# 1 Introduction or Why?

Discrete Systems

Dwight Day

This text is intended to cover the primary topics in Digital Signal Processing (DSP), or Discrete Signal Processing.  It is not intended to be complete or comprehensive, rather to focus on those topics that have proven most pertinent in using DSP to collect and analyze data.  Although the data processed with these techniques vary, our primary goal is to work on the classic sampled data system as is shown in Figure 1.1.

In the above figure the input is a continuous signal or voltage, which might represent the output of a sensor detecting a position, speed, etc.  For the computer to work with this signal it must be converted to digital numbers.  This is achieved by the use of an Analog to Digital Converter (ADC), which sends out a stream of N-bit digital number at discrete points in time.  This stream of numbers is depicted as a string of values which are discrete both in time and value.  Once these samples have been processed, they are sent out as an altered stream of numbers.  Note the output may be at a different time and value resolution, M-bit.  Finally the output can be changed into a voltage via a Digital to Analog Converter (DAC).  The most notable feature of the reconstructed signal is its blocky nature.

The text will be organized in a fashion to address the various parts of this system, but not in a direct sequence since some background is needed to address some topics.  Chapter 2 will be about the basic models for sampling, discuss techniques for characterizing the spectrum of a sampled signal and introduce the problems associated with send the data out via a DAC.  Chapter 3 will develop the z transform and its application to difference equations, which is the classic technique used to analyze and characterize the processing of discrete data.  The  Chapter 4 will describe the processing of data, or what is commonly called filtering of data, which will encompass classic filters, but also discuss some nonlinear and intuitive based filters.  Chapter 5 will use the filtering techniques developed in Chapter 4 to address the reconstruction of the outputs.   Chapter 6 will address the application and computation of the Discrete Fourier Transform (DFT).  Chapter 7 will describe the application of adaptive filtering.

As stated previously this is far from a comprehensive text. It should however prepare the student to understand the basics of the field of DSP, be able apply DSP to the most basic systems and learn about any new DSP system them may encounter.