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barcode generator for ssrs Therefore, X ( k ) may be computed from the N/2point DFTs G ( k ) and H ( k ) . Note that because in Software
Therefore, X ( k ) may be computed from the N/2point DFTs G ( k ) and H ( k ) . Note that because Code 128 Code Set C Recognizer In None Using Barcode Control SDK for Software Control to generate, create, read, scan barcode image in Software applications. Encode Code 128A In None Using Barcode generation for Software Control to generate, create ANSI/AIM Code 128 image in Software applications. and it is only necessary to form the products W ; H ( k ) for k = 0. 1, . . . , N/2  1. The complex exponentials multiplying H ( k ) in Eq. ( 7 . 3 ) are called twiddle factors. A block diagram showing the computations that are necessary for the first stage of an eightpoint decimationintime FFT is shown in Fig. 72. If N / 2 is even, g ( n ) and h ( n ) may again be decimated. For example, G ( k ) may be evaluated as follows: Scanning Code 128 Code Set B In None Using Barcode recognizer for Software Control to read, scan read, scan image in Software applications. Encoding ANSI/AIM Code 128 In C# Using Barcode creation for Visual Studio .NET Control to generate, create Code 128A image in Visual Studio .NET applications. II=O
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As before, this leads to
where the first term is the N /4point DFT of the even samples of ~ ( n )and the second is the N/4point DFT of , the odd samples. A block diagram illustrating this decomposition is shown in Fig. 73. If N is a power of 2, the decimation may be continued until there are only twopoint DFTs of the form shown in Fig. 74. Fig. 73. Decimation of the fourpoint DFT into two twopoint DFTs in the decimationintime FFT.
CHAP. 71
THE FAST FOURIER TRANSFORM
The basic computational unit of the FFT, shown in Fig. 75(a), is called a hurterjy. This structure may be simplified by factoring out a term W h from the lower branch as illustrated in Fig. 7  5 ( h ) . The factor that remains =  1. A complete eightpoint radix2 decimationintime FFT is shown in Fig. 76. is w:'~
Fig. 75. ( a ) The butterfly, which is the basic computational clement of the FFT algorith; ( b )A simplitied butterfly. with only one complex multiplication. Fig. 76. A complete eightpoint radix:! decrmationinlime FFT.
Computing an Npoint DFT using a radix2 decimationintime FFT is much more efficient than calculating the DFT directly. For example, if N = 2", there are log, N = v stages of computation. Because each stage requires N / 2 complex multiplies by the twiddle factors W h and N complex additions. there are a total of N logz N complex multiplications' and N log2 N complex additions. From the structure of the decimationintime FFT algorithm, note that once a butterfly operation has been performed on a pair of complex numbers, there is no need to save the input pair. Therefore, the output pair may be stored in the same registers as the input. Thus, only one array of size N is required, and it is said that the computations may be performed in place. To perform the computations in place, however, the input sequence x ( n ) must be stored (or accessed) in nonsequential order as seen in Fig. 7  6 . The shufling of the input sequence that takes place is due to the successive decimations of .u(n). The ordering that results corresponds to a bitreversed indexing of the original sequence. In other words, if the index n is written in binary form, the order in which in the input sequence must be accessed is found by reading the binary representation for n in reverse order as illustrated ill the table below for N = 8: h he number of multiplications is actually a bit less than this because some of the twiddle factors are equal to THE FAST FOURIER TRANSFORM
[CHAP. 7
Binary
BitReversed Binary
Alternate forms of FFT algorithms may be derived from the' decimationintime FFT by manipulating the flowgraph and rearranging the order in which the results of each stage of the computation are stored. For example, the nodes of the flowgraph may be rearranged so that the input sequence x(n) is in normal order. What is lost with this reordering, however, is the ability to perform the computations in place. 7.2.2 DecimationinFrequency FFT
Another class of FFT algorithms may be derived by decimating the output sequence X(k) into smaller and smaller subsequences. These algorithms are called decimationinfrequency FFTs and may be derived as follows. Let N be a power of 2, N = 2". and consider separately evaluating the evenindex and oddindex samples of X(k). The even samples are X(2k) =

