Q.1 Which transform converts a continuous‑time function into the complex frequency domain?
Fourier Transform
Z-Transform
Laplace Transform
Discrete Fourier Transform
Explanation - The Laplace Transform maps a continuous‑time signal into the complex s‑plane (s = σ + jω), whereas the Z‑Transform is for discrete‑time signals.
Correct answer is: Laplace Transform
Q.2 In the Laplace domain, what does the term 's' represent?
Sampling period
Complex frequency variable
Signal amplitude
Time delay
Explanation - In the Laplace Transform, s = σ + jω is the complex frequency variable.
Correct answer is: Complex frequency variable
Q.3 Which property of the Laplace Transform states that L{f(t−a)u(t−a)} = e^(−as)F(s)?
Time shifting
Frequency shifting
Scaling
Convolution
Explanation - The time‑shift property shows that delaying a function by 'a' results in multiplication by e^(−as).
Correct answer is: Time shifting
Q.4 What is the inverse Z‑Transform of X(z) = 1/(1−z^(-1))?
u[n]
δ[n]
(-1)^n
n
Explanation - The inverse Z‑Transform yields the unit‑step sequence u[n] = 1 for n ≥ 0.
Correct answer is: u[n]
Q.5 If x(t) = e^(−2t)u(t), what is its Laplace Transform X(s)?
1/(s+2)
1/(s−2)
2/(s+2)
1/(s^2+2s)
Explanation - The Laplace Transform of e^(−at)u(t) is 1/(s + a).
Correct answer is: 1/(s+2)
Q.6 Which region of convergence (ROC) is associated with a right‑hand-sided (causal) sequence?
Outside the outermost pole
Inside the innermost pole
Between two poles
Entire z‑plane
Explanation - Causal sequences have ROCs outside the outermost pole of the Z‑Transform.
Correct answer is: Outside the outermost pole
Q.7 What is the Laplace Transform of a unit impulse δ(t)?
0
1
s
e^(-s)
Explanation - The Laplace Transform of δ(t) is 1 for all s in the ROC.
Correct answer is: 1
Q.8 The Z‑Transform of the sequence x[n] = a^n u[n] is:
1/(1−az^(-1))
z/(z−a)
1/(1−a z)
z/(1−a z)
Explanation - Using the definition, X(z) = Σ a^n z^(-n) = 1/(1−a z^(-1)) = z/(z−a).
Correct answer is: z/(z−a)
Q.9 Which transform is suitable for analyzing linear time‑invariant systems in the frequency domain?
Laplace Transform
Z-Transform
Both Laplace and Z-Transform
None of the above
Explanation - Both transforms are used to analyze LTI systems: Laplace for continuous‑time, Z‑Transform for discrete‑time.
Correct answer is: Both Laplace and Z-Transform
Q.10 For the function f(t) = sin(ω0 t)u(t), its Laplace Transform is:
ω0/(s^2 + ω0^2)
s/(s^2 + ω0^2)
1/(s^2 + ω0^2)
ω0^2/(s^2 + ω0^2)
Explanation - The Laplace Transform of sin(ωt)u(t) is ω/(s^2 + ω^2).
Correct answer is: ω0/(s^2 + ω0^2)
Q.11 If X(z) = (1−z^(−1))/(1−2z^(−1)+z^(−2)), what is the system's impulse response h[n]?
δ[n] − 2δ[n−1] + δ[n−2]
δ[n] + 2δ[n−1] + δ[n−2]
δ[n] − δ[n−1]
2δ[n] − δ[n−1]
Explanation - The numerator indicates a difference of unit impulses; the denominator expands to h[n] as shown.
Correct answer is: δ[n] − 2δ[n−1] + δ[n−2]
Q.12 The Laplace Transform of t^n u(t) is:
n! / s^(n+1)
1 / s^n
n! s^n
s^(n+1) / n!
Explanation - L{t^n u(t)} = n! / s^(n+1) for Re{s} > 0.
Correct answer is: n! / s^(n+1)
Q.13 Which of the following is NOT a property of the Z‑Transform?
Linearity
Time shifting
Convolution in time domain
Derivative property
Explanation - The derivative property belongs to the Laplace Transform; the Z‑Transform deals with difference equations, not derivatives.
Correct answer is: Derivative property
Q.14 The Laplace Transform of a zero‑order hold output for a unit step input applied at t = 0 is:
1/s^2
1/s
1/(s−1)
s/(s^2+1)
Explanation - A zero‑order hold of a step yields a ramp, whose Laplace Transform is 1/s^2.
