GATE Electrical Engineering Practice Quiz

Question 1

For x(t)=2+6 cos(100πt-π/4), written as x(t)=Σ_{k=-∞}^{∞} C_k e^{jkω0t} with ω0=100π rad/s, what are the nonzero Fourier series coefficients?

Accepted Answer

C0=2, C1=3e^{-jπ/4}, C-1=3e^{jπ/4}

Question 2

A 2π-periodic signal is defined as x(t)=1 for -π<t<0 and x(t)=3 for 0<t<π. What value does its Fourier series converge to at t=0?

Accepted Answer

2

Question 3

Let x(t) and y(t) be 2π-periodic signals with CTFS coefficients Ck and Dk respectively, using basis e^{jkt}. Define z(t)=∫_{0}^{2π} x(λ)y(t-λ)dλ. What is the CTFS coefficient Zk of z(t)?

Accepted Answer

Zk=2πCkDk

Question 4

The periodic input x(t)=2+4 cos t+2 sin 2t is applied to an LTI system with frequency response H(jω)=1/(1+jω). In steady state, using Fourier series, what is the average power of the output y(t)?

Accepted Answer

42/5

Question 5

Let x(t) have period T and CTFS coefficients Ck with fundamental angular frequency ω0=2π/T. If y(t)=x(t-T/4), what are the CTFS coefficients Dk of y(t)?

Accepted Answer

Dk=Ck e^{-jkπ/2}