Q.1 What is the primary purpose of the Discrete Fourier Transform (DFT)?
To convert a time‑domain signal into its frequency‑domain representation
To compress a signal for storage
To filter out high‑frequency components
To convert a signal from analog to digital
Explanation - The DFT transforms a finite set of time‑domain samples into a set of complex numbers that represent the amplitude and phase of sinusoidal components at discrete frequencies.
Correct answer is: To convert a time‑domain signal into its frequency‑domain representation
Q.2 Which of the following expressions correctly represents the DFT of a sequence \(x[n]\) of length \(N\)?
X[k] = \sum_{n=0}^{N-1} x[n] e^{-j2\pi kn/N}
X[k] = \sum_{n=0}^{N-1} x[n] e^{+j2\pi kn/N}
X[k] = \prod_{n=0}^{N-1} x[n] e^{-j2\pi kn/N}
X[k] = \sum_{n=0}^{N-1} x[n] \cos(2\pi kn/N)
Explanation - The standard DFT formula includes a negative exponential in the sum to produce the frequency‑domain spectrum.
Correct answer is: X[k] = \sum_{n=0}^{N-1} x[n] e^{-j2\pi kn/N}
Q.3 What does the Fast Fourier Transform (FFT) algorithm primarily reduce compared to the direct DFT computation?
Memory usage by half
Computational complexity from \(O(N^2)\) to \(O(N \log N)\)
Number of input samples
The size of the output spectrum
Explanation - FFT algorithms, such as the radix‑2 Cooley–Tukey, reorganize computations to achieve \(O(N \log N)\) operations instead of the \(O(N^2)\) operations required by the naive DFT.
Correct answer is: Computational complexity from \(O(N^2)\) to \(O(N \log N)\)
Q.4 Which property of the DFT indicates that a time‑shift of a sequence results in a phase shift of its spectrum?
Linearity
Time‑Shift Property
Convolution Property
Frequency‑Shift Property
Explanation - The Time‑Shift Property states \(x[n-n_0] \leftrightarrow X[k] e^{-j2\pi kn_0/N}\); a shift in time produces a linear phase shift in frequency.
Correct answer is: Time‑Shift Property
Q.5 In an FFT implementation, why is it advantageous to have the input length \(N\) be a power of two?
It reduces memory usage
It allows the use of radix‑2 algorithms that split the transform into log₂N stages
It increases the frequency resolution
It simplifies windowing functions
Explanation - Radix‑2 FFT algorithms recursively split the DFT into smaller transforms; this requires \(N = 2^m\) to work efficiently.
Correct answer is: It allows the use of radix‑2 algorithms that split the transform into log₂N stages
Q.6 Which of the following best describes the main output of a complex‑valued FFT?
Real‑valued amplitude spectrum only
Imaginary‑valued frequency bins only
Both magnitude and phase information for each frequency bin
Time‑domain waveform
Explanation - A complex FFT gives \(X[k] = A_k e^{j\phi_k}\), providing both magnitude \(|X[k]|\) and phase \(\angle X[k]\) for each bin.
Correct answer is: Both magnitude and phase information for each frequency bin
Q.7 Which statement about the symmetry of the DFT of a real‑valued sequence is true?
The spectrum is symmetric about the DC bin.
The spectrum has even symmetry: \(X[k] = X^*[N-k]\).
The magnitude spectrum is always zero.
The phase spectrum is always zero.
Explanation - For real input, the DFT satisfies Hermitian symmetry: \(X[k] = X^*[N-k]\), meaning magnitude is symmetric and phase is antisymmetric.
Correct answer is: The spectrum has even symmetry: \(X[k] = X^*[N-k]\).
Q.8 What is the effect of zero‑padding a time‑domain signal before performing an FFT?
It changes the frequency resolution by interpolating between bins
It increases the amplitude of all frequency components
It reduces the computational complexity
It removes spectral leakage completely
Explanation - Zero‑padding extends the signal length, yielding more FFT points which effectively interpolate the spectrum, improving apparent frequency resolution.
Correct answer is: It changes the frequency resolution by interpolating between bins
Q.9 Which of these window functions is commonly used to reduce spectral leakage in FFT analysis?
Rectangular window
Hanning window
Zigzag window
Binary window
Explanation - The Hanning (or Hann) window tapers the ends of the signal, reducing discontinuities that cause spectral leakage.
