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ssrs 2d barcode CHAP. 81 in Software
CHAP. 81 Recognizing Code 128 Code Set B In None Using Barcode Control SDK for Software Control to generate, create, read, scan barcode image in Software applications. Generating Code 128 Code Set C In None Using Barcode printer for Software Control to generate, create Code128 image in Software applications. IMPLEMENTATION OF DISCRETETIME SYSTEMS
Scan Code 128 Code Set C In None Using Barcode reader for Software Control to read, scan read, scan image in Software applications. Encode ANSI/AIM Code 128 In Visual C# Using Barcode maker for .NET Control to generate, create Code 128A image in VS .NET applications. where G(z) is the secondorder system shown below.
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Decode UPC A In VB.NET Using Barcode recognizer for .NET Control to read, scan read, scan image in .NET framework applications. Barcode Creator In Java Using Barcode generator for Java Control to generate, create barcode image in Java applications. Substituting the second equation into the first gives
Finally, from the last difference equation, we have
Therefore. and for H(z) we have
For stability, it is necessary and sufficient that the coefficients [a(2) +a(4)] and [a(l) triangle (see Chap. 5 ) , which requires that + a(3)] lie within the stability
326 8.16 IMPLEMENTATION OF DISCRETETIME SYSTEMS
[CHAP. 8
Find the system function of the following network: x(n) This system is a feedback network that has the following form: x(n) G(z) If2
Therefore, the system function is
With
we have
Find the system function of the following network: x(n) The system function of this network may be found by writing down the difference equations corresponding to each adder and solving these equations using ztransforms. A simpler approach, however, is to redraw the network as follows, CHAP. 81
IMPLEMENTATION O F DISCRETETIME SYSTEMS
which we recognize as a cascade of three secondorder networks. Therefore, the system function is the product of the system functions of each network in the cascade, and we have Consider the network in the figure below. Redraw the flowgraph as a cascade of secondorder sections in transposed direct form 11. To implement lhis system as a cascade of secondorder transformed direct form 1 networks, we must first find the 1 system function corresponding to this network. Note that this network is of the form shown in the following figure: HI(:) Hs(z) XOI) Hz() ~ ( n ) where
Therefore, or
 0.2:r1
+ 0.8:r2)( 1 + 0
. 2 ~ 0.82I) ~ ~
Therefore, the desired network is as shown in the following figure: .~(II) IMPLEMENTATION OF DISCRETETIME SYSTEMS
[CHAP. 8
A digital oscillator has a unit sample response
The system function of this oscillator is
(a) Draw a direct form I1 network for this oscillator, and show how the sinusoid may be generated with no input and the appropriate set of initial conditions. (b) In applications involving the modulation of sinusoidal carrier signals in phase quadrature, it is necessary to generate the sinusoids Beginning with a system that has a unit sample response h ( n ) = e J n q u ( n ) , separate the difference equation for this system into its real and imaginary parts, and draw a network that will generate these signals when initialized with the appropriate set of initial conditions. (a) A direct form I1 network for the oscillator is as follows: sin wl
With the input x ( n ) = &n). the response is y(n) = sin[(n registers corresponding to the delays are initialized so that + I)w]for n
2 0. Equivalently, if the storage
the zeroinput response will be a sinusoid of frequency wo.
( h ) A complex exponential sequence y(n) = e J n q u(n) is generated by the difference equation
with the initial condition y( I ) = e  j q Writing this difference equation in terms of its real and imaginary parts, we have This equation is equivalent to the following pair of coupled difference equations, which are formed from the real part and the imaginary part of the equation: CHAP. 81
IMPLEMENTATION OF DISCRETETIME SYSTEMS A network that implements this pair of equations is shown below. .yi(n) = sinnwo
COS Wg
The initial condition required to generate the desired output is yc  I ) = e  J q , or
Implement the system
as a parallel network of firstorder direct form structures.
Factoring the denominator of the system function, we find
To implement H ( z ) as a parallel network of firstorder filters, we must express H ( z ) as a sum of firstorder factors using a partial fraction expansion. Because the order of the numerator is equal to the order of the denominator, this expansion will contain a constant term, To find the value of C. we divide the numerator polynomial by the denominator as follows: Therefore, C = 2, and we may write H ( z ) as follows: Finally, with
we have, for the coefficients A and B, IMPLEMENTATION OF DISCRETETIME SYSTEMS
[CHAP. 8
Thus, and the parallel network for this system is as shown below.
82 .1 The system function of a discretetime system is
Draw a signal flowgraph of this system using a cascade of secondorder systems in direct form 11, and write down the set of difference equations that corresponds to this implementation. Expressing H ( z ) as a product of two secondorder systems, we have H ( z )= + 22I + z=  z1 + 2z' + z
+ 122 8 1 + 221 + 3
which leads to the following cascade implementation for H(z): CHAP. 81
IMPLEMENTATION OF DISCRETETIME SYSTEMS
With w(n), vl(n). and v2(n) as labeled in the figure above, the set of difference equations for this network is:

