By Cong, Fengyu; Lyytinen, Heikki; Ristaniemi, Tapani
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Extra info for Advanced signal processing on brain event-related potentials : filtering ERPs in time, frequency and space domains sequentially and simultaneously
1 Definition of DFT Given length-T discrete sequence x(t), t = 0, 1, 2, . . , T − 1, the K -point (K ≥ T ) DFT is defined as follows: T −1 X(k)= x(t) · e− j 2π K t , k (2-5) t =0 where k = 0, 1, 2, . . 718……). 2 Interpreting DFT using correlation Equation (2-5) can be written as T −1 X(k) = rαβk − j · rαθk = T −1 α(t) · βk (t) − j · t =0 k e− j 2π K t = cos 2π k k t − j · sin 2π t K K α(t) · θk (t), (2-6) t =0 = βk (t) − j · θk (t), (2-7) where α(t) = x(t), βk (t) = cos(2π Kk t), and θk (t) = sin(2π Kk t).
7 shows the magnitude responses of the wavelet filter frequency responses of wavelet filters when the detail coefficients at each of some levels are used for the signal reconstruction. Obviously, the frequency contents of the detail coefficients of some levels largely overlap with each other and the degrees of overlapping of the different wavelets can be different as well. This result indicates that the previous frequency division approach by the DWT is not precise. Indeed, wavelet decomposition based on the DWT is similar to a digital filter bank that includes many LP and HP digital filters.
B) Phase response. The sampling frequency of the signal is 200 Hz, and the pass band is from 40 to 70 Hz. March 4, 2015 5:46 Advanced Signal Processing on Brain ERPs 9 x 6in b1907-ch02 page 23 Wavelet Filter Design Based on Frequency Responses 23 the input signal. In actual case, such difference results in different filtering performance. Hereinafter, we use “DFT filter” to denote the DFT-based digital filter for simplicity. 4 Problems of the DFT filter and their corresponding solutions As mentioned earlier, Fourier transform is the correlation between the signal and the cosine and the sine functions at a given frequency.
Categories: Biomedical Engineering