BEE Signals & Systems 2 — Questions and Answers
Question 1: A discrete-time signal x[n] is defined as x[n] = u[n] - u[n-4]. What is the duration of this signal?
- 3 samples
- 4 samples (Correct answer)
- 5 samples
- Infinite
Correct answer: 4 samples
x[n] is nonzero for n = 0, 1, 2, 3, giving exactly 4 samples.
Question 2: The Z-transform of δ[n - k] for k > 0 is:
- z^k
- z^(-k) (Correct answer)
- 1
- k·z^(-1)
Correct answer: z^(-k)
By the time-shifting property of the Z-transform, δ[n-k] ↔ z^(-k).
Question 3: Which condition must hold for a system described by H(z) to be causal and stable?
- All poles inside the unit circle (Correct answer)
- All zeros outside the unit circle
- All poles outside the unit circle
- ROC excludes the unit circle
Correct answer: All poles inside the unit circle
A causal, stable discrete-time LTI system requires all poles to lie strictly inside the unit circle.
Question 4: The energy of the complex exponential signal x(t) = e^(jω₀t) over one period T₀ is:
- 0
- T₀ (Correct answer)
- 1/T₀
- ∞
Correct answer: T₀
|e^(jω₀t)|² = 1, so integrating over one period T₀ yields energy = T₀.
Question 5: The cross-correlation of two signals x(t) and y(t) evaluated at τ = 0 equals:
- The convolution of x(t) and y(t)
- The inner product ∫x(t)y(t)dt (Correct answer)
- The autocorrelation of x(t)
- The power spectral density
Correct answer: The inner product ∫x(t)y(t)dt
R_xy(0) = ∫x(t)y(t)dt, which is the inner product of x and y.
Question 6: A signal x(t) has Fourier transform X(jω). What is the Fourier transform of x(t - t₀)?
- X(jω)·e^(jωt₀)
- X(jω)·e^(-jωt₀) (Correct answer)
- X(j(ω - t₀))
- e^(jωt₀)/X(jω)
Correct answer: X(jω)·e^(-jωt₀)
The time-shift property gives FT{x(t-t₀)} = e^(-jωt₀)·X(jω).
Question 7: The Nyquist sampling theorem states that to perfectly reconstruct a bandlimited signal with maximum frequency f_max, the sampling rate must be:
- At least f_max
- Greater than f_max
- At least 2·f_max (Correct answer)
- Exactly f_max/2
Correct answer: At least 2·f_max
The sampling rate must be at least twice the highest frequency component to avoid aliasing.
BEE Signals & Systems 2
A discrete-time signal x[n] is defined as x[n] = u[n] - u[n-4].
What is the duration of this signal?