also decomposed into orthonormal zero-mean components. In the output, if a band-limited target signal is present, the respective single subband component (or two components in marginal cases) containing the target signal presents a non-zero mean. The presence of a non-zero mean componen (s) in this canonical form is tested using a well-known Generalized Likelihood Ratio (GLR) solution ($F$-test), which is based on the ratio between the output power of one (or two) subband(s) and the average output power of the other subbands (estimating the noise variance). Comparing to a threshold, a Constant False Alarm Rate (CFAR) detector is constructed. Since the target signal's central frequency is unknown, the proper subband(s) is selected as the one(or two) maximizing the $F$-test statistic and a GLR test, namely a Wavelet Detector (WD), is obtained. It turns out that the performance of WD depends on the frequency of the signal. For instance, a lowpass signal is detected better than a bandpass signal by this detector. To overcome this problem, the frequency band, where the signal may exist, is estimated, and the signal is down-converted such that the detection is always accomplished at the lowest subband in the new detector, a Modified WD (MWD). The performance of the proposed methods is evaluated in solving two well-known problems, compared with the existing DFT detector. A sinusoid with unknown amplitude, phase and frequency is detected by these detectors as an approximately band-limited signal. The proposed detectors are also applicable for the detection of a signal composed of a white component and an approximately band limited component. A sinusoid, with unknown phase and frequency and Rayleigh-distributed amplitude, is also detected as such a signal.