7. Signals, Systems and Frequency Domain Analysis

7. Signals, Systems and Frequency Domain Analysis

• 7.1 Fundamentals of Signal and Systems

  • Signal Classification: continuous-time vs. discrete-time, periodic vs. aperiodic, energy vs. power, even vs. odd, orthogonal, causal/anticausal/noncausal

  • Signal Transformations: time shifting, time scaling, time reversal

  • Standard Signals: unit impulse, unit step, unit ramp, exponential, signum

  • System properties (linearity, time-invariance, causality, stability, memory)

  • Linear Time-Invariant (LTI) systems

  • Convolution integral (continuous) and convolution sum (discrete)

• 7.2 Laplace Transforms

  • Definition and properties

  • Transforms of common functions

  • Inverse Laplace transform techniques: partial fraction expansion, Heaviside’s theorem

  • Application to solving linear differential equations

  • Transfer function and frequency response from Laplace domain

• 7.3 Fourier Series and Transforms

  • Continuous-Time (CT) Fourier Series for periodic signals

  • Fourier Integral and the Continuous-Time Fourier Transform (CTFT) for aperiodic signals

  • Forward and inverse Fourier transforms, key properties

  • Parseval's theorem (power/energy conservation)

  • Discrete-Time Fourier Series (DTFS) and Discrete-Time Fourier Transform (DTFT) properties

• 7.4 Z-Transform and Digital Systems

  • Sampling theorem, aliasing, signal reconstruction, zero-order hold

  • Z-transform: definition, region of convergence (ROC), properties

  • Inverse Z-transform methods

  • System response and transfer function H(z) in the Z-domain

  • Sinusoidal steady-state response

  • Pole-zero relationships and stability analysis

  • Introduction to Discrete Fourier Transform (DFT) and Fast Fourier Transform (FFT)

• 7.5 Application of Frequency Domain Analysis

  • Stability analysis using frequency domain methods

  • Spectral analysis of signals

  • Introduction to spectrum sensing

  • Correlation (auto-correlation, cross-correlation)

  • System design considerations in the frequency domain

• 7.6 Filters

  • Filter applications and ideal filter characteristics

  • Digital vs. Analog filters

  • Active vs. Passive filters and their frequency responses

  • Analog filter approximations: Butterworth and Chebyshev

  • Digital Filter Fundamentals:

    • Finite Impulse Response (FIR) filters

    • Infinite Impulse Response (IIR) filters