To determine the inverse z-transform \(\text{x[n]}\) from the given z-transform \(\text{X(z)}\), we will utilize the fundamental definition of the z-transform along with the known z-transform properties of the impulse function.

The provided z-transform is:

\[ \text{X(z)} = 5z^2 + 4z^{-1} + 3 \]

The Region of Convergence (ROC) is specified as \(\text{0 < |z| < \infty}\). This ROC is characteristic of finite-duration signals, which consist of a finite number of non-zero samples.

Z-transform Principles and Impulse Functions

The z-transform for a discrete-time signal \(\text{x[n]}\) is mathematically defined as a power series:

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

To find the inverse z-transform \(\text{x[n]}\), we need to identify the discrete-time sequence whose z-transform matches each term in the given \(\text{X(z)}\). Key properties of the z-transform for the unit impulse function \(\delta[n]\) and its shifts are essential:

• The z-transform of a unit impulse occurring at \(\text{n=0}\) is: \( \text{Z}\{\delta[n]\} = 1 \)
• For a right-shifted impulse, where \(k\) is a positive integer: \( \text{Z}\{\delta[n-k]\} = z^{-k} \)
• For a left-shifted impulse, where \(k\) is a positive integer: \( \text{Z}\{\delta[n+k]\} = z^{k} \)

The given ROC, \(\text{0 < |z| < \infty}\), confirms that \(\text{x[n]}\) is a finite-duration sequence, which means it has non-zero values only for a limited range of \(n\).

Inverse Z-transform Derivation

We can systematically determine the inverse z-transform for each individual term in the given \(\text{X(z)}\) by applying the properties of the z-transform:

• Term 1: \(5z^2\)

  Comparing the term \(z^2\) with the property \(z^{k}\), we deduce that \(k=2\). Therefore, this term corresponds to an impulse function shifted to the left by 2 units, scaled by 5.

  \( \text{Z}\{5\delta[n+2]\} = 5z^2 \)

• Term 2: \(4z^{-1}\)

  By comparing the term \(z^{-1}\) with the property \(z^{-k}\), we find that \(k=1\). This indicates an impulse function shifted to the right by 1 unit, scaled by 4.

  \( \text{Z}\{4\delta[n-1]\} = 4z^{-1} \)

• Term 3: \(3\)

  The constant term \(3\) corresponds directly to an impulse function at \(\text{n=0}\), scaled by 3, based on the property \( \text{Z}\{\delta[n]\} = 1 \).

  \( \text{Z}\{3\delta[n]\} = 3 \)

Combining for the Complete Inverse Transform

Due to the linearity property of the z-transform, the inverse z-transform \(\text{x[n]}\) of the sum of terms is the sum of the inverse z-transforms of each individual term:

\[ \text{x[n]} = \text{Z}^{-1}\{5z^2 + 4z^{-1} + 3\} \]

\[ \text{x[n]} = \text{Z}^{-1}\{5z^2\} + \text{Z}^{-1}\{4z^{-1}\} + \text{Z}^{-1}\{3\} \]

Substituting the inverse transforms we determined for each term:

\[ \text{x[n]} = 5\delta[n+2] + 4\delta[n-1] + 3\delta[n] \]

Rearranging the terms for clarity, for example, by the position of the impulse in time:

\[ \text{x[n]} = 5\delta[n+2] + 3\delta[n] + 4\delta[n-1] \]

This expression represents the inverse z-transform \(\text{x[n]}\) of the given \(\text{X(z)}\).