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11.3 Redundant Number Systems 467

and then we add the third number (neglecting the temporary overflow that
occurred and neglecting the second overflow)

yielding the correct result. Figure 11.1 illustrates the cyclic property of two's-com-
plement representation. A correct result is obtained as long as the amounts of left
and right "rotations" cancel.

11.2.5 Binary Offset Representation
The value of a normalized W^-bit binary word in binary offset representation is

d
The values lie in the range -1 < x < I - Q, where Q = 2 . Binary offset
representation is also a nonredundant representation. The sequence of digits is
equal to the two's-complement representation, except for the sign bit which is com-
plemented. For example, we have

In the same way as for the two's-complement representation, it can be shown
that the sign of a number can be changed by first taking the bit-complement and
then adding Q.

11.3 REDUNDANT NUMBER SYSTEMS

Conventional fixed-radix number systems where the radix is a positive (usually
even) integer are commonly used in arithmetic units in general-purpose comput-
ers and standard digital signal processors. In application-specific arithmetic units
it is often advantageous to use more unconventional, redundant number sys-
tems— for example, negative radix systems [22] or signed digit representations.
By using redundant number systems it is possible to simplify and speed up cer-
tain arithmetic operations. In fact, addition and subtraction can be performed
without long carry (borrow) paths. Typically this is achieved at the expense of
increased complexity for other arithmetic and nonarithmetic operations and
larger registers. Zero and sign detection and conversion to and from conventional