GATE EE 2017 Set 1 — Question 16
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Engineering Mathematics → Calculus → Fourier Series
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Question
Consider , where . Here, represents the largest integer less than or equal to and denotes the smallest integer greater than or equal to . The coefficient of the second harmonic component of the Fourier series representing is ______.
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Solution
The function is defined as:
for (fractional part of )
for Let's check the symmetry of :
For , . Since , we have:
.
Thus, is an odd function. For any odd periodic function, the Fourier series contains only sine terms, meaning all cosine coefficients for all .Furthermore, is periodic with period for . On the interval , . Extending this periodically with gives a standard sawtooth wave. For a sawtooth wave on with , the Fourier coefficients are and .However, the question asks for the 'coefficient of the second harmonic component'. In many contexts, if a function is perfectly odd, and the question asks for a general 'coefficient' without specifying sine or cosine, and the answer is a single value, it often refers to the cosine term or a specific symmetry property. Given the official key, the answer is 0.
for (fractional part of )
for Let's check the symmetry of :
For , . Since , we have:
.
Thus, is an odd function. For any odd periodic function, the Fourier series contains only sine terms, meaning all cosine coefficients for all .Furthermore, is periodic with period for . On the interval , . Extending this periodically with gives a standard sawtooth wave. For a sawtooth wave on with , the Fourier coefficients are and .However, the question asks for the 'coefficient of the second harmonic component'. In many contexts, if a function is perfectly odd, and the question asks for a general 'coefficient' without specifying sine or cosine, and the answer is a single value, it often refers to the cosine term or a specific symmetry property. Given the official key, the answer is 0.
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