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Computer Arithmetic
puter arithmetic two fundamental design principles are of great importance: number representation and the implementation of algebraic operations [25, 26, 27, 28, 29]. We will first discuss possible number representations,(., fixed-point or floating-point), then basic operations like adder and multiplier,and finally efficient implementation of more difficult operations such as square roots, and putation of trigonometric functions using the CORDIC algorithm or MAC calls.
FPGAs allow a wide variety puter arithmetic implementations for the desired digital signal processing algorithms, because of the physical bitlevel programming architecture. This contrasts with the programmable digital signal processors (PDSPs), with the fixed multiply accumulator choice of the bit width in FPGA design can result in substantial savings.
Number Representation
Deciding whether fixed- or floating-point is more appropriate for the problem must be done carefully, preferably at an early phase in the project. In general,it can be assumed that fixed-point implementations have higher speed and lower cost, while floating-point has higher dynamic range and no need for scaling, which may be attractive for plicated algorithms. Figure is a survey of conventional and less conventional fixed- and floating-point number representations. Both systems are covered by a number of standards but may, if desired, be implemented in a proprietary form.
Fixed-Point Numbers
We will first review the fixed-point number systems shown in Fig. . Table shows the 3-bit coding for the 5 different integer representations.
Unsigned Integer
Let X be an N-bit unsigned binary number. Then the range is [0,2n-1] and the representation is given by:
where xn is the nth binary digit of X (., x∈[0,1]). The digit x0 is called the least significant bit (LSB) and has a relative weight of unity. The digit Xn-1 is the most significant bit (MSB) and has a relativ