124k views
3 votes
Determine the z-transform, included the region of convergence for the following sequences: 1. x(n)= ()u(n) 2. x(n)= ()u(-n) 71 3. x(n)= = u(n) - 2" u(-n-1)

User Traendy
by
7.9k points

1 Answer

3 votes

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)) +

User Sidharth Ramesh
by
8.1k points