Correct answer is: 1/s^2
Q.15 The Z‑Transform of the sequence x[n] = (−1)^n u[n] is:
1/(1+z^(−1))
1/(1−z^(−1))
1/(1−z)
1/(1+z)
Explanation - Summing (−1)^n z^(−n) gives 1/(1+z^(−1)) for |z| > 1.
Correct answer is: 1/(1+z^(−1))
Q.16 Which equation represents the convolution property in the Laplace domain?
L{f(t)g(t)} = F(s) + G(s)
L{f(t)g(t)} = F(s) G(s)
L{f(t)g(t)} = F(s) / G(s)
L{f(t)g(t)} = F(s) * G(s)
Explanation - Multiplication in the Laplace domain corresponds to convolution in time.
Correct answer is: L{f(t)g(t)} = F(s) G(s)
Q.17 For a causal system, the ROC of its Z‑Transform includes:
Inside the unit circle
On the unit circle only
Outside the outermost pole
All of the above
Explanation - Causal systems have ROCs that extend outward from the outermost pole to infinity.
Correct answer is: Outside the outermost pole
Q.18 The Laplace Transform of a derivative f′(t)u(t) is:
sF(s)
F(s) − f(0)
sF(s) − f(0)
F(s)/s
Explanation - The differentiation property: L{f′(t)} = sF(s) − f(0).
Correct answer is: sF(s) − f(0)
Q.19 What is the Z‑Transform of the sequence x[n] = n u[n]?
z / (z − 1)^2
z^2 / (z − 1)^2
1 / (z − 1)^2
z / (z + 1)^2
Explanation - Using the formula for the first derivative of 1/(1−z^(−1)).
Correct answer is: z / (z − 1)^2
Q.20 Which of the following is an example of a bilateral Laplace Transform?
L{u(t)}
L{δ(t)}
∫_{−∞}^{∞} f(t) e^(−st) dt
Z{δ[n]}
Explanation - Bilateral transforms integrate over the entire real axis, not just t≥0.
Correct answer is: ∫_{−∞}^{∞} f(t) e^(−st) dt
Q.21 The Laplace Transform of f(t) = e^(−a t) sin(ωt)u(t) is:
(ω)/( (s + a)^2 + ω^2 )
(s + a)/( (s + a)^2 + ω^2 )
ω/( (s − a)^2 + ω^2 )
s/( (s + a)^2 + ω^2 )
Explanation - Standard transform: e^(−at) sin(ωt) => ω / ((s + a)^2 + ω^2).
Correct answer is: (ω)/( (s + a)^2 + ω^2 )
Q.22 Which of the following is the Z‑Transform of a unit impulse δ[n]?
0
1
z
1/z
Explanation - The Z-Transform of δ[n] is 1 for all z ≠ 0.
Correct answer is: 1
Q.23 What is the inverse Laplace Transform of X(s) = 1/(s^2 + 1)?
cos(t)
sin(t)
t cos(t)
t sin(t)
Explanation - L{sin(t)} = 1/(s^2 + 1).
Correct answer is: sin(t)
Q.24 The Z‑Transform of the sequence x[n] = (−1)^n is:
1/(1 + z^(−1))
1/(1 − z^(−1))
1/(1 + z)
1/(1 − z)
Explanation - Summing (−1)^n z^(−n) gives 1/(1 + z^(−1)).
Correct answer is: 1/(1 + z^(−1))
Q.25 Which transform is commonly used to solve linear differential equations in control systems?
Fourier Transform
Z-Transform
Laplace Transform
Hilbert Transform
Explanation - Laplace Transform converts differential equations into algebraic equations.
Correct answer is: Laplace Transform
Q.26 If X(z) has a pole at z = 0.5, the system is:
Stable
Unstable
Marginally stable
None of the above
Explanation - For a discrete system, poles inside the unit circle (|z|<1) imply stability.
Correct answer is: Stable
Q.27 The Laplace Transform of a unit step u(t) is:
1/s
s
0
δ(t)
Explanation - L{u(t)} = 1/s for Re{s} > 0.
Correct answer is: 1/s
Q.28 What is the Z‑Transform of x[n] = n^2 u[n]?
z(1+z)/(z−1)^3
z(1−z)/(z−1)^3
z/(z−1)^3
z^2/(z−1)^3
Explanation - Using higher‑order sum formulas: Σ n^2 z^(−n) = z(1+z)/(z−1)^3.
Correct answer is: z(1+z)/(z−1)^3
Q.29 In the Laplace domain, convolution of f(t) and g(t) corresponds to:
Addition
Multiplication
Division
Differentiation
Explanation - Convolution in time ↔ multiplication in Laplace domain.
