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SIGNALS AND SYSTEMS
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1.3 DISCRETE-TIME SYSTEMS
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A discrete-time system is a mathematical operator or mapping that transforms one signal (the input) into another signal (the output) by means of a fixed set of rules or operations. The notation T [ - ]is used to represent a general system as shown in Fig. 1-4, in which an input signal x(n) is transformed into an output signal y(n) through the transformation T [ . ] . The input-output properties of a system may be specified in any one of a number of different ways. The relationship between the input and output, for example, may be expressed in terms of a concise mathematical rule or function such as
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It is also possible, however, to describe a system in terms of an algorithm that provides a sequence of instructions or operations that is to be applied to the input signal, such as yl(n) = 0.5yl(n - 1) 0.25x(n)
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+ y2(n) = 0.25y2(n - 1) + 0.5x(n) ys(n) = 0.4y3(n - 1) + 0.5x(n) + y(n) = Y I ( ~ ) y2(n) + ydn)
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In some cases, a system may conveniently be specified in terms of a table that defines the set of all possible input-output signal pairs of interest.
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Fig. 1-4. The representation of a discrete-timesystem as a transformation T [ . ] that maps an input signal x ( n ) into an output signal y(n).
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Discrete-time systems may be classified in terms of the properties that they possess. The most common properties of interest include linearity, shift-invariance, causality, stability, and invertibility. These properties, along with a few others, are described in the following section.
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System Properties
Memoryless System
The first property is concerned with whether or not a system has memory.
Definition: A system is said to be memoryless if the output at any time n = no depends only on the input at time n = no.
In other words, a system is memoryless if, for any no, we are able to determine the value of y(no) given only the value of x(no).
EXAMPLE 1.3.1 The system
y(n) = x 2 b )
is memoryless because y(no)depends only on the value of x ( n ) at time no. The system
y(n) = x(n)
+ x(n - I)
on the other hand, is not memoryless because the output at time no depends on the value of the input both at time no and at time no - 1.
Additivity
SIGNALS AND SYSTEMS
[CHAP. 1
An additive system is one for which the response to a sum of inputs is equal to the sum of the inputs individually. Thus,
Definition: A system is said to be additive if
T [ x l ( n )+ x2(n)I = T [ x ~ ( n ) l T[x2(n)l
for any signals X I (n) and x2(n).
Homogeneity
A system is said to be homogeneous if scaling the input by a constant results in a scaling of the output by the same amount. Specifically,
Definition: A system is said to be homogeneous if
T [cx(n)]= cT [x(n)]
for any complex constant c and for any input sequence x(n).
EXAMPLE 1.3.2 The system defined by
~ ( n= )
x(n - 1 )
x2(n)
This system is, however, homogeneous because, for an input c x ( n ) the output is
On the other hand, the system defined by the equation
y(n) =x(n)
+x'(n - 1 )
- I ) ] + [xp(n) + x ; ( n - l ) ]
is additive because
[ x ~ ( n ) + x ~ ( n +I[ X I @ )
- 1) + x A n - I l l *
= [xl(n) + x f ( n
However, this system is not homogeneous because the response to c x ( n ) is
T [ c x ( n ) ]= c x ( n )
+ c*x*(n - 1) + cx*(n - 1)
which is not the same as
cT[x(n)] = cx(n)
Linear Systems
A system that is both additive and homogeneous is said to be linear. Thus,
Definition: A system is said to be linear if
T [ a m ( n ) am(n)l =alT[x~(n)l+ azT[xAn)l
for any two inputs xl(n) and x2(n) and for any complex constants a1 and a2.
CHAP. 11
SIGNALS AND SYSTEMS
Linearity greatly simplifies the evaluation of the response of a system to a given input. For example, using the decomposition for x ( n ) given in Eq. (1.4),and using the additivity property, it follows that the output y ( n ) may be written as
y(n)=T[x(n)]=T
k=-m
x(k)S(n-k)
Because the coefficients x ( k ) are constants, we may use the homogeneity property to write
k=-m
T[x(k)S(n-k)]
~ ( n= )
k=-ca
T[x(k)G(n- k)l =
k=-m
x(k)T[S(n- k)]
If we define h k ( n ) to be the response of the system to a unit sample at time n = k ,
h k ( n ) = T [S(n - k ) ]
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