Model-based approach to envelope and positive instantaneous frequency estimation of signals with speech applications

An analytic signal s(t) is modeled over a T second duration by a pole-zero model by considering its periodic extensions. This type of representation is analogous to that used in discrete-time systems theory, where the periodic frequency response of a system is characterized by a finite number of poles and zeros in the z-plane. Except, in this case, the poles and zeros are located in the complex-time plane. Using this signal model, expressions are derived for the envelope, phase, and the instantaneous frequency of the signal s(t). In the special case of an analytic signal having poles and zeros in reciprocal complex conjugate locations about the unit circle in the complex-time plane, it is shown that their instantaneous frequency (IF) is always positive. This result paves the way for representing signals by positive envelopes and positive IF (PIF). An algorithm is proposed for decomposing an analytic signal into two analytic signals, one completely characterized by its envelope and the other having a positive IF. This algorithm is new and does not have a counterpart in the cepstral literature. It consists of two steps. In the first step, the envelope of the signal is approximated to desired accuracy using a minimum-phase approximation by using the dual of the autocorrelation method of linear prediction, well known in spectral analysis. The criterion that is optimized is a waveform flatness measure as opposed to the spectral flatness measure used in spectral analysis. This method is called linear prediction in spectral domain (LPSD). The resulting residual error signal is an all-phase or phase-only analytic signal. In the second step, the derivative of the error signal, which is the PIF, is computed. The two steps together provide a unique AM-FM or minimum-phase/all-phase decomposition of a signal. This method is then applied to synthetic signals and filtered speech signals

On representing signals using only timing information

It is well known that only a special class of bandpass signals, called real-zero (RZ) signals can be uniquely represented (up to a scale factor) by their zero crossings, i.e., the time instants at which the signals change their sign. However, it is possible to invertibly map arbitrary bandpass signals into RZ signals, thereby, implicitly represent the bandpass signal using the mapped RZ signal’s zero crossings. This mapping is known as real-zero conversion (RZC). In this paper a class of novel signal-adaptive RZC algorithms is proposed. Specifically, algorithms that are analogs of well-known adaptive filtering methods to convert an arbitrary bandpass signal into other signals, whose zero crossings contain sufficient information to represent the bandpass signal’s phase and envelope are presented. Since the proposed zero crossings are not those of the original signal, but only indirectly related to it, they are called hidden or covert zero crossings (CoZeCs). The CoZeCs-based representations are developed first for analytic signals, and then extended to real-valued signals. Finally, the proposed algorithms are used to represent synthetic signals and speech signals processed through an analysis filter bank, and it is shown that they can be reconstructed given the CoZeCs. This signal representation has potential in many speech applications

Harmonic Sum-based Method for Heart Rate Estimation using PPG Signals Affected with Motion Artifacts

Wearable photoplethysmography has recently become a common technology in heart rate (HR) monitoring. General observation is that the motion artifacts change the statistics of the acquired PPG signal. Consequently, estimation of HR from such a corrupted PPG signal is challenging. However, if an accelerometer is also used to acquire the acceleration signal simultaneously, it can provide helpful information that can be used to reduce the motion artifacts in the PPG signal. By dint of repetitive movements of the subjects hands while running, the accelerometer signal is found to be quasi-periodic. Over short-time intervals, it can be modeled by a finite harmonic sum (HSUM). Using the HSUM model, we obtain an estimate of the instantaneous fundamental frequency of the accelerometer signal. Since the PPG signal is a composite of the heart rate information (that is also quasi-periodic) and the motion artifact, we fit a joint HSUM model to the PPG signal. One of the harmonic sums corresponds to the heart-beat component in PPG and the other models the motion artifact. However, the fundamental frequency of the motion artifact has already been determined from the accelerometer signal. Subsequently, the HR is estimated from the joint HSUM model. The mean absolute error in HR estimates was 0.7359 beats per minute (BPM) with a standard deviation of 0.8328 BPM for 2015 IEEE Signal Processing cup data. The ground-truth HR was obtained from the simultaneously acquired ECG for validating the accuracy of the proposed method. The proposed method is compared with four methods that were recently developed and evaluated on the same dataset

