Chapter 7 Filter Design Techniques - Prince of Songkla...
Transcript of Chapter 7 Filter Design Techniques - Prince of Songkla...
![Page 1: Chapter 7 Filter Design Techniques - Prince of Songkla ...fivedots.coe.psu.ac.th/Software.coe/240-381/slide/DSPCh7.pdf · Filter Design Techniques Revise 11/10/2004 Page 1 ... - The](https://reader036.fdocuments.us/reader036/viewer/2022062504/5a9e26fd7f8b9a39338db6fe/html5/thumbnails/1.jpg)
Filter Design Techniques
Revise 11/10/2004
Page 1
Chapter 7
Filter Design Techniques
![Page 2: Chapter 7 Filter Design Techniques - Prince of Songkla ...fivedots.coe.psu.ac.th/Software.coe/240-381/slide/DSPCh7.pdf · Filter Design Techniques Revise 11/10/2004 Page 1 ... - The](https://reader036.fdocuments.us/reader036/viewer/2022062504/5a9e26fd7f8b9a39338db6fe/html5/thumbnails/2.jpg)
Filter Design Techniques
Dig
ital S
igna
l Pro
cess
ing
Revise 11/10/2004
Page 2
Outline
7.0 Introduction7.1 Design of Discrete Time IIR Filters7.2 Design of FIR Filters
![Page 3: Chapter 7 Filter Design Techniques - Prince of Songkla ...fivedots.coe.psu.ac.th/Software.coe/240-381/slide/DSPCh7.pdf · Filter Design Techniques Revise 11/10/2004 Page 1 ... - The](https://reader036.fdocuments.us/reader036/viewer/2022062504/5a9e26fd7f8b9a39338db6fe/html5/thumbnails/3.jpg)
Filter Design Techniques
Dig
ital S
igna
l Pro
cess
ing
Revise 11/10/2004
Page 3
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
![Page 4: Chapter 7 Filter Design Techniques - Prince of Songkla ...fivedots.coe.psu.ac.th/Software.coe/240-381/slide/DSPCh7.pdf · Filter Design Techniques Revise 11/10/2004 Page 1 ... - The](https://reader036.fdocuments.us/reader036/viewer/2022062504/5a9e26fd7f8b9a39338db6fe/html5/thumbnails/4.jpg)
Filter Design Techniques
Dig
ital S
igna
l Pro
cess
ing
Revise 11/10/2004
Page 4
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
![Page 5: Chapter 7 Filter Design Techniques - Prince of Songkla ...fivedots.coe.psu.ac.th/Software.coe/240-381/slide/DSPCh7.pdf · Filter Design Techniques Revise 11/10/2004 Page 1 ... - The](https://reader036.fdocuments.us/reader036/viewer/2022062504/5a9e26fd7f8b9a39338db6fe/html5/thumbnails/5.jpg)
Filter Design Techniques
Dig
ital S
igna
l Pro
cess
ing
Revise 11/10/2004
Page 5
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
![Page 6: Chapter 7 Filter Design Techniques - Prince of Songkla ...fivedots.coe.psu.ac.th/Software.coe/240-381/slide/DSPCh7.pdf · Filter Design Techniques Revise 11/10/2004 Page 1 ... - The](https://reader036.fdocuments.us/reader036/viewer/2022062504/5a9e26fd7f8b9a39338db6fe/html5/thumbnails/6.jpg)
Filter Design Techniques
Dig
ital S
igna
l Pro
cess
ing
Revise 11/10/2004
Page 6
If a effective continuous-time system has the frequency response.
Basic system for discrete-time filtering of continuous-time signals.
![Page 7: Chapter 7 Filter Design Techniques - Prince of Songkla ...fivedots.coe.psu.ac.th/Software.coe/240-381/slide/DSPCh7.pdf · Filter Design Techniques Revise 11/10/2004 Page 1 ... - The](https://reader036.fdocuments.us/reader036/viewer/2022062504/5a9e26fd7f8b9a39338db6fe/html5/thumbnails/7.jpg)
Filter Design Techniques
Dig
ital S
igna
l Pro
cess
ing
Revise 11/10/2004
Page 7
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 :
![Page 8: Chapter 7 Filter Design Techniques - Prince of Songkla ...fivedots.coe.psu.ac.th/Software.coe/240-381/slide/DSPCh7.pdf · Filter Design Techniques Revise 11/10/2004 Page 1 ... - The](https://reader036.fdocuments.us/reader036/viewer/2022062504/5a9e26fd7f8b9a39338db6fe/html5/thumbnails/8.jpg)
Filter Design Techniques
Dig
ital S
igna
l Pro
cess
ing
Revise 11/10/2004
Page 8
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
![Page 9: Chapter 7 Filter Design Techniques - Prince of Songkla ...fivedots.coe.psu.ac.th/Software.coe/240-381/slide/DSPCh7.pdf · Filter Design Techniques Revise 11/10/2004 Page 1 ... - The](https://reader036.fdocuments.us/reader036/viewer/2022062504/5a9e26fd7f8b9a39338db6fe/html5/thumbnails/9.jpg)
Filter Design Techniques
Dig
ital S
igna
l Pro
cess
ing
Revise 11/10/2004
Page 9
(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.
