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barcode lib ssrs Consider the fourthorder combfilter that has a system function in Software
Consider the fourthorder combfilter that has a system function ANSI/AIM Code 128 Reader In None Using Barcode Control SDK for Software Control to generate, create, read, scan barcode image in Software applications. Code 128C Drawer In None Using Barcode printer for Software Control to generate, create Code 128 Code Set C image in Software applications. (a) Draw a polezero diagram for H (z). USS Code 128 Scanner In None Using Barcode decoder for Software Control to read, scan read, scan image in Software applications. Code 128A Maker In Visual C#.NET Using Barcode generation for Visual Studio .NET Control to generate, create USS Code 128 image in .NET framework applications. (b) Find the value for A so that the peak gain of the filter is equal to 2.
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IMPLEMENTATION OF DISCRETETIME SYSTEMS
(c) With a system function
[CHAP. 8
we may implement this system using two multiplies as shown in the figure below.
Note, however, that the multiplier may be shared as follows: a(n) The difference equations for this network are
The system function of an allpass filter has the form
The symmetry that exists between the numerator and denominator polynomials allows for special structures that are more efficient than direct form realizations in terms of the number of multiplies required to compute each output value y(n ). (a) Consider the firstorder allpass filter, H(z) = +azI
where a! is real. Find an implementation for this system that requires two delays but only one multiplication. (h) For a secondorder allpass filter with h ( n ) real, the system function has the form
where a and B are real. Derive a structure that implements this system using four delays but only two multiplies. (a) The direct form realization of a firstorder allpass filter requires two multiplies and one delay as shown in the figure below. CHAP. 81
IMPLEMENTATION OF DISCRETETIME SYSTEMS To see how the two multiplies may be combined, consider the difference equation for this system: Therefore, only one multiplication is necessary if we form the difference x ( n )  y ( n  I ) prior to multiplying by a. Thus, we have the structure illustrated in the figure below that has two delays but only one multiplication. Because this structure requires an extra delay compared to direct form, this structure is not canonic. ( b ) As with the firstorder allpass filter. we may find a twomultiplier realization ol'a secondorder allpass filter by combining together terms in the difference equation for the allpass filter as follows: Thus, only two multiplications are required if we can form the differencess(n  1 )  y ( n  1 ) and . r ( n )  y ( n 2) prior to performing any multiplications. A structure that accomplishes this is given in the figure below. Note that with the additional delays, two multiplications are saved compared to a direct form implementation. Lattice Filters
8.24 Sketch a lattice filter implementation of the FIR filter
To implement this system using a lattice filter structure, we must find the reflection coefficients that generate the polynomial H(z). First, however, it is necessary to normalize H ( z ) so that the first coefficient is unity: Now, with we see that
IMPLEMENTATION OF DISCRETETIME SYSTEMS Next. we generate the secondorder system H r ( z )using the stepdown recursion: [CHAP. 8
Therefore, r2 = 0.1905. Finally, we have
and, therefore, I1 = 0.4. Thus, the lattice filter structure is as shown below.
Shown in the figure below is an FIR lattice filter.
((1) Find the system function A ( = ) = F ( z ) / X ( z ) relating the input x ( n ) to the output f ( n ) . Does this system have minimum phase (0) for
(b) Repeat part
the system function relating x ( n ) to ~ ( n ) . ( a ) To find the system function relating .u(tz) to f ' ( t l ) , we use the stepup recursion. Using the vector form of the recursion, we have for the coefficients u l ( k ) Then, with Tz = 0.4, for a (/,), have we
CHAP. 81 Finally, with
IMPLEMENTATION O F DISCRETETIME SYSTEMS
r3= 0.2. we have
Thus.
A(:) = 1 + 0.78;' + 0.54z' + 0.2;' This system will have minimum phase if the zeros of A(z) are inside this unit circle. Although this could be determined by factoring A(z), because the reflection coefficients used to generate A ( : ) are bounded by I In magnitude. it follows that A ( z ) has minimum phase. ( b ) The system function A'(z) = G(.)/X ( z ) is related to X ( z ) as follows: all Because the zeros of A1(z)are formed by flipping the zeros of A ( )about the unit c~rcle. of the zeros of A'(:) will be o~ctside unit circle and thus will not have minimum phase. the

