3.3 Introduction to digital signal processing

Not only do we have analog signals --- signals that are real- or complex-valued functions of a continuous variable such as timeor space --- we can define digital ones as well. Digital signals are sequences , functions defined only for the integers. We thus use the notation $s(n)$ to denote a discrete-time one-dimensional signal such as a digital musicrecording and $s(m, n)$ for a discrete-"time" two-dimensional signal like a photo taken with a digital camera. Sequences are fundamentallydifferent than continuous-time signals. For example, continuity has no meaning for sequences.

Despite such fundamental differences, the theory underlying digital signal processing mirrors that for analog signals:Fourier transforms, linear filtering, and linear systems parallel what previous chapters described. These similaritiesmake it easy to understand the definitions and why we need them, but the similarities should not be construed as "analogwannabes." We will discover that digital signal processing is not an approximation to analog processing. We must explicitly worry about the fidelity of converting analogsignals into digital ones. The music stored on CDs, the speech sent over digital cellular telephones, and the video carried bydigital television all evidence that analog signals can be accurately converted to digital ones and back again.

The key reason why digital signal processing systems have a technological advantage today is the computer : computations, like the Fourier transform, can be performed quicklyenough to be calculated as the signal is produced,