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barcode lib ssrs Therefore the system is additive. It is not homogeneous, however, because in Software
Therefore the system is additive. It is not homogeneous, however, because Code 128A Decoder In None Using Barcode Control SDK for Software Control to generate, create, read, scan barcode image in Software applications. Code 128A Creator In None Using Barcode encoder for Software Control to generate, create Code 128B image in Software applications. 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 Decoding Code 128A In None Using Barcode recognizer for Software Control to read, scan read, scan image in Software applications. Making Code 128B In Visual C# Using Barcode encoder for .NET Control to generate, create Code128 image in Visual Studio .NET applications. Therefore, this system is not homogeneous. This system is, however, additive because
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Code 128C Printer In VB.NET Using Barcode drawer for .NET framework Control to generate, create Code 128 Code Set C image in .NET framework applications. Painting Code 39 In None Using Barcode maker for Software Control to generate, create USS Code 39 image in Software applications. (a) Give an example of a system that is homogeneous but not additive.
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~ ( n= ) cx(nl)cx(n) cx(n
+ I) x(nI)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 shiftinvariant: (a) Let y ( n ) be the response of the system to an arbitrary input x ( n ) . To test for shiftinvariance 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 shiflinvariant.
( 6 ) This system is a special case of a more general system that has an inputoutput description given by where f ( n ) is a shiftvarying gain. Systems of this form are always shiftvarying 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 shiftvarying. (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 shiftinvariant.
SIGNALS AND SYSTEMS
[CHAP. 1
(d) This system is shiftvarying, 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(n2). the response will be yl(n) = xl(n2)= 6(n22) = 0, which is not equal to y(n  2). Therefore, the system is shiftvarying. (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 shiftinvariant. (f) This system may easily be shown to be shiftvarying 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 shiftvarying.
A linear discretetime 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 shiftinvariant. (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 shiftinvariant. To verify this, let y(n) be the response of the system to x(n):

