PRACTICAL MATLAB® FOR ENGINEERS PRACTICAL MATLAB

Analog and Digital Filters 589

R.6.112 Recall that an IIR fi lter can be defi ned in the time domain by a difference equation.

For example, let the system differential equation be given by

y(n) − 0.02y(n − 1 ) + 0.009y(n − 2 ) = 0.1x(n) + 0.2x(n − 1 ) + 0.3x(n − 2 )

where x represents the input sequence and y is its output sequence.

Observe that the output sequence y(n) is a function of the input sequence x(n) as

well as the previous (delayed) output sequences.

R.6.113 An FIR fi lter can be defi ned by a difference equation with a fi nite input sequence x.

For example, let the system differential equation be given by

y(n) = 0.036x(n) − 0.02x(n − 1 ) + x(n − 2 )

Observe that the output sequence y(n) is a function of its input sequence x(n)

(scaled and delayed).

R.6.114 Some practical IIR design considerations and observations are summarized as

follows:

a. There is a ripple problem to consider.

b. There is no guarantee that the design leads to a stable implementation.

c. There is a stability problem to consider.

d. Differences may exist among the different types of IIR fi lters.

e. When the magnitude response is the main concern IIR is generally a good

choice.

f. IIR fi lters require 5–10 times less coeffi cients than the equivalent FIR

implementation.

R.6.115 An FIR fi lter is a fi lter that has a fi nite response for n < N and a linear phase, where

N is the length of the fi lter response. Such fi lters generally require N constant

multipliers, N − 1 two port input adders and N stages of delays resulting in shift

register–type structure.

R.6.116 All FIR fi lters present linear phase and an impulse response with some sort of

symmetry that satisfi es the general relation given by

h(n) = ±h(N − n)

This sequence (h(n)) presents an even symmetry sequence for the positive case

and an odd symmetry for the negative case, assuming the sequence h(n) consists

of real coeffi cients.

R.6.117 The symmetry condition for the impulse response is a necessary condition for the

linear phase.

R.6.118 The symmetry condition depends on the length of h(n) and whether N is odd or

even.

R.6.119 Four types of symmetry conditions for FIR fi lters with real coeffi cients and

lengths N result. They are summarized as follows:

a. Ty p e 1. Symmetric impulse response with N = odd.

b. Ty p e 2. Symmetric impulse response with N = even.

c. Ty p e 3. Asymmetric impulse response with N = odd.

d. Ty p e 4. Asymmetric impulse response with N = even.