Difficulty: Hard
For Fs=16 kHz and N=512, what is DFT bin spacing?
A 15.625 Hz
B 62.5 Hz
C 512 Hz
D 31.25 Hz
Answer: D. 31.25 Hz. DFT line spacing is Fs/N: 16,000/512=31.25 Hz.
Doubling record duration changes DFT bin spacing how?
A It doubles bin spacing
B It halves bin spacing
C It leaves spacing fixed
D It removes Nyquist
Answer: B. It halves bin spacing. DFT bin spacing is 1/T, so doubling the record duration halves the spacing.
Convolution lengths 600 and 425 need what minimum FFT size?
A 512 points
B 600 points
C 1024 points
D 2048 points
Answer: C. 1024 points. Linear convolution needs at least 600+425−1=1024 points; that is already a power of two.
Sampling at 10 kHz folds a 7 kHz tone to what frequency?
A 3 kHz
B 2 kHz
C 5 kHz
D 7 kHz
Answer: A. 3 kHz. The nearest sampled alias is |7−10|=3 kHz.
Radix-2 FFT stages for length 1024 number how many?
A 9 stages
B 10 stages
C 11 stages
D 512 stages
Answer: B. 10 stages. A radix-2 FFT uses log2(N) stages; log2(1024)=10.
With N=64, how many radix-2 butterflies are computed?
A 64 butterflies
B 128 butterflies
C 192 butterflies
D 384 butterflies
Answer: C. 192 butterflies. The count is (N/2)log2N=32×6=192 butterflies.
Decimation-in-time radix-2 implementations commonly reorder which data?
A Output in Gray-code order
B Coefficients in decimal order
C Stages in random order
D Input in bit-reversed order
Answer: D. Input in bit-reversed order. A common in-place DIT flow graph bit-reverses the input before computing the output in natural order.
In a common decimation-in-frequency FFT, which data are bit-reversed?
A Output ordering
B Input ordering
C Twiddle signs
D Sampling instants
Answer: A. Output ordering. A common DIF implementation accepts natural-order input and produces bit-reversed output.
The bilinear transform maps stable analog poles where?
A Outside unit circle
B Onto the real axis
C Inside unit circle
D To infinity
Answer: C. Inside unit circle. The bilinear map sends the stable left half of the s-plane into the interior of the z-plane unit circle.
Prewarping compensates for which bilinear-transform effect?
A Spectral leakage
B Quantization noise
C Circular aliasing
D Frequency warping
Answer: D. Frequency warping. Prewarping selects analog critical frequencies so they map to desired digital edge frequencies.
Impulse invariance can introduce what in the digital response?
A Spectral aliasing
B Frequency warping only
C Quantizer overload
D Linear phase
Answer: A. Spectral aliasing. Sampling the analog impulse response causes shifted copies of its spectrum to overlap.
Coefficient rounding in an IIR can move what across the stability boundary?
A Input sample times
B Poles
C DFT bin labels
D ADC thresholds
Answer: B. Poles. Small denominator-coefficient changes can move poles and may make a stable IIR unstable.
Recursive fixed-point filters may show zero-input limit cycles due to what?
A Window sidelobes
B Input aliasing
C Quantization feedback
D DFT leakage
Answer: C. Quantization feedback. Quantization inside feedback can sustain a nonzero output even after the input becomes zero.
Scaling internal signals before arithmetic mainly reduces what risk?
A Overflow
B Spectral leakage
C Frequency warping
D Time reversal
Answer: A. Overflow. Proper scaling keeps intermediate values within the representable numeric range.
Saturating arithmetic handles overflow by doing what?
A Wrapping modulo range
B Adding a random bit
C Resetting the filter
D Clamping to limits
Answer: D. Clamping to limits. Saturation clips an out-of-range result to the largest or smallest representable value.
Wraparound arithmetic handles overflow by what behavior?
A Clamping at maximum
B Modulo rollover
C Linear interpolation
D Averaging adjacent values
Answer: B. Modulo rollover. In fixed-width arithmetic, wraparound returns to the opposite end of the representable range.
High-order IIR filters are often implemented as what?
A Second-order sections
B One long FIR
C Single lookup table
D Rectangular windows
Answer: A. Second-order sections. Cascaded second-order sections improve numerical robustness compared with one high-order polynomial form.
All-pass filters preserve what magnitude?
A Zero at DC
B Linear with frequency
C Unity at every frequency
D Infinite at resonance
Answer: C. Unity at every frequency. An ideal all-pass filter changes phase while maintaining unit magnitude.
Minimum-phase stable filters place their zeros where?
A Outside unit circle
B Only at origin
C On every frequency
D Inside unit circle
Answer: D. Inside unit circle. A causal stable minimum-phase digital system has its zeros inside the unit circle.
Equiripple FIR design seeks what passband/stopband error pattern?
A One-sided ripple only
B Equal peak ripple
C No transition band
D Randomized error
Answer: B. Equal peak ripple. Equiripple designs minimize the maximum error with equal-magnitude ripples in constrained bands.
Elliptic filters are known for what tradeoff?
A No ripple anywhere
B Passband ripple only
C Ripple in both bands
D Stopband ripple only
Answer: C. Ripple in both bands. Elliptic designs permit ripple in both passband and stopband to achieve a steep transition for a given order.
Only a few DFT bins are needed. Which algorithm can be efficient?
A Goertzel algorithm
B Overlap-save method
C Bilinear transform
D Hamming window
Answer: A. Goertzel algorithm. Goertzel computes selected DFT components efficiently when only a small number of bins are required.
Basic radix-2 FFT supports which transform sizes?
A Odd primes only
B Powers of two
C Multiples of three
D Any real number
Answer: B. Powers of two. The radix-2 decomposition repeatedly halves the data, so N is a power of two.
Which result distinguishes bilinear mapping from impulse invariance?
A No frequency warping
B No poles
C No phase response
D No frequency aliasing
Answer: D. No frequency aliasing. The bilinear transformation maps the frequency axis one-to-one, avoiding aliasing, although it introduces warping.
Stable minimum-phase systems admit what kind of inverse?
A Unstable causal inverse
B Noncausal FIR inverse
C Stable causal inverse
D No inverse
Answer: C. Stable causal inverse. For a stable causal minimum-phase system, the inverse is also causal and stable.