Electronics and Communication Engineering (ECE) Exam > Electronics and Communication Engineering (ECE) Questions > Which of the following cannot be the Fourier ...
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Which of the following cannot be the Fourier series expansion of a periodic signal?
- a)x(t) = 2 cos t + 3 cos 3t
- b)x(t) = 2 cospt + 7 cos t
- c)x(t) = cos t + 0.5
- d)x(t) = 2 cos 1.5pt + sin 3.5pt
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
Which of the following cannot be the Fourier series expansion of a per...
x(t) = cos t + 0.5
not satisfies the Dirichlet condition.
The integration of constant term is ∞.
Most Upvoted Answer
Which of the following cannot be the Fourier series expansion of a per...
Fourier Series Expansion
The Fourier series expansion is used to represent a periodic signal as a sum of harmonically related sinusoids. It is given by the equation:
x(t) = a0 + ∑(n=1 to infinity) (an cos(nω0t) + bn sin(nω0t))
where a0, an, and bn are the Fourier coefficients, ω0 is the fundamental frequency, and t is time.
Cannot be the Fourier series expansion
Option C, x(t) = cos(t) + 0.5, cannot be the Fourier series expansion of a periodic signal because it contains a constant term (0.5) which does not oscillate with time. The Fourier series expansion represents a periodic signal as a sum of harmonically related sinusoids, and therefore, should not have a constant term.
Other Options
Option A, x(t) = 2 cos(t) + 3 cos(3t), can be represented as the Fourier series expansion:
x(t) = 2 cos(t) + 3/2 cos(2t) - 1/2 cos(4t)
Option B, x(t) = 2 cos(pt) + 7 cos(t), can be represented as the Fourier series expansion:
x(t) = 2 cos(pt) + 7/2 cos(2t) - 3/2 cos(3t) + 1/2 cos(4t) - 1/2 cos(5t) + ...
Option D, x(t) = 2 cos(1.5pt) + sin(3.5pt), can be represented as the Fourier series expansion:
x(t) = √5 cos(0.5t - θ)
where θ = arctan(7/3) and the Fourier coefficients are given by:
a0 = 0
an = (-1)n+1 √5/(nπ) sin(nθ) for n = 1, 2, 3, ...
bn = (-1)n+1 √5/(nπ) cos(nθ) for n = 1, 2, 3, ...
Conclusion
Option C cannot be the Fourier series expansion of a periodic signal because it contains a constant term, which violates the condition of representing a periodic signal as a sum of harmonically related sinusoids.
The Fourier series expansion is used to represent a periodic signal as a sum of harmonically related sinusoids. It is given by the equation:
x(t) = a0 + ∑(n=1 to infinity) (an cos(nω0t) + bn sin(nω0t))
where a0, an, and bn are the Fourier coefficients, ω0 is the fundamental frequency, and t is time.
Cannot be the Fourier series expansion
Option C, x(t) = cos(t) + 0.5, cannot be the Fourier series expansion of a periodic signal because it contains a constant term (0.5) which does not oscillate with time. The Fourier series expansion represents a periodic signal as a sum of harmonically related sinusoids, and therefore, should not have a constant term.
Other Options
Option A, x(t) = 2 cos(t) + 3 cos(3t), can be represented as the Fourier series expansion:
x(t) = 2 cos(t) + 3/2 cos(2t) - 1/2 cos(4t)
Option B, x(t) = 2 cos(pt) + 7 cos(t), can be represented as the Fourier series expansion:
x(t) = 2 cos(pt) + 7/2 cos(2t) - 3/2 cos(3t) + 1/2 cos(4t) - 1/2 cos(5t) + ...
Option D, x(t) = 2 cos(1.5pt) + sin(3.5pt), can be represented as the Fourier series expansion:
x(t) = √5 cos(0.5t - θ)
where θ = arctan(7/3) and the Fourier coefficients are given by:
a0 = 0
an = (-1)n+1 √5/(nπ) sin(nθ) for n = 1, 2, 3, ...
bn = (-1)n+1 √5/(nπ) cos(nθ) for n = 1, 2, 3, ...
Conclusion
Option C cannot be the Fourier series expansion of a periodic signal because it contains a constant term, which violates the condition of representing a periodic signal as a sum of harmonically related sinusoids.
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Community Answer
Which of the following cannot be the Fourier series expansion of a per...
B
for fourier series to be possible signal should be periodic.
Here when you calculate periodicity of signal you will get irrational (T1/T2). Therefore non periodic.
for fourier series to be possible signal should be periodic.
Here when you calculate periodicity of signal you will get irrational (T1/T2). Therefore non periodic.
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Which of the following cannot be the Fourier series expansion of a periodic signal?a)x(t) = 2 cos t + 3 cos 3tb)x(t) = 2 cos pt + 7 cos tc)x(t) = cos t + 0.5d)x(t) = 2 cos 1.5pt + sin 3.5 ptCorrect answer is option 'C'. Can you explain this answer?
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