1. Introduction to Digital Signal Processing
Byeong Gi Lee

2. Chapter1. INTRODUCTION to DSP
1.1 Analog vs. Digital
1.2 Applications
1.3 Why Digital?
1.4 Digital Signal Processing
1.5 Course Description
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3. Analog : voice, audio, video, ….
1.1. Analog vs. Digital
i) Signal
Analog : voice, audio, video, ….
Digital : digitized analog signal, data
ii) Processing
Analog : passive/active filtering
AM, FM, PM modulation
Fourier, Laplace transform
Digital : FIR/IIR filtering
AM, windowing
Discrete Fourier transform, z-transform
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4. 1.1. Analog vs. Digital (cont’d)
iii) System
Analog : R, L, C, Op-amp, switch, …
differential equation
Digital : adder, multiplier, memory, …
difference equation
iv) Theory
Circuit theory
DSP theory
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5. 1.2. Applications Information Processing system Recognition Signal
- radar, sonar, seismic, …
Storage Media
Storage
Processing system
Display
Transmission
Communications
Information
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6. 1.2. Applications (cont’d)
i) Processing
- filtering, modulation, transform, deconvolution
- A/D, D/A conversion, coding
ii) Storage
- LP, tape (analog)
- CD, DVD (digital)
iii) Transmission
- FDM, FDMA, TDMA (analog)
- TDM(PCM), CDMA (digital)
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7. - Environmental change!
1.3. Why Digital?
- Environmental change!
Global communication - noise immunity
Multimedia communication - integration
Networking - encryption, packetizing
Wireless, mobile - encryption, compression
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8. 1.4. Digital Signal Processing
Computer-aided approximation
Exact self-containing processing
Processing complexity
FAST-ENOUGH
Computing capability
Implementation means
* invention of FFT,
Cooley Tukey, 1965
DSP is realizable (real-time processing)
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9. It is time, time is matured for “DSP-only”!
1.4. Digital Signal Processing (cont’d)
It is time, time is matured for “DSP-only”!
Theoretical support - DSP theory
Environmental demand
Microelectronics support
- processing + storage + logic devices
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10. 1.5. Course Description
Objective : To study the theoretical fundamentals on Digital Signal Processing and the mathematical foundations for sampling, discrete-time Fourier Transform, filtering, fast computation techniques and confirm them through computer programming.
Text : Discrete-Time Signal Processing, 2nd ed., A. V. Oppenheim & R. W. Schafer, Prentice-Hall
Reference : Digital Signal Processing, 2nd ed., Sanjit K. Mitra, McGraw-Hill
Homepage : tsp.snu.ac.kr
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11. 1.5. Course Description (cont’d) Fall 2003
Week 1 Chap 1. Introduction, Chap 2. Discrete-time signals and systems Week 2 Chap 2. Discrete-time signals and systems (CHUSEOK) Week 3 Chap 3. z-transform Week 4 Chap 4. Sampling & Discrete- and continuous-time signal processing Week 5 Chap 5. Frequency response of LTI systems Week 6 Chap 5. All-pass and Minimum-phase system Week 7    Midterm (Univ. Anniversary, Student Festival) Week 8    Chap 6. Basic structure for LTI systems Week 9    Chap 6. FIR & IIR systems, Chap 7. FIR & IIR filter design Week 10  Chap 7. FIR & IIR filter design Week 11  Chap 8. Discrete Fourier Transform Week 12  Chap 8. Discrete Fourier Transform Week 13  Chap 9. Fast transform computation Week 14  Overall Review and Problem Solving (GLOBECOM) Week 15  Final Exam
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12. Chapter2. DISCRETE-TIME SIGNALS AND SYSTEMS
2.0 Introduction
2.1 Discrete-Time Signals : Sequences
2.2 Discrete-Time Systems
2.3 Linear Time-Invariant Systems
2.4 Properties of Linear Time-Invariant Systems
2.5 Linear Constant-Coefficient Difference Equations
2.6 Frequency-Domain Representation
2.7 Representation of Sequences of the Fourier Transform
2.8 Symmetry Properties of the Fourier Transform
2.9 Fourier Transform Theorems
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13. 2.1.Discrete - Time Signals
x[n]= x(t)|t=nT
n : -1,0,1,2,…
T: sampling period
x(t) : analog signal
i) unit impulse signal(sequence)
d[n] = , n=0
0, n0
ii) unit step sequence
u[n] = , n0
0, n0
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14. iii) exponential/sinusoidal sequence x[n]= Aej(won+), Acos(won+)
- not necessarily periodic in n with period 2p/wo
- periodic in n with period N (discrete number)
for woN=2pk or wo = 2pk/N
[note] x(t)= Ae j(wo t +)
is periodic in t with period T= 2p/wo (continuous value)
iv) general expression
x[n] = S x[k]d[n-k]
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15. 2.2Discrete-Time Systems
x[n]
y[n]
System : signal processor
i) memoryless or with memory
y[n] = f(x[n]), y[n]=f(x[n-k]) with delay
ii) linearity
x1[n]y1[n]
x2[n]y2[n]
a1x1[n] + a2x2[n] a1y1[n] + a2 y2[n]
- e.g. T[a1x1[n] + a2x2[n]] = T[a1x1[n]] + T[a2x2[n]]
= a1T[x1[n]] + a2T[x2[n]]
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16. x[n]  y[n]  x[n-n0]  y[n-n0]
iii) time-invariance
x[n]  y[n]  x[n-n0]  y[n-n0]
- e.g., T[x [n-n0]] = T[x [n]] | n  n-no
- e.g., d[n]  h[n]  d[n-k]  h[n-k]
- counter-example : decimator T[ ] = x[Mn]
iv) causality
y[n] for n=n1, depends on x[n] for nn1 only
- counter-example : y[n] = x[n+1] - x[n]
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17. bounded input yields bounded output(BIBO)
v) stability
bounded input yields bounded output(BIBO)
|x[n]| <  for all n  |y[n]| <  for all n
- counter-example : y[n] = S u[k] = 0, n<0
n+1, n0
unbounded ( no fixed value By exists that keeps
y[n]  By <  .) n
k=-
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18. = S x[k]T[d[n-k]] = S x[k]h[n-k] = x[n]*h[n] = S x[n-r]h[r]
2.3 Linear Time-Invariant Systems
d[n]
h[n]
LTI
T[d[n]] : impulse response
x[n]
y[n]
In general, let x[n] = S x[k]d[n-k] 
k=-
y[n] = T[ S x[k]d[n-k]] k
( by linearity)
= S x[k]T[d[n-k]] k
( by time-invariance)
= S x[k]h[n-k]
coefficient 
= x[n]*h[n]
= S x[n-r]h[r]
r=-
Convolution!
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19. S h[k] x[n-k] = S h[k] x[n-k] In summary, h[n] x[n] y[n]
y[n] = x[n]*h[n]
LTI
h[n] : unique characteristic of the LTI system
- causal LTI system  
S h[k] x[n-k]
k=0
y[n] =
S h[k] x[n-k] =
k=-
[note] h[n] = T[d[n]] = 0 n<0. as
d[n] = , n=0
0, n0
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20. S h[k] x[n-k] |  S | x[n-k] | • | h[k] | S | h[k] | S | h[k] |
- Stable LTI System
S h[k] x[n-k] |  
S | x[n-k] | • | h[k] | 
|y[n]| = |
k=-
k=- 
S | h[k] |
 Bx
 By
< 
k=- 
Therefore,
S | h[k] |
< 
k=-
In fact, this is necessary and sufficient condition for stability of a BIBO system.
( You prove it! )(*1)
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21. - Example of non-LTI system - Decimator
x[n]
Decimator M
y[n] = x[Mn] ? ~
x[n]
= x[n-1] ~
y[n]
= y[n-1] = x[M[n-1]]
M=3
y[n] = x[Mn] ~
y[n]
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22. y[n]
y[n-1] 
No!
= x[Mn-1]
 x[M[n-1]] = y[n-1] ~
y[n]
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23. 2.4 Properties of LTI System
h[n]
x[n]
y[n] = x[n]*h[n]
LTI
i) parallel connection
h[n] = h1[n] + h2[n]
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24. ii) cascade connection
h[n] = h1[n]*h2[n]
=h2[n]* h1[n]
[note] distinctive feature of digital LTI system (*2)
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25. S ak y[n-k] = S br x[n-r] = S br x[n-r]
2.5 Linear Difference Equations
LTI
x[n]
y[n] N
S ak y[n-k]
k = 0 M
= S br x[n-r]
r = 0
i) Case 1 : N=0  FIR System
(set a0 =1, for convenience)
y[n]
For impulse input, x[n]=d[n], the response is
h[n]= 0, n<0 or n>M
br 0 n  M finite impulse response! M
= S br x[n-r]
r = 0
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26. = S br x[n-r] S ak y[n-k] ii) Case 2 : N  0  IIR System
(set a0 =1, for convenience)
y[n]
e.g., set N=1 (lst order), and a1 = -a
 y[n] = b0 x[n] + ay[n-1] M
= S br x[n-r]
r = 0 N
S ak y[n-k]
k = 1 -
For impulse input x[n] = d[n], the response is
1) If assume a causal system, i.e., y[n]=0 n<0.
y[0] = b0 d[0] + ay[-1] = b0
y[1] = b0 d[1] + ay[0] = ab0
• • •
y[n] = b0 d[n] + ay[n-1] = anb0
h[n] = anb0u[n]
infinite impulse response!
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27. 2) If assume an anti-causal system, i.e., y[n]=0 n>0.
y[n-1] = a-1(-b0 d[n] + y[n])
y[0] = a-1(-b0 d[1] + y[1]) = 0
y[-1] = a-1(-b0 d[0] + y[0]) = a-1b h[n] = -anb0u[-n-1]
• • •
y[-n] = a-1(- b0 d[n+1] + y[n+1]) = -anb0
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28. A y[n] = S h[k] ejw(n-k) ( S h[k] e-jwk )ejwn =H(ejw) ejwn
2.6 Frequency-Domain Representation
• Linear System x
y = Ax A x ^ ^
y = Ax=lx
scalar, eigenvalue
for eigenvector input x ^
• LTI System
x[n]
y[n]=x[n]*h[n]
h[n]
ejwn
y[n]=ejwn*h[n]
= H(ejw)ejwn
Fourier transform 
y[n] = S h[k] ejw(n-k)
k = -  =
( S h[k] e-jwk )ejwn =H(ejw) ejwn
k = -
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29. S h[k] e-jwk • Fourier Transform H(ejw) = p h[k] = 1/2p H(ejw) ejwkdw
H(ejw) =
h[k] = 1/2p H(ejw) ejwkdw p -p
You prove this! (*3)
• Condition for existence of FT
| X(ejw) | <   S | x[n] | < 
“ absolutely summable”
(BIBO stable condition)
• Real - imaginary
H(ejw) = HR(ejw) + j HI(ejw)
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30. H(ejw) = | H(ejw) |ej H(ejw)
• Magnitude-phase
H(ejw) = | H(ejw) |ej H(ejw)
(e.g.) ideal delay system
x[n]
y[n] = x[n-nd]
h[n]
ejwn
y[n] = ejw(n-nd) = H(ejw) ejwn
H(ejw) = e-jwnd
HR(ejw) = coswnd HI(ejw) = - sinwnd
| H(ejw) | = 1  H(ejw) = -wnd
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31. x[n] = Acos(w0n + j) = (A/2) ejjejw0n + (A/2) e-jje-jw0n
(e.g.) sinusoidal input
x[n] = Acos(w0n + j) = (A/2) ejjejw0n + (A/2) e-jje-jw0n
y[n] = H(ejw0) (A/2) ejjejw0n + H(e-jw0) (A/2) e-jje-jw0n
= (A/2)(H(ejw0) ejjejw0n + H(e-jw0) e-jje-jw0n)
y[n] = H(ejw) ejwn
<1>
= (A/2){ (H(ejw0) ejjejw0n ) + (H(ejw0) ejjejw0n)* }
= ARe{H(ejw0) ejjejw0n}
= A | H(ejw0) |(cosw0n + j + q)
= A cos (w0(n-nd) + j)
<2>
<3>
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32. = ( S h*[n] e-jw0n)* = H*(ejw0)
<1> if h[n] real
h[n] = hR[n]+jhI[n] = hR[n] = h*[n]
H(e-jw0) =
S h[n] e-(-jw0n)
= ( S h*[n] e-jw0n)* = H*(ejw0) n n
<2> H(ejw) = | H(ejw) |ejq
<3> if ideal delay system
with | H(ejw) | = 1,  H(ejw) = q = -wnd
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33. (e.g.) ideal lowpass filter (LPF)
x[n]
y[n] 1
-wc wc
Hl(ejw) = e-jwnd |w| 
0. elsewhere wc
periodic with period 2p
Input x[n] = Acos(w0n + j)
output y[n] = Acos(w0(n-nd )+ j) , if
, otherwise wo
< wc
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34. = S ane-jwn = S (ae-jw)n (e.g.) Fourier transform of anu[n] |a|<1
X(ejw)  
= S ane-jwn = S (ae-jw)n =
n = 0
n = 0
(e.g.) inverse Fourier transform of ideal LPF
hl [k] = e-jwnd ejwndw wc
-wc
sinw0[n-nd ]
p[n-nd ] =
- <n < 
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35. - Data length vs. Spectrum Change
- Gibb’s phenomenon (page 52)
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36. - Error reduces in RMS sense but not in Chebyshev sense.
 Limitation of rectangular windowing
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37. 2.8 Symmetry Properties (table 2.1)
i) even / odd
e : conjugate symmetric  even
o : conjugate anti-symmetric  odd
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38. ii) real/imaginary
iii) conjugation/reversal S
= ( S )*
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39. iv) real/imaginary - even/odd
v) for real x[n]
real  even
imaginary  odd
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40. (e.g.) x[n] = anu[n] |a|<1, real (example 2.25)
even
odd
even
odd
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41. Dashed line : a = 0.5 Solid line : a = 0.9
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42. 2.9 Fourier Transform Theorems (table 2.2)
i) linearity
ii) time shifting
iii) frequency shifting
iv) time reversal
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43. v) differentiation in frequency
vi) Parserval’s relation
vii) convolution relation
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44. viii) modulation/windowing relation
ix) fundamental functions
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45. 1. 0.
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46. H.W. of Chapter 2
Matlab: [1] Consider the following discrete-time systems characterized by the difference equations:
y[n]=0.5x[n]+0.27x[n-1]+0.77x[n-2]
Write a MATLAB program to compute the output of the above systems for an input x[n]=cos(20n/256)+cos(200n/256), with 0n<299 and plot the output.
[2] The Matlab command y=impz(num,den,N) can be used to compute the first N samples of the impulse response of the causal LTI discrete-time system.
Compute the impulse response of the system described by
y[n]-0.4y[n-1]+0.75y[n-2]
=2.2403x[n] x[n-1] x[n-2]
and plot the output using the stem function.
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47. H.W. of Chapter 2 Text: [3]2-15 [4]2-30 [5]2-42 [6]2-56 [7]2-58
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48. Chapter4. Sampling of Continuous-Time Signals
1. A/D Conversion and Sampling
2. Reconstruction of a Bandlimited Signals
3. Continuous-Discrete Frequency Characteristics
4. Digital Processing of Continuous-Time Signals
5. Changing the Sampling Rate Using Discrete-Time Processing
6. Multirate Signal Processing
7. Practical Considerations in A/D and D/A Conversions
8. Multirate Processing for A/D and D/A Conversions
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49. 1. Analog-to-Digital Conversion and Sampling
A-to-D Conversion
D-to-A Conversion
Conti
Disc
DSP D C
Sampling
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50. Ideal C/D Conversion
Ideal D/C conversion
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51. Sampling <Input> <Impulse train> <Impulsed input>
Sampling frequency
<Impulsed input>
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52. Nyquist Sampling Theorem
<output> T If
is an ideal LPF
Nyquist Sampling Theorem
When xc (t) is bandlimited s.t.
If we do sampling by taking
Then xc (t) is uniquely reconstructed from x[n]= xc (nT)
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53. Illustration of Sampling1 t t
0 T ... t
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54. t
What if
Aliasing
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55. Illustration of aliasing
Sampling period
(note, to avoid aliasing, )
 (*1000)
Then the reconstructed output has component
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56. Original signal
Aliased signal
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57. 2. Reconstruction of Band-limited Signal
(note 1) T
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58. (note 1) t T 2T
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59. 3. Continuous/Discrete Frequency Characteristics
vs.
 
