barcode generator for ssrs THE DFT in Software

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THE DFT
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[CHAP. 6
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Fig. 6-3. Sampling the z-transform at eight equally spaced points around the unit circle.
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Substituting this expression for Y (k) into Eq. (6.12) gives
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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 overlap-add, and the second is overlap-save.
CHAP. 61
THE DFT
Overlap-Add
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 overlap-add method, x ( n ) is partitioned into nonoverlapping subsequences of length M as illustrated in Fig. 6-4. Thus, x ( n ) may be written as a sum of shifted finite-length 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 N-point DFTs of xi(n)and h(n).The reason for the name overlap-add 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. 6-4. Partitioning a sequence into subsequences of length M for the ovelap-add method of block convolulion.
Overlap-Save
THE DFT
[CHAP. 6
The second way that the DFT may be used to perform linear convolution is to use the overlap-save 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 finite-length sequences of lengths L and N, respectively, the linear convolution y ( n ) is a finite-length sequence of' lengths N L - I . Therefore, assuming that N > L, if we perform an N-point 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. 6-5). 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:
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