John E. Hershey

Architecture of the fast Walsh-Hadamard and fast Fourier transforms with charge transfer devices

This paper presents a novel approach for charge transfer device (CTD) implementation of the fast Walsh-Hadamard and fast Fourier transforms. This is achieved by first expressing the Hadamard matrix as a power of a matrix, which allows for efficient CTD implementation. These results are then used in CTD implementation of fast Fourier transforms. The errors which accrue on using CTDs for implementing the Walsh-Hadamard transform are discussed in terms of a dispersion estimate.

Random and Pseudorandom Sequences

The study of random sequences is required by many aspects of data transmission. Synchronization and privacy are but two of these. In this chapter we examine the behavior of random sequences and conclude with a study of the m-sequence—an often used approximation to a random sequence.

The Natural Numbers and Their Primes

The natural numbers are the positive integers 1,2,3,…. Perhaps the hoariest of mathematical jokes is that all the natural numbers are interesting! The proof is by induction and contradiction. Consider that the first natural number, unity, divides all the natural numbers. It is therefore “interesting.” Now let k be the least natural number that is not “interesting.” Is it then not “interesting” that k is the first noninteresting of the natural numbers? Ergo, all natural numbers are interesting.

Network Reliability and Survivability

The ability of a network to withstand component failure, subversion, or openly hostile actions is inextricably linked to its topology as well as its individual component survivabilities. The most basic of the topological questions is node “connectivity.” Node connectivity means simply the linking together of centers that can originate or receive messages by means of pathways or, as they are more appropriately termed, “links.” We will at first assume only two-way or bidirectional links which are appropriate models, for example, for microwave links that use the same physical antenna for transmission and reception through polarization or frequency diversity. We further restrict our attention, for now, to those cases that allow a maximum of one link to be placed between any two nodes.

The Channel and Error Control

Our information bits are transported from sender to receiver over a medium which we call a channel. The bits, or groupings of bits, are represented by waveforms and then either put directly onto a medium, as is the case with many local area networks, or sent through a modem (modulator/ demodulator) which generates and resolves waveforms designed for use on a specific channel. A communications engineer characterizes the behavior, not of bits through the channel, but of waveforms and signal processing algorithms that are used to decide which particular symbol is being sent. The need for this arises, of course, because of noise on the channel. Here we are concerned only with the bit stream as it is presented to us after the modem, after all the decisions regarding received waveform processing have been made.

Basic Concepts in Matrix Theory

In this chapter we consider some of the basic concepts associated with matrix theory and its applications to digital communications, signal processing, and many others.

Matrix Equations and Transformations

In this chapter we consider solutions of a linear set of equations using some of the concepts discussed in Chapter 4. In addition, we introduce the concepts of a vector space, rank of a matrix, and so on. We end the chapter with a discussion of various transformations that are popular in the digital signal processing area.

Data—Its Representation and Manipulation

In this chapter we are concerned with the preliminaries of representing information, or data, using binary units or bits. We start with a most basic concept—number systems. The number systems considered are those common ones of “normal binary representation,” negabinary, and Gray coding. We also introduce a less well known “mixed-radix system,” based on the factorials. This representation will be of use to us later on when we look at combinatorics.

Applications of Matrices to Discrete Data System Analysis

In the last three chapters we discussed various aspects of matrix theory. In this chapter we apply some of these concepts to discrete data systems. In the process we show that matrix analysis is a logical tool to use for solving discrete data problems. The major topics of this chapter are convolution, deconvolution, and difference equations. In an earlier chapter we pointed out that matrix transformations can be visualized as converting one set of numbers to another set of numbers. For discrete data systems, we need to identify the order of these numbers, that is, the time these numbers appear before we process. In the next section we discuss discrete signals in terms of some special functions.

Space Division Connecting Networks

If nothing ever changed, this would be a simpler world; not a particularly interesting world but a simpler one. Consider that there are n nodes. These nodes may represent people, computers, ports, or anything that wishes to send and receive information. If node i always wants to communicate with node j, node k with l, and so on, then our data engineering task is simple. We connect i with j, k with l, and so on for the other pairs. But suppose i tired of j and k with l and that i wished to discourse with k andj with l. If so, we would have to disconnect these nodes and then reconnect them. This is motivation for considering the network shown in Figure 13.1.