Week 2 System Functionals Primitive Systems Example Addition Cascade Feedback Sometimes we want to speak of the relationship between x and y without a particular x and y in mind We can do this by de ning a function x x 6 We can do the same thing with LTI systems by de ning system functionals We will refer to Y X as the system functional It characterises the operation of a system independent of the particular input and output signals involved It is functional in the sense that it is itself an operation which can be applied to arbitrary input signals We can specify system functions for each of our system primitives A gain element is governed by operator equation Y kX for constant k so its system functional is A delay element is governed by operator equation Y RX so its system functional is Combining System Functionals We have three basic composition operations sum cascade and feedback This PCAP system as our previous ones have been is compositional in the sense that whenever we make a new system function out of existing ones it is a system functional on its on right which can be an element in further compositions The system functional of the sum of two systems is the sum of their systems The system functional of the cascade of two systems is the product of their system functionals There are several ways of connection systems in feedback We study a particular case of negative feedback combination which results in a classical formula called Black s formula Predicting System Behavior We will see how we can use properties of the system function to predict how the system will behave in the long term and for any input We can provide a general characterization of the long term behavior of the output as increasing or decreasing with constant or alternating sign for any nite input to the system We will begin by studying the unit sample response of systems and then generalize to more general inputs First Order Systems Systems that only have forward connection can only have a nite response The output signal will only have a nite number on non zero samples Systems with feedback have surprisingly different characters Finite inputs can result in persistent response meaning they can result in output signals with in nitely many non zero samples The qualitative long term behavior of this output is generally independent of the particular input given to the system for any nite input We will consider the class of rst order systems in which the denominator of the system function is a rst order polynomial First order polynomial means it only involves R but not R or other higher powers of R Example Y X pRY Y 1 pR X Y X 1 pR Deriving a system functional we get H Y X 1 1 pR System responses can be characterized by a single number called the pole which is the base of the geometric sequence The value of the pole p determines the nature and rate of growth If p 1 the magnitude increases to in nite and the sign alternates If 1 p 0 the magnitude decreases and the sign alternates If 0 p 1 the magnitude decreases monotonically If p 0 the magnitude increases monotonically to in nity Any system for which the output increases to in nity whether monotonically or not is called unstable Systems whose output magnitude decreases or stays constant are called stable Second Order Systems Additive Decomposition Complex Poles We will call these persistent long term behaviors of a signal modes For a xed p the rst order system only exhibited one mode but different values of p resulted in very different modes Second order systems are characterized by a system functional whose denominator polynomial is second order We will nd that it is the mode whose pole has the largest magnitude that governs the long term behavior of the system They will generally exhibit two modes We will call this pole the dominant pole Another way to try to decompose the system is as the sum of two simpler systems In this case we seek H1 and H2 such that H1 H2 H Difference equations that represent physical systems have real valued coef cents but they might still have complex poles Polar Representation of Complex Numbers Sometimes it is easier to think about a complex number as a bj where a rcos and b rsin so that the magnitude r sometimes written as a bj is de ned as r a b and that the angle is de ned as tan b a
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