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