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