奥本海姆版信号与系统ppt

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
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
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
n0>0
Delay
x(t) and x(t-t0), or x[n] and x[n-n0]:

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
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)
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 )
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
T 6 , T 8
1 2
x(t) is periodic. Its period is T 24 The smallest multiples of T1 and T2 in common

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)
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
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
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
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.

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
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
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
compress]
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：
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


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
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.
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

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
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
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
(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.

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

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.