Time-Frequency Distribution for Undersampled Non-stationary Signals using Chirp-based Kernel

Abstract

Missing samples and randomly sampled nonstationary signals give rise to artifacts that spread over both the time-frequency and the ambiguity domains. These two domains are related by a two-dimensional Fourier transform. As these artifacts resemble noise, the traditional reduced interference signal-independent kernels, which belong to Cohen's class, cannot mitigate them efficiently. In this paper, a novel signal-independent kernel in the ambiguity domain is proposed. The proposed method is based on three important facts. Firstly, any windowed non-stationary signal can be approximated as a sum of chirps. Secondly, in the ambiguity domain, any chirp resides inside certain regions, which just occupy half of the ambiguity plane. Thirdly, the missing data artifacts always appear along the Doppler axis where the chirps auto-terms do not appear. Therefore, we propose using a chirp-based fixed kernel on windowed non-stationary signals in order to remove half of the noise-like artifacts in the ambiguity domain and compensate for the missing data effect located along the Doppler axis. It is shown that our method outperforms other reduced interference time-frequency distributions.

A Chirp-based, Adaptive, Signal-dependent Reduced Interference Distribution for Limited Data

“…This approach applies the chirp's properties in the ambiguity domain [15] and the algorithm of RGK to obtain a new kernel that is appropriate for chirps in the cases of full and incomplete data. The principle of RGK is that it keeps the magnitude of the kernel in the ambiguity domain large wherever that of the signal's AF is large, regardless of whether the peaks correspond to auto-terms or undesired terms ( [11], [12]).…”

“…Therefore, the RGK will be wrongly guided to take this region, which leads to noisy TFRs. Fortunately, for any chirps, the autoterms always reside inside a half of the ambiguity domain, |φ| ≤ π/4 and 3π/4 ≤ φ ≤ 5π/4, which excludes the Doppler axis [15]. Therefore, the other half of the ambiguity domain can be removed without causing any loss of auto-terms.…”

“…The above kernel is basically applied when the input signals are chirps and not for other types of non-stationary signals. Nevertheless, according to [14], [15], the frequency law of any non-stationary windowed signal can be approximated as a sum of chirps. Thus we can use the aforementioned fixed kernel for each windowed signal.…”

“…The current-time slice of the TFR is computed as follows: 4)) to create the incomplete data to be processed. To get a visual comparison, the other three methods, Choi-Williams, the AOK, and the fixed chirp-based TFR [15], are simulated with the same signals. The resulting images are normalized and transfered to the energy versions to display.…”

“…The method relies on three features. Firstly, according to [15], a chirp's auto-terms always reside in only a fixed half of the ambiguity domain. Thus, for chirp signals, the other half of the ambiguity domain could be eliminated without any concern for loss of the signal's auto-terms.…”