Multiplier-less Multiplication by Constants - Digital Design of Signal Processing Systems: A Practical Approach - page 263

Solution: The five coefficients of the filter in double-precision floating-point using the
MATLAB fir1 function are:

h[n] ¼ [0.024553834015017
0.234389464237986
0.482113403493995
0.234389464237986
0.024553834015017]

Converting h[n] into Q1.15 format gives:

*2 15 h½n¼ round ðh½n Þ¼½ 805 7680 15798 7680 805

Binary representation of the coefficients is:
16 0 b 0000 0011 0010 0101

16 0 b 0001 1110 0000 0000
16 0 b 001 1110 1101 10110
16 0 b 0001 1110 0000 0000

16 0 b 0000 0011 0010 0101
Converting the coefficients into CSD representation gives:

000001010 0100 101
001000100 0000 000
01000010 0100 101
001000100 0000 000
000001010 0100 101

Keeping a maximum of four non-zero CSDs in each coefficient results in:

00000101001001
00100010000000000
0100001001001
0010001000000000
00000101001001

The sixteen PPs are as follows:

y½n¼
ðx½n 2 5 x½n 2 7 þx½n 2 10 þx½n 2 13 Þ

2 2 2 6 þðx½n 1 x½n Þ

2 1 2 6 2 9 2 12 þðx½n 2 x½n 2 x½n 2 x½n 2 Þ

2 2 2 6 þðx½n 3 x½n 3 Þ

2 5 2 7 2 10 2 13 þðx½n 4 x½n 4 þx½n 4 þx½n 4 Þ

For sing-extension elimination, five CVs for each multiplier are computed and added to form the
GCV. The computed CV 0 for the first coefficient is shown in Figure 6.6, and so:

CV 0 ¼ 32 0 b 1111 1010 1101 1100 0000 0010 0000 0000