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Filtering Impulse Data
Published in Donald E. Struble, John D. Struble, Automotive Accident Reconstruction, 2020
Donald E. Struble, John D. Struble
The bilinear transform13 is a useful approximation for converting continuous-time filters (represented in Laplace s space) into discrete time filters (represented in z space), and vice versa. We start with the previously mentioned substitution: z=esT=e(sT/2+sT/2)=esT/2esT/2=esT/2e−sT/2
Digital Filters
Published in Samir I. Abood, Digital Signal Processing, 2020
Digital IIR filters are derived from their analog counterparts. There are several common types of analog filters: Butterworth which has maximally flat pass-bands in filters of the same order, Chebyshev which have a ripple in the pass-band, and elliptic filters which are equi-ripple in both the pass-band and the stop-band. Our strategy will be to design the filter in the analog domain and then transform the filter to the digital domain. We can derive this transformation by recalling the relationship between the Laplace transform and the z-transform: This transformation is known as the bilinear transform.
z-Transform
Published in David C. Swanson, ®, 2011
The literature on mapping between the s-plane and z-place often cites the use of a “bilinear transform” which maps the entire left-hand s-plane inside the unit circle on the z-plane. While this is mathematically satisfying because the stable part of the s-plane maps to the inside of the unit circle, it is not very practical because the Nyquist sample rate maps to an infinite analog frequency. This is why our presentation above examines a linear frequency mapping using z = esT allowing us to ignore the frequency response beyond the Nyquist rate (helped by a good antialiasing filter and a little oversampling). To be complete, we briefly present the bilinear transformation to explain how this popular technique significantly warps the frequency mapping at the higher end of the digital frequency spectrum. The bilinear mapping is defined by () z=1+(T/2)s1−T/2s
Fractional interpolation and multirate technique based design of optimum IIR integrators and differentiators
Published in International Journal of Electronics, 2021
Om Prakash Goswami, Tarun K. Rawat, Dharmendra K. Upadhyay
Conventionally, the digital integrators have been designed by inverting the transfer function of the first-order bilinear transform. Since the bilinear transform maps the analog domain to the digital domain without losing their stability. The transfer function of the bilinear transform-based integrator is given by (Dyer & Dyer, 2000; Oppenheim et al., 1999).