Lecture 13 - Lecture 13 Digital IIR Filter Design Lecturer Dr ALI AL-ATABY [email protected] Contents Digital IIR Filter Design Introduction From

Lecture 13 - Lecture 13 Digital IIR Filter Design Lecturer...

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Lecture 13: Digital IIR Filter Design Lecturer: Dr ALI AL-ATABY [email protected]
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Contents Digital IIR Filter Design Introduction From s-plane to z-plane : 1. Impulse Invariant Method 2. Bilinear Transform. Design case study: bilinear transform applied to a notch filter. Biquadratic filter structure. Notch filter by direct pole/zero placement. Summary
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Digital IIR Filter Design Introduction Usually, an IIR digital filter H ( z ) is designed from an analogue filter H a ( s ). The basic idea behind the conversion of H a ( s ) into H ( z ) is to apply a mapping from the s -domain to the z -domain so that essential properties of the analogue frequency response are preserved. Thus, mapping function should be such that: 1. Imaginary j Ω axis in the s -plane be mapped in to the unit circle of the z -plane. 2. A stable analogue transfer function be mapped into a stable digital transfer function.
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Mapping from s-domain to z-domain When the Laplace transform is performed on a discrete-time signal (with each element of the discrete-time sequence attached to a correspondingly delayed unit impulse), the result is precisely the z- transform of the discrete-time sequence with the substitution of: sT e z snT n st n a st n st a n n d e nT h dt e nT t nT h s H dt e nT t nT h dt e t h s H z n h z H 0 0 0 0 0 0 0 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ] [ ) (
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Analogue to Digital Methods of Approximating 1) Impulse invariant method 2) Bilinear transform method 3) Matched z-transform (pole-zero mapping) Poles and zeros of the analogue filter are mapped to the poles and zeros of the digital filter using the mapping. sT e z
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Impulse Invariant Method 1. A suitable analogue transfer function H(s) is designed. 2. Its impulse response h(n) is obtained using the inverse Laplace transform. 3. The h(n) is suitably sampled to produce h(nT) 4. The desired H(z) is then obtained by z-transforming h(nT) .
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Impulse Invariant Method-Example I p is a pole in the s-domain which has now been transformed to a pole e pT in z-domain. pt Ce p s C L s H L t h p s C s H 1 1 )] ( [ ) ( ) ( pnT nT t Ce t h nT h ) ( ) ( 1 0 0 1 ) ( ) ( z e C z Ce z n h z H pT n n pnT n n
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Impulse Invariant Method-Example II Using the impulse invariant method obtain the transfer function H(z) of the digital filter which approximates the following normalised analogue transfer function: This is the 2 nd order Butterworth LPF filter. The sampling frequency is 1.28 kHz and the 3 dB cut off frequency should be 150 Hz. 1 2 1 ) ( 2 s s s H
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Impulse Invariant Method-Example II- Cont’d Before applying the impulse invariant method, we need to frequency scale the normalised transfer function. This is achieved by replacing s by s / α , where α =2 π ×150=942.4778 [radian].
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