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CHAPTER 1
DISCRETE-TIME SIGNALS AND SYSTEMS
1.1 INTRODUCTION
A discrete-time signal is a sequence of values usually representing the behavior of a physical phenomenon. In electrical engineering problems, those values are samples of a continuous-time-varying electrical signal taken at uniform rate, called sampling rate or sampling frequency. The inverse of the sampling rate is called sampling interval or sampling period. Figure 1.1 shows the graphical representations of a continuous-time signal and its discrete-time uniformly sampled version. For the latter, time is normalized by the sampling period becoming the index n. This can be stated as:
where T is the sampling period.
In practice, continuous-time or analog signals are electrical events (voltage or current) representing the behavior of some physical phenomenon such as speech or temperature as a function of time. Devices known as transducers are utilized to convert physical variations (of pressure or temperature, for example) into voltage or electrical current changes, thus creating an electrical signal. To be digitally processed, electrical signals have to be sampled, time discretized, quantized, and encoded, thus becoming a digital signal.
Therefore, a digital signal is a discrete-time signal, as represented in Figure 1.1(b), for which the amplitude is quantized, that is, it is constrained to assume values in a finite set. Quantization is usually accomplished by rounding or truncating the amplitude sample to the nearest value in the discrete set. It is always present in digital signal processing, as samples must be stored in finite length registers. Chapter 3 analyzes the sampling and quantization processes.
Figure 1.1. (a) Continuous-time signal s(t), (b) Discrete-time signal sd(n) = s(nT).
In this chapter, we present the main properties of discrete-time signals, demonstrate how those signals are affected by operations on the independent variable, and introduce some signals that are important in digital signal processing.
1.2 PROPERTIES OF DISCRETE-TIME SIGNALS
Most properties of continuous-time (analog) signals are also common to discrete-time (digital) signals. In this section, we review some relevant properties to digital signal processing.
1.2.1 PERIODICITY
A discrete-time signal is periodic if there exists an integer N such that:
for any value of n. The integer N is called the period of the signal.
Figure 1.2. Discrete-time periodic signal (segment) with period N0.
Equation 1.2 holds for any integer multiple of N. Figure 1.2 shows a periodic signal where N0 is the smallest value that period N can assume, called fundamental period. Thus, this signal is periodic for any period N = kN0, k integer. Equation 1.2, thus, generalizes to:
1.2.2 POWER AND ENERGY
The energy of a discrete-time signal x(n) is defined as:
The average ...
