Q.1 Which of the following best describes a discrete‑time signal?
A continuous function of time
A sequence of numbers indexed by integers
A physical wave
A random variable
Explanation - A discrete‑time signal is defined only at integer time indices and is represented by a sequence of samples.
Correct answer is: A sequence of numbers indexed by integers
Q.2 If x[n] = 3, what is its value at n = 5?
3
5
8
0
Explanation - The signal x[n] = 3 is constant for all n, so its value is always 3 regardless of n.
Correct answer is: 3
Q.3 Which of these is NOT a common type of discrete‑time signal?
Periodic
Aperiodic
Band‑limited
Continuous
Explanation - Discrete‑time signals are defined at discrete times; continuous signals are defined over all real numbers.
Correct answer is: Continuous
Q.4 What does the notation x[n-k] represent?
Shifted right by k samples
Shifted left by k samples
Amplitude scaled by k
Frequency shifted by k
Explanation - x[n-k] is the same sequence shifted to the right (delayed) by k samples.
Correct answer is: Shifted right by k samples
Q.5 Which operation would make a signal longer by adding zeros at the end?
Decimation
Upsampling
Zero‑padding
Convolution
Explanation - Zero‑padding appends zeros to the end of a signal, increasing its length without changing its content.
Correct answer is: Zero‑padding
Q.6 How many samples does a sequence have if it is defined for n = 0, 1, 2, 3, 4?
4
5
6
10
Explanation - The indices 0 through 4 inclusive give 5 samples.
Correct answer is: 5
Q.7 Which of these signals is periodic?
x[n] = n
x[n] = (-1)^n
x[n] = 1/(n+1)
x[n] = 0
Explanation - (-1)^n repeats every two samples, making it periodic; the others do not repeat.
Correct answer is: x[n] = (-1)^n
Q.8 What does the term 'sampling period' refer to in discrete‑time signals?
Time between two consecutive samples
Total duration of the signal
Frequency of the signal
Amplitude of the signal
Explanation - The sampling period is the interval in seconds between successive samples of a continuous‑time signal.
Correct answer is: Time between two consecutive samples
Q.9 Which of these is an example of an aperiodic signal?
x[n] = sin(πn/4)
x[n] = 1/(1 + n^2)
x[n] = 0.5^n
x[n] = cos(πn)
Explanation - The function 1/(1 + n^2) does not repeat for any integer period, so it is aperiodic.
Correct answer is: x[n] = 1/(1 + n^2)
Q.10 What is the effect of multiplying a signal by 2?
Double the time axis
Halve the signal
Double the amplitude
Shift the signal by 2 samples
Explanation - Multiplication by 2 scales the signal's amplitude by a factor of 2.
Correct answer is: Double the amplitude
Q.11 Which operation adds two sequences together?
Convolution
Correlation
Addition
Subtraction
Explanation - Simple addition of corresponding samples is the straightforward way to add two sequences.
Correct answer is: Addition
Q.12 In a sequence x[n], what does the index 'n' represent?
Amplitude
Time step
Frequency
Phase
Explanation - The integer index n indicates the discrete time step at which the signal is sampled.
Correct answer is: Time step
Q.13 Which of these is a linear system property?
Non‑invertibility
Time‑variance
Time‑invariance
Non‑causality
Explanation - A linear system is time‑invariant if shifting the input results in an identical shift in the output.
Correct answer is: Time‑invariance
Q.14 What is the result of convolving a signal x[n] with a delta function δ[n-k]?
x[n-k]
x[n+k]
δ[n]
Zero
Explanation - Convolving with a shifted delta function delays the signal by k samples.
Correct answer is: x[n-k]
Q.15 Which of the following statements about causality is true?
A system can produce future output based on past input only.
A causal system depends on future input.
Causality is unrelated to time.
All systems are causal.
Explanation - A causal system's output at time n depends only on inputs at times ≤ n, not on future inputs.
Correct answer is: A system can produce future output based on past input only.
Q.16 What does the z‑transform of a sequence x[n] convert?
Time domain to frequency domain
Sequence to a polynomial
Signal to a matrix
None of the above
Explanation - The z‑transform represents a discrete sequence as a power series (or rational function) in the complex variable z.
