IF and GD estimation from evolutionary spectrum
Introduction
There are two basic approaches for estimation of instantaneous frequency (IF) from time–frequency distributions (TFD): peak detection and moment methods [1]. Performance of peak detection method is good at high signal-to-noise ratios (SNR). However, it deteriorates at low SNRs or when IF is a nonlinear function. The moment method may be used for both IF and group delay (GD) estimation from TFDs.
In this work, we use two estimators of the evolutionary spectrum for IF and GD estimation. The estimators are the evolutionary periodogram (EP) and the transitory evolutionary periodogram (TEP). We use the first moments of the EP and the TEP for the estimation of IF and GD. It is shown that IF and GD may be obtained from a sample of evolutionary autocorrelation and spectral correlation of a signal, respectively.
The paper is organized as follows. In Section 2, we present a review of the evolutionary spectral estimators, the EP and the TEP. We consider IF estimation and GD estimation in 3 IF estimation, 4 GD estimation, respectively. In Section 5, we propose modifications to the estimators to improve their performances. Finally, in Section 6, we present experimental results.
Section snippets
Evolutionary spectral estimation
According to the Wold–Cramer decomposition, a nonstationary process {x[n]} may be represented as [5], [7]where Z(ω) is an incrementally orthogonal process and H(n,ω) is a slowly time-varying function. In the above equation, {x[n]} is represented as the integral sum of complex exponentials with time-varying, complex, and random amplitudes.
Then, the (oscillatory) evolutionary spectral density function may be defined as [5], [7]
For a signal x[n], 0⩽n⩽N
IF estimation
Although there are many methods for IF estimation, we will consider only the moment method involving TFDs. The moment method uses the first moment of a given TFD, , at each time instant [1], [2], [8]Since the discrete-time TFDs are periodic in frequency, circular moment is used instead of linear moment in the above equation [2], [8].
In order to estimate IF from the EP, (4) is substituted into (11) and the result is (see Appendix A)
GD estimation
Group delay may be estimated by taking the first moment of TFDs along the time axis [1]However, both the EP (with Fourier functions) and the TEP may be considered as periodic in n because of their special form. In this case, we can define the normalized GD estimator as the first circular moment of as
GD estimate from the EP may be obtained by substituting (3) into (15):
Modified estimators
All of the above estimators involve terms which may be classified as forward or backward correlations. In order to improve their performances, they may be replaced with central correlations. Thus, the modified estimators may be defined as follows.
The modified EP–IF estimator:
The modified TEP–IF estimator:
The modified EP–GD estimator:
The modified TEP–GD
Experimental results
In this section, we present some experimental results. We first consider a constant-amplitude linear FM chirp signal. The signal length, N=128 and the instantaneous frequency, , increases linearly from 0 to 0.5. The instantaneous frequency is estimated using the equations presented in the previous sections for both the EP and the TEP with M=11. The Fourier functions are used in the simulations but other basis functions produce similar results. In Fig. 1(a), we show the IF estimates at
Conclusions
We presented estimators of IF and GD obtained from the first moments of two evolutionary spectral estimators, the evolutionary periodogram and the transitory evolutionary periodogram. We showed that IF estimators could be obtained from a sample of the evolutionary autocorrelation and GD estimators from a sample of the evolutionary spectral correlation. Finally, we presented experimental results illustrating performances of the estimators.
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