EE 20. Structure and Interpretation of Systems and Signals
Course objectives: This course introduces mathematical modeling techniques used in the study of signals and systems. Its intention is to promote rigorous thinking and mathematical intuition about, and an appreciation for a multidisciplinary study of, signals systems through precise modeling.
- Mathematical Foundations - Sets and functions; Fundamentals of mathematical logic; Complex algebra (including complex-valued functions); Basic linear algebra (matrix-vector manipulations, eigenanalysis of 2x2 matrices; Vector-space concepts (e.g., axioms describing vector spaces, basis, dimension, inner products, orthogonal basis expansions).
- Signals - Signals as functions: continuous time, discrete time; Signals as vectors in an appropriately-defined vector space having an appropriately-defined inner product; Orthogonality of signals; Two-dimensional space continuum; Discrete-space: discrete time and continuous space, discrete time and mixed space, discrete time and space, discrete events, discrete events and discrete time; Sequences.
- Sinusoids - Periodic signals, sinusoids-phase and amplitude, complex numbers, complex signals, complex exponentials; Phasors, amplitude modulation, frequency modulation.
- Spectrum - Summing sinusoids, approximating periodic signals, harmonics and musical sounds, beat notes, two-dimensional sinusoids; Approximating images.
- Sampling - Analog-to-digital conversion, aliasing, downsampling, digital-to-analog conversion, upsampling, oversample CD players.
- Systems - Filters, Running average filter, Two-dimensional running average filter, FIR filters; Linearity; Time invariance; Causality; Memory; Impulse response; Convolution.
- Frequency Response and Filtering - Sinusoidal input, complex sinusoidal input, transfer function; Filtering audio signals, blurring and sharpening images.
- Fourier Series and Transforms - Fourier series using vector-space concepts; Signals viewed as vectors; Harmonically-related sinusoids and complex exponentials are viewed as orthogonal vectors in an appropriately-defined vector space; Discrete Fourier series (DFS/DFT); Continuous-time Fourier series (FS); Discrete-time Fourier transform (DTFT); Continuous-time Fourier transform (CTFT).
- Z Transform - Unit delay; Polynomials to represent signals; Z transform as an operator; Convolution; Poles and zeroes and their relationship to frequency response (e.g., digital lowpass, bandpass, notch, and comb filters).
- Spectrum Analysis - Spectra of periodic and aperiodic signals; Time and frequency sampling; Amplitude modulation; Spectrograms.
- Nonlinear Systems - First- and second-order continuous-time and discrete-time autonomous systems: equilibrium point identification, stability analysis of equilibrium points, sketching vector fields, sketching phase portraits; Frequency modulation; synthesis of musical sounds; hard limiting; Edge detection; Feedback; Fractals and chaos; Noise in musical sounds and images.
- State Machines - Events, I/O traces; State and finite state; Moore machines; Deadlock freedom; Trace language of a state machine; Hiding of events
- Relating State - Concrete and abstract system descriptions; Nondeterminism; Safety requirements: The simulation relation; Simulation implies trace inclusion, layers of abstraction.
- Composing State - Parallel components, I/O synchronization, interleaving of internal events; Product state space, intersection of trace languages; Composition preserves simulation, control, state equivalence; State minimization, machine equivalence, bisimilarity implies simulation; Composition preserves bisimilarity.
- Discrete Events - Time stamps, non-uniform sampling, pulse position modulation; Zeno signals, communicating automata, communication networks; Speech and video on communication networks.