Digital Signal Processing Reference
In-Depth Information
These values represent numbers in the range of zero and smallest normalized number on the
number line:
(
ð
s
2
e
127
1
:ðÞ
1
normalized
;
0
< e <
255
value
¼
0
:ðÞ
denormalized
; e ¼
0
; m >
0
:
ð
s
2
e
126
The example below illustrates normalized and denormalized numbers.
Example: Assume a floating-point number is represented as an 8-bit number. There is one sign bit
and 4 and 3 bits, respectively, are allocated to store exponent and mantissa. By traversing through
the entire range of the number, limits for denormalized and normalized values can be easily marked.
The valuewhere the e-field is all zeros and them-field is non-zero represents denormalized values. This
value does not assume an implied1 in themantissa. For a normalizedvalue an implied 1 is assumed and
a bias of
þ
7 is added in the true exponent to get the e-field. Therefore for a normalizedvalue this bias is
subtracted from the e-field to get the actual exponent. These numbers can be represented as:
ð
s
2
e
7
1
:ðÞ
normalized
;
0
< e <
7
(
value
¼
2
6
ð
s
1
:ðÞ
0
denormalized
; e ¼
0
; m >
0
:
The ranges of positive denormalized and normalized numbers for 8-bit floating-point format are
given in Table 3.5.
Floating-point representationworkswell where variables and results of computationmay vary over
a large dynamic range. In signal processing, this usually is not the case. In the initial phase of
algorithm development, though, before the ranges are conceived, floating-point format gives comfort
to the developer as one can concentrate more on the algorithmic correctness and less on implementa-
tion details. If there are no strict requirements on numerical accuracy of the computation, performing
Table 3.5
Various 8-bit floating-point numbers
s
E
m
Exp
Value
Denormalized
numbers
0
0000
000
-
0
0
:
125
2
6
0
0000
001
6
Closest to zero
0
:
5
2
6
0
0000
010
6
...
0
0
:
75
2
6
0000
110
6
0
:
875
2
6
0
0000
111
6
Largest denormalized value
1
:
0
2
6
0
0001
000
6
Smallest normalized value
1
:
125
2
6
Normalized
numbers
0
0001
001
6
...
0
1
:
75
2
1
0110
110
1
1
:
875
2
1
1
0
0110
111
0
0111
000
0
1
1
:
125
2
0
0
0111
001
0
...
0
2
7
1110
111
7
1
:
875
Largest normalized value
0
1111
000
1
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