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Chapter 1:
Classical Hadamard Matrices and Arrays

In this chapter, we introduce the primary nonsinusoidal orthogonal transforms, such as Hadamard, Haar, etc., which are extensively reported in the literature. The basic advantages of the Hadamard transform (HT) are as follows:

• Multiplication by HT involves only an algebraic sign assignment.

• Digital circuits can generate Hadamard functions because they assume only the values +1 and −1.

• Computer processing can be accomplished using addition, which is very fast, rather than multiplication.

• The continence case of these systems is very good for representing piecewise constants or continuous functions.

• The simplicity and efficiency of the transforms is found in a variety of practical applications. These include, for example, digital signal and image processing, such as compact signal representation, filtering, coding, data compression, and recognition; speech and biomedical signal analysis; and digital communication. A prime example is the code division multiple access system (CDMA) cellular standard IS-95, which uses a 64-Hadamard matrix in addition to experimental and combinatorial designs, digital logic, statistics, error-correcting codes, masks for spectroscopic analysis, encryption, and several other fields. Among other possible applications, which can be added to this list, are analysis of stock market data, combinatorics, error-correcting codes, spreading sequences, experimental design, quantum computing, environmental monitoring, chemistry, physics, optics, combinatorial designs, and geophysical analysis. In this chapter, we introduce the commonly used Sylvester, Walsh-Hadamard, Walsh, Walsh-Paley, and complex HTs.

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