# Cascade and Parallel realizations structures for Higher order IIR Filters.

Software and s/w Development

Dec 1, 2013 (4 years and 6 months ago)

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DSP Notes; Implementation Issues in IIR Filters

Page
1

of
4

Richard Hayes, Dept Control Systems and Electrical Engineering.

Cascade and Parallel realizations structures for Higher order
IIR Filters.

The Nth order TF is factored into N/2 products or sums of second order sections (+ 1
first order if the order of the original filter is odd).

Second order sections, the practica
l building blocks for IIR realization structures
include the canonic and direct forms shown below:

Canonic section most popular good round off noise property (see later) and
minimum number of delays. Susceptible to internal overflow therefore must
use sca
ling

Direct form does not require scaling.

DSP Notes; Implementation Issues in IIR Filters

Page
2

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4

Richard Hayes, Dept Control Systems and Electrical Engineering.

The transposes of the second order sections are shown below. The transposes have
different finite wordlength properties.

DSP Notes; Implementation Issues in IIR Filters

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4

Richard Hayes, Dept Control Systems and Electrical Engineering.

Serial or Cascade Implementations.

Factor the Numerator and Denominator into
Second order factors. (Why Second
Order?)

How to pair numerator with denominator factors

1
2
1
2
1
2
2
1
2
1
1
2
2
1
2
1
D
D
N
N
or
D
D
N
N
or
D
D
N
N
or
D
D
N
N

Many possible orderings. How Many?

Rule of Thumb:

Pair
N
i

with
D
j

if the zeros of
N
i

are closest to the poles of
D
j

to avoid a large
amplitud
e response at the frequency corresponding to the pole

And

Place the section with the poles closest to the unit circle last
.

Parallel Realizations.

Order of sections is NOT important.

Scaling is easier. Independe
ntly carried out for each section.

SNR’s are

comparable to the best cascade realizations.

Zero’s of Parallel are more sensitive to coefficient quantization errors.

Cascade Realizations are more popular

More coefficients are simple integers (0,1,2)

Software packages produce cascade filters.

2
1
s
s

s
1
b
o1

)
(
n
w
)
(
n
x
)
(
n
y
1
1
s

s
1
b
1
o
s
1
b
o

s
1
b
1
o
s
1
b
o

s
1
b
o
s
1
b
o

)
(
n
w
)
(
n
x
)
(
n
y
s
1
b
1
o
s
1
b
o

s
1
b
1
o
s
1
b
o

1
1
s

s
1
b
11

s
1
b
21

s
2
b
22

s
2
b
22

s
2
b
o2

DSP Notes; Implementation Issues in IIR Filters

Page
4

of
4

Richard Hayes, Dept Control Systems and Electrical Engineering.

EXAMPL
E.

Determine the transfer functions for the cascade and Parallel structure for the filter

1 2 3
1 2 3
0.1432(1 3 3 )
( )
1 0.18 0.342 0.0615
z z z
H z
z z z
  
  
  

  

Using MATLAB

b=[1 3 3 1];a=[1
-
.1801 .3419
-
.0165];

roots(a)

sos=tf2sos(b,a,'up')

sos =

Numerators

Denominators

a
0

a
1

a
2

b
0

b
1

b
2

First
Section

1.0000

1.0000

0

1.0000

-
0.0492

0

Second
Section

1.0000

2.0000

1.0000

1.0000

-
0.1309

0.3355

Thus
H(z)

can be written as

1 1 2
1 1 2
1 1 2
( ).
1 0.492 1 0.1309 0.3355
z z z
H z
z z z
  
  
  

  

Parallel. Use Partial fraction expansion to obtain the coefficients of t
he parallel
implementation of the filter.

EXERCISE.

Write MATLAB program to implement both filters based on the following code for a
single 2
nd

order
section implemented in direct form. Rewrite the program to
implement the cascade filter in canonic form.