Correct answer is: Multiplication
Q.30 The bilateral Laplace Transform of f(t) = e^(at) is:
1/(s−a)
s/(s−a)
1/(s+a)
s/(s+a)
Explanation - Bilateral transform integrates over all t, giving 1/(s−a) for Re{s} > a.
Correct answer is: 1/(s−a)
Q.31 Which of the following statements about the Z‑Transform is TRUE?
It can only be applied to finite sequences.
The ROC must always include the unit circle.
It converts time‑domain sequences to frequency‑domain representation.
It is identical to the Fourier Transform.
Explanation - The Z‑Transform maps discrete‑time sequences to the complex z‑plane, providing a frequency domain view.
Correct answer is: It converts time‑domain sequences to frequency‑domain representation.
Q.32 The Laplace Transform of f(t) = cos(ωt)u(t) is:
s/(s^2 + ω^2)
ω/(s^2 + ω^2)
1/(s^2 + ω^2)
s/(s^2 - ω^2)
Explanation - Standard transform: L{cos(ωt)} = s/(s^2 + ω^2).
Correct answer is: s/(s^2 + ω^2)
Q.33 The Z‑Transform of the sequence x[n] = (−1)^n u[n] is:
1/(1+z^(−1))
1/(1−z^(−1))
1/(1+z)
1/(1−z)
Explanation - Summing (−1)^n z^(−n) yields 1/(1+z^(−1)).
Correct answer is: 1/(1+z^(−1))
Q.34 What does the term 'region of convergence' (ROC) refer to in the Z‑Transform?
Area where the sequence is defined
Set of z values for which the transform converges
The radius of the unit circle
The time range of the input
Explanation - ROC is the set of z where the Z‑Transform integral/sum exists.
Correct answer is: Set of z values for which the transform converges
Q.35 Which of these is NOT a valid Laplace Transform?
1/(s+2)
s/(s^2 + 1)
e^(−s)
1/(s^2 - 4)
Explanation - e^(−s) is not a standard Laplace Transform of a time‑domain function.
Correct answer is: e^(−s)
Q.36 The Z‑Transform of the sequence x[n] = 1 for 0 ≤ n ≤ N and 0 otherwise is:
(1−z^(−(N+1)))/(1−z^(−1))
(1−z^N)/(1−z)
(1−z^(N+1))/(1−z)
(1−z^N)/(1−z^(−1))
Explanation - This is a finite geometric series summed over N+1 terms.
Correct answer is: (1−z^(−(N+1)))/(1−z^(−1))
Q.37 What is the Laplace Transform of f(t) = t e^(−t)u(t)?
1/(s+1)^2
1/(s+1)
(s+1)/s^2
s/(s+1)^2
Explanation - Using L{t e^(−at)} = 1/(s+a)^2.
Correct answer is: 1/(s+1)^2
Q.38 If X(z) = 1/(1−0.8z^(−1)), what is the impulse response h[n]?
0.8^n u[n]
0.8^(n+1) u[n]
0.8^(−n) u[n]
u[n] − 0.8^n u[n]
Explanation - Inverse Z‑Transform of a simple pole yields an exponential sequence.
Correct answer is: 0.8^n u[n]
Q.39 The Laplace Transform of f(t) = e^(at)u(t) is:
1/(s−a)
1/(s+a)
s/(s−a)
s/(s+a)
Explanation - Standard transform: e^(at) => 1/(s−a).
Correct answer is: 1/(s−a)
Q.40 In a Z‑Transform, the term z^(−1) is often associated with:
Time advancement
Time delay
Amplitude scaling
Frequency shift
Explanation - Multiplication by z^(−1) corresponds to a unit‑sample delay in time domain.
Correct answer is: Time delay
Q.41 The Laplace Transform of f(t) = u(t) − u(t−T) is:
1/s − e^(−Ts)/s
e^(−Ts)/s
1/(s^2)
1/(s+T)
Explanation - It represents a rectangular pulse of duration T.
Correct answer is: 1/s − e^(−Ts)/s
Q.42 Which of the following is a property of the Laplace Transform regarding scaling in time?
L{f(at)} = aF(s)
L{f(at)} = (1/a)F(s/a)
L{f(at)} = F(as)
L{f(at)} = aF(as)
Explanation - Time scaling: f(at) → (1/a)F(s/a).
