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(a) With a direct form implementation of the FIR filter H (2). the decimator is as shown below. in Software
(a) With a direct form implementation of the FIR filter H (2). the decimator is as shown below. Code128 Scanner In None Using Barcode Control SDK for Software Control to generate, create, read, scan barcode image in Software applications. Code 128 Code Set C Creation In None Using Barcode printer for Software Control to generate, create USS Code 128 image in Software applications. Because we need N multiplies and N  1 adds to find each value of w(n),and because only one value of y(n) is computed for every M values of w ( n ) ,M N multiplies and M ( N  1) adds are performed for each value of y ( n ) . USS Code 128 Scanner In None Using Barcode reader for Software Control to read, scan read, scan image in Software applications. Encode USS Code 128 In Visual C# Using Barcode maker for .NET Control to generate, create Code 128B image in .NET applications. (b) Because the downsampler only saves one out of every M values of w(n), the decimator may be implemented more efficiently by only evaluating those values of w ( n ) that are passed through the downsampler. This may be accomplished by embedding the downsampler within the FIR filter as illustrated below. Make Code128 In Visual Studio .NET Using Barcode encoder for ASP.NET Control to generate, create Code 128 image in ASP.NET applications. Encode Code128 In Visual Studio .NET Using Barcode maker for .NET framework Control to generate, create Code128 image in VS .NET applications. CHAP. 81
Draw Code 128 In Visual Basic .NET Using Barcode generator for VS .NET Control to generate, create ANSI/AIM Code 128 image in VS .NET applications. Generating Code 128 Code Set A In None Using Barcode creation for Software Control to generate, create USS Code 128 image in Software applications. IMPLEMENTATION OF DISCRETETIME SYSTEMS
Paint GTIN  128 In None Using Barcode encoder for Software Control to generate, create EAN / UCC  13 image in Software applications. Code 39 Extended Maker In None Using Barcode generator for Software Control to generate, create Code 39 image in Software applications. Now. because only one out of every M input samples is multiplied by h ( k ) , this implementation only requires N multiplies and N  1 adds to compute each value of y ( n ) . Thus. the number of multiplies and adds has been reduced by a factor of M. Paint UPC Symbol In None Using Barcode generation for Software Control to generate, create UPCA image in Software applications. Encoding Bar Code In None Using Barcode drawer for Software Control to generate, create bar code image in Software applications. (c) If H ( z ) is an IIR filter, it is not possible, in general, to commute the downsampling operation with branch Leitcode Drawer In None Using Barcode printer for Software Control to generate, create Leitcode image in Software applications. Data Matrix 2d Barcode Drawer In ObjectiveC Using Barcode generation for iPhone Control to generate, create Data Matrix 2d barcode image in iPhone applications. operations as was done with the FIR filter. For example, if
Bar Code Encoder In Java Using Barcode drawer for Java Control to generate, create bar code image in Java applications. Code 128A Encoder In None Using Barcode generator for Microsoft Word Control to generate, create Code128 image in Word applications. we have the system illustrated below.
USS Code 39 Reader In None Using Barcode recognizer for Software Control to read, scan read, scan image in Software applications. EAN13 Supplement 5 Generator In Visual Studio .NET Using Barcode drawer for .NET framework Control to generate, create EAN13 Supplement 5 image in VS .NET applications. However, in order to evaluate a given value of w ( n ) . the previous value, w ( n  I), must be known. Therefore, the downsampler cannot be commuted with any branch operations within the filter, because this would discard values of w ( n ) that are required to compute future values. On the other hand, consider the direct form I1 implementation of Generate DataMatrix In C# Using Barcode printer for VS .NET Control to generate, create DataMatrix image in .NET applications. ECC200 Drawer In ObjectiveC Using Barcode encoder for iPad Control to generate, create Data Matrix image in iPad applications. as illustrated below.
Because
w ( n ) = b(O)v(n) +b(l)u(n  1 ) the downsampler may be commuted with the branch operations that form the multiplications by b ( 0 ) and b(1) as illustrated in the following figure: To compute each value of y ( n ) , this structure requires that we find M values of v ( n ) , which requires M multiplies and M adds, and it requires two multiplies and one add to find y ( n ) from u(n). Thus, the total number of computations is M 2 multiplications and M I additions. The direct form 1 structure is the only one that 1 allows for a savings in computation. For direct form I, transposed direct form 1, and transposed direct form 11. the downsampler cannot be commuted with any branch operations. IMPLEMENTATION O F DISCRETETIME SYSTEMS
[CHAP. 8
The previous problem examined the simplifications that are possible in implementing a decimator. Similar savings are possible for the interpolator shown in the figure below. Because the upsampler inserts L  I zeros between each sample of x ( n ) , assume that H ( z ) is the system function of an FIR filter, and use the fact that many of the values of w ( n ) are equal to zero to derive a more efficient implementation of this system. A direct implementation of the cascade of an upsampler with an FIR filter using the transposed direct form is illustrated in the figure below. Note that the evaluation of each value of y ( n ) requires N multiplications and N  I additions. However, only one out of every L values that are being multiplied by the coefficients h ( n ) is nonzero. Therefore, it is more efficient to modify the structure so that the filtering is performed prior to the insertion of zeros. With the transposed direct form structure, we may commute the upsampler with the branch multiplies as illustrated in the following figure: With this simplification, only N multiplies and N
1 adds are required for every L output values.
CHAP. 81
IMPLEMENTATION OF DISCRETETIME SYSTEMS
Structures for IIR Systems
Consider the causal linear shiftinvariant filter with system function
Draw a signal flowgraph for this system using
(a) Direct form I (b) Direct form 11
(c) A casczlde of first and secondorder systems realized in direct form I1 (4 A cascade of first and secondorder systems realized in transposed direct form I1

