RFX144V24-S23 and RFX96V24-S23 Modem Designer’s Guide
6-2
1070
Filter 1 has a transfer function:
H
1
(Z) = 2(
α
0
+
α
1
Z
-1
+
α
2
Z
-2
)/(1 - 2
β
1
Z
-1
- 2
β
2
Z
-2
)
(Eq. 1)
Filter 2 has a transfer function:
H
2
(Z) = 2(
α
’
0
+
α
’
1
Z
-1
+
α
’
2
Z
-2
)/(1 - 2
β
’
1
Z
-1
- 2
β
’
2
Z
-2
)
(Eq. 2)
The energy averaging filter has a transfer function:
H
3
(Z) =
α
"/(1 -
β
”Z
-1
)
(Eq. 3)
The output of the energy averager is fed to a threshold comparator that sets interface memory bit FR1, FR2, or FR3 to a 1 if
the output is equal to or greater than the Tone Detector threshold (default value = 1/8 of full scale), otherwise, the bits are
set to a 0.
Filters 1 and 2 have frequency response as shown in Figure 6-2. When cascaded, they form a bandpass filter with a narrow
bandwidth as shown in Figure 6-3.
Given the transfer functions H
1
(Z) and H
2
(Z), an analytical method is provided to compute their coefficients for any frequency
in the 400 Hz - 3 kHz band. First, consider H1(Z). This transfer function can be rewritten as:
H
1
(Z) = 2(
α
0
Z
2
+
α
1
Z +
α
2
)/(Z
2
- 2
β
1
Z
−
2
β
2
)
(Eq. 4)
which has a conjugate pair of poles:
P
1
=
β
1
+
√
(
β
1
2 + 2
β
2
)
and
P
2
=
β
1
-
√
(
β
1
2 + 2
β
2
)
Upon power up, these poles lie on a circle of radius 0.994030884 on the Z-plane. The radius of the tone detector circle was
chosen so that each filter has a high Q without being unstable (i.e., poles must lie inside the unit circle for stability). Figure 6-
4 shows a Z-plane pole-zero diagram for an arbitrary conjugate pole pair on the tone detector circle. The angle
θ
= 360
ƒO/ƒS, where ƒO is the desired center frequency and ƒS is the sampling rate.
The following equations are derived from the angle and magnitude of the position vector pointing to a pole pair located at the
desired angle:
cos
-1
(
β
1
/r) =
θ
= 360° x ƒ
O
/ƒ
S
(Eq. 5)
√
[
β
1
2 + (-
β
1
2 - 2
β
2
)]= r = 0.994030884
(Eq. 6)
Solving for
β
1 and
β
2:
β
1
= r cos (360° x ƒ
O
/ƒ
S
)
(Eq. 7)
β
2
= - r
2
/2
(Eq. 8)
In deriving these equations, only H1(Z) was considered. However, the tone detector consists of two identical filters in
cascade. Referring to Figure 6-3, shifting filter 1 and filter 2 above and below the desired center frequency, a response with
the desired bandwidth is achieved. Furthermore, since the
α
1,
α
'1,
α
2, and
α
'2 coefficients default to zero upon power-up,
α
0 controls the amplitude response, and one may set
α
0 =
α
'0 to uniformly raise or lower the overall cascade response.
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