1Which of the following defines the Bilateral Laplace Transform of a continuous-time signal ?
A.
B.
C.
D.
Correct Answer:
Explanation:The bilateral (or two-sided) Laplace transform integrates from to . The integral from $0$ to defines the Unilateral Laplace transform.
Incorrect! Try again.
2In the Laplace transform variable , what does represent?
A.The angular frequency
B.The phase angle
C.The real part (attenuation/growth factor)
D.The sampling period
Correct Answer: The real part (attenuation/growth factor)
Explanation: is a complex frequency variable where is the real part (Neper frequency) and is the imaginary part (angular frequency).
Incorrect! Try again.
3What is the relationship between the Laplace Transform and the Fourier Transform ?
A.The Fourier transform is the Laplace transform evaluated at .
B.The Fourier transform is the Laplace transform evaluated along the imaginary axis, i.e., , provided the ROC includes the axis.
C.The Fourier transform is the derivative of the Laplace transform.
D.There is no mathematical relationship between them.
Correct Answer: The Fourier transform is the Laplace transform evaluated along the imaginary axis, i.e., , provided the ROC includes the axis.
Explanation:The Fourier transform is a special case of the Laplace transform where the real part , assuming the Region of Convergence (ROC) includes the imaginary axis.
Incorrect! Try again.
4What defines the Region of Convergence (ROC) for a Laplace Transform?
A.The set of all for which is finite.
B.The set of values of for which the Fourier Transform exists.
C.The set of values of for which the integral converges.
D.The values of where the denominator of is zero.
Correct Answer: The set of values of for which the integral converges.
Explanation:The ROC consists of the values of the complex variable for which the Laplace transform integral converges absolutely.
Incorrect! Try again.
5Which of the following is a property of the ROC for rational Laplace transforms?
A.The ROC always includes the origin.
B.The ROC may contain poles.
C.The ROC consists of strips parallel to the axis and does not contain any poles.
D.The ROC is always a circular region.
Correct Answer: The ROC consists of strips parallel to the axis and does not contain any poles.
Explanation:For rational Laplace transforms, the ROC is bounded by poles (but contains none) and forms vertical strips in the s-plane parallel to the imaginary axis.
Incorrect! Try again.
6Calculate the Laplace transform of the unit impulse function, .
A.
B.
C.
D.
Correct Answer:
Explanation:Using the sifting property of the impulse function in the integral: .
Incorrect! Try again.
7What is the Laplace transform of the unit step function ?
A.
B.
C.
D.
Correct Answer:
Explanation: for .
Incorrect! Try again.
8What is the ROC for the signal ?
A.
B.
C.
D.
Correct Answer:
Explanation:For a right-sided signal , the Laplace transform converges when the real part of is greater than the pole at .
Incorrect! Try again.
9What is the Laplace transform of ?
A.
B.
C.
D.
Correct Answer:
Explanation:Algebraically, results in the same rational expression as the causal counterpart, but the ROC is to the left of the pole () because the signal is left-sided.
Incorrect! Try again.
10For a system to be stable, the ROC of its transfer function must:
A.Include the origin.
B.Include the axis.
C.Be entirely in the right-half plane.
D.Not include infinity.
Correct Answer: Include the axis.
Explanation:BIBO stability requires that the integral of the impulse response be absolutely convergent, which corresponds to the ROC including the axis ().
Incorrect! Try again.
11For a causal LTI system with a rational transfer function , the ROC is:
A.To the left of the leftmost pole.
B.To the right of the rightmost pole.
C.A strip between two poles.
D.The entire s-plane.
Correct Answer: To the right of the rightmost pole.
Explanation:A causal system has a right-sided impulse response . Consequently, its ROC must be a right-half plane region defined by , where is the real part of the rightmost pole.
Incorrect! Try again.
12What is the Laplace transform of ?
A.
B.
C.
D.
Correct Answer:
Explanation:Using Euler's identity , the transform is .
Incorrect! Try again.
13If , what is the Laplace transform of ?
A.
B.
C.
D.
