chapter 7 filter design techniques - prince of songkla...

Filter Design Techniques

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Chapter 7



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Outline

7.0 Introduction7.1 Design of Discrete Time IIR Filters7.2 Design of FIR Filters


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7.0 Introduction

Filter is a system that passes certain frequency components and totally rejects all others, but in a broader context any system that modifies certain frequencies relative to others is called a filter.

Definition of Filter


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1 The specification of the desired properties of the system.2 The approximation of the specification using a causal discrete-time system. 3 The realization of the system.

In this chapter, we focus on the second step.

The Design of Filter


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When a discrete-time filter is to be used for discrete-time processing of continuous-time filter and the effective continuous-time filter are typically given in the frequency domain.

The relationship between specifications of the discrete-time filter and the effective continuous-time filter


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If a effective continuous-time system has the frequency response.

Basic system for discrete-time filtering of continuous-time signals.


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In such cases, it is straightforward to convert from specifications on the effective continuous-time filter through the relation ω = ΩT.That is, H(ejω) is specified over one period by the equation :


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Consider a discrete-time filter that is to be used to lowpass filter a continuous-time signal using the basic configuration. Specifically, we want the overall system to have the following properties when the sampling rate is 104 samples/s (T=10-4 s) : (1) The gain |Heff(jΩ)| should be within ∓0.01 (0.086dB) of unity (zero dB) in the frequency band 0 ≤ Ω ≤ 2π(2000).

Example


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(2) The gain should be no greater than 0.001 (-60dB) in the frequency band 2π(3000) ≤ Ω

Such a set of lowpasss pecifications on |Heff(jΩ)| can be depicted where the limits of tolerable approximation error are indicated by the shaded horizontal lines.


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7.1 Design of Discrete-time IIR Filters form Continuous-Time Filters

The Transformation of a continuous-time filter into a discrete-time filter meeting prescribed specifications. The Reasons for Using this Method: - The art of continuous-time IIR filter design has developed and many results can be used. - Many continuous-time IIR filter design methods have relatively simple closed form design formulas, therefore it is easy to carry out.


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- The standard approximation methods for continuous-time IIR filters can not be directly used in discrete-time filter design.3. Processes of design: - Specifications transformation; - Continuous-time filter design; - Mapping continuous-time filter into discrete-time filter (From s-plane to z- plane) .


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7.1.1 Filter Design by Impulse Invariance

If hc(t) is the impulse response of continuous-time filter, and hc(nTd) is equally spaced samples of it.

The frequency response :


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If the continuous-time filter is bandlimited, so that

then


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Assume that the specifications for the designed discrete-time filter are shown in next slide with ,δ1 = 0.10875, δ2= 0.17783, ωp = 0.2π and ωs = 0.3π. Τhe maximum gain in stopband is -15dB (20log10 0.17783), The maximum deviation of 1dB below 0dB gain in passband (20log10(1) – 20log10(1-0.10875) =-1 dB). In this case the band pass tolerance is between 1- δ1 and 1.

Example


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The impulse invariance transformation from CT to DT :


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Example

Consider the design of a lowpass discrete-time filter by applying impulse invariance to an appropriate Butterworth continuous-time filter. The specifications for the discrete-time filter are :


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Choose Td =1 so that ω=Ω

Continuous-time Butterworth filter with magnitude function |Hc(jΩ)|


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Let Ωp = 0.2π and Ωs = 0.3π

The magnitude squared function of a Btterworth filter


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So that the filter design process consists of determining the parameters N and Ωc to meet the desired specification.


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Since N must be integer N=6 substuting N=6 in equation slide 26. We have Ωc = 0.7032


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Find the poles :


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7.1.2 Bilinear Transformation

In order to avoid the aliasing in impulse invariance, we introduce another method of transformation bilinear transformation, which use an algebraic transform between the variables s and z. This transform is


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In the transformation, -∞ ≤ Ω ≤ ∞ maps onto -π ≤ ω ≤ π ,the transformation between the continuous-time and discrete-time frequency variables must be nonlinear. Therefore the use of this technique is restricted to the situation where the corresponding warping of the frequency axis is acceptable.


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To develop the properties of the algebraic transformationWe solve for z to obtain :

Substituting s = σ+jΩ, we obtain :


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If σ<0 then |z|<1 for any value of Ω. Similarly, If σ>0 then |z|>1for all Ω.That is if a pole of Hc(s) is in the left−half s-plane, its image in the z-plane will be inside the unit circle. Therefore causal stable continuous-time filters map into causal stable discrete-time filters.


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To show that the jΩ-axis of the s-plane maps onto the unit circle, we substitute s=jΩ :

It is clear that |z| =1 for all value of s on the jΩ-axis


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To derive the relationship between the variables ω and Ω, we substituting z= ejω.

or


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Equating real and imaginary parts on both sides leads to the relations σ=0

or


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The bilinear transformation avoids the problem of aliasing encountered with the use of impulse invariance because it maps the entire imaginary axis of the s-plane onto the unit circle in the z-plane. The price paid for this, however, is the nonlinear compression the frequency axis.


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If we transform a lowpass filter from continuous-time form into discrete-time form, the warping of bilinear transformation can be demonstrated in next slide.


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If the critical frequencies (such as the passband and stopband edge frequencies) of continuous-time filter are prewaped according the equation


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then when the continuous-time filter is transformed to the discrete-time filter the discrete-time filter will meet the desired specifications.


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Example

Consider the specification on the discrete-time filter :

Using the bilinear transformation, the critical frequencies of the discrete-time filter must be prewarped to the corresponding continuous-time frequency


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For convenience we choose Td=1, since Butterworth filter has a monotonic magnitude response, so from above equations we obtain :


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The form of the magnitude-squared function for the Butterworth filter is :

Solving for N and Ωc, we obtain :

and


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The result are N = 5.30466, and take N=6, substituting N = 6 and Ωc = 0. 7662.

For this value of Ωc, the passband specifications are exceeded and the stopband specifications are met exactly.

N =log [1/0.17821/1/0.8921]

2 log [ tan 0.15/ tan 0.1]


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In the s-plane, the 12 poles are uniformly distributed in angle on a circle of radius 0.76622.


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The system function of the continuous-time filter by selecting the left-plane poles is


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At ω=0.2π, the log magnitude is -0.56dB, and at ω=0.3π, log magnitude is exactly -15dB.

The magnitude, log magnitude, and group delay of the frequency response of the discrete-time filter are shown in next slides


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From above example, we know Nth-order Butterworth filter has the following form


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Homework

We wish to design a lowpass digital filter to meet the specifications :

δ1 = 0.01, δ2 = 0.001, ωp = 0.4π and ωs = 0.6π.


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1. Butterworth filter design by impulse invariance.2. Butterworth filter design by bilinear transformation3. Chebyshev filter design by bilinear transformation

chapter 7 filter design techniques - prince of songkla...

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