Short Notes for Signals & Systems (GATE EE ) - Signals and Systems - Electrical

Making short notes should begin as soon as you decide to take the GATE exam. Concepts, formulas, shortcuts, exceptional situations, tricky details and the standard approach to frequently asked questions should all be captured in these revision notes. At EduRev, we advise making your own concise notes and we also provide precise topic-wise notes covering the above items so that you do not forget important points on the final day before the exam. To begin your preparation, follow the curated resources and summaries listed in the relevant sections below.

1. Introduction to Signals & Systems

Signals carry information and systems process signals. A signal is any quantity that varies with one or more independent variables (commonly time). A system maps an input signal to an output signal according to some rule. Understanding classification, basic operations and representations of signals and systems is foundational for analysis in time and frequency domains.

Key ideas and definitions you should remember:

• Continuous-time vs discrete-time signals: continuous signals are defined for every real time t, discrete signals are defined only at integer sample indices n.
• Deterministic vs random signals: deterministic signals can be exactly described by a formula; random (stochastic) signals are described statistically.
• Even/odd decomposition: any signal $x(t) = x_e(t) + x_o(t) where x_e(t) = 0.5[x(t)+x(-t)], x_o(t) = 0.5[x(t)-x(-t)].$
• Periodic signals: x(t+T)=x(t) for all t; discrete periodic if $x[n+N]=x[n].$
• Energy and power signals: energy =$ ∫|x(t)|^2 dt$ (or Σ for discrete), power = time-average of $|x|^2$; a signal cannot be both finite-energy and finite non-zero power.
• System properties: linearity, time-invariance, causality, stability, memoryless.
• Linear Time-Invariant (LTI) systems: completely characterised by the impulse response h(t) (continuous) or h[n] (discrete) and the convolution operation $y(t)=x(t)*h(t) or y[n]=Σ x[k]h[n-k].$
• Shifting and scaling operations: x(t-t0), x(at) (continuous) and x[n-n0], x[an] (where meaningful) - pay attention to how scaling changes support and sampling.

The following curated resources expand these basic topics and give worked examples and slides.

• Description of Systems
• Properties of Systems
• Representation of Continuous & Discrete, Time Signals -1
• Representation of Continuous & Discrete, Time Signals - 2
• PPT: Introduction to Signal & Systems
• Introduction to Signal & Systems
• Description of Signals
• Discrete Time Convolution
• PPT: Discrete Time Convolution
• Linear Shift Invariant Systems
• Classification of Systems
• Continuous Time LTI Systems
• Properties of LTI System
• Signal & System Formulas for GATE EE Exam
• Shifting & Scaling Properties of Signals

2. Sampling Theorem

Sampling converts a continuous-time signal into a discrete-time sequence by taking values at regular intervals. The central result is the Nyquist-Shannon sampling theorem: a band-limited signal with maximum frequency component ωm (or fm) can be perfectly reconstructed from samples if the sampling frequency fs satisfies fs > 2fm (the Nyquist rate). If sampling is below this rate, aliasing occurs - high-frequency components fold into low frequencies and information is lost.

Important practical points:

• Ideal reconstruction uses an ideal low-pass filter (sinc interpolation). Real filters approximate this and introduce distortion near band edges.
• Realistic sampling includes effects of aperture, sample-and-hold, and practical anti-aliasing filtering before sampling.
• When sampling periodic or non-band-limited signals, care is needed: oversampling and prefiltering reduce aliasing and ease reconstruction.

Study these links for derivations, examples of aliasing and realistic sampling models.

• Sampling - Sampling & Reconstruction
• Sampling & Reconstruction of Band Limited Signals
• Study Notes for Linear Time Invariant System & Sampling Theorem
• PPT: Sampling & Reconstruction
• Realistic Sampling of Signals
• Aliasing (Under Sampling)

3. Fourier Series in Signals & Systems

The Fourier series represents a periodic signal as a sum of harmonically related complex exponentials or sines and cosines. For a periodic continuous-time signal x(t) with period T0, the complex exponential Fourier series is:

x(t)=Σ_{k=-∞}^{∞} C_k e^{j k ω0 t} where ω0 = 2π/T0 and C_k are the Fourier series coefficients.

Key points to remember:

• Coefficients C_k are obtained by integrating x(t) times e^{-j k ω0 t} over one period.
• Convergence depends on signal properties (Dirichlet conditions). Gibbs phenomenon occurs for discontinuities.
• Parseval's theorem relates average power to the sum of squared magnitudes of coefficients.

Refer to the linked notes for worked examples and coefficient calculations for common waveforms.

