The autocorrelation function R_x[n] of a discrete signal satisfies:
Answer: B
Autocorrelation is an even function: R_x[n] = E[x[m]x[m+n]] = E[x[m]x[m-n]] = R_x[-n].
Q.162Medium
In a Butterworth filter design, increasing the filter order primarily:
Answer: C
Higher-order filters provide steeper roll-off (slope = -20N dB/decade for Butterworth). They reduce transition width and are more complex computationally.
Q.163Medium
The group delay of a filter is defined as:
Answer: B
Group delay τ_g(ω) = -dφ(ω)/dω represents the delay of signal envelope.
Q.164Medium
Which window function has the narrowest main lobe but highest side lobes?
Answer: A
Rectangular window has the narrowest main lobe (~4π/N) but -13 dB side lobe level. Other windows trade main lobe width for lower side lobes.
Q.165Medium
For a real signal x[n], if X[k] is its DFT, then X[N-k] equals:
Answer: B
For real signals, DFT exhibits conjugate symmetry: X[N-k] = X*[k].
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Q.166Medium
An analog signal with bandwidth 10 kHz is sampled at 30 kHz. What is the aliased frequency of a 25 kHz component?
Answer: A
Aliased frequency = |f_original - k×f_s| where k is chosen to bring frequency into [-f_s/2, f_s/2]. For 25 kHz with f_s=30 kHz: |25-30| = 5 kHz.
Q.167Medium
For a causal discrete-time LTI system with H(z) = 1/(1-0.5z⁻¹), the system is:
Answer: B
Pole at z=0.5 (inside unit circle) makes the system stable. The denominator has z⁻¹ term, making it IIR (infinite impulse response).
Q.168Medium
A real continuous-time signal has Fourier transform X(f) with magnitude |X(f)| symmetric about f=0. Which property is satisfied?
Answer: A
For a real signal, X(f) exhibits Hermitian symmetry: X(-f) = X*(f). Magnitude symmetry |X(-f)| = |X(f)| indicates even symmetry in x(t).
Q.169Medium
The pole-zero diagram of a causal system shows poles at z = 0.3 and z = 0.7, with a zero at z = 0. The system is:
Answer: B
Zero at z=0 (DC) causes attenuation at low frequencies. Poles near z=1 allow high frequencies to pass, characteristic of a high-pass filter.
Q.170Medium
The inverse Fourier transform of X(f) = δ(f-f₀) + δ(f+f₀) is:
Answer: A
IFT{δ(f-f₀)} = e^(j2πf₀t) and IFT{δ(f+f₀)} = e^(-j2πf₀t). Sum = 2cos(2πf₀t), but accounting for delta properties, result is cos(2πf₀t).
Q.171Medium
For a finite impulse response (FIR) filter of length M=5, the maximum linear phase is achieved when:
Answer: A
Linear phase FIR filters require symmetric (even) or antisymmetric (odd) impulse responses about the center. For length 5, center is n=2.
Q.172Medium
A system has step response s(t) = 1 - e^(-2t)u(t). Its impulse response h(t) is:
