barcode generator for ssrs Therefore, X ( k ) may be computed from the N/2-point DFTs G ( k ) and H ( k ) . Note that because in Software

Generating Code 128 Code Set B in Software Therefore, X ( k ) may be computed from the N/2-point DFTs G ( k ) and H ( k ) . Note that because

Therefore, X ( k ) may be computed from the N/2-point DFTs G ( k ) and H ( k ) . Note that because
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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 eight-point decimation-in-time FFT is shown in Fig. 7-2. If N / 2 is even, g ( n ) and h ( n ) may again be decimated. For example, G ( k ) may be evaluated as follows:
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THE FAST FOURIER TRANSFORM
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Fig. 7-2. An eight-point decimation-in-time FFT algorithm after the first decimation.
As before, this leads to
where the first term is the N /4-point DFT of the even samples of ~ ( n )and the second is the N/4-point DFT of , the odd samples. A block diagram illustrating this decomposition is shown in Fig. 7-3. If N is a power of 2, the decimation may be continued until there are only two-point DFTs of the form shown in Fig. 7-4.
Fig. 7-3. Decimation of the four-point DFT into two two-point DFTs in the decimation-in-time FFT.
CHAP. 71
THE FAST FOURIER TRANSFORM
The basic computational unit of the FFT, shown in Fig. 7-5(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 eight-point radix-2 decimation-in-time FFT is shown in Fig. 7-6. is
w:'~
Fig. 7-5. ( a ) The butterfly, which is the basic computational clement of the FFT algorith-; ( b )A simplitied butterfly. with only one complex multiplication.
Fig. 7-6. A complete eight-point radix-:! decrmation-in-lime FFT.
Computing an N-point DFT using a radix-2 decimation-in-time 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 decimation-in-time 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 bit-reversed 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
Bit-Reversed Binary
Alternate forms of FFT algorithms may be derived from the' decimation-in-time 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 Decimation-in-Frequency 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 decimation-in-frequency FFTs and may be derived as follows. Let N be a power of 2, N = 2". and consider separately evaluating the even-index and odd-index samples of X(k). The even samples are
X(2k) =
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