Correct answer is: L{f(at)} = (1/a)F(s/a)
Q.43 What is the Z‑Transform of x[n] = 2^n u[n]?
z/(z−2)
2z/(z−2)
z/(z−1/2)
2z/(z−1/2)
Explanation - Sum of 2^n z^(−n) gives z/(z−2).
Correct answer is: z/(z−2)
Q.44 Which transform is used to analyze frequency content of discrete signals?
Laplace Transform
Z-Transform
Fourier Transform
Hilbert Transform
Explanation - Z‑Transform provides frequency domain for discrete‑time signals.
Correct answer is: Z-Transform
Q.45 The Laplace Transform of the derivative f′′(t) is:
s^2 F(s)
s^2 F(s) − s f(0) − f′(0)
s^2 F(s) − f(0)
s^2 F(s) + f(0)
Explanation - Second‑order differentiation property.
Correct answer is: s^2 F(s) − s f(0) − f′(0)
Q.46 Which of the following is the Z‑Transform of the sequence x[n] = δ[n] − δ[n−1]?
1 − z^(−1)
1 + z^(−1)
z^(−1) − 1
z^(−1) + 1
Explanation - Direct substitution into definition: X(z) = 1 − z^(−1).
Correct answer is: 1 − z^(−1)
Q.47 The Laplace Transform of a unit step multiplied by e^(−at) is:
1/(s+a)
1/(s−a)
s/(s+a)
s/(s−a)
Explanation - L{e^(−at)u(t)} = 1/(s + a).
Correct answer is: 1/(s+a)
Q.48 Which property states that multiplication by n in time domain corresponds to differentiation in Z‑domain?
Time‑shift
Frequency‑shift
Time‑scaling
Differentiation
Explanation - In Z‑transform, n·x[n] ↔ −z dX(z)/dz.
Correct answer is: Differentiation
Q.49 The Z‑Transform of the sequence x[n] = (−1)^n n u[n] is:
z(1+z)/(z−1)^3
−z(1+z)/(z+1)^3
z(1−z)/(z−1)^3
−z(1−z)/(z+1)^3
Explanation - Combining (−1)^n with n yields a negative sign and pole at −1.
Correct answer is: −z(1+z)/(z+1)^3
Q.50 For a continuous‑time system with transfer function G(s) = 1/(s+1), the step response is:
1 − e^(−t)
e^(−t)
t e^(−t)
1 + e^(−t)
Explanation - Inverse Laplace of G(s)/s gives 1 − e^(−t).
Correct answer is: 1 − e^(−t)
Q.51 What is the Laplace Transform of f(t) = t^2 u(t)?
2/s^3
1/s^3
2/s^2
1/s^2
Explanation - L{t^n} = n!/s^(n+1) → 2!/s^3 = 2/s^3.
Correct answer is: 2/s^3
Q.52 The Z‑Transform of a sequence that starts at n = 0 and is zero elsewhere is called a:
Causal sequence
Anti‑causal sequence
Two‑sided sequence
Periodic sequence
Explanation - Causal sequences are defined for n ≥ 0.
Correct answer is: Causal sequence
Q.53 The Laplace Transform of a unit ramp r(t) = t u(t) is:
1/s^2
1/s
s
s^2
Explanation - A ramp has transform 1/s^2.
Correct answer is: 1/s^2
Q.54 Which transform would you use to find the frequency response of a digital filter defined by its difference equation?
Laplace Transform
Z-Transform
Fourier Transform
Hilbert Transform
Explanation - Digital filters are analyzed using the Z‑Transform.
Correct answer is: Z-Transform
Q.55 If X(s) = (s+1)/(s^2 + 2s + 2), what is the inverse Laplace transform?
e^(−t) cos(t)
e^(−t) sin(t)
e^(−t) + t e^(−t)
t e^(−t)
Explanation - Partial fraction yields cos(t) component.
Correct answer is: e^(−t) cos(t)
Q.56 The Laplace Transform of f(t) = u(t−2) is:
e^(−2s)/s
e^(2s)/s
e^(−2s)
1/(s−2)
Explanation - Delay property: L{u(t−T)} = e^(−Ts)/s.
Correct answer is: e^(−2s)/s
Q.57 The Z‑Transform of the sequence x[n] = δ[n] + δ[n−1] is:
1 + z^(−1)
1 − z^(−1)
z + 1
z − 1
Explanation - Direct substitution into definition.
Correct answer is: 1 + z^(−1)
Q.58 Which of the following represents the convolution property of the Laplace Transform?
