GATE 2017 EE – Question 26

Signals and Systems · Fourier series representation of continuous and discrete time periodic signals, Sampling theorem · 1 mark · Numerical answer

Consider $g(t) = \begin{cases} t - \lfloor t \rfloor, & t \geq 0 \\ t - \lceil t \rceil, & \text{otherwise} \end{cases}$, where $t \in \mathbb{R}$.

Here, $\lfloor t \rfloor$ represents the largest integer less than or equal to $t$ and $\lceil t \rceil$ denotes the smallest integer greater than or equal to $t$. The coefficient of the second harmonic component of the Fourier series representing $g(t)$ is ________.

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Correct answer: -0.17 to -0.15

Explanation

The sawtooth $t - \lfloor t \rfloor$ has period 1 and the Fourier series $\frac{1}{2} - \sum_{k \geq 1}\frac{\sin(2\pi k t)}{\pi k}$. The coefficient of the second harmonic ($k = 2$) is $-\frac{1}{2\pi} = -0.159$. For $t < 0$ the function $t - \lceil t \rceil$ is the same sawtooth shifted down by 1, so $g(t)$ is not periodic, which is why the official key gives marks to all candidates.