Z-Transform Solved Problems

Introduction

Working through solved problems is the most effective way to build confidence with the Z-Transform. This collection covers the essential problem types you will encounter in examinations and practical applications: computing forward transforms, finding inverse transforms through partial fractions and long division, analyzing system stability, and determining transfer functions from difference equations. Each solution includes detailed steps so you can follow the reasoning process.

Problem 1: Forward Z-Transform of a Combined Sequence

Find the Z-Transform of $x[n] = (0.8)^n u[n] + 3(0.5)^n u[n]$.

Solution: By the linearity property, we can transform each term separately:

$$\mathcal{Z}\{(0.8)^n u[n]\} = \frac{z}{z-0.8}, \quad |z| > 0.8$$

$$\mathcal{Z}\{3(0.5)^n u[n]\} = \frac{3z}{z-0.5}, \quad |z| > 0.5$$

Therefore: $$X(z) = \frac{z}{z-0.8} + \frac{3z}{z-0.5}$$

Combining over a common denominator: $$X(z) = \frac{z(z-0.5) + 3z(z-0.8)}{(z-0.8)(z-0.5)} = \frac{z^2 - 0.5z + 3z^2 - 2.4z}{(z-0.8)(z-0.5)}$$

$$X(z) = \frac{4z^2 - 2.9z}{z^2 - 1.3z + 0.4}$$

The ROC is $|z| > 0.8$ (outside the outermost pole), since both component signals are causal.

Problem 2: Inverse Z-Transform by Partial Fractions

Find $x[n]$ given $X(z) = \frac{3z}{(z-0.25)(z-0.5)}$ with ROC: $|z| > 0.5$.

Solution: The standard approach is to expand $X(z)/z$ in partial fractions.

Step 1: Form $X(z)/z$: $$\frac{X(z)}{z} = \frac{3}{(z-0.25)(z-0.5)}$$

Step 2: Decompose into partial fractions: $$\frac{3}{(z-0.25)(z-0.5)} = \frac{A}{z-0.25} + \frac{B}{z-0.5}$$

Step 3: Find coefficients using the cover-up method: $$A = \frac{3}{0.25-0.5} = \frac{3}{-0.25} = -12$$

$$B = \frac{3}{0.5-0.25} = \frac{3}{0.25} = 12$$

Step 4: Reconstruct $X(z)$: $$X(z) = \frac{-12z}{z-0.25} + \frac{12z}{z-0.5}$$

Step 5: Invert using the standard pair $\frac{z}{z-a} \leftrightarrow a^n u[n]$ (for ROC outside the pole):

$$x[n] = [-12(0.25)^n + 12(0.5)^n]u[n]$$

Verification: At $n=0$: $x[0] = -12 + 12 = 0$. Check using initial value theorem: $x[0] = \lim_{z\to\infty} X(z) = \lim_{z\to\infty} \frac{3z}{z^2 - 0.75z + 0.125} = 0$. Confirmed.

Problem 3: System Transfer Function and Stability

Given the difference equation: $y[n] - 0.9y[n-1] + 0.2y[n-2] = x[n]$

Find the transfer function $H(z)$ and determine whether the system is stable.

Solution:

Step 1: Take the Z-Transform of both sides (assuming zero initial conditions): $$Y(z)[1 - 0.9z^{-1} + 0.2z^{-2}] = X(z)$$

Step 2: Form the transfer function: $$H(z) = \frac{Y(z)}{X(z)} = \frac{1}{1-0.9z^{-1}+0.2z^{-2}}$$

Multiply numerator and denominator by $z^2$: $$H(z) = \frac{z^2}{z^2-0.9z+0.2}$$

Step 3: Find the poles by factoring the denominator: $$z^2 - 0.9z + 0.2 = (z-0.4)(z-0.5)$$

Poles are at $z = 0.4$ and $z = 0.5$.

Step 4: Check stability. Both poles satisfy $|z| < 1$:

• $|0.4| = 0.4 < 1$ ✓
• $|0.5| = 0.5 < 1$ ✓

Conclusion: The system is BIBO stable because all poles lie inside the unit circle.

Problem 4: Inverse Z-Transform by Long Division

Find the first several values of $x[n]$ for $X(z) = \frac{z}{z^2-3z+2}$.

