奥本海姆版信号与系统ppt

2
1
shift
f (t )
2 1
1 t
2
2
0
Scaling
Scaling
2
reversal
t
f (t )
2 1
shift
2 1
f (1 t )
f (1 3t )
1
t
0 1
1 0
1
2
2
1
0 1
t
1
2

1 3
0 2
t
3
f (3t )
f (1 3t )
Scaling
1
1 3
2
shift
Example: ( See page before )
(3) Sequence-representation for discretetime signals:
x[n]={-2 1 3 2 1 –1} or x[n]=(-2 1 3 2 1 –1)
3
Note:

Since many of the concepts associated with continuous and discrete signals are similar (but not identical), we develop the concepts and techniques in parallel.
Chapter1 Signals And Systems
崔琳莉
Contents

Description of signals
Transformations of the independent variable Some basic signals Systems and their mathematical models Basic systems properties

t2
t1
n2
x(t ) dt
2
2
1 t 2 t1

t2
t1
x(t ) dt
2
n n1
x[n]
1 2 x[n] n2 n1 1 n n1
n2
We will frequently find it convenient to consider signals that take on complex values.
3 1. x(t ) A sin t 8
It is periodic signal. Its period is T=16/3.
cos t , t 0 2. x(t ) It is not periodic. 0, t 0
1 1 3. x(t ) A cos t B sin t 3 4
Instantaneous power: 1 2 R i (t ) p(t ) v(t ) i(t ) v (t ) R i 2 (t ) R _ v(t ) Let R=1Ω, so p(t ) i 2 (t ) v 2 (t ) x 2 (t )
+
Energy : t1 t t2


1.1 Continuous-Time and Discrete-Time Signals
1.1.1 Examples and Mathematical Representation
A. Examples (1) A simple RC circuit
Source voltage Vs and Capacitor voltage Vc

t2
t1
p(t )dt

t2
t1
v (t )dt
2

t2
t1
x 2 (t )dt
t2
1 Average Power: t 2 t1

t2
t1
1 p (t )dt t 2 t1

t1
x 2 (t )dt
Definition:
Total Energy Continuous-Time: (t1 t t2 ) Discrete-Time: (n1 n n2 ) Average Power
There are many other signals classification:


Analog vs. Digital Periodic vs. Aperiodic Even vs. Odd Deterministic vs. Random ……
1.1.2 Signal Energy and Power
t
8
4 2
12
4
Time Scaling
x(at) ( a>0 )
Stretch if a<1
Compressed
if a>1
How about the discrete-time signal?
Generally,
time scaling only for continuous time signals
1.2 Transformation of the Independent Variable
1.2.1 Examples of Transformations 1. Time Shift x(t-t0), x[n-n0]
t0<0
Advance
Time Shift
Leabharlann Baidu
n0>0
Delay
x(t) and x(t-t0), or x[n] and x[n-n0]:
x[n]
x[n]
x[2n]
x[n]
x[2n]
2 2 2
x[n/2]
n 0 1 2 3 4 5 6 This is also called decimation of signals. （信号的抽取）
Example
x(t)
1 0 1
t
Solution 1： Solution 2：
Solution 1：

B. Two basic types of signals 1. Continuous-Time signal
t: continuous time x(t):continuum of value
A
x(t)
x(t)
t
0
1
t
x(t) Asin(t ),tR
at , t 0 x(t) e 0 , t 0
t
2
1
0
2 3
reversal
t
1 3
－2 0 3
1.2.2 Periodic Signals
A periodic signal x(t) (or x[n]) has
the property that there is a positive value of T (or integer N) for which : x(t)=x(t+T) , for all t x[n]=x[n+N], for all n
If a signal is not periodic, it is called
aperiodic signal.
Examples of periodic signals
CT: x(t)=x(t+T)
DT:
x[n]=x[n+N]
Periodic Signals
of x(t) (x[n]) is the smallest positive value
x(t)
1
x(t-1/2)
t
1
x(3t-1/2)
t
1
t
0
1
0 1/2
3/2
0 1/6 1/2
Solution 2：
x(t)
1
x(3t) t
1
x(3t-1/2) t
1
t
0
1
0
1/3
0 1/6 1/2
Example
f (t 1)
2
f(t) f(1-3t)
reversal
t 1
1
0
f (1 t )
x(t)
1s 8k
Sampling
x[n]
Why DT?
C. Representation
(1) Function Representation
Example: x(t) = cos0t x[n] = cos0n x(t) = ej0t x[n] = ej0n
(2) Graphical Representation
With these definitions, we can identify three important class of signals——

a. finite total energy
E
E P lim 0 T 2T
b. finite average power
P
(2) An automobile
Force f from engine Retarding frictional force ρV Velocity V
(3) A Speech Signal
(4) A Picture
(5) vital statistics(人口统计)
Note

In this book, we focus on our attention on signals involving a single independent variable. For convenience, we will generally refer to the independent variable as time, although it may not in fact represent time in specific applications.

0
2. Discrete-Time signal
n: discrete time x[n]: a discrete set of values (sequence)
Example1: 1990-2002年的某村农民的年平均收入
Example2: x[n] is sampled from x(t)
when t n
Total Energy
E lim
T
T
T
N
x(t ) dt
2
2


x(t ) dt
2
2
E lim
N n N
x[n]

T T

n
x[n]

Average Power
P lim
x(t)
0

t
Note: the difference between x(-t) and –x(t)
x(-t) ??? -x(t)
3. Time Scaling
x(at) (a>0)
8
4 2
4
x(t)
t
12
4
stretch
4
x(t/2)
t
8
4 2
12
4
compressed
4
x(2t)

2. Time Reversal x(-t), x[-n]
——Reflection of x(t) or x[n]
2. Time Reversal x(-t), x[-n]
——Reflection of x(t) or x[n]
a mirror
Time Reversal
x[n]
x[-n]
Looking for mistakes

They are identical in shape If t0>0, x(t-t0) represents a delay n0>0, x[n-n0] represents a delay If t0<0, x(t-t0) represents an advance n0<0, x[n-n0] represents an advance
T 6 , T 8
1 2
x(t) is periodic. Its period is T 24 The smallest multiples of T1 and T2 in common
The fundamental period T0 (N0)
of T(or N) for which the equation holds.
Note: x(t)=C is a periodic signal, but its fundamental period is undefined.
Examples of periodic signals
E ,
(if
P 0, then
E lim P )
T T
T
c. infinite total energy, infinite average power
P
Read textbook P71: MATHEMATICAL REVIEW
Homework： P57--1.2
T
1 2T
x(t ) dt
2
P lim
N
N 1 2 N x[n] 2 N 1 n
Note:

It is important to remember that the terms ―Power‖ and ―energy‖ are used here independently of the quantities actually are related to physical energy.