signals and systems filter design. part iii design
TRANSCRIPT
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Signals and SystemsFilter Design
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Part III
Design
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Filter Design Techniques
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Discrete-time filtersDiscrete-time filters
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Discrete-time IIR filter
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Specifications for DT filters
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Specifications for DT filters in Log domain
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A Design Example
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Discrete-time IIR filter design is done using analog filter techniques:
1. Analog IIR filter design methods have simple closed form solutions;
2. Design examples have existed for years.3. Direct design of IIR filters has traditionally
been avoided4. Direct design of FIR filters is possible.
Discrete-time IIR filter
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Discrete-time IIR filter Design Flow
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Discrete-time IIR filter Design
1. Poles on the jΩ axis in the s-plane correspond to poles on the unit circle in the z-plane.
2. Poles in the left half of the s-plane correspond to poles inside the unit circle in the z-plane.
Hence stable and causal continuous-time filters will produce stable and causal discrete-time filters.
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Traditional Analog Filter Design
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Traditional Analog Filter Design
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Butterworth Design
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Butterworth Design
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Chebyshev filters
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Chebyshev filters
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Chebyshev filters
Chebyshev filters
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Elliptic filters
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Example
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Filter Design Techniques
Impulse InvarianceBilinear Transformation
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The design technique is as follows: (1) Perform a partial fractions
expansion on H(s). (2) Transform each pole into its -
transform equivalent. (3) Combine the terms into a single
polynomial.
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Impulse Invariance
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Butterworth Design
To get a stable and causal filter,
choose Hc(s) to implement the poles in the left-hand plane.
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Butterworth Filter
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Butterworth Filter-Impulse Invariance
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Butterworth Filter-Impulse Invariance
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Example: Impulse Invariance
Take T = 1, value of T will not change the discrete-time filter results.)
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Bilinear Transformation
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Bilinear TransformTo avoid aliasing, we need a one-to-one mapping
from the s-plane to the z-plane.
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Bilinear Transform: Freq axis
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Bilinear TransformationBilinear Transformation Transformation is unaffected by
scaling. Consider inverse transformation with scale factor equal to unity
For
and so
ssz
11
oo js
22
222
)1()1(
)1()1(
oo
oo
oo
oo zjj
z
10 zo10 zo10 zo
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Bilinear TransformationBilinear Transformation
Mapping of s-plane into the z-plane
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Bilinear Transformation
Nonlinear mapping introduces a distortion in the frequency axis called frequency warping
Effect of warping shown below
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Bilinear Transformation (Graphical Translation)
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1. Perform frequency prewarp to obtain the corresponding analog filter specs (pick any T)
2. Design the analog filter Hc(s) using any one of the analog filter prototypes.
3. Transform Hc(s) to H(z).
Bilinear Transform: Design Procedure
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Example
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Bilinear Transform: Ex.
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Bilinear Transform
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FIR Filter Design
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Windowing Principal
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Windowing: Frequency Interpretation
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Windowing Effects
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Rectangular Window
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Common Windows
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Common window
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Effect of Windowing
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Windows Freq Domain
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Other Windows in Feq Domain
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Comparison
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Kaiser Method
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Kaiser
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Kaiser
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Kaiser
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Marks McClellan Algo
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Parks McClellan Algorithm
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Butterworth Approx. in MATLAB
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Butterworth Approximation
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Chebyshev Approximation
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Elliptic Approximation in MATLAB
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Elliptic Approximation