Correct Answer:
Explanation:The time-shifting property states that delaying a signal by corresponds to multiplying its Laplace transform by .
Incorrect! Try again.
14If , what is the Laplace transform of ?
A.
B.
C.
D.
Correct Answer:
Explanation:This is the s-domain shifting (or frequency shifting) property. Multiplying by an exponential in time results in a shift in the s-domain.
Incorrect! Try again.
15What is the Laplace transform of the derivative , assuming is continuous and causal ()?
A.
B.
C.
D.
Correct Answer:
Explanation:Differentiation in the time domain corresponds to multiplication by in the s-domain (assuming zero initial conditions).
Incorrect! Try again.
16Which property states that ?
A.Frequency shifting property
B.Differentiation property
C.Convolution property
D.Scaling property
Correct Answer: Convolution property
Explanation:Convolution in the time domain corresponds to multiplication in the Laplace (s) domain.
Incorrect! Try again.
17If , then is:
A.
B.
C.
D.
Correct Answer:
Explanation:Differentiation in the s-domain property: multiplying by in the time domain corresponds to the negative derivative with respect to in the frequency domain.
Incorrect! Try again.
18The Initial Value Theorem states that is valid only if:
A.The system is unstable.
B. is strictly proper.
C. is a strictly proper rational function (degree of numerator < degree of denominator).
D.The ROC includes the origin.
Correct Answer: is a strictly proper rational function (degree of numerator < degree of denominator).
Explanation:For the Initial Value Theorem to hold, the degree of the numerator of must be strictly less than the degree of the denominator.
Incorrect! Try again.
19The Final Value Theorem applies only if:
A.All poles of are in the right-half plane.
B.All poles of are in the left-half plane.
C.The system is causal.
D.There are no zeros in the system.
Correct Answer: All poles of are in the left-half plane.
Explanation:The Final Value Theorem is only valid if the signal settles to a constant or zero, which requires the poles of to have negative real parts (located in the Left Half Plane).
Incorrect! Try again.
20What corresponds to a pole of a system transfer function ?
A.A value of where .
B.A value of where .
C.A value of where the ROC ends.
D.The value of the impulse response at .
Correct Answer: A value of where .
Explanation:Roots of the denominator polynomial of are called poles. At these values, the transfer function becomes infinite.
Incorrect! Try again.
21In the geometric evaluation of the Fourier transform from the pole-zero plot, the magnitude is proportional to:
A.The sum of vectors from poles to the point .
B.The product of lengths of vectors from zeros to divided by the product of lengths of vectors from poles to .
C.The product of lengths of vectors from poles to divided by the product of lengths of vectors from zeros to .
D.The distance from the origin to .
Correct Answer: The product of lengths of vectors from zeros to divided by the product of lengths of vectors from poles to .
Explanation:Geometric evaluation treats terms and as vectors. The magnitude is evaluated at .
Incorrect! Try again.
22If a pole is located exactly on the axis at , what happens to the Fourier Transform magnitude at frequency ?
A.It becomes 0.
B.It becomes 1.
C.It approaches infinity.
D.It remains constant.
Correct Answer: It approaches infinity.
Explanation:Since the distance from the pole to the evaluation point is zero, and pole vector lengths are in the denominator of the magnitude calculation, the magnitude approaches infinity.
Incorrect! Try again.
23Which of the following describes the method of Partial Fraction Expansion?
A.Approximating an integral using rectangles.
B.Decomposing a complex rational function into a sum of simpler terms to perform the Inverse Laplace Transform.
C.Expanding a signal into a Taylor series.
D.Multiplying polynomials to find the total response.
Correct Answer: Decomposing a complex rational function into a sum of simpler terms to perform the Inverse Laplace Transform.
Explanation:Partial Fraction Expansion breaks into simple terms (like ) whose inverse transforms are known standard pairs (like ).
Incorrect! Try again.
24Find the Inverse Laplace transform of , assuming .
A.
B.
C.
D.
Correct Answer:
Explanation:Using the linearity property and the standard pair for causal ROC.
Incorrect! Try again.
25What is the ROC for a signal that is of finite duration (time-limited)?
