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[SOWHAT0819]

Given the sequence representation of the discrete-time signal: 𝑥(𝑛) = {−6, 4, 6, 8, 6, −6, 2, −2, 0, −6, −4} arrow point at positive 2
a. Sketch the function x Draw the following signals:

c. 𝑦(𝑛) = 𝑥(−2𝑛) /2 , . What arrow points at and need a complete solution

d.y = x(n-2/2)

 

[🤖]

To sketch the function \( x = \{-6, 4, 6, 8, 6, -6, 2, -2, 0, -6, -4\} \), we plot the values of the sequence at each integer value of \( n \):

1. \( n = 0 \): \( x(0) = -6 \)
2. \( n = 1 \): \( x(1) = 4 \)
3. \( n = 2 \): \( x(2) = 6 \)
4. \( n = 3 \): \( x(3) = 8 \)
5. \( n = 4 \): \( x(4) = 6 \)
6. \( n = 5 \): \( x(5) = -6 \)
7. \( n = 6 \): \( x(6) = 2 \)
8. \( n = 7 \): \( x(7) = -2 \)
9. \( n = 8 \): \( x(8) = 0 \)
10. \( n = 9 \): \( x(9) = -6 \)
11. \( n = 10 \): \( x(10) = -4 \)

Now, to sketch the signal \( x \), we can plot these points on a graph with the x-axis representing \( n \) and the y-axis representing the values of \( x \).

Moving on to parts c and d:

c. To find \( y = x(-2n) / 2 \), we substitute \( n \) for \( -2n \) in the sequence:

\( y = x(-2n) / 2 = x(-2 \cdot n) / 2 = x(-2n) = \{ x(0), x(-2), x(-4), x(-6), x(-8), x(-10) \} \)

Now, we can find the values of \( y \) at each of these points:

1. \( n = 0 \): \( y(0) = x(0) = -6 \)
2. \( n = -1 \): \( y(-1) = x(2) = 6 \)
3. \( n = -2 \): \( y(-2) = x(4) = 6 \)
4. \( n = -3 \): \( y(-3) = x(6) = 2 \)
5. \( n = -4 \): \( y(-4) = x(8) = 0 \)
6. \( n = -5 \): \( y(-5) = x(10) = -4 \)

Thus, the sequence representation of \( y \) is \( \{-6, 6, 6, 2, 0, -4\} \). The arrow points at the value of \( y \) when \( n = 0 \), which is -6.

d. To find \( y = x(n-2) / 2 \), we shift the sequence \( x \) to the right by 2 units and then scale it by a factor of 1/2:

\( y = x(n-2) / 2 = \{ x(-2), x(-1), x(0), x(1), x(2), x(3), x(4), x(5), x(6), x(7), x(8) \} / 2 \)

Now, we can find the values of \( y \) at each of these points:

1. \( n = -2 \): \( y(-2) = x(-4) / 2 = 6 / 2 = 3 \)
2. \( n = -1 \): \( y(-1) = x(-3) / 2 = 8 / 2 = 4 \)
3. \( n = 0 \): \( y(0) = x(-2) / 2 = 6 / 2 = 3 \)
4. \( n = 1 \): \( y(1) = x(-1) / 2 = -6 / 2 = -3 \)
5. \( n = 2 \): \( y(2) = x(0) / 2 = 4 / 2 = 2
6. \( n = 3 \): \( y(3) = x(1) / 2 = 6 / 2 = 3
7. \( n = 4 \): \( y(4) = x(2) / 2 = 8 / 2 = 4
8. \( n = 5 \): \( y(5) = x(3) / 2 = 6 / 2 = 3
9. \( n = 6 \): \( y(6) = x(4) / 2 = -6 / 2 = -3
10. \( n = 7 \): \( y(7) = x(5) / 2 = 2 / 2 = 1
11. \( n = 8 \): \( y(8) = x(6) / 2 = -2 / 2 = -1

Thus, the sequence representation of \( y \) is \( \{3, 4, 3, -3, 2, 3, 4, 3, -3, 1, -1\} \).