4. Use the backward difference for the derivative to convert the following analog filter with system function (a) H(s) = 1 s+5 (b) H(s) = 1 s 2+4 (c) H(s) = 1 (s+0.2)2+1
Question
- Use the backward difference for the derivative to convert the following analog filter with system function (a) H(s) = 1 s+5 (b) H(s) = 1 s 2+4 (c) H(s) = 1 (s+0.2)2+1
Solution
The backward difference method is a finite difference method used to numerically approximate derivatives of functions. It's often used in digital signal processing to convert continuous-time systems (analog filters) to discrete-time systems (digital filters).
Here's how you can use the backward difference method to convert the given analog filters:
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The backward difference approximation of the derivative is given by: dy/dt ≈ (y(t) - y(t - T)) / T, where T is the sampling period.
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Replace every instance of s in your analog filter transfer function with (1 - z^-1) / T.
(a) For H(s) = 1 / (s + 5), the digital filter transfer function becomes H(z) = 1 / ((1 - z^-1) / T + 5).
(b) For H(s) = 1 / (s^2 + 4), the digital filter transfer function becomes H(z) = 1 / (((1 - z^-1) / T)^2 + 4).
(c) For H(s) = 1 / ((s + 0.2)^2 + 1), the digital filter transfer function becomes H(z) = 1 / ((((1 - z^-1) / T) + 0.2)^2 + 1).
Remember that these are approximations and the accuracy depends on the choice of the sampling period T. The smaller the T, the more accurate the approximation.
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