1. EE313 Linear Systems and Signals Fall 2010 Initial conversion of content to PowerPoint by Dr. Wade C. Schwartzkopf Prof. Brian L. Evans Dept. of Electrical and Computer Engineering The University of Texas at Austin Difference Equations

2. 22 - 2 Linear Difference Equations Discrete-time LTI systems can sometimes be characterized by difference equations y[n] = (1/2) y[n-1] + (1/8) y[n-2] + x[n] Taking z-transform of difference equation gives description of system in z-domain  x[n]x[n]y[n]y[n] Unit Delay 1/2 1/8 + + + y[n-1] y[n-2]

3. 22 - 3 Advances and Delays Sometimes differential equations will be presented as unit advances rather than delays y[n+2] – 5 y[n+1] + 6 y[n] = 3 x[n+1] + 5 x[n] One can make a substitution that reindexes the equation so that it is in terms of delays Substitute n with n-2 to yield y[n] – 5 y[n-1] + 6 y[n-2] = 3 x[n-1] + 5 x[n-2] Before taking the z-transform, recognize that we work with time n  0 so u[n] is often implied y[n-1] → y[n-1] u[n]  y[n-1] u[n-1]

4. 22 - 4 Example System described by a difference equation y[n] – 5 y[n-1] + 6 y[n-2] = 3 x[n-1] + 5 x[n-2] y[-1] = 11/6, y[-2] = 37/36 x[n] = 2 -n u[n]

5. 22 - 5 Transfer Functions Previous example describes output in time domain for specific input and initial conditions It is not a general solution, which motivates us to look at system transfer functions. In order to derive the transfer function, one must separate “Zero state” response of the system to a given input with zero initial conditions “Zero input” response to initial conditions only

6. 22 - 6 Transfer Functions Consider zero-state response LTI properties → all initial conditions are zero Causality → initial conditions are with respect to index 0 LTI + causality → y[-n] = 0 and x[-n] = 0 for all n > 0 Write general N th order difference equation

7. 22 - 7 BIBO Stability Given H(z) and X(z), compute output Y(z) = H(z) X(z) Product is only valid for values of z in region of convergence for H(z) and region of convergence for X(z) Since H(z) is ratio of two polynomials, roots of denominator polynomial (called poles) control where H(z) may blow up H(z) can be represented as a series Series converges when poles lie inside (not on) unit circle Corresponds to magnitudes of all poles being less than 1 System is said to be bounded-input bounded-output stable H(z)H(z) Y(z)Y(z)X(z)X(z)

8. 22 - 8 Relation between h[n] and H(z) Either can be used to describe an LTI system Having one is equivalent to having the other since they are a z-transform pair By definition, impulse response, h[n], is y[n] = h[n] when f[n] =  [n] Z{h[n]} = H(z) Z{  [n]}  H(z) = H(z) · 1 h[n]  H(z) Since discrete-time signals can be built up from unit impulses, knowing the impulse response completely characterizes the LTI system