Q.1 What is the Fourier series representation of a real, even, periodic function?
A sine series only
A cosine series only
Both sine and cosine series
No series representation
Explanation - A real, even function contains only cosine terms in its Fourier series because cosine is even and sine is odd.
Correct answer is: A cosine series only
Q.2 Which of the following is a property of the Fourier transform?
Time shift in time domain results in multiplication by a complex exponential in frequency domain
Time shift in time domain results in a time shift in frequency domain
Frequency shift results in time shift in time domain
None of the above
Explanation - The time shift property states that shifting a signal in time multiplies its Fourier transform by e^{-jωt0}.
Correct answer is: Time shift in time domain results in multiplication by a complex exponential in frequency domain
Q.3 The Dirichlet conditions ensure convergence of a Fourier series at which points?
Only at points of continuity
At all points except jump discontinuities
At all points including discontinuities
No convergence guarantee
Explanation - Dirichlet conditions guarantee the Fourier series converges to the average of left and right limits at jump discontinuities.
Correct answer is: At all points except jump discontinuities
Q.4 In the Fourier transform, what does the variable 'ω' represent?
Frequency in Hz
Angular frequency in rad/s
Wavenumber
Time shift
Explanation - The symbol ω is the angular frequency (2π times the ordinary frequency), measured in radians per second.
Correct answer is: Angular frequency in rad/s
Q.5 Which transform pair is used for discrete time signals?
Fourier Transform and Laplace Transform
Fourier Transform and Z-transform
Z-transform and Laplace Transform
Fourier Transform only
Explanation - The Z-transform handles discrete-time signals, while the Laplace transform is used for continuous-time.
Correct answer is: Z-transform and Laplace Transform
Q.6 What is the main difference between Fourier series and Fourier transform?
Fourier series is for continuous signals, Fourier transform for discrete signals
Fourier series represents periodic functions; Fourier transform represents aperiodic functions
Fourier transform is only for real signals
Fourier series uses complex exponentials; Fourier transform uses sine and cosine only
Explanation - Fourier series decomposes periodic signals into harmonics, while the Fourier transform analyzes aperiodic signals.
Correct answer is: Fourier series represents periodic functions; Fourier transform represents aperiodic functions
Q.7 The Fourier transform of a rectangular pulse of width T and amplitude A is:
A·T·sinc(ωT)
A·sin(ωT)/ω
A·cos(ωT)
A/T·sinc(ω/T)
Explanation - The transform of a rectangular pulse yields A·T·sinc(ωT), where sinc(x)=sin(x)/x.
Correct answer is: A·T·sinc(ωT)
Q.8 Which of the following statements about the Nyquist frequency is true?
It is half the sampling rate
It is equal to the sampling rate
It is twice the sampling rate
It has nothing to do with sampling
Explanation - The Nyquist frequency is f_s/2, the highest frequency that can be accurately sampled without aliasing.
Correct answer is: It is half the sampling rate
Q.9 If a signal x(t) has a Fourier transform X(jω), what is the Fourier transform of its derivative dx/dt?
jωX(jω)
X(jω)/jω
-jωX(jω)
jX(jω)/ω
Explanation - Differentiation in time corresponds to multiplication by jω in the frequency domain.
Correct answer is: jωX(jω)
Q.10 Which function has a Fourier transform that is also a Gaussian?
Rectangular function
Sinc function
Gaussian function
Exponential decay
Explanation - A Gaussian in time domain transforms to a Gaussian in frequency domain, a unique property.
Correct answer is: Gaussian function
Q.11 What does the convolution theorem state about Fourier transforms?
Fourier transform of a product is the sum of transforms
Fourier transform of a sum is the product of transforms
Fourier transform of a convolution is the product of transforms
Fourier transform of a convolution is the sum of transforms
Explanation - Convolution in time corresponds to multiplication in frequency.
Correct answer is: Fourier transform of a convolution is the product of transforms
Q.12 The Dirac delta function δ(t) has a Fourier transform:
0
1
∞
δ(ω)
Explanation - The Fourier transform of δ(t) is 1 for all frequencies.
Correct answer is: 1
Q.13 Which of the following is the inverse Fourier transform formula for continuous signals?
x(t) = (1/2π)∫ X(jω) e^{jωt} dω
x(t) = ∫ X(jω) e^{-jωt} dω
x(t) = 2π∫ X(jω) e^{jωt} dω
x(t) = (1/π)∫ X(jω) e^{-jωt} dω
Explanation - The inverse Fourier transform uses a 1/2π factor and a positive exponent.
