Difficulty: Medium
What is the period of x[n]=cos(πn/3)?
A 6 samples
B 12 samples
C 3 samples
D 4 samples
Answer: A. 6 samples The normalized frequency is 1/6 of 2π, so six integer samples complete a cycle.
For x[n]=[1,1] and h[n]=[1,1], what is their convolution?
A [1,1]
B [2,2]
C [1,2,1]
D [1,0,1]
Answer: C. [1,2,1] The overlapping products produce 1, then 2, then 1.
Fundamental angular spacing for period T₀ equals which value?
A ω₀=1/T₀²
B ω₀=πT₀
C ω₀=T₀/2π
D ω₀=2π/T₀
Answer: D. ω₀=2π/T₀ Harmonics are spaced by 2π/T₀ radians per second.
Delaying x(t) by τ multiplies X(jω) by what factor?
A Adds τ to X
B Multiplies by e⁻ʲωτ
C Reverses X
D Multiplies by jω
Answer: B. Multiplies by e⁻ʲωτ A delay x(t−τ) gives e⁻ʲωτX(jω) under the standard Fourier convention.
Multiplying x(t) by eʲω₀ᵗ shifts its spectrum by how much?
A By ω₀
B By 2π only
C By τ seconds
D Nowhere
Answer: A. By ω₀ Complex modulation shifts X(jω) to X(j(ω−ω₀)).
Differentiating x(t) multiplies its Fourier transform by what?
A ω²X(jω)
B −X(jω)
C X(jω)/jω
D jωX(jω)
Answer: D. jωX(jω) Differentiation in time multiplies the Fourier transform by jω.
Under the Fourier convention used here, time convolution transforms to what?
A X(jω−ω₀)
B X(jω)+H(jω)
C X(jω)H(jω)
D X(jω)/H(jω)
Answer: C. X(jω)H(jω) Time-domain convolution becomes multiplication of the two Fourier transforms.
Parseval’s relation equates signal energy in which domains?
A Input and output power
B Time and frequency energy
C Period and frequency
D Impulse and step
Answer: B. Time and frequency energy Parseval’s relation equates total signal energy in time and frequency domains.
For a>0, what is the bilateral Laplace transform of e⁻ᵃᵗu(t)?
A s+a, Re(s)>0
B 1/s², all s
D 1/(s+a), Re(s)>−a
Answer: D. 1/(s+a), Re(s)>−a Integrating the right-sided exponential gives 1/(s+a) with convergence for Re(s)>−a.
The unit step u(t) transforms to what, including its ROC?
A 1/s, Re(s)>0
B s, Re(s)<0
C δ(s)
D 1/(s+1), all s
Answer: A. 1/s, Re(s)>0 The right-sided unit step transforms to 1/s in the right half-plane.
For x[n]=aⁿu[n], what is its Z-transform and ROC?
A z−a, all z
B 1/(1−az⁻¹), |z|>|a|
C 1/(1+az), |z|<|a|
D 1/(z−a), |z|<1
Answer: B. 1/(1−az⁻¹), |z|>|a| Summing the right-sided geometric series gives this expression and exterior ROC.
Right-sided rational sequences have their ROC outside which pole?
A At z=0 only
B Only on poles
C Outside outermost pole
D Inside innermost pole
Answer: C. Outside outermost pole A right-sided sequence has a region of convergence exterior to its outermost pole.
Delaying a sequence by n₀ samples multiplies X(z) by which factor?
A Replaces z by n₀
B Adds n₀ to X(z)
C Multiplies by n₀z
D Multiplies by z⁻ⁿ⁰
Answer: D. Multiplies by z⁻ⁿ⁰ A delay of n₀ samples multiplies the Z-transform by z⁻ⁿ⁰.
Discrete-time LTI BIBO stability requires which sum to be finite?
A Σ|h[n]| is finite
B Σh[n] is zero
C h[n] is periodic
D h[0] equals one
Answer: A. Σ|h[n]| is finite Absolute summability of the discrete-time impulse response guarantees BIBO stability.
Half-wave symmetry eliminates which Fourier-series harmonics?
A Every harmonic
B Fundamental only
C Even harmonics
D Odd harmonics
Answer: C. Even harmonics A half-wave-symmetric periodic signal has no DC or even harmonics; its nonzero harmonic terms are odd.
Frequency response H(eʲΩ) is evaluated on which contour in the z-plane?
A Real axis only
B Unit circle
C Pole locations
D Origin only
Answer: B. Unit circle Frequency response is H(eʲΩ), which evaluates H(z) on the unit circle.
The system function H(s)=1/(s+2) has its pole where?
A s=−2
B s=0
C s=2
D s=−1/2
Answer: A. s=−2 The denominator is zero at s=−2.
With zero initial conditions, dy/dt+ay=x(t) gives which transfer function?
A s/(s+a)
B 1/(s−a)
C s+a
D 1/(s+a)
Answer: D. 1/(s+a) Taking the Laplace transform with zero initial state gives (s+a)Y(s)=X(s).
Period T₀ sets Fourier-series spacing to which angular frequency?
A ω₀=πT₀
B ω₀=T₀/2π
C ω₀=2π/T₀
D ω₀=1/T₀²
Answer: C. ω₀=2π/T₀ Harmonics are spaced by the fundamental angular frequency 2π/T₀.
Integer-period repetition occurs when Ω₀/(2π) is what?
A Ω₀ is irrational
B Ω₀/(2π) rational
C Amplitude is nonzero
D Phase is zero
Answer: B. Ω₀/(2π) rational The frequency ratio to 2π must be rational for integer-period repetition.
Pure delay y(t)=x(t−τ) corresponds to which h(t)?
A u(t−τ)
B τδ(t)
C δ(t+τ)
D δ(t−τ)
Answer: D. δ(t−τ) A pure delay produces an impulse delayed by the same amount.
Evaluating H(s) at s=jω describes which system response?
A Frequency response
B Step response only
C Time reversal
D Impulse area
Answer: A. Frequency response Evaluating H(s) at s=jω gives the continuous-time frequency response when it exists.
Complex exponentials are eigenfunctions of LTI systems; which input has that form?
A δ(t) only
B u(t) only
C eʲω₀ᵗ
D t²
Answer: C. eʲω₀ᵗ An LTI system scales a complex exponential by H(jω₀) without changing its form.
The even part of x(t) is computed how from its time reverse?
A x′(t)
B [x(t)+x(−t)]/2
C x(t)x(−t)
D [x(t)−x(−t)]/2
Answer: B. [x(t)+x(−t)]/2 The even component is the average of the signal and its time reversal.
Convolution order can be exchanged because of which property?
A Homogeneity
B Periodicity
C Causality
D Commutativity
Answer: D. Commutativity Convolution satisfies x*h=h*x.