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The Fourier series expansion of a periodic function f(t) contains only odd harmonics of sine waves. It is clearly say that funciton f(t) is an
- a)Even function with half-wave symmetry.
- b)Odd function with half-wave symmetry.
- c)Even function without half-wave symmetry
- d)Odd function without half-wave symmetry
Correct answer is option 'B'. Can you explain this answer?
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The Fourier series expansion of a periodic function f(t) contains only...
Explanation:
The Fourier series expansion of a periodic function f(t) represents the function as a sum of sine and cosine waves with different frequencies (harmonics). The expansion can be written as:
f(t) = a0 + ∑[an*cos(nωt) + bn*sin(nωt)]
where a0, an, and bn are the coefficients of the expansion, ω is the fundamental frequency, and n is the harmonic number.
The given statement says that the Fourier series expansion of the function f(t) contains only odd harmonics of sine waves. This means that all the coefficients for the cosine terms (an) are zero, and only the sine terms (bn) are present.
Odd Function:
An odd function is a function that satisfies the property f(-t) = -f(t), which means that the function is symmetric with respect to the origin (0,0). If a function contains only odd harmonics, it can be concluded that the function is odd.
Half-Wave Symmetry:
Half-wave symmetry refers to the property of a function where its shape repeats itself every half cycle. In other words, the function is symmetric about the midpoint of each half-cycle. This symmetry can be observed when the function is plotted.
Explanation of the Correct Answer:
The given statement says that the Fourier series expansion of the function f(t) contains only odd harmonics of sine waves. Since the function contains only sine terms, this implies that the coefficients for the cosine terms (an) are zero. Therefore, the function is odd.
Additionally, the function is periodic, and the presence of only odd harmonics indicates that the function has half-wave symmetry. This means that the function repeats itself every half cycle, which is a characteristic of odd functions.
Hence, the correct answer is option 'B' - Odd function with half-wave symmetry.
The Fourier series expansion of a periodic function f(t) represents the function as a sum of sine and cosine waves with different frequencies (harmonics). The expansion can be written as:
f(t) = a0 + ∑[an*cos(nωt) + bn*sin(nωt)]
where a0, an, and bn are the coefficients of the expansion, ω is the fundamental frequency, and n is the harmonic number.
The given statement says that the Fourier series expansion of the function f(t) contains only odd harmonics of sine waves. This means that all the coefficients for the cosine terms (an) are zero, and only the sine terms (bn) are present.
Odd Function:
An odd function is a function that satisfies the property f(-t) = -f(t), which means that the function is symmetric with respect to the origin (0,0). If a function contains only odd harmonics, it can be concluded that the function is odd.
Half-Wave Symmetry:
Half-wave symmetry refers to the property of a function where its shape repeats itself every half cycle. In other words, the function is symmetric about the midpoint of each half-cycle. This symmetry can be observed when the function is plotted.
Explanation of the Correct Answer:
The given statement says that the Fourier series expansion of the function f(t) contains only odd harmonics of sine waves. Since the function contains only sine terms, this implies that the coefficients for the cosine terms (an) are zero. Therefore, the function is odd.
Additionally, the function is periodic, and the presence of only odd harmonics indicates that the function has half-wave symmetry. This means that the function repeats itself every half cycle, which is a characteristic of odd functions.
Hence, the correct answer is option 'B' - Odd function with half-wave symmetry.
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The Fourier series expansion of a periodic function f(t) contains only odd harmonics of sine waves. It is clearly say that funciton f(t) is ana)Even function with half-wave symmetry.b)Odd function with half-wave symmetry.c)Even function without half-wave symmetryd)Odd function without half-wave symmetryCorrect answer is option 'B'. Can you explain this answer?
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