Biosignal Processing - Introduction to Biomedical Engineering - page 713

EXAMPLE PROBLEM 11.23
Find the impulse response for the digital filter

1
3 x ð k Þþ

1
3 x ð k

1
3 x ð k

y ð k Þ¼ 1 Þþ 2 Þ

Solution
Assume the system is at rest before input begins—that is, y ( n ) ¼ 0 for n < 0.

1
3 d

1
3 d

1
3 d

ð 2 Þþ ð 3 Þþ ð 4 Þ¼ 0 þ 0 þ 0 ¼ 0

y ð 2 Þ¼

1
3 dð

1
3 dð

1
3 dð

y ð 1 Þ¼ 1 Þþ 2 Þþ 3 Þ¼ 0 þ 0 þ 0 ¼ 0

1
3 dð

1
3 dð

1
3 dð

1
3 þ

1
3

y ð 0 Þ¼ 0 Þþ 1 Þþ 2 Þ¼ 0 þ 0 ¼

1
3 dð

1
3 dð

1
3 dð

1
3 þ

1
3

y ð 1 Þ¼ 1 Þþ 0 Þþ 1 Þ¼ 0 þ 0 ¼

1
3 d

1
3 d

1
3 d

1
3 ¼

1
3

y ð 2 Þ¼ ð 2 Þþ ð 1 Þþ ð 0 Þ¼ 0 þ 0 þ

1
3 dð

1
3 dð

1
3 dð

y ð 3 Þ¼ 3 Þþ 2 Þþ 1 Þ¼ 0 þ 0 þ 0 ¼ 0

1
3 dð

1
3 dð

1
3 dð

y ð 4 Þ¼ 4 Þþ 3 Þþ 2 Þ¼ 0 þ 0 þ 0 ¼ 0

...

y ð k Þ¼ 0; k 3

This is an FIR filter with only three nonzero coefficients.

IIR filters are particularly useful for simulating analog systems. The main advantage
of an IIR filter is that the desired job can usually be accomplished with fewer filter
coefficients than for an FIR filter; in other words, IIR filters tend to be more efficient. The
main disadvantage of an IIR filter is that signals may be distorted in an undesirable way.
FIR filters can be designed with symmetry to prevent undesired signal distortion. Methods
for dealing with the distortion problem in FIR filters are outside our discussions here.
Digital filters, as the name implies, are most often designed to perform specific
“filtering” operations: low-pass filters, high-pass filters, band-pass filters, bandstop
filters, notch filters, and so on. However, digital filters can be used to simulate most
analog systems—for example, to differentiate and to integrate. Many textbooks have
been written on digital filter design. The key components of the process are illustrated
following.