信号与系统

cos(

4
n)
cos(
n 6
)
cos( 15 8 n)
Figure 1.6 Two periodic sinusoidal sequence and an aperiodic sinusoidal sequence
1.5.4 The Discrete-Time Unit Impulse and Unit Step Sequences （单位冲激序列和单位阶跃序列）
t 2 t1

t2
p ( t ) dt
t1
t 2 t1

t2
v ( t )dt
2
t1
Generally, the total energy over the time interval continuous-time signal: x ( t ) dt
t2 2 t1
t1 t t 2
P lim
1 2T
T

1
T
T
x ( t ) dt
2
P lim
N
2 N 1 n N

N
x[ n ]
2
in continuous time and discrete time, respectively.
With these definitions, we can identify three important
C e
x(t)
rt
x(t)
Figure 1.3 (a) Growing sinusoidal signal , x ( t ) Ce rt cos( 0 t ) （增长的） (b) decaying sinusoid , x ( t ) Ce rt cos( 0 t ) r<0. （衰减的）
X(-t/3+2) 1
0
1
2
t
X(t+2) 1 -2/3 -1/3 0
(a)
X(t/3+2) 1
-2
-1
0 X(t)
1 Fra Baidu bibliotek 4/3 5/3 2
X(-t/3) 1
-6
-5 -4
0
X(t)
X(t/3) 1 0 4 5 6
1 0 4/3 5/3 2
1.4 Even-odd Decomposition of a Signal
1.2.3 Determinate and Random Signals （确定性信号和随机信号） • A determinate signal ——x(t)
• A random signal —— cannot find a function to represent it
1.2.4 Energy and Power Signals
unit impulse (or unit sample): （单位样值信号） unit step sequence:
[n]
1
0, n 0 [n] 1, n 0
n 0
0, u[n ] 1,
n 0
u[n ]
1
n
0
0
n
[n k ]
1 n 0 k
delayed unit sample:
Ⅰ.Real Exponential Signal: if C﹑a are real
Ⅱ. Periodic Complex Exponential and Sinusoidal Signals : * if C is real and a purely imaginary j 0 t x ( t ) Ce periodic complex exponential
classes of signals:
• Energy signals • Power signals • Signals with neither finite total energy nor finite average power
1.3 Transformations of the Independent Variable of Signals
in a
Similarly, the total energy in a discrete-time signal x[n] over the time interval n 1 n n 2 : x[ n ]
n2 2 n1
The total energy over an infinite interval is,
Ⅲ.General Complex Exponential Signals: * both C and a are complex numbers C C e j , a r j 0
Ce
at
Ce
rt
cos( 0 t ) j C e
rt
sin( 0 t )
1.2.2 Periodic and Aperiodic Signals （周期和非周期信号） • For a continuous-time signal x(t) x(t) = x(t + T) for all values of t. • For a discrete-time signal x[n] x[n] = x[n + N] for all values of n. In this case, we say that x(t)(x[n] ) is periodic with period T(N).
C C e , e
j
j 0
C
n
C
n
cos( 0 n ) j C
n
sin( 0 n )
1
0 1
1
Figure 1.4
1 0
n
the real exponential signal x [ n ] C
（能量信号和功率信号） The instantaneous power is p ( t ) v ( t ) i ( t ) 1 v 2 ( t )
R
The total energy is

t2
p ( t ) dt
1
t1

t2
1 R
1
v ( t )dt
1 R
2
t1
The average power is
x (t) = Ev{x(t)} + Od {x(t)}
even part of x (t)
odd part of x (t)
Ev {x(t)}= [ x (t) + x (-t) ]
2 2
1 1
Od {x(t)}= [ x (t) - x (-t) ]
x (-t) = Ev{x(-t)} + Od {x(-t)} =Ev{x(t)} -
Representing Signals Graphically
X(t)
t 0
(a)
X[n] X[0] X[4]
-4
-3
-1
-2
0 1 2 3 4 5
n
X[-1] (b) Figure 1.1 Graphical representations of (a) continuous-time and (b) discrete-time signals
Example 1.1 Determine the fundamental period of the signal x(t) = 2cos(10πt+1)-sin(4πt-1).
From trigonometry, we know that the fundamental period of cos(10πt+1) is T1=1/5, and sin(4πt-1) is T2=1/2. What about the fundamental period of x(t)? The answer is if there is a rational T, and it is the lowest common multiple（最小公倍数） of T1 and T2, then we say that x(t) is periodic with fundamental period T, or else, x(t) is aperiodic. For the x(t) in this example, the lowest common multiple of 0.2 and 0.5 is unit 1, and it is rational, so that the fundamental period of x(t) is 1.
• time shift（时移）: x(t-t0) or x[n-n0] • time reversal（反褶） : x(-t) or x[-n] • time scaling（尺度变换） : x(at) or x[an]
Example 1.2 Given a signal x(-t/3+2), shown in Fig. 1.2(a), draw the graph of x(t).
1.2 CLASSIFICATION OF SIGNALS
1.2.1. Continuous-time and Discrete-time Signals （连续时间和离散时间信号） • continuous-time signals’ independent variable is continuous : x(t) • discrete-time signals are defined only at discrete times : x[n]
r>0;
1.5.2 Discrete-Time Complex Exponential and Sinusoidal signals(sequences)（序列）:
x[ n ] C
n
Ⅰ.Real Exponential Signals: if C﹑a are real
Ⅱ. General Complex Exponential Signals * both C and a are complex numbers
CHAPTER 1 Signals and Systems 1. Signals
1.1 Definition and Mathematical Representation of Signals
• Signals: physical phenomena or physical quantities, which change with time or space. • Functions of one or more independent variables. example: x(t)
Od {x(t)}
1.5 SEVERAL BASIC SIGNALS
1.5.1 Continuous-time Complex Exponential （复 指数函数） and Sinusoidal Signals（正弦信号） complex exponential signal : x ( t ) Ce at
Relationship between unit sample and unit step sequence the unit sample is the first difference（差分） of the unit step sequence [ n ] u [ n ] u [ n 1]
in continuous time,
E lim
in discrete time,
E lim
T

T
T
x ( t ) dt
x[ n ]
2
2



x ( t ) dt
2
2
N
n N

N




x[ n ]
The time-average power over an infinite interval:
1
n
1
n
Figure 1.5
(a) Growing sinusoidal sequence (b) Decaying sinusoidal sequence
1.5.3 Periodicity Property of Discrete-Time Complex Exponentials（离散正弦序列的周期性）
the
unit step sequence is the running sum of the unit sample
u[n ]
m
[m ]
n
u[n ]
[n k ]
k 0

sampling property （抽样性质）of the unit sample: