Modern communication systems and digital signal processing (signal modeling), image compression and image encoding,3 and digital signal processing systems are heavily reliant on statistical techniques to recover information in the presence of noise and interference. One of the mathematical structures used to achieve this goal is the Hadamard matrix. Historically, Plotkin18 first showed the error-correcting capabilities of codes generated from Hadamard matrices. Later, Bose and Shrikhande found the connection between Hadamard matrices and symmetrical block code designs. In this chapter, we will discuss some of these applications in error-control coding and in CDMAs.
13.1 Hadamard Matrices and Communication Systems
13.1.1 Hadamard matrices and error-correction codes
The storage and transmission of digital data lies at the heart of modern computers and communications systems. When a message is transmitted, it has the potential to become scrambled by noise. The goal of this section is to provide a brief introduction to the basic definitions, goals, and constructions in coding theory. We describe some of the classical algebraic constructions of error-correcting codes, including the Hadamard codes. The Hadamard codes are relatively easy to decode; they are the first large class of codes to correct more than a single error. A Hadamard code was used in the Mariner and Voyager space probes to encode information transmitted back to the Earth when the probes visited Mars and the outer planets of the solar system from 1969 to 1976. Mariner 9 was a space shuttle whose mission was to fly to Mars and transmit pictures back to Earth. Fig. 13.1 is one of the pictures transmitted by 9. With Mariner 5, six-bit pixels were encoded using 32-bit long Hadamard code that could correct up to seven errors.