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258 Chapter 6 IMPLEMENTATION OF DISCRETE-TIME FILTERS
Since the sign bits were the same, an overflow is generated and the result is
invalid.
4. 40 + 20 − 30
This is an example of more than one addition or subtraction. Since the final
result is well within the range, the overflow can be ignored—that is,
which is a correct result.
5. −40 − 20 + 30
In this case, we have
40 + 20 + (100 − 30) = 40 + 20 + 70 = 130 ≡ 30
(100 − 40) + (100 − 20) + 30 = 60 + 80 + 30 = 170 ≡−30
in the sign-magnitude format, which is, again, a correct result.
□
MATLAB Implementation Using (6.34), one can develop the
MATLAB function, TensComplement, which obtains ten’s-complement
format representation. It is similar to the TwosComplement function and
is explored in Problem P6.25.
Advantages of two’s-complement format Using the results of the
Example 6.13, we now state the benefits of the two’s-complement format.
These also hold (with obvious modifications) for the ten’s-complement
format.
1. It provides for all 2 B+1 distinct representations for a B-bit fractional
representation. There is only one representation for zero.
2. This complement is compatible with our notion of negation: the complement
of a complement is the number itself.
3. It unifies the subtraction and addition operations (subtractions are
essentially additions).
4. In a sum of more than two numbers, the internal overflows do not affect
the final result so long as the result is within the range (i.e., adding
two positive numbers gives a positive result, and adding two negative
numbers gives a negative result).
Hence in most A/D converters and processors, negative numbers are represented
using two’s-complement format. Almost all current processors
implement signed arithmetic using this format and provide special functions
(e.g., an overflow flag) to support it.
Excess-2 B−1 format This format is used in describing the exponent
of floating-point arithmetic; hence it is briefly discussed here. In excess-
2 B−1 signed format (also known as a biased format), all positive and
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