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262 Chapter 6 IMPLEMENTATION OF DISCRETE-TIME FILTERS
function sm2oc converts a decimal sign-magnitude fraction into its one’scomplement
format, while the function oc2sm performs the inverse operation.
These functions are explored in Problem P6.24. Similarly, MATLAB
functions sm2tc and tc2sm convert a decimal sign-magnitude fraction
into its two’s-complement format and vice versa, respectively; they are
explored in Problem P6.25.
6.6.3 FLOATING-POINT ARITHMETIC
In many applications, the range of numbers needed is very large. For
example, in physics one might need, at the same time, the mass of the sun
(e.g., 2.10 30 kg) and the mass of the electron (e.g., 9.10 −31 kg). These two
numbers cover a range of over 10 60 .For fixed-point arithmetic, we would
need 62-digit numbers (or 62-digit precision). However, even the mass of
the sun is not accurately known with a precision of 5 digits, and there is
almost no measurement in physics that could be made with a precision of
62 digits. One could then imagine making all calculations with a precision
of 62 digits and throwing away 50 or 60 of them before printing out the
final results. This would be wasteful of both CPU time and memory space.
So what is needed is a system for representing numbers in which the range
of expressible numbers is independent of the number of significant digits.
Decimal numbers The floating-point representation for a decimal
number x is based on expressing the number in the scientific notation:
x = ±M × 10 ±E
where M is called the mantissa and E is the exponent. However, there
are different possible representations of the same number, depending on
the actual position of the decimal point—for example,
1234 = 0.1234 × 10 4 =1.234 × 10 3 =12.34 × 10 2 = ···
To fix this problem, a floating-point number is always stored using
a unique representation, which has only one nonzero digit to the left
of the decimal point. This representation of a floating-point number is
called a normalized form. The normalized form of the preceding number
is 1.234 × 10 3 ,because it is the only representation resulting in a unique
nonzero digit to the left of the decimal point. The digit arrangement for
the normalized form is given by
sign of M
ˆx =
↓
± x xx ···x
} {{ }
N-bit M
sign of E
↓
± xx ···x
} {{ }
L-bit E
(6.41)
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