Correct answer is: Sequence to a polynomial
Q.17 Which of these signals is stable?
x[n] = n
x[n] = (0.5)^n u[n]
x[n] = 2^n u[n]
x[n] = 1/(n+1)
Explanation - A stable system has bounded output for bounded input; (0.5)^n decays, so it remains bounded.
Correct answer is: x[n] = (0.5)^n u[n]
Q.18 What is the main difference between a continuous‑time and a discrete‑time Fourier transform?
Continuous‑time uses sine, discrete‑time uses cosine
Continuous‑time integrates over time, discrete‑time sums over samples
Discrete‑time uses the Laplace transform
They are identical
Explanation - The CTFT involves integration over a continuous variable, while the DTFT involves an infinite sum over discrete samples.
Correct answer is: Continuous‑time integrates over time, discrete‑time sums over samples
Q.19 A digital filter is described by y[n] = 0.5x[n] + 0.5x[n-1]. What type of filter is it?
Low‑pass
High‑pass
Band‑stop
Band‑pass
Explanation - A simple moving‑average filter smooths the signal, acting as a low‑pass filter.
Correct answer is: Low‑pass
Q.20 The z‑transform of x[n] = u[n] (the unit step) is?
1/(1 - z^{-1})
1/(1 + z^{-1})
z/(z - 1)
1 - z^{-1}
Explanation - The sum of an infinite geometric series gives the z‑transform as 1/(1 - z^{-1}) for |z|>1.
Correct answer is: 1/(1 - z^{-1})
Q.21 Which of the following is a necessary condition for a system to be LTI?
Causality
BIBO stability
Time invariance
All of the above
Explanation - Linearity and time invariance together define an LTI system; causality and stability are separate properties.
Correct answer is: Time invariance
Q.22 If the impulse response h[n] = δ[n] - 0.5δ[n-1], what is the frequency response H(e^{jω})?
1 - 0.5e^{-jω}
1 + 0.5e^{-jω}
0.5 - e^{-jω}
e^{-jω} - 0.5
Explanation - Taking the DTFT of each impulse gives H(e^{jω}) = 1 - 0.5e^{-jω}.
Correct answer is: 1 - 0.5e^{-jω}
Q.23 What does the term 'aliasing' refer to in digital signal processing?
Overlapping of frequency components due to undersampling
The creation of harmonics during filtering
Noise in a digital signal
A type of phase distortion
Explanation - Aliasing occurs when a signal is sampled below its Nyquist rate, causing high‑frequency components to masquerade as lower frequencies.
Correct answer is: Overlapping of frequency components due to undersampling
Q.24 Which of the following is NOT a valid property of the DTFT?
Periodicity of 2π
Linearity
Time‑shift
Real‑valuedness for real sequences
Explanation - The DTFT of a real sequence is generally complex and only symmetric in magnitude, not real‑valued.
Correct answer is: Real‑valuedness for real sequences
Q.25 For a causal system, which region of convergence (ROC) of its z‑transform includes which region?
Outside the outermost pole
Inside the innermost pole
Between poles
Anywhere
Explanation - A causal system's ROC is the exterior of the outermost pole, extending to infinity.
Correct answer is: Outside the outermost pole
Q.26 What does the unit delay operator z^{-1} do to a sequence x[n]?
Advances the sequence by one sample
Delays the sequence by one sample
Multiplies amplitude by -1
Reverses the sequence
Explanation - Multiplying by z^{-1} in the z‑domain corresponds to shifting the sequence one sample to the right in time.
Correct answer is: Delays the sequence by one sample
Q.27 Which of these statements about a system’s impulse response h[n] is true?
It determines the system’s output for any input
It is always zero for n < 0
It is always symmetric
It is independent of the input
Explanation - For an LTI system, the output is the convolution of the input with the impulse response.
Correct answer is: It determines the system’s output for any input
Q.28 Which of the following sequences is BIBO stable?
h[n] = 2^n u[n]
h[n] = (-1)^n u[n]
h[n] = (0.8)^n u[n]
h[n] = n u[n]
Explanation - The magnitude of (0.8)^n decays to zero; thus the sum over all n converges, giving stability.
Correct answer is: h[n] = (0.8)^n u[n]
Q.29 The DTFT of x[n] = (0.5)^n u[n] is?
1/(1 - 0.5e^{-jω})
1/(1 + 0.5e^{-jω})
0.5/(1 - e^{-jω})
e^{-jω}/(1 - 0.5e^{-jω})
Explanation - The DTFT of a decaying exponential is the sum of a geometric series with ratio 0.5e^{-jω}.