![Page 10: Chapter 7 Filter Design Techniques - Prince of Songkla ...fivedots.coe.psu.ac.th/Software.coe/240-381/slide/DSPCh7.pdf · Filter Design Techniques Revise 11/10/2004 Page 1 ... - The](https://reader036.fdocuments.us/reader036/viewer/2022062504/5a9e26fd7f8b9a39338db6fe/html5/thumbnails/10.jpg)
Filter Design Techniques
Dig
ital S
igna
l Pro
cess
ing
Revise 11/10/2004
Page 10
![Page 11: Chapter 7 Filter Design Techniques - Prince of Songkla ...fivedots.coe.psu.ac.th/Software.coe/240-381/slide/DSPCh7.pdf · Filter Design Techniques Revise 11/10/2004 Page 1 ... - The](https://reader036.fdocuments.us/reader036/viewer/2022062504/5a9e26fd7f8b9a39338db6fe/html5/thumbnails/11.jpg)
Filter Design Techniques
Dig
ital S
igna
l Pro
cess
ing
Revise 11/10/2004
Page 11
![Page 12: Chapter 7 Filter Design Techniques - Prince of Songkla ...fivedots.coe.psu.ac.th/Software.coe/240-381/slide/DSPCh7.pdf · Filter Design Techniques Revise 11/10/2004 Page 1 ... - The](https://reader036.fdocuments.us/reader036/viewer/2022062504/5a9e26fd7f8b9a39338db6fe/html5/thumbnails/12.jpg)
Filter Design Techniques
Dig
ital S
igna
l Pro
cess
ing
Revise 11/10/2004
Page 12
![Page 13: Chapter 7 Filter Design Techniques - Prince of Songkla ...fivedots.coe.psu.ac.th/Software.coe/240-381/slide/DSPCh7.pdf · Filter Design Techniques Revise 11/10/2004 Page 1 ... - The](https://reader036.fdocuments.us/reader036/viewer/2022062504/5a9e26fd7f8b9a39338db6fe/html5/thumbnails/13.jpg)
Filter Design Techniques
Dig
ital S
igna
l Pro
cess
ing
Revise 11/10/2004
Page 13
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.
![Page 14: Chapter 7 Filter Design Techniques - Prince of Songkla ...fivedots.coe.psu.ac.th/Software.coe/240-381/slide/DSPCh7.pdf · Filter Design Techniques Revise 11/10/2004 Page 1 ... - The](https://reader036.fdocuments.us/reader036/viewer/2022062504/5a9e26fd7f8b9a39338db6fe/html5/thumbnails/14.jpg)
Filter Design Techniques
Dig
ital S
igna
l Pro
cess
ing
Revise 11/10/2004
Page 14
- 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) .
