tailieunhanh - Biosignal and Biomedical Image Processing phần 5

Tham khảo tài liệu 'biosignal and biomedical image processing phần 5', kỹ thuật - công nghệ, kĩ thuật viễn thông phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 152 Chapter 6 of the instantaneous autocorrelation function but only along the T . lag dimension. The result is a function of both frequency and time. When the onedimensional power spectrum was computed using the autocorrelation function it was common to filter the autocorrelation function before taking the Fourier transform to improve features of the resulting power spectrum. While no such filtering is done in constructing the Wigner-Ville distribution all of the other approaches apply a filter in this case a two-dimensional filter to the instantaneous autocorrelation function before taking the Fourier transform. In fact the primary difference between many of the distributions in Cohen s class is simply the type of filter that is used. The formal equation for determining a time-frequency distribution from Cohen s class of distributions is rather formidable but can be simplified in practice. Specifically the general equation is p t f JJJg v T ej2 v -Tlx u 2T x u - 2T e-2nfdv du dT 8 where g v T provides the two-dimensional filtering of the instantaneous autocorrelation and is also know as a kernel. It is this filter-like function that differentiates between the various distributions in Cohen s class. Note that the rest of the integrand is the Fourier transform of the instantaneous autocorrelation function. There are several ways to simplify Eq. 8 for a specific kernel. For the Wigner-Ville distribution there is no filtering and the kernel is simply 1 . g v T 1 and the general equation of Eq. 8 after integration by dv reduces to Eq. 9 presented in both continuous and discrete form. tt W t f j e-j2nfTx t - 2 x t - 2 dT 9a -tt tt W n m 2Xe 21inmlNx n k x n - k 9b k -tt tt W n m X e-2nnmlNRx n k FFTk Rx n k 9c m -tt Note that t nTs and f ml NTs The Wigner-Ville has several advantages over the STFT but also has a number of shortcomings. It greatest strength is that produces a remarkably good picture of the time-frequency structure Cohen 1992 . It also has .