1 Signal and System
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
1,Signals and Systems
1 Signal and System
(2) An automobile
Force f from engine
Retarding frictional force ρV
Velocity V
1 Signal and System
(3) A Speech Signal
1 Signal and System
(4) A Picture
1 Signal and System
(5) Vertical Wind Profile
1 Signal and System
B,Types of Signals
(1) Continuous-time Signal
1 Signal and System
(2) Discrete-time Signal
1 Signal and System
C,Representation
(1) Function Representation
Example,x(t) = cos?0t
x(t) = ej?0t
(2) Graphical Representation
Example,( See page before )
1 Signal and System
1.1.2 Signal Energy and Power
A,Energy (Continuous-time)
Instantaneous power:
)()(1)()()( 22 tiRtvRtitvtp
Let R=1Ω,so
p(t)=i2(t)=v2(t)=x2(t)
1 Signal and System
Energy over t1? t? t2:
212121 )()()( 22 tttttt dttxdttvdttp
Total Energy:


2
1
)(2lim t
tT
dttxE
Average Power:



T
TT
dttxTP )(2 1 2l i m
1 Signal and System
B,Energy (Discrete-time)
Instantaneous power,][)( 2 nxnp?
Energy over n1? n? n2:
2
1
][2
n
nn
nxE
Total Energy,


n
nxE ][2
Average Power:


N
NnN
nxNP ][12 1 2lim
1 Signal and System
C,Finite Energy and Finite Power Signal
Finite Energy Signal,
dttxE )(2


n
nxE ][2
Finite Power Signal,


N
NnN
nxNP ][12 1 2lim


T
TT
dttxTP )(2 1 2lim
( P? 0 )
( E )
1 Signal and System
1.2 Transformations of the Independent Variable
1.2.1 Examples of Transformations
A,Time Shift
Right shift,x(t-t0)
x[n-n0] (Delay)
Left shift,x(t+t0)
x[n+n0] (Advance)
1 Signal and System
Examples
1 Signal and System
B,Time Reversal
x(-t) or x[-n],Reflection of x(t) or x[n]
1 Signal and System
C,Time Scaling
x(at) ( a>0 )
Stretch if a<0
Compressed if a>0
Example 1.1
1 Signal and System
1.2.2 Periodic Signals
Definition,There is a posotive value of T which,
x(t)=x(t+T),for all t
x(t) is periodic with period T,
T? Fundamental Period
For Discrete-time period signal:
x[n]=x[n+N] for all n
N? Fundamental Period
1 Signal and System
Examples of periodic signal
1 Signal and System
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]
Even-Odd Decomposition:
)]()([21)()}({ txtxtxtxEv e
)]()([21)()}({ txtxtxtxOd o
]}[][{21][]}[{ nxnxnxnxEv e
]}[][{21][]}[{ nxnxnxnxOd o
or:
1 Signal and System
Examples
1 Signal and System
1.3 Exponential and Sinusoidal signal
1.3.1 Continuous-time Complex Exponential
and Sinusoidal Signals
A,Real Exponential Signals
x(t)= C eat ( C,a are real value)
1 Signal and System
B,Periodic Complex Exponential and
Sinusoidal Signals
(1) x(t) = e j?0t
(2) x(t) = Acos(?0t+?)
(3) x(t) = e jk?0t
All x(t) satisfy for x(t) = x(t+T),and T=2?/?0
So x(t) is periodic.
1 Signal and System
Euler’s Relation,
e j?0t = cos?0t + sin?0t
and cos?0t = (e j?0t + e -j?0t ) / 2
sin?0t = (e j?0t - e -j?0t ) / 2
We also have
tjjtjj eeAeeAtA 00
22)c o s ( 0

1 Signal and System
C,General Complex Exponential Signals
x(t) = C e jat,
in which C = |C| ej?,a = r + j?0
So
x(t) = |C| ej? eat ej?0t
= |C| eat ej(?0t+? )
= |C| eat cos(?0t+?) + j |C| eat sin(?0t+?)
1 Signal and System
Signal waves
1 Signal and System
1.3.2 Discrete-time Complex Exponential
and Sinusoidal Signals
Complex Exponential Signal (sequence),
x[n] = C?n or x[n] = C e?n
1 Signal and System
A,Real Exponential Signal
Real Exponential Signal
x[n] = C?n
(a)?>1
(b) 0<?<1
(c) -1<?<0
(d)?<-1
1 Signal and System
B,Sinusoidal Signals
Complex exponential:
x[n] = e j?0n
= cos?0n + jsin?0n
Sinusoidal signal:
x[n] = cos(?0n+?)
1 Signal and System
C,General Complex Exponential Signals
Complex Exponential Signal:
x[n] = C?n
in which C = |C| ej?,? = |?|ej?0 (polar form)
then
x[n]=|C| |?|ncos(?0n +?)+j|C| |?|nsin(?0n +?)
1 Signal and System
Real or Imaginary of Signal
1 Signal and System
1.3.3 Periodicity Properties of Discrete-time
Complex Exponentials
Continuous-time,e j?0t,T=2?/?0
Discrete-time,e j?0n,N=?
Calculate period,
By definition,e j?0n = e j?0(n+N)
thus e j?0N = 1 or?0N = 2? m
So N = 2?m/?0
Condition of periodicity,2?/?0 is rational
1 Signal and System
Periodicity Properties
1 Signal and System
1.4 The Unit Impulse and Unit Step Functions
1.4.1 The Discrete-time Unit Impulse and Unit
Step Sequences
(1) Unit Sample(Impulse):