Correct answer is: x(t) = (1/2π)∫ X(jω) e^{jωt} dω
Q.14 Which of the following statements about the Fourier series coefficients a_n for a real signal is false?
a_{-n} = a_n^*
If the signal is even, a_n are real
If the signal is odd, a_n are imaginary
All coefficients are zero for a constant signal
Explanation - A constant signal has a non-zero zero-frequency component (a_0); other harmonics are zero.
Correct answer is: All coefficients are zero for a constant signal
Q.15 What is the bandwidth of a signal that has a Fourier transform X(jω) = 1 for |ω| ≤ 5 rad/s and 0 elsewhere?
5 rad/s
10 rad/s
2.5 rad/s
15 rad/s
Explanation - Bandwidth is the difference between the highest and lowest non-zero frequency components: 5 - (-5) = 10 rad/s.
Correct answer is: 10 rad/s
Q.16 The Fourier series of a square wave with amplitude A and duty cycle 50% has:
All odd harmonics with amplitudes (4A)/(πn)
All even harmonics with amplitudes (4A)/(πn)
Only the fundamental frequency
All harmonics with equal amplitude
Explanation - A square wave contains only odd harmonics with amplitude decreasing as 1/n.
Correct answer is: All odd harmonics with amplitudes (4A)/(πn)
Q.17 Which integral represents the Fourier transform of a function f(t)?
∫ f(t) e^{-jωt} dt
∫ f(t) e^{jωt} dt
∫ f(t) sin(ωt) dt
∫ f(t) cos(ωt) dt
Explanation - The standard definition uses the exponential with a negative sign in the integrand.
Correct answer is: ∫ f(t) e^{-jωt} dt
Q.18 When a signal is multiplied by a sinusoid of frequency ω0, in the frequency domain this operation is equivalent to:
A shift by ω0
A scaling by ω0
A convolution with a delta at ω0
A convolution with a sinc function
Explanation - Modulation in time results in frequency shifting in the frequency domain.
Correct answer is: A shift by ω0
Q.19 What is the effect of time dilation t → a t on the Fourier transform X(jω)?
X(jω) → X(jω/a)
X(jω) → aX(jω)
X(jω) → X(jaω)
X(jω) → X(jω)/a
Explanation - Time scaling compresses or stretches the frequency axis inversely.
Correct answer is: X(jω) → X(jω/a)
Q.20 Which of the following is a property of the Hilbert transform in relation to the Fourier transform?
Multiplication by j for all frequencies
Multiplication by j for positive frequencies and -j for negative
Adding a constant to the phase
It has no effect on the magnitude spectrum
Explanation - The Hilbert transform introduces a 90° phase shift that depends on frequency sign.
Correct answer is: Multiplication by j for positive frequencies and -j for negative
Q.21 In Fourier analysis, the term 'aliasing' refers to:
Distortion due to low sampling rate
The spreading of a signal in frequency domain
The superposition of harmonics
None of the above
Explanation - Aliasing occurs when the sampling rate is insufficient to capture high-frequency components.
Correct answer is: Distortion due to low sampling rate
Q.22 Which of the following is a window function used to reduce spectral leakage?
Rectangular window
Hamming window
Square window
Linear window
Explanation - The Hamming window smooths the edges to reduce leakage compared to a rectangular window.
Correct answer is: Hamming window
Q.23 What is the Fourier transform of e^{-a|t|} for a > 0?
2a/(a^2 + ω^2)
a/(a^2 + ω^2)
2/(a^2 + ω^2)
a/(a^2 - ω^2)
Explanation - The transform of a double-sided exponential decays as 1/(a^2+ω^2) multiplied by 2a.
Correct answer is: 2a/(a^2 + ω^2)
Q.24 Which transform is most suitable for analyzing signals whose frequency content changes over time?
Fourier Transform
Laplace Transform
Wavelet Transform
Z-transform
Explanation - Wavelets provide time–frequency localization, unlike the global Fourier transform.
Correct answer is: Wavelet Transform
Q.25 For a periodic function with period T, the fundamental angular frequency is:
2π/T
π/T
T/2π
T/π
Explanation - Fundamental frequency ω0 = 2π/T.
Correct answer is: 2π/T
Q.26 The Parseval's theorem relates which two quantities?
