signals and systems filter design. part iii design
TRANSCRIPT
Signals and SystemsFilter Design
Part III
Design
Filter Design Techniques
Discrete-time filtersDiscrete-time filters
Discrete-time IIR filter
Specifications for DT filters
Specifications for DT filters in Log domain
A Design Example
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
Discrete-time IIR filter Design Flow
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.
Traditional Analog Filter Design
Traditional Analog Filter Design
Butterworth Design
Butterworth Design
Chebyshev filters
Chebyshev filters
Chebyshev filters
Chebyshev filters
Elliptic filters
Example
Filter Design Techniques
Impulse InvarianceBilinear Transformation
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.
Impulse Invariance
Butterworth Design
To get a stable and causal filter,
choose Hc(s) to implement the poles in the left-hand plane.
Butterworth Filter
Butterworth Filter-Impulse Invariance
Butterworth Filter-Impulse Invariance
Example: Impulse Invariance
Take T = 1, value of T will not change the discrete-time filter results.)
Bilinear Transformation
Bilinear TransformTo avoid aliasing, we need a one-to-one mapping
from the s-plane to the z-plane.
Bilinear Transform: Freq axis
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
Bilinear TransformationBilinear Transformation
Mapping of s-plane into the z-plane
Bilinear Transformation
Nonlinear mapping introduces a distortion in the frequency axis called frequency warping
Effect of warping shown below
Bilinear Transformation (Graphical Translation)
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
Example
Bilinear Transform: Ex.
Bilinear Transform
FIR Filter Design
Windowing Principal
Windowing: Frequency Interpretation
Windowing Effects
Rectangular Window
Common Windows
Common window
Effect of Windowing
Windows Freq Domain
Other Windows in Feq Domain
Comparison
Kaiser Method
Kaiser
Kaiser
Kaiser
Marks McClellan Algo
Parks McClellan Algorithm
Butterworth Approx. in MATLAB
Butterworth Approximation
Chebyshev Approximation
Elliptic Approximation in MATLAB
Elliptic Approximation