barcode lib ssrs Therefore the system is additive. It is not homogeneous, however, because in Software

Encoder Code128 in Software Therefore the system is additive. It is not homogeneous, however, because

Therefore the system is additive. It is not homogeneous, however, because
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unless c is real. Thus, this system is nonlinear. For an input x(n), this system produces an output that is the conjugate symmetric part of x(n). If c is a complex constant, and if the input to the system is xl(n) = cx(n), the output is
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Therefore, this system is not homogeneous. This system is, however, additive because
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A linear system is one that is both homogeneous and additive.
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(a) Give an example of a system that is homogeneous but not additive.
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(b) Give an example of a system that is additive but not homogeneous.
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There are many different systems that are either homogeneous or additive but not both. One example of a system that is homogeneous but not additive is the following:
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~ ( n= ) x(n
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Specifically, note that if x(n) is multiplied by a complex constant c, the output will be
~ ( n= ) cx(n-l)cx(n) cx(n
+ I)
x(n-I)x(n) x(n
+ 1)
which is c times the response to x(n). Therefore, the system is homogeneous. On the other hand, it should be clear that the system is not additive because, in general,
{xl(n - 1)
x ~ ( n 1) +xAn
+ X Z (- l)J(x~(n) xz(n)I ~ + + +I)
x ~ ( n l)x~(n) x,(n 1)
+ xdn - l)xz(n) xz(n + 1)
CHAP. I]
SIGNALS AND SYSTEMS
An example of a system that is additive but not homogeneous is
Additivity follows from the fact that the imaginary part of a sum of complex numbers is equal to the sum of imaginary parts. This system is not homogeneous, however, because
Determine whether or not each of the following systems is shift-invariant:
(a) Let y ( n ) be the response of the system to an arbitrary input x ( n ) . To test for shift-invariance we want to compare the shifted response y ( n - n o ) with the response of the system to the shifted input .r(n - nu). With
we have. for the shifted response.
Now, the response of the system to x l ( n ) = x ( n - n o ) is
Because y l ( n ) = y ( n
- n o ) , the system is shifl-invariant.
( 6 ) This system is a special case of a more general system that has an input-output description given by
where f ( n ) is a shift-varying gain. Systems of this form are always shift-varying provided f ( n ) is not a constant. To show this, assume that f ( n ) is not constant and let n I and nz be two indices for which f ( n , ) # f ( n z ) . With an input . r l ( n ) = S(n - n l ) , note that the output y l ( n ) is
If, on the other hand, the input is x 2 ( n ) = 6 ( n - n 2 ) , the response is
Although .t.,(n) and x Z ( n ) differ only by a shift, the responses y l ( n ) and y 2 ( n ) differ by a shift and a change in amplitude. 'Therefore, the systcm is shift-varying. (c) Let
be the response of the system to an arbitrary inpul .r(n). The response of the system to the shifted input
. r l ( n ) = x(n - no) is
Because this is equal to v ( n - n o ) , the system is shift-invariant.
SIGNALS AND SYSTEMS
[CHAP. 1
(d) This system is shift-varying, which may be shown with a simple counterexample. Note that if x(n) = S(n), the response will be y(n) = 6(n). However,ifxl(n) = 6(n-2). the response will be yl(n) = xl(n2)= 6(n2-2) = 0, which is not equal to y(n - 2). Therefore, the system is shift-varying.
(e) With y(n) the response to x(n), note that for the input xl(n) = x(n
N), the output is
which is the same as the response tox(n). Because yl (n) # y(n- N), ingeneral, this system isnot shift-invariant.
(f) This system may easily be shown to be shift-varying with a counterexample. However, suppose we use the direct approach and let x(n) be an input and y(n) = x(-n) be the response. If we consider the shifted input, x l (n) = x(n - no), we find that the response is
However, note that if we shift y(n) by no,
which is not equal to yl (n). Therefore, the system is shift-varying.
A linear discrete-time system is characterized by its response h k ( n ) to a delayed unit sample S(n - k). For each linear system defined below, determine whether or not the system is shift-invariant.
(a) hk(n) = ( n - k)u(n - k )
(6) hk(n) = S(2n - k )
k - 1)
5u(n
k even k odd
(a) Note that hk(n)is a function of n - k . This suggests that the system is shift-invariant. To verify this, let y(n) be the response of the system to x(n):
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