C/D
- (i)
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60. -(ii)
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61. (i)+(ii)
- (iii)
In summary, 1
normalized
frequency!
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62. 4. Digital Processing of Analog Signal
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63. or equivalently, take
Then,
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64. 5. Changing the Sampling Rate
Decimation
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65. (e.g) If set p 1 , w < H ( e ) = M , otherwise Then 1 Y ( e ) = X (j w ) = M ,
otherwise
Then 1 Y ( e j w ) = X ( e j w / M ), w
< p M
(e.g)
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66. Interpolation
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67. Anti- imaging filter If set p L , w < H ( e ) = L , otherwise Thenj w ) = L
Anti-
imaging
filter ,
otherwise
Then Y ( e j w ) = LX ( e j w L ), w
< p / L
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68. M/L Sampling Rate Conversion
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69. Illustration of Decimation and Interpolation
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70. 6. Multirate Signal Processing
- Interchange of Filtering and Up/Down-Sampling
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71. Polyphase Decomposition
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72. K K
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73. Polyphase Implementation of Decimation Filters
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74. Polyphase Implementation of Decimation Filters
Likewise ( you derive it! )
What is the computational saving?
1. Before : KM multiplications/sample-out
2. After : K multiplications/sample-out (1/M reduction!)
You show it!
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75. 7. Practical Considerations
(1) Prefiltering – Anti-aliasing filtering
- analog filter,
costly to implement sharp analog filters
 use over-sampling technique
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76. (2) A/D Conversion
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77. -Signal-to-Quantizing Noise Ratio
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78. (3) D/A Conversion
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79. BGL/SNU

