Suppose there are several quantization points in an IIR filter
structure. By our simplifying assumptions about quantization error
and Parseval's theorem, the quantization noise variance
σ
y,i
2
σ
y,i
2
at the output of the filter from the iith
quantizer is

σ
y,i
2=12π∫−ππ|
H
i
w|2S
S
n
i
wd
w
=
σ
n
i
22π∫−ππ|
H
i
w|2d
w
=
σ
n
i
2∑
n
=−∞∞
h
i
2n
σ
y,i
2
1
2
w
H
i
w
2
S
S
n
i
w
σ
n
i
2
2
w
H
i
w
2
σ
n
i
2
n
h
i
n
2

(1)
where

σ
n
i
2
σ
n
i
2
is the variance of the quantization error at the

iith quantizer,

S
S
n
i
w
S
S
n
i
w
is the power spectral density of that quantization error, and

H
H
i
w
H
H
i
w
is the transfer function from the

iith
quantizer to the output point.
Thus for

PP independent quantizers in the
structure, the total quantization noise variance is

σ
y
2=12π∑
i
=1P
σ
n
i
2∫−ππ|
H
i
w|2d
w
σ
y
2
1
2
i
P
1
σ
n
i
2
w
H
i
w
2
Note that in general, each

H
i
w
H
i
w
, and thus the variance at the output due to each quantizer,
is different; for example, the system as seen by a quantizer at the
input to the first delay state in the Direct-Form II IIR filter
structure to the output, call it

n
4
n
4
, is

with a transfer function

H
4
z=z-21+
a
1
z-1+
a
2
z-2
H
4
z
z
-2
1
a
1
z
a
2
z
-2
which can be evaluated at

z=ejw
z
w
to obtain the frequency response.

A general approach to find
H
i
w
H
i
w
is to write state equations for the equivalent
structure as seen by
n
i
n
i
, and to determine the transfer function according to
Hz=CzI−A-1B+d
H
z
C
z
I
A
B
d
.

The above figure illustrates the quantization points in a
typical implementation of a Direct-Form II IIR
second-order section.
What is the total variance of the output error due to all of the
quantizers in the system?

By making the assumption that each
Q
i
Q
i
represents a noise source that is white,
independent of the other sources, and additive,

the variance at the output is the sum of the variances at
the output due to each noise source:

σ
y
2=∑
i
=14
σ
y
,
i
2
σ
y
2
i
1
4
σ
y
,
i
2
The variance due to each noise source at the output
can be determined from

12π∫−ππ|
H
i
w|2
S
n
i
wd
w
1
2
w
H
i
w
2
S
n
i
w
; note that

S
n
i
w=
σ
n
i
2
S
n
i
w
σ
n
i
2
by our assumptions, and

H
i
w
H
i
w
is the transfer function

*from the noise source to
the output*.