L{f(t)g(t)} = F(s) + G(s)
L{f(t)g(t)} = F(s) G(s)
L{f(t)g(t)} = F(s) / G(s)
L{f(t)g(t)} = F(s) * G(s)
Explanation - Convolution in time domain ↔ multiplication in Laplace domain.
Correct answer is: L{f(t)g(t)} = F(s) G(s)
Q.59 The Laplace Transform of f(t) = e^(at) sin(ωt)u(t) is:
ω / ((s−a)^2 + ω^2)
ω / ((s+a)^2 + ω^2)
ω / (s^2 + ω^2)
ω / ((s+a)^2 − ω^2)
Explanation - Standard form with shift by +a.
Correct answer is: ω / ((s+a)^2 + ω^2)
Q.60 Which statement correctly describes the ROC for a bilateral Z‑Transform of a finite‑length sequence?
ROC is the entire z‑plane except z = 0
ROC is the entire z‑plane except the poles
ROC includes the unit circle
ROC is only the unit circle
Explanation - Finite sequences are absolutely summable; ROC includes |z| > 0, hence the unit circle.
Correct answer is: ROC includes the unit circle
Q.61 The Laplace Transform of f(t) = e^(−t) cos(2t)u(t) is:
(s+1)/( (s+1)^2 + 4)
(s+1)/( (s−1)^2 + 4)
(s+1)/( (s+1)^2 − 4)
(s−1)/( (s+1)^2 + 4)
Explanation - Apply standard formula with a = 1, ω = 2.
Correct answer is: (s+1)/( (s+1)^2 + 4)
Q.62 The Z‑Transform of a right‑handed exponential sequence x[n] = a^n u[n] is:
1/(1−a z^(−1))
1/(1−a z)
z/(z−a)
z/(z−1/a)
Explanation - Standard result from summing geometric series.
Correct answer is: z/(z−a)
Q.63 Which transform can be used to analyze a signal that is defined for negative and positive time?
Unilateral Laplace Transform
Bilateral Laplace Transform
Z-Transform
Fourier Transform
Explanation - Bilateral transform integrates from −∞ to +∞, suitable for two‑sided signals.
Correct answer is: Bilateral Laplace Transform
Q.64 The Laplace Transform of a delayed impulse δ(t−T) is:
e^(−Ts)
e^(Ts)
s e^(−Ts)
s e^(Ts)
Explanation - Delay property: L{δ(t−T)} = e^(−Ts).
Correct answer is: e^(−Ts)
Q.65 The Z‑Transform of the sequence x[n] = n u[n] is:
z/(z−1)^2
1/(z−1)^2
z^2/(z−1)^2
z/(z+1)^2
Explanation - Derivative of 1/(1−z^(−1)).
Correct answer is: z/(z−1)^2
Q.66 The Laplace Transform of f(t) = cosh(at)u(t) is:
(s)/(s^2 − a^2)
s/(s^2 + a^2)
(s)/(s^2 + a^2)
s/(s^2 − a^2)
Explanation - cosh(at) transforms to s/(s^2 − a^2).
Correct answer is: s/(s^2 − a^2)
Q.67 What is the Z‑Transform of a unit sample at n = 3, i.e., x[n] = δ[n−3]?
z^(−3)
z^3
z^(−1)
z^1
Explanation - General rule: δ[n−k] ↔ z^(−k).
Correct answer is: z^(−3)
Q.68 Which property of the Laplace Transform corresponds to scaling the time axis by a factor 'a' (a > 0)?
Time shifting
Frequency shifting
Time scaling
Amplitude scaling
Explanation - f(at) ↔ (1/a)F(s/a).
Correct answer is: Time scaling
Q.69 The Laplace Transform of f(t) = e^(−at) cos(ωt)u(t) is:
(s + a)/((s + a)^2 + ω^2)
(s − a)/((s − a)^2 + ω^2)
(s + a)/((s − a)^2 + ω^2)
(s − a)/((s + a)^2 + ω^2)
Explanation - Standard result with shift a.
Correct answer is: (s + a)/((s + a)^2 + ω^2)
Q.70 Which of the following is the Laplace Transform of a unit step function delayed by 5 seconds?
e^(−5s)/s
e^(5s)/s
1/(s+5)
s/(s+5)
Explanation - Delay property applied to u(t).
Correct answer is: e^(−5s)/s
Q.71 The Z‑Transform of the sequence x[n] = (−1)^n u[n] is:
1/(1 + z^(−1))
1/(1 − z^(−1))
1/(1 + z)
1/(1 − z)
Explanation - Summing yields 1/(1 + z^(−1)).
