1. 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

2. SignalsSignals
Signals are functions of independent variables that
carry information.
FUNCTIONS OF TIME AS SIGNALS
6552111 Signals and Systems6552111 Signals and Systems
Sopapun Suwansawang 2
Figure : Domain, co-domain, and range of a real function of continuous time.
)(tfv =

3. SignalsSignals
For example:
Electrical signals voltages and currents
in a circuit
Acoustic signals audio or speech
6552111 Signals and Systems6552111 Signals and Systems
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

4. There are two types of signals:
Continuous-time signals (CT) are functions
of a continuous variable (time).
Discrete-time signals (DT) are functions of
6552111 Signals and Systems6552111 Signals and Systems
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

5. CT and DT SignalsCT and DT Signals
6552111 Signals and Systems6552111 Signals and Systems
CT DT
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Signal such as :)(tx ),...(),...,(),( 10 ntxtxtx
or in a shorter form as :
,...,...,,
],...[],...,1[],0[
10 nxxx
nxxx
or

6. where we understand that
6552111 Signals and Systems6552111 Signals and Systems
)(][ nn txnxx ==
and 's are called samples and the time interval
between them is called the sampling interval. When
nx
CT and DT SignalsCT and DT Signals
6Sopapun Suwansawang
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

7. 6552111 Signals and Systems6552111 Signals and Systems



<
≥
=
0,0
0,8.0
)(
t
t
tx
t



<
≥
=
0,0
0,8.0
][
n
n
nx
n
CT and DT SignalsCT and DT Signals
7Sopapun Suwansawang
)(tx
t
0
1
0
1
][nx
n
1 2 3 4 5

8. 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)
6552111 Signals and Systems6552111 Signals and Systems
CT and DT SignalsCT and DT Signals
value of the sequence. (see Example 1)
2. We can also explicitly list the values of the
sequence. (see Example 2)
8Sopapun Suwansawang

9. 6552111 Signals and Systems6552111 Signals and Systems
DT SignalsDT Signals




≥
== 




 0
][
2
1
n
xnx
n
n
Example 1
9Sopapun Suwansawang


 <

00 n
...},
8
1
,
4
1
,
2
1
,1{}{ =nx

10. 6552111 Signals and Systems6552111 Signals and Systems
DT SignalsDT Signals
Example 1: Continue
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11. DT SignalsDT Signals
6552111 Signals and Systems6552111 Signals and Systems
Example 2
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12. DT SignalsDT Signals
6552111 Signals and Systems6552111 Signals and Systems
The sequence can be written as
Example 2 : continue
,...}0,0,2,0,1,0,1,2,2,1,0,0{...,}{ =nx
12Sopapun Suwansawang
}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.

13. Example 3 Given the continuous-time signal
specified by
DT SignalsDT Signals
6552111 Signals and Systems6552111 Signals and Systems


 ≤≤−−
=
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
13Sopapun Suwansawang


otherwise0

14. Solve :
Ts=0.25 s,
Ts=1 s,
DT SignalsDT Signals
6552111 Signals and Systems6552111 Signals and Systems





=
↑
,...0,25.0,5.0,75.0,1,75.0,5.0,25.0,0...,][nx



= ,...0,1,0...,][nxTs=1 s,
14Sopapun Suwansawang





=
↑
,...0,1,0...,][nx

15. Analog signals
6552111 Signals and Systems6552111 Signals and Systems
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.
15Sopapun Suwansawang
the continuous-time signal x(t) is called an analog
signal.

16. CT and DT
6552111 Signals and Systems6552111 Signals and Systems
Digital SignalsDigital Signals
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17. CT
Binary signal Multi-level signal
6552111 Signals and Systems6552111 Signals and Systems
Digital SignalsDigital Signals
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18. 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
6552111 Signals and Systems6552111 Signals and Systems
Periodic SignalsPeriodic Signals
for all t and any integer m.The fundamental period
T0 of x(t) is the smallest positive value of T
18Sopapun Suwansawang
)()( mTtxtx +=
0
0
2
ω
π
=T

19. Fundamental frequency
6552111 Signals and Systems6552111 Signals and Systems
Periodic SignalsPeriodic Signals
0
0
1
T
f = Hz
Fundamental angular frequency
19Sopapun Suwansawang
0
00
2
2
T
f
π
πω == rad/sec