Correct answer is: Hanning window
Q.10 In the context of FFT, what does 'bit‑reversal' refer to?
Reversing the sign of the FFT output
Reordering the input indices based on the reverse binary representation
Swapping real and imaginary parts of each sample
Reversing the frequency bins
Explanation - Bit‑reversal reorders input samples so that butterfly operations can be performed in place during radix‑2 FFT stages.
Correct answer is: Reordering the input indices based on the reverse binary representation
Q.11 Which property of the DFT states that convolution in the time domain equals multiplication in the frequency domain?
Convolution Property
Correlation Property
Time‑Shift Property
Duality Property
Explanation - The Convolution Property: if \(y[n] = x[n] * h[n]\), then \(Y[k] = X[k] H[k]\).
Correct answer is: Convolution Property
Q.12 What is the computational complexity of a radix‑4 FFT algorithm for an input length \(N = 4^m\)?
O(N log₂N)
O(N log₄N)
O(N)
O(N²)
Explanation - Radix‑4 FFT still has overall \(O(N \log N)\) complexity (with base‑2 logs), but each stage processes four points at once, reducing constant factors.
Correct answer is: O(N log₂N)
Q.13 Which of the following best describes the 'windowing' effect when performing a DFT on a finite signal?
It increases the bandwidth of the signal
It acts like a convolution in frequency, causing sidelobes
It removes all high‑frequency content
It has no effect on the spectrum
Explanation - Multiplying a signal by a finite window is equivalent to convolving its spectrum with the window's spectrum, creating sidelobes and leakage.
Correct answer is: It acts like a convolution in frequency, causing sidelobes
Q.14 The term 'spectral leakage' is most commonly associated with which phenomenon in FFT analysis?
Insufficient frequency resolution due to a short data record
Using an inappropriate window function
Aliasing from undersampling
Errors in the FFT algorithm
Explanation - Spectral leakage occurs when a frequency component is not aligned with an FFT bin, causing its energy to spread to adjacent bins.
Correct answer is: Insufficient frequency resolution due to a short data record
Q.15 Which of the following is a true statement about the Nyquist-Shannon sampling theorem in relation to FFT?
The FFT can reconstruct a signal if the sampling rate is less than twice the maximum frequency
The FFT cannot be applied to signals sampled below the Nyquist rate
The FFT is unaffected by undersampling if zero‑padding is used
The FFT requires oversampling to work properly
Explanation - If a signal contains frequency components above half the sampling rate, aliasing occurs and the FFT cannot recover the original frequencies.
Correct answer is: The FFT cannot be applied to signals sampled below the Nyquist rate
Q.16 What does the 'complexity factor' of an FFT algorithm refer to?
The number of arithmetic operations per output point
The amount of memory needed for the transform
The ratio of FFT to DFT execution times
The maximum input length it can handle
Explanation - Complexity factor is the number of complex multiplies and adds needed per output sample; for radix‑2 FFT it's roughly \(5/2 \log₂N\).
Correct answer is: The number of arithmetic operations per output point
Q.17 Which of the following best explains why the FFT algorithm is preferred in real‑time audio processing?
It can handle arbitrary sample sizes efficiently
It provides exact spectral values for all frequencies
It reduces computation time from quadratic to logarithmic complexity
It eliminates the need for windowing functions
Explanation - Real‑time audio requires quick processing; FFT's \(O(N \log N)\) operations make it feasible compared to \(O(N^2)\) DFT.
Correct answer is: It reduces computation time from quadratic to logarithmic complexity
Q.18 Which of the following is NOT a typical application of the FFT?
Spectral analysis of signals
Solving linear convolution problems
Compressing images via JPEG
Simulating mechanical vibrations
Explanation - While FFT can analyze vibration spectra, directly simulating mechanical vibrations usually involves differential equations and other methods; FFT is not the primary tool.
Correct answer is: Simulating mechanical vibrations
Q.19 When performing an FFT on a real‑valued sequence, what can be said about the negative‑frequency bins?
They contain identical magnitude but opposite phase to the positive‑frequency bins
They are always zero
They are irrelevant and can be discarded
They are identical to the positive‑frequency bins
Explanation - For real input, the spectrum is Hermitian: \(X[N-k] = X^*[k]\), so magnitude is symmetric and phase is negative.