![Page 15: Chapter 7 Filter Design Techniques - Prince of Songkla ...fivedots.coe.psu.ac.th/Software.coe/240-381/slide/DSPCh7.pdf · Filter Design Techniques Revise 11/10/2004 Page 1 ... - The](https://reader036.fdocuments.us/reader036/viewer/2022062504/5a9e26fd7f8b9a39338db6fe/html5/thumbnails/15.jpg)
Filter Design Techniques
Dig
ital S
igna
l Pro
cess
ing
Revise 11/10/2004
Page 15
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 :
![Page 16: Chapter 7 Filter Design Techniques - Prince of Songkla ...fivedots.coe.psu.ac.th/Software.coe/240-381/slide/DSPCh7.pdf · Filter Design Techniques Revise 11/10/2004 Page 1 ... - The](https://reader036.fdocuments.us/reader036/viewer/2022062504/5a9e26fd7f8b9a39338db6fe/html5/thumbnails/16.jpg)
Filter Design Techniques
Dig
ital S
igna
l Pro
cess
ing
Revise 11/10/2004
Page 16
If the continuous-time filter is bandlimited, so that
then
![Page 17: Chapter 7 Filter Design Techniques - Prince of Songkla ...fivedots.coe.psu.ac.th/Software.coe/240-381/slide/DSPCh7.pdf · Filter Design Techniques Revise 11/10/2004 Page 1 ... - The](https://reader036.fdocuments.us/reader036/viewer/2022062504/5a9e26fd7f8b9a39338db6fe/html5/thumbnails/17.jpg)
Filter Design Techniques
Dig
ital S
igna
l Pro
cess
ing
Revise 11/10/2004
Page 17
![Page 18: Chapter 7 Filter Design Techniques - Prince of Songkla ...fivedots.coe.psu.ac.th/Software.coe/240-381/slide/DSPCh7.pdf · Filter Design Techniques Revise 11/10/2004 Page 1 ... - The](https://reader036.fdocuments.us/reader036/viewer/2022062504/5a9e26fd7f8b9a39338db6fe/html5/thumbnails/18.jpg)
Filter Design Techniques
Dig
ital S
igna
l Pro
cess
ing
Revise 11/10/2004
Page 18
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
![Page 19: Chapter 7 Filter Design Techniques - Prince of Songkla ...fivedots.coe.psu.ac.th/Software.coe/240-381/slide/DSPCh7.pdf · Filter Design Techniques Revise 11/10/2004 Page 1 ... - The](https://reader036.fdocuments.us/reader036/viewer/2022062504/5a9e26fd7f8b9a39338db6fe/html5/thumbnails/19.jpg)
Filter Design Techniques
Dig
ital S
igna
l Pro
cess
ing
Revise 11/10/2004
Page 19
![Page 20: Chapter 7 Filter Design Techniques - Prince of Songkla ...fivedots.coe.psu.ac.th/Software.coe/240-381/slide/DSPCh7.pdf · Filter Design Techniques Revise 11/10/2004 Page 1 ... - The](https://reader036.fdocuments.us/reader036/viewer/2022062504/5a9e26fd7f8b9a39338db6fe/html5/thumbnails/20.jpg)
Filter Design Techniques
Dig
ital S
igna
l Pro
cess
ing
Revise 11/10/2004
Page 20
![Page 21: Chapter 7 Filter Design Techniques - Prince of Songkla ...fivedots.coe.psu.ac.th/Software.coe/240-381/slide/DSPCh7.pdf · Filter Design Techniques Revise 11/10/2004 Page 1 ... - The](https://reader036.fdocuments.us/reader036/viewer/2022062504/5a9e26fd7f8b9a39338db6fe/html5/thumbnails/21.jpg)
Filter Design Techniques
Dig
ital S
igna
l Pro
cess
ing
Revise 11/10/2004
Page 21
The impulse invariance transformation from CT to DT :
![Page 22: Chapter 7 Filter Design Techniques - Prince of Songkla ...fivedots.coe.psu.ac.th/Software.coe/240-381/slide/DSPCh7.pdf · Filter Design Techniques Revise 11/10/2004 Page 1 ... - The](https://reader036.fdocuments.us/reader036/viewer/2022062504/5a9e26fd7f8b9a39338db6fe/html5/thumbnails/22.jpg)
Filter Design Techniques
Dig
ital S
igna
l Pro
cess
ing
Revise 11/10/2004
Page 22
![Page 23: Chapter 7 Filter Design Techniques - Prince of Songkla ...fivedots.coe.psu.ac.th/Software.coe/240-381/slide/DSPCh7.pdf · Filter Design Techniques Revise 11/10/2004 Page 1 ... - The](https://reader036.fdocuments.us/reader036/viewer/2022062504/5a9e26fd7f8b9a39338db6fe/html5/thumbnails/23.jpg)
Filter Design Techniques
Dig
ital S
igna
l Pro
cess
ing
Revise 11/10/2004
Page 23
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 :
![Page 24: Chapter 7 Filter Design Techniques - Prince of Songkla ...fivedots.coe.psu.ac.th/Software.coe/240-381/slide/DSPCh7.pdf · Filter Design Techniques Revise 11/10/2004 Page 1 ... - The](https://reader036.fdocuments.us/reader036/viewer/2022062504/5a9e26fd7f8b9a39338db6fe/html5/thumbnails/24.jpg)
Filter Design Techniques
Dig
ital S
igna
l Pro
cess
ing
Revise 11/10/2004
Page 24
Choose Td =1 so that ω=Ω
Continuous-time Butterworth filter with magnitude function |Hc(jΩ)|
![Page 25: Chapter 7 Filter Design Techniques - Prince of Songkla ...fivedots.coe.psu.ac.th/Software.coe/240-381/slide/DSPCh7.pdf · Filter Design Techniques Revise 11/10/2004 Page 1 ... - The](https://reader036.fdocuments.us/reader036/viewer/2022062504/5a9e26fd7f8b9a39338db6fe/html5/thumbnails/25.jpg)
Filter Design Techniques
Dig
ital S
igna
l Pro
cess
ing
Revise 11/10/2004
Page 25
Let Ωp = 0.2π and Ωs = 0.3π
The magnitude squared function of a Btterworth filter
![Page 26: Chapter 7 Filter Design Techniques - Prince of Songkla ...fivedots.coe.psu.ac.th/Software.coe/240-381/slide/DSPCh7.pdf · Filter Design Techniques Revise 11/10/2004 Page 1 ... - The](https://reader036.fdocuments.us/reader036/viewer/2022062504/5a9e26fd7f8b9a39338db6fe/html5/thumbnails/26.jpg)
Filter Design Techniques
Dig
ital S
igna
l Pro
cess
ing
Revise 11/10/2004
Page 26
So that the filter design process consists of determining the parameters N and Ωc to meet the desired specification.
