Questions
6 questions in major PSU papers
Difficulty
Medium-Hard
Importance
Core — never skip
Overview
Networks, Signals, and Systems forms the mathematical backbone of Electrical and Electronics engineering, focusing on analyzing circuit behaviors and signal processing properties. Mastering this topic is essential for PSU exams as it provides the analytical tools required to solve complex transfer function and transient analysis problems efficiently.
Fourier and Laplace Transforms
These integral transforms convert time-domain functions into frequency or complex-frequency domains, simplifying differential equations into algebraic expressions. They are vital for identifying system stability and frequency response characteristics.
- Fourier Series: x(t) = sum of harmonics
- Fourier Transform: X(jω) = ∫x(t)e^(-jωt)dt
- Laplace Transform: X(s) = ∫x(t)e^(-st)dt
- Region of Convergence (ROC) is critical for existence
- Final Value Theorem: lim(t→∞)x(t) = lim(s→0)sX(s)
LTI Systems and Convolution
Linear Time-Invariant (LTI) systems are characterized by their impulse response, which fully defines the system's output for any input. Convolution is the mathematical operation used to compute this output in the time domain.
- y(t) = x(t) * h(t)
- System is stable if impulse response h(t) is absolutely integrable
- System is causal if h(t) = 0 for t < 0
- Commutative, distributive, and associative properties of convolution
- Discrete convolution: y[n] = x[n] * h[n]
Z-Transform and Discrete Systems
The Z-Transform acts as the discrete-time counterpart to the Laplace Transform, used extensively in digital signal processing. It helps in analyzing the stability and pole-zero locations of digital filters.
- X(z) = Σx[n]z^(-n)
- ROC must be a ring in the z-plane
- Pole-zero plot determines stability
- Unit circle stability criterion for discrete systems
- Time-shifting property: x[n-k] ↔ z^(-k)X(z)
Network Theorems
Network theorems provide simplified methods to analyze complex electrical circuits without solving massive systems of nodal or mesh equations. These are frequent sources of calculation-based questions in PSU papers.
- Thevenin's Theorem: Vth in series with Rth
- Norton's Theorem: IN in parallel with RN
- Superposition Theorem for linear circuits
- Maximum Power Transfer: RL = Rth
- Reciprocity and Millman’s Theorems
Formula Sheet
y(t) = ∫ x(τ)h(t-τ)dτ
H(s) = Y(s)/X(s)
H(z) = Y(z)/X(z)
X(e^jω) = Σx[n]e^(-jωn)
P = V^2 / (4Rth)
Parseval’s Theorem: ∫|x(t)|^2 dt = (1/2π)∫|X(jω)|^2 dω
Exam Tip
Focus on memorizing the standard Transform pairs and properties instead of deriving them, as most PSU questions test direct application or boundary conditions.
Common Mistakes
- Ignoring the ROC (Region of Convergence) when determining the inverse Laplace or Z-transform, leading to ambiguous results.
- Confusing the stability criteria for continuous time (left half of s-plane) versus discrete time (inside unit circle in z-plane).
- Applying Network Theorems to non-linear components or sources.
More Revision Notes
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