Correct answer is: They contain identical magnitude but opposite phase to the positive‑frequency bins
Q.20 In an FFT, the term 'butterfly' refers to:
A specific window shape used to reduce leakage
A graphical representation of the frequency spectrum
A computational pattern that combines pairs of data points
The final output of the transform
Explanation - Butterfly operations are the core computational step in FFTs where two inputs are combined to produce two outputs.
Correct answer is: A computational pattern that combines pairs of data points
Q.21 What is the main advantage of the split‑radix FFT algorithm compared to the radix‑2 algorithm?
It requires fewer memory accesses
It uses only real‑valued operations
It reduces the number of complex multiplications
It can handle non‑power‑of‑two lengths directly
Explanation - Split‑radix combines radix‑2 and radix‑4 stages, achieving slightly fewer complex multiplies than pure radix‑2.
Correct answer is: It reduces the number of complex multiplications
Q.22 Which property of the DFT allows the computation of the DFT of a sequence and its complex conjugate reverse to be derived from each other?
Duality Property
Convolution Property
Shift Property
Symmetry Property
Explanation - The duality property states that the DFT of the reversed conjugate sequence equals the conjugate of the original DFT reversed.
Correct answer is: Duality Property
Q.23 Which of the following describes the 'frequency resolution' of an \(N\)-point FFT performed on a signal sampled at \(f_s\) Hz?
f_s / N
N / f_s
f_s * N
N * f_s
Explanation - Frequency resolution \(\Delta f = f_s / N\) is the spacing between adjacent FFT bins.
Correct answer is: f_s / N
Q.24 Which of the following is a common trick to improve the performance of FFT on modern processors?
Using a larger input size
Avoiding cache‑friendly memory access patterns
Exploiting SIMD vector instructions
Computing the FFT in double precision only
Explanation - SIMD (Single Instruction Multiple Data) allows processing multiple data points simultaneously, speeding up FFT operations.
Correct answer is: Exploiting SIMD vector instructions
Q.25 What does the 'windowing effect' do to the DFT spectrum of a signal that contains a sinusoid not aligned with an FFT bin?
It sharpens the main lobe only
It introduces sidelobes around the main frequency bin
It eliminates the sinusoid entirely
It doubles the amplitude of the sinusoid
Explanation - Mis‑aligned sinusoids cause spectral leakage, creating sidelobes in the DFT output.
Correct answer is: It introduces sidelobes around the main frequency bin
Q.26 Which of the following best defines the 'spectral leakage' phenomenon in the context of FFT?
Loss of high‑frequency components due to insufficient sampling
Smearing of spectral energy across adjacent bins when the signal is not periodic within the FFT window
An error caused by using an inadequate number of bits for quantization
The effect of windowing that increases the main lobe width
Explanation - Spectral leakage occurs because the DFT assumes the signal repeats outside the window; non‑periodic components leak into neighboring bins.
Correct answer is: Smearing of spectral energy across adjacent bins when the signal is not periodic within the FFT window
Q.27 In an FFT, what is the role of the 'twiddle factor'?
It adjusts the windowing function
It represents the complex exponential multipliers used in butterfly computations
It controls the length of the output spectrum
It determines the sampling rate of the input signal
Explanation - Twiddle factors are \(W_N^{kn} = e^{-j2\pi kn/N}\) used to mix signals in each butterfly.
Correct answer is: It represents the complex exponential multipliers used in butterfly computations
Q.28 Which of the following is a key assumption in the derivation of the DFT formula?
The input sequence is infinite and periodic
The input sequence is finite and zero‑outside the considered interval
The input sequence contains only real numbers
The input sequence is continuous in time
Explanation - The DFT assumes the sequence is defined only over \(N\) samples; outside this interval it's implicitly zero (periodic extension).
Correct answer is: The input sequence is finite and zero‑outside the considered interval
Q.29 What does the term 'spectral centroid' refer to in signal analysis using FFT?
The highest frequency component in the signal
The frequency at which the magnitude spectrum has its maximum
A weighted average frequency that indicates the center of mass of the spectrum
The lowest frequency component in the signal
Explanation - Spectral centroid is \(\frac{\sum f_k |X[k]|}{\sum |X[k]|}\) and indicates perceived pitch or timbre.
