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Determine which of the following sinusoids are periodic and compute their fundamental period.

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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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