20. A discrete-time signal x[n] is periodic if there
exists a positive integer N for which
6552111 Signals and Systems6552111 Signals and Systems
Periodic SignalsPeriodic Signals
][][ mNnxnx +=
for all n and any integer m.The fundamental period N0 of
x[n] is the smallest positive integer N
20Sopapun Suwansawang
0
0
2
Ω
=
π
N

21. Any sequence which is not periodic is
called a non-periodic (or aperiodic)
sequence.
6552111 Signals and Systems6552111 Signals and Systems
NonperiodicNonperiodic SignalsSignals
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22. 6552111 Signals and Systems6552111 Signals and Systems
Periodic SignalsPeriodic Signals
CT
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DT

23. Example 3 Find the fundamental frequency
in figure below.
6552111 Signals and Systems6552111 Signals and Systems
Periodic SignalsPeriodic Signals
23Sopapun Suwansawang
Hz
T
f
4
11
0
0 ==.sec40 =T
(sec.)

24. Exercise Determine whether or not each of the
following signals is periodic. If a signal is periodic,
determine its fundamental period.
6552111 Signals and Systems6552111 Signals and Systems
Periodic SignalsPeriodic Signals
ttx )
4
cos()(.1
π
+=
24Sopapun Suwansawang
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
+=
=
=
+=
+=

25. Solve EX.1
6552111 Signals and Systems6552111 Signals and Systems
Periodic SignalsPeriodic Signals
1
4
cos)
4
cos()( 00 =→





+=+= ω
π
ω
π
tttx
ππ 22
25Sopapun Suwansawang
π
π
ω
π
2
1
22
0
0 ===T
x(t) is periodic with fundamental period T0 = 2π.

26. Solve EX.2
6552111 Signals and Systems6552111 Signals and Systems
Periodic SignalsPeriodic Signals
)()(
4
sin
3
cos)( 21 txtxtttx +=+=
ππ
( ) .6
2
cos3/cos)( 111 ==→==
π
ωπ Ttttxwhere
26Sopapun Suwansawang
( )
.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

27. 6552111 Signals and Systems6552111 Signals and Systems
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
−=
−=
27Sopapun Suwansawang
A signal x(t) or x[n] is referred to as an odd signal if
][][
)()(
nxnx
txtx
−=−
−=−

28. 6552111 Signals and Systems6552111 Signals and Systems
Even and Odd Signals:Even and Odd Signals:
28Sopapun Suwansawang

29. 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,
6552111 Signals and Systems6552111 Signals and Systems
Even and Odd Signals:Even and Odd Signals:
)()()( txtxtx oe +=
29Sopapun Suwansawang
][][][ nxnxnx oe
oe
+=
{ }
{ }][][
2
1
][
)()(
2
1
)(
nxnxnx
txtxtx
e
e
−+=
−+= { }
{ }][][
2
1
][
)()(
2
1
)(
nxnxnx
txtxtx
o
o
−−=
−−=
even part odd part

30. 6552111 Signals and Systems6552111 Signals and Systems
Even and Odd Signals:Even and Odd Signals:
Example 4 Find the even and odd components of
the signals shown in figure below
30Sopapun Suwansawang
Solve even part { })()()(2 tftftfe −+=
2fe(t)

31. Example 4 : continue
Odd part
6552111 Signals and Systems6552111 Signals and Systems
Even and Odd Signals:Even and Odd Signals:
{ })()()(2 tftftfo −−=
2fo(t)
31Sopapun Suwansawang
2fo(t)

32. Example 4 : continue
Check
)()()( tftftf oe +=
6552111 Signals and Systems6552111 Signals and Systems
Even and Odd Signals:Even and Odd Signals:
{ })()(
2
1
)( tftftfo −−={ },)()(
2
1
)( tftftfe −+=
32
)()()( tftftf oe +=
Sopapun Suwansawang
2
fo(t)
fe(t)

33. 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
6552111 Signals and Systems6552111 Signals and Systems
Even and Odd Signals:Even and Odd Signals:
(even)(even)=even
(even)(odd)=odd
(odd)(even)=odd
(odd)(odd)=even
33Sopapun Suwansawang

34. 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.
6552111 Signals and Systems6552111 Signals and Systems
Even and Odd Signals:Even and Odd Signals:
Let )()()( txtxtx =
34Sopapun Suwansawang
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 ==−−=−−=−

