What is the formula for Z-transform of na)𝑧(𝑧+1)(z+1)z​ b)𝑧(𝑧−1)2(z−1) 2 z​ c)𝑧(𝑧+1)2(z+1) 2 z​ d)𝑧(𝑧−1)(z−1)z​

To find the Z-transform formula for the given expressions, we can analyze each expression one at a time. The Z-transform of a sequence $x[n]$ is defined as:

$X(z) = \sum_{n=-\infty}^{\infty} x[n] z^{-n}$

A. For the expression $a) \quad \frac{z}{(z+1)z}$

1. Break Down the Problem:

   • The expression is $\frac{z}{(z+1)z}$.
   • Simplifying it gives $\frac{1}{z+1}$.

2. Relevant Concepts:

   • The Z-transform of $a[n] = (-1)^n u[n]$ is known to be $\frac{1}{1 + z}$.

3. Analysis and Detail:

   • Here, $u[n]$ is the unit step function.

4. Verify and Summarize:

   • Therefore, the Z-transform matches the known transform.

Final Answer:

The Z-transform for $\frac{z}{(z+1)}$ is: $X(z) = \frac{1}{1 + z}$

B. For the expression $b) \quad \frac{z}{(z-1)^2 z}$

1. Break Down the Problem:

   • The expression simplifies to $\frac{1}{(z-1)^2}$.

2. Relevant Concepts:

   • The Z-transform of $a[n] = n u[n]$ is known to be $\frac{z}{(z-1)^2}$.

3. Analysis and Detail:

   • The Z-transform corresponds to the sequence of $n$ scaled by the unit step function.
   • The inverse transform of $\frac{1}{(z-1)^2}$ corresponds to $n u[n]$.

4. Verify and Summarize:

   • Hence, $\frac{1}{(z-1)^2}$ confirms the known relationship.

Final Answer:

The Z-transform for $\frac{z}{(z-1)^2}$ is: $X(z) = \frac{1}{(z-1)^2}$

C. For the expression $c) \quad \frac{z}{(z+1)^2 z}$

1. Break Down the Problem:

   • The expression simplifies to $\frac{1}{(z+1)^2}$.

2. Relevant Concepts:

   • The Z-transform of $a[n] = n (-1)^n u[n]$ is known to be $\frac{z}{(z+1)^2}$.

3. Analysis and Detail:

   • The result corresponds to the Z-transform of the geometric sequence scaled by the unit step function.

4. Verify and Summarize:

   • Same consistency as previous transformations.

Final Answer:

The Z-transform for $\frac{z}{(z+1)^2}$ is: $X(z) = \frac{1}{(z+1)^2}$

D. For the expression $d) \quad \frac{z}{(z-1)(z-1)z}$

1. Break Down the Problem:

   • The expression simplifies to $\frac{1}{(z-1)^2}$.

2. Relevant Concepts:

   • This again corresponds to the Z-transform of $n u[n]$ scaled by $z$.

3. Analysis and Detail:

   • The expressions for Z-transforms agree with linear transformations.

4. Verify and Summarize:

   • Consistent with known Z-transform properties.

Final Answer:

The Z-transform for $\frac{z}{(z-1)^2}$ is: $X(z) = \frac{1}{(z-1)^2}$

In summary, these questions revolve around recognizing Z-transform properties and simplifying given expressions.