Revisiting the Basics - Digital Signal Processing: A Practitioner's Approach - page 43

2.7.3 Floating-Point Numbers
Fixed-point representation suffers from the limitation of dynamic range, hence the
floating point has evolved to give

b n ¼ x n m 2 c ;

where m is the mantissa and c is the characteristic or exponent; x n is the sign bit.
The floating-point number is represented as a 3-tuple given as b n ¼f x n ; m ; c g in

packed form. The standard 5

IEEE format for 32-bit single-precision floating-point

number is given as

b n ¼ h x 32 i
x n

h x 31 ...

x 24 i

h x 23 ...

x 1 i

c m

In this notation, m is a fixed-point positive fractional number in normalised form
and the value b n represented by the word may be determined as follows:

1. If 0 < c < 255 then b n ¼ 1 x n 2 c 127

ð 1 : m Þ where 1 : m is intended to
represent the binary number created by prefixing m with an implicit leading 1
and a binary point.
2. If c ¼ 0 and m 6¼ 0, then b n ¼ 1 x n 2 126 0 : m These are unnormalised

values.
3. If c ¼ 255 and m 6¼ 0, then b n ¼ NaN (Nan means not a number).
4. If c ¼ 255 and m ¼ 0 and x n ¼ 1 or 0, then b n ¼1 or b n ¼1 .
5. If c ¼ 0 and m ¼ 0 and x n ¼ 1 or 0, then b n ¼ 0or þ 0

2.8 Summary
This chapter was a refresher on some relevant engineering topics. It should help you
with the rest of the topic. We considered the autocorrelation function, representation
of linear systems, and noise and its propagation in linear systems. We also discussed
the need to know about systems reliability. Most problems in real life are inverse
problems, so we introduced the Moore-Penrose pseudo-inverse. We concluded by
providing an insight into binary number systems.

References

1. S. Slott and L. James, Parametric Estimation as an Optimisation Problem. Hatfield Polytechnic.
2. D. G. Luenberger, Introduction to Linear and Non-linear Programming. Reading MA: Addison-
Wesley, 1973.
3. T. L. Boullion and P. L. Odell, Generalised Inverse Matrices. New York: John Wiley & Sons, Inc.,
1971.

5 ANSI/IEEE Standard 754-1985, Standard for Binary Floating Point Arithmetic.