80. 8. Multirate Processing for A/D and D/A Conversions
(1)Oversampling-Decimation based A/D Conversion
(2) Interpolation - Low-order Reconstruction based D/A conversion
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81. Oversampling-Decimation based A/D Conversion
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82. Interpolation - Low-order Reconstruction based D/AC
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83. 5.Linear Time-Invariant System
5.1 The Frequency Response of LTI systems
5.2 System Functions for Systems Characterized by Linear
Constant-Coefficient Difference
5.3 Frequency Response for Rational System Functions
5.4 Relationship between Magnitude and Phase
5.5 All-Pass Systems
5.6 Minimum-Phase Systems
5.7 Linear Systems with Generalized Linear Phase
5.8 Summary
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84. 5.1 Frequency Response (e.q) Frequency selective filter - ideal
( : delay,centerpoint of sync function)
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85. Group dalay
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86. 5.2 System Functions for Constant Coefficient Systems
Stable if
Causal if Roc includes
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87. Inverse system, stable if all poles and zeros, inside the uc
minimum-phase system
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88. -FIR vs IIR
FIR part
IIR part
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89. 5.3 Frequency Response of Rational System Functions
(note) arg : continuous phase
ARG : its principal value in
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90. Check how they change as r and vary.
(example)
Check how they change as r and vary.
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91. 5.4 Relationship between Magnitude and PhaseF F
(complex cepstrum of x[n])
from Eqs. (11.28) and (11.29) (pp. 781)
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92. ※Relationship between real part and imaginary part of complex sequence (single-side band)
single-side band sequence
(complex sequence) F or
Hilbert Transform
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93. Illustration of decomposition of a one-sided Fourier transform
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94. Inverse Hilbert transform
xr[n]
xr[n]
x[n]
Hilbert transformer
xi[n]
impulse response of an ideal Hilbert transformer
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95. 5.5 Allpass System
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96. - 2nd order allpass function
- Nth order allpass function ( real-coeff)
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97. 5.6 Minimum phase System
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98. BGL/SNU

99. Sequences all having the same frequency response magnitude
( zeros are at all combinations of 0.9ej0.6 and 0.8ej0.8 and their reciprocals)
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100. since
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101. - Frequency response Compensation
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102. 5.7 Generalized Linear Phase System
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103. BGL/SNU

