Arithmetic Circuits for DSP Applications

Pramod Kumar Meher, Thanos Stouraitis, Pramod Kumar Meher, Thanos Stouraitis

• Publisher: Wiley-IEEE Press
• Year: 2017
• ISBN:
  9781119206798
• Edition: 1
• Topic: Technology & Engineering
• Subtopic: Signals & Signal Processing
• Index: Technology & Engineering

A comprehensive guide to the fundamental concepts, designs, and implementation schemes, performance considerations, and applications of arithmetic circuits for DSP

Arithmetic Circuits for DSP Applications is a complete resource on arithmetic circuits for digital signal processing (DSP). It covers the key concepts, designs and developments of different types of arithmetic circuits, which can be used for improving the efficiency of implementation of a multitude of DSP applications. Each chapter includes various applications of the respective class of arithmetic circuits along with information on the future scope of research. Written for students, engineers, and researchers in electrical and computer engineering, this comprehensive text offers a clear understanding of different types of arithmetic circuits used for digital signal processing applications.

The text includes contributions from noted researchers on a wide range of topics, including a review of circuits used in implementing basic operations like additions and multiplications; distributed arithmetic as a technique for the multiplier-less implementation of inner products for DSP applications; discussions on look up table-based techniques and their key applications; CORDIC circuits for calculation of trigonometric, hyperbolic and logarithmic functions; real and complex multiplications, division, and square-root; solution of linear systems; eigenvalue estimation; singular value decomposition; QR factorization and many other functions through the use of simple shift-add operations; and much more. This book serves as a comprehensive resource, which describes the arithmetic circuits as fundamental building blocks for state-of-the-art DSP and reviews in - depth the scope of their applications.

1
Basic Arithmetic Circuits

Oscar Gustafsson and Lars Wanhammar

Division of Computer Engineering, Linköping University, Linköping, Sweden

1.1 Introduction

General-purpose DSP processors, application-specific processors [1], and algorithm-specific processors are used to implement different types of DSP systems or subsystems. General-purpose DSP processors are programmable and therefore, provide maximum flexibility and reusability. They are typically used in applications involving complex and irregular algorithms while application-specific processors provide lower unit cost and higher performance for a specific application, particularly when the volume of production is high. The highest performance and lowest unit cost is obtained by using algorithm-specific processors. The drawback is the restricted or even lack of flexibility, and very often the nonrecurring engineering (NRE) cost could be very high.

The throughput requirement in most real-time DSP applications is generally fixed, and there is no advantage of an implementation with throughput than that design to minimize the chip area, and power consumption. Now in a CMOS implementation with higher throughput than required, it is possible to reduce the power consumption by lowering the supply voltage and operating the system at lower frequency [2].

1.2 Addition and Subtraction

The operation of adding two or more numbers is the most fundamental arithmetic operation, since most other operations in one or another way are based on addition. The operands of concern here are either two's-complement or unsigned representation.

Most DSP applications use fractional arithmetic instead of integer arithmetic [3]. The sum of two w-bit numbers is a (w + 1)-bit number while the product of two w-bit binary numbers is a 2w-bit number. In many cases and always in recursive algorithms the resulting number needs to be quantized to a w-bit number. Hence, the question is which bits of the result are to be retained. In fractional arithmetic, the input operands as well as the result are interpreted as being in the range [0, 1], that is,

(1.1)

for unsigned numbers and in the range [-1, 1], that is,

(1.2)

for signed numbers in two's-complement representation. For convenience, we let w denote the number of fractional bits and one additional bit is used for a signed representation.

We use the graphic representation shown in Figure 1.1 to represent the operands and the sum bits with the most significant bit to the left.

Figure 1.1 Addition of two binary numbers.

1.2.1 Ripple-Carry Addition

Ripple-carry addition is illustrated in Figure 1.2. A ripple-carry adder performs addition of two numbers; adds the bits of the same significance and the carry-bit from the previous stage sequentially using a full adder (FA), and propagates the carry-bit to the next stage. Obviously, the addition takes w addition cycles, where duration of each clock cycle is the time required by an FA to complete the addition of three bits. This type of adder can add both unsigned and two's-complement numbers.

Figure 1.2 Ripple-carry adder

The major drawback with the ripple-carry adder is that the worst-case delay is proportional to the word length. Also, typically, the ripple-carry adder produces many glitches since the full adder cells need to wait for the correc...