Lecture123

Signals and SystemsSignals and Systems
6552111 Signals and Systems6552111 Signals and Systems
Sopapun Suwansawang
Lecture #1
1
Lecture #1
Elementary Signals and Systems
Week#1-2

SignalsSignals
Signals are functions of independent variables that
carry information.
FUNCTIONS OF TIME AS SIGNALS
Sopapun Suwansawang 2
Figure : Domain, co-domain, and range of a real function of continuous time.
)(tfv =

SignalsSignals
For example:
Electrical signals voltages and currents
in a circuit
Acoustic signals audio or speech
Acoustic signals audio or speech
signals (analog or digital)
Video signals intensity variations in an
image
Biological signals sequence of bases in
a gene
Sopapun Suwansawang 3

There are two types of signals:
Continuous-time signals (CT) are functions
of a continuous variable (time).
Discrete-time signals (DT) are functions of
SignalsSignals
Discrete-time signals (DT) are functions of
a discrete variable; that is, they are
defined only for integer values of the
independent variable (time steps).
4Sopapun Suwansawang

CT and DT SignalsCT and DT Signals
CT DT
Signal such as :)(tx ),...(),...,(),( 10 ntxtxtx
or in a shorter form as :
,...,...,,
],...[],...,1[],0[
10 nxxx
nxxx
or

where we understand that
)(][ nn txnxx ==
and 's are called samples and the time interval
between them is called the sampling interval. When
nx
between them is called the sampling interval. When
the sampling intervals are equal (uniform sampling),
then
n
)()(][ snTtn nTxtxnxx
s
=== =
where the constant is the sampling intervalsT




<
≥
=
0,0
0,8.0
)(
t
t
tx
t



<
≥
=
0,0
0,8.0
][
n
n
nx
n
)(tx
t
0
1
0
1
][nx
n
1 2 3 4 5

A discrete-time signal x[n] can be defined in two
ways:
1. We can specify a rule for calculating the nth
value of the sequence. (see Example 1)
value of the sequence. (see Example 1)
2. We can also explicitly list the values of the
sequence. (see Example 2)

DT SignalsDT Signals




≥
== 




 0
][
2
1
n
xnx
n
n
Example 1


 <

00 n
...},
8
1
,
4
1
,
2
1
,1{}{ =nx

Example 1: Continue

Example 2

The sequence can be written as
Example 2 : continue
,...}0,0,2,0,1,0,1,2,2,1,0,0{...,}{ =nx
}2,0,1,0,1,2,2,1{}{ =nx
We use the arrow to denote the n = 0 term. We shall use the
convention that if no arrow is indicated, then the first term
corresponds to n = 0 and all the values of the sequence are
zero for n < 0.

Example 3 Given the continuous-time signal
specified by


 ≤≤−−
=
otherwise
tt
tx
0
111
)(
Determine the resultant discrete-time sequence
obtained by uniform sampling of x(t) with a
sampling interval of 0.25 s


otherwise0

Solve :
Ts=0.25 s,
Ts=1 s,





=
↑
,...0,25.0,5.0,75.0,1,75.0,5.0,25.0,0...,][nx



= ,...0,1,0...,][nxTs=1 s,





=
↑
,...0,1,0...,][nx

Analog signals
Analog and Digital SignalsAnalog and Digital Signals
If a continuous-time signal x(t) can take on any
value in the continuous interval (-∞∞∞∞ , +∞∞∞∞), then
the continuous-time signal x(t) is called an analog
Digital signals
A signal x[n] can take on only a finite number of
distinct values, then we call this signal a digital
signal.
the continuous-time signal x(t) is called an analog
signal.

CT and DT
Digital SignalsDigital Signals

CT
Binary signal Multi-level signal
Digital SignalsDigital Signals

Intuitively, a signal is periodic when it repeats
itself.
A continuous-time signal x(t) is periodic if there
exists a positive real T for which
Periodic SignalsPeriodic Signals
for all t and any integer m.The fundamental period
T0 of x(t) is the smallest positive value of T
)()( mTtxtx +=
0
0
2
ω
π
=T

Fundamental frequency
0
0
1
T
f = Hz
Fundamental angular frequency
0
00
2
2
T
f
π
πω == rad/sec

A discrete-time signal x[n] is periodic if there
exists a positive integer N for which
][][ mNnxnx +=
for all n and any integer m.The fundamental period N0 of
x[n] is the smallest positive integer N
0
0
2
Ω
=
π
N

Any sequence which is not periodic is
called a non-periodic (or aperiodic)
sequence.
NonperiodicNonperiodic SignalsSignals

CT
DT

Example 3 Find the fundamental frequency
in figure below.
Hz
T
f
4
11
0
0 ==.sec40 =T
(sec.)