Signal energy in time domain and frequency domain
Signal amplitude and phase
Time and frequency resolution
Sampling rate and bandwidth
Explanation - Parseval's theorem states that the total energy is preserved in both domains.
Correct answer is: Signal energy in time domain and frequency domain
Q.27 Which of the following signals has a Fourier series with only even harmonics?
A square wave
A sine wave
An even function symmetric about the origin
An odd function symmetric about the origin
Explanation - Even functions contain only cosine (even) terms, which correspond to even harmonics.
Correct answer is: An even function symmetric about the origin
Q.28 The 'sinc' function is defined as:
sin(πt)/(πt)
sin(t)/t
sin(πt)/t
sin(t)/(πt)
Explanation - sinc(t) = sin(πt)/(πt) is the normalized sinc function.
Correct answer is: sin(πt)/(πt)
Q.29 Which of these is NOT a characteristic of the Laplace transform?
Handles initial conditions
Used for differential equations
Transforms to complex frequency domain
Only works for periodic signals
Explanation - The Laplace transform applies to both periodic and aperiodic signals.
Correct answer is: Only works for periodic signals
Q.30 In the context of Fourier analysis, what is 'spectral leakage'?
Energy leaking from one frequency to another due to windowing
Signal energy lost during transmission
Loss of spectral data due to poor resolution
All of the above
Explanation - Spectral leakage is caused by truncating or windowing a signal.
Correct answer is: Energy leaking from one frequency to another due to windowing
Q.31 Which of the following correctly represents the convolution property of the Fourier transform?
F{x(t)·y(t)} = F{x(t)} + F{y(t)}
F{x(t)∗y(t)} = F{x(t)}·F{y(t)}
F{x(t)∗y(t)} = F{x(t)} + F{y(t)}
F{x(t)·y(t)} = F{x(t)}·F{y(t)}
Explanation - Convolution in time equals multiplication in frequency.
Correct answer is: F{x(t)∗y(t)} = F{x(t)}·F{y(t)}
Q.32 The frequency domain representation of a time-shifted signal x(t - t0) is:
X(jω) e^{-jωt0}
X(jω) e^{jωt0}
X(jω + t0)
X(jω - t0)
Explanation - Time delay introduces a linear phase term e^{-jωt0}.
Correct answer is: X(jω) e^{-jωt0}
Q.33 What is the main benefit of using the Discrete Fourier Transform (DFT) over the Continuous Fourier Transform?
It can handle infinite-length signals
It provides exact time resolution
It is computationally efficient for sampled data
It has no scaling factor
Explanation - DFT works with discrete, finite-length data and can be computed via FFT.
Correct answer is: It is computationally efficient for sampled data
Q.34 Which of the following is a property of the Fourier transform of a real-valued even function?
X(jω) is purely imaginary
X(jω) is purely real and even
X(jω) is complex conjugate symmetric
None of the above
Explanation - The Fourier transform of a real even function is real and even.
Correct answer is: X(jω) is purely real and even
Q.35 What does the 'bandwidth' of a filter describe?
The number of poles
The frequency range where the filter passes signals
The maximum amplitude of the filter
The time delay of the filter
Explanation - Bandwidth is the width of the frequency range the filter allows.
Correct answer is: The frequency range where the filter passes signals
Q.36 Which of these is a commonly used sampling theorem?
Nyquist–Shannon
Heisenberg
Euler
Zeno
Explanation - Nyquist–Shannon theorem dictates the minimum sampling rate to avoid aliasing.
Correct answer is: Nyquist–Shannon
Q.37 The Fourier transform of cos(ω0t) is:
π[δ(ω-ω0)+δ(ω+ω0)]
π[δ(ω-ω0)-δ(ω+ω0)]
δ(ω-ω0)+δ(ω+ω0)
0
Explanation - Cosine transforms into two delta functions at ±ω0, each scaled by π.
Correct answer is: π[δ(ω-ω0)+δ(ω+ω0)]
Q.38 In the context of signal processing, what does 'smoothing' refer to?
Increasing the high-frequency components
Applying a low-pass filter
Amplifying the signal
None of the above
Explanation - Smoothing removes high-frequency variations, often using a low-pass filter.
Correct answer is: Applying a low-pass filter
Q.39 Which of these is a requirement for the existence of the Fourier transform of a signal?
