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barcode generator for ssrs The sequence that has an Npoint DFT equal to Y(k) = H(k)X(k) is in Software
The sequence that has an Npoint DFT equal to Y(k) = H(k)X(k) is Decoding Code 128 In None Using Barcode Control SDK for Software Control to generate, create, read, scan barcode image in Software applications. Code 128 Code Set A Encoder In None Using Barcode generation for Software Control to generate, create Code 128C image in Software applications. CHAP. 61
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Print Code 39 In None Using Barcode creator for Font Control to generate, create Code 39 image in Font applications. Barcode Maker In .NET Using Barcode maker for ASP.NET Control to generate, create bar code image in ASP.NET applications. Let y(n) be Ihe linear convolution of the two finitelength sequences, h ( n )and r ( n ) ,of length N , and let y N ( n )be the Npoint circular convolution
Derive the following relationship between y(n) and yN(n): There are several ways toderive this relationship. One is to examine what happens when the DTFT of y(n) is sampled. Alternatively, this result may be derived from a systems point of view as follows. First, note that yN(n) is equal to one period of the linear. convolution of the finitelength sequence h(n) with the periodic sequencei(n): If we let
then the periodic sequence .f(n) is formed by linearly convolving x ( n ) with pN(n): Therefore, the Npoint circular convolution may be written as
THE DFT which is illustrated in the figure below.
[CHAP. 6
Because the first three systems are linear and shiftinvariant. the order of these systems may be interchanged as illustrated in the following figure: However, note that the output of the second filter, y(n), is the linear convolution of h(n) with x(n). This sequence is then convolved with pN(n), which gives the periodic sequence This sequence is then multiplied by the rectangular window R N ( n ) , which is the relationship that was to be established.
How may we compute the Npoint DFT of two realvalued sequences, xl ( n ) and x2(n),using one N point DFT The DFTs of two realvalued sequences may be fbund from one N point DFT as follows. First, we form the Npoint complex sequence After finding the Npoint DFTofx(n), we extract X , ( k )and X2(k)from X(k) by exploiting the symmetry properties of the DFT. Specifically, recall that the DFT of a realvalued sequence is conjugate symmetric, and the DFT of an imaginary sequence is conjugate antisymmetric, Therefore, because X(k) = Xl(k) with X I(k) the DFT of a realvalued sequence, then
+ X (k) which is the conjugate symmetric part of X(k). Similarly, because Xz(k) is the DFT of an imaginary sequence, Xz(k) = i [ ~ ( k X*((N  k ) ) ~ l ) which is the conjugate antisymmetric part of X(k). CHAP. 61
THE DFT
Let x l ( n ) and x z ( n ) be Npoint sequences with Npoint DFTs X l ( k ) and X z ( k ) , respectively. Find an expression for the Npoint DFT of the product x ( n ) = xl(n).rz(n)in terms of X 1 ( k ) and X 2 ( k ) . Just as we have seen with the DTFT, there is a duality in the DFT properties. We have seen, for example, that multiplying a sequence by a complex exponential results in a circular shift of the DFT coefficients. Similarly, multiplying the DFT coefficients by a complex exponential results in a circular shift of the sequence. Therefore, because multiplying DFTcoefficients corresponds to a circular convolution of the sequences, we expect that the multiplication of two sequences would result in the circular convolution of their DFTs. To establish this property, we begin by noting that Because we would like to express X ( k ) in terms of X I ( k )and X 2 ( k ) ,we substitute the following expression for x z ( n ) into Eq. (6.21): The result is
Interchanging the order of the summations, this becomes
Recognizing that the second sum is X I( ( k  we have
Therefore, X ( k ) is I /N times the circular convolution of X I ( k ) with X z ( k ) : If x l ( n ) and q ( n ) are Npoint sequences with Npoint DFTs X I ( k ) and X (k), respectively, show that This result is easy to derive if we use the properties of the DFT thar we already have. First, note that if X ( k ) is the Npoint DFT of x ( n ) = x l ( n ) x ; ( n ) , then Second, note that the DFT of x;(n) is X;((k)),v, Finally. recall that if x ( n ) = .ul(n)x;(n), the Npoint DFT of x ( n ) is I/N times the circular convolution of X l ( k ) and X ; ( (  k ) ) , ~(see Prob. 6.19):

