Signals and Systems

Summary

The course presents the theory of signal transformations and linear time-invariant systems, preparing for courses on signal processing, systems theory, and telecommunication.

Starting from complex function theory, this course develops the mathematical description of signals and linear time-invariant (LTI) systems by means of the Laplace and Fourier transforms. In this description, signals are represented by sums of complex exponentials, being the 'eigenfuctions' of LTI systems. It follows that the effect of an LTI system on a signal (convolution) can equivalently be described by a product in the Laplace or Fourier domain. The implications are profound and form the basis of a large part of electrical engineering (and other engineering studies). This course is the basis for follow-up courses such as Digital Signal Processing.

The course covers the Laplace, Fourier and z-transform, and presents the relations between signals in time domain and frequency domain, first for time-continuous (analog) signals, and then for time-discrete (digital) signals.

The course also covers the basics of analog filter design (analog filter functions, IIR filter design, Butterworth and Chebyshev filters), digital filter design via transformation of analog filters to digital filters (impulse invariance, bilinear transform, frequency transformations), and simple digital filter structures.

The three course labs in Part 2 cover convolution, frequency domain, and filter design. They use Python programming and Google colab scripts.

Part 1: continuous-time signals

• Linear time-invariant systems (LTI), eigenfunctions and eigenvalues of an LTI system, Dirac delta function, Heaviside unit step function. Convolution.

• Laplace transform, region of convergence. Properties. Convoluton and product, inverse transform.

• Periodic signals. Fourier series, line spectrum.

• Fourier transform (continuous time).

• Transformations of a pulse, sinc function, sign function, step function. Duality of the Fourier transform. Shift and modulation, Parseval's theorem.

Part 2: discrete-time signals

• Ideal sampling. impulse train, periodic spectrum. Sampling theorem (Nyquist, Shannon). Ideal reconstruction

• Discrete signals and systems, convolution.

• Z-transform

• Realizations (direct form for FIR, IIR)

• Fourier transform (discrete time)

• Filter design (analog and digital), frequency transformations.

Part 2 also contains 3 course labs that train you in the use of convolution, frequency domain (Fourier transform) and filter design. The course labs are based on Python programming.