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 .