Solution: Rewrite in terms of $z^{-1}$: $$X(z) = \frac{z^{-1}}{1-3z^{-1}+2z^{-2}}$$

Perform polynomial long division of $z^{-1}$ by $(1-3z^{-1}+2z^{-2})$:

Division step 1: $z^{-1} \div 1 = z^{-1}$ → first term Remainder: $z^{-1} - z^{-1}(1-3z^{-1}+2z^{-2}) = 3z^{-2} - 2z^{-3}$

Division step 2: $3z^{-2} \div 1 = 3z^{-2}$ Remainder: $3z^{-2} - 3z^{-2}(1-3z^{-1}+2z^{-2}) = 9z^{-3} - 6z^{-4}$

Division step 3: $(9z^{-3}-2z^{-3}) = 7z^{-3} \div 1 = 7z^{-3}$

Continuing this process: $x[0] = 0, \; x[1] = 1, \; x[2] = 3, \; x[3] = 7, \; x[4] = 15, \ldots$

The pattern is $x[n] = 2^n - 1$ for $n \geq 1$, which can be verified by partial fraction expansion:

$$\frac{X(z)}{z} = \frac{1}{(z-1)(z-2)} = \frac{-1}{z-1} + \frac{1}{z-2}$$

$$X(z) = \frac{-z}{z-1} + \frac{z}{z-2}$$, giving $x[n] = (-1 + 2^n)u[n] = (2^n - 1)u[n]$

Problem 5: Complex Pole Stability Analysis

Given: $H(z) = \frac{z+1}{z^2+1.5z+0.56}$

Determine if the system is stable and find the impulse response characteristics.

Solution:

Step 1: Find the poles using the quadratic formula: $$z = \frac{-1.5 \pm \sqrt{1.5^2 - 4(0.56)}}{2} = \frac{-1.5 \pm \sqrt{2.25 - 2.24}}{2} = \frac{-1.5 \pm \sqrt{0.01}}{2}$$

$$z = \frac{-1.5 \pm 0.1}{2}$$

So: $z_1 = \frac{-1.4}{2} = -0.7$ and $z_2 = \frac{-1.6}{2} = -0.8$

Step 2: Check magnitudes:

• $|z_1| = 0.7 < 1$ ✓
• $|z_2| = 0.8 < 1$ ✓

Conclusion: The system is BIBO stable. Since both poles are real and negative, the impulse response will exhibit alternating-sign behavior (oscillation with decay).

Problem 6: Z-Transform with ROC Ambiguity

Given: $X(z) = \frac{z}{z-2}$. Find $x[n]$ for (a) ROC: $|z| > 2$, and (b) ROC: $|z| < 2$.

Solution:

(a) ROC: $|z| > 2$ implies a right-sided (causal) signal: $$x[n] = 2^n u[n]$$

This is an exponentially growing causal sequence.

(b) ROC: $|z| < 2$ implies a left-sided (anti-causal) signal: $$x[n] = -2^n u[-n-1]$$

This is a decaying sequence for negative time. Note that the algebraic expression for $X(z)$ is identical in both cases — only the ROC distinguishes the two entirely different sequences.

Problem 7: Transfer Function from Block Diagram

A system has the difference equation $y[n] = x[n] + 2x[n-1] - 0.5y[n-1] + 0.25y[n-2]$.

Find $H(z)$, its poles and zeros, and determine stability.

Solution:

Taking the Z-Transform: $$Y(z) = X(z) + 2z^{-1}X(z) - 0.5z^{-1}Y(z) + 0.25z^{-2}Y(z)$$

$$Y(z)[1 + 0.5z^{-1} - 0.25z^{-2}] = X(z)[1 + 2z^{-1}]$$

$$H(z) = \frac{1 + 2z^{-1}}{1 + 0.5z^{-1} - 0.25z^{-2}} = \frac{z^2 + 2z}{z^2 + 0.5z - 0.25}$$

Zeros: $z(z+2) = 0$ → zeros at $z = 0$ and $z = -2$

Poles: Using the quadratic formula on $z^2 + 0.5z - 0.25 = 0$: $$z = \frac{-0.5 \pm \sqrt{0.25 + 1}}{2} = \frac{-0.5 \pm 1.118}{2}$$

$z_1 = 0.309$ and $z_2 = -0.809$

Since $|0.309| < 1$ and $|-0.809| = 0.809 < 1$, the system is stable.

Key Problem-Solving Strategies

1. For forward transforms: Use linearity and the standard table of Z-Transform pairs
2. For inverse transforms: Always try partial fractions via $X(z)/z$ first; use long division for verification
3. For stability: Find all poles of $H(z)$ and verify $|p_i| < 1$ for each pole
4. Long division provides a direct numerical check — the coefficients are the sequence values
5. ROC matters: The same $X(z)$ with different ROCs yields completely different sequences
6. Transfer function: Take Z-Transform of the difference equation assuming zero initial conditions
7. Complex poles: Convert to polar form $z = re^{j\theta}$ to identify oscillation frequency and decay rate