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Difference Equations

Difference Equations. 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 ]. .

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Difference Equations

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  1. Difference Equations

  2. Linear Difference Equations • Discrete-timeLTI systemscan sometimes becharacterizedby differenceequations 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]  y[n] + UnitDelay + 1/2 y[n-1] UnitDelay 1/8 y[n-2]

  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. 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. 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. 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 Nth order difference equation

  7. X(z) H(z) Y(z) BIBO Stability • Given H(z) and X(z), computeoutput 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

  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] = d[n] Z{h[n]} = H(z) Z{d[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

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