• Study Notes for Fourier Series Representation of Continuous Periodic Signals -1

4. Fourier Transform in Signals & Systems

The Fourier transform (FT) generalises the Fourier series to aperiodic signals and gives a continuous frequency-domain representation. For a continuous-time signal x(t) the FT is X(ω)=∫_{-∞}^{∞} x(t)e^{-jωt} dt. The inverse transform reconstructs x(t) from X(ω).

Important concepts and properties:

• Linearity, time-shifting, frequency shifting, time-scaling, convolution in time ↔ multiplication in frequency and vice versa.
• For periodic signals, the FT is an impulse train (discrete lines) at harmonics of the fundamental frequency; for aperiodic signals, the FT is continuous.
• Discrete-Time Fourier Transform (DTFT) is the Fourier transform for discrete-time signals; it is periodic in frequency (2π periodic).
• Use FT to analyse LTI systems: the frequency response H(ω) is FT of impulse response h(t); y(t) ↔ X(ω)H(ω).

Study the listed materials for derivations, properties and examples of FT and DTFT, including inverse transforms and transform pair tables.

• Introduction to Transformation of Signals
• Fourier Transform
• Fourier Transform & Its Properties
• Fourier Transform as a System
• Fourier Transform of Periodic Signals & Some Basic Properties of Fourier Transform
• PPT: Fourier Transform
• Properties of Fourier Transform
• Discrete Time Fourier Transform & Its Properties
• Inverse Discrete Time Fourier Transform

5. Laplace Transform in Signals & Systems

The Laplace transform extends the Fourier transform to handle signals that grow or decay exponentially and facilitates solving linear differential equations with initial conditions. The bilateral Laplace transform is X(s)=∫_{-∞}^{∞} x(t) e^{-s t} dt where s=σ+jω.

Key concepts:

• Region of convergence (ROC): set of s for which the integral converges; ROC determines causality and stability when combined with pole-zero locations.
• Pole-zero representation: rational Laplace transforms are expressed as ratios of polynomials; poles locate natural modes, zeros indicate frequency cancellations.
• Inverse Laplace transform techniques include partial fraction expansion and use of transform tables. For LTI systems, H(s) = Laplace{h(t)} gives system behaviour in s-domain; stability requires ROC including jω axis for BIBO stability.

Use the linked notes for properties, transforms of common signals and example problems involving circuit and system analysis.

• Laplace Transform
• Study Notes for Laplace Transform & Their Properties
• PPT: Laplace Transform & Its Applications

6. Z-Transform in Signals & Systems

The Z-transform is the discrete-time counterpart of the Laplace transform. For a sequence x[n], the bilateral Z-transform is X(z)=Σ_{n=-∞}^{∞} x[n] z^{-n}. The Z-transform is invaluable for analysing discrete-time LTI systems, difference equations, stability and causality.

Important properties and uses:

• Region of convergence (ROC): determined by the magnitude of z; ROC shape (exterior of outermost pole, interior of innermost pole, or annulus) indicates causality and stability.
• Relation to DTFT: the DTFT is the Z-transform evaluated on the unit circle (z = e^{jω}), provided the ROC includes the unit circle.
• Inverse Z-transform: obtained by power series expansion, partial fraction expansion or contour integration. Rational system functions are commonly used to represent LTI systems.
• Analysis of difference equations uses H(z)=Y(z)/X(z); poles of H(z) determine natural response and stability.

Study the links for ROC examples, inverse transforms, rational functions and LTI system analysis in discrete time.

• Z-Transform & Region of Convergence
• Study Notes For Z-Transform
• PPT: Z-Transform & Region of Convergence
• Properties of Laplace & Z-Transform
• Inverse Laplace & Z-Transform
• Rational System Functions
• Inverse Laplace & Z-Transform of Rational Functions
• Analysis of LTI Systems with Rational System Functions

Final advice for making effective short notes:

• Record definitions, standard transforms and transform pairs, common Fourier/Laplace/Z-transform results, and key properties in one place.
• Maintain a page of common formulas and tricks (convolution patterns, time/frequency-shift rules, ROC rules for stability/causality).
• Include one worked example per concept (e.g., compute FS coefficients, perform DTFT, solve difference equation using Z-transform, sample-and-reconstruct example) with essential steps and final result.
• Use these notes for quick revision; practice problem-solving to reinforce concepts and spot typical exam traps (aliasing, ROC misinterpretation, Gibbs phenomenon, pole-zero placement consequences).

If you follow the conceptual summaries above, and study the linked materials for detailed derivations and examples, your short notes will become a reliable revision resource for both understanding and fast recall.