By Frank Fahy, John Walker
Complicated purposes in Noise and Vibration provides remedies of acoustic and vibration phenomena including summaries of the most recent wisdom, research and techniques at present on hand for facing useful difficulties.
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Additional resources for Advanced Applications in Acoustics, Noise and Vibration
The ‘tan’ function serves to compress the complete frequency axis in the analogue domain onto the circumference of the unit disc in the z-plane. This compression distorts the frequency axis in a non-linear manner. 52). 52) with d = c . This process adjusts the transition points in a filter to allow for the distortion that is introduced by the bilinear transform and is called ‘pre-warping’. To illustrate the need for pre-warping, consider the problem of designing a digital filter consisting of four pass bands, each of width 100 Hz, centred on 100, 300, 500 and 700 Hz and using a sampling rate of 1600 Hz.
Fourier transform of a sequence. 32) where, as before, it is assumed that M is even. R. K. 6 Frequency responses of low-pass filters designed with different spectral windows. 34) k=0 If this approach is applied directly to a problem, the results are identical to those obtained using the windowing method with a rectangular window. 22)). 35) Signal processing techniques 17 This equation relates the frequency response H f for any frequency f to the sampled frequency point H k . 7. Note how the solid line (describing H f is constrained to pass through each of the circles indicating H k .
This shows the equivalence of the MEM approach and AR processes. 3 MEM and AR processes The MEM spectrum is equivalent to the spectrum of a white excited AR filter. 62). 62). 62) by multiplying by x n−k for k > 0 and taking expectations. 63) Signal processing techniques 33 Knowledge of R 0 R p − 1 and the ai is sufficient to go on generating R n for n > 0. 5. 62) as one that identifies x n with a value that is predicted from the previous values of the process with an error e n = u n , and the coefficients ai ap define a p-point prediction filter with u n = x n + a1 x n − 1 + · · · + ap x n − p , so that the prediction error filter has coefficients 1 a1 ap (Makhoul 1975).