Signal must be periodic
Signal must be absolutely integrable
Signal must be bounded
Signal must be differentiable
Explanation - Absolute integrability ensures the Fourier transform exists.
Correct answer is: Signal must be absolutely integrable
Q.40 The term 'alias' in digital signal processing means:
The original signal
A misrepresented higher frequency component
A low-frequency component
None of the above
Explanation - Alias is a high frequency appearing as a lower frequency due to undersampling.
Correct answer is: A misrepresented higher frequency component
Q.41 Which operation in the time domain corresponds to scaling the frequency axis by a factor of 2?
Time scaling by 1/2
Time scaling by 2
Frequency scaling by 1/2
Frequency scaling by 2
Explanation - Compressing time by 1/2 stretches frequency by 2.
Correct answer is: Time scaling by 1/2
Q.42 The 'frequency resolution' of an FFT depends on:
Sampling frequency only
FFT length only
Both sampling frequency and FFT length
None of the above
Explanation - Resolution Δf = fs/N, so both affect it.
Correct answer is: Both sampling frequency and FFT length
Q.43 Which of the following is a property of the magnitude spectrum of a real-valued even function?
It is odd
It is even and real
It is complex
It has no symmetry
Explanation - The magnitude of a real even function’s transform is real and even.
Correct answer is: It is even and real
Q.44 What is the Fourier transform of a unit step function u(t)?
πδ(ω) + 1/jω
πδ(ω) - 1/jω
1/(jω)
0
Explanation - The transform includes a delta term and a principal value 1/jω.
Correct answer is: πδ(ω) + 1/jω
Q.45 In Fourier analysis, the 'spectral density' is related to:
Signal amplitude
Signal power spectrum
Signal phase
Sampling rate
Explanation - Spectral density describes how power is distributed across frequencies.
Correct answer is: Signal power spectrum
Q.46 Which of the following is a valid property of the Dirac delta function?
δ(t) = 0 for t=0
∫ δ(t) dt = ∞
δ(t-a) = δ(a-t)
δ(t) = 1 for all t
Explanation - The delta function is symmetric: δ(t-a) = δ(a-t).
Correct answer is: δ(t-a) = δ(a-t)
Q.47 Which of the following transforms is most appropriate for analyzing non-stationary signals?
Fourier Transform
Laplace Transform
Wavelet Transform
Z-transform
Explanation - Wavelet transform provides time-frequency analysis suitable for non-stationary signals.
Correct answer is: Wavelet Transform
Q.48 The property that the Fourier transform of a derivative equals jω times the transform of the original signal is known as:
Time-shift
Frequency-shift
Differentiation
Integration
Explanation - Differentiation in time domain translates to multiplication by jω in frequency domain.
Correct answer is: Differentiation
Q.49 Which of these is true for an odd function f(t)?
Fourier transform is real
Fourier transform is imaginary
Fourier transform is complex conjugate symmetric
Fourier transform is zero
Explanation - An odd real function transforms into a purely imaginary spectrum.
Correct answer is: Fourier transform is imaginary
Q.50 Which of the following statements about the continuous-time Fourier transform (CTFT) is correct?
It only applies to periodic signals
It is defined for any integrable signal
It can handle only real-valued signals
It has no inverse
Explanation - CTFT applies to any absolutely integrable signal, not just periodic or real.
Correct answer is: It is defined for any integrable signal
Q.51 The 'time-limited' function e^{-t^2} has a Fourier transform that is also:
Another Gaussian
A sinc function
A delta function
Zero for all ω
Explanation - A Gaussian in time remains Gaussian in frequency (self-Fourier property).
Correct answer is: Another Gaussian
Q.52 What does the 'sampling interval' T_s represent?
1/sampling frequency
Sampling frequency
Time shift
Frequency shift
Explanation - T_s = 1/f_s is the time between consecutive samples.
Correct answer is: 1/sampling frequency
Q.53 The Fourier series of a sinusoid has how many non-zero terms?
Infinite
One
Two
None
Explanation - A sinusoid produces two non-zero complex exponentials at ±frequency.
Correct answer is: Two
Q.54 Which of the following describes the 'Fourier pair' of a low-pass filter?
Rectangular impulse response and sinc frequency response
Gaussian impulse response and Gaussian frequency response
Sinc impulse response and rectangular frequency response
Exponential impulse response and Lorentzian frequency response
Explanation - An ideal low-pass filter has a rectangular frequency response and sinc impulse response.