![Page 27: Chapter 7 Filter Design Techniques - Prince of Songkla ...fivedots.coe.psu.ac.th/Software.coe/240-381/slide/DSPCh7.pdf · Filter Design Techniques Revise 11/10/2004 Page 1 ... - The](https://reader036.fdocuments.us/reader036/viewer/2022062504/5a9e26fd7f8b9a39338db6fe/html5/thumbnails/27.jpg)
Filter Design Techniques
Revise 11/10/2004
Page 27
![Page 28: Chapter 7 Filter Design Techniques - Prince of Songkla ...fivedots.coe.psu.ac.th/Software.coe/240-381/slide/DSPCh7.pdf · Filter Design Techniques Revise 11/10/2004 Page 1 ... - The](https://reader036.fdocuments.us/reader036/viewer/2022062504/5a9e26fd7f8b9a39338db6fe/html5/thumbnails/28.jpg)
Filter Design Techniques
Dig
ital S
igna
l Pro
cess
ing
Revise 11/10/2004
Page 28
Since N must be integer N=6 substuting N=6 in equation slide 26. We have Ωc = 0.7032
![Page 29: Chapter 7 Filter Design Techniques - Prince of Songkla ...fivedots.coe.psu.ac.th/Software.coe/240-381/slide/DSPCh7.pdf · Filter Design Techniques Revise 11/10/2004 Page 1 ... - The](https://reader036.fdocuments.us/reader036/viewer/2022062504/5a9e26fd7f8b9a39338db6fe/html5/thumbnails/29.jpg)
Filter Design Techniques
Dig
ital S
igna
l Pro
cess
ing
Revise 11/10/2004
Page 29
![Page 30: Chapter 7 Filter Design Techniques - Prince of Songkla ...fivedots.coe.psu.ac.th/Software.coe/240-381/slide/DSPCh7.pdf · Filter Design Techniques Revise 11/10/2004 Page 1 ... - The](https://reader036.fdocuments.us/reader036/viewer/2022062504/5a9e26fd7f8b9a39338db6fe/html5/thumbnails/30.jpg)
Filter Design Techniques
Dig
ital S
igna
l Pro
cess
ing
Revise 11/10/2004
Page 30
![Page 31: Chapter 7 Filter Design Techniques - Prince of Songkla ...fivedots.coe.psu.ac.th/Software.coe/240-381/slide/DSPCh7.pdf · Filter Design Techniques Revise 11/10/2004 Page 1 ... - The](https://reader036.fdocuments.us/reader036/viewer/2022062504/5a9e26fd7f8b9a39338db6fe/html5/thumbnails/31.jpg)
Filter Design Techniques
Dig
ital S
igna
l Pro
cess
ing
Revise 11/10/2004
Page 31
Find the poles :
![Page 32: Chapter 7 Filter Design Techniques - Prince of Songkla ...fivedots.coe.psu.ac.th/Software.coe/240-381/slide/DSPCh7.pdf · Filter Design Techniques Revise 11/10/2004 Page 1 ... - The](https://reader036.fdocuments.us/reader036/viewer/2022062504/5a9e26fd7f8b9a39338db6fe/html5/thumbnails/32.jpg)
Filter Design Techniques
Revise 11/10/2004
Page 32
![Page 33: Chapter 7 Filter Design Techniques - Prince of Songkla ...fivedots.coe.psu.ac.th/Software.coe/240-381/slide/DSPCh7.pdf · Filter Design Techniques Revise 11/10/2004 Page 1 ... - The](https://reader036.fdocuments.us/reader036/viewer/2022062504/5a9e26fd7f8b9a39338db6fe/html5/thumbnails/33.jpg)
Filter Design Techniques
Dig
ital S
igna
l Pro
cess
ing
Revise 11/10/2004
Page 33
![Page 34: Chapter 7 Filter Design Techniques - Prince of Songkla ...fivedots.coe.psu.ac.th/Software.coe/240-381/slide/DSPCh7.pdf · Filter Design Techniques Revise 11/10/2004 Page 1 ... - The](https://reader036.fdocuments.us/reader036/viewer/2022062504/5a9e26fd7f8b9a39338db6fe/html5/thumbnails/34.jpg)
