Linear Canonical Correlation And Its Application For Detection Of Lfm Signals

Linear canonical transform(LCT) is a new potent tool to analyze the nonstationary signals. Many operations, such as the Fourier transform(FT), the fractional Fourier transform(FRFT), the Fresnel transform, and scaling operations can be deemed to be the special cases of the LCT. In past decades, as nonstationary signal processing based on the conventional FT has certain limitations, the LCT has attracted much attention in the signal processing community. Compared to the FRFT with one extra degree of freedom and the FT without any parameter, the LCT is more flexible due to its extra three degrees of freedom and has been used in filter design, time-frequency analysis, image encryption, modulation as well as multiplexing in communications, etc. The above mentioned applications demonstrate the ability as well as the potential of the LCT in signal processing fields.Linear frequency modulated(LFM) signals are frequently encountered in applications such as radar and sonar due to the high range resolution and the immunity from interference, consequently the detection of LFM signals is much important. In recent years, several timefrequency analysis based methods for the LFM signal detection have been proposed. Since the ambiguity function(AF) of the LFM signal distributes as a straight line passing through the origin of the ambiguity plane, the detection statistic using Radon-ambiguity transform(RAT) can be used as an effective method to detect the LFM signal, especially in the case that the chirp rate is the only parameter of interest, but in the case of long signals detection, the computational burden severely hinders the usefulness of this method.In this paper, a LCT-shift operator is proposed which is inspired by the operator methods and the fact that the LCT generalizes the conventional FT as well as the FRFT. This LCTshift operator generalizes the time, frequency and fractional operators into the LCT domains. As the correlation operation, which is fundamental in the theory of linear time-invariant(LTI) systems, can be derived by using the well-known time-shift operator. In consideration of this, a new form of the correlation operation in the LCT domains is defined using the proposed LCT-shift operator. The relationship between the AF and the proposed linear canonical autocorrelation is also established, which is shown that the linear canonical autocorrelation corresponds to radial slices of the AF in the ambiguity plane. Based on this relationship, an application of the fast linear canonical autocorrelation for detection and parameter estimation with respect to the chirp rates of LFM signals corrupted by noise is proposed. Finally, the validity of the proposed method is verified by simulation results.