Correct answer is: Sinc impulse response and rectangular frequency response
Q.55 In a discrete-time system, the z-transform variable 'z' is related to frequency by:
z = e^{jω}
z = jω
z = ω
z = 1/jω
Explanation - On the unit circle z = e^{jω} maps z-domain to frequency.
Correct answer is: z = e^{jω}
Q.56 The convolution of two time-domain signals results in:
Addition in frequency domain
Multiplication in frequency domain
Subtraction in frequency domain
Division in frequency domain
Explanation - Convolution in time domain corresponds to multiplication in frequency domain.
Correct answer is: Multiplication in frequency domain
Q.57 A signal x(t) = e^{-αt}u(t) has a Fourier transform:
1/(α + jω)
α/(α^2 + ω^2)
1/(α - jω)
α/(α^2 - ω^2)
Explanation - The unilateral exponential transforms to 1/(α + jω).
Correct answer is: 1/(α + jω)
Q.58 Which of the following is true for a purely imaginary Fourier transform?
The time domain signal is even
The time domain signal is odd
The time domain signal is constant
The time domain signal is zero
Explanation - A purely imaginary transform corresponds to a real odd time-domain signal.
Correct answer is: The time domain signal is odd
Q.59 In the context of Fourier series, the coefficient a0 represents:
The average value of the function
The maximum amplitude
The fundamental frequency
Zero for all periodic functions
Explanation - a0 is the DC component, equal to the average of the periodic signal.
Correct answer is: The average value of the function
Q.60 Which of the following best describes the 'band-limited' signal?
Signal with infinite bandwidth
Signal that contains only one frequency
Signal that has non-zero components only in a finite frequency band
Signal that is zero for all frequencies
Explanation - Band-limited means the spectrum is zero outside a specified band.
Correct answer is: Signal that has non-zero components only in a finite frequency band
Q.61 The 'Hilbert transform' of a real signal yields a:
Real part only
Imaginary part only
Analytic signal
Zero signal
Explanation - Combining a signal with its Hilbert transform gives the analytic signal.
Correct answer is: Analytic signal
Q.62 In Fourier analysis, the term 'frequency resolution' refers to:
The width of a single frequency bin
The total bandwidth of the system
The sampling frequency
The number of harmonics
Explanation - Frequency resolution is Δf = fs/N in an FFT.
Correct answer is: The width of a single frequency bin
Q.63 What is the effect of multiplying a signal by a cosine in time domain?
Frequency shift by the cosine frequency
Amplitude scaling
Time scaling
No effect
Explanation - Modulation shifts the spectrum by the cosine's frequency.
Correct answer is: Frequency shift by the cosine frequency
Q.64 The 'Fourier integral' representation is used for:
Periodic signals only
Aperiodic signals only
Both periodic and aperiodic signals
None of the above
Explanation - The Fourier integral generalizes to aperiodic signals; Fourier series is for periodic.
Correct answer is: Both periodic and aperiodic signals
Q.65 In the context of digital signal processing, 'zero padding' primarily affects:
Amplitude of the signal
Time resolution
Frequency resolution
Sampling rate
Explanation - Zero padding increases FFT points, refining frequency bin width.
Correct answer is: Frequency resolution
Q.66 The Fourier transform of a causal exponential e^{-αt}u(t) with α>0 is:
1/(α + jω)
α/(α^2 + ω^2)
1/(α - jω)
α/(α^2 - ω^2)
Explanation - The unilateral exponential transforms to 1/(α + jω).
Correct answer is: 1/(α + jω)
Q.67 Which of the following is NOT a property of the Fourier transform?
Linearity
Time-shifting
Nonlinearity with multiplication
Frequency-shifting
Explanation - Fourier transform is linear; multiplication in time corresponds to convolution in frequency.
Correct answer is: Nonlinearity with multiplication
Q.68 What does the 'spectral centroid' measure?
Average frequency of a spectrum weighted by magnitude
Peak frequency of the spectrum
Bandwidth of the spectrum
Zero-crossing rate
Explanation - Spectral centroid indicates where the 'center of mass' of the spectrum lies.
Correct answer is: Average frequency of a spectrum weighted by magnitude
Q.69 The 'Z-transform' is a generalization of which continuous-time transform?
Fourier Transform
Laplace Transform
Wavelet Transform
Hilbert Transform
Explanation - Z-transform can be viewed as the Laplace transform evaluated on the unit circle for discrete signals.
