6 081 Spring Semester 2007 Lecture 6 Notes 1 MASSACHVSETTS INSTITVTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science 6 081 Introduction to EECS I Spring Semester 2007 Lecture 6 Notes Linear Systems and Z Transforms Zee secret is in zee transforms Difference Equations with Input So far we ve used difference equations to model the behavior of systems whose values at some time depend only on their own values at some previous time points But it is also important to consider systems that depend on an input value as well Let s get the idea by considering a very simple example to start Example 1 Let s think about a simple first order system with a constant input We can think for instance of a bank account like Zelda s or Oswald s into which a constant payment c is deposited each year We would model that system using the difference equation y n y n 1 c Because of the input c it is called a non homogenous difference equation Because it is so simple we can see what s going on just by expanding it out y n y n 1 c y n 2 c c y n 3 c c c n y 0 c n X i i 0 n y 0 c 1 n 1 1 That last step is the standard formula for the sum of a geometric series What will happen to this bank account as n goes to infinity It s clear that if 1 then the first term will go to positive or negative infinity and we needn t bother thinking about the second term However if 1 then as n goes to infinity the whole expression goes to c 1 So for example if Uncle Oswald lived forever with a 100 year being deposited into his account which as you may recall had a 5 management fee the steady state value of the account would be 100 1 0 95 2000 6 081 Spring Semester 2007 Lecture 6 Notes 2 More generally a linear difference equation with input can be described in the form M X ak y n k N X bl x n l 1 l 0 k 0 We can think about it as a process by which a sequence x n is transformed into a new sequence y n If x n 0 for all n then this is one of our old familiar homogeneous without input difference equations from last time but written slightly differently To convert back into that form we d have to say M X am i y n y n 1 am i 1 For the study of the behavior of more complex systems we ll find it algebraically easier to write as in equation 1 Even in more complex problems it will still be the case that the solution to the difference equation has two parts one of which depends on the initial conditions and one of which depends on the input The details of how to derive a complete closed form solution are cool but more detail than we want to get into in this course We are going to continue to concentrate on the qualitative behavior of the system in particular understanding whether or not it will be stable Definition 1 A system is bounded input bounded output BIBO stable if whenever x n is bounded for all n then y n is also bounded for all n If the natural frequencies roots of the characteristic polynomial i associated with the difference equation are all such that i 1 then the associated system is BIBO stable So our first step in understanding the behavior of a system with or without input is to understand its natural frequencies In the format of equation 1 the characteristic polynomial is M X ai i 0 i 0 Abstraction and modularity We ve introduced two kinds of objects in our informal discussion above sequences and transformations on sequences As we build complex control or signal processing systems or wish to analyze Aunt Zelda s secret financial empire of linked companies investments and bank accounts we need to develop a system of modularity and abstraction so we can put small pieces together into a clearly understood and analyzable system We will restrict our attention to a limited but powerful class of sequences which are defined by linear difference equations with input We can define a set of primitive operations on those sequences which guarantee that if the input sequences are describable in terms of linear difference equations then so are the results Addition y n x1 n x2 n Scaling y n kx n Shifting back y n x n 1 6 081 Spring Semester 2007 Lecture 6 Notes 3 Shifting forward y n x n 1 We will call operations on sequences system functions If these were the only operations we could do we wouldn t be able to get very far In fact we can take these primitive operations and combine them using two primary methods of combination Cascade Feeding the output of one system function into the input of another Parallel sum Feeding the input into two different system functions adding the results and letting that be the output See pictures in lecture slides In the next sections we ll be able to define this all much more formally The important thing here is that when we combine system functions we get another system function and that system function has the property that the relationship between the input sequence and the output sequence is describable by a linear difference equation Z Transforms In the Coyote and Roadrunner example from the last lab we had to play with two coupled linear difference equations We made it all work out but it was a lot of algebra We could think of that as a cascade of two systems with the output of one the coyote population being treated as input to the other the roadrunner population and vice versa As we want to build ever more complex systems the algebra will get even more complicated and not be any fun Remember how we made multiplication of complex numbers a lot easier by changing to the complex exponential representation It turns out that we can make operations on signals and system functions a lot easier by changing their representation using something called the z transform The z transform representation of a linear system is no weaker or stronger than the difference equation representation each linear difference equation has exactly one representation as a z transform and vice versa It s easier to calculate values of the system using the difference equation representation and easier to combine sequences and operate on them using the z transform representation So Here we go The official definition Definition 2 Let x n be the coefficients of a power series in a variable called z The bilateral Z transform of x n is the function X z X x n z n n What is this about If we picked a particular z then this would just be a number which summed up the power …
View Full Document
Unlocking...