Wavelet estimation of cyclospectra the case of unknown cyclic frequency
Sami Touati · 2003
In this paper, we propose wavelet esti- mators of cyclospectra of a zero-mean almost cyclo- stationary signal in the case of unknown cyclic fre- quencies. In this context, we will show that wavelet- thresholding estimators lead to improved perfor- mances compared to traditional (linear) ones. 'These results are confirmed by a simulation example. I. DEFINITION OF THE ESTIMATOR AND CONVERGENCE RESULTS We consider a real discrete-time (almost) cyclostationary sig- nal {Xt},. Since {Xt}, is real, the set A of its cyclic frequen- cies is symmetric with respect to 0, and its cyclospectra g7(v), 9 E A, satisfy the following symmetries : given by : yt + 0.2%-1 + 0.9yt-2 = Et + 0.5~t--2, and {et}, (2,) are independent Gaussian zero-mean white noises with variance 1. The cyclic-frequencies of {X,} are {-2f,0,2f}, with f = 0.125 in this example. The theoretical cyclospectra are given in Fig. l(a) and l.(b) (the 2f-component on the right). We used the Symmlet 10 basis (the least asymmetric). Wavelet-thresholding was performed for the levels j = 3, 4, 5, and the coefficients from scales j > 5 were set to zero. In Fig. 1. we show one realization of the wavelet-estimation (l(c) and l.(d)) and kernel-estimation (l(e) and l.(f)), with optimally chosen bandwidth (h = O.ll), of the cyclospectra. The wavelet estimator better captures the peaks and it is also slightly better in smooth parts. We also estimated the aver- aged NMSE. The results are provided in the figure captions. (4 (b) 2.51 I 21 I I 0' I -0.5 0 0.5 -0.5 0 0.5 0.5'