Digital Signal Processing Reference
In-Depth Information
1
z
k
2
−
k
in a similar way as for division.
More precisely, the iteration for square root extraction is
i
k
partially computed square root
Z
i
=
∑
=
z
i
2
−
i
r
i
=
2
r
i
−
1
−
z
i
(
2
z
i
+
)
(38)
where
Z
0
=
0.
Schemes similar restoring, nonrestoring, and SRT division can be defined. For the
quotient digit selection scheme similar to SRT division the square root is restricted
to 1
/
2
≤
Z
<
1, which corresponds to 1
/
4
≤
X
<
1. The selection rule is then
⎧
⎨
⎩
/
≤
<
11
2
r
i
−
1
2
z
i
=
−
/
<
<
/
.
(39)
0
1
2
r
i
−
1
1
2
−
1
−
2
≤
r
i
−
1
≤−
2
5
Floating-Point Representation
Floating-point numbers consists of two parts, the mantissa (or significand),
M
,and
the exponent (or characteristic),
E
, with a number,
X
, represented as
Mb
E
X
=
(40)
where
b
is the base of the exponent. For ease of exposition we assume
b
2. With
floating-point numbers we obtain a larger dynamic range, but at the same time a
lower precision compared to a fixed-point number system using the same number
of bits.
Both the exponent and the mantissa are typically signed integer or fractional
numbers. However, their representation are often not two's complement. For the
mantissa it is common to use a sign-magnitude representation, i.e., use a separate
sign-bit,
S
, and represent an unsigned mantissa magnitude with the remaining bits.
For the exponent it is common to use excess-
k
, i.e., add
k
to the exponent to obtain
an unsigned number.
=
5.1
Normalized Representations
A general floating-point representation is redundant since
M
2
2
E
+
1
M
2
E
=
(41)
