This set of Signals & Systems Multiple Choice Questions & Answers (MCQs) focuses on “Trigonometric Fourier Series”.

1. The Fourier series coefficient of time domain signal x (t) is X[k] = jδ[k-1] – jδ[k+1] + δ[k+3] + δ[k-3], the fundamental frequency of the signal is ω=2π. The signal is ___________

a) 2(cos 3πt – sin πt)

b) -2(cos 3πt – sin πt)

c) 2(cos 6πt – sin 2πt)

d) -2(cos 6πt – sin 2πt)

2. The unit impulse response of the system c (t) = -4e^{-t} + 6e^{-2t} for t>0. The step response of the same system for t ≥ 0 is _______________

a) -3e^{-2t} – 4e^{-t} + 1

b) -3e^{-2t} + 4e^{-t} – 1

c) -3e^{-2t} – 4e^{-t} – 1

d) 3e^{-2t} + 4e^{-t} – 1

3. The Fourier transform of sin(2πt) e-t u (t) is ____________

a) 12j(11+j(ω−2π)+11+j(ω+2π))

b) 12j(11+j(ω−2π)–11+j(ω+2π))

c) 12j(11+j(ω+2π)–11+j(ω−2π))

d) 1j(11+j(ω+2π)–11+j(ω−2π))

4. Given a real valued function x (t) with period T. Its trigonometric Fourier series expansion contains no term of frequency ω = 2π (2k)T; where, k = 1, 2….. Also no terms are present. Then, x(t) satisfies the equation ____________

a) x (t) = x (t+T) = -x (t + T2)

b) x (t) = x (t+T) = x (t + T2)

c) x (t) = x (t-T) = -x (t – T2)

d) x (t) = x (t-T) = x (t – T2)

5. The lengths of two discrete time sequence x_{1}[n] and x_{2}[n] are 5 and 7 respectively. The maximum length of a sequence x_{1}[n] * x_{2}[n] is ____________

a) 5

b) 6

c) 7

d) 11

6. If the FT of x(t) is 2ω sin(πω), then the FT of e^{j5t} x(t) is ____________

a) 2ω−5 sin(πω)

b) 2ω+5 sin{π(ω-5)}

c) 2ω+5 sin(πω)

d) 2ω−5 sin{π(ω-5)}

7. Given a periodic rectangular waveform of frequency 1 kHz, symmetrical about t=0 and having a pulse width of 500 µs and amplitude 5 V. The Fourier series is ___________

a) 2.5 + 3.18 cos (2π × 10^{3} t) – 1.06 cos (6π × 10^{3} t)

b) 2.5 – 3.18 cos (2π × 10^{3} t) – 1.06 cos (6π × 10^{3} t)

c) 2.5 + 3.18 cos (2π × 10^{3} t) + 1.06 cos (6π × 10^{3} t)

d) 2.5 – 3.18 cos (2π × 10^{3} t) + 1.06 cos (6π × 10^{3} t)

8. A waveform is given by v(t) = 10 sin2π 100 t. The magnitude of the second harmonic in its Fourier series representation is ____________

a) 0 V

b) 20 V

c) 100 V

d) 200 V

9. A signal x (t) has its FT as X (f). The inverse FT of X(3f +2) is _____________

a) 12x(t2)ej3πt

b) 13x(t3)e−j4πt/3

c) 3x(3t)e^{-j4πt}

d) x(3t + 2)

10. The first two components of trigonometric Fourier series of the given signal is ____________

a) 0, 4/π

b) 4/π, 0

c) 0, -4/π

d) -4/π, 1

11. A linear phase channel with phase delay T_{p} and group delay T_{g} must have ____________

a) T_{p} = T_{g} = constant

b) T_{p} ∝ f and T_{g} ∝ f

c) T_{p} = constant and T_{g} ∝ f

d) T_{p} ∝ f and T_{g} = constant

12. The impulse response of a continuous time system is given by h(t) = δ(t-1) + δ(t-3). The value of the step response at t=2 is _______________

a) 0

b) 1

c) 2

d) 3

13. The convolution of two continuous time signals x(t) = e^{-t} u(t) and h(t) = e^{-2t} u(t) is ________________

a) (e^{t} – e^{2t}) u (t)

b) (e^{-2t} – e^{-t}) u (t)

c) (e^{-t} – e^{-2t}) u (t)

d) (e^{-t} + e^{-2t}) u (t)

14. The z-transform of cos(π3 n) u[n] is __________

a) z2(2z−1)(z2−z+1), 0<|z|<1

b) z2(2z−1)(z2−z+1), |z|>1

c) z2(1−2z)(z2−z+1), 0<|z|<1

d) z2(1−2z)(z2−z+1), |z|>1

15. The Laplace transform of the function cosh^{2}(t) is ____________

a) s2+2s(s2+4)

b) s2−2s(s2−4)

c) s2−2s(s2+4)

d) s2+2s(s2−4)

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