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Course overview
This course develops the mathematical framework for analyzing signals and linear time-invariant systems. Students begin by classifying signals and understanding system properties, then move into convolution and impulse response, Fourier series, Fourier transform, and finally Laplace and Z-transforms with their applications to filtering and stability. After completing the course, students can compute convolution integrals and sums, apply transform techniques to find system output for arbitrary inputs, determine system stability from pole locations, and interpret frequency spectra.
Details
Target audience: Second-year B.Tech ECE students taking Signals and Systems as a core course, and GATE aspirants who need a structured review of transform methods.
Duration: 11 to 14 hours across 5 modules
Prerequisites
- Calculus: integration by parts and improper integrals
- Complex numbers and Euler's formula
- Basic differential equations
Learning outcomes
- Classify any given signal as energy or power, periodic or aperiodic, causal or non-causal
- Compute continuous-time and discrete-time convolution for standard signal pairs
- Derive Fourier series coefficients for periodic signals and sketch the spectrum
- Apply Fourier transform properties to find the transform of composite signals without direct integration
- Find the inverse Laplace transform using partial fraction decomposition and interpret pole locations for stability
- Compute the Z-transform and its inverse, and determine stability of discrete-time systems from pole locations
Modules
- Continuous-time and discrete-time signal definitions
- Energy and power signal classification
- Basic signal operations: shifting, scaling, and reversal
- Elementary signals: unit step, unit impulse, ramp, and sinusoid
- Even and odd decomposition of signals
- System properties: linearity, time-invariance, causality, stability
- Impulse response and system characterization
- Continuous-time convolution integral
- Discrete-time convolution sum
- BIBO stability condition via impulse response
- Fourier series coefficients for periodic signals
- Dirichlet conditions and convergence
- Continuous-time Fourier transform and inverse
- Fourier transform properties: linearity, shifting, scaling, duality
- Parseval's theorem and energy spectral density
- Bilateral and unilateral Laplace transform
- Region of convergence and pole-zero plots
- Laplace transform properties and pairs
- Inverse Laplace via partial fractions
- Transfer function and system analysis in s-domain
- Z-transform definition and region of convergence
- Z-transform properties and common pairs
- Inverse Z-transform using partial fractions and long division
- Stability analysis using Z-domain poles
- DTFT from Z-transform and frequency response of discrete systems