Answer:
To determine the z-transform and the region of convergence (ROC) for the given sequences, we can use the standard formulas and properties of the z-transform. Let's calculate the z-transform and ROC for each sequence:
Sequence: x(n) = (0.8)^n * u(n)
The z-transform of the sequence x(n) is given by:
X(z) = Σ [x(n) * z^(-n)]
Since the sequence is right-sided (nonzero only for n ≥ 0), we have:
X(z) = Σ [(0.8)^n * u(n) * z^(-n)]
= Σ [(0.8/z)^n * u(n)]
The sum can be evaluated using the formula for the sum of a geometric series:
X(z) = 1 / (1 - 0.8/z) (using the formula for a geometric series)
Simplifying, we get:
X(z) = z / (z - 0.8)
The region of convergence (ROC) is the set of values of z for which the z-transform converges. For this sequence, the ROC includes all values of z for which the magnitude of z is greater than 0.8. In other words:
|z| > 0.8
Sequence: x(n) = (0.8)^n * u(-n)
The sequence x(n) is left-sided (nonzero only for n ≤ 0). To find the z-transform, we need to rewrite the sequence in terms of n ≥ 0.
x(n) = (0.8)^n * u(-n)
= (0.8)^(-n) * u(n)
Now, we can calculate the z-transform:
X(z) = Σ [x(n) * z^(-n)]
= Σ [(0.8)^(-n) * u(n) * z^(-n)]
= Σ [(0.8/z)^n * u(n)]
Using the formula for the sum of a geometric series, we have:
X(z) = 1 / (1 - (0.8/z))
Simplifying further:
X(z) = z / (z - 0.8)
The ROC for this sequence is the set of values of z for which the magnitude of z is less than 0.8. In other words:
|z| < 0.8
Sequence: x(n) = u(n) - 2^(n+1) * u(-n-1)
We can split the sequence into two parts:
x(n) = u(n) - 2^(n+1) * u(-n-1)
= u(n) - 2 * 2^n * u(-n-1)
The first part, u(n), is a right-sided step sequence with a z-transform of 1 / (1 - z^(-1)) and an ROC of |z| > 1.
The second part, 2 * 2^n * u(-n-1), is a left-sided exponential sequence scaled by 2. To find its z-transform, we can rewrite it as:
2 * 2^n * u(-n-1) = 2 * (2/z)^(-n-1) * u(n)
The z-transform of this exponential sequence is given by:
2 * (2/z)^(-n-1) * u(n) = 2 * z^(n+1) * u(n)
The z-transform of the entire sequence is the sum of the z-transforms of the individual parts:
X(z) = 1 / (1 - z^(-1)) +