A.The right half plane.
B.The left half plane.
C.The entire s-plane, except possibly or .
D.It has no ROC.
Correct Answer: The entire s-plane, except possibly or .
Explanation:Finite duration signals have Laplace transforms that converge everywhere in the complex plane, except potentially at the origin or infinity depending on causality and anticausality components.
Incorrect! Try again.
26The transfer function of a system is given by . If (Impulse input), then is:
A.The step response.
B.The impulse response.
C.The frequency response.
D.Zero.
Correct Answer: The impulse response.
Explanation:If the input is an impulse, , so . The inverse transform is the impulse response.
Incorrect! Try again.
27Determine the transfer function for the differential equation: .
A.
B.
C.
D.
Correct Answer:
Explanation:Taking the Laplace transform: .
Incorrect! Try again.
28A system has a pole at . If the system is causal, it is:
A.Stable.
B.Unstable.
C.Marginally stable.
D.Oscillatory.
Correct Answer: Unstable.
Explanation:For a causal system, the ROC is to the right of the rightmost pole (). This ROC does not include the axis (), so the system is unstable.
Incorrect! Try again.
29Which software function is commonly used to compute the roots of the denominator polynomial (poles) given the coefficients?
A.
B.
C.
D.
Correct Answer:
Explanation:In software like MATLAB or Python (SciPy), roots() calculates the roots of a polynomial provided as a vector of coefficients.
Incorrect! Try again.
30How is a transfer function represented numerically in simulation software?
A.As a string "2s+1 / s^2+3s+2"
B.As two vectors: num = [2, 1] and den = [1, 3, 2]
C.As a matrix of size 2x2
D.By its impulse response only
Correct Answer: As two vectors: num = [2, 1] and den = [1, 3, 2]
Explanation:Rational transfer functions are typically represented by two arrays/vectors containing the coefficients of the numerator and denominator polynomials in descending powers of .
Incorrect! Try again.
31What is the Laplace transform of the ramp function ?
A.
B.
C.
D.
Correct Answer:
Explanation:Since and multiplication by corresponds to , then .
Incorrect! Try again.
32Which of the following indicates a system is marginally stable?
A.Multiple order poles on the axis.
B.Simple poles on the axis and all other poles in LHP.
C.Poles in the RHP.
D.Zeros in the RHP.
Correct Answer: Simple poles on the axis and all other poles in LHP.
Explanation:Simple (non-repeated) poles on the imaginary axis result in constant or oscillating (non-decaying, non-growing) responses, indicating marginal stability.
Incorrect! Try again.
33The Laplace transform of is:
A.
B.
C.
D.
Correct Answer:
Explanation:Applying the s-shifting property () to the transform of which is .
Incorrect! Try again.
34A system is described by . The system is:
A.Underdamped
B.Critically Damped
C.Overdamped
D.Undamped
Correct Answer: Critically Damped
Explanation:The denominator factors as . It has a repeated real pole at , which corresponds to a critically damped system.
Incorrect! Try again.
35What is the Laplace transform of the rectangular pulse ?
A.
B.
C.
D.
Correct Answer:
Explanation:By linearity: .
Incorrect! Try again.
36Which software command is typically used to plot the poles and zeros of a system in the complex plane?
A.
B.
C. or
D.
Correct Answer: or
Explanation:zplane (typically for Z-transform but used for pole-zero plots) or pzmap (Pole-Zero Map) are standard commands to visualize poles and zeros.
Incorrect! Try again.
37If a system has a zero in the Right Half Plane (RHP), it is called:
A.Unstable
B.Non-causal
C.Minimum phase
D.Non-minimum phase
Correct Answer: Non-minimum phase
Explanation:A stable causal system with zeros in the right half plane is referred to as a non-minimum phase system.
Incorrect! Try again.
38What is the region of convergence for if the signal is right-sided?
A.
B.
C.
D.
Correct Answer:
Explanation:For a right-sided signal, the ROC is to the right of the rightmost pole. The poles are at -1 and -3. The rightmost is -1.
