DAFX: Digital Audio Effects, Second Edition

1.3 Fundamentals of Digital Signal Processing

The fundamentals of digital signal processing consist of the description of digital signals—in the context of this book we use digital audio signals—as a sequence of numbers with appropriate number representation and the description of digital systems, which are described by software algorithms to calculate an output sequence of numbers from an input sequence of numbers. The visual representation of digital systems is achieved by functional block diagram representations or signal flow graphs. We will focus on some simple basics as an introduction to the notation and refer the reader to the literature for an introduction to digital signal processing [ME93, Orf96, Zöl05, MSY98, Mit01].

1.3.1 Digital Signals

The digital signal representation of an analog audio signal as a sequence of numbers is achieved by an analog-to-digital converter (ADC). The ADC performs sampling of the amplitudes of the analog signal x(t) on an equidistant grid along the horizontal time axis and quantization of the amplitudes to fixed samples represented by numbers x(n) along the vertical amplitude axis (see Fig. 1.11). The samples are shown as vertical lines with dots on the top. The analog signal x(t) denotes the signal amplitude over continuous time t in micro seconds. Following the ADC, the digital (discrete time and quantized amplitude) signal is a sequence (stream) of samples x(n) represented by numbers over the discrete time index n. The time distance between ...