Digital Signal Processing Reference
In-Depth Information
Decimator
w
()
y
()
x
()
m-bit SDM ADC
oversampling
Anti-aliasing LPF
M
f
s
'
f
2
f
Oversampling rate
s
max
Noise shaping
filter frequency
response
In-band
quantization
noise without
SDM
j
1
e
In-band
quantization
noise
P
()
2
q
f
s
fH
()
f
s
2
f
s
2
f
max
f
max
Quantization
noise to be
shaped and
filtered with
SDM
FIGURE 12.31
Noise shaping of quantization noise for SDM ADC.
Using the Maclaurin series expansion and neglecting the higher-order terms due to the small value of
U
max
, we yield
!
2
1
þ
ðjUÞ
1
!
þ
ðjUÞ
1
e
jU
¼
1
þ
/
z
jU
2
!
Applying this approximation to Equation
(12.27)
leads to
U
max
2
q
2
p
jjUj
2
q
3
p
U
s
dU ¼
s
2
3
max
Shaped-in-band noise power
z
(12.28)
U
max
After simple algebra, we have
2
f
max
f
s
3
2
f
max
f
s
3
2
2
q
Shaped-in-band noise power
z
p
s
2
2
2
2
m
12
¼
p
3
$
A
(12.29)
3
If we let the shaped-in-band noise power equal the quantization noise power from the regular ADC
using a minimum sampling rate, we have
2
f
max
f
s
3
2
2
2
2
m
12
2
12
$
2
2
n
p
3
$
A
¼
A
(12.30)
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