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Discrete-time IIR filter
Discrete-time IIR filter design is done using analog filter techniques:analog filter techniques:
1. Analog IIR filter design methods have simple closed form solutions;closed form solutions;
2. Design examples have existed for years.3 Direct design of IIR filters has traditionally3. Direct design of IIR filters has traditionally
been avoided4 Direct design of FIR filters is possible4. Direct design of FIR filters is possible.
Discrete-time IIR filter Design
1. Poles on the jΩ axis in the s-plane correspond to j p ppoles on the unit circle in the z-plane.2. Poles in the left half of the s-plane correspond to p ppoles inside the unit circle in the z-plane.Hence stable and causal continuous-time filters will produce stable and causal discrete-time filters.
Filter Design TechniquesFilter Design Techniques
Impulse InvarianceImpulse InvarianceBilinear Transformation
The design technique is as follows:(1) Perform a partial fractions(1) Perform a partial fractions expansion on H(s).(2) T f h l i t it(2) Transform each pole into its -transform equivalent.(3) Combine the terms into a single polynomial.
Butterworth Design
To get a stable and causal filter, choose Hc(s) to implement the poles in the left-hand plane.c( ) p p p
Example: Impulse Invariance
Take T = 1, value of T will not change the discrete-time filt lt )filter results.)
Bilinear TransformTo avoid aliasing we need a one to one mappingTo avoid aliasing, we need a one-to-one mapping from the s-plane to the z-plane.
Bilinear TransformationBilinear TransformationTransformation is unaffected by scaling. Consider inverse gtransformation with scale factor equal to unity sz += 1
1q yFor
s−1oo js Ω+σ=
22)1()1( j Ω++Ω++ σσ22
2
)1()1(
)1()1(
oo
oo
oo
oo zjjz
Ω+−Ω++=⇒
Ω−−Ω++=
σσ
σσ
and so10 =→= zoσ10 <→< zσ 10 <→< zoσ10 >→> zoσ
Bilinear Transformation
Nonlinear mapping introduces a distortion in the frequency axis calleddistortion in the frequency axis called frequency warpingEff t f i h b lEffect of warping shown below
Bilinear Transform: Design Procedure
1. Perform frequency prewarp to obtain the corresponding analog filter specs (pick any T) p g g p (p y )
2. Design the analog filter Hc(s) using any one of the analog filter prototypes.
3. Transform Hc(s) to H(z).