Correct answer is: Laplace Transform
Q.70 The 'frequency response' of a system is:
The output amplitude and phase as functions of frequency
The time-domain impulse response
The input signal
The system's power consumption
Explanation - Frequency response shows how a system modifies signals across frequencies.
Correct answer is: The output amplitude and phase as functions of frequency
Q.71 Which of the following best describes a 'continuous-time Fourier transform' (CTFT)?
A transform for discrete signals
A transform for continuous signals
A transform only for real signals
A transform only for complex signals
Explanation - CTFT analyzes continuous-time signals in the frequency domain.
Correct answer is: A transform for continuous signals
Q.72 Which property states that multiplying a signal by a constant scales its Fourier transform by the same constant?
Time scaling
Frequency scaling
Linearity
Time shifting
Explanation - Fourier transform is linear; scaling time domain scales frequency domain accordingly.
Correct answer is: Linearity
Q.73 The 'bandwidth' of an ideal low-pass filter with cutoff frequency ωc is:
ωc
2ωc
π/ωc
0
Explanation - The bandwidth is defined as the range of frequencies passed, i.e., from 0 to ωc.
Correct answer is: ωc
Q.74 Which of the following transforms is most suitable for signals with finite support?
Fourier Transform
Laplace Transform
Z-transform
Wavelet Transform
Explanation - Fourier Transform handles finite support signals effectively; other transforms are better for specific cases.
Correct answer is: Fourier Transform
Q.75 A function that is its own Fourier transform is called a:
Eigenfunction
Self-Fourier
Fourier pair
Unitary function
Explanation - A self-Fourier function remains unchanged under the Fourier transform.
Correct answer is: Self-Fourier
Q.76 In the Fourier series, the coefficients for an odd function are:
All real
All imaginary
All zero
All complex
Explanation - The Fourier series of a real odd function contains only sine terms, which are imaginary in complex form.
Correct answer is: All imaginary
Q.77 Which of the following is a common use of the Fourier transform in engineering?
Solving differential equations
Analyzing audio signals
Designing filters
All of the above
Explanation - The Fourier transform is widely used for solving differential equations, signal analysis, and filter design.
Correct answer is: All of the above
Q.78 Which of these signals has a symmetric magnitude spectrum but an asymmetric phase spectrum?
Pure cosine
Pure sine
Even function
Odd function
Explanation - A cosine has symmetric magnitude but zero phase, while a sine has symmetric magnitude and a 90° phase shift.
Correct answer is: Pure cosine
Q.79 The 'zero-frequency' component of a Fourier transform corresponds to the:
High-frequency content
Average value of the signal
Maximum amplitude
Minimum amplitude
Explanation - The DC component at ω=0 equals the time-average of the signal.
Correct answer is: Average value of the signal
Q.80 In Fourier analysis, a 'delta train' is a series of:
Rectangular pulses
Sine waves
Dirac delta functions
Gaussian pulses
Explanation - A delta train consists of repeated Dirac delta spikes at regular intervals.
Correct answer is: Dirac delta functions
Q.81 Which of the following describes the 'frequency resolution' of an FFT with N points and sampling frequency fs?
Δf = fs/N
Δf = N/fs
Δf = fs
Δf = 1/N
Explanation - Frequency resolution is sampling frequency divided by number of points.
Correct answer is: Δf = fs/N
Q.82 The Fourier transform of a real, even function f(t) is:
Purely real and even
Purely imaginary and odd
Complex conjugate symmetric
Zero
Explanation - Real even functions yield real even spectra.
Correct answer is: Purely real and even
Q.83 The 'sampling theorem' states that a signal can be perfectly reconstructed from samples if the sampling rate:
Is equal to the signal's maximum frequency
Is greater than twice the maximum frequency
Is less than the signal's maximum frequency
Has no restrictions
Explanation - According to Nyquist–Shannon, fs > 2f_max.
Correct answer is: Is greater than twice the maximum frequency
Q.84 Which of the following statements about the Fourier transform of a band-limited function is correct?
It is infinite in time
It has a finite support in time
It has infinite support in frequency
It is zero outside the band
Explanation - Band-limited means its frequency representation is zero beyond a certain band.
Correct answer is: It is zero outside the band
Q.85 Which transform pair is used for solving partial differential equations in the context of signal processing?
