Heuristic Optimal Design of Multiplier-less Digital Filter - Advances in Heuristic Signal Processing and Applications - page 243

11.3 CSD-Based Genetic Algorithm

In CSD format, two consecutive digits of a binary coded number cannot be '1'
simultaneously. In other words, the product of every two consecutive digits must be
zero, i.e., b n ×

b n + 1 . With this feature, fewer bits of nonzero values will appear in the
CSD-coded parameters of a system, and hence fewer shifting and adding operations
are required for the computation of the system output. Therefore, the CSD coding of
system parameters can accelerate the operation speed [ 9 ]. Since the CSD format of
a parameter is destroyed when it is evolved by GA, in the past research, examining
after each evolution whether the CSD format remained or not was necessary. A new
evolution process will be active, if the CSD format is destroyed by the previous
evolution. However, this method will waste much evolution time. Therefore, in this
section, in order to decrease the evolution time of GA, a CSD-based crossover and
a CSD-based mutation are introduced to keep the CSD code structure during the
evolution process of GA.

11.3.1 Definition of the Fitness Function

In order to identify the quality of a chromosome in GA, the fitness function is usually
used to evaluate each chromosome. Different fitness functions are used for different
environments. It is important to define the fitness function for a chromosome. In this
chapter, the fitness function for the digital filter design is defined as follows. First,
the error function is defined as:

Ω |

2

H(Ω) − H I (Ω) |

E p = , 0 ≤ Ω ≤ π (11.5)

N s

in which H I (Ω) is the desired frequency response and N S is the sampling point,
Ω

2 π f s is the digital frequency, f is the analog frequency, f s is the sampling
frequency. The fitness function for a chromosome p is then defined as:

=

1
E p +

fitness(p) =

1 .

(11.6)

11.3.2 CSD-Based Crossover

Concerning the crossover of the chromosomes with the CSD code format, the CSD
structure should be retained after the crossover. Usually, crossover will result in a
non-CSD structure and then a renewal of the chromosome is required. In the past
research, one had to examine the structure of the chromosome after crossover. If a
non-CSD structure had been found, the chromosome was renewed. If the bit length