104. BGL/SNU

105. BGL/SNU

106. BGL/SNU

107. BGL/SNU

108. BGL/SNU

109. Chapter 6. Structures for Discrete-Time Systems
6.1 Block Diagram Representation of Linear Constant-Coefficient Difference
Equations
6.2 Signal Flow Graph Representation of Linear Constant-Coefficient
Difference Equations
6.3 Basic Structures for IIR Systems
6.4 Transposed Forms
6.5 Basic Network Structures for FIR Systems
6.6 Overview of Finite-Precision Numerical Effects
6.7 The Effects of Coefficient Quantization
6.8 Effects of Round-off Noise in Digital Filters
6.9 Zero-Input Limit Cycles in Fixed-Point Realizations of IIR Digital Filters
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110. 1. Digital circuits vs. Analog circuits
(1) Expression
(2) Circuit Elements + -
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111. Discrete – time Continuous - time
DIGITAL
ANALOG
(3) Time - Domain
Discrete – time Continuous - time
(4) Transform - Domain
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112. Conversion via KCL, KVL and branch equations
DIGITAL
ANALOG
(5) Signal Flow Graph
Straightforward
Conversion via KCL, KVL and branch equations
( no delay-free loop allowed)
( no improper Source-sink )
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113. Examples
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114.  Mason’s rule for signal flow graph
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115. 2. Circuit Structure for IIR System
(1) Direct Form
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116. BGL/SNU

117. - Second order factored form - pole-zero pairing
(2) Cascade Form
- Second order factored form
- pole-zero pairing
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118. - Partial fraction expansion
(3) Parallel Form
- Partial fraction expansion
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119. - Reverse the flow of a structure,
(4) Transposed Form
- Reverse the flow of a structure,
then you will get the identical transfer function
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120. 3. Circuit Structures for FIR Systems
(1) Direct Form x ( n - N + 1 )
Z-1
Z-1
Z-1
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121. (3) Parallel Form (Frequency Sampling)
(2) Cascade Form
(3) Parallel Form (Frequency Sampling) 1 N - 1 2 p - j h ( n ) = H ( k ) W - kn
IDFT ( W = e N ) N N N k =
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122. BGL/SNU

123. (4) Linear Phase FIR Structure= h ( M - n ) h ( ) = h ( 1 ) = h ( M ) h ( M - 1 ) h ( ) = h ( M )
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124. (5) Linear Phase FIR Structure in Quad Form
zeros :
factors :
coefficients in A
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125.  IIR Structure of special interestB
coefficients in
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126. * Observation
grid denser
grid uniform
top and
bottom
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127. (1) Effects of pole-clustering
4. Effects of Coefficient Quantization
(1) Effects of pole-clustering
- after quantization
- sensitivity of (in denominator) on H(z) is larger
than that of :
Change poles changed denominator of H(z) changed
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128. Therefore, a small change in could cause a large change in
Let
Then
Therefore, a small change in could cause a large change in
when is close to , (or when poles are clustered), causing
a large change in H(z) or its frequency spectrum
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129. Effects of quantization on the zeros of a 27th order polynomial P(z)
Illustration:
Effects of quantization on the zeros of a 27th order polynomial P(z)
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130. (2) pole-zero pairing to realize low-sensitivity 2nd order cascade
Due to the quantization effect discussed above, it is desirable to
separate out closely located poles into different 2nd order blocks.
Jackson (1970, 1986) : How to pair poles and zeros
1) The pole that is closest to the unit circle should be paired
with the zero that is closest to it in the z-plane
2) This rule should be repeatedly applied until all the poles
and zeros have been paired
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131. 3) The resulting second order sections should be ordered
according to the closeness of the poles to the unit circle,
either in increasing closeness to the unit circle or in its
reverse order
a11
a21
b11
b21
(1)
(2)
(3)
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132. Illustration: a 12th order IIR bandpass filter
- with the cutoff frequencies at 0.3pi, 0.4pi
- stopband attenuation of -40dB
Cascaded Structure
Direct-form Structure
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133. (A) Arrangement 1: direct-form implementation
(3) Grid granularity and Filter Structure
(A) Arrangement 1: direct-form implementation
(A)
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134. Illustration: zero locations for direct-form implementation
4 bit quantization
7 bit quantization
Put Figure 6.42 (a) below
Put Figure 6.42 (b) below
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135. (B) Arrangement 2: coupled-form implementation
( Note) the cross points are
physically realizable pole points
-guantization moves a pole
from one crosspoint to another
-if poles are on the upper region
take arrangement 1 (structure (A)),
otherwise, take arrangement 2 (structure (B))
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136. Illustration: zero locations for coupled-form implementation
4 bit quantization
7 bit quantization
Put Figure 6.44 (a) below
Put Figure 6.44 (b) below
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137. Y(n) = ay(n-1) - x(n) -> y(n) = Q[ay(n-1)] - x(n)
5 Limit Cycle
Y(n) = ay(n-1) - x(n) -> y(n) = Q[ay(n-1)] - x(n)
y[n]
y[n]
x[n]
x[n] a a
z-1
Q[]
z-1
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138. Chapter 7. Filter Design Techniques
7.1 Introduction - Digital Filter Design
7.2 IIR Filter Design by Impulse Invariance
7.3 IIR Filter Design by Bilinear Transformation
7.4 FIR Filter Design by Windowing
7.5 Kaiser Window based FIR Filter Design
7.6 Approximation based Optimal FIR Filter Design
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139. 1. Introduction -- Digital Filter Design
(1) Frequency selective filters : spectral shapers
lowpass highpass bandpass bandstop
(2)Filter Design Techniques
IIR : - mapping from analog filters
- impulse invariance
FIR: - windowing
- equiripple design
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140. In some IIR filter design
(3)Filter Specification(LPF)
In some IIR filter design 1 2 3 1 2 3
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141. 2. IIR Filter Design by Impulse Invariance
(1) Design Concept
- Utilize existing analog filter design technique
- Convert analog impulse response into digital impulse
response h[n] by taking samples
- Take
Then by fundamental sampling property, we get
- If analog filter were bandlimited,
Then
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142. - But the issue is that the above assumption is not likely,
(2) Aliasing Problem
- But the issue is that the above assumption is not likely,
so aliasing is inevitable in reality
aliasing
Therefore this design technique is useful only when
designing a narrowband sharp lowpass filters
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143. (3) Parameter Conversion
- Let the analog filter has the partial-fraction expansion
- After sampling, h [ n ] = T h
(nT ) = d c d
Therefore,
pole at in is mapped to
Pole at
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144. * Impulse Invariance Filter Design Procedure
1) Given specification in domain.
2) Convert it into specification in domain
3) Design analog filter meeting the specification 
4) Convert it into digital filter function H(z)
by putting
[ 5) Implement it in 2nd order cascade form]
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145. Design Example 1 2
* Choose Td=1 3
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146. * You plot the pole locations in the z-plans!4
* You plot the pole locations in the z-plans!
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147. 3. IIR Filter Design by Bilinear Transformation
(1) Design Concept
- s-plane to z-plane conversion
any mapping than maps stable region is s-plane (left half plane)
to stable region in z-plane (inside u.c) ? or
bilinear transform!
* Td inserted for convention may put to any convenient value
for practical use.

