Bài giảng Digital signal processing - Chapter 0: Introduction - Nguyen Thanh Tuan

Click to edit Master subtitle style Nguyen Thanh Tuan, M.Eng. 
Department of Telecommunications (113B3) 
Ho Chi Minh City University of Technology 
Email: nttbk97@yahoo.com 
 Introduction 
 Chapter 0 
Digital Signal Processing 
 A signal is defined as any physical quantity that varies with time, 
space, or any other independent variable(s). 
1. Signal and System 
2 
 Speech, image, video and electrocardiogram signals are information-bearing 
signals. 
Mathematically, we describe a signal as a function of one or more 
independent variables. 
 Examples: 
( ) 110sin(2 50 )x t t 
2( , ) 3 2 10I x y x xy y 
 A system is defined as a physical device that performs any operation 
on a signal. 
 A filter is used to reduce noise and interference corrupting a desired 
information-bearing signal. 
Introduction 
Digital Signal Processing 
 Signal processing is to pass a signal through a system. 
1. Signal and System 
3 
 A digital system can be implemented as a combination of 
hardware and software (program, algorithm). 
Introduction 
Digital Signal Processing 
Multichannel and Multidimensional signals 
2. Classification of Signals 
4 
 Signals which are generated by multiple sources or multiple sensors 
can be represented in a vector form. Such a vector of signals is 
referred to as a multichannel signals 
 Ex: 3-lead and 12-lead electrocardiograms (ECG) are often used in practice, 
which results in 3-channel and 12-channel signals. 
 A signal is called M-dimensional if its value is a function of M 
independent variable 
 Picture: the intensity or brightness I(x,y) at each point is a function of 2 
independent variables 
 TV picture is 3-dimensional signal I(x,y,t) 
Introduction 
Digital Signal Processing 
Continuous-time versus discrete-time signal 
2. Classification of Signals 
5 
 Signals can be classified into four different categories depending on 
the characteristics of the time variable and the values they take. 
Introduction 
 Time 
Amplitude 
 Continuous Discrete 
Continuous 
 Analog signal 
 Discrete signal 
Discrete 
 Quantized signal 
 Digital signal 
t 
x(t) 
n 
x(n) 
n 
xQ(n) 
000 
001 
010 
011 
100 
101 
110 
111 
t 
xQ(t) 
Digital Signal Processing 
3. Basic elements of a DSP system 
6 
Most of the signals encountered in science and engineering are 
analog in nature. To perform the processing digitally, there is a need 
for an interface between the analog signal and the digital processor. 
Fig 0.1: Analog signal processing 
Fig 0.2: Digital signal processing 
Introduction 
Xử lý tín hiệu số Xử lý số tín hiệu 
Digital Signal Processing 
 Telephony: transmission of information in 
digital form via telephone lines, modem 
technology, mobile phone. 
4. DSP applications-Communications 
7 Introduction 
 Encoding and decoding of the 
information sent over physical 
channels (to optimize 
transmission, to detect or 
correct errors in transmission) 
Digital Signal Processing 
4. DSP applications-Radar and Sonar 
8 Introduction 
 Target detection: 
position and 
velocity estimation 
 Tracking 
Digital Signal Processing 
 Analysis of biomedical signals, diagnosis, patient monitoring, 
preventive health care, artificial organs. 
4. DSP applications-Biomedical 
9 Introduction 
 Examples: 
 Electrocardiogram (ECG) signal provides 
information about the condition of the 
patient’s heart. 
 Electroencephalogram (EEG) signal 
provides information about the 
activity of the brain. 
Digital Signal Processing 
Noise reduction: reducing 
background noise in the 
sequence produced by a sensing 
device (a microphone). 
4. DSP applications-Speech 
10 Introduction 
 Speech recognition: 
differentiating between various 
speech sounds. 
 Synthesis of artificial speech: 
text to speech systems. 
Digital Signal Processing 
 Content based image retrieval: 
browsing, searching and retrieving 
images from database. 
