Advanced Applications in Acoustics, Noise and Vibration by Frank Fahy, John Walker

By Frank Fahy, John Walker

Complex functions in Noise and Vibration provides remedies of acoustic and vibration phenomena including summaries of the most recent wisdom, research and techniques presently on hand for facing functional difficulties.

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If we can find the filter H f that achieves this, then the output power of the filter is the power of the signal in the ‘close’ vicinity of f . Note that only by division by a bandwidth is it a density function. 76) The problem is to find h such that the output variance of the filter E y n 2 is minimised, subject to the constraint that the transfer function of the filter is unity at frequency f Hz, that is, H f = 1. 78) t where ef = 1 ej2 f ej2 2f · · · ej2 L−1 f and † denotes the conjugate transpose.

Underestimates of p tend to lead to ‘oversmoothed’ spectra, and overestimates should be avoided since (i) computation should be minimised, (ii) ill-conditioning of the equations increases with numbers of poles and (iii) spurious ripples occur in the spectrum. Several authors have considered the problem of order determination, and the most successful and widely used method of order determination can be attributed to Akaike (1974) (though there are others (like Gustaffson and Hjalmarsson 1995)). The basic idea on which the criteria are based use the ‘error variance’ which is monitored at each iteration.

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.

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