Correct answer is: A weighted average frequency that indicates the center of mass of the spectrum
Q.30 Which of the following best describes the 'overlap‑add' method in the context of FFT-based convolution?
Dividing the input into overlapping segments, computing FFTs, and summing the results to produce the output
Adding a constant offset to the FFT output to reduce noise
Using overlapping window functions to improve spectral leakage
Adding the input to a zero‑filled buffer before computing FFT
Explanation - Overlap‑add splits long convolution into short blocks, uses FFT on each, and sums overlapping parts to get the full output.
Correct answer is: Dividing the input into overlapping segments, computing FFTs, and summing the results to produce the output
Q.31 Which algorithmic approach is used to compute the FFT for a length that is a product of two co‑prime numbers \(N = N_1 N_2\)?
Radix‑2 algorithm
Cooley‑Tukey mixed‑radix algorithm
Divide‑and‑conquer algorithm
Recursive convolution algorithm
Explanation - The mixed‑radix Cooley‑Tukey algorithm handles composite lengths by splitting into smaller transforms of size \(N_1\) and \(N_2\).
Correct answer is: Cooley‑Tukey mixed‑radix algorithm
Q.32 If an FFT of length \(N=1024\) is performed on a signal sampled at \(f_s=44.1\) kHz, what is the frequency resolution?
43.07 Hz
44.1 Hz
21.6 Hz
1024 Hz
Explanation - Resolution \(\Delta f = f_s/N = 44100/1024 \approx 43.07\) Hz.
Correct answer is: 43.07 Hz
Q.33 Which of the following best describes 'zero‑padding' in the FFT process?
Adding zeros to the beginning of the sequence only
Adding zeros to the end of the sequence only
Appending zeros to extend the sequence length before computing FFT
Replacing all samples with zeros to reduce computational load
Explanation - Zero‑padding increases the number of FFT points, thereby refining the interpolation of the spectrum.
Correct answer is: Appending zeros to extend the sequence length before computing FFT
Q.34 Which property of the DFT is used to demonstrate that the DFT of a sinusoid at one of its bin frequencies yields a single non‑zero coefficient?
Linearity
Orthogonality of complex exponentials
Time‑Shift Property
Convolution Property
Explanation - Complex exponentials are orthogonal over the interval; a sinusoid at a bin frequency aligns with one basis function, producing one non‑zero coefficient.
Correct answer is: Orthogonality of complex exponentials
Q.35 Which of the following best explains why the FFT of real signals requires only half of the output bins to be stored for efficient processing?
Because the spectrum is symmetric and redundant information is present
Because real signals have only positive frequencies
Because the FFT algorithm discards the negative frequency bins
Because the output bins are always zero for real inputs
Explanation - Hermitian symmetry means \(X[k]\) for \(k>N/2\) is the complex conjugate of \(X[N-k]\); only half the bins are needed to reconstruct the full spectrum.
Correct answer is: Because the spectrum is symmetric and redundant information is present
Q.36 Which of the following statements about the 'FFT shift' operation in MATLAB is true?
It multiplies the FFT output by -1 to correct phase
It shifts the zero‑frequency component to the center of the output array
It performs the inverse FFT
It doubles the frequency resolution
Explanation - FFT shift reorders the output so that negative frequencies appear on the left, positive on the right, centering DC.
Correct answer is: It shifts the zero‑frequency component to the center of the output array
Q.37 What is the effect of applying a Hann window before computing an FFT on a pure sinusoid at frequency \(f_0\) that is not an FFT bin?
It eliminates spectral leakage entirely
It reduces spectral leakage but does not eliminate it completely
It increases spectral leakage
It has no effect on spectral leakage
Explanation - Windowing reduces sidelobes, mitigating leakage, but if \(f_0\) is not aligned, leakage still occurs.
Correct answer is: It reduces spectral leakage but does not eliminate it completely
Q.38 Which of the following best explains why the FFT algorithm can be parallelized efficiently on GPUs?
Because the FFT algorithm is inherently sequential
Because it requires only scalar operations
Because it consists of many independent butterfly operations that can be executed in parallel
Because GPUs only support real‑valued operations
Explanation - Butterfly operations involve pairs of points that are independent, making them suitable for massively parallel execution.
