Q5Advanced Engineering Mathematics

The problem dictates the calculation of the discrete Z-transform for the double-sided sequence $x_n=a^{|n|}$, which extends infinitely in both the positive and negative temporal directions. The fundamental definition for a two-sided Z-transform is the infinite bilateral series $Z\{x_n\}=X(z)={\sum }_{n=-\infty }^{\infty }{x}_n{z}^{-n}$. Because the sequence involves an absolute value operation on the index $n$, the mathematical behavior of the sequence fundamentally changes at the origin. Therefore, to evaluate the summation, we must meticulously split it into two independent, one-sided summations: one for negative indices and one for non-negative indices.

1. Decomposing the Bilateral Summation

We mathematically partition the infinite summation around $n=0$:

For the right-hand summation (where $n\ge 0$), the absolute value is redundant since $n$ is inherently positive, so $|n|=n$. The sum simplifies directly to ${\sum }_{n=0}^{\infty }{a}^n{z}^{-n}$.

For the left-hand summation (where $n<0$), the absolute value mathematically equates to negation, so $|n|=-n$. To make the indices positive and standard, we perform a substitution of variables. We let an arbitrary dummy variable $m=-n$. As $n$ travels from $-1$ down to $-\infty$, the new index $m$ will correspondingly travel from $1$ up to $\infty$.

2. Evaluating the Infinite Geometric Series

We now have two distinct infinite geometric series to evaluate. The formula for the sum of an infinite geometric progression ${\sum }_{k=K}^{\infty }{r}^k$ is $\frac{\text{First Term}}{1-\text{Common Ratio}}$, strictly provided that the absolute value of the common ratio $|r|<1$. This strict mathematical condition defines the system's Region of Convergence (ROC).

Evaluating the Positive-Time Summation:

This specific summation only converges if $|a{z}^{-1}|<1$, which algebraically implies $|z|>|a|$.

Evaluating the Negative-Time Summation:

This summation only converges if $|az|<1$, which algebraically implies $|z|<1/|a|$.

3. System Synthesis and ROC Intersections

The complete, overarching transform $X(z)$ only mathematically exists in regions where both individual summations simultaneously converge. Therefore, the total Region of Convergence is the exact intersection of the two individual constraints: $|a|<|z|<1/|a|$. (Note: For this annular region to exist, we must strictly have $|a|<1/|a|$, implying $|a|<1$).

We now algebraically combine the evaluated closed-form sums to construct the final expression:

To provide the most robust final answer, we synthesize this into a single cohesive fraction by finding a common denominator $(1-az)(z-a)$:

This elegantly compact rational function, constrained perfectly within the annular region of convergence $|a|<|z|<1/|a|$, rigorously represents the complete two-sided Z-transform of the symmetric exponential sequence.