Correct answer is: 1/(1 - 0.5e^{-jω})
Q.30 Which of the following describes a linear time‑invariant (LTI) system?
Output scales linearly with input but can vary with time
Output does not scale linearly but is invariant over time
Output scales linearly with input and shifts in time cause corresponding output shifts
Output is nonlinear and time‑variant
Explanation - Linearity ensures scaling and superposition; time invariance means a time shift in the input produces an equal shift in output.
Correct answer is: Output scales linearly with input and shifts in time cause corresponding output shifts
Q.31 The system y[n] = x[n] * h[n] is called what operation?
Correlation
Convolution
Differentiation
Integration
Explanation - The output of a linear time‑invariant system is the convolution of the input with the system’s impulse response.
Correct answer is: Convolution
Q.32 Which condition ensures BIBO stability for an LTI system?
Impulse response h[n] is absolutely summable
Impulse response h[n] is bounded
System has at least one pole inside the unit circle
System is causal
Explanation - For BIBO stability, the sum of |h[n]| over all n must be finite.
Correct answer is: Impulse response h[n] is absolutely summable
Q.33 If the z‑transform of a sequence x[n] has ROC |z| > 2, what can be said about the sequence?
It is anti‑causal
It is right‑sided (causal)
It is two‑sided
It is finite‑length
Explanation - ROC outside the outermost pole indicates a causal (right‑sided) sequence.
Correct answer is: It is right‑sided (causal)
Q.34 Which of the following is the frequency response of a first‑order low‑pass filter H(z) = (1 - a z^{-1})/(1 - b z^{-1}) evaluated on the unit circle?
H(e^{jω}) = (1 - a e^{-jω})/(1 - b e^{-jω})
H(e^{jω}) = (1 + a e^{-jω})/(1 + b e^{-jω})
H(e^{jω}) = (1 - a e^{jω})/(1 - b e^{jω})
H(e^{jω}) = (1 + a e^{jω})/(1 + b e^{jω})
Explanation - Substitute z = e^{jω} to get the frequency response on the unit circle.
Correct answer is: H(e^{jω}) = (1 - a e^{-jω})/(1 - b e^{-jω})
Q.35 A digital filter with transfer function H(z) = (1 - 0.5 z^{-1})/(1 - 0.9 z^{-1}) is stable or unstable?
Stable
Unstable
Marginally stable
Cannot be determined
Explanation - The pole at 0.9 lies inside the unit circle, satisfying the stability condition for causal LTI systems.
Correct answer is: Stable
Q.36 What is the relationship between the DTFT X(e^{jω}) and the z‑transform X(z) of a sequence x[n]?
X(e^{jω}) = X(z) evaluated at z = e^{jω}
X(e^{jω}) = X(z) evaluated at z = 1
X(e^{jω}) = X(z) evaluated at z = 0
X(e^{jω}) = X(z) evaluated at z = ∞
Explanation - The DTFT is a special case of the z‑transform when the complex variable z lies on the unit circle.
Correct answer is: X(e^{jω}) = X(z) evaluated at z = e^{jω}
Q.37 Which of the following best describes the unit‑sample function u[n]?
u[n] = 1 for n = 0, 0 otherwise
u[n] = 0 for n = 0, 1 otherwise
u[n] = 1 for n ≥ 0, 0 for n < 0
u[n] = 0 for n ≥ 0, 1 for n < 0
Explanation - The unit‑step (unit‑sample) function turns on at n = 0 and stays on for all future samples.
Correct answer is: u[n] = 1 for n ≥ 0, 0 for n < 0
Q.38 Which property of a sequence is represented by its symmetry x[n] = x[-n]?
Antisymmetry
Even symmetry
Odd symmetry
Complex conjugate symmetry
Explanation - Even symmetry means the sequence is mirrored around the origin, so x[n] = x[-n].
Correct answer is: Even symmetry
Q.39 What does the 'conjugate symmetry' property of the DTFT imply for real sequences?
X(e^{jω}) = X(e^{-jω})
X(e^{jω}) = X^{*}(e^{-jω})
X(e^{jω}) = -X(e^{-jω})
X(e^{jω}) = 0
Explanation - For real‑valued sequences, the DTFT satisfies conjugate symmetry: the value at -ω is the complex conjugate of the value at ω.