Fourier Transform and Z-transform
Laplace Transform and Fourier Transform
Fourier Transform and Hilbert Transform
Wavelet Transform and Fourier Transform
Explanation - Laplace is used for time-domain PDEs; Fourier for frequency-domain analysis.
Correct answer is: Laplace Transform and Fourier Transform
Q.86 Which of the following best describes an 'analytic signal'?
A real signal with zero imaginary part
A complex signal with only positive frequency components
A signal whose Fourier transform is zero
A signal with only negative frequencies
Explanation - An analytic signal has no negative frequency content, constructed via the Hilbert transform.
Correct answer is: A complex signal with only positive frequency components
Q.87 Which of the following statements about the Fourier transform of a delta function is correct?
It is zero for all frequencies
It is a constant value of 1 for all frequencies
It is an impulse at zero frequency
It is a sinc function
Explanation - The transform of δ(t) is 1, constant across frequency.
Correct answer is: It is a constant value of 1 for all frequencies
Q.88 The 'frequency domain' representation of a signal is obtained by applying:
Differential equation
Fourier transform
Integral transform
None of the above
Explanation - Fourier transform converts time domain signals to frequency domain.
Correct answer is: Fourier transform
Q.89 The 'time-frequency resolution' of a spectrogram is affected by:
The window size only
The window size and overlap
The sampling rate only
None of the above
Explanation - Window size determines time resolution; overlap affects frequency resolution.
Correct answer is: The window size and overlap
Q.90 Which of these is a correct representation of the Fourier series coefficients for a square wave?
An infinite series of even harmonics
An infinite series of odd harmonics with amplitude 1/n
A single harmonic at the fundamental frequency
Zero for all harmonics
Explanation - Square waves contain only odd harmonics decaying as 1/n.
Correct answer is: An infinite series of odd harmonics with amplitude 1/n
Q.91 The 'time-domain' representation of a frequency-shifted signal is:
Amplitude scaled
Frequency multiplied
Phase shifted linearly
Time dilated
Explanation - Frequency shifting adds a linear phase term in time.
Correct answer is: Phase shifted linearly
Q.92 In Fourier analysis, what does the term 'sinc' refer to?
A function that decays exponentially
A function defined as sin(x)/x
A function defined as sin(πx)/(πx)
A function that is constant
Explanation - The normalized sinc function is sin(πx)/(πx).
Correct answer is: A function defined as sin(πx)/(πx)
Q.93 Which of the following transforms is primarily used for discrete-time signals?
Laplace Transform
Z-transform
Fourier Transform
Wavelet Transform
Explanation - Z-transform handles discrete-time sequences.
Correct answer is: Z-transform
Q.94 What is the effect of applying a low-pass filter to a high-frequency signal?
Increase the amplitude of high frequencies
Attenuate high frequencies, preserve low frequencies
Reverse the signal
No effect
Explanation - Low-pass filters allow low frequencies to pass while attenuating higher ones.
Correct answer is: Attenuate high frequencies, preserve low frequencies
Q.95 Which of the following is true about the Fourier transform of a real, odd function?
It is purely real and even
It is purely imaginary and odd
It is complex and symmetric
It is zero
Explanation - The transform of a real odd function yields a purely imaginary odd spectrum.
Correct answer is: It is purely imaginary and odd
Q.96 The 'time delay' property in Fourier analysis states that a delay of t0 in time corresponds to:
Multiplication by e^{-jωt0} in frequency
Multiplication by e^{jωt0} in frequency
A shift in frequency by t0
A scaling in frequency by t0
Explanation - Time delay introduces a linear phase term e^{-jωt0}.
Correct answer is: Multiplication by e^{-jωt0} in frequency
Q.97 In Fourier series, the coefficient a_n for a cosine term corresponds to:
The amplitude of the nth harmonic
The phase of the nth harmonic
The frequency of the nth harmonic
The period of the nth harmonic
Explanation - Coefficients a_n directly represent amplitudes of cosine components.
Correct answer is: The amplitude of the nth harmonic
Q.98 What is the relationship between the Fourier transform of a real, even function and its conjugate symmetry property?
X(jω) = X(jω)^*
X(jω) = X(-jω)^*
X(jω) = -X(-jω)
X(jω) = X(ω)
Explanation - Real even functions yield spectra that are conjugate symmetric about the origin.
Correct answer is: X(jω) = X(-jω)^*
Q.99 Which of the following best describes a 'band-limited' signal?
