A.S. Kayhan

Evolutionary periodogram for nonstationary signals

Presents a novel estimator for the time-dependent spectrum of a nonstationary signal. By modeling the signal, at any given frequency, as having a time-varying amplitude accurately represented by an orthonormal basis expansion, the authors are able to compute a minimum mean-squared error estimate of this time-varying amplitude. Repeating the process over all frequencies, they obtain a power distribution as a function of time and frequency that is consistent with the Wold-Cramer evolutionary spectrum. Based on the model assumptions, the authors develop the evolutionary periodogram (EP) for nonstationary signals, an estimator analogous to the periodogram used in the stationary case. They also derive the time-frequency resolution of the new estimator. The approach is free of some of the drawbacks of the bilinear distributions and of the short-time Fourier transform spectral estimates. It is guaranteed to produce nonnegative spectra without the cross-term behavior of the bilinear distributions, and it does not require windowing of data in the time domain. Examples illustrating the new estimator are given. >

Data-adaptive evolutionary spectral estimation

We present a novel data-adaptive estimator for the evolutionary spectrum of nonstationary signals. We model the signal at a frequency of interest as a sinusoid with a time-varying amplitude, which is accurately represented by an orthonormal basis expansion. We then compute a minimum mean-squared error estimate of this amplitude and use it to estimate the time-varying spectrum at that frequency, all while minimizing the interference from the signal components at other frequencies. Repeating the process over all frequencies, we obtain a power distribution that is consistent with the Wold-Cramer evolutionary spectrum and reduces to Capons (1969) method for the stationary case. Our estimator possesses desirable properties in terms of time-frequency resolution and positivity and is robust in the spectral estimation of noisy nonstationary data. We also propose a new estimator for the autocorrelation of nonstationary signals. This autocorrelation estimate is needed in the data-adaptive spectral estimation. We illustrate the performance of our estimator using simulation examples and compare it with the recently presented evolutionary periodogram and the bilinear time-frequency distribution with exponential kernels. >

Spatial evolutionary spectrum for DOA estimation and blind signal separation

We combine the concepts of evolutionary spectrum and array processing. We present a cross-power stationary periodogram for both direction-of-arrival (DOA) estimation and blind separation of nonstationary signals. We model the nonstationary signals received by each array sensor as a sum of complex sinusoids with time-varying amplitudes. These magnitudes carry information about the DOA that may also be time-varying. We first estimate the time-varying amplitudes using estimators obtained by minimizing the mean-squared error. Then using the estimated time-varying amplitudes, we estimate the evolutionary cross-power distributions of the sensor. Next, using cross-power estimates at time-frequency points interest, we estimate the DOAs using one of the existing methods. If the directions are time varying, we choose time-frequency points around the time of interest to estimate spontaneous source locations. If the sources are stationary, time-frequency points of interest can be combined for the estimation of fixed directions. Whitening and subspace methods used to find the mixing matrix and separate nonstationary signals received by the array. We present examples illustrating the performance of the proposed algorithms.

Spectral estimation of ARMA processes using ARMA-cepstrum recursion

In this letter, the spectral estimation problem of a stationary autoregressive moving average (ARMA) process is considered, and a new method for the estimation of the MA part is proposed. A simple recursion relating the ARMA parameters and the cepstral coefficients of an ARMA process is derived and utilized for the estimation of the MA parameters. The method requires neither any initial estimates nor fitting of a large order AR model, both of which require further a priori knowledge of the signal and increase the computational complexity. Simulation results illustrating the performance of the new method are also given.

Difference equation representation of chirp signals and instantaneous frequency/amplitude estimation

We present a time-varying coefficient difference equation representation for sinusoidal signals with time-varying amplitudes and frequencies. We first obtain a recursive equation for a single chirp signal. Then, using this result, we obtain time-varying coefficient difference equation representations for signals composed of multiple chirp signals. We analyze these equations using the skew-shift operators. We show that the phases of the poles of the difference equations produce instantaneous frequencies (IF), and the magnitudes are proportional to the ratio of successive values of the instantaneous amplitudes (IA). Then algorithms are presented for the estimation of instantaneous frequencies and instantaneous amplitudes for multicomponent signals composed of chirps using the difference equation representation. The first algorithm we propose is based on the skew-shift operators. Next we derive the conditions under which we can use the so-called frozen-time approach. We propose an algorithm for IF and IA estimation based on the frozen-time approach. Then we propose an automatic signal separation method. Finally, we apply the proposed algorithms to single and multicomponent signals and compare the results with some existing methods.

Evolutionary maximum entropy spectral analysis

We extend maximum entropy (ME) spectral analysis to non-stationary signals using the theory of the Wold-Cramer evolutionary spectrum. The evolutionary maximum entropy (EME) problem reduces to the fitting of a time-varying autoregressive model to the Fourier coefficients of the evolutionary spectrum. The model parameters are efficiently found by means of the Levinson algorithm. In the non-stationary case it is not the autocorrelation function that provides the appropriate data for the EME analysis, but rather the Fourier coefficients of the evolutionary spectrum. An estimator of these coefficients is proposed. By means of examples we show the EME estimator provides higher frequency resolution and better sidelobe behavior than existing estimators of the evolutionary spectrum.<<ETX>>

Spectral estimation of nonstationary ARMA processes using the evolutionary cepstrum

The spectral estimation problem of nonstationary autoregressive moving-average (ARMA) processes is considered and a new method is proposed for the estimation of the time-varying spectral content of such signals. The proposed method can be viewed as an extension of the estimator proposed earlier by the authors to the time-varying case. The AR part of the model is estimated by solving the time-varying modified Yule-Walker equations using an estimated time-varying autocorrelation function. An evolutionary cepstrum estimator is proposed, which is then used in a simple recursion to obtain the MA parameters of the model.

IF and GD estimation from evolutionary spectrum

In this paper, we present estimators of instantaneous frequency (IF) and group delay (GD) obtained from dual evolutionary spectral estimators, the evolutionary periodogram and the transitory evolutionary periodogram. We obtain the IF and the GD estimators using the first moments. We show that IF and GD may be estimated using a sample of evolutionary time and spectral correlation functions, respectively. Examples illustrating the performances of the estimators are presented.

Representation and analysis of complex chirp signals

Abstract In this paper, we present a difference equation model with time-varying coefficients for complex-valued multicomponent chirp signals. This model may be applied to real-valued signals as a special case. We show the relation between chirp parameters, i.e., instantaneous frequency (IF) and amplitude (IA), and difference equation coefficients. We present a recursive method for the estimation of the signal poles from which IF and IA are obtained. We show that the estimates may be improved by proper choice of the initial values. We present examples to illustrate the performance of the proposed algorithms.

Evolutionary maximum entropy spectral analysis of chirps in noise

Abstract In this manuscript, we analyze chirps in noise using the evolutionary maximum entropy spectral estimator (EMES). First, brief descriptions of evolutionary spectrum and its estimators, the evolutionary periodogram (EP), the smoothed evolutionary periodogram (SEP) and the EMES are given. Then, chirps in noise are analyzed with the EMES. It is shown that the EMES parameters are weighted functions of frequency components around the time of analysis. Later, the relation between bandwidth, SNR and number of basis functions is given for the case of a single pure sinusoid.