Correct answer is: 1/(1 + z^(−1))
Q.72 The Laplace Transform of f(t) = t e^(−2t)u(t) is:
1/(s+2)^2
2/(s+2)^2
1/(s+2)
2/(s+2)
Explanation - Formula: L{t e^(−at)} = 1/(s + a)^2.
Correct answer is: 1/(s+2)^2
Q.73 Which transform is typically used to solve linear difference equations in digital signal processing?
Laplace Transform
Z-Transform
Fourier Transform
Hilbert Transform
Explanation - Z‑Transform converts difference equations to algebraic equations.
Correct answer is: Z-Transform
Q.74 The Z‑Transform of the sequence x[n] = 2^n u[n] is:
z/(z−2)
2z/(z−2)
z/(z−1/2)
2z/(z−1/2)
Explanation - Sum of 2^n z^(−n) = z/(z−2).
Correct answer is: z/(z−2)
Q.75 Which of the following best describes the Laplace Transform?
It transforms signals from time to frequency domain using sinusoidal basis functions.
It transforms signals from time to complex frequency domain using exponential basis functions.
It transforms discrete signals to the z‑plane.
It is only used for steady‑state analysis.
Explanation - Laplace uses e^(−st) as kernel.
Correct answer is: It transforms signals from time to complex frequency domain using exponential basis functions.
Q.76 What is the Laplace Transform of f(t) = (1/2) e^(−3t)u(t) + (1/2) e^(−5t)u(t)?
1/(s+3) + 1/(s+5)
1/(2(s+3)) + 1/(2(s+5))
1/(2(s+3)) + 1/(s+5)
1/(s+3) + 1/(2(s+5))
Explanation - Linear combination with 0.5 coefficients.
Correct answer is: 1/(2(s+3)) + 1/(2(s+5))
Q.77 The ROC for a right‑handed sequence with a pole at z = 0.5 is:
0 < |z| < 0.5
|z| > 0.5
|z| < 0.5
All z except 0.5
Explanation - Causal sequence → ROC outside the outermost pole.
Correct answer is: |z| > 0.5
Q.78 Which of the following is a property of the Z‑Transform regarding time reversal?
X(z) = X(−z)
X(z) = X(1/z)
X(z) = z X(z)
X(z) = (1/z) X(z)
Explanation - Time reversal: x[−n] ↔ X(1/z).
Correct answer is: X(z) = X(1/z)
Q.79 The Laplace Transform of f(t) = u(t) − e^(−t)u(t) is:
1/s − 1/(s+1)
1/s + 1/(s+1)
1/(s+1) − 1/s
1/(s+1) + 1/s
Explanation - Difference of two transforms.
Correct answer is: 1/s − 1/(s+1)
Q.80 The Z‑Transform of the sequence x[n] = sin(πn/4) u[n] is:
z/(z^2 + 2z + 1)
z/(z^2 − 2z + 1)
z/(z^2 + 1)
z/(z^2 − 1)
Explanation - Use identity sin(θ) → (z^2 - 1)/(z^2 + 1).
Correct answer is: z/(z^2 + 2z + 1)
Q.81 Which transform is used to analyze the frequency response of a continuous‑time system?
Laplace Transform
Z-Transform
Fourier Transform
Hilbert Transform
Explanation - Continuous‑time frequency response is obtained from G(s) evaluated on the jω axis.
Correct answer is: Laplace Transform
Q.82 The Laplace Transform of f(t) = t sin(t)u(t) is:
(2s)/(s^2 + 1)^2
(2s)/(s^2 - 1)^2
s/(s^2 + 1)^2
s/(s^2 - 1)^2
Explanation - Use differentiation with respect to s.
Correct answer is: (2s)/(s^2 + 1)^2
Q.83 The Z‑Transform of a sequence defined as x[n] = 1 for 0 ≤ n ≤ 4 and 0 otherwise is:
(1−z^(−5))/(1−z^(−1))
(1−z^5)/(1−z)
(1−z^(−5))/(1−z)
(1−z^5)/(1−z^(−1))
Explanation - Finite geometric series sum over 5 terms.
Correct answer is: (1−z^(−5))/(1−z^(−1))
Q.84 Which of the following is the Laplace Transform of a unit step multiplied by e^(−2t)u(t)?
1/(s+2)
1/(s−2)
s/(s+2)
s/(s−2)
Explanation - Direct application of the first‑order exponential transform.
Correct answer is: 1/(s+2)
Q.85 The Z‑Transform of the sequence x[n] = n^2 u[n] is:
z(1+z)/(z−1)^3
z(1−z)/(z−1)^3
z^2/(z−1)^3
z/(z−1)^3
Explanation - Higher‑order sum formula.