Filter Design Techniques
Dig
ital S
igna
l Pro
cess
ing
Revise 11/10/2004
Page 34
![Page 35: Chapter 7 Filter Design Techniques - Prince of Songkla ...fivedots.coe.psu.ac.th/Software.coe/240-381/slide/DSPCh7.pdf · Filter Design Techniques Revise 11/10/2004 Page 1 ... - The](https://reader036.fdocuments.us/reader036/viewer/2022062504/5a9e26fd7f8b9a39338db6fe/html5/thumbnails/35.jpg)
Filter Design Techniques
Dig
ital S
igna
l Pro
cess
ing
Revise 11/10/2004
Page 35
![Page 36: Chapter 7 Filter Design Techniques - Prince of Songkla ...fivedots.coe.psu.ac.th/Software.coe/240-381/slide/DSPCh7.pdf · Filter Design Techniques Revise 11/10/2004 Page 1 ... - The](https://reader036.fdocuments.us/reader036/viewer/2022062504/5a9e26fd7f8b9a39338db6fe/html5/thumbnails/36.jpg)
Filter Design Techniques
Dig
ital S
igna
l Pro
cess
ing
Revise 11/10/2004
Page 36
![Page 37: Chapter 7 Filter Design Techniques - Prince of Songkla ...fivedots.coe.psu.ac.th/Software.coe/240-381/slide/DSPCh7.pdf · Filter Design Techniques Revise 11/10/2004 Page 1 ... - The](https://reader036.fdocuments.us/reader036/viewer/2022062504/5a9e26fd7f8b9a39338db6fe/html5/thumbnails/37.jpg)
Filter Design Techniques
Dig
ital S
igna
l Pro
cess
ing
Revise 11/10/2004
Page 37
![Page 38: Chapter 7 Filter Design Techniques - Prince of Songkla ...fivedots.coe.psu.ac.th/Software.coe/240-381/slide/DSPCh7.pdf · Filter Design Techniques Revise 11/10/2004 Page 1 ... - The](https://reader036.fdocuments.us/reader036/viewer/2022062504/5a9e26fd7f8b9a39338db6fe/html5/thumbnails/38.jpg)
Filter Design Techniques
Dig
ital S
igna
l Pro
cess
ing
Revise 11/10/2004
Page 38
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
![Page 39: Chapter 7 Filter Design Techniques - Prince of Songkla ...fivedots.coe.psu.ac.th/Software.coe/240-381/slide/DSPCh7.pdf · Filter Design Techniques Revise 11/10/2004 Page 1 ... - The](https://reader036.fdocuments.us/reader036/viewer/2022062504/5a9e26fd7f8b9a39338db6fe/html5/thumbnails/39.jpg)
Filter Design Techniques
Dig
ital S
igna
l Pro
cess
ing
Revise 11/10/2004
Page 39
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.
![Page 40: Chapter 7 Filter Design Techniques - Prince of Songkla ...fivedots.coe.psu.ac.th/Software.coe/240-381/slide/DSPCh7.pdf · Filter Design Techniques Revise 11/10/2004 Page 1 ... - The](https://reader036.fdocuments.us/reader036/viewer/2022062504/5a9e26fd7f8b9a39338db6fe/html5/thumbnails/40.jpg)
Filter Design Techniques
Dig
ital S
igna
l Pro
cess
ing
Revise 11/10/2004
Page 40
To develop the properties of the algebraic transformationWe solve for z to obtain :
Substituting s = σ+jΩ, we obtain :
![Page 41: Chapter 7 Filter Design Techniques - Prince of Songkla ...fivedots.coe.psu.ac.th/Software.coe/240-381/slide/DSPCh7.pdf · Filter Design Techniques Revise 11/10/2004 Page 1 ... - The](https://reader036.fdocuments.us/reader036/viewer/2022062504/5a9e26fd7f8b9a39338db6fe/html5/thumbnails/41.jpg)
Filter Design Techniques
Dig
ital S
igna
l Pro
cess
ing
Revise 11/10/2004
Page 41
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.
