(Solved): 4. The following figure shows the linear convolution \( x_{3}[n]=x_{1}[n] * x ...

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4. The following figure shows the linear convolution \( x_{3}[n]=x_{1}[n] * x_{2}[n] \) for \( L=8 \) and \( P=4 \). We calculate 15 -point DFTs \( X_{1}[k] \) and \( X_{2}[k] \), and perform \( X_{3}[k]=X_{1}[k] X_{2}[k] \) (a) (10 points) Form \( \tilde{X}_{3}[k] \) with period \( N=15 \) and perform IDFS to obtain \( \tilde{x}_{3}[n] \). Draw \( \tilde{x}_{3}[n] \) for \( -2 \leq n \leq 16 \). (b) (15 points) Shift \( x_{1}[n] \) to the right by 2 sampling points to form \( x_{1}^{\prime}[n]=\left\{\begin{array}{ll}1 & \text { for } 2 \leq n \leq 6 \\ 2 & \text { for } 7 \leq n \leq 9 \\ 0 & \text { otherwise }\end{array}\right. \). Calculate 11 -point DFTs \( X_{3}^{\prime}[k]=X_{1}^{\prime}[k] X_{2}[k] \) and find \( x_{3}^{\prime}[n] \) by IDFT. Draw \( x_{3}^{\prime}[n] \) for \( 0 \leq n \leq 10 \). Hint: Use Item 5 of Table 8.2. (a) (c) TABLE 8.2 SUMMARY OF PROPERTIES OF THE DFT