Spectral And Temporal Zero-Crossings-Based Signal Analysis
We consider real zero-crossing analysis of the real/imaginary parts of the spectrum, namely, spectral zero-crossings (SZCs). The two major contributions are to show that: (i) SZCs provide enable temporal localization of transients; and (b) SZCs are suitable for modeling transient signals. We develop a spectral dual of Kedem’s result linking temporal zero-crossing rate (ZCR) to the spectral centroid. The key requirement is stationarity, which we achieve through random-phase modulations of the time-domain signal. Transient signals are not amenable to modelling in the time domain since they are bursts of energy localized in time and lack structure. We show that the spectrum of transient signals have a rich modulation structure, which leads to an amplitude-modulation – frequency-modulation (AM-FM) model of the spectrum. We generalize Kedem’s arc-cosine formula for lags greater than one. For the specific case of a sinusoid in white Gaussian noise, He and Kedem devised an iterative filtering algorithm, which leads to a contraction mapping. An autoregressive filter of order one is employed and the location of the pole is the parameter that is updated based on the filtered output. We use the higher-order property, which relates the autocorrelation to the expected ZCR of the filtered process, between lagged ZCR and higher-lag autocorrelation to develop an iterative higher-order autoregressive-filtering scheme, which stabilizes the ZCR and consequently provides robust estimates of the autocorrelation at higher lags. Next, we investigate ZC properties of critically sampled outputs of a maximally decimated M-channel power complementary analysis filterbank (PCAF) and derive the relationship between the ZCR of the input Gaussian process at lags that are integer multiples of M in terms of the subband ZCRs. Based on this result, we propose a robust autocorrelation estimator for a signal consisting of a sum of sinusoids of fixed amplitudes and uniformly distributed random phases. Robust subband ZCRs are obtained through iterative filtering and the subband variances are estimated using the method-of-moments estimator. We compare the performance of the proposed estimator with the sample auto-correlation estimate in terms of bias, variance, and mean-squared error, and show through simulations that the performance of the proposed estimator is better than the sample auto- correlation for medium to low SNR. We then consider the ZC statistics of the real/imaginary parts of the discrete Fourier spectrum. We introduce the notion of the spectral zero-crossing rate (SZCR) and show that, for transients, it gives information regarding the location of the transient. We also demonstrate the utility of SZCR to estimate interaural time delay between the left and right head-related impulse responses. The accuracy of interaural time delay plays a vital role in binaural synthesis and a comparison of the performance of the SZCR estimates with that of the cross-correlation estimates illustrate that spectral zeros alone contain enough information for accurately estimating interaural time delay. We provide a mathematical formalism for establishing the dual of the link between zero-crossing rate and spectral centroid. Specifically, we show that the expected SZCR of a stationary spectrum is a temporal centroid. For a deterministic sequence, we obtain the stationary spectrum by modulating the sequence with a random phase unit amplitude sequence and then computing the spectrum. The notion of a stationary spectrum is necessary for deriving counterparts of the results available in temporal zero-crossings literature. The robustness of location information embedded in SZCR is analyzed in presence of a second transient within the observation window, and also in the presence of additive white Gaussian noise. A spectral-domain iterative filtering scheme based on autoregressive filters is presented and improvement in the robustness of the location estimates is demonstrated. As an application, we consider epoch estimation in voiced speech signals and show that the location information is accurately estimated using spectral zeros than other techniques. The relationship between temporal centroid and SZCR also finds applications in frequency-domain linear prediction (FDLP), which is used in audio compression. The prediction coefficients are estimated by solving the Yule-Walker equations constructed from the spectral autocorrelation. We use the relationship between the spectral autocorrelation and temporal centroid to obtain the spectral autocorrelation directly by time-domain windowing without explicitly computing the spectrum. The proposed method leads to identical results as the standard FDLP method but with reduced computational load. We then develop a SZCs-based spectral-envelope and group-delay (SEGD) model, which finds applications in modelling of non-stationary signals such as Castanets. Taking into account the modulation structure and spectral continuity, local polynomial regression is performed to estimate the GD from the real spectral zeros. The SE is estimated based on the phase function computed from the estimated GD. Since the GD estimate is parametric, the degree of smoothness can be controlled directly. Simulation results based on synthetic transient signals are presented to analyze the noise-robustness of the SE-GD model. Applications to castanet modeling, transient compression, and estimation of the glottal closure instants in speech are shown.
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