Correct answer is: Because it consists of many independent butterfly operations that can be executed in parallel
Q.39 Which of the following is a property that allows the DFT to convert convolution in the time domain into multiplication in the frequency domain?
Associativity
Duality
Convolution Theorem
Distributivity
Explanation - The Convolution Theorem states \(x[n] * h[n] \leftrightarrow X[k]H[k]\).
Correct answer is: Convolution Theorem
Q.40 Which of the following is an advantage of using a real‑to‑complex FFT over a complex‑to‑complex FFT when processing real‑valued signals?
It requires no additional memory
It outputs a full complex spectrum
It halves the number of arithmetic operations
It avoids the need for a window function
Explanation - Real‑to‑complex FFT exploits Hermitian symmetry to reduce computational load by about half.
Correct answer is: It halves the number of arithmetic operations
Q.41 Which of the following is true regarding the phase of an FFT output when the input signal has a linear time shift of \(n_0\) samples?
Phase remains unchanged
Phase is multiplied by \(-n_0\)
Phase is shifted by a linear ramp \(-2\pi k n_0 / N\)
Phase becomes zero for all bins
Explanation - Time shift results in a linear phase change across frequency bins.
Correct answer is: Phase is shifted by a linear ramp \(-2\pi k n_0 / N\)
Q.42 Which of the following best describes the 'FFT of a windowed signal' compared to the FFT of the unwindowed signal?
It always has higher amplitude peaks
It has reduced spectral leakage but a wider main lobe
It eliminates all noise components
It shifts all frequencies downward
Explanation - Windowing trades off between leakage reduction and main lobe width (resolution).
Correct answer is: It has reduced spectral leakage but a wider main lobe
Q.43 Which of the following statements about the Cooley‑Tukey FFT algorithm is incorrect?
It requires the input length to be a power of two for radix‑2
It achieves \(O(N \log N)\) complexity
It uses butterfly operations for in‑place computation
It can handle any arbitrary input length without modification
Explanation - The classic Cooley‑Tukey algorithm works for lengths that are products of small primes; arbitrary lengths require padding or mixed‑radix methods.
Correct answer is: It can handle any arbitrary input length without modification
Q.44 In which scenario would a 'complex‑to‑complex FFT' be preferable over a 'real‑to‑complex FFT'?
When the input signal is purely real
When the output spectrum is required in real form only
When the input contains complex samples or both real and imaginary components
When the signal is a pure sinusoid
Explanation - Real‑to‑complex optimizations exploit symmetry; for complex input, full complex FFT is needed.
Correct answer is: When the input contains complex samples or both real and imaginary components
Q.45 Which of the following best describes the 'bit‑reversal permutation' used in radix‑2 FFT implementations?
A method to reverse the bit order of each sample's amplitude
A rearrangement of indices that places the data in an order suitable for butterfly stages
A technique to double the FFT output resolution
An approach to correct phase errors
Explanation - Bit‑reversal reorders the input so that butterflies operate on adjacent elements in each stage.
Correct answer is: A rearrangement of indices that places the data in an order suitable for butterfly stages
Q.46 Which of the following is a direct consequence of the 'zero‑padding' technique on the FFT output?
It increases the dynamic range of the output spectrum
It changes the amplitude of the existing frequency components
It adds additional frequency bins with zero values
It improves frequency resolution without altering the bin spacing
Explanation - Zero‑padding extends the sequence with zeros, creating more FFT points and interpolating the spectrum.
Correct answer is: It adds additional frequency bins with zero values
Q.47 Which of the following best defines the 'time‑domain window function' in FFT analysis?
A function applied to the frequency domain to remove noise
A function used to taper the edges of the time‑domain signal to reduce discontinuities
A method for increasing the number of samples in the time domain
A process for converting a real signal into a complex one
Explanation - Window functions shape the time‑domain signal to minimize spectral leakage when computing its FFT.
Correct answer is: A function used to taper the edges of the time‑domain signal to reduce discontinuities
Q.48 What does the 'conjugate symmetry' property of the DFT imply for a real‑valued input sequence?
The DFT output is purely real
The DFT output satisfies \(X[N-k] = X^*[k]\)
All frequency bins have equal magnitude
The DFT output is zero for all odd \(k\)
Explanation - Conjugate symmetry ensures the spectrum is Hermitian for real input.
