Difficulty: Easy & Basic
Consider an N-point DFT of a finite block. What is assumed beyond the block?
A A random process
B An infinite trend
C One period repeated
D A continuous waveform
Answer: C. One period repeated. The DFT treats its N samples as one period of a periodic sequence.
Multiplying two N-point DFTs corresponds to which operation?
A N-point circular convolution
B Linear convolution
C Pointwise addition
D Time reversal
Answer: A. N-point circular convolution. The DFT multiplication property yields circular convolution in the time domain.
Real-valued input produces which DFT symmetry?
A Odd symmetry
B No symmetry
C Purely real bins
D Conjugate symmetry
Answer: D. Conjugate symmetry. For real samples, positive- and negative-frequency DFT bins are complex conjugates.
Shifting a sequence circularly changes its DFT by what?
A Constant offset
B Linear phase factor
C Magnitude scaling
D Frequency reversal
Answer: B. Linear phase factor. A circular time shift multiplies each DFT bin by a phase factor without changing its magnitude.
Parseval’s relation connects which quantities?
A Time and frequency energy
B Poles and zeros
C Input and output delay
D Sampling and bit depth
Answer: A. Time and frequency energy. Parseval’s relation equates energy measured in the time and frequency domains, subject to DFT scaling.
Difference equations become algebraic relations under which transform?
A Laplace transform
B Hilbert transform
C Cosine transform
D Z-transform
Answer: D. Z-transform. The Z-transform converts shifts in a linear constant-coefficient difference equation into powers of z.
Under the bilateral Z-transform, δ[n] maps to what?
A z
B Zero everywhere
C One everywhere
D z inverse
Answer: C. One everywhere. Only the n=0 term contributes, so the transform of δ[n] is 1.
Right-sided rational sequences with poles have an ROC where?
A Inside innermost pole
B Outside outermost pole
C Between every pole
D At infinity only
Answer: B. Outside outermost pole. A right-sided rational sequence has a region of convergence extending outward from its outermost pole.
Conversely, where is the ROC of a left-sided rational sequence with poles?
A Outside outermost pole
B On the unit circle
C Inside innermost pole
D At every pole
Answer: C. Inside innermost pole. A left-sided rational sequence has an ROC extending inward from its innermost pole.
No pole location can belong to what?
A A zero set
B A frequency bin
C A numerator
D The ROC
Answer: D. The ROC. The Z-transform cannot converge at a pole, so every pole lies outside the ROC.
Negative-frequency bins of an even-length DFT usually appear where?
A Below zero
B Above N/2
C At every even bin
D Only at index zero
Answer: B. Above N/2. For even N, bins k>N/2 represent wrapped negative frequencies.
Because discrete-time sequences use integer indices, the DTFT repeats every what?
A 2π radians
B π/2 radians
C One hertz
D N samples
Answer: A. 2π radians. Discrete-time frequency responses repeat every 2π radians per sample.
Time reversal of a periodic sequence reverses which DFT ordering?
A Bit depth order
B Amplitude scale
C Pole radius
D Frequency-bin order
Answer: D. Frequency-bin order. Reversing a length-N sequence maps bin k to the corresponding wrapped bin −k.
Odd symmetry in a real sequence produces what DFT form?
A Real and even
B Real and odd
C Purely imaginary, odd
D Complex and arbitrary
Answer: C. Purely imaginary, odd. Under the N-point DFT’s circular symmetry convention, a real odd-symmetric sequence has an imaginary odd spectrum.
DFT linearity preserves which input operation?
A Same weighted sums
B Only products
C Only time shifts
D Magnitude clipping
Answer: A. Same weighted sums. A weighted sum of input sequences transforms to the same weighted sum of their DFTs.
For a bin-centered complex exponential, where is its DFT nonzero?
A At every bin
B At its matching bin
C Only at DC
D Only at Nyquist
Answer: B. At its matching bin. A complex sinusoid exactly at DFT bin m concentrates its energy at bin m.
Windowing a finite record helps reduce what?
A Aliasing
B Quantization
C Saturation
D Spectral leakage
Answer: D. Spectral leakage. A tapered window reduces leakage caused by abruptly truncating a record.
Abrupt record truncation corresponds most closely to which window?
A Hamming window
B Hann window
C Rectangular window
D Bartlett window
Answer: C. Rectangular window. A rectangular window keeps the record unchanged and cuts it off abruptly at its endpoints.
Compared with a rectangular window, which often lowers sidelobes?
A Triangular window
B Hamming window
C Rectangular window
D Impulse window
Answer: B. Hamming window. A Hamming window lowers sidelobes, usually at the cost of a wider mainlobe.
Lower sidelobes commonly come with what tradeoff?
A Wider mainlobe
B Narrower mainlobe
C Sharper transition
D No leakage
Answer: A. Wider mainlobe. Many windows reduce sidelobes by broadening the mainlobe.
Cross-correlation can help estimate what?
A Absolute amplitude
B Sampling rate
C Relative signal delay
D Filter order
Answer: C. Relative signal delay. A cross-correlation peak indicates the relative alignment or delay between two signals.
Autocorrelation compares a signal with what?
A Its derivative
B Its quantizer
C A random phase
D Its delayed copy
Answer: D. Its delayed copy. Autocorrelation measures similarity between a signal and shifted versions of itself.
At zero frequency, which DFT bin is used?
A X[1]
B X[0]
C X[N/2]
D X[N−1]
Answer: B. X[0]. The zero-frequency, or DC, component is the k=0 DFT bin.
When N is even, where is the Nyquist bin?
A k equals N/2
B k equals N−1
C k equals one
D k equals zero
Answer: A. k equals N/2. The Nyquist-frequency bin is k=N/2 for an even-length DFT.
Sampling the DTFT at N equally spaced frequencies gives which transform?
A The Laplace transform
B The Hilbert transform
C The Z-plane ROC
D The N-point DFT
Answer: D. The N-point DFT. The N-point DFT samples the DTFT at N uniformly spaced frequencies.