# Mathematical Preliminaries

• Jan W. M. Bergmans
Chapter

## Abstract

Digital transmission and recording systems all convey discrete-time data sequences across channels with a continuous-time nature. Conversions between the discrete-time and continuous-time domains are key operations in such systems, as are linear filtering operations in both domains. It is, therefore, appropriate that we set out with an introduction to some of the mathematical concepts and tools that are central to all of these transformations. In doing so we will assume that the reader has a basic working knowledge of the theory of signal analysis in both domains. For this reason we recapitulate this theory only cursorily, aiming in part to familiarize the reader with our notation. The borderline between both domains, on the other hand, is relatively under-exposed in most textbooks on signal analysis and will therefore receive more detailed attention. Section 1.2 is concerned with deterministic signals and sequences, with an emphasis on the spectral effects of linear filtering, sampling and linear modulation. A similar account is given in Section 1.3 for stochastic sequences and signals. The remaining sections are concerned with various other subjects that are of interest to later chapters. These are probability density functions (Section 1.4), the arithmetic, geometric and harmonic averages of spectra (Section 1.5), properties of discrete-time minimum-phase functions (Section 1.6), and the method of Lagrange multipliers (Section 1.7). Appendices 1A and 1B provide a brief survey of properties of the Fourier transforms of continuous-time signals and discrete-time sequences, along with a summary of basic Fourier transform pairs.

## Keywords

Impulse Response Power Spectral Density Autocorrelation Function Decision Feedback Equalizer Spectral Effect
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