Signal Theory

**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

- 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
- 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
- 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
- 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
- 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

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