Correct answer is: z(1+z)/(z−1)^3
Q.86 In the Laplace domain, the term 'e^(−Ts)' is associated with:
Time delay of T seconds
Time scaling by T
Frequency shift by T
Amplitude scaling by T
Explanation - Delay property: multiplication by e^(−Ts).
Correct answer is: Time delay of T seconds
Q.87 What is the inverse Laplace transform of X(s) = 1/(s^2 + 9)?
sin(3t)
cos(3t)
t sin(3t)
t cos(3t)
Explanation - Standard transform: L{sin(ωt)} = ω/(s^2 + ω^2).
Correct answer is: sin(3t)
Q.88 Which property of the Z‑Transform states that multiplication by z^(−1) corresponds to a one‑sample delay?
Time shifting
Frequency shifting
Scaling
Differentiation
Explanation - Multiplication by z^(−1) delays the sequence by one sample.
Correct answer is: Time shifting
Q.89 The Laplace Transform of the derivative of a function f(t) is:
sF(s)
F(s)/s
sF(s) − f(0)
F(s) + f(0)
Explanation - Standard differentiation property.
Correct answer is: sF(s) − f(0)
Q.90 The Z‑Transform of a unit step u[n] is:
1/(1−z^(−1))
1/(1+z^(−1))
1/(1−z)
1/(1+z)
Explanation - Sum of a geometric series.
Correct answer is: 1/(1−z^(−1))
Q.91 Which of the following represents the Laplace Transform of a system with transfer function G(s) = (s+2)/(s^2 + 4s + 5)?
G(s) = (s+2)/(s^2 + 4s + 5)
G(s) = (s+2)/((s+2)^2 + 1)
G(s) = (s+2)/((s+2)^2 + 1)
G(s) = (s+2)/((s+2)^2 + 1)
Explanation - Complete the square on the denominator.
Correct answer is: G(s) = (s+2)/((s+2)^2 + 1)
Q.92 The Laplace Transform of f(t) = t^3 u(t) is:
6/s^4
3/s^4
6/s^3
3/s^3
Explanation - n! = 3! = 6, n = 3, so 6/s^(3+1).
Correct answer is: 6/s^4
Q.93 Which transform is used to compute the frequency spectrum of a discrete signal?
Laplace Transform
Z-Transform
Fourier Transform
Hilbert Transform
Explanation - Z‑Transform provides frequency content for discrete signals.
Correct answer is: Z-Transform
Q.94 The Z‑Transform of the sequence x[n] = δ[n] + 2δ[n−1] is:
1 + 2z^(−1)
1 − 2z^(−1)
z + 2z^(−1)
z − 2z^(−1)
Explanation - Direct substitution into definition.
Correct answer is: 1 + 2z^(−1)
Q.95 Which of the following is NOT a valid Z‑Transform?
z/(z−1)
1/(z^2 + 1)
e^(−z)
1/(1−z^(−1))
Explanation - e^(−z) is not a standard Z‑Transform of a sequence.
Correct answer is: e^(−z)
Q.96 The Laplace Transform of f(t) = cos(2t)u(t) is:
s/(s^2 + 4)
s/(s^2 − 4)
4/(s^2 + 4)
4/(s^2 − 4)
Explanation - Standard result for cosine.
Correct answer is: s/(s^2 + 4)
Q.97 What is the inverse Laplace transform of X(s) = s/(s^2 + 1)?
cos(t)
sin(t)
t cos(t)
t sin(t)
Explanation - L{cos(t)} = s/(s^2 + 1).
Correct answer is: cos(t)
Q.98 The Z‑Transform of a two‑sided exponential sequence x[n] = (0.5)^|n| is:
(1−0.5^2)/(1−0.5(z + z^(−1)))
(1−0.5^2)/(1−0.5(z + z^(−1)))
(1−0.5^2)/(1−0.5(z + z^(−1)))
(1−0.5^2)/(1−0.5(z + z^(−1)))
Explanation - Sum of two geometric series for n≥0 and n<0.
Correct answer is: (1−0.5^2)/(1−0.5(z + z^(−1)))
Q.99 The Laplace Transform of f(t) = e^(−3t) sin(2t)u(t) is:
2/((s+3)^2 + 4)
2/((s−3)^2 + 4)
2/((s+3)^2 − 4)
2/((s−3)^2 − 4)
Explanation - Shift by +3 and standard sine transform.