A signal that occupies infinite bandwidth
A signal with nonzero spectral content only within a finite frequency range
A signal that has a single frequency component
A signal that is zero for all time
Explanation - Band-limited signals have zero energy outside a specific frequency band.
Correct answer is: A signal with nonzero spectral content only within a finite frequency range
Q.100 Which of the following is true about the Fourier transform of a Gaussian function?
It is zero everywhere
It is also a Gaussian function
It is a sinc function
It has infinite bandwidth
Explanation - The Fourier transform of a Gaussian is another Gaussian, preserving shape.
Correct answer is: It is also a Gaussian function
Q.101 The 'Hilbert transform' introduces a phase shift of:
0 degrees
45 degrees
90 degrees
180 degrees
Explanation - Hilbert transform shifts phase by ±90° across all frequencies.
Correct answer is: 90 degrees
Q.102 What is the Fourier transform of a time-limited rectangular pulse of width T and amplitude 1?
T sinc(ωT/2)
T sinc(ωT)
sinc(ωT)
T sinc(ω/2T)
Explanation - The transform is T sinc(ωT/2), accounting for the pulse width.
Correct answer is: T sinc(ωT/2)
Q.103 Which of the following describes the 'sampling theorem' in Fourier analysis?
Samples must be taken at random intervals
Sampling rate must be at least twice the maximum frequency
Sampling rate can be arbitrarily low
Sampling does not affect the signal
Explanation - Nyquist–Shannon theorem dictates fs ≥ 2·f_max.
Correct answer is: Sampling rate must be at least twice the maximum frequency
Q.104 What does the 'Fourier integral' generalize?
Fourier series for nonperiodic signals
Laplace transform for periodic signals
Z-transform for continuous signals
Wavelet transform for discrete signals
Explanation - Fourier integral extends Fourier series to aperiodic signals.
Correct answer is: Fourier series for nonperiodic signals
Q.105 Which of the following statements is true regarding the Fourier transform of an even function?
It is purely imaginary
It is purely real
It is zero
It has no symmetry
Explanation - Even functions yield purely real Fourier spectra.
Correct answer is: It is purely real
Q.106 The 'bandwidth' of a filter is measured in:
Time (seconds)
Frequency (Hz or rad/s)
Amplitude (dB)
Phase (radians)
Explanation - Bandwidth quantifies the frequency range the filter allows.
Correct answer is: Frequency (Hz or rad/s)
Q.107 Which of the following transforms is a discrete counterpart of the continuous-time Fourier transform?
Laplace Transform
Z-transform
Discrete Fourier Transform (DFT)
Wavelet Transform
Explanation - DFT is the discrete version of the Fourier transform for sampled signals.
Correct answer is: Discrete Fourier Transform (DFT)
Q.108 Which of these best describes a 'frequency shift' in the Fourier domain?
Time scaling of the signal
Multiplication by a complex exponential
Convolution with a delta function
Multiplication by a constant
Explanation - Frequency shift corresponds to multiplying by e^{jω0t} in time domain, or a shift in frequency domain.
Correct answer is: Multiplication by a complex exponential
Q.109 The 'inverse Fourier transform' of X(jω) is given by:
(1/2π)∫ X(jω) e^{jωt} dω
(1/2π)∫ X(jω) e^{-jωt} dω
∫ X(jω) e^{jωt} dω
∫ X(jω) e^{-jωt} dω
Explanation - The inverse transform formula uses a 1/(2π) factor and a positive exponential.
Correct answer is: (1/2π)∫ X(jω) e^{jωt} dω
Q.110 Which property of Fourier transform says that time reversal of a signal leads to:
Conjugation of the spectrum
Inversion of the spectrum
Reversal of the spectrum
No change
Explanation - Time reversal x(-t) corresponds to X(-jω) in the frequency domain.
Correct answer is: Reversal of the spectrum
Q.111 The 'phase spectrum' of a real, even function is:
Zero for all frequencies
Constant zero or π
Symmetric around zero
All random
Explanation - Even real signals have zero phase across all frequencies.
Correct answer is: Zero for all frequencies
Q.112 Which of the following describes a 'Gaussian filter' in the frequency domain?
A rectangular shape
A sinc shape
A Gaussian shape
An exponential shape
Explanation - Gaussian filters have a Gaussian spectral shape, smoothing signals smoothly.
Correct answer is: A Gaussian shape