chapter 1 signals and systems

2
∫
∞
∞
−∞
x ( t ) dt
x[ n ]
2
2
lim ∑
x[ n ]
2
=
n=− N
n = −∞
∑
Average Power
P∞ =
P∞ =
lim
lim
T→∞
1 2T
∫
T
−T
x ( t ) dt
2
N →∞
1 2N + 1
n=− N
∑
N
x[ n ]
2
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. With these definitions, we can identify three important class of signals:
1.3.1 Continuous-time Complex Exponential and Sinusoidal Signals The continuous-time complex exponential signal is of the form
1.1.2 Signal Energy and Power
R &nstantaneous power: 1 2 p(t) = v(t) ⋅ i(t) = v (t) = R ⋅ i2 (t) R Let R=1Ω, so p(t) = i 2 (t) = v2 (t) = x2 (t)
Example
Solution 1：
Solution 2：
Example
f (t + 1)
2 1
f (t) f (1-3t)
reversal
t 1
f (1 − t )
2 1 t −1
0
shift
f (t )
2 1 −1 0 t 1 2
−2
0
2
Scaling Scaling
2 1
t
reversal
−2 −1
f (− t )
2 1
shift
2 1
f (1 − t )
f (1 − 3t )
0 1
t
−1
0 1
2
−
1 3
0 2
t
3
f (3t )
f (1 + 3t )
Scaling
1
−
1 3
2
shift
t
2 1 t
－2 0 3
1 3
reversal
0
2 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. The fundamental period T0 (N0) of x(t) (x[n]) is the smallest positive value of T(or N) for which the equation holds.
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
0
2. Discrete-Time signal
n: discrete time x[n]: consequence series
Example1: 1990-2002年的某村农民的年平均收入
1600
800
1990
2002
Example2: x[n] is sampled from x(t)
Note:
An odd signal must necessarily be 0 at t=0, or n=0.
Even-Odd Decomposition: Any signal can be broken into a sum of two signals, one of which is even and another is odd.
1 Ev { x ( t )} = x e ( t ) = [ x ( t ) + x ( − t )] 2 1 Od { x ( t )} = x o ( t ) = [ x ( t ) − x ( − t )] 2 1 Ev { x[ n ]} = x e [ n ] = { x [ n ] + x [ − n ]} 2 1 Od { x [ n ]} = x o [ n ] = { x [ n ] − x [ − n ]} 2
Energy : t1≤ t ≤ t2
∫
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:
(2) Graphical Representation
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
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
3. Time Scaling x(at) ( a>0 ) Stretch Compressed if a<1 if a>1
Note:
Generally, Time scaling only for continuous time signals
This is also called decimation of signals. （信号的抽 取）
We will frequently find it convenient to consider signals that take on complex values.
when − ∞ < t < ∞
Total Energy
E∞ =
E∞ =
−∞ < n < ∞
T
lim ∫
T→∞
N →∞
−T
N
x ( t ) dt =
6. x[n] = cos 3π n
It is periodic with period N=16.
8
1
cosπt
0
-1 -20 1
-15
-10
-5
0
5
10
15
20
cos2t
0
-1 -20 2
-15
-10
-5
0
5
10
15
20
cosπt +cos2t
0
-2 -20
-15
-10
-5
0
5
10
15
20
1.2.3 Even and Odd Signals Even signal: x(-t)=x(t) or x[-n]=x[n] Odd signal : x(-t)=-x(t) or x[-n]=-x[n]
or:
Example of the even-odd decompositon
Example of the even-odd decompositon
Homework：P57--1.6 1.9 1.10 1.21(a)(b)(c)(d) 1.22(a)(b)(g) 1.23 1.24
1.3 Exponential and Sinusoidal Signals
Chapter1 Signals And Systems
Contents
Description of signals Transformations of the independent variable Some basic signals Systems and their mathematical models Basic systems properties
E ∞ = ∞, P∞ = ∞
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
n0>0
Delay
2. Time Reversal x(-t), x[-n] ——Reflection of x(t) or x[n]
B. Two basic types of signals 1. Continuous-Time signal t: continuous time