Page 487 - DSP Integrated Circuits

472 Chapter 11 Processing Elements

RNS number it has to be converted to a normal number representation. RNS is
therefore not suitable for use in recursive loops. Division is possible only in special
cases and would seriously complicate the system. Further, the dynamic range
must then be kept under strict control. Overflow would spoil the results. The RNS
method is effective only for special applications— e.g., nonrecursive algorithms.

11.5 BIT-PARALLEL ARITHMETIC

"Heigh-ho, Heigh-ho, it's off to work we go...
Larry Morey and Frank Churchill

In this section we will discuss methods to implement
arithmetic operations such as addition, subtraction,
and multiplication using bit-parallel methods. These
operations are simple to implement in two's-comple- Figure 11.2 Full-adder
ment representation, since they are independent of
the signs of the numbers involved. Hence, two's-com-
plement representation is used predominantly. However, in many cases it is
advantageous to exploit other number systems to improve the speed and reduce
the required chip area and power consumption.
The basic computation element for addition, subtraction, and multiplication is a
full-adder (FA). It accepts three binary inputs: A, B, and D, called addend, augend,
and carry-in, respectively. The two outputs are the sum, S, and the carry-out, C. A
symbol for the full-adder is shown in Figure 11.2. We have for the full-adder

These expressions can be modified to be more suitable for implementation in
CMOS with XOR gates:

11.5.1 Addition and Subtraction

x
Two binary numbers, x = Oco«#i*2--- Wd-l) andy = (yo<yiy2--- ywd-l\ can be added
bit-serial in O(W^) steps and, by using more advanced techniques and bit-parallel
operation, the number of steps, or gate levels, can be reduced to only O(log(W^)).
Notice, however, that the times required for a computational step may differ
between different addition algorithms, since the switching time also depends on
gate loads. In practice the addition time for one algorithm with more steps may