Pipelining, Retiming, Look-ahead Transformation and Polyphase Decomposition - Digital Design of Signal Processing Systems: A Practical Approach - page 326

Both these expressions achieve the objective of augmenting eight registers in the feedback loop,
but the transfer function using (7.9) requires three multipliers whereas the transfer function
using (7.10) requires eight multipliers.

7.6 Look-ahead Transformation for Generalized IIR Filters
The look-ahead transformation techniques can be extended for pipelining any Nth-order generalized
IIR filter. All these general techniques multiply and divide the transfer function of the IIR filter by
a polynomial that adds more registers in the critical loop or all loops of an IIR filter. These registers
are then moved to reduce the IPB of the graph representing the IIR filter.
An effective technique is cluster look-ahead (CLA) transformation [9-11]. This technique adds
additional registers bymultiplying and dividing the transfer function by a polynomial 1 þ P i¼ 1 c i z i

and then solving the product in the denominator 1 þ P i¼ 1 a i z i

1 þ P i¼ 1 c i z i to force the first

M coefficients to zero:

X M
i¼

1 þ

1 c i z i

1 þ

HðÞ¼ X N

i¼

X M
i¼

1 þ

1 a i z i 1 c i z i

ð 7 : 11 Þ

X M
i¼ 1 c i z i

1 þ

¼ X N þM

i¼M d i z i

1 þ

This technique does not guarantee stability of the resultant IIR filter. To solve this, scatter-
edcluster look-ahead (SCA) is proposed. This adds, for every pole, an additional M - 1 poles and
zeros such that the resultant transfer function is stable. The transfer function H(z) is written as a
function of z M that adds M registers for every register in the design. These registers are thenmoved to
reduce the IPB of the dataflow graph representing the IIR filter. For conjugate poles at re jy , that

represents a second-order section ð 1 2 r cos yz 1

2

z 2 Þ . Constraining M to be a power of 2, and

applying SCA using expression (7.10), the transfer function corresponding to this pole pair changes
to:

þ r

1 þ re jy z 1

Y log 2 M
i¼

2 i 2 i

1 þ re jy z 1

1

HðÞ¼ 1

Þ M Þ M 1 re jy z 1 ð re jy z 1 ð

ð 7 : 12 Þ

Y log 2 M
i¼

2 i cos 2 i yz 2 i

z 2 i þ 1

1 þ 2 r

þ r

2 i þ 1

1

¼

ð 1 2 r M cos Mð z M þ r

2 M z 2 M

Þ

which is always stable.
There are other transformations. For example, a generalized cluster look-ahead (GCLA) does not
force the first M coefficients in the expression to zero; rather it makes them equal to 1 or some
signed power of 2 [11]. This helps in achieving a stable pipelined digital IIR filter.