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Signals and Systems
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1.1 INTRODUCTION
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In this chapter we begin our study of digital signal processing by developing the notion of a discrete-time signal and a discrete-time system. We will concentrate on solving problems related to signal representations, signal manipulations, properties of signals, system classification, and system properties. First, in Sec. 1.2 we define precisely what is meant by a discrete-time signal and then develop some basic, yet important, operations that may be performed on these signals. Then, in Sec. 1.3 we consider discrete-time systems. Of special importance will be the notions of linearity, shift-invariance, causality, stability, and invertibility. It will be shown that for systems that are linear and shift-invariant, the input and output are related by a convolution sum. Properties of the convolution sum and methods for performing convolutions are then discussed in Sec. 1.4. Finally, in Sec. 1.5 we look at discrete-time systems that are described in terms of a difference equation.
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1.2 DISCRETE-TIME SIGNALS
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A discrete-time signal is an indexed sequence of real or complex numbers. Thus, a discrete-time signal is a function of an integer-valued variable, n, that is denoted by x(n). Although the independent variable n need not necessarily represent "time" (n may, for example, correspond to a spatial coordinate or distance), x(n) is generally referred to as a function of time. A discrete-time signal is undefined for noninteger values of n. Therefore, a real-valued signal x(n) will be represented graphically in the form of a lollipop plot as shown in Fig. 1- I. In
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Fig. 1-1. The graphical representation of a discrete-time signal x ( n ) .
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some problems and applications it is convenient to view x(n) as a vector. Thus, the sequence values x(0) to x(N - 1) may often be considered to be the elements of a column vector as follows:
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Discrete-timesignals are often derived by sampling a continuous-timesignal, such as speech, with an analogto-digital (AID) converter.' For example, a continuous-time signal x,(t) that is sampled at a rate of fs = l/Ts samples per second produces the sampled signal x(n), which is related to xa(t) as follows:
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Not all discrete-timesignals, however, are obtained in this manner. Some signals may be consideredto be naturally occurring discrete-time sequences because there is no physical analog-to-digital converter that is converting an
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Analog-to-digital conversion will be discussed in Chap. 3.
SIGNALS AND SYSTEMS
[CHAP. 1
analog signal into a discrete-time signal. Examples of signals that fall into this category include daily stock market prices, population statistics, warehouse inventories, and the Wolfer sunspot number^.^
Complex Sequences
In general, a discrete-time signal may be complex-valued. In fact, in a number of important applications such as digital communications, complex signals arise naturally. A complex signal may be expressed either in terms of its real and imaginary parts,
or in polar form in terms of its magnitude and phase,
The magnitude may be derived from the real and imaginary parts as follows:
whereas the phase may be found using arg{z(n)) = tan-' ImMn)) Re(z(n))
If z(n) is a complex sequence, the complex conjugate, denoted by z*(n), is formed by changing the sign on the imaginary part of z(n):
1.2.2 Some Fundamental Sequences
Although most information-bearing signals of practical interest are complicated functions of time, there are three simple, yet important, discrete-time signals that are frequently used in the representation and description of more complicated signals. These are the unit sample, the unit step, and the exponential. The unit sample, denoted by S(n), is defined by S(n) =
n=O otherwise
and plays the same role in discrete-time signal processing that the unit impulse plays in continuous-time signal processing. The unit step, denoted by u(n), is defined by u(n) = and is related to the unit sample by
n 1 0 otherwise
Similarly, a unit sample may be written as a difference of two steps:
2 ~ h Wolfer sunspot number R was introduced by Rudolf Wolf in 1848 as a measure of sunspot activity. Daily records are available back e to 1818 and estimates of monthly means have been made since 1749. There has been much interest in studying the correlation between sunspot activity and terrestrial phenomena such as meteorological data and climatic variations.
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