GATE EE 2023 Set 1 — Question 49
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Signals & Systems → Signal Representation → Periodic & Aperiodic Signals
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Question
The period of the discrete-time signal described by the equation below is N = \text{__________} (Round off to the nearest integer).
Correct answer
48 to 48
Solution
The given discrete-time signal is .
For a discrete-time sinusoidal signal , the fundamental period is given by , where is the smallest integer such that is an integer. This means must be a rational number.For the first sinusoidal component, .
The period is found such that is an integer.
. The smallest integer for which this is an integer is .For the second sinusoidal component, .
The period is found such that is an integer.
. The smallest integer for which this is an integer is .The fundamental period of the overall signal is the Least Common Multiple (LCM) of the individual fundamental periods and .
.
To find LCM(16, 6):
Prime factorization of 16:
Prime factorization of 6:
LCM(16, 6) = .Therefore, the period of the discrete-time signal is .
Rounding off to the nearest integer, the answer is .
For a discrete-time sinusoidal signal , the fundamental period is given by , where is the smallest integer such that is an integer. This means must be a rational number.For the first sinusoidal component, .
The period is found such that is an integer.
. The smallest integer for which this is an integer is .For the second sinusoidal component, .
The period is found such that is an integer.
. The smallest integer for which this is an integer is .The fundamental period of the overall signal is the Least Common Multiple (LCM) of the individual fundamental periods and .
.
To find LCM(16, 6):
Prime factorization of 16:
Prime factorization of 6:
LCM(16, 6) = .Therefore, the period of the discrete-time signal is .
Rounding off to the nearest integer, the answer is .
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