Incorrect! Try again.
39The time scaling property states that for is:
A.
B.
C.
D.
Correct Answer:
Explanation:Standard property: Time compression by results in frequency expansion by and magnitude scaling by .
Incorrect! Try again.
40Inverse Laplace Transform of is:
A.
B.
C.
D.
Correct Answer:
Explanation:The impulse function transforms to the constant 1.
Incorrect! Try again.
41Which of the following is true regarding the Inverse Laplace Transform contour integral?
A.It is evaluated along the real axis.
B.It is evaluated along a line where is in the ROC.
C.It is a circle around the origin.
D.It is always zero.
Correct Answer: It is evaluated along a line where is in the ROC.
Explanation:The inverse formula involves an integral , where the path of integration is a vertical line within the ROC.
Incorrect! Try again.
42If represents an LTI system, the step response in the s-domain is:
A.
B.
C.
D.
Correct Answer:
Explanation:The step response is the output when input is . . Thus .
Incorrect! Try again.
43Geometric evaluation of phase: The phase of is calculated as:
A.Sum of angles of zero vectors + Sum of angles of pole vectors.
B.Sum of angles of zero vectors - Sum of angles of pole vectors.
C.Sum of angles of pole vectors - Sum of angles of zero vectors.
D.Product of all angles.
Correct Answer: Sum of angles of zero vectors - Sum of angles of pole vectors.
Explanation:The total phase is the sum of phases of the numerator terms (zeros) minus the sum of phases of the denominator terms (poles).
Incorrect! Try again.
44What is the Laplace transform of where is a positive integer?
A.
B.
C.
D.
Correct Answer:
Explanation:This is a standard transform pair derived from repeated differentiation of the frequency domain representation of .
Incorrect! Try again.
45For a system to be both causal and stable, all poles of its transfer function must lie:
A.On the axis.
B.In the right half of the s-plane.
C.In the left half of the s-plane.
D.At the origin.
Correct Answer: In the left half of the s-plane.
Explanation:Causality implies ROC is to the right of the rightmost pole. Stability implies ROC includes the axis. For both to be true, the rightmost pole must be to the left of the axis (i.e., in the LHP).
Incorrect! Try again.
46In a pole-zero plot, if a pole is very close to the axis, the frequency response will exhibit:
A.A notch (dip).
B.A sharp peak (resonance).
C.Zero magnitude.
D.Constant magnitude.
Correct Answer: A sharp peak (resonance).
Explanation:Proximity to a pole makes the denominator of the transfer function small, resulting in a large magnitude (peak) at that frequency.
Incorrect! Try again.
47Which Laplace property explains why unstable systems cannot be analyzed using the Fourier Transform?
A.Linearity.
B.Time Shifting.
C.ROC definition.
D.Differentiation.
Correct Answer: ROC definition.
Explanation:The Fourier Transform exists only if the ROC of the Laplace Transform includes the axis. Unstable causal systems have poles in the RHP, so their ROC (right of poles) does not include the axis.
Incorrect! Try again.
48The unilateral Laplace transform is primarily used for:
A.Analyzing non-causal systems.
B.Solving differential equations with non-zero initial conditions.
C.Steady-state frequency analysis.
D.Discrete-time signal analysis.
Correct Answer: Solving differential equations with non-zero initial conditions.
Explanation:The property allows the unilateral transform to incorporate initial conditions directly into the algebraic solution.
Incorrect! Try again.
49If , what is the final value using the Final Value Theorem?
A.
B.1
C.2
D.Infinity
Correct Answer: 2
Explanation:FVT: . (Valid because poles of are at -1, which is in LHP).
Incorrect! Try again.
50The Laplace transform is a transformation from:
A.Time domain to Frequency domain (complex).
B.Frequency domain to Time domain.
C.Discrete time to Continuous time.
D.Time domain to Z-domain.
Correct Answer: Time domain to Frequency domain (complex).
Explanation:It maps a function of time to a function of a complex variable (complex frequency).
Incorrect! Try again.
Give Feedback
Help us improve by sharing your thoughts or reporting issues.