This article presents a novel algorithm based on the cross-Wigner-Ville Distribution (XWVD) for optimum phase estimation within the class of phase shift keying signals. The proposed method is a special case of the general class of cross time-frequency distributions, which can represent the phase information for digitally phase modulated signals, unlike the quadratic time-frequency distributions. An adaptive window kernel is proposed where the window is adjusted using the localized lag autocorrelation function to remove most of the undesirable duplicated terms. The method is compared with the S-transform, a hybrid between the short-time Fourier transform and wavelet transform that has the property of preserving the phase of the signals as well as other key signal characteristics. The peak of the time-frequency representation is used as an estimator of the instantaneous information bearing phase. It is shown that the adaptive windowed XWVD (AW-XWVD) is an optimum phase estimator as it meets the Cramer-Rao Lower Bound (CRLB) at signal-to-noise ratio (SNR) of 5 dB for both binary phase shift keying and quadrature phase shift keying. The 8 phase shift keying signal requires a higher threshold of about 7 dB. In contrast, the S-transform never meets the CRLB for all range of SNR and its performance depends greatly on the signal's frequency. On the average, the difference in the phase estimate error between the Stransform estimate and the CRLB is approximately 20 dB. In terms of symbol error rate, the AW-XWVD outperforms the S-transform and it has a performance comparable to the conventional detector. Thus, the AW-XWVD is the preferred phase estimator as it clearly outperforms the S-transform.
Constellation diagram is a graphical representation of a signal in the complex coordinate plane. From the representation, the various class of digitally phase modulated signal such as BPSK, QPSK, 8PSK, 16QAM and 64QAM can be recognized. Constellation diagram has been used for modulation classification in spectrum monitoring, surveillance, cognitiveradio, and electronic warfare. Conventionally signal constellation diagram is generated using an IQ demodulator. In this paper, we present a new algorithm to generate the signal constellation diagram using the cross time-frequency distribution (XTFD). From the resulting cross time-frequency representation (XTFR), signal parameters such as the carrier frequency, instantaneous energy, instantaneous information bearing phase (IIBP) and symbol rate can be estimated directly. The estimated parameters are then used to generate the constellation diagram. The advantages of the proposed method are that it does not need the signal to be demodulated and it is more robust to noise. Thus, the resulting constellation diagram generated using the XTFD has outperformed the conventional IQ demodulator method for minimum SNR of 3dB.
Phase Shift-Keying (PSK) modulation has been applied widely in most of the data communication system nowadays due to its noise immunity and bandwidth efficiency capability. Analysis of digital modulation is an important component of spectrum monitoring. Furthermore, it has been identified recently that spectrum sensing is part of the function in cognitive radio. Works on bilinear time-frequency distribution (TFD) previously does not represent the phase information of a signal and it is incapable to analyze PSK type of signals. The cross TFD method proposed overcomes this problem, and is able to analyze signal carrying phase information like the class of PSK signals. From the cross time-frequency representation (TFR), signal parameters like the subcarrier frequency, instantaneous phase and bit duration can be estimated. The choice of the window width has significant effect on the accuracy of the phase estimation as well as the frequency resolution of the Time Frequency Representation (TFR). It is shown that the cross TFD with appropriate window width can give a complete representation of a phase modulated signal.
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