Finite Impulse Response Filter Design - Digital Signal Processing: Fundamentals and Applications - page 221

h ( n )

n

0

FIGURE 7.3
Impulse response of an ideal digital lowpass filter.

The obtained filter is a noncausal z-transfer function of the FIR filter, since the filter transfer function
contains terms with the positive powers of z, which in turn means that the filter output depends on the
future filter inputs. To remedy the noncausal z-transfer function, we delay the truncated impulse
response hðnÞ by M samples to yield the following causal FIR filter:

HðzÞ¼b 0 þ b 1 z 1 þ / þ b 2 M ð 2 MÞz 2 M

(7.11)

Table 7.1 Summary of Ideal Impulse Responses for Standard FIR Filters

Filter Type Ideal Impulse Response hðnÞ (noncausal FIR coefficients)

8
<
:

U c
p

for n ¼ 0

Lowpass: hðnÞ¼

0 M n M

sin ðU c nÞ

np

for n s

<
:

p U c
p

for n ¼ 0

Highpass: hðnÞ¼ M n M

sin ðU c nÞ

np

for n s 0

<
:

U H U L
p

for n ¼ 0

Bandpass: hðnÞ¼ M n M

sin ðU H nÞ

np

sin ðU L nÞ

np

for n s 0

8
<
:

p U H þ U L
p

for n ¼ 0

Bandstop: hðnÞ¼ M n M

sin ðU H nÞ

np

sin ðU L nÞ

np

þ for n s 0

Causal FIR filter coefficients: shifting hðnÞ to the right by M samples.
Transfer function:
HðzÞ¼b 0 þ b 1 z 1 þ b 2 z 2 þ / b 2 M z 2 M
where b n ¼ hðn MÞ,n ¼ 0 ; 1 ; / ; 2 M.