Exercise Determine whether or not each of the
following signals is periodic. If a signal is periodic,
determine its fundamental period.
ttx )
4
cos()(.1
π
+=
nnnx
nnx
enx
tttx
tttx
nj
4
sin
3
cos][.6
4
1
cos][.5
][.4
2sincos)(.3
4
sin
3
cos)(.2
4
)4/(
ππ
ππ
π
+=
=
=
+=
+=

Solve EX.1
1
4
cos)
4
cos()( 00 =→





+=+= ω
π
ω
π
tttx
ππ 22
π
π
ω
π
2
1
22
0
0 ===T
x(t) is periodic with fundamental period T0 = 2π.

Solve EX.2
)()(
4
sin
3
cos)( 21 txtxtttx +=+=
ππ
( ) .6
2
cos3/cos)( 111 ==→==
π
ωπ Ttttxwhere
( )
.8
4/
2
sin)4/sin()(
.6
3/
cos3/cos)(
222
111
==→==
==→==
π
π
ωπ
π
ωπ
Ttttx
Ttttxwhere
numberrationalais
T
T
4
3
8
6
2
1 ==
x(t) is periodic with fundamental period T0 = 4T1=3T2=24.
Note : Least Common Multiplier of (6,8) is 24

Even and Odd Signals:Even and Odd Signals:
A signal x(t) or x[n] is referred to as an even signal if
][][
)()(
nxnx
txtx
−=
−=
A signal x(t) or x[n] is referred to as an odd signal if
][][
)()(
nxnx
txtx
−=−
−=−

Any signal x(t) or x[n] can be expressed as a
sum of two signals, one of which is even and
one of which is odd.That is,
)()()( txtxtx oe +=
][][][ nxnxnx oe
oe
+=
{ }
{ }][][
2
1
][
)()(
2
1
)(
nxnxnx
txtxtx
e
e
−+=
−+= { }
{ }][][
2
1
][
)()(
2
1
)(
nxnxnx
txtxtx
o
o
−−=
−−=
even part odd part

Example 4 Find the even and odd components of
the signals shown in figure below
Solve even part { })()()(2 tftftfe −+=
2fe(t)

Odd part
{ })()()(2 tftftfo −−=
2fo(t)
2fo(t)

Check
)()()( tftftf oe +=
{ })()(
2
1
)( tftftfo −−={ },)()(
2
1
)( tftftfe −+=
32
)()()( tftftf oe +=
Sopapun Suwansawang
2
fo(t)
fe(t)

Note that the product of two even signals or of
two odd signals is an even signal and that the
product of an even signal and an odd signal is an
odd signal.
(even)(even)=even
(even)(even)=even
(even)(odd)=odd
(odd)(even)=odd
(odd)(odd)=even

Example 5 Show that the product of two even
signals or of two odd signals is an even signal
and that the product of an even and an odd
signaI is an odd signal.
Let )()()( txtxtx =
Let )()()( 21 txtxtx =
If x1(t) and x2(t) are both even, then
)()()()()()( 2121 txtxtxtxtxtx ==−−=−
If x1(t) and x2(t) are both even, then
)()()())()(()()()( 212121 txtxtxtxtxtxtxtx ==−−=−−=−

A deterministic signal is a signal in which each
value of the signal is fixed and can be
determined by a mathematical expression, rule,
or table. Because of this the future values of the
signal can be calculated from past values with
Deterministic and Random Signals:Deterministic and Random Signals:
signal can be calculated from past values with
complete confidence.
A random signal has a lot of uncertainty about
its behavior. The future values of a random
signal cannot be accurately predicted and can
usually only be guessed based on the averages
of sets of signals