Correct answer is: X(e^{jω}) = X^{*}(e^{-jω})
Q.40 Which of the following is a correct representation of a discrete‑time Gaussian pulse?
x[n] = e^{-n^2}
x[n] = e^{-(n-5)^2}
x[n] = n^2
x[n] = sin(n)
Explanation - The Gaussian pulse is centered at n = 5 and decays symmetrically around that center.
Correct answer is: x[n] = e^{-(n-5)^2}
Q.41 The ZOH (Zero‑Order Hold) reconstruction method samples a continuous‑time signal. Which of the following best describes its output in the frequency domain?
Ideal low‑pass filter
Rectangular pulse shaping
Impulse train
Band‑stop filter
Explanation - ZOH holds each sample value constant over the sampling interval, producing a rectangular pulse shape in time, which corresponds to sinc‑shaped spectrum in frequency.
Correct answer is: Rectangular pulse shaping
Q.42 The aliasing effect is most severe when the sampling frequency Fs is:
Much greater than the highest signal frequency
Equal to twice the highest signal frequency (Nyquist rate)
Less than twice the highest signal frequency
Exactly equal to the signal frequency
Explanation - Sampling below the Nyquist rate causes frequency components to overlap and distort.
Correct answer is: Less than twice the highest signal frequency
Q.43 What is the primary advantage of using the FFT over the direct DFT calculation?
Higher computational complexity
Exact frequency resolution
Reduced computational complexity from O(N^2) to O(N log N)
Better memory usage
Explanation - The FFT algorithm exploits symmetry to compute the DFT efficiently, reducing required operations.
Correct answer is: Reduced computational complexity from O(N^2) to O(N log N)
Q.44 A system with impulse response h[n] = δ[n] - δ[n-1] corresponds to what type of filter?
High‑pass
Low‑pass
Band‑stop
Band‑pass
Explanation - Subtracting the delayed sample removes low‑frequency components, acting as a high‑pass filter.
Correct answer is: High‑pass
Q.45 Which of the following is NOT a typical application of discrete‑time signal processing?
Digital audio equalization
Image compression
Analog circuit design
Wireless communication modulation
Explanation - Discrete‑time processing deals with digital signals; analog circuit design is a separate domain.
Correct answer is: Analog circuit design
Q.46 In the context of DSP, what does 'decimation' refer to?
Increasing the sampling rate
Reducing the sampling rate by an integer factor
Shifting the frequency spectrum
Adding noise to a signal
Explanation - Decimation subsamples the signal to lower the effective sampling rate.
Correct answer is: Reducing the sampling rate by an integer factor
Q.47 The Nyquist–Shannon sampling theorem requires the sampling frequency Fs to satisfy which condition for perfect reconstruction?
Fs > 2 × maximum frequency component
Fs > maximum frequency component
Fs = 2 × minimum frequency component
Fs < maximum frequency component
Explanation - To avoid aliasing, Fs must be greater than twice the highest frequency present in the signal.
Correct answer is: Fs > 2 × maximum frequency component
Q.48 Which of the following is a property of the ideal sinc interpolation function?
It has infinite support and zero at all integer multiples other than zero
It has finite support and non‑zero at all points
It is non‑linear
It is a constant function
Explanation - The sinc function is infinite in time but perfectly interpolates at integer sample points.
Correct answer is: It has infinite support and zero at all integer multiples other than zero
Q.49 In filter design, the 'Butterworth' filter is chosen for?
Sharp cutoff with ringing
Flat frequency response in the passband
Linear phase
Maximum attenuation
Explanation - A Butterworth filter has a maximally flat magnitude response within its passband.
Correct answer is: Flat frequency response in the passband
Q.50 The z‑transform of a right‑sided exponential sequence x[n] = a^n u[n] converges if |z| > |a|. Which statement about its ROC is true?
ROC includes the origin
ROC excludes the origin
ROC is a circle with radius |a|
ROC is a line
Explanation - For a causal exponential, the ROC lies outside the outermost pole, i.e., |z| > |a|, forming an annulus.
Correct answer is: ROC is a circle with radius |a|
Q.51 Which of the following correctly expresses the convolution sum for discrete signals x[n] and h[n]?
y[n] = Σ_k x[k] h[n-k]
y[n] = Σ_k x[n-k] h[k]
y[n] = x[n] * h[n]
Both A and B
Explanation - Both expressions are equivalent formulations of discrete convolution.