Correct answer is: The DFT output satisfies \(X[N-k] = X^*[k]\)
Q.49 Which of the following statements best explains the effect of 'overlap‑save' method in FFT-based convolution?
It reduces the number of FFT computations by reusing previous results
It adds extra samples at the beginning of each block to maintain continuity
It uses overlapping data blocks and discards the first part of each convolution result
It is identical to the overlap‑add method
Explanation - Overlap‑save processes blocks that overlap by \(L-1\) samples (filter length minus one) and throws away the initial samples that contain aliasing.
Correct answer is: It uses overlapping data blocks and discards the first part of each convolution result
Q.50 Which property of the DFT allows a single sinusoidal component at a frequency corresponding to an FFT bin to appear as a single non‑zero coefficient in the spectrum?
Orthogonality of basis functions
Linearity
Conjugate symmetry
Frequency‑shift property
Explanation - Complex exponentials form an orthogonal set; a sinusoid at bin frequency matches exactly one basis function.
Correct answer is: Orthogonality of basis functions
Q.51 Which of the following best describes the 'frequency resolution' improvement obtained by zero‑padding an \(N\)-point FFT to \(M\) points where \(M>N\)?
It halves the frequency resolution
It doubles the frequency resolution
It provides \(\Delta f = f_s / M\), thus finer resolution
It has no effect on frequency resolution
Explanation - Zero‑padding increases the number of FFT bins, reducing bin spacing and providing better interpolation.
Correct answer is: It provides \(\Delta f = f_s / M\), thus finer resolution
Q.52 Which of the following is a common use of the FFT in digital communications?
Modulating data onto an analog carrier
Demodulating phase‑shift keying signals
Generating random noise
Amplifying weak signals
Explanation - FFT helps compute the spectrum of received signals to detect modulation patterns like PSK or QAM.
Correct answer is: Demodulating phase‑shift keying signals
Q.53 In the FFT algorithm, which of the following statements is true about the 'twiddle factor' \(W_N^k = e^{-j2\pi k/N}\)?
It changes value only after each stage
It is constant for all \(k\)
It is pre‑computed and reused across butterfly stages
It is always equal to 1
Explanation - Twiddle factors depend on \(k\) and \(N\); they are calculated once and reused to save computation.
Correct answer is: It is pre‑computed and reused across butterfly stages
Q.54 Which of the following best explains the relationship between the length of the FFT \(N\) and the number of complex multiplications required for a radix‑2 FFT?
It requires exactly \(N\) complex multiplications
It requires \(N \log_2 N\) complex multiplications
It requires \((N/2) \log_2 N\) complex multiplications
It requires \(N^2\) complex multiplications
Explanation - Each of the \(\log_2 N\) stages processes \(N/2\) butterfly operations, each involving one complex multiplication.
Correct answer is: It requires \((N/2) \log_2 N\) complex multiplications
Q.55 Which of the following is a characteristic of the FFT algorithm that makes it well‑suited for hardware implementation?
It requires large amounts of memory
It can be parallelized at the butterfly level
It uses only scalar operations
It demands sequential execution
Explanation - Butterflies can be executed concurrently, making FFT efficient on FPGA and ASIC designs.
Correct answer is: It can be parallelized at the butterfly level
Q.56 Which of the following best describes the effect of applying a rectangular window to a signal before computing its FFT?
It eliminates spectral leakage entirely
It results in the same spectrum as zero‑padding
It yields maximum main‑lobe width but minimum sidelobes
It produces a spectrum with infinite sidelobes
Explanation - A rectangular window has the narrowest main lobe but highest sidelobe levels, leading to strong spectral leakage.
Correct answer is: It yields maximum main‑lobe width but minimum sidelobes
Q.57 Which of the following is NOT a valid statement about the 'FFT shift' operation in MATLAB?
It moves the zero‑frequency component to the center of the output array
It swaps the first and second halves of the FFT output
It is implemented by the function `fftshift`
It does not alter the magnitude of the spectrum
Explanation - FFT shift only reorders bins; it does not change magnitudes or phases.
Correct answer is: It does not alter the magnitude of the spectrum
Q.58 Which of the following is the most significant factor determining the computational load of an FFT for a given signal length?