0,1
0,0][
n
nn?
1 Signal and System
Unit Step Function:


0,1
0,0][
n
nnu
(2) Relation Between Unit Sample and Unit Step


n
m
mnu
nunun
][][
]1[][][
or

0
][][
k
knnu?
1 Signal and System
(3) Sampling Property of Unit Sample




m
knkxnx
nnnxnnnx
nxnnx
][][][
][][][][
][]0[][][
000


1 Signal and System
Illustration of Sampling
1 Signal and System
1.4.2 The Continuous-time Unit Step and Unit
Impulse Functions
(1) Unit Step Function:


0,1
0,0)(
t
ttu
1 Signal and System
Unit Impulse Function:



1)(
0,
0,0
)( dtta n d
t
t
t
1 Signal and System
(2) Relation Between Unit Impulse and Unit Step

t
dtu
dt
tdu
t

)()(
)(
)(
1 Signal and System
(3) Sampling Property of?(t)
)()()(
)()()()(
)()0()()(
00
000
txdttttx
tttxtttx
txttx






Example 1.7
1 Signal and System
1.5 Continuous-time and Discrete-time System
Definition:
(1) Interconnection of Component,device,
subsystem…,(Broadest sense)
(2) A process in which signals can be
transformed,(Narrow sense)
Representation of System:
(1) Relation by the notation
][][
)()(
nynx
tytx
L
L


1 Signal and System
(2) Pictorial Representation
Continuous-time
system
x(t) y(t)
Discrete-time
system
x[n] y[n]
1 Signal and System
1.5.1 Simple Example of systems
Example 1.8:
RC Circuit in Figure 1.1,Vc(t)? Vs(t)
)(1)(1)( tvRCtvRCdt tdv scc
RC Circuit
(system)
vs(t) vc(t)
1 Signal and System
Example 1.10:
Balance in a bank account from month
to month:
balance --- y[n]
net deposit --- x[n]
interest --- 1%
so y[n]=y[n-1]+1%?y[n-1]+x[n]
or y[n]-1.01y[n-1]=x[n]
Balance in bank
(system)
x[n] y[n]
1 Signal and System
1.5.2 Interconnections of System
(1) Series(cascade) interconnection
1 Signal and System
(2) Parallel interconnection
Series-Parallel interconnection
1 Signal and System
(3) Feed-back interconnection
1 Signal and System
Example of Feed-back interconnection
1 Signal and System
1.6 Basic System Properties
1.6.1 Systems with and without Memory
Memoryless system,It’s output is dependent only
on the input at the same time.
Features,No capacitor,no conductor,no delayer.
Examples of memoryless system:
y(t) = C x(t) or y[n] = C x[n]
Examples of memory system:
)()(2)( txtydt tdy
or y[n]-0.5y[n-1]=2x[n]
1 Signal and System
1.6.1 Invertibility and Inverse Systems
Definition:
(1) If system is invertibility,then an inverse system
exists.
(2) An inverse system cascaded with the original
system,yields an output equal to the input.
1 Signal and System
1 Signal and System
1.6.3 Causality
Definition:
A system is causal If the output at any time
depends only on values of the input at the
present time and in the past.
For causal system,if x(t)=0 for t<t0,there must
be y(t)=0 for t<t0.
( nonanticipative )
Memoryless systems are causal.
1 Signal and System
x(t)
y(t)
t1
t2
1 Signal and System
1.6.4 Stability
Definition:
Small inputs lead to responses that don not
diverge.
Finite input lead to finite output:
if |x(t)|<M,then |y(t)|<N,
Examples:
Stable pendulum
Motion of automobile
1 Signal and System
Example 1.13
1 Signal and System
1.6.5 Time Invariance
Definition:
Characteristics of the system are fixed over
time.
Time invariant system,
If x(t) y(t),then x(t-t0) y(t-t0),
Example 1.14 1.15 1.16
1 Signal and System
x(t)
y(t)
x(t-t0)
y(t-t0)
1 Signal and System
1.6.6 Linearity
Definition:
The system possesses the important property
of superposition:
(1) Additivity property:
The response to x1(t)+x2(t) is y1(t)+y2(t),
(2) Scaling or homogeneity property:
The response to ax1(t) is ay1(t),
(where a is any complex constant,a?0,)
1 Signal and System
L
x1(t)
x2(t)
y1(t)
y2(t)
a x1(t)
x1(t) +x2(t)
ax1(t) +bx2(t)
a y1(t)
y1(t) +y2(t)
ay1(t) +by2(t)
Represented in block-diagram:
Example 1.17 1.18
1 Signal and System
LTI System
LTI
x(t) y(t)
x(t-t0)
ax(t) +bx(t-t0)
y(t-t0)
ay(t) +by(t-t0)
Linear and Time-invariant system
Problems:
1.14 1.15 1.16 1.17 1.23 1.24 1.31