BEE Signals & Systems 4 — Questions and Answers
Question 1: The convolution of u(t) with itself, u(t) * u(t), results in:
- u(t)
- t·u(t) (Correct answer)
- δ(t)
- t²·u(t)/2
Correct answer: t·u(t)
Convolving two unit step functions yields the ramp signal r(t) = t·u(t).
Question 2: The power spectral density (PSD) of a wide-sense stationary (WSS) random process is the Fourier transform of its:
- Probability density function
- Mean function
- Autocorrelation function (Correct answer)
- Cross-correlation function
Correct answer: Autocorrelation function
By the Wiener-Khinchin theorem, the PSD is the Fourier transform of the autocorrelation function.
Question 3: A system is memoryless if its output at time t depends only on:
- Past inputs
- Future inputs
- The input at the same instant t (Correct answer)
- The derivative of the input
Correct answer: The input at the same instant t
A memoryless system's output depends solely on the present input value, with no dependency on past or future.
Question 4: The Laplace transform of the unit ramp r(t) = t·u(t) is:
- 1/s
- 1/s² (Correct answer)
- s
- 1/(s+1)
Correct answer: 1/s²
L{t·u(t)} = 1/s², valid for Re{s} > 0.
Question 5: Gibb's phenomenon occurs when reconstructing a signal from its Fourier series near a:
- Smooth maximum
- Zero crossing
- Discontinuity (Correct answer)
- Periodic repeat
Correct answer: Discontinuity
Gibbs phenomenon produces approximately 9% overshoot near jump discontinuities regardless of how many harmonics are included.
Question 6: The transfer function of an ideal differentiator in the frequency domain is:
- 1/jω
- jω (Correct answer)
- 1/(jω)²
- jω/(1+jω)
Correct answer: jω
Differentiation in time corresponds to multiplication by jω in the frequency domain.
Question 7: Downsampling a discrete-time signal x[n] by factor M produces y[n] = x[Mn]. This operation in the frequency domain causes the spectrum to be:
- Compressed by M
- Expanded by M with aliasing if not bandlimited (Correct answer)
- Shifted by M
- Multiplied by M
Correct answer: Expanded by M with aliasing if not bandlimited
Downsampling by M expands the spectrum by M and introduces aliasing unless the signal is bandlimited to π/M.
BEE Signals & Systems 4
The convolution of u(t) with itself, u(t) * u(t), results in: