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264 Chapter 6 IMPLEMENTATION OF DISCRETE-TIME FILTERS
However, the most widely used format for the mantissa is the signmagnitude
one.
□ EXAMPLE 6.17 Consider a 32-bit floating-point word with the following arrangement:
ˆx = ± xx ···x
} {{ }
8-bit E
Determine the decimal equivalent of
1x ···x
} {{ }
23-bit M
01000001111000000000000000000000
Solution
Since the exponent is 8-bit, it is expressed in excess-2 7 or in excess-128 format.
Then the bit pattern can be partitioned into
ˆx =
Sign
↓
0 10000011
} {{ }
E=131
11000000000000000000000
} {{ }
M=2 −1 +2 −2
The sign bit is 0, which means that the number is positive. The exponent code is
131, which means that its decimal value is 131 − 128 = 3. Thus, the bit pattern
represents the decimal number ˆx =+ ( 2 −1 +2 −2) (2 3 )=2 2 +2 1 =6. □
□ EXAMPLE 6.18 Let ˆx = −0.1875. Represent ˆx using the format given in (6.42), in which B = 11,
L =4(for a total of 16 bits), and sign-magnitude format is used for the mantissa.
Solution
We can write
ˆx = −0.1875 = −0.75 × 2 −2
Hence the exponent is −2, the mantissa is 0.75, and the sign is negative. The
4-bit exponent, in excess-8 format, is expressed as 8 − 2=6orwith bit pattern
0110. The mantissa is expressed as 11000000000. Since ˆx is negative, the bit
pattern is
ˆx ≡ 1011011000000000
The advantages of the floating-point representation are that it has
a large dynamic range and that its resolution, defined as the interval
between two consecutive representable levels, is proportional to the magnitude.
The disadvantages include no representation for the number 0 and
the fact that the arithmetic operations are more complicated than their
fixed-point representations.
□
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