4. DSP applications-Image Processing 
11 Introduction 
 Image enhancement 
 Compression: reducing the 
redundancy in the image data to 
optimize transmission/storage 
Digital Signal Processing 
 Generation, storage and transmission 
of sound, still images, motion 
pictures. 
4. DSP applications-Multimedia 
12 Introduction 
 Digital TV 
 Video conference 
Digital Signal Processing 
The Journey 
13 Introduction 
“Learning digital signal processing is not 
something you accomplish; 
it’s a journey you take”. 
R.G. Lyons, Understanding Digital Signal Processing 
Digital Signal Processing 
5. Advantages of digital 
 over analog signal processing 
14 
 A digital programmable system allows flexibility in reconfiguring 
the DSP operations simply by changing the program. 
 A digital system provides much better control of accuracy 
requirements. 
 Digital signals are easily stored. 
 DSP methods allow for implementation of more sophisticated 
signal processing algorithms. 
 Limitation: Practical limitations of DSP are the quantization errors 
and the speed of A/D converters and digital signal processors -> 
not suitable for analog signals with large bandwidths. 
Introduction 
Digital Signal Processing 
Course overview 
15 Introduction 
 Chapter 0: Introduction to Digital Signal Processing (3 periods) 
 Chapter 7: Fourier transform and FFT algorithm (6 periods) 
 Chapter 1: Sampling and Reconstruction (6 periods) 
 Chapter 3: Analysis of linear time invariant systems (LTI) (6 periods) 
 Chapter 4: Finite Impulse Response and convolution (3 periods) 
 Chapter 5: Z-transform and its applications (6 periods) 
 Chapter 6: Transfer function and filter realization (3 periods) 
 Chapter 8: FIR and IIR filter designs (6 periods) 
 Chapter 2: Quantization (3 periods) 
 Review and mid-term exam: 3 periods 
Digital Signal Processing 
 Text books: 
[1] S. J. Orfanidis, Introduction to Signal Processing, Prentice-
Hall Publisher 2010. 
[2] J. Proakis, D. Manolakis, Digital Signal Processing, Macmillan 
Publishing Company, 1989. 
References 
16 Introduction 
 Reference books: 
[3] V. K. Ingle, J. Proakis, Digital Signal Processing Using Matlab, 
Cengage Learning, 3 Edt, 2011. 
Digital Signal Processing 
Learning outcomes 
17 Introduction 
 Understand how to convert the analog to digital signal 
 Be able to design and implement FIR and IIR filters. 
 Have a thorough grasp of signal processing in linear time-invariant 
systems. 
 Understand the z-transform and Fourier transforms in analyzing the 
signal and systems. 
Digital Signal Processing 
Assessment 
18 Introduction 
Mid-term test: 20% 
 Homework: 20% 
 Final exam: 60% 
 Bonus: added to 
Test and Homework 
Test and 
Homework 
(40%) 
Final 
exam 
(60%) 
Final 
Mark 
(100%) 
0.0 7.5 4.50 4.5 
2.5 6.0 4.60 4.5 
3.0 6.0 4.80 5.0 
4.0 5.5 4.90 5.0 
5.5 4.5 4.90 5.0 
6.0 4.0 4.80 5.0 
7.0 3.5 4.90 5.0 
7.5 3.0 4.80 5.0 
7.0 3.0 4.60 4.5 
10.0 2.5 5.50 2.5 
10.0 4.00 Absent 
Digital Signal Processing 
Assessment 
19 Introduction 
Điểm ghi trên Bảng điểm kiểm tra, Bảng điểm 
thi và Bảng điểm tổng kết được làm tròn đến 
0,5. (từ 0 đến dưới 0,25 làm tròn thành 0; từ 0,25 
đến dưới 0,75 làm tròn thành 0,5; từ 0,75 đến 
dưới 1,0 làm tròn thành 1,0) 
Nếu điểm thi nhỏ hơn 3 và nhỏ hơn điểm tổng 
kết tính từ các điểm thành phẩn (kể cả điểm thi) 
thì lấy điểm thi làm điểm tổng kết. 
