 Home
 Products
 Integration
 Tutorial
 Barcode FAQ
 Purchase
 Company
barcode generator for ssrs THE DFT in Software
THE DFT Code 128B Scanner In None Using Barcode Control SDK for Software Control to generate, create, read, scan barcode image in Software applications. Code 128 Code Set B Maker In None Using Barcode encoder for Software Control to generate, create USS Code 128 image in Software applications. [CHAP. 6
Code 128 Reader In None Using Barcode decoder for Software Control to read, scan read, scan image in Software applications. Code 128 Code Set A Generator In C#.NET Using Barcode generation for Visual Studio .NET Control to generate, create Code 128B image in VS .NET applications. Fig. 63. Sampling the ztransform at eight equally spaced points around the unit circle.
Code128 Creator In VS .NET Using Barcode printer for ASP.NET Control to generate, create Code 128 Code Set A image in ASP.NET applications. Creating Code 128 Code Set B In Visual Studio .NET Using Barcode creation for Visual Studio .NET Control to generate, create Code 128B image in Visual Studio .NET applications. Unit circle
Code128 Generator In Visual Basic .NET Using Barcode printer for .NET framework Control to generate, create Code 128 Code Set B image in VS .NET applications. Print EAN 13 In None Using Barcode generation for Software Control to generate, create GS1  13 image in Software applications. Substituting this expression for Y (k) into Eq. (6.12) gives
Bar Code Drawer In None Using Barcode printer for Software Control to generate, create bar code image in Software applications. Code 128 Code Set A Drawer In None Using Barcode generation for Software Control to generate, create Code128 image in Software applications. x()e12r/X/N
Data Matrix 2d Barcode Creation In None Using Barcode generator for Software Control to generate, create Data Matrix image in Software applications. Bar Code Maker In None Using Barcode drawer for Software Control to generate, create barcode image in Software applications. I=m
Making MSI Plessey In None Using Barcode creation for Software Control to generate, create MSI Plessey image in Software applications. Read ECC200 In Java Using Barcode recognizer for Java Control to read, scan read, scan image in Java applications. ej2rnk/N
Encoding UPCA In Java Using Barcode generator for Eclipse BIRT Control to generate, create UPC Symbol image in BIRT reports applications. Reading EAN128 In C# Using Barcode recognizer for Visual Studio .NET Control to read, scan read, scan image in Visual Studio .NET applications. I=m
Decoding EAN13 In C# Using Barcode reader for VS .NET Control to read, scan read, scan image in .NET framework applications. Bar Code Scanner In Visual C# Using Barcode reader for Visual Studio .NET Control to read, scan read, scan image in Visual Studio .NET applications. x(l)(= Bar Code Printer In Java Using Barcode printer for BIRT Control to generate, create bar code image in BIRT reports applications. Making Data Matrix 2d Barcode In None Using Barcode maker for Office Excel Control to generate, create Data Matrix image in Excel applications. X =o
cj2n+/)k/N
The term in brackets is equal to N when I = n Therefore, + mN where m is an integer, and it is equal to zero otherwise.
and it follows that y ( n ) is formed by aliasing x ( n ) in time.
6.6 LINEAR CONVOLUTION USING THE DFT
The DFT provides a convenient way to perform convolutions without having to evaluate the convolution sum. Specifically, if h(n) is N 1 points long and x ( n ) is N2 points long. h ( n ) may be linearly convolved with x ( n ) as follows: Pad the sequences h ( n ) and x ( n ) with zeros so that they are of length N 2 N I
+ N2  1. 2. Find the N point DFTs of h ( n ) and x ( n ) . 3. Multiply the DFTs to form the product Y ( k ) = H (k)X ( k ) . 4. Find the inverse DFT of Y (k). It would appear that there is considerably more effort involved in performing convolutions using DFTs. However, significant computational savings may be realized with this approach if the DFTs are computed efficiently. As we will see in Chap. 7, the fast Fourier transform (FFT) provides such an algorithm. In spite of its computational advantages, there are some difficulties with the DFT approach. For example, if x ( n ) is very long, we must commit a significant amount of time computing very long DFTs and in the process accept very long processing delays. In some cases, it may even be possible that x ( n ) is roo long to compute the DFT. The solution to these problems is to use block convolution, which involves segmenting the signal to be filtered, x ( n ) , into sections. Each section is then filtered with the FIR filter h(n), and the filtered sections are pieced together to form the sequence y(n). There are two block convolution techniques. The first is overlapadd, and the second is overlapsave. CHAP. 61
THE DFT
OverlapAdd
Let x ( n ) be a sequence that is to be convolved with a causal FIR filter h(n)of length L: Assume that x ( n ) = 0 for n i0 and that the length of x ( n ) is much greater than L . In the overlapadd method, x ( n ) is partitioned into nonoverlapping subsequences of length M as illustrated in Fig. 64. Thus, x ( n ) may be written as a sum of shifted finitelength sequences of length M, x ( n )= x r i ( n  M i ) 1 =O where
xi(n) = x(n+Mi) n = 0 . I . . . . , M 1 else
Therefore, the linear convolution of x ( n ) with h(n) is
where yi(n) is the linear convolution of x;(n)with lz(n), Because each sequence yi(n) is of length N = L M  I . it may be found by multiplying the Npoint DFTs of xi(n)and h(n).The reason for the name overlapadd is that, for each i. the sequences yi(n)and y ; + ~ ( n ) overlap at (N  M) points and, in performing the sum in Eq. (6.14). these overlapping points are udded. Fig. 64. Partitioning a sequence into subsequences of length M for the ovelapadd method of block convolulion. OverlapSave
THE DFT
[CHAP. 6
The second way that the DFT may be used to perform linear convolution is to use the overlapsave method. This method takes advantage of the fact that the aliasing that occurs in circular convolution only affects a portion of the sequence. For example, if .r(n) and h ( n ) are finitelength sequences of lengths L and N, respectively, the linear convolution y ( n ) is a finitelength sequence of' lengths N L  I . Therefore, assuming that N > L, if we perform an Npoint circular convolution of x ( n ) with h ( n ) , Because y(n N ) is the only term that is aliased into the interval 0 5 n 5 N  1, and because y ( n N) only overlaps the first L  1 values of y ( n ) , the remaining values in the circular convolution will not be aliased. In other words, the first L  1 values of the circular convolution are not equal to the linear convolution, whereas the last M = N  L I values are the same (see Fig. 65). Thus, with the appropriate partitioning of the input sequence x ( n ) . linear convolution may be performed by piecing together circular convolutions. The procedure is as follows:

