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

image

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A signal image is periodic with period ‘N’ image, if and if

image is a ratio of integers (rational).

 N is the time period.

(a) Consider the signal.

image

The value of image is,

image

Consider the expression for image.

image  …… (1)

Here, image is ratio of integers.

Hence, the signal image is periodic.

From equation (1),

image

Therefore, the signal image  has fundamental period image.

(b)

Consider the signal.

image

The value of image is,

image

Consider the expression for image.

image …… (2)

Here,  image is ratio of integers.

Hence, the signal image  is periodic.

From equation (2),

N = 7

Therefore, the signal image  has fundamental period image.

(c) Consider the signal.

image

The value of image is,

image

Consider the expression for .

image …… (3)

Here,  image is ratio of integers.

Hence, the signal image is periodic.

From equation (3),

N = 2

Therefore, the signal  image has fundamental period image.

(d) Consider the signal.

image

The value of image is,

image

Consider the expression for image.

image

Here, image is not rational.

Hence, the signal  image is not -periodic.

(e) Consider the signal.

image

The value of  image is,

image

Consider the expression for image .

image

image  …… (4)

Here, image is ratio of integers.

Hence, the signal image is periodic.

From equation (4),

N = 10.

Therefore, the signal  image has fundamental period image.

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