Digital Signals and Systems - Digital Signal Processing: Fundamentals and Applications - page 57

CHAPTER

3

Digital Signals and Systems

CHAPTER OUTLINE

3.1 Digital Signals ............................................................................................................................... 57
3.1.1 Common Digital Sequences..........................................................................................58
3.1.2 Generation of Digital Signals ........................................................................................61
3.2 Linear Time-Invariant, Causal Systems............................................................................................. 63
3.2.1 Linearity .....................................................................................................................63
3.2.2 Time Invariance ..........................................................................................................65
3.2.3 Causality ....................................................................................................................66
3.3 Difference Equations and Impulse Responses................................................................................... 67
3.3.1 Format of the Difference Equation ................................................................................67
3.3.2 System Representation Using Its Impulse Response .......................................................68
3.4 Bounded-In and Bounded-Out Stability ............................................................................................. 71
3.5 Digital Convolution ......................................................................................................................... 72
3.6 Summary ....................................................................................................................................... 79

OBJECTIVES:
This chapter introduces notations for digital signals and special digital sequences that are widely used in
this topic. The chapter continues to study some properties of linear systems such as time invariance, BIBO
(bounded-in and bounded-out) stability, causality, impulse response, difference equations, and digital
convolution.

3.1 DIGITAL SIGNALS
In our daily lives, analog signals appear in forms such as speech, audio, seismic, biomedical, and
communications signals. To process an analog signal using a digital signal processor, the analog signal
must be converted in to a digital signal, that is, analog-to-digital conversion (ADC) must take place, as
discussed in Chapter 2. Then the digital signal is processed via digital signal processing (DSP)
algorithm(s).
A typical digital signal xðnÞ is shown in Figure 3.1 , where both the time and the amplitude of the
digital signal are discrete. Notice that the amplitudes of the digital signal samples are given and
sketched only at their corresponding time indices, where xðnÞ represents the amplitude of the n th
sample and n is the time index or sample number. From Figure 3.1 , we learn that

xð 0 Þ : zeroth sample amplitude at sample number n ¼ 0,
xð 1 Þ : first sample amplitude at sample number n ¼ 1,