6 01 Fall Semester 2007 Lecture Notes 1 MASSACHVSETTS INSTITVTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science 6 01 Introduction to EECS I Fall Semester 2007 Lecture Notes Difference Equations 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 01 Fall Semester 2007 Lecture Notes 2 More generally a linear difference equation with input can be described in the form K X ak y n k k 0 L X bl x n l 1 l 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 K X aK 1 y n k y n ak k 1 For the study of the behavior of more complex systems we ll find it algebraically easier to write difference equations in the form of equation 1 For difference equations with inputs natural frequencies play an important role and can be used to compute solutions The general solution to a linear difference equation has two parts one of which depends only 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 systems described by difference equations in particular understanding whether or not a given system will be stable in the sense made precise by the definition below Definition 1 A system is bounded input bounded output BIBO stable if x n being bounded for all n necessarily implies that y n will also be bounded for all n For linear difference equations 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 determine the magnitude of the system s natural frequencies In the format of equation 1 the natural frequencies are the roots of the characteristic polynomial K X ak k 0 k 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 thoses which are solutions to difference equations with input sequences which are bounded We can start by defining a set of primitive operations on sequences ones which guarantee that there is a difference equation that relates the given input usually denoted as x n to the final output usually denoted y n These primative operations are 6 01 Fall Semester 2007 Lecture Notes 3 Addition y n x1 n x2 n Scaling y n kx n Shifting back y n x n 1 Shifting forward y n x n 1 Note that the general difference equation in 1 can be generated by a combination of scaling shifting and adding Regardless of whether we are referring to one of the primative operations or to a general difference equation we think of a system as taking an input sequence and producing an output sequence When representing more complicated systems two primary methods of combination for systems are quite helpful Cascade Using the output sequence of one system as the the input to another system Parallel sum Summing the sequences generated by two different systems to generate an output In the next sections we ll be able to define this all much more formally The important thing here is that when we combine systems we get another system and that system 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 sequences a lot easier represent by changing the sequence representation using something called the z transform also known as generating functions in much of the computer science literature The z transform representation of a sequence is no weaker or stronger than the sequence representation a sequence has has exactly one representation as a z transform and every power series representation of a z transform corresponds to exactly one sequence It s easier to
View Full Document
Unlocking...