Digital Signal Processing Reference
In-Depth Information
Fig. 4
Overflow
characteristic of two's
complement arithmetic
X
Q
1-
Q
X
-2
-1
1
2
-1
Fig. 5
Overflow
characteristic of saturation
arithmetic
X
Q
1-
Q
X
-2
-1
1
2
-1
arithmetic is not really needed in good filter algorithms, since filter structures with
low coefficient sensitivity also utilize the available number range efficiently.
Fixed-point arithmetic is predominately used in application-specific ICs since the
required hardware is much simpler, faster, and consumes less power compared with
floating-point arithmetic. We will therefore focus on fixed-point arithmetic.
1.6.1
Overflow Characteristics
A two's complement representation of negative numbers is usually employed in dig-
ital hardware. The overflow characteristic of the two's complement representation
As discussed earlier, the largest and smallest numbers in two's complement
representation are 1
−
Q
and
−
1, respectively. A two's complement number,
X
,
larger than 1
−
Q
will be interpreted as
X
−
2, while a number slightly smaller than
−
2. Hence, very large overflows errors are incurred.
A common scheme to reduce the size of overflow errors and mitigate their harmful
effect is to limit numbers outside the normal range to either the largest or smallest
representable number. This scheme is referred to as
saturation arithmetic
and is
Many standard signal processors provide addition and subtraction instructions
with inherent saturation. Another saturation scheme, which may be simpler to
implement in hardware, is to invert all bits when overflow occurs. Using fixed-point
arithmetic the signal levels need to be adjusted at the input of multiplications with
non-integer coefficients. Note that, as discussed earlier, a sum of several numbers,
using two's complement representation, may have intermediate overflows as long as
the final value (sum) is within the valid range.
1 will be interpreted as
X
+
