Heuristic Optimal Design of Multiplier-less Digital Filter - Advances in Heuristic Signal Processing and Applications - page 242

Fig. 11.2 Hybrid code of the
real number 0.125

The value of a real number with the structure of the hybrid code in Fig. 11.1 can
then be derived as:

m

2 − k i C = S k i × (11.2)

i = 1

where C represents the value of the hybrid-coded coefficients of a system. A simple
example of the hybrid code is shown as follows. A real number 0 . 125 is coded by
the hybrid code and is decoded as 0 . 125 2 − 2 2 − 3 . Figure 11.2 shows the hybrid

= −

code of this example.

11.2.2 New Hybrid Code Method

The disadvantage of a binary coding parameter is that the truncation error exists
between the original value and the binary coding value. This will cause the design
parameters to be far from the optimal solution. This problem, of course, can be im-
proved by increasing the bit length of the binary code. However, this will waste more
memory space. In this section, a more precise hybrid code method, Accumulation
Hybrid Code (AHC), is introduced without increasing the bit length of the binary
code. Hence, a closer to the optimal solution for the designed parameters can be
obtained without occupying more memory space. The main feature of AHC is that
the exponent in ( 11.2 ) is calculated by accumulating the prior k i and is derived by
the following Eq. ( 11.3 )[ 4 ]:

i

a i = k l , (11.3)

l = 1

m

2 − a j . C = S k j × (11.4)

j = 1

According to the example in Fig. 11.2 ,thevalueof C can be calculated by using
AHC as 2 − 2

2 − 5

0 . 21875. Comparing this value to the value obtained by the
traditional hybrid code method, we can see that the precision of the proposed AHC
is about 10 − 5 while that of the traditional hybrid code method is about 10 − 3 . So,
a more precise result can be expected when AHC is used for designing the system
parameters. As a result, a solution closer to the optimal solution can be found.

− =