Advanced Signals and Systems Discrete Systems Gerhard Schmidt - PowerPoint PPT Presentation

Advanced Signals and Systems – Discrete Systems Gerhard Schmidt Christian-Albrechts-Universität zu Kiel Faculty of Engineering Institute of Electrical and Information Engineering Digital Signal Processing and System Theory Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems

Contents of the Lecture Entire Semester:  Introduction  Discrete signals and random processes  Spectra  Discrete systems  Idealized linear, shift-invariant systems  Hilbert transform  State-space description and system realizations  Generalizations for signals, systems, and spectra Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems Slide IV-2

Contents of this Part Discrete Systems  System description  System classification  Discrete linear systems and their response to deterministic signals  Stability of linear systems  Special symmetries for real-valued systems  Discrete linear systems and their responses to stochastic processes Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems Slide IV-3

Discrete Systems System Description – Part 1 Basics: The most general description of systems defines a system as on operator Typically we will use the following graphical description for systems System Beyond the input and the output signal vector also internal signals can contribute to the system output. These internal “system states” will be denoted as a so -called state vector with state variables that contain the complete information on the system state in a non-redundant form . Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems Slide IV-4

Discrete Systems System Description – Part 2 Basics (continued): It is also possible to describe a system by means of its state-space description . Usually two equations are used for that purpose:  The output equation , that describes how the output signal vector is generated using the current input vector and the current state vector  and the state equation , that describes how the new state vector is generated using the current state vector and the current input vector Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems Slide IV-5

Discrete Systems Contents of the Section „System Classification “  System description  System classification  Continuous, discrete, and digital systems  One- and multi-dimensional systems  Deterministic and stochastic systems  Passivity  Dynamic and memoryless systems  Real and complex systems  Causality  Linearity  Shift or time invariance  Stability  Discrete linear systems and their response to deterministic signals  Stability of linear systems  ... Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems Slide IV-6

Discrete Systems System Classification – Part 1 Continuous, discrete, and digital systems: In our notation of the last slides, the description of discrete systems respectively produce, upon a discrete input signal , only a discrete output signal . In contrast to that also continuous state-space descriptions exist. Since such descriptions are based on differential equations (instead of difference equations) the derivative of the state vector is utilized in the state equation: Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems Slide IV-7

Discrete Systems System Classification – Part 2 Continuous, discrete, and digital systems (continued): In discrete systems the state variables usually describe the contents of memory elements. In continuous systems usually energy-storing elements are described by the state variable. Some general comments :  A system taking a number of input sequences and producing a number of output sequences (and ) may, of course, be generally seen as a computing system .  However, digital computers necessarily work with quantized (limited word length) signals and variables . These digital systems are a special type of discrete systems .  Here (in this lecture) we will deal only with discrete systems – digital systems are treated in the lecture “Advanced digital signal processing”. Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems Slide IV-8

Discrete Systems System Classification – Part 3 One-dimensional and multi-dimensional systems: Within the introduction slides of this lecture we had already explained the difference between scalar and vector signals on the one hand and one- and multi-dimensional signals on the other hand: E.g. speed of an object (x, y, and z-direction) E.g. brightness of a picture ( ) For systems we can apply the same definition. However, as mentioned before, in this section we will focus first on one-dimensional systems – multi-dimensional extensions will be covered briefly at the end of this part. Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems Slide IV-9

Discrete Systems System Classification – Part 4 Deterministic and stochastic systems: If is fixed (not necessarily constant) in all aspects (structure, parameters), then is deterministic . If at least one detail of is random-like (like one coefficient), then is stochastic . Consequence: For a stochastic system , is stochastic even for a deterministic input signal (while for a deterministic system, is deterministic for a deterministic input signal)! We will focus in the remaining slides on deterministic systems. Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems Slide IV-10

Discrete Systems System Classification – Part 5 Passivity: In the second part of this lecture (“Signals and Stochastic Processes”) we have introduced the instantaneous or local energy: This quantity is used to determine passive systems. For a system with one input and one output the system is passive if the instantaneous power of the output signal is always smaller or equal than the instantaneous input power: For passive systems with more than one input or output we define passivity as Please note that both definitions must be fulfilled for all ! Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems Slide IV-11

Discrete Systems System Classification – Part 6 Passivity (continued): As consequences we can conclude that …  … no signal sources are allowed inside a passive system and  … if all input signals are zero up to a certain index than also all output signals of a passive system must be zero Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems Slide IV-12

Discrete Systems System Classification – Part 7 Dynamic and memoryless systems: Definition of a dynamic system: A system is called a dynamic system if does not only depend on , but also on . Consequences:  Dynamic systems must contain some “ storage ” of values beyond the index . The storage may concern  values for (“ left ” of , for representing a time index this means “ before “, meaning storage of the past , i.e. memory in the usual sense).  values for (“ right ” of , for representing a time index this means “ after “, meaning storage of the future , which is possible if is not related to real-time). In contrast to that: Systems without storage or memory (“non -dynamic" systems) are called memoryless . Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems Slide IV-13

Discrete Systems System Classification – Part 8 Real-valued and complex-valued systems: If all elements of the input signal vector of a system are real and also all elements of the output signal vector are real then the system is called a real-valued (or real) system . Otherwise we call the system to be complex-valued (or simply complex). Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Systems Slide IV-14