Signal and Systems
Chapter 1: Introduction
H. F. Francis Lu
2011 Spring: Signal and Systems
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H. F. Francis Lu
1 / 67
Signal and Systems
Signal
A signal is formally defined as a function of one or more
variables that conveys information on the nature of a physical
phenomenon.
System
A system is formally defined as an entity that manipulates one
or more signals to accomplish a function, thereby yielding new
signals.
2011 Spring: Signal and Systems
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H. F. Francis Lu
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Sec. 1.4 Classification of Signals
2011 Spring: Signal and Systems
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Continuous Time Signals
Definition 1
A signal x
(
t
)
is said to be a
continuous time
signal if it is
defined for all time t.
2011 Spring: Signal and Systems
Ver. 2011.02.20
H. F. Francis Lu
4 / 67
Discrete Time Signals
Definition 2
A discretetime signal is defined only at discrete instants of
time.
A discretetime signal is often derived from a continuous
time signal by sampling it at a uniform rate.
For example, given a continuous time signal
x
(
t
)
, let
T
s
be
the sampling period. Then the discrete time signal is given by
x
[
n
]
=
x
(
nT
s
)
,
n
=
0
,
±
1
,
±
2
,
2011 Spring: Signal and Systems
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H. F. Francis Lu
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Even and Odd Signals
Definition 3
A signal is said to be an even signal if
x
(
t
)
=
x
(

t
)
,
if continuous time
x
[
n
]
=
x
[

n
]
,
if discrete time
and is said to be an odd signal if
x
(
t
)
=

x
(

t
)
,
if continuous time
x
[
n
]
=

x
[

n
]
,
if discrete time
2011 Spring: Signal and Systems
Ver. 2011.02.20
H. F. Francis Lu
6 / 67
Example
1 (p.18)
Consider the signal
x
(
t
)
=
sin
π
t
T
,

T
≤
t
≤
T
0
,
otherwise
Is the signal x
(
t
)
even or odd?
2011 Spring: Signal and Systems
Ver. 2011.02.20
H. F. Francis Lu
7 / 67
EvenOdd Decomposition of Signals
Given an arbitrary signal
x
(
t
)
, we can decompose
x
(
t
)
into
x
(
t
)
=
x
e
(
t
)
+
x
o
(
t
)
where
x
e
(
t
)
is even and
x
o
(
t
)
is odd, according to
x
e
(
t
)
=
1
2
[
x
(
t
)
+
x
(

t
)]
x
o
(
t
)
=
1
2
[
x
(
t
)

x
(

t
)]
The decomposition is similar for discrete time signals
x
[
n
]
.
Proof.
Straightforward.
2011 Spring: Signal and Systems
Ver. 2011.02.20
H. F. Francis Lu
8 / 67
Example
2 (p.19)
Decompose the signal
x
(
t
)
=
e

2
t
cos
(
t
)
into even and odd.
Sol.
x
e
(
t
)
=
1
2
[
x
(
t
)
+
x
(

t
)]
=
cosh
(
2
t
)
cos
t
x
o
(
t
)
=
1
2
[
x
(
t
)

x
(

t
)]
=

sinh
(
2
t
)
cos
t
2011 Spring: Signal and Systems
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H. F. Francis Lu
9 / 67
Hermitian Symmetric or Complex Symmetric
Definition 4
Let x
(
t
)
(and the same for x
[
n
]
) be a complexvalued signal.
We say x
(
t
)
is
Hermitian symmetric
if
x
(
t
)
=
x
*
(

t
)
“Hermitian symmetric” is called “conjugate symmetric” in the book,
but the latter name is less typical.
Proposition 1
x
(
t
)
=
a
(
t
)
+
ı
b
(
t
)
, where a
(
t
)
and b
(
t
)
are realvalued
functions, is Hermitian symmetric iff a
(
t
)
is even and b
(
t
)
is
odd.
2011 Spring: Signal and Systems
Ver. 2011.02.20
H. F. Francis Lu
10 / 67
Example
3
Show that
x
(
t
)
=
exp
(
ı
2
π
t
)
is Hermitian symmetric.
Sol.
x
(
t
)
=
cos
(
2
π
t
)
+
ı
sin
(
2
π
t
)
x
*
(

t
)
=
cos
(

2
π
t
)

ı
sin
(

2
π
t
)
=
cos
(
2
π
t
)
+
ı
sin
(
2
π
t
)
=
x
(
t
)
2011 Spring: Signal and Systems
Ver. 2011.02.20
H. F. Francis Lu
11 / 67
Periodic Signals
Definition 5
A signal x
(
t
)
(and the same for x
[
n
]
) is called a periodic
signal if there exists a positive constant T such that
x
(
t
)
=
x
(
t
+
T
)
for all t
∈
R
. The smallest such value, say
T
min
is called