Correct answer is: 2/((s+3)^2 + 4)
Q.100 The Z‑Transform of x[n] = u[n] − u[n−3] is:
1 − z^(−3)
1 + z^(−3)
z^(−3) − 1
z^(−3) + 1
Explanation - Difference of two step sequences.
Correct answer is: 1 − z^(−3)
Q.101 Which of the following best describes the ROC for a two‑sided sequence with poles at z = 2 and z = 0.5?
0.5 < |z| < 2
|z| > 2
|z| < 0.5
All |z| except 0.5 and 2
Explanation - ROC lies between the poles for a two‑sided sequence.
Correct answer is: 0.5 < |z| < 2
Q.102 The Laplace Transform of f(t) = e^(−t)u(t) is:
1/(s+1)
1/(s−1)
s/(s+1)
s/(s−1)
Explanation - Standard exponential transform.
Correct answer is: 1/(s+1)
Q.103 What is the Z‑Transform of the sequence x[n] = n! u[n]?
1/(1−z^(−1))^2
1/(1−z^(−1))^3
1/(1−z^(−1))^n
1/(1−z^(−1))^2
Explanation - n! sequence corresponds to second‑order pole at z = 1.
Correct answer is: 1/(1−z^(−1))^2
Q.104 The Laplace Transform of a unit step delayed by 2 seconds is:
e^(−2s)/s
e^(2s)/s
1/(s+2)
s/(s+2)
Explanation - Delay property.
Correct answer is: e^(−2s)/s
Q.105 Which of the following is the Z‑Transform of the sequence x[n] = (−1)^n u[n]?
1/(1 + z^(−1))
1/(1 − z^(−1))
1/(1 + z)
1/(1 − z)
Explanation - Summing the alternating sequence yields this expression.
Correct answer is: 1/(1 + z^(−1))
Q.106 The Laplace Transform of f(t) = e^(−2t) cos(3t)u(t) is:
(s+2)/((s+2)^2 + 9)
(s+2)/((s−2)^2 + 9)
(s+2)/((s+2)^2 − 9)
(s−2)/((s+2)^2 + 9)
Explanation - Standard formula with shift by +2.
Correct answer is: (s+2)/((s+2)^2 + 9)
Q.107 The Z‑Transform of x[n] = (3^n)u[n] is:
z/(z−3)
3z/(z−3)
z/(z−1/3)
3z/(z−1/3)
Explanation - Geometric series sum with a = 3.
Correct answer is: z/(z−3)
Q.108 Which property of the Laplace Transform indicates that scaling the amplitude of a signal by a constant 'k' scales its transform by the same constant?
Linearity
Time shifting
Frequency shifting
Scaling in time
Explanation - Multiplying the input by k multiplies the transform by k.
Correct answer is: Linearity
Q.109 The Laplace Transform of f(t) = e^(−5t) sin(4t)u(t) is:
4/((s+5)^2 + 16)
4/((s−5)^2 + 16)
4/((s+5)^2 − 16)
4/((s−5)^2 − 16)
Explanation - Standard sine transform with exponential shift.
Correct answer is: 4/((s+5)^2 + 16)
Q.110 The Z‑Transform of the sequence x[n] = (−1)^n n u[n] is:
−z/(z+1)^2
z/(z−1)^2
−z/(z−1)^2
z/(z+1)^2
Explanation - Combining (−1)^n with n introduces a pole at −1.
Correct answer is: −z/(z+1)^2
Q.111 Which transform is used to analyze the transfer function of a digital filter defined by H(z) = (z^2 − 0.5z + 0.25)/(z^2 + 0.5z + 0.25)?
Laplace Transform
Z-Transform
Fourier Transform
Hilbert Transform
Explanation - Digital filter analysis uses the Z‑Transform.
Correct answer is: Z-Transform
Q.112 The Laplace Transform of f(t) = u(t) − e^(−t)u(t) is:
1/s − 1/(s+1)
1/s + 1/(s+1)
1/(s+1) − 1/s
1/(s+1) + 1/s
Explanation - Difference of transforms of step and delayed exponential.
Correct answer is: 1/s − 1/(s+1)
Q.113 What is the inverse Laplace transform of X(s) = s/(s^2 + 9)?
cos(3t)
sin(3t)
t cos(3t)
t sin(3t)
Explanation - Standard cosine transform.
Correct answer is: cos(3t)
Q.114 Which of the following is NOT a valid Laplace Transform?
1/(s+2)
s/(s^2 + 1)
e^(−s)
1/(s^2 + 1)
Explanation - e^(−s) is not a transform of a time‑domain function.
Correct answer is: e^(−s)