Accurate Frequency Estimation Using an All-Pole Filter with Mostly Zero Coefficients

An all-pole Lth order filter with only M (M \u3c Z) nonzero coefficients is used to estimate accurately the frequencies of M closely spaced sinusoidal signals in noise. The method eliminates redundancy in die linear prediction equations which caused the spurious spectral peaks observed in previously known methods [3], [4]. Since most filter coefficients are zero, the computational effort required is small. Copyright © 1982 by The Institute of Electrical and Electronics Engineers, Inc

Inverse signal approach to computing the envelope of a real valued signal

We address the problem of estimating the envelope of a real-valued signal, s(t), that is observed for a duration of T seconds. We model s(t) using a Fourier series, by considering periodic extensions of the signal. By using an analog of the autocorrelation method of linear prediction on the Fourier coefficients of s(t), the envelope of the signal is estimated without explicitly computing the analytic signal through Hilbert transformation. Using this method the envelope of a non-stationary signal can be computed by processing the signal through a sliding T-second window

On instantaneous frequency of multicomponent signals

Instantaneous frequency (IF) of analytic signals has been extensively studied in the literature. The general perception is that it is meaningful only for narrowband or monocomponent signals. In this paper, new insights into IF\u27s of multicomponent signals are provided; it is argued that an IF\u27s erratic behavior is dictated by the proximity of the signal\u27s complex zeros to the unit circle and not by the band occupancy of the signal\u27s spectrum. A connection between product representation of signals and well-known ideas in linear systems literature are first established. Closed-form expressions for IF\u27s of signals consisting of multiple complex sinewaves are then derived; it is shown that there exists a one-to-one correspondence between signals\u27 Fourier coefficients and their IF\u27s. Using them, IF\u27s of some simple signals are first studied. Next, signals for which IF\u27s tend to be impulsive are addressed. This is followed by discussions on intensity-weighted IF and signals having positive IF\u27s. While IF and log-envelope are known to be time functions that typically have infinite spectral bandwidths, it is pointed out that their filtered versions sufficiently characterize a signal. Finally, issues related to computation of digital IF are addressed. ©[997 ]£££

Encoding bandpass signals using level crossings: A model-based approach

A new approach to representing a time-limited, and essentially bandlimited signal x(t), by a set of discrete frequency/time values is proposed. The set of discrete frequencies is the set of frequency locations at which (real and imaginary parts of) the Fourier transform of x(t) cross certain levels and the set of discrete time values corresponds to the traditional level crossings of x(t). The proposed representation is based on a simple bandpass signal model called a Sum-of-Sincs (SOS) model, that exploits our knowledge of the bandwidth/timewidth of x(t). Given the discrete fequency/time locations, we can reconstruct the x(t) by solving a least-squares problem. Using this approach, we propose an analysis/synthesis algorithm to decompose and represent composite signals like speech

A parametric modeling approach to hubert transformation

A parametric approach to compute the Hilbert transform (HT) of a real signal is proposed. A scaled version of the real signal is raised to the exponent of Napier\u27s base, and an all-pole signal model is fitted to approximate the resulting signal as though it were the envelope. An error criterion that flattens this envelope is then minimized over a set of complex Fourier coefficients. The resulting approximation is guaranteed to be a minimum-phase (MinP) signal. Since a MinP signal has the property that the logarithm of its envelope and its phase are related by HT, it follows that the resulting phase of MinP approximation yields the desired HT

A new real-zero conversion algorithm

The odd-order real zeros of a real-valued signal, s(t), correspond to its conventional or overt zerocrossings. The overt zero-crossings can be easily determined. But the complex zeros of s(t) are not easy to determine. This suggests the possibility of converting s(t) by using some invertible transformation, into a different signal, whose real zero-crossings alone determine s(t) completely. This process is known as realzero conversion. The real zero-crossings of the transformed signal are termed Covert Zero-Crossings (CoZeCs). Currently known real-zero conversion algorithms are impractical. In this paper we present a new adaptive algorithm for converting a band-pass signal into a pair of signals whose real zero-crossings essentially determine its envelope (up to a scale factor) and phase (up to a frequency translation)