A system is described by y[n] = 0.8y[n-1] + x[n]. The steady-state gain is:
Answer: D
DC gain H(z)|_{z=1} = 1/(1-0.8) = 5. Alternatively, setting y[n] = y[n-1] = y_ss: y_ss = 0.8y_ss + x_ss gives gain = 5.
Q.42Easy
Which of the following is a property of the Discrete-Time Fourier Transform (DTFT)?
Answer: B
The DTFT X(e^jω) is periodic with period 2π because e^j(ω+2π)n = e^jωn.
Q.43Medium
An IIR filter has poles at z = 0.6e^j0.4π and z = 0.6e^-j0.4π. The filter order is:
Answer: B
Two distinct poles indicate a second-order filter. Complex conjugate pairs count as one pole pair.
Q.44Medium
For a linear-phase FIR filter, the impulse response h[n] must satisfy:
Answer: C
Linear phase requires either symmetric (Type I, II) or antisymmetric (Type III, IV) impulse response.
Q.45Easy
A signal x[n] has Z-transform X(z) = 1 + 2z^-1 + 3z^-2 for |z| > 0. The signal values are:
Answer: A
By definition, X(z) = Σx[n]z^-n. Comparing coefficients: x[0]=1, x[1]=2, x[2]=3.
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Q.46Medium
In a multirate DSP system, downsampling by factor M introduces aliasing if the signal is not:
Answer: D
To prevent aliasing in downsampling, the signal must be bandlimited to π/M and preceded by an anti-aliasing lowpass filter.
Q.47Easy
The convolution of x[n] = {1, 2, 3} and h[n] = {1, 1} yields a sequence of length:
Answer: C
Length of convolution = M + N - 1 = 3 + 2 - 1 = 4.
Q.48Medium
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.49Easy
A signal has power spectral density S_x(f) that is nonzero only for |f| ≤ 2 kHz. The minimum sampling frequency is:
Answer: B
By Nyquist theorem, f_s ≥ 2×f_max = 2×2 = 4 kHz.
Q.50Easy
A finite-duration signal x[n] has length N=8. Its DFT X[k] has length:
Answer: B
DFT length equals the input signal length unless zero-padding is applied. For N-point signal, DFT yields N frequency bins.
Q.51Medium
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.52Medium
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.53Medium
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.54Medium
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].
Q.55Medium
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.56Easy
The Z-transform of x[n] = n·(0.8)^n u[n] is:
Answer: A
Using the property that Z{n·a^n·u[n]} = az/(z-a)², with a=0.8, we get X(z) = 0.8z/(z-0.8)².
Q.57Easy
A signal x(t) = sin(2πf₀t) is sampled at fs = 5f₀. What is the Nyquist sampling rate required?
Answer: B
Nyquist rate = 2 × maximum frequency = 2f₀. The given sampling rate (5f₀) exceeds this, ensuring no aliasing.
Q.58Medium
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.59Easy
The energy of signal x[n] = {1, 2, 1, -1} over the given support is:
