System Design Flow and Fixed-point Arithmetic - Digital Design of Signal Processing Systems: A Practical Approach - page 126

3.8.4 Folded FIR Filters
From the hardware perspective, a multiplier takes more area than an adder. FIR filter design using
the MATLAB filter design tool are symmetric or anti-symmetric and mathematically can be
represented as:

hMn ½ ¼ h½ for n ¼ 0 ; 1 ; 2 ; ...; M

This characteristic of the FIR filter can be effectively used to reduce the number of multipliers in
mapping these designs in HW. The design can be folded to use this symmetry. For example, if the
filter has four symmetric coefficients then convolution summation is written as:

y½¼h 0 x n þh 1 x n 1 þh 1 x n 2 þh 0 x n 3

The terms in this summation can be regrouped to reduce the number of multiplications from four
to two. The new summation is:

y½¼h 0 x n þx n 3 ð Þþh 1 x n 1 þx n 2 ð Þ

The generalized folded flow graph that uses symmetry of the coefficients in an FIR is shown in
Figure 3.31 for even and odd number of coefficients.

z -1 z -1 z -1 z -1

x[n]

z -1
h 1

z -1
h 2

z -1 z -1

h

m

h

h 0

m

2

- 1

2

y[n]

(a)

z -1 z -1 z -1 z -1

x[n]

z -1

z -1
h 1

z -1
h 2

z -1

z -1

h 0

h

y[n]

m - 3

h

m - 1

2

2

(b)

Figure 3.31 Symmetry of coefficients reduces the number of multipliers to almost half. (a) Folded
design with odd number of coefficients. (b) Folded design with even number of coefficients