BMEN
Lecture 14 - IIR Filter Design via Bilinear Transformation(4) (2).pptx

# Lecture 14 - IIR Filter Design via Bilinear Transformation(4) (2).pptx

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BMEN 3350 – Biomedical Component and System Design Lecture 14 IIR filter design via bilinear transformation.

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BMEN 3350 – Lecture 14 S-domain vs. Z-domain In order to convert an analog filter into a discrete-time IIR filter, we must convert the s-domain transfer function into a z-domain transfer function.
BMEN 3350 – Lecture 14 Bilinear Transformation - Derivation Consider a simple analog integrator: We can derive the transfer function by setting up a node-voltage equation in the Laplace domain: + 0 0 From this, it is clear that integration in the s-domain takes the general form: where A is a constant.

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BMEN 3350 – Lecture 14 Bilinear Transformation - Derivation Consider a simple digital integrator based on the trapezoidal rule: Z -1 + x[n] dt/2 + Z -1 Trapezoidal Rule:
BMEN 3350 – Lecture 14 Bilinear Transformation - Derivation Z -1 + x[n] dt/2 + Z -1 The z-transform of this system is given by: And the transfer function is given by: =

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BMEN 3350 – Lecture 14 Bilinear Transformation - Derivation Integrator in s-domain: = Integrator in z-domain: From this, it is clear that, to transform between the s and z-domains, we must make the substitution: This substitution is known as bilinear transformation, and converts between a continuous-time filter and discrete-time filter with approximately the same response to all input functions.
BMEN 3350 – Lecture 14 Bilinear Transformation – Mapping When |z| < 1, s < 0 When |z| = 1, s = 0 When |z| > 1, s > 0 When s = 0, z = 1 (frequency = 0 Hz i.e. DC) When s = ∞, z = -1 (highest allowable frequency, i.e. )

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BMEN 3350 – Lecture 14 Example #1 Design a digital IIR filter with the same behavior as the following analog high pass filter. Assume a sampling rate of 100 kS/s.
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