Digital Signal Processing Reference
In-Depth Information
At the central office, the received DTMF tones are detected through the digital filters and some logic
operations are used to decode the dialed signal consisting of 852 Hz and 1,209 Hz as key “7”. The
frequencies defined for each key are in
Figure 8.48
.
8.11.1
Single-Tone Generator
Now, let us look at a digital tone generator whose transfer function is obtained from the z-transform
function of a sinusoidal sequence sin
ðn
U
0
Þ
as
z
1
sin
U
0
1
2
z
1
cos
U
0
þ z
2
z
sin
U
0
HðzÞ¼
2
z
cos
U
0
þ
1
¼
(8.65)
2
z
where
U
0
is the normalized digital frequency. Given the sampling rate of the DSP system and the
frequency of the tone to be generated, we have the relationship
U
0
¼
2
pf
0
=f
s
(8.66)
Applying the inverse z-transform to the transfer function leads to the difference equation
yðnÞ¼
sin
U
0
xðn
1
Þþ
2 cos
U
0
yðn
1
Þyðn
2
Þ
(8.67)
since
ðHðzÞÞ ¼ Z
1
z
sin
U
0
Z
1
¼
sin
ð
U
0
nÞ¼
sin
ð
2
pf
0
n=f
s
Þ
2
z
2
z
cos
U
0
þ
1
which is the impulse response. Hence, to generate a pure tone with an amplitude of
A
, an impulse
function
xðnÞ¼AdðnÞ
must be used as the input to the digital filter, as illustrated in
Figure 8.49
.
Now, we illustrate implementation. Assuming that the sampling rate of the DSP system is 8,000
Hz, we need to generate a digital tone of 1 kHz. Then we compute
U
0
¼
2
p
1
;
000
=
8
;
000
¼ p=
4
;
sin
U
0
¼
0
:
707107
;
and
2 cos
U
0
¼
1
:
414214
The required filter transfer function is determined as
0
:
707107
z
1
1
1
:
414214
z
1
HðzÞ¼
þ z
2
is the generated tone of 1 kHz, and the bottom plot shows its spectrum. The corresponding MATLAB
code is in Program 8.18.
xn A n
()
()
z
z
1
sin
cos
Tone
yn A
Hz
()
0
1
2
12
z
()
sin(
2
fn f un
s
/ )()
0
0
FIGURE 8.49
Single-tone generator.
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