Digital Signal Processing 
Timetable 
20 Introduction 
Time Class 
Monday 
(T1-3) 
DD13BK01-A02 
314B1 
Tuesday 
(T7-9) 
DD13KSTD 
206B1 
Wednesday 
(T10-12) 
DD13LT04-A04 
303B1 
Digital Signal Processing 
Review of complex number 
21 Introduction 
cosx r  
siny r  
 Rectangular form: 
 Real part: 
 Imaginary part: 
 Polar form: 
 Absolute value (modulus, magnitude): 
 Argument (angle): 
cos sinie i    Euler’s formula: 
ire  z r  
x iy z
2 2| |r x y z
1arg( ) tan
y
x
 z
Cartesian 
coordinates 
Polar 
coordinates 
Argand diagram 
(−π , π] 
Digital Signal Processing 
Review of periodic signals 
22 Introduction 
 Definition: x(t) = x(t + T) t 
 Fundamental period (cycle duration): smallest T 
Ordinary frequency: f = 1/T (cps or Hz) --> F 
 Radial (angular) frequency:  = 2 f (rad/s) -->  
Digital Signal Processing 
Review of special functions 
23 Introduction 
 Rectangular (rect) 
 Unnormalized: 
 Normalized: 
 Cardinal sine (sinc) 
Digital Signal Processing 
Review of special functions 
24 Introduction 
Dirac delta: 
 Properties: 
Digital Signal Processing 
Review of special functions 
25 Introduction 
Dirac comb (impulse train, sampling function): 
 Properties: 
Digital Signal Processing 
Review of spectral analysis 
26 Introduction 
 Periodic signal: Fourier series (line spectrum) 
 Aperiodic signal: Fourier transform 
Digital Signal Processing 
Review of Fourier transforms 
27 Introduction 
0 0 0
1
cos(2 ) [ ( ) ( )]
2
FTF t F F F F   
0 0 0
1
sin(2 ) [ ( ) ( )]
2
FTF t j F F F F   
Digital Signal Processing 
Review of Fourier transform properties 
28 Introduction 
 Linear (superposition): 
Delay: 
 Convolution: 
Digital Signal Processing 
Review of trigonometric formulas 
29 Introduction 
1
cos( )cos( ) [cos( ) cos( )]
2
a b a b a b 
1
sin( )sin( ) [cos( ) cos( )]
2
a b a b a b 
1
sin( )cos( ) [sin( ) sin( )]
2
a b a b a b 
Digital Signal Processing 
Review of Poisson summation formula 
30 Introduction 
 Statement: 
 Condition: 
Digital Signal Processing 
Review of convolution and correlation 
31 Introduction 
 Convolution: 
 Correlation: 
 Auto-correlation: 
Digital Signal Processing 
Review of analog linear time-invariant system 
32 Introduction 
0( ) cos(2 )x t A F t  
( )x t Analog LTI system 
h(t) 
H(F) 
( ) ( ) ( )y t x t h t 
( )X F ( ) ( ) ( )Y F X F H F 
 Linear: 
 Time-invariant: 
 Impulse response: 
 Frequency response: 
 Amplitude (magnitude): |H(F)| 
 Phase: arg{H(F)} 
0 0 0( ) | ( ) | cos(2 arg{ ( )})y t A H F F t H F  
Digital Signal Processing 
Review of analog filters 
33 Introduction 
 Decibel: |A|dB = 20log10|A| 
 Logarithmic scales: 
 Decade: decades = log10(F2/F1) 
 Octave: octaves = log2(F2/F1) 
 Cut-off (-3dB) frequency 
 Bandwidth 
Digital Signal Processing 
Example of octave scale 
34 Introduction 
 An 88-key piano in twelve-tone equal temperament, with the octaves 
numbered and Middle C (cyan) and A440 (yellow) highlighted. 