Signal amplitude
Sampling rate
Length of the FFT \(N\)
Number of window functions used
Explanation - Computational complexity scales with \(N \log N\); larger \(N\) means more operations.
Correct answer is: Length of the FFT \(N\)
Q.59 Which of the following best describes the 'complex conjugate symmetry' property of the DFT of a real‑valued input?
The spectrum is symmetric about the DC bin
The imaginary part is always zero
The spectrum satisfies \(X[k] = X^*[N-k]\)
All spectral components are purely real
Explanation - This conjugate symmetry property leads to redundancy in the spectrum.
Correct answer is: The spectrum satisfies \(X[k] = X^*[N-k]\)
Q.60 Which of the following best explains why the FFT algorithm is called 'Fast'?
It uses faster arithmetic units
It reduces the number of arithmetic operations from quadratic to logarithmic
It can only process small data sizes
It approximates the DFT by a simple formula
Explanation - FFT reduces computational complexity from \(O(N^2)\) to \(O(N\log N)\).
Correct answer is: It reduces the number of arithmetic operations from quadratic to logarithmic
Q.61 In the context of the DFT, what is meant by the term 'frequency bin'?
A small segment of the input signal
A discrete frequency component represented by a single DFT coefficient
The portion of the output spectrum with zero magnitude
The maximum amplitude in the spectrum
Explanation - Each DFT output coefficient corresponds to a specific frequency bin \(k f_s/N\).
Correct answer is: A discrete frequency component represented by a single DFT coefficient
Q.62 Which of the following is a direct benefit of using a 'real‑to‑complex FFT' over a 'complex‑to‑complex FFT' when the input is real?
Higher precision in the output spectrum
Fewer required FFT points
Lower computational cost and memory usage
It removes the need for windowing
Explanation - Real‑to‑complex FFT exploits symmetry, reducing computations by about 50%.
Correct answer is: Lower computational cost and memory usage
Q.63 Which of the following statements correctly describes the 'overlap‑add' technique for large‑length convolution?
It splits the input into non‑overlapping blocks and sums the outputs
It processes overlapping blocks of input and adds the convolution results of each block
It adds an overlap of the output to the input signal
It is used exclusively for real‑to‑complex FFTs
Explanation - Overlap‑add overlaps input blocks by \(N-L\) samples and sums the overlapping FFT results to produce the final output.
Correct answer is: It processes overlapping blocks of input and adds the convolution results of each block
Q.64 Which of the following best explains why spectral leakage occurs when performing an FFT on a finite-length signal?
The FFT algorithm is inherently inaccurate
The signal is assumed to be periodic outside its observation window, leading to discontinuities at the boundaries
The input samples are quantized
The FFT uses an approximated exponential kernel
Explanation - Assuming periodicity causes abrupt changes at the window edges, spreading energy across frequencies.
Correct answer is: The signal is assumed to be periodic outside its observation window, leading to discontinuities at the boundaries
Q.65 Which of the following best describes the effect of using a Hann window on the main‑lobe width of an FFT spectrum?
It narrows the main‑lobe width
It widens the main‑lobe width
It has no effect on the main‑lobe width
It eliminates the main‑lobe entirely
Explanation - Hann window reduces sidelobes at the cost of a broader main‑lobe.
Correct answer is: It widens the main‑lobe width
Q.66 In which scenario would a 'mixed‑radix FFT' be preferred over a standard radix‑2 FFT?
When the transform length is a power of two
When the transform length is a prime number
When the transform length is composite with small factors other than two
When the input signal is purely real
Explanation - Mixed‑radix FFT handles lengths that factor into small primes efficiently.
Correct answer is: When the transform length is composite with small factors other than two
Q.67 Which of the following statements best describes the 'FFT algorithm's' memory access pattern?
It is random and unpredictable
It accesses data in a strictly linear fashion
It requires frequent jumps between distant memory addresses, which can be a bottleneck on some architectures
It never accesses memory after the first stage
Explanation - Butterfly operations involve non‑adjacent data access, impacting cache performance.
Correct answer is: It requires frequent jumps between distant memory addresses, which can be a bottleneck on some architectures
Q.68 Which of the following best describes the effect of zero‑padding on the amplitude of FFT bins?