![Page 42: Chapter 7 Filter Design Techniques - Prince of Songkla ...fivedots.coe.psu.ac.th/Software.coe/240-381/slide/DSPCh7.pdf · Filter Design Techniques Revise 11/10/2004 Page 1 ... - The](https://reader036.fdocuments.us/reader036/viewer/2022062504/5a9e26fd7f8b9a39338db6fe/html5/thumbnails/42.jpg)
Filter Design Techniques
Dig
ital S
igna
l Pro
cess
ing
Revise 11/10/2004
Page 42
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
![Page 43: Chapter 7 Filter Design Techniques - Prince of Songkla ...fivedots.coe.psu.ac.th/Software.coe/240-381/slide/DSPCh7.pdf · Filter Design Techniques Revise 11/10/2004 Page 1 ... - The](https://reader036.fdocuments.us/reader036/viewer/2022062504/5a9e26fd7f8b9a39338db6fe/html5/thumbnails/43.jpg)
Filter Design Techniques
Dig
ital S
igna
l Pro
cess
ing
Revise 11/10/2004
Page 43
To derive the relationship between the variables ω and Ω, we substituting z= ejω.
or
![Page 44: Chapter 7 Filter Design Techniques - Prince of Songkla ...fivedots.coe.psu.ac.th/Software.coe/240-381/slide/DSPCh7.pdf · Filter Design Techniques Revise 11/10/2004 Page 1 ... - The](https://reader036.fdocuments.us/reader036/viewer/2022062504/5a9e26fd7f8b9a39338db6fe/html5/thumbnails/44.jpg)
Filter Design Techniques
Dig
ital S
igna
l Pro
cess
ing
Revise 11/10/2004
Page 44
Equating real and imaginary parts on both sides leads to the relations σ=0
or
![Page 45: Chapter 7 Filter Design Techniques - Prince of Songkla ...fivedots.coe.psu.ac.th/Software.coe/240-381/slide/DSPCh7.pdf · Filter Design Techniques Revise 11/10/2004 Page 1 ... - The](https://reader036.fdocuments.us/reader036/viewer/2022062504/5a9e26fd7f8b9a39338db6fe/html5/thumbnails/45.jpg)
Filter Design Techniques
Dig
ital S
igna
l Pro
cess
ing
Revise 11/10/2004
Page 45
![Page 46: Chapter 7 Filter Design Techniques - Prince of Songkla ...fivedots.coe.psu.ac.th/Software.coe/240-381/slide/DSPCh7.pdf · Filter Design Techniques Revise 11/10/2004 Page 1 ... - The](https://reader036.fdocuments.us/reader036/viewer/2022062504/5a9e26fd7f8b9a39338db6fe/html5/thumbnails/46.jpg)
Filter Design Techniques
Dig
ital S
igna
l Pro
cess
ing
Revise 11/10/2004
Page 46
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.
![Page 47: Chapter 7 Filter Design Techniques - Prince of Songkla ...fivedots.coe.psu.ac.th/Software.coe/240-381/slide/DSPCh7.pdf · Filter Design Techniques Revise 11/10/2004 Page 1 ... - The](https://reader036.fdocuments.us/reader036/viewer/2022062504/5a9e26fd7f8b9a39338db6fe/html5/thumbnails/47.jpg)
Filter Design Techniques
Dig
ital S
igna
l Pro
cess
ing
Revise 11/10/2004
Page 47
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.
