New version page

MIT 6 01 - Linear Systems and Z Transforms

Documents in this Course
Op-Amps

Op-Amps

8 pages

Op-Amps

Op-Amps

6 pages

Syllabus

Syllabus

14 pages

Planning

Planning

14 pages

Load more
Upgrade to remove ads
Upgrade to remove ads
Unformatted text preview:

6.081, Spring Semester, 2007—Lecture 6 Notes 1MASSACHVSETTS INSTITVTE OF TECHNOLOGYDepartment of Electrical Engineering and Computer Science6.081—Introduction to EECS ISpring Semester, 2007Lecture 6 NotesLinear Systems and Z TransformsZee secret is in zee transforms.Difference Equations with InputSo far, we’ve used difference equations to model the behavior of systems whose values at some timedepend only on their own values at some previous time points. But it is also important to considersystems 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, forinstance, of a bank account, like Zelda’s or Oswald’s, into which a constant payment c is depositedeach year.We would model that system using the difference equationy[n] = αy[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] = αy[n − 1] + c= α(αy[n − 2] + c) + c= α(α(αy[n − 3] + c) + c) + c. . .= αny[0] + cnXi=0αi= αny[0] + c1 − αn+11 − α.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 |α| > 1 then thefirst term will go to positive or negative infinity, and we needn’t bother thinking about the secondterm. However, if |α| < 1, then as n goes to infinity, the whole expression goes to c/(1 − α).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 be100/(1 − 0.95) = 2000.6.081, Spring Semester, 2007—Lecture 6 Notes 2More generally, a linear difference equation with input can be described in the formMXk=0aky[n + k] =NXl=0blx[n + l] . (1)We can think about it as a process by which a sequence x[n] is transformed into a new sequencey[n]. If x[n] = 0 for all n, then this is one of our old familiar homogeneous (without input) differenceequations from last time, but written slightly differently. To convert back into that form, we’d haveto sayy[n] = −MXi=1am−iamy[n − 1] .For the study of the behavior of more complex systems, we’ll find it algebraically easier to write asin equation 1.Even in more complex problems, it will still be the case that the solution to the difference equationhas two parts, one of which depends on the initial conditions, and one of which depends on theinput. The details of how to derive a complete closed-form solution are cool, but more detail thanwe want to get into in this course. We are going to continue to concentrate on the qualitativebehavior 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] isbounded, for all n, then y[n] is also bounded for all n.If the natural frequencies (roots of the characteristic polynomial) λiassociated with the differenceequation are all such that |λi| < 1, then the associated system is BIBO-stable. So, our first stepin understanding the behavior of a system, with or without input, is to understand its naturalfrequencies. In the format of equation 1, the characteristic polynomial isMXi=0aiλi= 0 .Abstraction and modularityWe’ve introduced two kinds of objects in our informal discussion above: sequences and transforma-tions on sequences. As we build complex control or signal-processing systems, or wish to analyzeAunt Zelda’s secret financial empire of linked companies, investments, and bank accounts, we needto develop a system of modularity and abstraction so we can put small pieces together into a clearlyunderstood and analyzable system.We will restrict our attention to a limited but powerful class of sequences, which are defined by lineardifference 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 cantake 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, andletting that be the output(See pictures in lecture slides.) In the next sections, we’ll be able to define this all much moreformally.The important thing here is that when we combine system functions, we get another system func-tion, and that system function has the property that the relationship between the input sequenceand the output sequence is describable by a linear difference equation.Z TransformsIn the Coyote and Roadrunner example from the last lab, we had to play with two coupled lineardifference equations. We made it all work out, but it was a lot of algebra. We could think of thatas a cascade of two systems, with the output of one (the coyote population) being treated as inputto the other (the roadrunner population) and vice versa. As we want to build ever more complexsystems, 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 complexexponential representation? It turns out that we can make operations on signals and systemfunctions a lot easier by changing their representation, using something called the z transform. Thez-transform representation of a linear system is no weaker or stronger than the difference equationrepresentation: each linear difference equation has exactly one representation as a z transform, andvice 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 bilateralZ-transform of x[n] is


View Full Document
Download Linear Systems and Z Transforms
Our administrator received your request to download this document. We will send you the file to your email shortly.
Loading Unlocking...
Login

Join to view Linear Systems and Z Transforms and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view Linear Systems and Z Transforms 2 2 and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?