It scales all amplitudes by the padding factor
It leaves the amplitude of the original bins unchanged but adds zeros at the new bins
It reduces the amplitude of all bins
It doubles the amplitude of all bins
Explanation - Zero‑padding only adds zeros to the time‑domain signal; the original sample values remain unchanged.
Correct answer is: It leaves the amplitude of the original bins unchanged but adds zeros at the new bins
Q.69 Which of the following best describes the 'frequency‑shift property' of the DFT?
Shifting the input sequence in time introduces a phase shift in the spectrum
Multiplying the input sequence by a complex exponential shifts its spectrum in frequency
Adding a constant to the input shifts the spectrum by a constant frequency
The spectrum remains unchanged under any input transformation
Explanation - The frequency‑shift property states \(x[n] e^{j2\pi f_0 n} \leftrightarrow X[k - f_0]\).
Correct answer is: Multiplying the input sequence by a complex exponential shifts its spectrum in frequency
Q.70 In the context of FFT, what does the term 'aliasing' refer to?
When two frequency components produce identical FFT bins
When high‑frequency components fold back into the lower frequency range due to insufficient sampling
When the FFT output is misinterpreted as time domain
When the FFT algorithm produces an incorrect magnitude
Explanation - Aliasing arises when the sampling rate is below twice the highest frequency component.
Correct answer is: When high‑frequency components fold back into the lower frequency range due to insufficient sampling
Q.71 Which of the following best explains why the DFT is periodic in frequency?
Because the exponential basis functions repeat every \(N\) points
Because the FFT algorithm only works for periodic signals
Because real signals are always periodic
Because the sampling theorem guarantees periodicity
Explanation - The complex exponentials \(e^{-j2\pi kn/N}\) satisfy \(W_N^{k+N} = W_N^k\), making the DFT inherently periodic.
Correct answer is: Because the exponential basis functions repeat every \(N\) points
Q.72 Which of the following statements accurately describes the relationship between the FFT length \(N\) and the maximum frequency that can be represented without aliasing?
Maximum frequency is \(f_s / N\)
Maximum frequency is \(f_s / 2\)
Maximum frequency is \(f_s\)
Maximum frequency is \(2 f_s / N\)
Explanation - The Nyquist frequency \(f_s/2\) is the highest representable frequency irrespective of \(N\).
Correct answer is: Maximum frequency is \(f_s / 2\)
Q.73 Which of the following best describes the purpose of the 'FFT length' when performing spectral analysis?
To determine the number of output frequency bins and their spacing
To set the sampling frequency of the input signal
To decide how many zeros to pad the input
To define the window function to use
Explanation - FFT length \(N\) sets the number of bins \(N\) and the frequency resolution \(\Delta f = f_s/N\).
Correct answer is: To determine the number of output frequency bins and their spacing
Q.74 Which of the following best describes the computational advantage of the FFT when applied to signals with length that is a product of small primes?
The FFT can be performed in linear time
The FFT algorithm becomes inapplicable
Mixed‑radix FFTs can be used to reduce the number of operations
The FFT requires a larger memory footprint
Explanation - Mixed‑radix algorithms exploit the factorization of \(N\) into small primes for efficient computation.
Correct answer is: Mixed‑radix FFTs can be used to reduce the number of operations
Q.75 Which of the following best describes the role of the 'complex conjugate symmetry' property in reducing the storage requirements for real‑valued FFT outputs?
It allows storing only the first \(N/2+1\) unique complex values
It doubles the memory needed
It eliminates the need for storing negative frequencies
It requires storing both real and imaginary parts for all bins
Explanation - Due to symmetry, the second half of the spectrum is the complex conjugate of the first, so only the first half plus DC and Nyquist need storage.
Correct answer is: It allows storing only the first \(N/2+1\) unique complex values
Q.76 Which of the following best explains why the FFT algorithm is not suitable for real‑time audio streaming with extremely low latency (e.g., <1 ms) on a low‑power microcontroller?
Because it requires a power‑of‑two input length
Because its computational load is too high for such short windows on low‑power devices
Because it can only process real signals
Because it cannot be implemented in C
Explanation - Even \(O(N \log N)\) operations may exceed the processing budget of low‑power hardware for very small \(N\).
Correct answer is: Because its computational load is too high for such short windows on low‑power devices