Correct answer is: Both A and B
Q.52 When designing a digital filter using the bilinear transform, which frequency mapping is applied?
Ω = (2/T) tan(ωT/2)
Ω = ω / T
Ω = ω T
Ω = arctan(ωT/2)
Explanation - The bilinear transform maps continuous frequencies Ω to discrete frequencies ω via the tangent function.
Correct answer is: Ω = (2/T) tan(ωT/2)
Q.53 The z‑transform of a sequence x[n] = n u[n] is?
z/(z-1)^2
1/(z-1)^2
z^2/(z-1)
z/(z-1)
Explanation - Using known transforms, the transform of n u[n] is z/(z-1)^2 for |z| > 1.
Correct answer is: z/(z-1)^2
Q.54 Which of the following is a necessary condition for a system to be both time‑invariant and causal?
Impulse response must be zero for n < 0
Impulse response must be zero for n > 0
Impulse response must be symmetric
Impulse response must be real‑valued
Explanation - Causality requires h[n] = 0 for n < 0; time invariance imposes no further restriction on h[n] shape.
Correct answer is: Impulse response must be zero for n < 0
Q.55 In the context of digital communications, what does the term 'symbol rate' refer to?
Rate of data bits per second
Rate of symbols per second
Rate of samples per second
Rate of packets per second
Explanation - Symbol rate, also called baud, measures how many modulation symbols are transmitted per second.
Correct answer is: Rate of symbols per second
Q.56 Which of the following is the correct relationship between the DTFT X(e^{jω}) and the autocorrelation sequence r_x[k]?
X(e^{jω}) = Σ_k r_x[k] e^{-jωk}
r_x[k] = Σ_ω X(e^{jω}) e^{jωk}
Both A and B
Neither A nor B
Explanation - The DTFT of the autocorrelation gives the power spectral density and vice versa.
Correct answer is: Both A and B
Q.57 Which of the following methods can be used to mitigate aliasing when undersampling a signal?
Increase the sampling rate
Apply a low‑pass anti‑aliasing filter before sampling
Use a high‑pass filter after sampling
Both A and B
Explanation - Increasing Fs or filtering out high frequencies before sampling prevents aliasing.
Correct answer is: Both A and B
Q.58 The bilinear transform maps the continuous‑time s‑plane to which region of the z‑plane?
Inside the unit circle
On the unit circle
Outside the unit circle
All of the above
Explanation - The bilinear transform maps the left half of the s‑plane to the interior of the unit circle in the z‑plane.
Correct answer is: Inside the unit circle
Q.59 In designing a digital filter with a desired frequency response H_d(e^{jω}), which algorithm is commonly used for optimization?
Least‑Squares
Maximum‑Entropy
Monte‑Carlo
Simulated Annealing
Explanation - Least‑squares optimization is widely used to approximate a desired response in filter design.
Correct answer is: Least‑Squares
Q.60 Which of the following expressions correctly represents the impulse response of a second‑order recursive filter defined by y[n] - 0.6y[n-1] + 0.8y[n-2] = x[n]?
h[n] = (0.6)^n u[n] - (0.8)^n u[n]
h[n] = ((0.6)^n - (0.8)^n) u[n] / (0.6 - 0.8)
h[n] = (0.6^n + 0.8^n) u[n]
h[n] = (0.6^n - 0.8^n) u[n]
Explanation - Solving the difference equation via z‑transform yields the closed‑form impulse response.
Correct answer is: h[n] = ((0.6)^n - (0.8)^n) u[n] / (0.6 - 0.8)
Q.61 What is the advantage of using a windowed‑sinc function for FIR filter design?
Reduces Gibbs phenomenon
Provides infinite impulse response
Guarantees linear phase
Ensures perfect reconstruction
Explanation - Windowing smooths the discontinuities in the ideal impulse response, mitigating overshoot.
Correct answer is: Reduces Gibbs phenomenon
Q.62 For a digital communication system using QAM modulation, what does the term 'constellation diagram' represent?
Plot of signal amplitude over time
Plot of signal frequency over time
Plot of possible symbol points in the complex plane
Plot of noise distribution
Explanation - A constellation diagram visualizes the amplitude and phase of modulation symbols.
Correct answer is: Plot of possible symbol points in the complex plane