Download Periodic signals

Periodic signals
September 6, 2016
A continuous time signal x(t) is period if and only if
x(t) = x(t + kT0 ),
where k is integer number and T0 is a real number. T0 is called the fundamental period. T0 is the smallest number greater than 0. kT0 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( π3 t)
x(t)
1
−4 −3 −2 −1
1
2
3
4
5
6
7
−1
The fundamental period of the signal T0 = 6.
2. x(t) = sin( √π2 t)
1
8
9 10 11 12 13 14 15 16 17 t
x(t)
1
−4 −3 −2 −1
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 t
−1
√
The fundamental period of the signal T0 = 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( π3 t) + cos( π2 t)
signals
1
−4 −3 −2 −1
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 t
−1
The first signal sin( π3 t) has the fundamental period equal with 6. The
second signal cos( π2 t) has the fundamental period equal with 4.
2
x(t)
2
1
−4 −3 −2 −1
−1
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 t
−2
What will be the period of the signal x(t)?
The first signal sin( π3 t) has the set of periods {6, 12, 18, 24, 30, . . .}.
The second signal cos( π2 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( 2π
t) + cos( 3π
t)
3
2
signals
1
−4 −3 −2 −1
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 t
−1
The first signal sin( 2π
t) has the fundamental period equal with 3. The
3
second signal cos( 3π
t)
has the fundamental period equal with 43 .
2
3
x(t)
2
1
−4 −3 −2 −1
−1
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 t
−2
The fundamental period of the signal x(t) will be the
4
9 4
LCM (3, ) = LCM ( , )
3
3 3
LCM (9, 4)
=
3
36
=
3
= 12
3. x(t) = sin( √π2 t) + cos( π2 t)
signals
1
−4 −3 −2 −1
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 t
−1
√
The first signal sin( √π2 t) has the fundamental period equal with 2 2.
The second signal cos( π2 t) has the fundamental period equal with 4.
4
x(t)
2
1
−4 −3 −2 −1
−1
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 t
−2
The fundamental period of the signal x(t) will be the
√
√
LCM (2 2, 4) = 2LCM ( 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( π3 t) cos( π2 t)
signals
1
−4 −3 −2 −1
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 t
−1
The first signal sin( π3 t) has the fundamental period equal with 6. The
second signal cos( π2 t) has the fundamental period equal with 4.
5
x(t)
2
1
−4 −3 −2 −1
−1
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 t
−2
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
π
π 1
sin( t) cos( t)
x(t) =
2
3
2
1
π
π
π
π =
sin( t + t) − sin( t − t)
2
3
2
3
2
1
π
5π
=
sin( t) − sin( t)
2
6
6
The first signal sin( 5π
t) has the fundamental period 12
and the second
6
5
π
signal sin( 6 t) has the fundamental period 12. Therefore, if we apply
the LCM on these two values we will get 12.
2. x(t) = cos( π4 t) cos( π4 t)
signals
1
−4 −3 −2 −1
1
2
3
4
−1
6
5
6
7
8
9 10 11 12 13 14 15 16 17 t
The first signal cos( π4 t) has the fundamental period equal with 8. The
second signal cos( π4 t) has the fundamental period equal with 8.
x(t)
1
−4 −3 −2 −1
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 t
−1
If we take the LCM of the two fundamental periods LCM (8, 8) = 8.
However this values does not represent the fundamental period.
1
π
π cos( t) cos( t)
2
4
4
π
π
π
π 1
cos( t − t) + cos( t + t)
=
2
4
4
4
4
1
π =
1 + cos( t)
2
2
x(t) =
x(t)
1
−4 −3 −2 −1
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 t
−1
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.
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A continuous time signal x[n] is period if and only if
x[n] = x[n + kN0 ],
where k is an integer number and N0 is an integer number. N0 is called the
fundamental period. N0 is the smallest number greater than 0. kN0 is called
the period of the signal.
The same things apply for the discrete time signals.
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