148. (2) Properties

149. * IIR Filter Design Procedure
Given specification in digital domain
Convert it into analog filter specification
Design analog filter (Butterworth, Chebyshov, elliptic):H(s)
Apply bilinear transform to get H(z) out of H(s) 1 2 3 4 3 2 4 1

150. Design Example (Butterworth Filter)
Given specification 1 2
Specification Conversion
(Set Td=1)
Butterworth filter design 3 N c j H 2 ) / ( 1 | W + =

151. Bilinear Transform 4

152. *Comparison of Butterworth, Chebyshev, elliptic filters
-Filter equations 1
Butterworth filter B
Chebyshev filter (type I) C 1
Chebyshev polynomial

153. *Comparison of Butterworth, Chebyshev, elliptic filters (Cont’d)
Chebyshev filter (type II) 1
Elliptic filter E 1
Jacobian elliptic function

154. *Comparison of Butterworth, Chebyshev, elliptic filters: Example
-Given specification
-Order
Butterworth Filter : N=14. ( max flat)
Chebyshev Filter : N= ( Cheby 1, Cheby 2)
Elliptic Filter : N= ( equiripple) B C E

155. -Pole-zero plot (analog)B C1 C2 E
-Pole-zero plot (digital) B C1 C2 E
(14)
(8)

156. -Magnitude
-Group delay C1 B B E C1 E C2 C2

157. 4. FIR Filter Design by Windowing
(1) Design Concept
- Given a desired frequency response
evaluate
- Then, is
the
desired
filter
coefficients.
However, is
infinitely
long, so
not
practical.
Therefore, take a
finite
segment of , or
such
that
the
resulting
frequency
spectrum
fall in
the
given
specification.
This process of getting out of is
called Windowing

158. (2) Rectangular Windowing

159. W ( e j ( w ) )

161. But M cannot improve this. (due to Gibb’s phenomena).
(3) Design Point e ( w ) :
depends on
the
attenuation of
the
peak
sidelobe of W ( e j w ).
But M cannot improve this. (due to Gibb’s phenomena).
Therefore, once a specific window is given, is fixed. e ( w ) D w :
depends on
the
width of
main
lobe of W ( e j w ).
This can be improved by increasing M.

162. (4) Commonly Used Windows
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163. BGL/SNU

164. Frequency Spectrum of Windows
(a) Rectangular, (b) Bartlett, (c) Hanning, (d) Hamming, (e) Blackman , (M=50)
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165. 5. Kaiser Window based FIR Filter Design
(1) Design Concept ( I
: 0th order modified Bessel function)
targets at limited duration in time and energy concentration at
low frequency
- compromisable. (choose appropriate )
Performance comparable to Hamming window ( when )
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166. Frequency Spectrum of Kaiser Window
(a) Window shape, M=20, (b) Frequency spectrum, M=20, (c) beta=6
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167. (2) Determination of Filter Order ( Kaiser, 1974 )① ② ③
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168. - Design Example : ① ② ③ ④
(Note)

169. BGL/SNU

170. 6. Approximation based Optimal FIR Filter Design
(1) Design Concept
- Linear phase filters possess the property
- More Generally, constant delay filters have the expression
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171. - Approximation error
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172. - Approximation ( Chebyshev)
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173. (2) Type I Lowpass Filter case
- desired :
- approximation
- weighting
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174. - Error function

175. (3) Type II Lowpass Filter case
- desired (original) :
- approximation
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176. - Weighting (modified)
- Desired (modified)
- error function
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177. BGL/SNU

178. (4) Alternation Theorem
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179. BGL/SNU

180. BGL/SNU

181. (5) Parks-McClellan Algorithm① ② ③ ④
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182. Remez Exchange Algorithm (1934) (multiple exchange)⑤
Remez Exchange Algorithm (1934) (multiple exchange) ① ②
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183. ③
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184. ④ ⑤ ⑥
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185. Selection of new extrema
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186. Remez Exchange Algorithm
Initial guess of
(L+2) extremal frequencies
Remez Exchange Algorithm
Calculate the optimum
 On extremal set
Interpolate through (L+1)
Points to obtain Ae(ej)
Best approximation
Calculate error E()
And find local maxima
Where | E()|>= 
Unchanged
Check whether the
Extremal points changed
Changed
More than
(L+2) extrema?
Yes
Retain (L+2)
Largest extrema No
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187. Design Examples ①
Kaiser, (1974)
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188. DSP Sample & Hold Comp. Reconst. filter②
DSP
Sample
& Hold
Comp.
Reconst.
filter
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189. BGL/SNU

190. H.W. of Chapter 7 Due 11/17 (Mon.)
[1] Design a IIR lowpass filter whose specification is the same as that given in Example 7.3 (page 454) except the passband and stopband edges are shifted to 0.7pi and 0.8pi respectively, using bilinear transform technique.
Text : [2] [3] 7.3    [4] 7.17   
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191. H.W. of Chapter 7 Due 11/24 (Mon.) [1] Design a Length-21 FIR Filter
Use the MATLAB command h=remez(20, [0,0.4,0.5,1], [1,1,0,0]) to design a length-21 filter with a passband from 0 to wp=0.4pi and a stopband from ws=0.5pi to pi with a desired response of 1 in the passband and zero in the stopband. Plot the impulse response, the zero locations, and the amplitude response. How many “ripples” are there? How many extremal frequencies are there (places where the ripples are the same maximum size)? How many “small ripples” are there that do not give extremal frequencies, and if there are any, are they in the passband or stopband? Are there zeros that do not contribute directly to a ripple? Most zero pairs off the unit circle in the z-plane cause a maximum-size ripple in the passband or stopband. Some cause only a “small ripple,” and some cause no ripple.
Text : [2] 7.28     [3] 7.23   [6] 7.36
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192. Chapter 8. The Discrete Fourier Transform
8.1 Laplace, z-, and Fourier Transforms
8.2 Fourier Transform
8.3 Fourier Series
8.4 Discrete Fourier Transform (DFT)
8.5 Properties of DFS/DFT
8.6 DFT and z-Transform
8.7 Linear Convolution vs. Circular Convolution
8.8 Discrete Cosine Transform(DCT)
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193. 1. Laplace, z-, Fourier Transforms
Analog systems
(continuous time)
Digital Systems
(discrete time)
H(s)
H(z)
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194. Laplace transform -z-transform LHP inside u.c Fouier transforms
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195. 2. Fourier Transform (1) continuous aperiodic signals conti aper
aper conti
x(t) 1 t
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196. (2) Discrete aperiodic signals
conti per
aper discr
x(n) 1 t ω

197. 3. Fourier Series (1) continuous periodic signals discrete aper
per conti
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198. (2) discrete periodic signals (*Discrete Fourier Series)
X(t) 1 k t T
(2) discrete periodic signals (*Discrete Fourier Series)
discrete per
per discre
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199. x[n] 1
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200. 4. Discrete Fourier Transform (DFT)
-For a numerical evaluation of Fourier transform and its
inversion, (i.e,computer-aided computation), we need discrete
expression of of both the time and the transform domain data.
-For this,take the advantage of discrete Fourier series(DFS, on page
4), in which the data for both domain are discrete and periodic.
discrete periodic
periodic discrete
-Therefore, given a time sequence x[n], which is
aperiodic and discrete, take the following approach.
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201. Mip
Top
Top
Mip
DFS
DFT
Reminding that, in DFS
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202. Define DFT as
(eq)
X[k]
x[n] 1 k n N N
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203. Graphical Development of DFT

204. DFS
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205. DFT
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206. 5. Property of DFS/DFT (8.2 , 8.6) (1) Linearity (2) Time shift
(3) Frequency shift
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207. (4) Periodic/circular convolution in time
(5) Periodic/circular convolution in frequency
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208. (6) Symmetry
DFS
DFT
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209. 6. DFT and Z-Transform (1) Evaluation of from
①If length limited in time, (I.e., x[n]=0, n<0, n>=N)
then
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210. ② What if x[n] is not length-limited? then aliasing unavoidable.… … … … … …

211. (2) Recovery of [or ] from (in the length-limited case)
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212. BGL/SNU

213. 7. Linear Convolution vs. Circular Convolution
(1) Definition
① Linear convolution
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214. Rectangular window of length N
② Circular convolution N
Rectangular window of length N
Periodic
convolution N
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215. (2) Comparison N
H[n] 2N 2N
Omit chap. 8.7

216. Test signal for computing DFT and DCT
8. Discrete cosine transform (DCT)
Definition
- Effects of Energy compaction
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Test signal for computing DFT and DCT

217. (a) Real part of N-point DFT; (b) Imaginary part of N-point DFT; (c) N-point DCT-2 of the test signal
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218. Comparison of truncation errors for DFT and DCT-2
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219. Appendix: Illustration of DFTs for Derived Signals
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220. BGL/SNU

221. BGL/SNU

222. BGL/SNU

223. BGL/SNU

224. Chapter 9. Computation of Discrete Fourier Transform
9.1 Introduction
9.2 Decimation-in-Time Factorization
9.3 Decimation-in-Frequency Factorization
9.4 Application of FFT
9.5 Fast Computation of DCT
9.6 Matrix Approach
9.7 Prime Factor Algorithm
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225. 1. Introduction
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226. BGL/SNU

227. BGL/SNU

228. - Example of fast computation
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229. BGL/SNU

