Difference Equations

1
Difference Equations
2
Linear Difference Equations

• Discrete-timeLTI systemscan becharacterizedby
  differenceequations
• yn (1/2) yn-1 (1/8) yn-2 fn
• Taking z-transform of difference equation gives
  description of system in z-domain

?
fn
yn

UnitDelay

1/2
yn-1
UnitDelay
1/8
yn-2
3
Advances and Delays

• Sometimes differential equations will be
  presented as unit advances rather than delays
• yn2 5 yn1 6 yn 3 fn1 5 fn
• One can make a substitution that reindexes the
  equation so that it is in terms of delays
• Substitute n with n-2 to yield
• yn 5 yn-1 6 yn-2 3 fn-1 5 fn-2
• Before taking the z-transform, recognize that we
  work with time n ? 0 so un is often implied
• yn-1 ? yn-1 un ? yn-1 un-1

4
Example

• System described by a difference equation
• yn 5 yn-1 6 yn-2 3 fn-1 5 fn-2
• y-1 11/6, y-2 37/36
• fn 2-n un

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 f-n 0 for all
  n gt 0
• Write general Nth order difference equation

7
BIBO Stability

• Given Hz and Fz, computeoutput Yz Hz
  Fz
• Product is only valid for values of z in region
  of convergence for Hz and regions of
  convergence for Fz
• Since Hz is ratio of two polynomials, roots of
  denominator polynomial (called poles) control
  where Hz may blow up
• Hz 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 stable

8
Relation between hn and Hz

• 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, hn, is
• yn hn when fn dn
• Zhn Hz Zdn ? Hz Hz 1
• hn ? Hz
• Since discrete-time signals can be built up from
  unit impulses, knowing the impulse response
  completely characterizes the LTI system