Discrete-Time Signals and Their Fourier Transforms

Abstract

So far in this book we have concentrated on the continuous-time signals. These included continuous periodic and aperiodic time signals and the corresponding Fourier series and transform representations. We divided the continuous signals based on their energy and power properties. In the case of periodic signals, the Fourier series coefficients are discrete. In this chapter, we will start with continuous signals and their sampled versions. The advances in computers and the ease in implementing discrete algorithms using personal computers (PCs) made this as an essential area every electrical engineer should be interested in. Most discrete-time signals come from sampling continuous signals, such as speech, seismic, sonar, images, biological, and other signals. These days, telephone along with a computer forms an integral part of most communication systems. The advances in telemetry allow us to monitor remotely located patients. The ease of processing discrete-time signals made discrete-time implementations of analog operations, such as filtering, made it very popular. The analog signals are first converted to digital signals by making use of a device referred to as an analog-to-digital (A/D) converter.The reverse process of reconstructing an analog signal from a digital signal is achieved by a device referred to as a digital-to-analog (D/A) converter.Obviously if the source is an analog device and the end user requires an analog signal, the use of a digital processor requires the A/D and the D/A converters. Although user signals are usually analog, there are many situations wherein the discrete-time signals are source signals. For example, the stock prices, the temperatures at a particular time in a city, grades of students and many others are digital in nature. The transform study of the discrete-time signals is basic to our study.