(SEM IV) THEORY EXAMINATION 2021-22 SIGNAL SYSTEM

SECTION–A — Conceptual Short Questions (20 Marks)

Section–A contains ten brief questions, each worth two marks. Despite the short format, every question covers a foundational concept from the entire Signal & Systems syllabus. The section begins by asking what a signal is, pushing students to describe how a signal represents a physical quantity varying with time and to provide a clear example such as audio or ECG signals. The next question requires drawing the discrete-time signal u(n)−u(n−3)u(n) - u(n-3)u(n)−u(n−3), which tests understanding of unit step functions and how they form finite-duration sequences.

A fundamental system-properties question checks whether the system y(t)=t⋅x(t)y(t) = t \cdot x(t)y(t)=t⋅x(t) is causal and time-variant. This demands evaluating whether system output depends on future inputs and whether shifting the input leads to a proportional shift in output. The section further includes the Nyquist theorem, which tests sampling fundamentals, and the sufficient conditions for existence of CTFT, which relate to absolute integrability. Questions such as finding the Z-transform of (1/2)nu(n)(1/2)^n u(n)(1/2)nu(n), determining the fundamental period of a multi-cosine signal, writing the expression for convolution integral, comparing CTFT and DTFT, and finding the Z-transform of u(n)u(n)u(n) ensure the student demonstrates both conceptual understanding and mathematical recall.

SECTION–B — Analytical & Derivation-Based Questions (30 Marks)

Section–B asks students to attempt any three out of five long questions. These questions focus heavily on transforms, signal decomposition, and frequency-domain properties. The first question requires stating and proving the frequency shifting theorem of CTFT, followed by an explanation of linearity. This ensures students understand essential Fourier transform properties and can present proofs.

Another question gives a continuous-time cosine signal and asks whether it is a power or energy signal, requiring understanding of energy integrals and power calculations for periodic signals. The section also includes questions on even–odd decomposition for both continuous and discrete signals, which checks the student’s understanding of symmetry properties.

The paper then moves toward transform computation: one question asks for the Fourier transform of x(t)=e−atu(t)x(t) = e^{-at}u(t)x(t)=e−atu(t), followed by sketching its magnitude and phase spectra. This requires using the standard exponential FT pair and demonstrating visual interpretation of spectra. The last option in this section asks to compute convolution of two decaying exponentials using Fourier transform, testing the student’s ability to use the convolution theorem effectively.

SECTION–C — Laplace Transform and System Analysis (10 Marks)

Section–C contains two alternative questions focused on Laplace-domain system analysis. The first option provides a Laplace transform

X(s)=2(s+4)(s−1)X(s) = \frac{2}{(s+4)(s-1)}X(s)=(s+4)(s−1)2​

and asks the student to compute its inverse Laplace in two different regions of convergence (ROC). Different ROCs imply different right-sided or left-sided time-domain signals, so the student must express the partial fraction expansion carefully and identify time-domain solutions for each ROC condition.

The second option asks to obtain the impulse response of a second-order differential equation using Laplace transform. This involves transforming the differential equation, solving for H(s)H(s)H(s), and then taking inverse Laplace to get h(t)h(t)h(t). This reinforces the connection between systems, differential equations, and Laplace-domain representations.

SECTION–D — Z-Transform Computations (10 Marks)

Section–D focuses on discrete-time system analysis using Z-transforms. One option requires deriving the Z-transform of x(n)=cos⁡(ω0n)u(n)x(n) = \cos(\omega_0 n) u(n)x(n)=cos(ω0​n)u(n) and sketching its ROC. This demands knowledge of Z-transform pairs for causal sinusoids. The other option asks for the inverse Z-transform of a rational expression

H(z)=0.2z(z+0.4)(z−0.2),ROC: ∣z∣>0.4.H(z)= \frac{0.2z}{(z+0.4)(z-0.2)}, \quad \text{ROC: } |z|>0.4.H(z)=(z+0.4)(z−0.2)0.2z​,ROC: ∣z∣>0.4.

This requires partial fraction decomposition and interpreting ROC to identify right-sided sequences.

 SECTION–E — Parseval’s Theorem & DTFT Analysis (10 Marks)

The first option in this section asks for statement and proof of Parseval’s theorem, a fundamental identity relating energy in time and frequency domain. Students must derive the expression using Fourier transform definitions. The alternative option presents a DTFT computation question involving two exponential sequences, one right-sided and one left-sided. The student must derive DTFT expressions, combine them, and interpret the final form.

SECTION–F — Convolution Integral & Convolution Sum (10 Marks)

Section–F tests the student’s competence with convolution operations. The first question asks for convolution of x(t)=e−2tu(t)x(t) = e^{-2t}u(t)x(t)=e−2tu(t) with h(t)=u(t)h(t) = u(t)h(t)=u(t), requiring graphical or analytical computation of convolution integral. The second option asks for convolution sum of x(n)=anu(n)x(n) = a^n u(n)x(n)=anu(n) and h(n)=u(n)h(n) = u(n)h(n)=u(n). This reinforces understanding of linear time-invariant (LTI) system behavior in both continuous and discrete domains.

SECTION–G — Sampling and Reconstruction (10 Marks)

The final section evaluates conceptual mastery over sampling theory. One option asks the student to state and prove the sampling theorem and discuss aliasing due to under-sampling. This requires showing mathematically that a bandlimited signal can be reconstructed perfectly if sampling frequency exceeds twice the maximum signal frequency.

The alternate option asks for explanation of signal reconstruction using interpolation, which involves low-pass filtering of discrete samples to recreate the continuous-time signal. This question tests understanding of the Whittaker–Shannon reconstruction formula and practical interpolation techniques.

FINAL SUMMARY — Complete Descriptive Overview

The SIGNAL SYSTEM exam paper is a balanced blend of theoretical concepts, mathematical analysis, and transform-domain problem-solving. Section–A evaluates foundational clarity, Section–B tests analytical proofs and transform properties, Section–C applies Laplace transform to system modeling, Section–D focuses on Z-transform analysis, Section–E assesses signal energy and DTFT skills, Section–F reinforces convolution mastery, and Section–G examines sampling concepts and reconstruction fundamentals. The paper ensures that students demonstrate both intuitive understanding and strong mathematical rigor across all essential topics of signals and systems.