Digital filters - Continuous and Discrete Time Signals and Systems - page 655

Example 14.9

h [ k ]

10 0.318 348 783 765 15

9,11 0.261 850 185 125 51

8,12 0.132 021 415 468 16

7,13 0.012 135 562 150 39

6,14 − 0 . 041 086 983 052 48

5,15 − 0 . 032 969 416 668 68

4,16 − 0 . 005 898 263 640 95

3,17 0.008 055 858 168 72

2,18 0.006 608 361 295 03

1,19 0.001 494 396 943 68

0, 20 − 0 . 001 385 507 671 95

Table 14.11. Impulse response of the FIR filter in Example 14.9 with 4-bit and 8-bit finite precisions

h [ k ]

k Exact 8-bit binary representation 4-bit precision 8-bit precision

10 0.318 348 783 765 15 0.010 100 01 0.3125 0.316 406 25

9, 11 0.261 850 185 125 51 0.010 000 11 0.25 0.261 718 75

8, 12 0.132 021 415 468 16 0.001 000 01 0.125 0.128 906 25

7, 13 0.012 135 562 150 39 0.000 000 11 0 0.011 718 75

6, 14 − 0 . 041 086 983 052 48 − 0 . 000 010 10 0 − 0 . 039 062 5

− 0 . 032 969 416 668 68 − 0 . 000 010 00 − 0 . 031 25

5, 15 0

− 0 . 005 898 263 640 95 − 0 . 000 000 01 − 0 . 003 906 25

4, 16 0

3, 17 0.008 055 858 168 72 0.000 000 10 0 0.007 812 5

2, 18 0.006 608 361 295 03 0.000 000 01 0 0.003 906 25

1, 19 0.001 494 396 943 68 0.000 000 00 0 0

0, 20 − 0 . 001 385 507 671 95 0.000 000 00 0 0

Calculate the filter coefficients with the significand restricted to a total of 7 bits
and where 1 bit is allocated for the sign. Plot the magnitude response of the
filter. Repeat for a 3-bit significand with 1 bit allocated for the sign.

Solution
The filter coefficients with the 4-bit and 8-bit finite-precision arithmetic are
shown in Table 14.11. We illustrate how we derived the result for the filter
coefficient h [10] = 0 . 318 348 783 765 15. The remaining entries can be derived
by following the procedure specified for h [10].