1. Applications of DSP Imaging Medical Imaging Bandwidth compression
sGraphic
Spectrum Analysis
Array Processors
Control and Guidance
Radar

2. Reason for Processing of signals
Signals are carriers of information
Useful and unwanted
Extracting, enhancing, storing and transmitting the useful information
How signals are being processed?---
Analog Signal Processing
Digital Signal Processing

3. DSP
PrF: antialiasing filtering
PoF: smooth out the staircase waveform

4. Comparison of DSP over ASP
-Advantages
Developed Using Software on Computer;
Working Extremely Stable;
Easily Modified in Real Time ;
Low Cost and Portable;
Flexible

5. Comparison of DSP over ASP Contd…
-Disdvantages
Lower Speed and Lower Frequency
Can not be used at Higher frequency
Skilled manpower is required
Weak Signals can not be able to process

6. The two categories of DSP Tasks
Signal Analysis:
Measurement of signal properties
Spectrum(frequency/phase) analysis
Target detection, verification, recognition
Signal Filtering
Signal-in-signal-out, filter
Removal of noise/interference
Separation of frequency bands

7. Digital Filter Specification
Digital Filter designed to pass signal components of certain frequencies without distortion.
The frequency response should be equal to the signal’s frequencies to pass the signal. (passband)
The frequency response should be equal to zero to block the signal. (stopband)

8. Basic Filter Types Low pass filters High Pass filters
Band pass filters
Band reject filters

9. Digital Filter Specification
4 Types

10. Digital Filter Specification Contd…
The magnitude response specifications are given some acceptable tolerances.

11. Digital Filter Specification Contd…
Transition band is specified between the passband and the stopband to permit the magnitude to drop off smoothly.
In Passband
In Stopband
Where δp and δs are peak ripple values, ωp are passband edge frequency and ωs are stopband edge frequency

12. Digital Filter Specification Contd…
Digital filter specification are often given in terms of loss function,
A(ω) = -20 log10 |G(ejω)|
Loss specification of a digital filter
Peak passband ripple, αp = -20 log10 (1 – δp) dB
Minimum stopband attenuation, αs = -20 log10 (δs) dB

13. Digital Filter Specification Contd…
The magnitude response specifications may be given in a normalized form.

14. Digital Filter Specification Contd…
In practice, passband edge frequency and stopband edge frequency are specified in Hz
For digital filter design, normalized bandedge frequencies need to be computed from specifications in Hz using

15. Digital Filter Specification Contd…
Example - Let kHz, kHz, and kHz
Then

16. Digital Filter Type
Objective of digital filter design is to develop a causal transfer function meeting the frequency response specification.
For IIR digital filter design

17. Digital Filter Type Contd…
For FIR digital filter design
The degree N of H(z) must be small, for a linear phase, FIR filter coefficient must satisfy the constraint

18. FIR FILTERS

19. FIR Filter Design by Window function technique
Simplest FIR the filter design is window function technique
An ideal frequency response may express
where

20. FIR Filter Design by Window function technique Contd…
To get this kind of systematic causal FIR to be approximate, the most direct method intercepts its ideal impulse response!

21. FIR Filter Design by Window function technique Contd…
1.Rectangular window
2.Triangular window (Bartett window)
1.main lobe越窄 resolution越高 side lobe 越低越好
2.統計上常用 resolution降一半 main lobe 變寬(trade off)
Main lobe變寬(trade off) side lobe降一半

22. FIR Filter Design by Window function technique Contd…
1.Rectangular window
2.Triangular window (Bartett window)

23. FIR Filter Design by Window function technique Contd…
3.HANN window
4.Hamming window
1.In fact, the length of window is M-1
2.main lobe和HANN差不多但side lobe降了10dB
3.Hamming 常用在語音處理

24. FIR Filter Design by Window function technique Contd…
3.HANN window
4.Hamming window

25. FIR Filter Design by Window function technique Contd…
5.Kaiser’s window
6.Blackman window
有參數可調,能得適當的組合

26. FIR Filter Design by Window function technique Contd…
5.Kaiser’s window
6.Blackman window

27. Window Table Type of the window Transition Bandwidth
Minor Lobe attenuation in dB
Rectangular
4π/M
-21
Triangu;ar
8π/M
-26
Hanning
-44
Hamming
-53
Blackmann
12π/M
-74
Kaiser
variable

28. Filter Design by Windowing
Simplest way of designing FIR filters
Method is all discrete-time no continuous-time involved
Start with ideal frequency response
Choose ideal frequency response as desired response
Most ideal impulse responses are of infinite length
The easiest way to obtain a causal FIR filter from ideal is
More generally
351M Digital Signal Processing

29. Rectangular Window Narrowest main lob 4/(M+1)
Sharpest transitions at discontinuities in frequency
Large side lobs
-21 dB
Large oscillation around discontinuities
Simplest window possible

30. Bartlett (Triangular) Window
Medium main lob
8/M
Side lobs
-25 dB
Hamming window performs better
Simple equation

31. Hanning Window Medium main lob 8/M Side lobs -44 dB
Hamming window performs better
Same complexity as Hamming

32. Hamming Window Medium main lob Good side lobs Simpler than Blackman
-53 dB
Simpler than Blackman

33. Blackman Window Large main lob Very good side lobs Complex equation
-73 dB
Complex equation

34. Lowpass filter Desired frequency response
Corresponding impulse response

35. Highpass filter
Corresponding impulse response

36. Bandpass Filter
The Impulse Response is

37. Bandreject Filter
The Impulse Response is

38. FIR Filter Design Procedure
Step1:Draw the response of the given problem.
Step2:Convert the Analog frequencies in to the Digital frequencies
Step3:Calculate the Transition Band width.
Step4:Calculate the order of the filter by equating the calculated Transitions band width to the transition band width in the table.

39. FIR Filter Design Procedure Contd…
Step5:Calculate the Ʈ parameter Ʈ =(M-1)/2
Step6:Choose the Window to be used by considering the attenuation.
Step7:Calculate ht(n)
Step8:Calculate w(n) for the choosen window.

40. FIR Filter Design Procedure Contd…
Step9:Then calculate h(n)=ht(n) x w(n)
Step10: For verifying the design use the equation for calculating the magnitude response and the frequency response.

41. Table: Frequency ResponseѠ Ø
Attenuation
20log
0.2π
0.4π
0.6π
0.8π π

42. Kaiser Window Filter Design Method
Parameterized equation forming a set of windows
Parameter to change main-lob width and side-lob area trade-off
I0(.) represents zeroth-order modified Bessel function of 1st kind

43. Determining Kaiser Window Parameters
Given filter specifications Kaiser developed empirical equations
Given the peak approximation error  or in dB as A=-20log10 
and transition band width
The shape parameter  should be
The filter order M is determined approximately by
After the kaiser window design follow the same procedure for the filter design

44. IIR FILTER

45. IIR Filter Design
The transfer function of the IIR Filters will be of the form

46. Commonly used analog IIR filters
Butterworth filter
Chebyshev filters

47. Butterworth filters
It is governed by the magnitude squared response

48. Butterworth filters-Properties
The response is maximally flat at the origin
Magnitude square is having a value of 0.5 at the cutoff frequency
It is a monotonically decreasing function beyond the cutoff frequency.

49. Butterworth Polynomial
Order
Butterworth polynomial 1
S+1 2
S2+√2 S+1 3
(S2+S+1)(S+1)
This Polynomial may be obtained by finding the roots for n is odd and even
Then by considering the left half side poles the butterworth polynomial may be constructed

50. Butterworth filter design
Step1:Find the order of the filter n=log[(10(k1/10)-1)/ (10(k2/10)-1)]/2log(Ω1/Ω2) Step2:Obtain the normalised transfer function Hn(s)=1/Bn(s) Step3:By substituting the value of s from the analog transformation Table the actual filter transfer function may be obtained

51. Analog Transformation
Filter Type
Normalised Response
Analog Transformation
Actual Response
Backward Equation
Low pass filter
S=S/ΩC
ΩS=Ω2/Ω1
High Pass Filter
S=ΩC /S

52. Chebyshev filter design- Some Prelimnaries
Relative linear scale
The lowpass filter specifications on the magnitude-squared response are given by
Where epsilon is a passband ripple parameter, Omega_p is the passband cutoff frequency in rad/sec, A is a stopband attenuation parameter, and Omega_s is the stopband cutoff in rad/sec.

53. Analog Filter response

54. Design Procedure for chebyshev filters
Step1: Calculate the order of the filter
\Where
g-[(A2-1)/ϵ2]1/2
Ωr= Ω2/Ω1
n=log[(g+(g2-1)1/2 ]/log(Ωr/(Ωr2-1)1/2
A=10-K2/20
Step2:Obtained the normalised transfer function Hn(s)=k/(Sn+bn-I Sn-1+…+b1S+b0)

56. Analog to Digital Conversion
Impulse Invariance Transformation
Bilinear Transformation

57. Impulse Invariance method
The most straightforward of these is the impulse invariance transformation
Let be the impulse response corresponding to , and define the continuous to discrete time transformation by setting
We sample the continuous time impulse response to produce the discrete time filter

58. Impulse Invariance method contd…
The impulse invariance transformation does map the -axis and the left-half s plane into the unit circle and its interior, respectively
Re(Z)
Im(Z) 1
S domain
Z domain

59. Impulse invariance method contd…
is expanded a partial fraction expansion to produce
We have assumed that there are no multiple poles
And thus
The impulse invariant transformation is not usually performed directly in the form of (2.1)
the parameters of H(z) may be obtained directly from H(s)

60. Impulse invariance method contd…
Hence it is sufficient if we substitute

61. Impulse invariance method contd…
Example:
Expanding in a partial fraction
expansion, it produce
The impulse invariant transformation
yields a discrete time design with the
system function

62. Bilinear transformation method
The most generally useful is the
bilinear transformation.
To avoid aliasing of the frequency response as encountered with the impulse invariance transformation.
We need a one-to-one mapping from the s plane to the z plane.
The problem with the transformation is many-to-one.

