A New Approach for Estimation of Instantaneous Mean Frequency of a Time-Varying Signal
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A New Approach for Estimation of Instantaneous Mean Frequency of a Time-Varying Signal Sridhar Krishnan Department of Electrical and Computer Engineering, Ryerson University, Toronto, ON, Canada M5B 2K3 Email: [email protected] Received 1 June 2004; Revised 30 December 2004; Recommended for Publication by John Sorensen Analysis of nonstationary signals is a challenging task. True nonstationary signal analysis involves monitoring the frequency changes of the signal over time (i.e., monitoring the instantaneous frequency (IF) changes). The IF of a signal is traditionally obtained by taking the first derivative of the phase of the signal with respect to time. This poses some difficulties because the derivative of the phase of the signal may take negative values thus misleading the interpretation of instantaneous frequency. In this paper, a novel approach to extract the IF from its adaptive time-frequency distribution is proposed. The adaptive time-frequency distribution of a signal is obtained by decomposing the signal into components with good time-frequency localization and by combining the Wigner distribution of the components. The adaptive time-frequency distribution thus obtained is free of crossterms and is a positive time-frequency distribution but it does not satisfy the marginal properties. The marginal properties are achieved by applying the minimum cross-entropy optimization to the time-frequency distribution. Then, IF may be obtained as the first central moment of this adaptive time-frequency distribution. The proposed method of IF estimation is very powerful for applications with low SNR. A set of real-world and synthetic signals of known IF dynamics is tested with the proposed method as well as with other common time-frequency distributions. The simulation shows that the method successfully extracted the IF of the signals. Keywords and phrases: instantaneous frequency, nonstationary signals, positive time-frequency distributions, matching pursuit, minimum cross-entropy optimization, average frequency.
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INTRODUCTION
The instantaneous frequency (IF) of a signal is a parameter of practical importance in situations such as seismic, radar, sonar, communications, and biomedical applications [1, 2, 3, 4]. In all these applications the IF describes some physical phenomenon associated with them. Like most other signal processing concepts, the IF of the signal was originally used in describing FM modulation in communications. In a typical radar application, the IF aids in the detection, tracking, and imaging of targets whose radial velocities change with time. When the radial velocity is not constant, the radar’s Doppler induced frequency has a nonstationary spectrum, which can be tracked by IF estimation techniques. Instantaneous frequency can also be used as an analysis tool in watermarking of multimedia data such as audio and image [5, 6]. In the multimedia security application, time-frequency distribution is used as a tool to embed and to detect the watermark message of the signals of interest. Also, in
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