Adaptive Window Zero-Crossing-Based Instantaneous Frequency Estimation
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Adaptive Window Zero-Crossing-Based Instantaneous Frequency Estimation S. Chandra Sekhar Department of Electrical Communication Engineering, Indian Institute of Science, Bangalore 560 012, India Email: [email protected]
T. V. Sreenivas Department of Electrical Communication Engineering, Indian Institute of Science, Bangalore 560 012, India Email: [email protected] Received 2 September 2003; Revised 2 March 2004 We address the problem of estimating instantaneous frequency (IF) of a real-valued constant amplitude time-varying sinusoid. Estimation of polynomial IF is formulated using the zero-crossings of the signal. We propose an algorithm to estimate nonpolynomial IF by local approximation using a low-order polynomial, over a short segment of the signal. This involves the choice of window length to minimize the mean square error (MSE). The optimal window length found by directly minimizing the MSE is a function of the higher-order derivatives of the IF which are not available a priori. However, an optimum solution is formulated using an adaptive window technique based on the concept of intersection of confidence intervals. The adaptive algorithm enables minimum MSE-IF (MMSE-IF) estimation without requiring a priori information about the IF. Simulation results show that the adaptive window zero-crossing-based IF estimation method is superior to fixed window methods and is also better than adaptive spectrogram and adaptive Wigner-Ville distribution (WVD)-based IF estimators for different signal-to-noise ratio (SNR). Keywords and phrases: zero-crossing, irregular sampling, instantaneous frequency, bias-variance tradeoff, confidence interval, adaptation.
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INTRODUCTION
Almost all information carrying signals are time-varying in nature. The adjective “time-varying” is used to describe an “attribute” of the signal that is changing/evolving in time [1]. For most signals such as speech, audio, biomedical, or video signals, it is the spectral content that changes with time. These signals contain time-varying spectral attributes which are a direct consequence of the signal generation process. For example, continuous movements of the articulators, activated by time-varying excitation, is the cause of the timevarying spectral content in speech signals [2, 3]. In addition to these naturally occurring signals, man-made modulation signals, such as frequency-shift keyed (FSK) signals used for communication [4] carry information in their time-varying attributes. Estimating these attributes of a signal is important both for extracting their information content as well as synthesis in some applications. Typical attributes of time-varying signals are amplitude modulation (AM), phase/frequency modulation (FM) of a sinusoid. Another time-varying signal model is the output of a linear system with time-varying impulse response. However, the simplest and fundamental signal processing model
for time-varying signals is an AM-FM combination [5, 6, 7] of the type s(t) = A(t) sin(φ(t)). Further, if the amplitude does not
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