Chap5 IEEE 802 pdf

Associate Professor

Dept. of Electrical and Electronic Engineering

University of Dhaka

Dr. Mohammad Junaebur Rashid (JR)

ICT3105

:

Digital Signal Processing

(3.0 Cr)

Course Teacher

Bangladesh University of Professionals

Lecture 21

• Digital Filters

Design Procedure

•

To fully design and implement a filter five steps are required:

(1)

Filter specification.

(2)

Coefficient calculation.

(3)

Structure selection.

(4)

Simulation (optional).

Lecture 21

Lecture 21

• Digital Filters

• The finite word-length effect is the deviation of FIR filter characteristic. If such

characteristic still meets the filter specifications, the finite word-length effects are

negligible. As a result of greater error in coefficients representation, the finite word-length

effects are more prominent in fixed-point arithmetic.

• These effects are more prominent for IIR filters for their feedback property than for FIR

filters. In addition, coefficient representation can cause IIR filters to become instable,

whereas it cannot affect FIR filters that way.

• FIR filters keep their linear phase characteristic after quantization. Because the coefficients

of a FIR filter with linear phase characteristic are symmetric, which means that the

corresponding pairs of coefficients will be quantized to the same value. It results in the

impulse response symmetry remaining unchanged.

• Finite word length causes some problems such as: Coefficient quantization errors; Sample

quantization errors (quantization noise); and Overflow errors.

Lecture 21

• Digital Filters

5 main stages of IIR filter design:

Stage 1: Filter Specification

Designer gives the function of the filter (e.g. lowpass) and the desired performance.

Stage 2: Approximation or Coefficient Calculation

The selection method to calculate the coefficients such that the specs in stage 1 are

satisfied.

Stage 3: Realisation

Converting of the transfer function into a suitable filter structure

Stage 4: Finite Wordlength Effects

Analysis of errors that will occur due to working with finite number of bits

Stage 5: Implementation

Lecture 21

• Digital Filters

As with most other engineering problems, the design of the digital IIR filter starts with an

explicit specification of the performance requirements. This includes:

1. Signal characteristics

Types of signal source and sink I/O interface, data rates and wordlengths

2. Frequency response characteristics

Desired amplitude, phase and tolerances Speed of operation

3. Manner of implementation

High level language routine in a computer, or DSP-based system

Choice of signal processor, Modes of filtering (real-time or batch)

4. Other design constraints

Cost

Permissible signal degradation through the filter

Lecture 21

• Digital Filters

• For frequency selective filters, such as low-pass and band-pass filters, the frequency

Lecture 21

• Design of IIR Filter

Notes for the tolerance scheme of

a filter:

• IIR filter: passband ripple is the

difference btw. the minimum and

maximum deviation in the

passband.

• FIR filter: passband ripple is the

difference btw. the ideal response

and the maximum (or minimum)

deviation in the passband.

• Thus, for IIR, when we say

passband ripple, what is meant is

Lecture 21

• Design of IIR Filter

• There are two different methods available for calculating the coefficients:

- Direct placement of poles and zeros

- Matched z-transform

- Using analogue filter design

Impulse invariant, Bilinear z-transform

Coefficient Calculation - Step 2

• This is one of the simplest method.

• There is a rich collection of prototype analogue filters with well-established analysis

methods.

• The method involves designing an analogue filter and then transforming it to a digital

filter.

Lecture 21

• Design of IIR Filter

• Placing a zero near or on the unit circle in the z-plane will minimize the transfer function at

this point.

• Because when a zero is placed at a given point on the z-plane, the frequency response will

be zero at the corresponding point.

• Placing a pole near or on the unit circle in the z-plane will maximize the transfer function at

this point.

• Because when a pole is placed at a given point on the z-plane, the frequency response will

be a peak at the corresponding point.

• To obtain real coefficients the poles and zeros must either be real or occur in complex

conjugate pairs.

Lecture 21

• Design of IIR Filter

• Zeros that are close to or on the circle produce troughs or minima.

• Poles that are close to the unit circle give rise to large peaks.

• Thus, by strategically placing poles and zeros on the z-plane, we can obtain simple lowpass,

Realization Structures - Step 3

• Direct Form I:

•

Difference equation:

•

This leads to the following structure…

Lecture 21

Realization Structures - Step 3

•

Direct Form I:

Lecture 21

Realization Structures - Step 3

•

Direct Form II canonic realization:

Lecture 21

Lecture 21

• Design of IIR Filter

Example:

A bandpass digital filter is required to meet the following specifications:

• Complete signal rejection at DC and 250 Hz

• A narrow passband centered at 125 Hz

• A 3 dB bandwidth of 10 Hz

• Assuming a Sampling Frequency of 500 Hz

Now, obtain:

• Pole-zero diagram, by suitably placing z-plane poles and zeros

• Transfer Function of the filter

• Coefficients

Lecture 21

• Design of IIR Filter

• First: Location of the poles and zeros on the z-plane?

• Since a complete rejection is required at 0 and 250 Hz, we need to place zeros at the

corresponding points on the z-plane.

• Angles of the zeros on the unit circle are:

[ θ = 2πf / F

]

• To have a passband at 125 Hz requires us to place poles. Now locations of poles on the unit

circle are:

• NB. Since this falls on the imaginary axis, to ensure that the coefficients are real, it is

Lecture 21

Lecture 21

• Design of IIR Filter

-Lecture 21

Lecture 21

• Design of IIR Filter

• Practical example of the bilinear transform method:

– The design of a digital filter to approximate a second order low-pass

analogue filter is required.

– The transfer function that describes the analogue filter is (This is an analog

BUTTERWORTH filter):

– The digital filter is required to have:

• Cut-off frequency =100 Hz

• Sampling frequency of 1 kHz.

Lecture 21

• Design of IIR Filter

Design a digital equivalent of a 2

order Butterworth LP filter with a cut-off freq

fc=100 Hz, Sampling frequency fs=1000 Hz.

The normalized cut off frequency of the digital filter

ω=2πf

/f

=2πf

100/1000=0.628

Now equivalent analog filter cut-off freq

Ω=ktan(ω/2)= 1.tan(0.628/2)=0.325 rads/sec

Example: Design of a LP filter design using Bilinear Transformation

• H(s) for a Butterworth filter

Lecture 21

• Design of IIR Filter

• Now convert the analog filter H(s) into equivalent digital filter H(z) by applying the bilinear

z-transform-{b

}Coefficients

{a

}coefficients

b

=0.067

b

=0.0135

a

= -1.1429

a

=0.4127

coefficients-Lecture 21

• Design of IIR Filter

Direct realization of the design

Lecture 21

• Design of IIR Filter

Matlab Code for the BZT Method