![Page 48: Chapter 7 Filter Design Techniques - Prince of Songkla ...fivedots.coe.psu.ac.th/Software.coe/240-381/slide/DSPCh7.pdf · Filter Design Techniques Revise 11/10/2004 Page 1 ... - The](https://reader036.fdocuments.us/reader036/viewer/2022062504/5a9e26fd7f8b9a39338db6fe/html5/thumbnails/48.jpg)
Filter Design Techniques
Revise 11/10/2004
Page 48
![Page 49: Chapter 7 Filter Design Techniques - Prince of Songkla ...fivedots.coe.psu.ac.th/Software.coe/240-381/slide/DSPCh7.pdf · Filter Design Techniques Revise 11/10/2004 Page 1 ... - The](https://reader036.fdocuments.us/reader036/viewer/2022062504/5a9e26fd7f8b9a39338db6fe/html5/thumbnails/49.jpg)
Filter Design Techniques
Dig
ital S
igna
l Pro
cess
ing
Revise 11/10/2004
Page 49
![Page 50: Chapter 7 Filter Design Techniques - Prince of Songkla ...fivedots.coe.psu.ac.th/Software.coe/240-381/slide/DSPCh7.pdf · Filter Design Techniques Revise 11/10/2004 Page 1 ... - The](https://reader036.fdocuments.us/reader036/viewer/2022062504/5a9e26fd7f8b9a39338db6fe/html5/thumbnails/50.jpg)
Filter Design Techniques
Dig
ital S
igna
l Pro
cess
ing
Revise 11/10/2004
Page 50
If the critical frequencies (such as the passband and stopband edge frequencies) of continuous-time filter are prewaped according the equation
![Page 51: Chapter 7 Filter Design Techniques - Prince of Songkla ...fivedots.coe.psu.ac.th/Software.coe/240-381/slide/DSPCh7.pdf · Filter Design Techniques Revise 11/10/2004 Page 1 ... - The](https://reader036.fdocuments.us/reader036/viewer/2022062504/5a9e26fd7f8b9a39338db6fe/html5/thumbnails/51.jpg)
Filter Design Techniques
Dig
ital S
igna
l Pro
cess
ing
Revise 11/10/2004
Page 51
then when the continuous-time filter is transformed to the discrete-time filter the discrete-time filter will meet the desired specifications.
![Page 52: Chapter 7 Filter Design Techniques - Prince of Songkla ...fivedots.coe.psu.ac.th/Software.coe/240-381/slide/DSPCh7.pdf · Filter Design Techniques Revise 11/10/2004 Page 1 ... - The](https://reader036.fdocuments.us/reader036/viewer/2022062504/5a9e26fd7f8b9a39338db6fe/html5/thumbnails/52.jpg)
Filter Design Techniques
Dig
ital S
igna
l Pro
cess
ing
Revise 11/10/2004
Page 52
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
![Page 53: Chapter 7 Filter Design Techniques - Prince of Songkla ...fivedots.coe.psu.ac.th/Software.coe/240-381/slide/DSPCh7.pdf · Filter Design Techniques Revise 11/10/2004 Page 1 ... - The](https://reader036.fdocuments.us/reader036/viewer/2022062504/5a9e26fd7f8b9a39338db6fe/html5/thumbnails/53.jpg)
Filter Design Techniques
Dig
ital S
igna
l Pro
cess
ing
Revise 11/10/2004
Page 53
For convenience we choose Td=1, since Butterworth filter has a monotonic magnitude response, so from above equations we obtain :
![Page 54: Chapter 7 Filter Design Techniques - Prince of Songkla ...fivedots.coe.psu.ac.th/Software.coe/240-381/slide/DSPCh7.pdf · Filter Design Techniques Revise 11/10/2004 Page 1 ... - The](https://reader036.fdocuments.us/reader036/viewer/2022062504/5a9e26fd7f8b9a39338db6fe/html5/thumbnails/54.jpg)
Filter Design Techniques
Dig
ital S
igna
l Pro
cess
ing
Revise 11/10/2004
Page 54
The form of the magnitude-squared function for the Butterworth filter is :
Solving for N and Ωc, we obtain :
and
![Page 55: Chapter 7 Filter Design Techniques - Prince of Songkla ...fivedots.coe.psu.ac.th/Software.coe/240-381/slide/DSPCh7.pdf · Filter Design Techniques Revise 11/10/2004 Page 1 ... - The](https://reader036.fdocuments.us/reader036/viewer/2022062504/5a9e26fd7f8b9a39338db6fe/html5/thumbnails/55.jpg)
Filter Design Techniques
Dig
ital S
igna
l Pro
cess
ing
Revise 11/10/2004
Page 55
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]
![Page 56: Chapter 7 Filter Design Techniques - Prince of Songkla ...fivedots.coe.psu.ac.th/Software.coe/240-381/slide/DSPCh7.pdf · Filter Design Techniques Revise 11/10/2004 Page 1 ... - The](https://reader036.fdocuments.us/reader036/viewer/2022062504/5a9e26fd7f8b9a39338db6fe/html5/thumbnails/56.jpg)
Filter Design Techniques
Dig
ital S
igna
l Pro
cess
ing
Revise 11/10/2004
Page 56
In the s-plane, the 12 poles are uniformly distributed in angle on a circle of radius 0.76622.
