GATE 2016 EE (EE2) – Question 20

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

A real periodic function satisfies $f(-x)=-f(x)$. Its general Fourier-series form is

$a_0+\sum_{k=1}^{\infty}a_k\cos(kx)$
$\sum_{k=1}^{\infty}b_k\sin(kx)$
$a_0+\sum_{k=1}^{\infty}a_{2k}\cos(kx)$
$\sum_{k=0}^{\infty}a_{2k+1}\sin((2k+1)x)$

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: (B) $\sum_{k=1}^{\infty}b_k\sin(kx)$

Explanation

The stated symmetry makes f odd. Over a symmetric period its average is zero, and its product with any cosine is odd, so every cosine coefficient vanishes.

Only sine terms remain. Odd symmetry does not restrict the sine harmonics to odd-numbered k; that needs an additional half-wave symmetry. Thus the general form is **B**.

More Fourier series representation of continuous and discrete time periodic signals, Sampling theorem questions

GATE 2016 EE (EE2) Q38 – The spectra X 1(),X 2() have edge bandwidths B1,B2 as shown. The product x 1(t)x 2(t)…
GATE 2026 EE Q36 – A time-limited waveform g(x) is specified as follows:…
GATE 2025 EE Q30 – A continuous time periodic signal x(t) is x(t)=1+22 t+24 t+26 t. If T is the period of…
GATE 2023 EE Q63 – A signal x(t)=2(180 t)(60 t) is sampled at 200 Hz and then passed through an ideal low…
GATE 2022 EE Q49 – The discrete time Fourier series representation of a signal x[n] with period N is written…
GATE 2019 EE Q38 – A periodic function f(t), with a period of 2, is represented as its Fourier series, f(t)=a…

All Fourier series representation of continuous and discrete time periodic signals, Sampling theorem questions →