Which window function has the BEST frequency resolution but WORST spectral leakage?
Answer: C
Rectangular window has narrowest main lobe (best resolution) but highest sidelobe (worst leakage)
Q.2Hard
Parseval's theorem states that energy in time domain equals energy in frequency domain. For discrete signals:
Answer: B
Parseval's theorem for discrete-time: ∑|x[n]|² = (21π)∫|X(e^jω)|²dω over 2π
Q.3Hard
For a stable IIR filter with difference equation y[n] = 0.5y[n-1] + x[n], the DC gain is:
Answer: C
DC gain = H(z=1) = 1/(1-0.5) = 01.5 = 2
Q.4Hard
The concept of 'group delay' in filters refers to:
Answer: B
Group delay τ_g(ω) = -dθ(ω)/dω represents dispersion in filter's phase response
Q.5Hard
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.
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Q.6Hard
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.7Hard
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.8Hard
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.9Hard
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.10Hard
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.11Hard
For a complex exponential signal e^(j2πf₀t) sampled at fs = 10 kHz with f₀ = 3 kHz, aliasing occurs at frequency:
Answer: B
Aliased frequency = |f₀ - kfs| = |3 - 10| = 7 kHz (for k=1). The signal appears as a 7 kHz component in the baseband.
Q.12Hard
A second-order system has poles at s = -1 ± j2. Its natural frequency ωn is:
Answer: C
For poles at σ ± jωd, ωn = √(σ² + ωd²) = √(1 + 4) = √5.
Q.13Hard
A causal stable filter has H(s) = (s+3)/((s+1)(s+2)). Using partial fractions, the impulse response contains:
A 1024-point FFT is computed on a signal. The frequency resolution is 0.1 Hz. What is the sampling frequency?
Answer: B
Frequency resolution Δf = fs/N, so fs = Δf × N = 0.1 × 1024 = 102.4 Hz.
Q.15Hard
For a linear time-invariant system with Laplace transform H(s) = 1/(s+2), determine the response to input x(t) = e^(-2t)×u(t):
Answer: A
X(s) = 1/(s+2). Y(s) = H(s)×X(s) = 1/[(s+2)²]. This is a repeated pole, inverse Laplace gives y(t) = t×e^(-2t)×u(t). This is a resonance condition in the system.