230. 2. Decimation-in-Time Factorization
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231. BGL/SNU

232. BGL/SNU

233. BGL/SNU

234. Decimation-in-time FFT flow graphs
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235. Decimation-in-time FFT flow graphs
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236. Decimation-in-time FFT flow graphs
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237. 3. Decimation-in-frequency Factorization
(Sande- Tuckey) FFT
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238. g(n) h(n) g[0] x[0] X[0] g[1] g[2] g[3] h[0] h[1] h[2] h[3] x[1] x[2]

239. Final flow graph x[0] X[0] BGL/SNU x[1] x[2] x[3] x[4] x[5] x[6] x[7]

240. -Remarks
# Stages
# butterflies
# computations
inplace computations
output data ordering : bit-reversed
-Question
The flow graph for D-I-F is obtained by reversing.
The direction of the flow graph for D-I-T. Why?
-Omit Sections
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241. 4. Applications of FFT (1) Spectrum Analysis -
is the spectrum of x[n] , n=0,1,…,N-1
- Inverse transform can be done through the same mechanism
i) Take the complex conjugate of X[k]
ii) Pass it through the FFT process,
But with one shift right(/2) operation at each stage
iii) Finally, take the complex conjugate of the result
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242. - Operation reduction :
(2) Convolution ( Filtering )
h[n]
x[n] y[n]
N N N
x[n]
N n
h[n]
N n
#computation(multi)?
1+2+…+N+N-1+…+1+0
=N2
y[n
N n

243. -Utilize FFT of 2N-point~
h[n]
N N n
x[n]
N N n
y[n]
N-2 2N n
R2N[n] 1
N-1 n ~
2N-pt DFTs
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244. X[k]
2N-pt
FFT
x[n]
Y[k]
2N-pt
IFFT
y[n]
2N-pt
FFT
h[n]
H[k] 2N
# operation (multi)
operation reduction :
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245. (3) Correlation /Power Spectrum
2N-point DFTs
# Operation :
Power spectrum P[k] = X[k] X*[k]
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246. $ Comparison of # computation
Direct
Computation
106 1M
250k
105
FFT-based
Convolution
Correlation
62.5k
104
35k
16k
16k
7.25k
FFT
103
3.3k 5k 2k 1k
102
0.45k 10 1 N
512
1024
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247. 5. Fast Computation of DCT
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248. - Example: Lee’s Algorithm (1984, IEEE Trans , ASSP, Dec) 1D x[0]
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249. - Example: 2D DCT Algorithm (1991, N.I.Cho and S.U.Lee)
Separable Transform
NxN 2D DCT = N 1-D DCT into row direction followed by
N 1-D DCT into column direction.
Totally 2N 1-D DCT (each N-point) are required.
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250. Fast Algorithm reduces the number of 1-D DCTs into N.
By using the trigonometric properties, 2D DCT is
decomposed into 1-D DCTs.

251. Signal flow graph of 2-D DCT
8x8 DCT
4x4 DCT

252. 6. Matrix Approach
· Decimation-in-time
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253.    
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254. · Decimation-in-frequency 
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255. · General expression for N=2 case

256. · Extension to general N (Cooley/Tuckey)
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257. · # computations (complex)
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258. 7. Prime Factor Algorithm (Thomas/Good)
(1) Basics from Number Theory
 Euler’s Phi function
 Euler’s Theorem If ( a , N ) = 1 ,
then a f ( N ) = 1
mod N . ( eg ) a = 5 , N = 6 , f ( N ) = 2 , a f ( N ) = 25 = 1
mod 6
 Chinese Remainder Theorem (CRT)

259.  Second Integer Representation (SIR)
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260. BGL/SNU

261. (2) Prime Factor Algorithm
Set
Then
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262. Therefore
Note that the only difference is in the “twiddle factor”
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263. (3) Comparison Example 12-Point DFT (N=12, p=3, q=4)
C/T : Cooley/Tuckey T/G : Thomas/Good
· Transform
· Index Mappings

264. · Diagram X ( k , k ) (0,0) (0,0) (0,0) 3 3 (1,0) (0,1) (0,1) 4 4 6 61 ) , ( n k x 1 2 ) , ( k x X ( k , k )
(0,0)
4pt DFT
3pt DFT 1
(0,0)
(0,0) 3 3
(1,0)
(0,1)
(0,1) 4 4 6 6
(2,0)
(0,2) 9 9
(3,0)
(0,3)
(0,2) 8 8
3pt DFT
(1,0) 1 9
(1,1) 5 1 4 1
(0,1)
4pt DFT
(1,0)
(1,2) 9 5 7 4
(1,1)
(1,1) 10 7
(2,1)
(1,2)
3pt DFT
(2,0) 2 6 1 10
(3,1)
(1,3)
(2,1) 6 10
(2,2) 10 2 8 2
(0,2)
4pt DFT
(2,0)
3pt DFT
(3,0) 3 3 11 5
(1,2)
(2,1)
(3,1) 7 7 2 8
(2,2)
(2,2) 5 11
(3,2)
(2,3)
(3,2) 11 11
T/G
C/T
C/T
C/T
T/G
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265. Radix-4 algorithm - Radix-2 algorithms: algorithms in textbook :
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266. - Radix-4 butterfly
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267. - Radix-4 butterfly -j -1 j -1 1 -1 j -1 -j
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