Heuristic Optimal Design of Multiplier-less Digital Filter - Advances in Heuristic Signal Processing and Applications - page 247

Table 11.1 Parameters of the

Parameter Value

example

order 28

number of k i 3

bit length of k i 5

δ p 0.05

δ s 00001

passband width 0 . 425 π

stopband width 0 . 425 π

transfer frequency band 0 . 15 π

where

10 − 3 , 10 − 2 , a 1 = 5 . 309 × a 2 = 7 . 114 ×

10 − 1 , 10 − 3 , a 3 =− 4 . 761 × 4 =− 2 . 66 ×

10 − 1 , 10 − 1 a 5 =− 5 . 941 × 6 =− 4 . 278 ×

in which δ p is the magnitude of the ripple in the passband of a target filter, and δ s
is the magnitude of the ripple in the stopband of a target filter. The order of the FIR
filter N is the smallest integer which is greater than N c .

11.4.3 Design Example of an FIR Filter [ 4 ]

Assume the parameters are δ p = 0 . 001, and that the frequency band

ΔF is 0 . 15 π .ByusingEq.( 11.14 ), we have N c =

0 . 05 and δ s =

28
as the order of the designed filter. For the design of the lowpass filter, we assume the
passband width is 0 . 425 π and the stopband width is 0 . 425 π . The parameters used
for this design are listed in Table 11.1 . According to the earlier described proce-
dures, in this section the verification of the proposed method and an efficiency test
are attempted on a symmetrical finite impulse digital lowpass filter of order 28. In
addition, the simulated results are also presented.
The parameters for GA in this example are listed in Table 11.2 . It is noted that
the bit length for each chromosome is 270 because there are 15 coefficients in a
chromosome and each coefficient contains three fields k 1 ,k 2 ,k 3 with a bit length of
6 for each field (including the sign bit).
The initial generation was simulated randomly with a CSD code. Thereafter, the
CSD-based GA introduced in Sect. 11.3 is then used to design the coefficients of
the filter. The flow chart for this example is as shown in Fig. 11.4 .
The mean square errors (MSEs) between the target filter and the designed filter
during the GA evolution from 100 to 100000 iterations are listed in Table 11.3 .
The designed coefficients with the CSD structure for the FIR filter are listed in

27 . 024775. Thus, we set N =