A signal x(t) is time-limited to 0 to T seconds. What is the minimum sampling rate required to avoid aliasing if its bandwidth is B Hz?
Answer: B
According to Nyquist sampling theorem, minimum sampling rate = 2×(maximum frequency) = 2B Hz.
Q.23Medium
An LTI system has impulse response h(t) = e^(-3t)u(t). Is this system stable?
Answer: A
For stability, ∫|h(t)|dt must be finite. Here, ∫₀^∞ e^(-3t)dt = 31, which is finite. System is stable.
Q.24Medium
The output of a system is y[n] = 0.5y[n-1] + x[n]. What is the transfer function H(z)?
Answer: A
From y[n] = 0.5y[n-1] + x[n], taking Z-transform: Y(z) = 0.5z⁻¹Y(z) + X(z). So H(z) = Y(z)/X(z) = 1/(1-0.5z⁻¹).
Q.25Medium
A sinusoidal signal x(t) = 5sin(2πf₀t + π/4) is sampled at fs = 10f₀. What is the Nyquist frequency?
Answer: C
Nyquist frequency = fs/2 = 10f₀/2 = 5f₀. The Nyquist rate is 2×(highest frequency) = 2f₀.
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Q.26Medium
Which window function provides the narrowest main lobe but highest side lobes in frequency domain?
Answer: B
Rectangular window has the narrowest main lobe (4π/N) but the highest side lobes (-13 dB). Other windows trade main lobe width for lower side lobes.
Q.27Medium
For a sequence x[n] = {1, 2, 1, -1}, what is the circular convolution with itself using 4-point DFT?
Answer: B
Circular convolution in time domain equals multiplication in frequency domain. y[n] = IDFT{X[k]×X[k]} = IDFT{X[k]²}.
Q.28Medium
A signal has autocorrelation R_x(τ) = Ae^(-2|τ|). What is its bandwidth (3dB) approximately?
Answer: A
Power spectral density S_x(ω) = Fourier transform of R_x(τ) = 4A/(4+ω²). At 3dB point: 4+ω² = 8, so ω ≈ 2 rad/s.
Q.29Medium
A filter has poles at z = 0.7 and z = 0.9. What can we infer about its stability and response?
Answer: B
For discrete-time systems, stability requires all poles inside the unit circle (|z| < 1). Both 0.7 and 0.9 satisfy this, so the filter is stable.
Q.30Medium
An analog filter has magnitude response |H(jω)| = 1/√(1+(ω/ωc)⁴). What is the order of the filter?
Answer: B
The denominator power indicates filter order: (ω/ωc)⁴ represents 4th-order filter response.
Q.31Medium
A real discrete signal x[n] has DTFT X(e^jω). What is the relationship between X(e^jω) and X(e^-jω)?
Answer: A
For real signals, the DTFT exhibits Hermitian symmetry: X(e^-jω) = X*(e^jω), meaning magnitude is even and phase is odd.
Q.32Hard
A linear phase FIR filter of length N has which characteristic for group delay?
Answer: A
Linear phase FIR filters have constant group delay = (N-1)/2 samples, which is why they're preferred for applications requiring minimal distortion.
Q.33Hard
A speech signal is band-limited to 4 kHz and sampled at 8 kHz. If 1000 samples are collected, what is the frequency resolution of its DFT?
Answer: C
Frequency resolution = fs/N = 10008000 = 8 Hz. This is the minimum frequency spacing in DFT.
Q.34Hard
A system with H(s) = (s+2)/((s+1)(s+3)) is analyzed. What is the steady-state response to u(t) = 10sin(2t)?
Answer: A
For sinusoidal input, steady-state output is |H(jω)|×A at angle ∠H(jω). Here ω=2, so output amplitude = |H(j2)|×10.
Q.35Hard
In multirate signal processing, if a signal is decimated by factor M and then interpolated by factor M, what is the output relationship to input?
Answer: B
Decimation-interpolation by same factor M recovers original signal only with ideal lowpass filters. Non-ideal filters cause distortion and aliasing.
Q.36Hard
A minimum-phase system has a zero at z = 2. Where should its mirror zero be placed for a linear-phase equivalent system?
Answer: A
For linear phase from minimum-phase, zeros must be placed symmetrically: if z₀ is a zero, then 1/z₀* must also be a zero. For z=2, mirror is z=21.
Q.37Hard
A signal undergoes spectral analysis using FFT with windowing. If spectral leakage is observed, which of the following is NOT a solution?
Answer: D
Spectral leakage is reduced by increasing observation window length, using better windows, or zero-padding FFT. Reducing fs would worsen aliasing and doesn't address leakage.
Q.38Easy
A continuous-time signal x(t) = 5cos(2π×100t) is sampled at 250 Hz. What is the Nyquist frequency required?
Answer: B
Nyquist frequency = 2 × maximum frequency = 2 × 100 = 200 Hz. Since sampling rate (250 Hz) > Nyquist frequency, no aliasing occurs.
Q.39Easy
For a causal LTI system with transfer function H(z) = 1/(1-0.5z^-1), the system is:
Answer: C
Pole is at z = 0.5, which lies inside the unit circle. For causality and stability, all poles must be inside |z| < 1.
Q.40Easy
The energy of a discrete-time signal x[n] = (0.5)^n u[n] is:
