← 9.08 Transient Response of Series & Parallel RLC Networks | Chapter 10 Map | 10.02 Z-Transform Properties & Standard Pairs →
10.01 Z-Transform: Definitions & Region of Convergence (ROC)
Alright — let’s cross the boundary from continuous-time systems into transform-domain discrete-time analysis! Just as the Laplace Transform generalized the Continuous-Time Fourier Transform (CTFT) to analyze unstable or growing continuous waveforms, the Z-Transform generalizes the Discrete-Time Fourier Transform (DTFT) to analyze discrete-time sequences by introducing a complex radial scaling factor.
By mapping discrete sequences from the time-domain index into the complex -plane, we transform linear difference equations into simple algebraic expressions. Let’s derive the fundamental definitions, prove how the continuous -plane maps to the discrete -plane, establish Region of Convergence (ROC) properties, and master the core “exam-killer” numericals to secure full marks on your exam!
1. Unilateral vs. Bilateral Z-Transform Definitions
Depending on the causal boundaries of your signal, the Z-transform is mathematically defined in two ways:
1.1 Bilateral (Two-Sided) Z-Transform
The bilateral Z-transform of an arbitrary, infinite-duration discrete-time signal is defined as:
where is a continuous complex variable represented in polar coordinates as: Here, is the radial distance (magnitude) from the origin of the complex -plane, and \omega = ngle z represents the angular frequency in radians per sample.
1.2 Unilateral (One-Sided) Z-Transform
The unilateral Z-transform is appropriate for causal signals and systems {where for all } and is heavily used to solve difference equations with non-zero initial conditions:
1.3 Structural Comparison
Let’s look at the key operational differences between these two domains:
| Feature | Bilateral (Two-Sided) Z-Transform | Unilateral (One-Sided) Z-Transform |
|---|---|---|
| Summation Limits | Sums over | Sums strictly over |
| Sequence Handling | Evaluates general causal, anticausal, or bilateral sequences. | Assumes causality {forces for }. |
| ROC Complexity | Must specify explicit ROC boundaries to ensure uniqueness. | The ROC is always the exterior of a circle (no need to specify). |
| Initial Conditions | Cannot handle systems with non-zero initial states. | Excellent for solving difference equations with initial conditions. |
| Laplace Counterpart | Analogous to the Bilateral Laplace transform. | Analogous to the Unilateral Laplace transform. |
2. Mathematical Derivation: Laplace-to-Z Mapping ()
NOTE
This derivation is a highly tested exam classic [PYQ 2024, 2019, 2017, 2015] commanding 6 to 8 marks. It mathematically bridges continuous-time and discrete-time transform spaces.
Let’s begin with a continuous-time signal . When we sample uniformly at intervals of seconds, we can model the physical sampled signal mathematically as a continuous-time multiplication of with an ideal periodic impulse train:
Now, let’s take the Unilateral Laplace transform of this continuous sampled impulse train :
ight] e^{-st} \, dt$$ Because the integral operator is linear, we can slide the integration inside the summation: $$X^*(s) = \sum_{n=0}^{\infty} x(nT_s) \int_{0^{-}}^{\infty} \delta(t - nT_s) e^{-st} \, dt$$ Applying the **Sifting Property** of the unit impulse function {$\int \delta(t - t_0)f(t)dt = f(t_0)$}: $$X^*(s) = \sum_{n=0}^{\infty} x(nT_s) e^{-n s T_s} \quad [148]$$ Let's look at this equation side-by-side with the definition of the unilateral Z-transform of the discrete-time sampled sequence $x[n] = x(nT_s)$: $$X(z) = \sum_{n=0}^{\infty} x[n] z^{-n} \quad [149]$$ By direct matching of the exponential terms: $$z^{-n} = e^{-n s T_s} \implies z = e^{s T_s} \quad [147]$$ ### 2.1 Mapping the Complex Plane Territories Let's analyze how this complex exponential relationship $z = e^{s T_s}$ maps the continuous complex $s$-plane ($s = \sigma + j\Omega$) into the discrete complex $z$-plane: $$z = e^{(\sigma + j\Omega)T_s} = e^{\sigma T_s} e^{j\Omega T_s} \quad [147]$$ Expressing this mapped point in polar coordinates $z = r e^{j\omega}$ reveals the matching relationships: 1. **Magnitude Mapping:** $r = |z| = e^{\sigma T_s}$ 2. **Frequency Mapping:** $\omega = ngle z = \Omega T_s$ This math yields three absolute geometric mappings: #### Case 1: The imaginary $j\Omega$-axis of the $s$-plane maps to the unit circle $|z| = 1$ * On the imaginary axis, the real attenuation factor is strictly zero ($\sigma = 0$). * Therefore, the magnitude of $z$ becomes: $$|z| = e^{0 \cdot T_s} = 1 \quad [147]$$ * As the continuous frequency $\Omega$ sweeps from $-\infty$ to $+\infty$, the mapped point wraps around the unit circle $|z| = 1$ in the $z$-plane periodically with period $2\pi/T_s$. #### Case 2: The Left-Half Plane (LHP) of the $s$-plane maps to the interior of the unit circle $|z| < 1$ * For stable systems, poles must lie in the LHP, meaning the real part is negative ($\sigma < 0$). * Therefore, the radial magnitude in the $z$-plane is bounded by: $$|z| = e^{\sigma T_s} < 1 \quad ( ext{since } \sigma < 0) \quad [147]$$ * This proves that the entire stable LHP of continuous-time systems maps directly **inside the unit circle** in the $z$-plane. #### Case 3: The Right-Half Plane (RHP) of the $s$-plane maps to the exterior of the unit circle $|z| > 1$ * Unstable system poles lie in the RHP, where the real growth factor is positive ($\sigma > 0$). * Therefore, the radial magnitude in the $z$-plane is: $$|z| = e^{\sigma T_s} > 1 \quad ( ext{since } \sigma > 0) \quad [147]$$ * This maps the unstable continuous territory **outside the unit circle**. ``` Continuous s-Plane Discrete z-Plane jΩ Im(z) ▲ ▲ LHP │ RHP _.-'''''''-._ ROC: Exterior (Stable) │ (Unstable) .-' _.._ '-. (Causal Stable) ◄─────────┼─────────► σ .' .' '. '. xxxxxxxxx │ Z(s)=eˢᵀ │ │ o │ │ xxxxxxxxx │ ────────►│◄────┼────*───┼───────┼► Re(z) Poles (x) │ │ │ z=0 │Pole(x)│ ◄── σ < 0 │ ──► σ > 0 '. '. .' .' |z|=1 │ '-._ '..' _.-' (Unit Circle) ▼ '-._____.-' ``` --- ## 3. Core Concept: The Region of Convergence (ROC) The **Region of Convergence (ROC)** of a Z-transform is defined as the set of values of $z$ in the complex $z$-plane for which the infinite summation converges to a finite, bounded value {i.e., $|X(z)| < \infty$}. If the summation diverges to infinity, the Z-transform does not exist for those values of $z$. ### 3.1 Mathematical Convergence Condition For $X(z)$ to converge, the series must be absolutely summable: $$\sum_{n=-\infty}^{\infty} |x[n] z^{-n}| < \infty \implies \sum_{n=-\infty}^{\infty} |x[n]| r^{-n} < \infty \quad [37, 153, 154]$$ This absolute convergence summation represents the Discrete-Time Fourier Transform (DTFT) of the exponentially weighted sequence $x[n]r^{-n}$. ### 3.2 The 8 Golden Properties of the ROC These properties are critical for identifying system boundaries during exam problems: 1. **The ROC is a ring or disk in the $z$-plane centered at the origin:** Since convergence depends strictly on $|z| = r$, the ROC is bounded by concentric circles. 2. **The ROC cannot contain any poles:** By definition, a pole is a root where $X(z) o \infty$. Since the ROC contains only points where $X(z)$ is finite, poles always form the physical boundaries of the ROC. 3. **ROC of a Finite-Duration Causal Sequence:** The ROC is the **entire $z$-plane except $z = 0$**. *{Explanation: If $x[n]$ exists only from $n = 0$ to $N-1$, $X(z) = x[0] + x[1]z^{-1} + \dots + x[N-1]z^{-N+1}$. The negative powers of $z$ blow up only when $z = 0$}.* 4. **ROC of a Finite-Duration Anticausal Sequence:** The ROC is the **entire $z$-plane except $z = \infty$**. *{Explanation: If $x[n]$ exists only for negative indices, $X(z) = x[-N]z^{N} + \dots + x[-1]z^1$. The positive powers of $z$ blow up only when $z o \infty$}.* 5. **ROC of a Finite-Duration Two-Sided Sequence:** The ROC is the **entire $z$-plane except $z = 0$ and $z = \infty$**. 6. **ROC of an Infinite-Duration Right-Sided (Causal) Sequence:** The ROC is the **exterior of a circle** $|z| > r_1$, extending out to infinity. 7. **ROC of an Infinite-Duration Left-Sided (Anticausal) Sequence:** The ROC is the **interior of a circle** $|z| < r_2$. 8. **ROC of an Infinite-Duration Two-Sided Sequence:** The ROC is an **open annular ring** $r_1 < |z| < r_2$, bounded by poles on both the inner and outer circles. --- ## 4. High-Yield Worked "Exam Killers" [PYQ Bank] Let's work through the highest-yielding Z-transform calculation problems step-by-step to guarantee maximum marks on your papers. ### 4.1 The Causal Window Sequence [PYQ 2025] > [!question] **2025 Exam Section B Q. 7b (05 Marks)** > > Find the Z-transform of the sequence $x[n] = [u[n] - u[n-5]]$ and specify its Region of Convergence (ROC). #### Step 1: Write down the sequence values The unit step difference represents a finite-duration causal pulse of length 5: $$x[n] = egin{cases} 1, & 0 \le n \le 4 \ 0, & ext{otherwise} \end{cases}$$ $$x[n] = \{1, 1, 1, 1, 1\}$$ #### Step 2: Apply the Z-transform definition $$X(z) = \sum_{n=0}^{4} x[n] z^{-n} = 1 + z^{-1} + z^{-2} + z^{-3} + z^{-4} \quad [74]$$ This is a finite geometric series. Applying the sum of a finite geometric progression ($S_N = rac{1 - q^N}{1 - q}$): $$X(z) = rac{1 - z^{-5}}{1 - z^{-1}} = rac{z^5 - 1}{z^4 (z - 1)} \quad [74]$$ #### Step 3: Determine the ROC and Pole-Zero behavior * **Zeros:** Roots of $z^5 - 1 = 0 \implies z = e^{j 2\pi k / 5}$ for $k = 0, 1, 2, 3, 4$. * **Poles:** There is a 4th-order pole at the origin $z = 0$, and a simple pole at $z = 1$. * **Pole-Zero Cancellation:** Notice that for $k=0$, the zero is $z = 1$. This zero exactly cancels the simple pole at $z = 