Deterministic and Random Signals:Deterministic and Random Signals:
Deterministic
Random

RightRight--Handed and LeftHanded and Left--Handed SignalsHanded Signals
A right-handed signal and left-handed signal are
those signals whose value is zero between a
given variable and positive or negative infinity.
Mathematically speaking,
A right-handed signal is defined as any signal
A right-handed signal is defined as any signal
where f(t) = 0 for
A left-handed signal is defined as any signal
where f(t) = 0 for
∞<< 1tt
−∞>> 1tt

RightRight--Handed and LeftHanded and Left--Handed SignalsHanded Signals
Right-Handed
1t
Left-Handed
1
1t

Causal vs.Anticausal vs. NoncausalCausal vs.Anticausal vs. Noncausal
Causal signals are signals that are zero for
all negative time.
Anticausal signals are signals that are zero
for all positive time.
for all positive time.
Noncausal signals are signals that have
nonzero values in both positive and
negative time.

Causal vs.Anticausal vs. NoncausalCausal vs.Anticausal vs. Noncausal
Causal
Anticausal
Noncausal

Energy and Power SignalsEnergy and Power Signals
Consider :
Rti
R
tv
titvtp
)(
)(
)()()(
2
2
=
=
⋅=
∫∫
∫
∞
∞−
∞
∞−
∞
∞−
==
=
)()()()(
1
)(
22
tdtitdtv
R
dttpE
Power Energy

Total energy E and average power P on a per-ohm
basis are
dttiE ∫
∞
∞−
= )(2
Joules
dtti
T
P
T
TT
∫
−∞→
∞−
= )(
2
1 2
lim Watts

For an arbitrary continuous-time signal x(t), the
normalized energy content E of x(t) is defined as
∫∫
−∞→
∞
∞−
==
T
TT
dttxdttxE
22
)()( lim
The normalized average power P of x(t) is
defined as
43
−∞→∞− TT
∫
−∞→
=
T
TT
dttx
T
P
2
)(
2
1
lim
Sopapun Suwansawang

Similarly, for a discrete-time signal x[n],
the normalized energy content E of x[n] is defined
as
∑∑
−=∞→
∞
−∞=
==
N
NnNn
nxnxE
22
][lim][
The normalized average power P of x[n] is defined
as
−=∞→−∞= NnNn
∑
−=∞→ +
=
N
NnN
nx
N
P
2
][
12
1
lim

x(t) (or x[n]) is said to be an energy signal (or
sequence) if and only if 0 < E < ∞∞∞∞, and P = 0.
x(t) (or x[n]) is said to be a power signal (or
sequence) if and only if 0 < P < ∞∞∞∞, thus
implying that E = ∞∞∞∞.
implying that E = ∞∞∞∞.
Note that a periodic signal is a power signal if
its energy content per period is finite, and then
the average power of this signal need only be
calculated over a period.
∫=
0
0
2
0
)(
1
T
dttx
T
P

Exercise Determine whether the following
signals are energy signals, power signals, or
neither.
1.
)cos()( 0 θω += tAtx1.
2.
3.
)cos()( 0 θω += tAtx
tj
eAtx 0
)( ω
=
)()( 3
tuetv t−
=

Solve Ex.1
The signal x(t) is periodic with T0=2π/ω0.
dttA
T
dttx
T
P
TT
∫∫ +==
00
0
0
22
00
2
0
)(cos
1
)(
1
θω
)cos()( 0 θω += tAtx
0000
dtt
T
A
P
T
∫ ++=
0
0
0
0
2
)22cos(1(
2
1
θω








++= ∫ ∫ dttdt
T
A
P
T T0 0
0
0
00
2
)22(cos1
2
θω
0
2
2
A
= ∞<
Thus, x(t) is power signal.

Solve Ex.2 tjAtAeAtx tj
00 sincos)( 0
ωωω
+==
The signal x(t) is periodic with T0=2π/ω0.
Note that periodic signals are, in general, power signals.
∫
T
21
∫
T
21
∫=
T
x dttx
T
P
0
2
)(
1
∫=
T
tj
dtAe
T 0
2
0
1 ω
2 2
2
0 0
2
1
T T
x
A A
A dt dt T
T T T
P A W
= = = ⋅
=
∫ ∫

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