Difficulty: Hard
Given a bandlimited signal with maximum frequency B, which sampling rate avoids aliasing?
A At least B/2
B Exactly B/2
C At most B
D At least 2B
Answer: D. At least 2B The sampling theorem requires a rate no lower than twice the highest frequency; practical systems use margin.
Sampling a 900 Hz tone at 1 kHz creates which baseband alias?
A 900 Hz
B 100 Hz
C 400 Hz
D 500 Hz
Answer: B. 100 Hz The sampled spectrum folds 900 Hz to |900−1000|=100 Hz.
Causal rational systems have their Z-transform ROC where?
A Outside outermost pole
B Inside innermost pole
C Between every pole pair
D Only on unit circle
Answer: A. Outside outermost pole A causal right-sided response has an exterior region of convergence.
Stability requires the DT LTI system’s ROC to include what?
A Origin only
B Positive real axis
C Unit circle
D Every pole
Answer: C. Unit circle Stability requires the unit circle to lie within the Z-transform ROC.
Under causality and rationality, stability requires poles in which region?
A Right half-plane
B Outside all finite values
C On imaginary axis
D Left half-plane
Answer: D. Left half-plane For a causal rational system, BIBO stability requires all poles in the open left half-plane.
Periodic CT signals produce what pattern in the Fourier transform?
A Impulse lines at harmonics
B Uniform white spectrum
C A single DC value
D A finite sequence
Answer: A. Impulse lines at harmonics Periodic signals have line spectra at integer multiples of their fundamental frequency.
With x(t)↔X(jω), compression x(at), a>0, gives which transform?
A aX(jω)
B (1/a)X(jω/a)
C X(j(ω−a))
D aX(jaω)
Answer: B. (1/a)X(jω/a) Time compression by a scales frequency by 1/a and amplitude by 1/a.
If x(t)↔X(jω), reversing time produces which spectrum?
A X(jω−1)
B −X(jω)
C X(−jω)
D jωX(jω)
Answer: C. X(−jω) Time reversal reflects the spectrum about zero frequency.
Differentiation in time introduces which frequency-domain factor?
A jω
B 1/(jω)
C −1
D ω²
Answer: A. jω Differentiation multiplies the Fourier transform by jω.
Left-sided rational sequences have their Z-transform ROC where?
A On every pole
B Only at infinity
C Outside outermost pole
D Inside innermost pole
Answer: D. Inside innermost pole A left-sided rational sequence has an interior ROC bounded by its innermost pole.
From y[n]−0.5y[n−1]=x[n], what is H(z) with zero initial state?
A 1−0.5z⁻¹
B z−0.5
C 1/(1−0.5z⁻¹)
D 1/(1+0.5z)
Answer: C. 1/(1−0.5z⁻¹) Taking the Z-transform yields (1−0.5z⁻¹)Y(z)=X(z).
Cascading two LTI blocks yields what overall transfer function?
A Difference of poles
B Product of transfer functions
C Sum of transfer functions
D Ratio of impulse areas
Answer: B. Product of transfer functions Cascaded LTI system functions multiply.
Parallel LTI blocks combine their transfer functions how?
A Sum of transfer functions
B Reciprocal of poles
C Convolution of inputs
D Product of transfer functions
Answer: A. Sum of transfer functions Parallel outputs add, so their transfer functions add.
Why does impulse-train sampling create shifted spectral replicas?
A Amplitude clipping
B Time reversal
C System feedback
D Impulse-train multiplication
Answer: D. Impulse-train multiplication Sampling multiplies the signal by an impulse train, creating frequency-domain replicas.
Before the ADC, which analog stage suppresses out-of-band energy?
A High-pass differentiator
B Anti-aliasing low-pass filter
C Band-stop after ADC
D Digital integrator
Answer: B. Anti-aliasing low-pass filter An analog low-pass filter suppresses frequencies that would alias into the sampled band.
CT convolution becomes which operation in the Fourier domain?
A Differentiation only
B Time shifting
C Multiplication
D Addition only
Answer: C. Multiplication The convolution theorem maps time convolution to multiplication in frequency.
Absolute summability of x[n] guarantees which transform converges on the unit circle?
A DTFT
B No transform
C Fourier series only
D Bilateral Laplace only
Answer: A. DTFT Absolute summability ensures convergence of the DTFT for all Ω.
Discrete-time sinusoid repetition requires Ω₀/(2π) to be what?
A Only Ω₀=π
B Irrational Ω₀/(2π)
C Any real Ω₀
D Rational Ω₀/(2π)
Answer: D. Rational Ω₀/(2π) A rational ratio gives a finite integer period.
At every frequency, what spectrum corresponds to δ(t)?
A δ(ω)
B jω
C 1
D 0
Answer: C. 1 The unit impulse has a flat, unit-valued Fourier spectrum.
Cosine modulation creates what pattern of shifted spectra?
A One doubled copy
B Two shifted half-amplitude copies
C Two time reversals
D One unshifted copy
Answer: B. Two shifted half-amplitude copies Cosine modulation adds half-amplitude copies centered at +ω₀ and −ω₀.
When h[n]=aⁿu[n], which condition ensures BIBO stability?
A |a|<1
B |a|=1
C Any real a
D |a|>1
Answer: A. |a|<1 The impulse response is absolutely summable only when the geometric ratio has magnitude below one.
For y[n]=x[n]−x[n−1], which digital operation is represented?
A Time reversal
B Running sum
C First difference
D Downsampling
Answer: C. First difference Subtracting the previous sample from the current sample forms the first difference.
Two CT periodic signals have a periodic sum when their period ratio is what?
A Infinite
B Always zero
C Irrational
D Rational
Answer: D. Rational Their sum is periodic when the ratio of their periods is rational.
Causality forces a DT LTI impulse response to be what for n<0?
A Σh[n]=0
B h[n]=0 for n<0
C h[n]=1 for n>0
D h[n] is even
Answer: B. h[n]=0 for n<0 A discrete-time LTI system is causal exactly when its impulse response vanishes for negative indices.
Cross-correlation between two signals measures what versus lag?
A Similarity versus lag
B Signal energy only
C System causality
D Spectral spacing
Answer: A. Similarity versus lag Cross-correlation measures how similar two signals are as one is shifted relative to the other.