barcode generator for ssrs Graphical Approach in Software

Generating Code 128 Code Set B in Software Graphical Approach

In addition to the direct method, convolutions may also be performed graphically. The steps involved in using the graphical approach are as follows: Plot both sequences, x(k) and h(k), as functions of k. 2. Choose one of the sequences, say h(k), and time-reverse it to form the sequence h(-k). 3. Shift the time-reversed sequence by n. [Note: If n > 0, this corresponds to a shift to the right (delay), whereas if n < 0, this corresponds to a shift to the left (advance).] 4. Multiply the two sequences x(k) and h(n - k) and sum the product for all values of k. The resulting value will be equal to y(n). This process is repeated for all possible shifts, n.

EXAMPLE 1.4.2 To illustrate the graphical approach to convolution, let us evaluate y ( n ) = x ( n ) * h ( n ) where x ( n ) and h ( n ) are the sequences shown in Fig. 1-6 ( a ) and (b), respectively.To perform this convolution, we follow the steps listed above:

Because x ( k ) and h ( k ) are both plotted as a function of k in Fig. 1-6 ( a ) and (b), we next choose one of the sequences to reverse in time. In this example, we time-reverse h ( k ) , which is shown in Fig. 1-6(c). Forming the product, x ( k ) h ( - k ) , and summing over k , we find that y ( 0 ) = 1. Shifting h ( k ) to the right by one results in the sequence h ( l - k ) shown in Fig. 1-6(d). Forming the product, x ( k ) h ( l - k ) , and summing over k , we find that y ( 1 ) = 3. Shifting h ( l - k ) to the right again gives the sequence h(2 - k ) shown in Fig. 1-6(e). Forming the product, x ( k ) h ( 2 - k ) , and summing over k , we find that y ( 2 ) = 6 . Continuing in this manner, we find that y ( 3 ) = 5. y ( 4 ) = 3, and y ( n ) = 0 for n > 4 . We next take h ( - k ) and shift it to the left by one as shown in Fig. 1-6 (f ). Because the product, x ( k ) h ( - 1 - k ) , is equal to zero for all k , we find that y(- I ) = 0. In fact. y ( n ) = 0 for all n < 0 .

5. 6.

A useful fact to remember in performing the convolution of two finite-length sequences is that if x ( n ) is of length L 1 and h ( n ) is of length L 2 , y ( n ) = x ( n ) * h ( n ) will be of length

Furthermore, if the nonzero values of x ( n ) are contained in the interval [ M,, N,] and the nonzero values of h ( n ) are contained in the interval [Mh, Nh], the nonzero values of y ( n ) will be confined to the interval [ M , Mh, N, Nh].

x(n) =

0 n 0

h(n) =

Because x ( n ) is zero outside the interval [ l o , 201, and h ( n ) is zero outside the interval [ - 5 , 51, the nonzero values of the convolution, y(n) = x ( n ) * h ( n ) ,will be contained in the interval [ 5 , 251.