35. 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
6552111 Signals and Systems6552111 Signals and Systems
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
35Sopapun Suwansawang

36. 6552111 Signals and Systems6552111 Signals and Systems
Deterministic and Random Signals:Deterministic and Random Signals:
Deterministic
36Sopapun Suwansawang
Random

37. 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
6552111 Signals and Systems6552111 Signals and Systems
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
37Sopapun Suwansawang
∞<< 1tt
−∞>> 1tt

38. RightRight--Handed and LeftHanded and Left--Handed SignalsHanded Signals
6552111 Signals and Systems6552111 Signals and Systems
Right-Handed
1t
38Sopapun Suwansawang
Left-Handed
1
1t

39. 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.
6552111 Signals and Systems6552111 Signals and Systems
for all positive time.
Noncausal signals are signals that have
nonzero values in both positive and
negative time.
39Sopapun Suwansawang

40. Causal vs.Anticausal vs. NoncausalCausal vs.Anticausal vs. Noncausal
6552111 Signals and Systems6552111 Signals and Systems
Causal
40Sopapun Suwansawang
Anticausal
Noncausal

41. Energy and Power SignalsEnergy and Power Signals
6552111 Signals and Systems6552111 Signals and Systems
Consider :
41Sopapun Suwansawang
Rti
R
tv
titvtp
)(
)(
)()()(
2
2
=
=
⋅=
∫∫
∫
∞
∞−
∞
∞−
∞
∞−
==
=
)()()()(
1
)(
22
tdtitdtv
R
dttpE
Power Energy

42. Total energy E and average power P on a per-ohm
basis are
Energy and Power SignalsEnergy and Power Signals
6552111 Signals and Systems6552111 Signals and Systems
dttiE ∫
∞
∞−
= )(2
Joules
42Sopapun Suwansawang
dtti
T
P
T
TT
∫
−∞→
∞−
= )(
2
1 2
lim Watts

43. For an arbitrary continuous-time signal x(t), the
normalized energy content E of x(t) is defined as
∫∫
−∞→
∞
∞−
==
T
TT
dttxdttxE
22
)()( lim
Energy and Power SignalsEnergy and Power Signals
6552111 Signals and Systems6552111 Signals and Systems
The normalized average power P of x(t) is
defined as
43
−∞→∞− TT
∫
−∞→
=
T
TT
dttx
T
P
2
)(
2
1
lim
Sopapun Suwansawang

44. Similarly, for a discrete-time signal x[n],
the normalized energy content E of x[n] is defined
as
Energy and Power SignalsEnergy and Power Signals
6552111 Signals and Systems6552111 Signals and Systems
∑∑
−=∞→
∞
−∞=
==
N
NnNn
nxnxE
22
][lim][
The normalized average power P of x[n] is defined
as
44Sopapun Suwansawang
−=∞→−∞= NnNn
∑
−=∞→ +
=
N
NnN
nx
N
P
2
][
12
1
lim

45. 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 = ∞∞∞∞.
Energy and Power SignalsEnergy and Power Signals
6552111 Signals and Systems6552111 Signals and Systems
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.
45Sopapun Suwansawang
∫=
0
0
2
0
)(
1
T
dttx
T
P

46. Exercise Determine whether the following
signals are energy signals, power signals, or
neither.
1.
Energy and Power SignalsEnergy and Power Signals
6552111 Signals and Systems6552111 Signals and Systems
)cos()( 0 θω += tAtx1.
2.
3.
46Sopapun Suwansawang
)cos()( 0 θω += tAtx
tj
eAtx 0
)( ω
=
)()( 3
tuetv t−
=

47. Solve Ex.1
The signal x(t) is periodic with T0=2π/ω0.
Energy and Power SignalsEnergy and Power Signals
6552111 Signals and Systems6552111 Signals and Systems
dttA
T
dttx
T
P
TT
∫∫ +==
00
0
0
22
00
2
0
)(cos
1
)(
1
θω
)cos()( 0 θω += tAtx
47Sopapun Suwansawang
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.

48. Energy and Power SignalsEnergy and Power Signals
6552111 Signals and Systems6552111 Signals and Systems
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
48Sopapun Suwansawang
∫=
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
= = = ⋅
=
∫ ∫