Periodic signals
September 6, 2016
A continuous time signal x ( t ) is period if and only if x ( t ) = x ( t + kT
0
) , where k is integer number and T
0 mental period.
T
0 is a real number.
T
0 is called the fundais the smallest number greater than 0.
kT
0 is called the period of the signal.
In this document we will focus on sin, cos and combinations of the two.
1.
x ( t ) = sin(
π
3 t ) x ( t )
1
− 4 − 3 − 2 − 1
− 1
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 t
The fundamental period of the signal T
0
= 6.
2.
x ( t ) = sin( √
2 t )
1

x ( t )
1
− 4 − 3 − 2 − 1
− 1
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 t
The fundamental period of the signal T
0
= 2
√
2.
Usually signals do not appear by themselves. Signals are composed of basic elementary signals. Given two periodic signals, what would be the period of the composition of the two signals.
First lets talk about addition of two period signals.
1.
x ( t ) = sin( π
3 t ) + cos( π
2 t ) signals
1
− 4 − 3 − 2 − 1
− 1
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 t
The first signal sin(
π
3 second signal cos(
π
2 t ) has the fundamental period equal with 6. The t ) has the fundamental period equal with 4.
2

x ( t )
2
1
− 4 − 3 − 2 − 1
− 1
− 2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 t
What will be the period of the signal x ( t )?
The first signal sin(
π
3 t ) has the set of periods { 6 , 12 , 18 , 24 , 30 , . . .
} .
The second signal cos(
π
2 t ) has the set of periods { 4 , 8 , 12 , 16 , 20 , 24 , . . .
} .
When adding the two signals up we need to perform an intersection of the two sets.
{ 6 , 12 , 18 , 24 , 30 , . . .
}∩{ 4 , 8 , 12 , 16 , 20 , 24 , . . .
} = { 12 , 24 , 36 , 48 , . . .
}
Therefore, the fundamental period of the signal x ( t ) will be the LCM (6 , 4) =
12.
2.
x ( t ) = sin(
2 π
3 t ) + cos(
3 π
2 t ) signals
1
− 4 − 3 − 2 − 1
− 1
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 t
The first signal sin( second signal cos(
3 π
2 π t ) has the fundamental period equal with 3. The
3 t ) has the fundamental period equal with
4
2 3
.
3

x ( t )
2
1
− 4 − 3 − 2 − 1
− 1
− 2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 t
The fundamental period of the signal x ( t ) will be the
LCM (3 ,
4
3
) = LCM (
9
3
,
4
3
)
LCM (9 , 4)
=
3
36
=
3
= 12
3.
x ( t ) = sin( √
2 t ) + cos(
π
2 t ) signals
1
− 4 − 3 − 2 − 1
− 1
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 t
The first signal sin( √
2
The second signal cos(
π
2 t ) has the fundamental period equal with 2
√
2.
t ) has the fundamental period equal with 4.
4

x ( t )
2
1
− 4 − 3 − 2 − 1
− 1
− 2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 t
The fundamental period of the signal x ( t ) will be the
LCM (2
√
2 , 4) = 2 LCM (
√
2 , 2)
The LCM is defined on rational numbers. Therefore in this case even though the signals are periodic, the sum of the signals won’t be periodic simply because the LCM does cannot find an integer number between an irrational and an integer number.
Secondly lets talk about the multiplication of two periodic signals.
1.
x ( t ) = sin(
π
3 t ) cos(
π
2 t ) signals
1
− 4 − 3 − 2 − 1
− 1
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 t
The first signal sin( second signal cos(
π
π t ) has the fundamental period equal with 6. The
3 t ) has the fundamental period equal with 4.
2
5

x ( t )
2
1
− 4 − 3 − 2 − 1
− 1
− 2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 t
LCM (6 , 4) = 12 this will give the period of the multiplication of the signals. It is not guaranteed to give the fundamental period
If we use trig identities we end up with
1 x ( t ) =
=
2
1
=
2
2
1 sin( sin( sin(
π t ) cos(
3
π t +
π
π
2 t ) t ) − sin(
3 2
π
2
5 π
6 t ) − sin(
π
6 t ) t −
π
3 t )
The first signal sin( signal sin(
π
5 π t ) has the fundamental period
6 the LCM on these two values we will get 12.
12 and the second
6 5 t ) has the fundamental period 12. Therefore, if we apply
2.
x ( t ) = cos(
π
4 t ) cos(
π
4 t ) signals
1
− 4 − 3 − 2 − 1
− 1
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 t
6

The first signal cos( second signal cos(
π
π t ) has the fundamental period equal with 8. The
4 t ) has the fundamental period equal with 8.
4 x ( t )
1
− 4 − 3 − 2 − 1
− 1
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 t
If we take the LCM of the two fundamental periods LCM (8 , 8) = 8.
However this values does not represent the fundamental period.
x ( t )
1
1 x ( t ) =
=
2
1
=
2
2
1 cos( cos(
π t ) cos(
4
π
4 t
1 + cos(
−
π
2 t )
π t )
π
4
4 t ) + cos(
π
4 t +
π
4 t )
− 4 − 3 − 2 − 1
− 1
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 t
From this we can conclude that the fundamental period is equal with
4.
It can easily be seen that in the case of signal multiplication there is no guarantee that the fundamental period can be given by the LCM function.
7

A continuous time signal x [ n ] is period if and only if x [ n ] = x [ n + kN
0
] , where k is an integer number and N
0 fundamental period.
N
0 the period of the signal.
is an integer number.
N is the smallest number greater than 0.
0 is called the kN
0 is called
The same things apply for the discrete time signals.
8