Institute of Multimedia Telecommunications
Signal Theory


Lecture



Lecturer:
dr inż. Tomasz Grajek (Ph.D. Eng.)

Initial requirements

  • Knowledge of trigonometry and analytic geometry, particularly operations on vectors.
  • Knowledge of the principles of differential and integral calculus. Proficiency in calculating definite integrals of trigonometric, exponential and power functions and polynomials. Proficiency in calculating the integrals of combined functions.
  • Good knowledge of complex numbers (Cartesian and polar form).
  • Function analysis. Efficient function plotting.
  • Knowledge of issues related to the sequences and series of numbers. The ability to determine the convergence of the series by various criteria. Ability to calculate the limits.
  • Knowledge of the basic laws of physics, especially related to the flow of electric current.

Signal Theory - course overview

  1. Fundamental concepts and measures
    • Signals and their models
    • Signal classes and examples
      • Continuous, discrete, analogue, quantized and digital signals
      • Periodic signals
      • Sinusoidal signals: real and complex
      • Non-periodic signals
    • Basic signal metrics
      • Amplitude
      • Mean value
      • Energy of a signal
      • Power of s signal
      • Effective value of a signal (RMS)
    • Energy signals vs power signals
    • Orthogonality. Orthogonal signals and vectors
    • Signal components
      • DC and AC signal components
      • Odd and even signal components
  2. Analysis of periodic signals using orthogonal series
    • Hilbert space
    • Orthogonal bases
    • Orthogonal series of functions
    • Trigonometric Fourier series
    • The influence of signal symmetries on the coefficients of the trigonometric Fourier series
    • Complex exponential Fourier series
    • The harmonic spectrum of a real signal
    • The relationship of the complex exponential and the trigonometric Fourier series
    • Linearity of Fourier series
    • The influence of signal symmetries on the coefficients of complex exponential Fourier series
    • The effect of signal shift in time on the complex exponential Fourier series
    • Spectrum of a product of two signals
    • Computing the power of a signal – the Parseval theorem
  3. Analysis of non-periodic signals. Fourier Transformation and Transform
    • An intuitive introduction
    • Definition
    • Fourier Transform vs Laplace Transform
    • The Magnitude Spectrum and Phase Spectrum
    • Symmetries of the Fourier Transform for real-valued signals
    • Special case of Fourier Transform for symmetrical signals
    • Theorems describing the properties of Fourier Transformation
      • Linearity
      • Shift theorem – shifting in time domain
      • Shifting in frequency domain (also known as modulation theorem)
      • Scaling theorem (also called the similarity theorem)
      • Time-frequency duality (also known as the symmetry theorem)
      • Derivative theorem (differentiation in time domain)
      • Integration theorem
    • Calculating energy of the signal from its Fourier transform. The Parseval's theorem
    • Generalization of the Fourier transformation for infinite energy signals
    • Fourier transform of a periodic signal
    • Calculating the power of a signal from its Fourier transform. The Parseval's theorem for power signals
  4. Processing of signals by linear and time invariant (LTI) systems
    • Introduction to LTI systems. Fundamental properties
    • Impulse response of an LTI system
    • Impulse response of a causal system
    • The response of an LTI system to arbitrary input
    • Properties of linear convolution
    • Frequency response
    • Determining the frequency response of an electronic circuit
    • Filters
  5. Sampling. Discrete-time signals
    • Introduction to discrete signals
    • Spectrum of a sampled signal
    • Spectral efect of sampling a continuous signal
    • Reconstruction of the continuous signal from its samples
    • Non-periodic and periodic discrete-time signals
    • Fourier transforms of discrete-time signals
    • Processing of discrete-time signals
    • Frequency response of discrete-time LTI systems

Basic bibliography

  1. A. Oppenheim, A. Wilsky, I. Young, Signals and Systems, Prentice Hall, 1996
  2. R.A. Gabel, R.A. Roberts, Signals and Linear Systems, Wiley, 1986
  3. B.P. Lathi, Linear Systems and Signals, Oxford University Press, 2004
  4. E. Kamen, Introduction to Signals and Systems, MacMillan, 1987