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barcode generator for ssrs (b) With x(n) = anu(n),the ztransform is in Software
(b) With x(n) = anu(n),the ztransform is ANSI/AIM Code 128 Scanner In None Using Barcode Control SDK for Software Control to generate, create, read, scan barcode image in Software applications. Code128 Creation In None Using Barcode creation for Software Control to generate, create Code 128A image in Software applications. and the ztransform of the autocorrelation sequence is R,(z) = Code 128 Code Set B Scanner In None Using Barcode recognizer for Software Control to read, scan read, scan image in Software applications. Code128 Generator In Visual C# Using Barcode printer for .NET Control to generate, create Code 128B image in .NET framework applications. I ( I  a z r 1 ) ( I az) Print Code 128C In .NET Using Barcode maker for ASP.NET Control to generate, create Code 128B image in ASP.NET applications. Generating Code 128 Code Set B In VS .NET Using Barcode generation for .NET Control to generate, create ANSI/AIM Code 128 image in VS .NET applications. lal < l z l < la I
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Generating Postnet 3 Of 5 In None Using Barcode creation for Software Control to generate, create USPS POSTNET Barcode image in Software applications. EAN13 Drawer In .NET Using Barcode drawer for .NET framework Control to generate, create EAN13 image in .NET applications. In many disciplines, differential equations play a major role in characterizing the behavior of various phenomena. Obtaining an approximate solution to a differential equation with the use of a digital computer requires that the differential equation be put into a form that is suitable for digital computation. This problem presents a transformation procedure that will convert a differential equation into a difference equation, which may then be solved by a digital computer. Consider a firstorder differential equation of the form GS1 128 Generation In ObjectiveC Using Barcode encoder for iPad Control to generate, create USS128 image in iPad applications. Barcode Creation In Visual C# Using Barcode creator for Visual Studio .NET Control to generate, create barcode image in Visual Studio .NET applications. where yu(0)=yo. Because numerical techniques are to be used, we will restrict our attention to investigating yu(t) at sampling instants nT where T is the sampling period. Evaluating the differential equation at t = n T , we have Scanning Code128 In VB.NET Using Barcode scanner for VS .NET Control to read, scan read, scan image in .NET framework applications. Draw Data Matrix ECC200 In None Using Barcode creator for Excel Control to generate, create ECC200 image in Microsoft Excel applications. CHAP. 41
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From calculus we know that the derivative of a function y,(t) at t = nT is simply the slope of the function at t = n T . This slope may be approximated by the relationship (a) Insert this approximation into the sampled differential equation above and find a difference equation that relates y(n) = y,(nT) and x ( n ) = x,(nT), and specify the appropriate initial conditions. ( b ) With x,(t) = u ( t ) and y,(O) = I, numerically solve the differential equation using the difference equation approximation obtained above. (c) Compare your approximation to the exact solution.
using the approximation
Y&W
 [ y , ( n ~ )  y,(nT  TI] T
we have
[ya(nT)  ya(nT  T)] + crya(nT) = .r,(nT) y,(O) = yo
If we let y(n) = ya(nT) and x(n) = xa(nT), With this becomes ~ ( n  ay(n ) 1) = a T x ( n ) ~ ( 0 = yo ) (b) Using the onesided ztransform to solve this difference equation, we have
We must now derive the initial condition on y(n) at time n =  1 from the initial condition at n = 0. From the difference equation, we have y(0)  ay(I) = aTx(0) With y(0) = 1 and x(0) = 1, the initial condition becomes With x,(t) = u(t) or x(n) = u(n), Therefore. using the given initial condition, we have
THE 2TRANSFORM
Performing a partial fraction expansion gives
[CHAP. 4
and we may find v(n) by taking the inverse ztransform: Because this may be written as
(c) The solution to the differential equation is a sum of two terms. The first is the homogeneous solution, which is yh(t)= A e  O 1 where A is a constant that is selected in order to satisfy the initial condition ~ ( 0  = 1. The second is the ) particular solution, which is Thus, the total solution is
Evaluating this a1 time I = 0, we see that in order to match the initial conditions. we must have
If we compare this to the approximation in part ( h ) ,note that if T
C  ~ ~ ) , _ , T=
< I, <
(e ) aT ,I
+( Y T )  ~
Supplementary Problems
z'kansforms 4.41
Find the 2transform of
CHAP. 41
THE zTRANSFORM
The ztransform of a sequence x(n) is
If the region of convergence includes the unit circle, find the DTFT of x(n) at w = r / 2 . Find the ztransform of each of the following sequences: (a) x(n) = (l)"u(n) ( b ) x(n) = ;u(n  I) (c) x(n) = z cosh (crn)u(n) Find the ztransform of the sequence
Find the ztransform of the sequence
How many different sequences have a ztransform given by
The sequence y(n) is formed from x(n) by
where X(z) = sinzr'. Find Y(z). If x(n) is an absolutely summable sequence with a rational ztransform that has poles at z = f and z = 2, what can be said about the extent of x(n) (i.e., finite in length, rightsided, etc.) Properties
A rightsided sequence x(n) has a ztransform X(z) given by
Find the values of x(n) for all n < 0. Use the ztransform to perform the convolution of the following two sequences: Evaluate the following summation: THE zTRANSFORM
Find the value of x(0) for the sequence that has a ztransform
[CHAP. 4
A rightsided sequence has a ztransform
Find the index a ~ the value of the first nonzero value of x ( n ) . d
Inverse zTransforms
Find the inverse ztransform of
4.55 4.56 4.57 Find the inverse ztransform of X(z) = coszp'. Assume that the ROC includes the unit circle, Izl = 1. Find the inverse ztransform of X(z) = e'. Assume that the ROC includes the unit circle, lzl = 1. Find the inverse ztransform of z5  3 X(z) = 1  zS lzl > I

