IF estimation of FSK signals using adaptive smoothed windowed cross Wigner–Ville distribution
Introduction
Frequency shift-keying (FSK) signals are widely used in digital communications [1], [2] and in Costas coded pulse signalling [3] for pulse compression radar. Spectrum monitoring [4, p. 235] in both regulatory organizations and military requires the detection and analysis of signals before the signals are classified by their modulation parameters such as modulation type, frequency, phase, and symbol duration. These signals are considered time-varying because the instantaneous frequency (IF) varies abruptly in both time and frequency according to the instantaneous information bearing (IIB) symbols carried by the signal. Time–frequency analysis [5, p. 9] offers advantages of over time-series analysis and power spectrum analysis by providing a complete representation of signals jointly in time and frequency. The problem of cross-terms is minimized using kernel functions in QTFD [5, p. 70] but the solution is signal dependent. In a non-cooperative environment where prior knowledge of the true signal characteristics is unknown, adaptive kernels are used before computing the TFD. Adaptive TFDs applied to multicomponent linear FM signals were reported in [6], [7], [8], [9] with further extension to digitally modulated signals reported in [10], [11]. Provided a correct reference signal is available, the cross time–frequency distribution (XTFD) [12] is a better IF estimator over the QTFD by approximately 3 dB. Furthermore, the XTFD can be applied in instantaneous information bearing phase estimation for phase shift keying (PSK) signals [13].
The concept of IF was introduced in [14] and the various algorithms for IF estimation is described in [15]. In general, the QTFD [15] is an optimum IF estimator for linear FM signals while higher order TFD [16] is optimum for nonlinear FM signals. Recent works on TFD and IF estimation described their use in radar, medical, and bio-acoustics applications. The adaptive fractional spectrogram [17] is used to estimate the IF of a multicomponent signal. The use of an optimal local window reduces variance in the estimation of IF, with improved performance over the B-distribution and the Wigner–Ville distribution, resulting in a signal-to-noise ratio (SNR) between −5 dB and 16 dB. Multi-window S-method [18] improves the energy concentration in the time–frequency plane by using the multi-order Hermite function as window function. The use of 5th order function was demonstrated to be sufficient to achieve a minimum mean square error (MSE) at SNR of 10 dB. A low complexity adaptive short-time Fourier transform (STFT) [19] features an adaptive window based on the chirp rate. Principle component analysis is used for estimating the frequency difference that is applied to a Gaussian kernel with time–frequency varying window width. The performance based on the MSE is better at SNR less than 2 dB compared to that of the other adaptive QTFD and STFT, but is inferior at high SNR. The D-distribution was introduced in [20] to estimate IF at the ending point of lag and resolve the inherent delay of half problem in the subsequent time delay estimation. The use of the S-method was shown to meet the theoretical limit at SNR above 5 dB and in general performs better than the Wigner distribution. An adaptive TFD based on autoregressive modelling was introduced in [8]. Since the model order is critical, the predictive least square (PLS) method by Rissanen is used to generate the TFR iteratively. The computational complexity is minimized by using efficient lattice filters. The resulting TFRs are comparable to the Wigner–Ville, spectrogram, and an adaptive optimum kernel distribution for synthetic and bio-acoustic signals at SNR 21 dB. Earlier work has been reported in [21] using TFD with a time-dependent two-sided linear predictor autoregressive model. Better performance was achieved in comparison with the TFD using the one-sided linear prediction approach and the Choi–Williams distribution at SNR of 10 dB. Most of these cited applications focus on the class of signals where the frequency varies linearly with time or follows a polynomial frequency law.
This paper presents a time–frequency analysis solution to the IF estimation for FSK class signals in the presence of noise followed by its performance evaluation. Unlike the signal used in [6], [7], [8], [9], FSK is the signals of interest where the IF changes abruptly in frequency and time according to the transmitted symbols. The signals tested are binary FSK (BFSK), 4FSK, and 8FSK signals. The main contribution of this paper is the adaptive smooth windowed cross Wigner–Ville distribution (ASW-XWVD), which features a separable kernel function with a time-dependent lag window and a time-smooth function. For FSK class signals, the cross bilinear product – the intermediate step before the time–frequency representation is computed – is mathematically modelled to identify the locations of the autoterms, cross terms, and duplicated terms. With this information, the adaptive procedure is designed to preserve the autoterms. The adaptation of the window width is obtained by performing the local lag autocorrelation (LLAC) function on the cross bilinear product and the time-smoothed function is adjusted based on the minimum frequency difference of the signal.
This correspondence is organized as follows. Section 2 first describes the signal models used in this paper. Section 3 presents the cross time–frequency distributions, the general equations for the cross bilinear product in the time–lag domain for autoterms, duplicated terms, and cross terms. In the same section, the ASW-XWVD together with the adaptation methodology used to estimate the kernel parameters for FSK signals is discussed. Next, we present the method for IF estimation from the peak of the ASW-XWVD. The Cramer–Rao lower bound (CRLB), which is used for bench marking purposes, is discussed in the following subsection. Section 4 presents the discrete-time implementation of the ASW-XWVD and the performance comparison between the optimal kernel and the adaptive kernel ASW-XWVD in the presence of noise. The performance of the ASW-XWVD, in terms of the variance in the IF estimates, is compared with the S-transform. Next, to validate the use of the proposed method in practice, field testing using real data communication signals is conducted. The conclusions are present in the last section.
Section snippets
Signal model and problem definition
In this paper, the FSK signals considered are BFSK, 4FSK and 8FSK, each of which is formed as a sum of N short duration complex exponential signals. The signal can be defined aswhere k represents the order of the IIB symbols, Ak represents the signal amplitude, fk is the subcarrier frequency, and Tb is the symbol duration. Parameter Ak is constant for the FSK signals. The box function, is defined as
In relation to the
IF estimation by adaptive smoothed windowed cross Wigner–Ville distribution
A new distribution known as the ASW-XWVD features a separable kernel that overcomes the problem of cross terms and duplicated terms interference similar to the one reported in [10], [26]. This section begins with the general formulation of the XTFD followed by the presentation of the signal characteristics in the time–lag domain. The location of the cross terms and duplicated terms are determined and then used as the basis to develop the adaptation procedure for the ASW-XWVD.
Results and discussion
This section first describes the TFR for a 4FSK signal using the optimal kernel ASW-XWVD and the adaptive kernel ASW-XWVD, followed by a comparison with a set performance criteria: main-lobe width (MLW), signal-to-cross terms ratio (SCR), peak-to-side lobe ratio (PSLR) and symbol duration (SD) estimate. The performance of the IF estimator based on ASW-XWVD is benchmarked with the CRLB. In the next subsection, the computation complexity of the proposed ASW-XWVD is presented. Finally, an analysis
Conclusion
Accurate time–frequency representation for FSK signals is obtained using the ASW-XWVD. The XTFD has a kernel function where the kernel parameters – window width and time smoothing function duration – are adaptively adjusted according to the signal parameters without any prior information on the signal. It is proven that the proposed ASW-XWVD outperforms the S-transform and is an optimum IF estimator as it meets the CRLB at very low SNR of −3 dB. Thus, this paper has provided a new insight on the
Acknowledgement
The authors thank Universiti Teknologi Malaysia (UTM) and Malaysia Ministry of Higher Education (MOHE) project number R.J130000.7823.4F188 for providing the resources and funding for this research. In addition, the authors thank the reviewers for their helpful comments and suggestions.
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