63. Bilinear transformation method Contd…
We could first use a one-to-one transformation from to , which compresses the entire s plane into the strip
Then could be transformed to z by
with no effect from aliasing.
優點:To avoid aliasing of the frequency response
缺點:It is nonlinear between discrete-time frequency and continuous-time frequency.
s domain
s’ domain

64. Bilinear transformation method Contd…
Hence by using this equation a digital transfer function may be obtained

65. Bilinear transformation method Contd…
The discrete-time filter design is obtained from the continuous-time design by means of the bilinear transformation
Unlike the impulse invariant transformation, the bilinear transformation is one-to-one, and invertible.
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DISP Lab

66. Filter Realization

67. Filter Structures Direct form I Direct form II Cascaded form
Parallel form

68. Basic elements of digital filter structures
Adder has two inputs and one output.
Multiplier (gain) has single-input, single-output.
Delay element delays the signal passing through it by one sample. It is implemented by using a shift register. a
Block diagram
Signal-flow graph
z-1 a
z-1
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69. Copyright © 2005. Shi Ping CUCa1
z-1 a2 b0 a1 a2 b0
z-1 1 2 5 3 4
Copyright © Shi Ping CUC

70. Introduction
The major factors that influence the choice of a specific structure
Computational complexity
refers to the number of arithmetic operations (multiplications, divisions, and additions) required to compute an output value y(n) for the system.
Memory requirements
refers to the number of memory locations required to store the system parameters, past inputs, past outputs, and any intermediate computed values.
Finite-word-length effects in the computations
refers to the quantization effects that are inherent in any digital implementation of the system, either in hardware or in software.

71. IIR Filter Structures The characteristics of the IIR filter
IIR filters have Infinite-duration Impulse Responses
The system function H(z) has poles in
可参考教材P75～76
The order of such an IIR filter is called N if aN≠0

72. Direct form
In this form the difference equation is implemented directly as given. There are two parts to this filter, namely the moving average part and the recursive part (or the numerator and denominator parts). Therefore this implementation leads to two versions: direct form I and direct form II structures

73. Copyright © 2005. Shi Ping CUC
Direct form I b1 b2 b0
z-1
bM-1 bM a1 a2
z-1
aN-1 aN
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74. Copyright © 2005. Shi Ping CUC
Direct form II
For an LTI cascade system, we can change the order of the systems without changing the overall system response b1 b2 b0
z-1
bM-1 bM a1 a2
aN-1 aN
z-1
直接II型比直接I型节省存储单元（软件实现），或节省寄存器（硬件实现）
但不论是直接I型还是直接II型，其共同的缺点是系数对滤波器的性能控制作用不明显，这是因为它们与系统函数的零极点关系不明确因而调整困难。
另外，这两种结构极点对系数的变化过于敏感，从而使系统频率响应对系数的变化过于灵敏，也就是对有限精度（有限字长）运算过于灵敏，容易出现不稳定或产生较大误差。
Copyright © Shi Ping CUC

75. Copyright © 2005. Shi Ping CUC
Cascade form
In this form the system function H(z) is written as a product of second-order sections with real coefficients
Copyright © Shi Ping CUC

76. Parallel form Structures
In this form the system function H(z) is written as a sum of sections using partial fraction expansion. Each section is implemented in a direct form. The entire system function is implemented as a parallel of every section.
Suppose M=N
当M<N，但M不等于N时，见教材P207（式5－7）
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77. Copyright © 2005. Shi Ping CUC
Example
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78. Copyright © 2005. Shi Ping CUC
-12.9
z-1
7/8
-3/32
-14.75
26.82 1
-1/2
24. 5
并联型可以用调整a1k和a2k的方法来单独调整一对极点的位置，但是不能单独调整零点的位置。
此外，并联型结构中，各并联基本节的误差相互没有影响，所以比级联型的误差一般来说要稍小一些，因此在要求准确的传输零点的场合下，宜采用级联型结构。
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79. Conclusions Discussed about the FIR filter design IIR Filter design
Realization of structures

80. References
[1]B. Jackson, Digital Filters and Signal Processing, Kluwer Academic Publishers 1986
[2]Dr. DePiero, Filter Design by Frequency Sampling, CalPoly State University
[3]W.James MacLean, FIR Filter Design Using Frequency Sampling
[5]Maurice G.Bellanger, Adaptive Digital Filters second edition, Marcel dekker 2001
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DISP Lab

81. References
[6] Lawrence R. Rabiner, Linear Program Design of Finite Impulse Response Digital Filters, IEEE 1972
[7] Terrence J mc Creary, On Frequency Sampling Digital Filters, IEEE 1972
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DISP Lab

82. THANK YOU