signal and system ch1-1 for students of NCNU EE

Signals and Systems
Jung-Shan LIN
Department of Electrical Engineering
National Chi Nan University
112 Second Semester

Discrete
Continuous
Periodic
Non-periodic
Fourier Transform (Chap. 4)
Fourier Series (Chap. 3)
Laplace Transform (Chap. 9)
Periodic
Non-periodic
Discrete-Time Fourier Transform
(Chap. 5)
Discrete-time Fourier Series
(Discrete Fourier Transform)
(Sec. 3-6)
Z-Transform (Chap. 10)
Transform
Methods
Time and Frequency characterization (Chap. 6)
Basic Concepts
Signals and Systems (Chap. 1)
Linear Time-Invariant Systems (Chap. 2)
Applications
Sampling (Chap. 7)
Communication (Chap. 8)
System Analysis
State-Space Model (Chap. 11)
Linear Feedback System (Chap. 12)
Structure of this
textbook

Chapter 1 Signals and Systems
Signal: Any variable that carries information
System: It processes input signals to produce output signals
Example: image, speech, music, video, temperature, pressure, GDP, light,
population, electrocardiography (ECG, from heart beat), …………….
System
Output
Signal
Input
Signal

Sec. 1.1 Continuous-Time & Discrete-Time Signals
Key concepts (i) continuous and discrete signals; (ii) energy; (iii) average power
signals
Discrete-time (DT)
signals
Continuous-time
(CT) signals
1.1.1 Examples and Mathematical Representation

For a continuous-time (CT) signal, the independent variable is always
enclosed by a parenthesis ()
Example: x(t), y(t), z(t), A(x, y), …….
For a discrete-time (DT) signal, the independent variable is always
enclosed by a brackets []
Example: x[n], y[n], z[n], A[m, n], …….

• Examples of continuous signals
(1) Circuits (voltage, current, electric charge)
(2) Motion (location, velocity, acceleration)

• Examples of continuous signals
(3) Speech (acoustic pressure)
(4) Others (temperature, pressure, light, ECG, …….)
Vowel: larger power
Consonant: smaller power

• Examples of discrete signals
(1) A monochromatic picture
117 151 112 136 110 95 166 131 125 159 114 130 159 133
164 209 173 171 168 126 225 189 184 219 172 180 206 188
154 151 143 174 156 160 186 179 163 193 167 161 167 168
173 179 172 182 195 181 187 205 183 205 196 186 180 193
155 188 149 145 181 131 166 185 166 181 164 164 166 171
176 176 113 178 169 159 177 184 171 182 148 159 174 164
162 198 141 164 202 149 181 142 176 182 189 136 165 176
143 143 148 127 132 97 177 152 160 74 163 119 162 143
62 51 40 32 95 82 28 39 75 35 60 58 77 28
26 67 14 34 3 49 22 69 48 26 81 67 49 91
176 134 129 97 185 120 212 160 173 139 197 166 207 192
243 255 250 255 254 242 242 255 254 222 255 253 239 252
248 255 226 238 255 255 249 255 255 255 230 252 255 255
243 238 255 253 245 247 255 238 255 244 255 237 249 241
white → 255
black → 0

(2) Stock market index
The weekly Dow-Jones stock market index

(3) Color images, digital video ……
(4) Samples of a continuous
Example: digital speech signal
the population recorded every month / every year
the temperature recorded every hour / every day

• Dimensionality of a Signal
Speech signal, circuit, voltage are one-dimensional (1D) signal
A monochromatic picture is a two-dimensional (2D) signal
A monochromatic video is a three-dimensional (3D) signal

• Signal Energy and Power in a Circuit System
1.1.2 Examples and Mathematical Representation
• Instantaneous power
• Total energy over a finite time interval
• Average power over a finite time interval

• Total energy over a finite time interval
• Time-averaged power over a finite time interval

• Total energy over an infinite time interval
• Time-averaged power over an infinite time interval

In the case of infinite time interval:
Finite total energy → zero average power
Nonzero average power → Infinite total energy

Sec. 1.2 Transformations of the Independent Variable
Key concepts (i) time shift;
(ii) time reversal;
(iii) time scaling;
(iv) periodic signal and its fundamental period;
(v) even and odd signals

(i) Time Shift:
1.2.1 Shifting, Reversal, and Scaling

(iii) Time Scaling:
In the discrete-time case, it is hard to define x[kn] if k is not an integer.

[Example 1.1]
Determine x(-t+1)
(I)
(II)

• If the order is wrong, the valid result cannot be obtained
(valid) (wrong)
Determine x(-t+1)
(I)
(II)

[Example 1.2]
Determine ( )
3 1
2
x t +
(I)
(II)

• If the order is wrong, the valid result cannot be obtained
Determine ( )
3 1
2
x t +
(valid) (wrong)
(I)
(II)

[Example 1.3] ( ) ( )
x t x t
 
→ +
➢ : linearly stretched
➢ : linearly compressed
➢ : time reversal
➢ : delayed time shift
➢ : advanced time shift
1
 
1
 
0
 
0
 
0
 

1.2.2 Periodic Signals
A CT Periodic Signal A DT Periodic Signal

Periodic Signals:
• A periodic signal is unchanged by a time shift of T or N
• They are also periodic with the following periods:
• 2T, 3T, 4T, …
• 2N, 3N, 4N, …
• T or N is called the fundamental period
denoted as T0 or N0
• If a signal is not periodic, we call it an aperiodic signal

[Example 1.4] Periodic and Aperiodic Signals
If ( ) ( )
1 cos
x t t
= for all t,
( ) ( )
1 1 2 ,
x t x t 
= + ( )
1
x t is periodic
If ( ) ( )
2 sin
x t t
= for all t,
( ) ( )
2 2 2 ,
x t x t 
= + ( )
2
x t is periodic
If
then x(t) is not periodic

1.2.3 Even and Odd Signals
even odd

• ANY signal can be broken into a sum of
an even signal and an odd signal

Even-odd decomposition of a DT signal:

signal and system ch1-1 for students of NCNU EE

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