PXR40RM Datasheet, PDF (1160/1434 Pages) Freescale Semiconductor, Inc – PXR40 Microcontroller

Decimation Filter
x(n)
B₀
+
+
y(n)
-1
Z
x(n-1)
B₁
+
-1
Z
x(n-2)
B₂
+
-1
Z
x(n-3)
B₃
+
-1
Z
x(n-4)
B₄
-1
Z
+
A₁
y(n-1)
-1
Z
+
A₂
y(n-2)
-1
Z
+
A₃
y(n-3)
-1
Z
A₄
y(n-4)
Figure 28-22. 1 x 4 Poles IIR Filter Functional Diagram
The generalized difference equation for the IIR filter of Figure 28-22 can be written as:
N
M
  ^{yn = Bixn – i + Ajyn – j}
i=0
j=1
Eqn. 28-1
where x(n) is the filter input at time n, y(n) is the filter output at time n, N is the number of feed-forward
filter coefficients minus one, B_i are the feed-forward filter coefficients, M is the number of feed-back filter
coefficients, and A_j are the feedback filter coefficients.
In order to optimize the hardware implementation, the coefficients must be scaled to a maximum range of
+1 to -1. Taking scaling into account, Equation 28-1 can be expressed as:
Eqn. 28-2
^N
M

  ^{yn}
=

S_

B_{----ixn – i +}
S
-^A----^j y  n
S
–
j



^{i = 0}
j=1

All coefficients are scaled down by S, and the output of the accumulator is multiplied by S.
28-46
PXR40 Microcontroller Reference Manual, Rev. 1
Freescale Semiconductor