2- Systems Classifications and Modeling.pdf

Signals and Systems
17EL (Section-I and II), Fall-2019
Lecture-2
Systems Classifications and
Modeling
Dr. Shoaib R. Soomro

▪ Systems
▪ System Classifications
▪ Linear and Nonlinear Systems
▪ Time variant and Time In-variant Systems
▪ Instantaneous and Dynamic Systems
▪ Casual and Non-casual Systems
▪ Lumped Parameter and Distributed Parameters Systems
▪ Continuous-Time and Discrete-Time Systems
▪ Analog and Digital Systems
▪ System Modeling
▪ Electrical Systems Model
▪ Mechanical System Model
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Contents
Dr. Shoaib R. Soomro, Fall 2019

▪ Systems are used to process signals to allow modification or extraction of additional information
from the signals.
▪ A system may consist of physical components (hardware realization) or of an algorithm that
computes the output signal from the input signal (software realization).
▪ A system can be conveniently illustrated by a "black box" with one set of accessible terminals
where the input variables x1(t), x2(t),...,xj(t) are applied and another set of accessible terminals
where the output variables y1(t), y2(t),..., yk(t) are observed.
▪ The study of systems consists of three major areas: mathematical modeling, analysis, and
design.
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Systems

▪ A system whose output is proportional to its input is an example of a linear system. But linearity
implies more than this; it also implies:
▪ Additivity property: that is, if several inputs are acting on a system, then the total effect on the
system due to all these inputs can be determined by considering one input at a time while assuming
all the other inputs to be zero. The total effect is then the sum of all the component effects.
▪ Homogeneity or Scaling property: States that for arbitrary real or imaginary number k,if an input is
increased k-fold, the effect also increases k-fold.
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Linear and Non-linear Systems

▪ Are the following systems linear?
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Linear and Non-linear Systems

▪ Systems whose parameters do not change with time are time-invariant (also constant-
parameter) systems.
▪ If the input is delayed by T seconds, the output is the same as before but delayed by T
(assuming initial conditions are also delayed by T).
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Time Variant and Time Invariant Systems

▪ A system is said to be instantaneous (or memoryless) if its output at any instant t depends, at
most, on the strength of its input(s) at the same instant t, and not on any past or future values of
the input(s).
▪ Otherwise, the system is said to be dynamic (or a system with memory).
▪ A system whose response at t is completely determined by the input signals over the past T
seconds is a finite-memory system with a memory of T seconds.
▪ Resistive circuits and amplifiers are instantaneous systems
▪ Inductive/Capacitive circuits and echo system are dynamic systems
▪ Instantaneous systems are a special case of dynamic systems.
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Instantaneous and Dynamic Systems

▪ A causal (also known as a physical or nonanticipative) system is one for which the output at any
instant to depends only on the value of the input x(t) for t ≤ to.
▪ In other words, the value of the output at the present instant depends only on the past and present
values of the input x(t), not on its future values.
▪ A system that violates the condition of causality is called a noncausal (or anticipative) system.
▪ Any practical system that operates in real time must necessarily be causal.
▪ We do not yet know how to build a system that can respond to future inputs (inputs not yet
applied).
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Causal and Noncausal Systems

▪ Why Study Noncausal Systems?
▪ The explanation may suggest that noncausal systems have no practical purpose.
▪ However, they are valuable in the study of systems for several reasons.
▪ Noncausal systems are realizable when the independent variable is other than "time"
(e.g.,space). Nontemporal systems, such as those occurring in optics, can be noncausal and
still realizable.
▪ Even for temporal systems, such as those used for signal processing, the study of noncausal
systems is important.
▪ In such systems we may have all input data prerecorded. (This often happens with speech
signals and video signals) In such cases, the input's future values are available to us.
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Causal and Noncausal Systems

▪ A lumped system is one in which the dependent variables of interest are a function of time
alone. In general, this will mean solving a set of ordinary differential equations (ODEs).
▪ A distributed system is one in which all dependent variables are functions of time and one or
more spatial variables. In this case, we will be solving partial differential equations (PDEs)
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Lumped and Distributed Parameters System

▪ Systems whose inputs and outputs are continuous-time signals are continuous-time systems.
▪ On the other hand, systems whose inputs and outputs are discrete-time signals are discrete-
time systems.
▪ A digital computer is a familiar example of this type of system. In practice, discrete-time signals
can arise from sampling continuous-time signals.
▪ A discrete-time signal may also be viewed as a sequence of numbers..., x[− 1], x[0], x[1], x[2],....
▪ Thus a discrete-time system may be seen as processing a sequence of numbers x[n] and
yielding as an output another sequence of numbers.
▪ Discrete-time signals arise naturally in situations that are inherently discrete time, such as
population studies, amortization problems, national income models, and radar tracking.
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Continuous-Time and Discrete-Time Systems

▪ A system whose input and output signals are analog is an analog system
▪ A system whose input and output signals are digital is a digital system.
▪ A digital computer is an example of a digital (binary) system (as well as a discrete-time system).
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Analog and Digital Systems

▪ Invertible and Noninvertible Systems
▪ If we can obtain the input x(t) back from the corresponding output y(t) by some operation, the system
is said to be invertible.
▪ If it is impossible to obtain the input from the output, and the system is noninvertible.
▪ Stable and Unstable Systems
▪ Stability can be internal or external. If every bounded input applied at the input terminal results in a
bounded output, the system is said to be stable externally.
▪ This type of stability is also known as the stability in the BIBO (bounded-input/bounded-output) sense.
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Other Systems Classifications

▪ A system description in terms of the measurements at the input and output terminals is called
the input-output description.
▪ Systems theory encompasses a variety of systems, such as electrical, mechanical, hydraulic,
acoustic, electromechanical, and chemical, as well as social, political, economic, and biological.
▪ The first step in analyzing any system is the construction of a system model, which is a
mathematical expression or a rule that satisfactorily approximates the dynamical behavior of
the system.
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System Model: Input-output Description

▪ In electrical systems, we must determine a satisfactory model for the voltage-current
relationship of each element, such as Ohm's law for a resistor.
▪ In addition, we must determine the various constraints on voltages and currents when several
electrical elements are interconnected.
▪ These are the laws of interconnection-the well-known Kirchhoff laws for voltage and current
(KVL and KCL).
▪ From all these equations, we eliminate unwanted variables to obtain equation(s) relating the
desired output variable(s) to the input(s).
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Electrical System Model

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Electrical System Model

▪ The basic elements used in modeling translational systems are ideal masses, linear springs,
and dashpots providing viscous damping.
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Mechanical System Model

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Mechanical System Model

▪ Signals and Systems, Symon Haykin, Chapter-1
▪ Signal Processing and Linear Systems, BP Lathi, Chapter-1
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