As far as the hardware is concerned, fixed-point number systems
represent data as

Inside Collection (Course): Digital Signal Processing: A User's Guide

Summary: Specialized DSP hardware typically uses fixed-point number representations for lower cost and complexity and greater speed. Interpretation of two's-complement binary numbers as signed fractions between -1 and 1 allows integer arithmetic to be used for DSP computations, but introduces quantization and overflow errors.

Fixed-point arithmetic is generally used when hardware cost, speed, or complexity is important. Finite-precision quantization issues usually arise in fixed-point systems, so we concentrate on fixed-point quantization and error analysis in the remainder of this course. For basic signal processing computations such as digital filters and FFTs, the magnitude of the data, the internal states, and the output can usually be scaled to obtain good performance with a fixed-point implementation.

As far as the hardware is concerned, fixed-point number systems
represent data as

For the purposes of signal processing, we often regard the
fixed-point numbers as binary fractions between

Consider the multiplication of two binary fractions

Consider the addition of two binary fractions;

There are thus two types of fixed-point error: roundoff error, associated with data quantization and multiplication, and overflow error, associated with data quantization and additions. In fixed-point systems, one must strike a balance between these two error sources; by scaling down the data, the occurence of overflow errors is reduced, but the relative size of the roundoff error is increased.

Since multiplies require a number of additions, they
are especially expensive in terms of hardware
(with a complexity proportional to
B
x
B
h
B
x
B
h
, where
B
x
B
x
is the number of bits in the data, and
B
h
B
h
is the number of bits in the filter coefficients).
Designers try to minimize both
B
x
B
x
and
B
h
B
h
, and often choose
B
x
≠
B
h
B
x
B
h
!

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Comments:"Doug course at UIUC using the TI C54x DSP has been adopted by many EE, CE and CS depts Worldwide "