Discrete and Digital Signals and Systems - Digital Signal Processing: An Introduction with MATLAB and Applications - page 124

This transfer function can be implemented as shown in Fig ( 2.40 ). If r is very close
to 1, then the 3-dB (half-power) bandwidth B f and the maximum gain G m of this
system are approximated, respectively, by:

B f 1 r
p

1
ð 1 r 2 Þ sin ð X o Þ

f s (Hz) and G m

Equivalently, the bandwidth can be written as:
B x = 2p.B f = 2(1 - r)f s (rad/s), (where x = 2pf, radian frequency), B m ¼
B f = f s ¼ 1 p ; (where m = f/f s , normalized frequency), and B X ¼ B x = f s ¼ 2 ð 1 r Þ;
(where X ¼ x = f s ¼ 2pf = f s ¼ 2pm ; normalized frequency).

2.6.9.5 A Digital DC Blocker

In some applications an undesirable DC voltage can appear in the information
signal. For example, in some audio applications a DC offset can be added to the
recoded sound from the microphone. The ADC may also add some unwanted DC
to the digitized signal. This DC component carries no information, and in audio
applications cannot even be heard. Nonetheless, in some cases it can hinder pro-
cessing and cause instabilities. For example, the DC component may drive the
signal outside the dynamic range of the processing system which will cause signal
clipping. Hence, DC removal is often desirable before other forms of processing
are pursued.
For DC removal, it is necessary for the magnitude response to be zero at
f = 0 Hz and unity elsewhere. This kind of frequency response can only be
approximated in practice. To block DC one can place a zero at z = 1, and to
ensure that there is approximately unity gain for non-zero frequencies, one addi-
tionally needs a pole very near to the zero at z = 1, and inside the unit circle. The
following transfer function has the necessary form:

H ð z Þ¼ 1 z 1

1 az 1 ;

a close to 1 :

ð 2 : 42 Þ

Fig. 2.41 A digital DC

x ( n ) y ( n )

Blocker

z −1 z −1

α

−1

Direct Form I

x ( n ) y ( n )

z −1

α

− 1

Direct Form II