UploadFile_6417

266 Chapter 6 IMPLEMENTATION OF DISCRETE-TIME FILTERS
±0, depending on the sign bit (called the soft zero). Thus 0 has two
representations.
• If E = 255 and M ̸= 0,then the representation is interpreted as a
not-a-number (abbreviated as NaN). MATLAB assigns a variable NaN
when this happens—e.g., 0/0.
• If E = 255 and M =0,then the representation is interpreted as ±∞.
MATLAB assigns a variable inf when this happens—e.g., 1/0.
□ EXAMPLE 6.19 Consider the bit pattern given in Example 6.17. Assuming IEEE-754 format,
determine its decimal equivalent.
Solution
The sign bit is 0 and the exponent code is 131, which means that the exponent
is 131 − 127 = 4. The significand is 1 + 2 −1 +2 −2 =1.75. Hence the bit pattern
represents
ˆx = +(1 + 2 −1 +2 −2 )(2 4 )=2 4 +2 3 +2 2 =28
which is different from the number in Example 6.17.
□
MATLAB employs the 64-bit double-precision IEEE-754 format for
all its number representations and the 80-bit temporary format for its internal
computations. Hence all calculations that we perform in MATLAB
are in fact floating-point computations. Simulating a different floatingpoint
format in MATLAB would be much more complicated and would
not add any more insight to our understanding than the native format.
Hence we will not consider a MATLAB simulation of floating-point arithmetic
as we did for fixed-point.
6.7 THE PROCESS OF QUANTIZATION AND ERROR
CHARACTERIZATIONS
From the discussion of number representations in the previous section, it
should be clear that a general infinite-precision real number must be assigned
to one of the finite representable number, given a specific structure
for the finite-length register (that is, the arithmetic as well as the format).
Usually in practice, there are two different operations by which this assignment
is made to the nearest number or level: the truncation operation
and the rounding operation. These operations affect the accuracy as well
as general characteristics of digital filters and DSP operations.
We assume, without loss of generality, that there are B +1 bits in
the fixed-point (fractional) arithmetic or in the mantissa of floating-point
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.