A signal
is periodic with period ‘N’
, if and if
is a ratio of integers (rational).
N is the time period.
(a) Consider the signal.

The value of
is,

Consider the expression for
.
…… (1)
Here,
is ratio of integers.
Hence, the signal
is periodic.
From equation (1),

Therefore, the signal
has fundamental period
.
(b)
Consider the signal.

The value of
is,

Consider the expression for
.
…… (2)
Here,
is ratio of integers.
Hence, the signal
is periodic.
From equation (2),
N = 7
Therefore, the signal
has fundamental period
.
(c) Consider the signal.

The value of
is,

Consider the expression for .
…… (3)
Here,
is ratio of integers.
Hence, the signal
is periodic.
From equation (3),
N = 2
Therefore, the signal
has fundamental period
.
(d) Consider the signal.

The value of
is,

Consider the expression for
.

Here,
is not rational.
Hence, the signal
is not -periodic.
(e) Consider the signal.

The value of
is,

Consider the expression for
.

…… (4)
Here,
is ratio of integers.
Hence, the signal
is periodic.
From equation (4),
N = 10.
Therefore, the signal
has fundamental period
.
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