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Question 1: Let x[n] be a discrete time sequence: x[n] = \begin{cases} (0.3)^n & 0 \le n \le 7 \\ 0 & \text{elsewhere} \end{cases} 1. Compute the 8-point DFT of x[n]. Plot the magnitude and phase [Figure 1.1]. 2. Compute the 16-point DFT of x[n]. Plot the magnitude and phase [Figure 1.2]. 3. Compare the graphs of 1 and 2 in one figure (use proper axis labels and titles) [Figure 1.3]. Question 2: To prove the Convolution property of the DFT, considering the following sequences: x[n] = [-3 5 8 6 2 2] y[n] = [1 1 4 2 0 0] 1. Perform the linear convolution of the two sequences z = x * y and show a figure [Figure 2.1] for all sequences x, y and z aligned vertically. 2. Compute the DFT of the two sequences x and y (name the result Xk and Yk respectively). 3. Find Zk by performing a point-by-point product of Xk and Yk. Then find the inverse DFT for the result sequence (name it zz). (fff 4. Compare the results of 1 with the result of 3 by computing the difference in sequence power by using: diff = sum(|zz - z|^2) Question 3: Given the system transfer function: $H(z) = \frac{1 - z^{-1}}{1 + 3z^{-1} + 2z^{-2}}$ 1. Compute the gain and Plot the Z-Plane graph of this system [Figure 3.1]. 2. Find the roots of the Residual Partial-Fraction Z-transform.

          Question 1: Let x[n] be a discrete time sequence:
x[n] = \begin{cases} (0.3)^n & 0 \le n \le 7 \\ 0 & \text{elsewhere} \end{cases}
1. Compute the 8-point DFT of x[n]. Plot the magnitude and phase [Figure 1.1].
2. Compute the 16-point DFT of x[n]. Plot the magnitude and phase [Figure 1.2].
3. Compare the graphs of 1 and 2 in one figure (use proper axis labels and titles) [Figure 1.3].
Question 2: To prove the Convolution property of the DFT, considering the following
sequences:
x[n] = [-3 5 8 6 2 2]
y[n] = [1 1 4 2 0 0]
1. Perform the linear convolution of the two sequences z = x * y and show a figure [Figure
2.1] for all sequences x, y and z aligned vertically.
2. Compute the DFT of the two sequences x and y (name the result Xk and Yk respectively).
3. Find Zk by performing a point-by-point product of Xk and Yk. Then find the inverse DFT
for the result sequence (name it zz). (fff
4. Compare the results of 1 with the result of 3 by computing the difference in sequence power
by using:
diff = sum(|zz - z|^2)
Question 3: Given the system transfer function:
$H(z) = \frac{1 - z^{-1}}{1 + 3z^{-1} + 2z^{-2}}$
1. Compute the gain and Plot the Z-Plane graph of this system [Figure 3.1].
2. Find the roots of the Residual Partial-Fraction Z-transform.

Question 1: Let x[n] be a discrete time sequence:
x[n] = (0.3)^n 0 ≤n ≤7
0 elsewhere
1. Compute the 8-point DFT of x[n]. Plot the magnitude and phase [Figure 1.1].
2. Compute the 16-point DFT of x[n]. Plot the magnitude and phase [Figure 1.2].
3. Compare the graphs of 1 and 2 in one figure (use proper axis labels and titles) [Figure 1.3].
Question 2: To prove the Convolution property of the DFT, considering the following
sequences:
x[n] = [-3 5 8 6 2 2]
y[n] = [1 1 4 2 0 0]
1. Perform the linear convolution of the two sequences z = x * y and show a figure [Figure
2.1] for all sequences x, y and z aligned vertically.
2. Compute the DFT of the two sequences x and y (name the result Xk and Yk respectively).
3. Find Zk by performing a point-by-point product of Xk and Yk. Then find the inverse DFT
for the result sequence (name it zz). (fff
4. Compare the results of 1 with the result of 3 by computing the difference in sequence power
by using:
diff = sum(|zz - z|^2)
Question 3: Given the system transfer function:
H(z) = (1 - z^-1)/(1 + 3z^-1 + 2z^-2)
1. Compute the gain and Plot the Z-Plane graph of this system [Figure 3.1].
2. Find the roots of the Residual Partial-Fraction Z-transform.

Added by Margarita I.

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