Digital Signal Processing Reference
In-Depth Information
X(z)
Y(z)
+
+
1/(1-
z
-1
)
+
z
-1
/(1-
z
-1
)
Q
-
-
Fig. 9
Block diagram of a second-order sigma-delta modulator
We can see that the input passes through the modulator delayed by a unit delay, and
the noise is filtered by a differentiator 1
z
−
1
. The differentiator has a zero at the
zero frequency (DC, direct current) and the response rises towards high frequencies.
The noise spectrum is therefore shaped by this transfer function. SNR again depends
on the oversampling ratio and is given by
−
=
.
−
.
+
(
)
.
SNR
SD
1
6
02
b
3
41
30 log
OSR
dB
(22)
Doubling the OSR improves SNR by 9 dB, corresponding to 1.5 bits.
3.2.3
Higher-Order Sigma-Delta Modulators
It includes two integrators, one of which is nondelaying (1
z
−
1
/
(
1
−
)
) and the other
one delaying (
z
−
1
z
−
1
/
(
1
−
)
). The output can be expressed as
z
−
1
X
z
−
1
2
N
Y
(
z
)=
(
z
)+(
1
−
)
(
z
)
.
(23)
The noise is therefore filtered by a double differentiator. The SNR is given by
SNR
SD
2
=
6
.
02
b
−
11
.
14
+
50 log
(
OSR
)
dB
.
(24)
Doubling the OSR improves SNR by 15 dB, corresponding to 2.5 bits.
The topology shown in Fig.
9
can be generalized to create an
M
th order sigma-
delta modulator, for which the output can be expressed as
z
−
1
X
z
−
1
M
N
Y
(
z
)=
(
z
)+(
1
−
)
(
z
)
.
(25)
Thus
M
times differentiation of the noise is achieved. An
M
th order sigma-delta
modulator improves the SNR by 6
M
+
3 dB for each doubling of the OSR, corre-
sponding to
M
5 bits. However, such higher-order modulators are problematic
from the stability point of view and alternative topologies have been developed, such
as multiple feedback, feedforward and cascaded structures.
A common approach is to construct a higher-order modulator as a cascade
of first- or second-order modulators. Such topologies are called multistage noise
+
0
.
