# Signals and Systems Lecture

• Instructor:  Prof. Dr.-Ing. Werner Henkel

This course offers a comprehensive exploration of signals and systems which is the key knowledge for almost all electrical engineering tasks. Continuous-time and discrete-time concepts/methods are developed in parallel, highlighting their similarities and differences. Introductory treatments of the applications of these basic methods in such areas as filtering, communication, sampling, discrete-time processing of continuous-time signals, and feedback, will be discussed.

To this end, all the major linear transforms are introduced, like Fourier series, Fourier transform, Laplace transform, unilateral Laplace transform, Discrete Fourier Transform, diagonalization of a convolutional Toeplitz matrix (eigenvalues/eigenvectors of a Toeplitz matrix), and z-transform. Additionally, Hilbert transform, delay and group delay, stability and minimum phase are discussed. Furthermore, AM and FM modulation is shortly introduced.

• The courses uses the following text book:

Alan V. Oppenheim, Alan S. Willsky, and S. Hamid Nawab, Signals and Systems, Pearson, 2nd ed., ISBN 978-93-325-5023-0

• Energy vs. power, even and odd signals, special functions, such as complex exponentials, cos, sin, Euler’s formula, discrete-time exponentials, unti impulse and unit step function (time continuous and time-discrete)
• Continuous- and discrete-time systems, memory, causality, BIBO stability, time invariance, linearity
• Convolution in continuous and discrete time, deconvolution, causality, linear differential equation for exemplary RL or RC circuit
• Fourier series, real and complex form, response of LTI system inside Fourier series
• Fourier transform
• Properties of the Fourier transform
• Excursion to cascade structure and analog filter design, introducing characteristic impedance for two-ports and transmission lines
• All-pass
• Inverse Fourier transform, especially partial fraction expansion.
• Function of homogeneous and particular solutions of a differential equation regarding the network and the source signal, discussion of poles and zeros
• Hilbert transform
• Sampling theorem
• Discrete-time Fourier transform, linking to z-transform, convolution as product of polynomials
• Laplace transform and linking to z-transform.
• Convergence fo discrete-time Fourier transform, Fourier transform of periodic signals, properties of discrete-time Fourier transform
• Excursion to analog modulation, AM, PM, FM (matching timing of Signals and Systems lab)
• Time and freqeuncy characterization of systems, especially delay and group delay
• Laplace transform in detail with region of convergence (RoC), exemplary functions and networks, inverse transform with partial fraction expansion, all-pass, minimum phase property, mentioning Heaviside extension (residue theorem), properties of the Laplace transform
• Links between homogeneous solution of a differential equation and the poles of the transfer function, initial and final value theorems
• Unilateral (one-sided) Laplace transform, transfer for circuit elements including their initial conditions, i.e., initial charges of a capacitor or initial currents at an inductor
• Introduction to z transform, RoC, transforms of zeros and poles between Laplace and z planes
• Convolution in matrix description, eigenvectors and eigenvalues of a convolutional Toeplitz matrix, diagonalization of a Toeplitz matrix by Fourier matrices