C D E F G A B 
Digital Signal Processing 
Bonus 1 
35 Introduction 
Write a program generating tones of an 88-key piano in twelve-tone 
equal temperament with A440 standard. 
Digital Signal Processing 
Bonus 2 
36 Introduction 
Write a program generating tones of a guitar with standard below. 
Digital Signal Processing 
Bonus 3 
37 Introduction 
Write a program plotting the waveform of signal below. 
Digital Signal Processing 
Bonus 4 
38 Introduction 
Write a program plotting the spectrum of signal below. 
Digital Signal Processing 
Greek alphabet 
39 Introduction 
Digital Signal Processing 
Portraits of Scientists and Inventors 
40 Introduction 
 René Descartes (1596-1650): French philosopher, mathematician 
and scientist. “Cogito, ergo sum” (“Tôi tư duy, vậy tôi tồn tại”). 
 Jean-Robert Argand (1768-1822): French amateur mathematician. 
 Jean-Baptiste Joseph Fourier (1768-1830): French mathematician 
and physicist. 
 Siméon Denis Poisson (1781-1840): French mathematician, 
geometer, and physicist. 
Digital Signal Processing 
Portraits of Scientists and Inventors 
41 Introduction 
Heinrich Rudolf Hertz (1857-1894) was a German physicist who 
first conclusively proved the existence of electromagnetic waves. 
 Alexander Graham Bell (1847-1922) was an eminent Scottish-
born scientist, inventor, engineer and innovator who is credited with 
inventing the first practical telephone. 
Digital Signal Processing 
Homework 1 
42 Introduction 
 For each case below, find the modulus and argument (both in radian 
and degree): 
1) –2 
2) –3i 
3) –2 – 3i 
4) –2 + 3i 
5) 2 – 3i 
6) 1/(2 – 3i) 
7) (2 – 3i)/i 
8) (2 – 3i)^2 
9) (2 – 3i) + 1/(2 – 3i) 
10) (2 – 3i).(–2 – 3i) 
11) (2 – 3i)/(–2 – 3i) 
12) (2 – 3i)/( 2 + 3i) 
Digital Signal Processing 
Homework 2 
43 Introduction 
 For each case below, find the modulus and argument (both in radian 
and degree): 
1) e^(i ) 
2) e^(i /2) 
3) e^(–i /2) 
4) e^(i /4) 
5) e^(i /2) + e^(i /4) 
6) 1/e^(i /4) 
7) e^(i /4) / e^(–i /4) 
8) e^(i /4) + e^(–i /4) 
9) e^(i /4) – e^(–i /4) 
10) 1 + e^(i /2) 
11) 1 – e^(i /2) 
12) (2 – 3i). e^(i /4) 
Digital Signal Processing 
Homework 3 
44 Introduction 
 For each case below, sketch the locus of z on the complex plane: 
1) |z| = 1 
2) |z – 2| = 1 
3) |z – 1| = 2 
4) |z – 1 – 2i| = 3 
5) |z| < 3 
6) |z| > 2 
7) 2 < |z| < 3 
8) |z -1| < 4 
9) |z -1| > 2 
10) 2 < |z -1| < 4 
11) z + z -1 ≠ ∞ 
12) 1 + z -2 ≠ ∞ 
Digital Signal Processing 
Homework 4 
45 Introduction 
 For each case below, sketch the waveform of the signal: 
1) x(t) = 4sin(2t) (t:s) 
2) x(t) = 4sin(2 t) (t:s) 
3) x(t) = 4cos(2 t) (t:s) 
4) x(t) = 4cos(10 t) (t:s) 
5) x(t) = 4cos(10 t) (t:ms) 
6) x(t) = 1 + 4cos(10 t) (t:s) 
7) x(t) = 4cos(2 t) + 4cos(10 t) (t:s) 
8) x(t) = 4sin2(2 t) (t:s) 
9) x(t) = 4sinc(2t) (t:s) 
10) x(t) = 4{(t – 3)/2} 
11) x(t) = k{4{(t – k5 – 3)/2}} 
12) x(t) = 4(t – 3) – 3(t + 4) 
Digital Signal Processing 
Homework 5 
46 Introduction 
 For each case below, plot the magnitude spectrum of the signal: 
1) A 
2) A.cos(2 Ft+) 
3) A.cos(2 Ft+) + B 
4) A.cos(2 F1t+1) + B.cos(2 F2t+2) 
5) A.cos(2 Ft+1) + B.cos(2 Ft+2) 
6) A.cos(2 Ft+1) + A.cos(2 Ft+2) 
7) A.cos(2 Ft+) + A.sin(2 Ft+) 
8) x(t) = 10 – 4cos6 t (t: ms) 
9) x(t) = 1 – 2cos6 t + 3sin14 t (t: ms) 
10) x(t) = 3cos103πt – 4sin104πt (t: s) 
11) x(t) = 14sin23 t + 3sin14 t (t: ms) 
12) x(t) = 4cos22πt – 10sin10πt (t: ms) 
Digital Signal Processing 
Homework 6 
47 Introduction 
 Suppose a filter has magnitude response as shown in figure below. 