![Page 57: Chapter 7 Filter Design Techniques - Prince of Songkla ...fivedots.coe.psu.ac.th/Software.coe/240-381/slide/DSPCh7.pdf · Filter Design Techniques Revise 11/10/2004 Page 1 ... - The](https://reader036.fdocuments.us/reader036/viewer/2022062504/5a9e26fd7f8b9a39338db6fe/html5/thumbnails/57.jpg)
Filter Design Techniques
Dig
ital S
igna
l Pro
cess
ing
Revise 11/10/2004
Page 57
The system function of the continuous-time filter by selecting the left-plane poles is
![Page 58: Chapter 7 Filter Design Techniques - Prince of Songkla ...fivedots.coe.psu.ac.th/Software.coe/240-381/slide/DSPCh7.pdf · Filter Design Techniques Revise 11/10/2004 Page 1 ... - The](https://reader036.fdocuments.us/reader036/viewer/2022062504/5a9e26fd7f8b9a39338db6fe/html5/thumbnails/58.jpg)
Filter Design Techniques
Dig
ital S
igna
l Pro
cess
ing
Revise 11/10/2004
Page 58
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
![Page 59: Chapter 7 Filter Design Techniques - Prince of Songkla ...fivedots.coe.psu.ac.th/Software.coe/240-381/slide/DSPCh7.pdf · Filter Design Techniques Revise 11/10/2004 Page 1 ... - The](https://reader036.fdocuments.us/reader036/viewer/2022062504/5a9e26fd7f8b9a39338db6fe/html5/thumbnails/59.jpg)
Filter Design Techniques
Dig
ital S
igna
l Pro
cess
ing
Revise 11/10/2004
Page 59
![Page 60: Chapter 7 Filter Design Techniques - Prince of Songkla ...fivedots.coe.psu.ac.th/Software.coe/240-381/slide/DSPCh7.pdf · Filter Design Techniques Revise 11/10/2004 Page 1 ... - The](https://reader036.fdocuments.us/reader036/viewer/2022062504/5a9e26fd7f8b9a39338db6fe/html5/thumbnails/60.jpg)
Filter Design Techniques
Dig
ital S
igna
l Pro
cess
ing
Revise 11/10/2004
Page 60
![Page 61: Chapter 7 Filter Design Techniques - Prince of Songkla ...fivedots.coe.psu.ac.th/Software.coe/240-381/slide/DSPCh7.pdf · Filter Design Techniques Revise 11/10/2004 Page 1 ... - The](https://reader036.fdocuments.us/reader036/viewer/2022062504/5a9e26fd7f8b9a39338db6fe/html5/thumbnails/61.jpg)
Filter Design Techniques
Dig
ital S
igna
l Pro
cess
ing
Revise 11/10/2004
Page 61
![Page 62: Chapter 7 Filter Design Techniques - Prince of Songkla ...fivedots.coe.psu.ac.th/Software.coe/240-381/slide/DSPCh7.pdf · Filter Design Techniques Revise 11/10/2004 Page 1 ... - The](https://reader036.fdocuments.us/reader036/viewer/2022062504/5a9e26fd7f8b9a39338db6fe/html5/thumbnails/62.jpg)
Filter Design Techniques
Dig
ital S
igna
l Pro
cess
ing
Revise 11/10/2004
Page 62
From above example, we know Nth-order Butterworth filter has the following form
![Page 63: Chapter 7 Filter Design Techniques - Prince of Songkla ...fivedots.coe.psu.ac.th/Software.coe/240-381/slide/DSPCh7.pdf · Filter Design Techniques Revise 11/10/2004 Page 1 ... - The](https://reader036.fdocuments.us/reader036/viewer/2022062504/5a9e26fd7f8b9a39338db6fe/html5/thumbnails/63.jpg)
Filter Design Techniques
Dig
ital S
igna
l Pro
cess
ing
Revise 11/10/2004
Page 63
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.
![Page 64: Chapter 7 Filter Design Techniques - Prince of Songkla ...fivedots.coe.psu.ac.th/Software.coe/240-381/slide/DSPCh7.pdf · Filter Design Techniques Revise 11/10/2004 Page 1 ... - The](https://reader036.fdocuments.us/reader036/viewer/2022062504/5a9e26fd7f8b9a39338db6fe/html5/thumbnails/64.jpg)
Filter Design Techniques
Dig
ital S
igna
l Pro
cess
ing
Revise 11/10/2004
Page 64
1. Butterworth filter design by impulse invariance.2. Butterworth filter design by bilinear transformation3. Chebyshev filter design by bilinear transformation