, = 10Jl"
27l" 21r
=-=1Orr
OJ
0
0
0
r(t + r,) = Sin[10ir(t +
,
itr ))
, = Sin(IO� -0JP ))= sin(l �m) cos(21r) + cos(l0m)sin(2n")
= s;�(1om)
:. x(t) = x(, + rJ
. X(t) is periodic·
.
x(t)=!OSin�-30) - � •
� Prove that x (t) is periodic
,X:
6
i1r
X(t + T0 = 1 OSin[l 0.1l"(t + )- 30]
)
· l01r
= lOSin[lOm + 21r - 30]
= I0Sin(I0m-30)
= X(t)
:. X(t + To)= X(t)
:. X(t) is periodic
IX(t�=V
°
0 =phase.,5hift =30
r·
The sum of N periodic continuous time signals is not necessarily periodic.
It is periodic with period T iff the following condition is satisfied� i.e.
F = Rational function =ratio of integers , 2 �; � N
"he period of the sum signals is given by:
I
.
.•.
. 11 2,r
l
w1ere:
T = peno
. d of t he sum s1gna 1 =6
{JJ 0
• I
'
1; = period of the first signal in the sum signal .
1P
1: = period of the i signal in the SU(Jl.
11
.
.
.
.
.
.
T
=· the ieast co1�·1mo � multiple (l.c.m) of the denomfoator of �
1
gnal .
1 = fundamental frequency of the sum si
·.· fo __
- T
l in3t
X(t) = trSinO. 6t - 3Cos1. St + 1S
0
EX :
For
c
Find the fundamental frequen y
Th� su� of two or mor� Sinusoi�s �ay _or �ay not be p�riodic, depending on
the relationships between therr respective penods or frequencies.
If the ratio of their periods can be expressed as a rational ,or their frequencies;
commensurable , if they have a common measure , that is there is w a number
contained in each an integer number of times
Thus w is.the largest such number,
tt1i= n 1 W
&
W2 =n 2 W
.
,
where: , n 1. and n2 are integers
O
0
0
0
I
= fundamental frequency = l
'
/
0
To
2,r
w 0 =To
·: r. = -1
,
f1
1
and r, = · ,
-.,· f2
, n2 ' . , .
n,f.
f, =. -Ti = -· :, - = -· =rahona l munber.
0
T,1.
f1
11Jo
nl
The sum of two signals is periodic iff :
;:T. =w 2 =-n,w = n2
- = rational number
.
n1 W" 11 1
W1
72
:. T1 11 1 = 1;n 2 = r,, =
period of the sum of two
0
signals ·
\ sin� 1
u
y
b
. r n al
.
.
·
• Y ou can determme the fundamental frequen�y Of. thesuins1gt corn 01on.
. .
egre•a faII
h
t
\'
na
great common d1v1so
ig
s
·
m
u
r
of the elements of the s .
sion o
. · ·
i
v
.
i
d
a
s
1
t
tI iviswn ( G .C.D.) is defined
as the largest number tlrn .
frequencies of the su1n sig·n�il .
I
FOR EXAMPLE
, 1. ., X(t) = Sin2t + Sin2m, is x(t) periodic ?
<J&·zn) have no G.C.D
� noL_�r0B
2. X(t) = nSinW-3Cosl3t + l lSi@ , is x(t) periodic?
J
1.5
0.6
flx_QJ
fix-
/'f
/ :.G.C.D.(9 =iU,.
15x0.1
30x0.1
5x0.3
?I0x0.3
:. X(t) is period_ ic
!'\ 2n ·2,r
:\...Li==-sec
o
w
,,
OJ'
,
&:. =���
,.,__
_ c_ yo
_ _f_t-he_s_u_m_s_ign, - -a-l x-(-t)....,l
_ n
'i
r:===- fundanieirta_l_fr_�_ue
The phasor signals (in time domain):
The sinosoidal signal is given by:
x(t) = A cos ( cot ± 0 � { e.. '\J r
OR
�>
x(t)=Asin(cQ±)
,,.
where:
A= peak value of x(t) , maximum value of x(t) or magnitude of x(t)
co = frequency in rad / sec.
8 = phase shift in rad.
e;:._�
0
Eulers theorem states that the complex exponential ej can be written as
Im
follows:
e O = cos e + j sin e
�os0:d�w
j sine
�"a = Im [ ej e )
o cos e
Re
A e1 ° = A cos 0 + j A sin 0
A eJ "'1 = cos O)t t j sin rot
where : j = ✓
-1 , j 2 = -1 .
j
we see that ei "'1 js a complex - valued function with real part cos rot
and Imaginary part sin rot . If we plot e1 'J) on a complex plane then t
is a variable parameter and co is constant then l' will rotate around a unit circle,
where: as shown in fig. ( ).
1
11
)
f
CX)
P (30) - Evaluate
f
e - a,
��
e-at
�
2
•
2
6(t - to)dt
t
•
J
8 ( t - (O)dt =
CX)
'\
i
t
e- a( +l�>
-�
i
�
L. .
8 ( t)dt
d [e -a(t+l0)
-.�- -
2
.. · dt
]
d ( e - a(t +20tt100)
_ -·- ) �0
dt
2
· d[
=--e
- dt
-at
1
e
-20a.t
e
-l00a.
)
t-o
= _ e -I00a i.[ -at -20at
.
. . . e . e • . I t=o
2
==_
dt .''
. ,
I
te �at
f-lOOa
l
e- OOa
= - 2at
[
e
J
-111 �
u ( t- lO)dt
.'. e
.
_,,,
2
•
=-
e-lOOa
d
-e
dt
(-20a.e -lOat j + e -�·oat (-2a.e -at )l =
I = 20ae
- at\-20at
(
t= lO
Or·: J x(t)'5(t- t,)dt = - ; (to)
CJ')
l
. .
ut
1
I 1=10
I - = 20ae
-20a. _ 2�
_ t)J t=0
-IOOa
l 10
-
t O
-1000
/
a1
== _ [ -2at e� .] 1=1
0
== 2oae-1001l 1 =IO = 20a.e-•ooo
Q
.., "so(
J
p (31) - Evaluate [
.
t) + e-(l-1)
.
j So(t)dt == 5
.....,
&
.....,
..,
..,
J 6(t)dt
e-
,2
•
o( t )] l=O dt
=5
-<O
· J Cos51tt6(t)dt
&
.
6( t) + Cos5mo( t) +
= Cqs. 21tt I 1=0 = Cos 9 = I
-<O
-<O
=5+e+l+0=6+e
P (33) - Sketch the following signals , and determine whether the sigm
power or energy or. neither .
I�- x (t) = A Sin t
��
\Jo
)
--"-'
�? < t < ?
b)-x t =A tJ(t+A)-u(t-A)]
c)-x(t)=c·• 111
d) - X t = ll (t)
e) - x ( t) = t u(t)
.1
4X
11,>0
4
a>O