Page 71 - Engineering Digital Design

42 CHAPTER 2 / NUMBER SYSTEMS, BINARY ARITHMETIC, AND CODES

Terminating a fraction conversion at n digits (to the right of the radix point) results in an
error or uncertainty. This error is given by

= r , <2-(n+/) r

where the quantity in brackets is less than (a- n + 1). Therefore, terminating a fraction
conversion at n digits from the radix point results in an error with bounds

n
0 < e < r- (a- n + 1). (2.9)
Equation (2.9) is useful in deciding when to terminate a fraction conversion.
Often, it is desirable to terminate a fraction conversion at n + 1 digits and then round off
to n from the radix point. A suitable method for rounding to n digits in radix r is:

Algorithm 2.4: Rounding Off to n Digits for Fraction Conversion in Radix r
Perform the fraction conversion to (n — 1) digits from the radix point, then drop the
M 1)
(n — 1) digit if «_( K+1) < r/2; add r~< ~' to the result if #_ (w _i) > r/2.

After rounding off to n digits, the maximum error becomes the difference between the
rounded result and the smallest value possible. By using Eq. (2.9), this difference is

n
nax = r (a_ n + l)-r "( a_,

= r-"|l-

Then, by rounding to n digits, there results an error with bounds

n
If a-( n+\) < r/2 and the (n + 1) digit is dropped, the maximum error is r . Note that for
N s —> NIQ —>• N r type conversions, the bounds of errors aggregate.
The fraction conversion methods given in Table 2.5 and Algorithms 2.3 and 2.4 are
illustrated by the following examples:

EXAMPLE 2.7 0.654 10 -> 7V 2 rounded to 8 bits:
•N s x r F I
0.654 x 2 0.308 1
0.308 x 2 0.616 0
0.616 x 2 0.232 1
0.232 x 2 0.464 0