Determine the expression (ignoring the phase) of the output signal 
and plot it’s magnitude response for each case of the input signal: 
1) x(t) = 2 
2) x(t) = 2cos(2 t) (t:ms) 
3) x(t) = 2cos(20 t) (t:ms) 
4) x(t) = 2cos(200 t) (t:ms) 
5) x(t) = 2cos(400 t) (t:ms) 
6) x(t) = 2cos2(400 t) (t:ms) 
7) x(t) = 2cos(200 t).sin(400 t) (t:ms) 
8) x(t) = 2cos(200 t) – 2cos(400 t) (t:ms) 
9) x(t) = 2cos(200 t) + 2sin(400 t) (t:ms) 
10) x(t) = 2cos(200 t) + 2sin(200 t) (t:ms) 
Digital Signal Processing 
Homework 7 
48 Introduction 
 Cho hệ thống tuyến tính bất biến có hàm truyền H(f) như hình: 
a) Xác định biểu thức đầy đủ của tín hiệu ngõ ra y(t) khi tín hiệu ngõ 
vào x(t) = 10cos2@πt – 30sin40πt (t:s). 
b) Xác định biểu thức đầy đủ của tín hiệu ngõ vào x(t) để tín hiệu 
ngõ ra y(t) = 10cos2@πt (t:s). 
Digital Signal Processing 
Homework 8 
49 Introduction 
 Cho các tín hiệu tương tự x1(t) = 2cos
22πt (t: s) và x2(t) = 6sin6πt 
+ 7cos7πt + 8sin8πt (t:s) lần lượt đi qua hệ thống tuyến tính bất 
biến có hàm truyền H(f) như hình: 
a) Xác định biểu thức (theo thời gian) của tín hiệu ngõ ra y1(t). 
b) Tính giá trị của tín hiệu ngõ ra y2(t = 0.125s). 
Digital Signal Processing 
Homework 9 
50 Introduction 
 Tìm giá trị đáp ứng biên độ |H(f)| tại các tần số sau: 
a) 1KHz. 
b) 3KHz. 
c) 4KHz. 
d) 5KHz. 
e) 8KHz. 
Digital Signal Processing 
Homework 10 
51 Introduction 
 Cho bộ lọc thông thấp có đáp ứng biên độ phẳng 0dB trong 
khoảng [0  4]KHz, suy giảm với độ dốc 12dB/octave trong 
khoảng [4  8]KHz và suy giảm với độ dốc 20dB/decade ngoài 
8KHz. Tìm giá trị đáp ứng biên độ của bộ lọc tại các tần số sau: 
a) 2KHz. 
b) 3KHz. 
c) 5KHz. 
d) 6KHz. 
e) 7KHz. 
f) 8KHz. 
g) 10KHz. 
h) 12KHz. 
i) 16KHz. 
j) 20KHz.