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Concept of information, its quantification and its signal coding. Signal types and main characteristics. What are systems, essential characteristics and the importance of their interface with the outside world. Invariant linear system (SLI), examples. Laplace and Fourier Transforms, properties, applications in Physics and Engineering. Description and analysis of SLI in time and frequency. Examples of electrical and optical systems. Causal SLI and Hilbert Transform. Discreet SLI. Random signals and SLI's. Feedback and control systems, introduction to nonlinear systems. An integrated perspective on signals and systems considering the comparative analysis of a simple electrical system (RLC circuit), a complex macroscopic system (the Earth System) and a biological system (the cell).

Identification of the macros objectives of the curricular unit by the exposition and framing around the following questions: what is information? How is it quantified? What are signals? How can they encode information? How are signals transmitted? What are hardware and software? What is a system? What are its main features? Is the interface of a system important? Is the universe a system?

Introduction to the Earth System and a biological system (cell).



2. GENERAL CONCEPTS

Signals. Signal types. Continuous and discret signals. Systems. Types of systems. Importance of linear and time invariant systems (SLIT) and their analysis. Reference to SLIT systems outside the time domain. The case of optical SLIT systems.


3.  LAPLACE AND FOURIER TRANSFORM

Importance of use of these mathematical tools in the context of studying SLIT systems. The formulation of eigenfunctions. Laplace transform. Convergence region, properties. Unilateral Laplace Transform and its importance in the context of causal systems. The fundamental principle of complex analysis. Circular integral involving singularities. Cauchy integrals and residual calculus. Inverse Laplace Transform. Calculation using the residual theorem and partial fraction expansion.

Fourier transform. The Fourier Transform as a particular case of the Laplace Transform. Fourier transform properties. Inverse Fourier Transform. Parseval's theorem. Passing band-time product. Reference to other transformations of interest in Physics and Engineering.


4.  SLIT SYSTEM ANALYSIS IN THE TEMPORAL DOMAIN

Description by linear differential equations with constant coefficients. Using the Laplace Transform to solve the differential equation of systems with initial conditions. Important examples in physics.

State space description. System state variables. System matrix, input and output matrices. Determination of these matrices from the coefficients of the differential equation. State equation and output equation. Transformation matrices between equivalent representations in state space. Controllable and observable systems.

Description by impulse response. Convolution operation. System impulse response. Physical interpretation. Series and parallel systems.


5.  SLIT SYSTEM ANALYSIS IN THE FREQUENCY DOMAIN 


Eigenfunctions of SLIT systems. Transfer function. Determination of the transfer function from the system differential equation. Relationship between the transfer function and the SLIT description in state space. Impact of initial conditions. Transfer function and impulse response of the SLIT system. Frequency response; amplitude and phase. Physical interpretation. Goat diagrams.

Example of systems with non-temporal frequencies; optical systems and spatial frequencies.


6.   CAUSALITY AND HILBERT'S TRANSFORM

Causal systems. Causal SLIT systems. Causal signals. Relationship between the real and imaginary parts of the causal signal Fourier transform. Hilbert transform. Consequences of this transform. Relationship between amplitude and phase of frequency response of a causal SLIT system.


7.   DISCRET SLIT SYSTEMS

Fourier transform of periodic signals. Properties. Periodic convolution. Sampling. Sampling theorem. Frequency domain sampling. Discreet signs. Fourier transform of discrete signals. Properties. Transform Z. Convergence region of Transform Z. Relationship of Transform Z to Laplace and Fourier transforms. Z transform theorems. Inverse Z transform. Z and pole diagrams in the Z plane.

Discrete time SLIT systems. Equations to linear differences with constant coefficients. Characteristic sequences (self-sequences). Transfer function; determination from difference equations. Description of discrete state-space SLIT systems. Discreet convolution and impulsive response. Z Transform convolution theorem.


8.  RANDOM SIGNS AND SLIT SYSTEMS

Random Signals. Expected and average value of a statistical set. Stationary and ergodic random processes. Real and complex signal correlation functions: autocorrelation, self-covariance, cross correlation, cross covariance. Power spectral densities. White noise.

SLIT systems response to random signals. Stationary and ergodicity. Linear average and autocorrelation of SLIT systems output. Cross correlation function between input and output of SLIT systems. Power spectral density and SLIT systems. SLIT systems output with white noise at input. Wiener filters.



9.   FEEDBACK AND CONTROL SYSTEMS

Feedback theory. Open loop and closed loop systems. Transfer function. Control principle. On-off control systems, proportional control systems, differential and integral control systems. Analysis and design of frequency and state space control systems. Stability criteria. Bases of digital control systems.

Introduction to nonlinear control systems.


10.  INFORMATION THEORY

General Aspects of Information Theory. Universe of messages. Message source entropy and connections to thermodynamic entropy. Flow of information. Scanning and coding. Shannon's theorem. Shannon-Hartley theorem for communication channels, consequences and applications.


11.   SOME THOUGHS

Some thoughs about Information, Signals, Systems, Life and Universe