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GATE 2015 EE – Question 38

Signals and Systems · Fourier series representation of continuous and discrete time periodic signals, Sampling theorem · 2 marks · Multiple choice

The signum function is given by

$$sgn(x) = \begin{cases} \frac{x}{|x|}; & x \neq 0 \\ 0; & x = 0 \end{cases}$$

The Fourier series expansion of $sgn(\cos(t))$ has

  1. only sine terms with all harmonics.
  2. only cosine terms with all harmonics.
  3. only sine terms with even numbered harmonics.
  4. only cosine terms with odd numbered harmonics.

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Show answer and explanation

Correct answer: (D) only cosine terms with odd numbered harmonics.

Explanation

$sgn(\cos t)$ is a square wave that is even about $t = 0$, so it contains only cosine terms. It also has half-wave symmetry, since shifting by half a period flips its sign, so only odd harmonics are present.