1$! * Therefore, the function has no poles outside the origin. * **Final ROC:** **The entire $z$-plane except $z = 0$**. This perfectly matches **Property 3** {finite causal sequence}. --- ### 4.2 Causal Decaying Exponential with Shifting [PYQ 2020] > [!question] **2020 Exam Section B Q. 6b (03 Marks)** > > Find the Z-transform of the discrete-time signal including the Region of Convergence: > $$x[n] = e^{-4n} u[n-1]$$ #### Step 1: Set up the Z-transform summation $$X(z) = \sum_{n=-\infty}^{\infty} e^{-4n} u[n-1] z^{-n} = \sum_{n=1}^{\infty} e^{-4n} z^{-n}$$ We can group the base terms inside the exponent: $$X(z) = \sum_{n=1}^{\infty} \left( e^{-4} z^{-1} ight)^n$$ #### Step 2: Adjust index to start from $m = 0$ Let $m = n - 1 \implies n = m + 1$: $$X(z) = \sum_{m=0}^{\infty} \left( e^{-4} z^{-1} ight)^{m+1} = \left( e^{-4} z^{-1} ight) \sum_{m=0}^{\infty} \left( e^{-4} z^{-1} ight)^m$$ #### Step 3: Evaluate the infinite sum Applying the infinite geometric series formula ($\sum_{m=0}^{\infty} q^m = rac{1}{1-q}$ for $|q| < 1$): $$X(z) = \left( e^{-4} z^{-1} ight) \left[ rac{1}{1 - e^{-4} z^{-1}} ight] = rac{e^{-4} z^{-1}}{1 - e^{-4} z^{-1}}$$ Multiplying the numerator and denominator by $z$: $$X(z) = rac{e^{-4}}{z - e^{-4}}$$ #### Step 4: Find the ROC The geometric series converges if and only if the common ratio has a magnitude less than 1: $$|e^{-4} z^{-1}| < 1 \implies e^{-4} |z|^{-1} < 1 \implies |z| > e^{-4}$$ * **Poles:** $z = e^{-4} pprox 0.0183$. * **Zeros:** No finite zeros (or a zero at infinity). * **ROC:** **$|z| > e^{-4}$** {exterior of a circle of radius $e^{-4}$}. --- ### 4.3 Left-Sided Anticausal Exponential Sequence [PYQ 2022] > [!question] **2022 Exam Section B Q. 5b (05 Marks)** > > Calculate the Z-transform and ROC of: > $$x[n] = \left(- rac{1}{3} ight)^n u[-n]$$ #### Step 1: Formulate the Z-transform summation The anticausal step $u[-n]$ is defined as $1$ for $n \le 0$ and $0$ for $n > 0$. Therefore: $$X(z) = \sum_{n=-\infty}^{0} \left(- rac{1}{3} ight)^n z^{-n}$$ #### Step 2: Use variable substitution to flip the index Let $k = -n \implies$ as $n o -\infty$, $k o +\infty$: $$X(z) = \sum_{k=0}^{\infty} \left(- rac{1}{3} ight)^{-k} z^{k} = \sum_{k=0}^{\infty} (-3)^k z^k = \sum_{k=0}^{\infty} (-3z)^k$$ #### Step 3: Evaluate the sum Applying the infinite geometric series formula: $$X(z) = rac{1}{1 - (-3z)} = rac{1}{1 + 3z}$$ #### Step 4: Determine the ROC The geometric series converges if and only if the absolute value of the ratio is less than 1: $$|-3z| < 1 \implies 3|z| < 1 \implies |z| < rac{1}{3}$$ * **Poles:** $z = - rac{1}{3}$. * **Zeros:** None (or a zero at infinity). * **ROC:** **$|z| < rac{1}{3}$** {interior of a circle of radius $1/3$, which matches Property 7 for anticausal sequences}. --- ### 4.4 Multi-Impulse Array with Time Shift [PYQ 2022, 2018] > [!question] **2022 / 2018 Exam Q. 5a (02 to 05 Marks)** > > Determine the Z-transform of the following signal and specify its ROC: > $$x[n] = \delta[n+1] + 3\delta[n] + 7\delta[n-3] - 3\delta[n-4]$$ #### Step 1: Take Z-transform of individual terms Using the definition of the Z-transform of an impulse $\mathcal{Z}\{\delta[n-k]\} = z^{-k}$: * $\mathcal{Z}\{\delta[n+1]\} = z^1 = z$ * $\mathcal{Z}\{3\delta[n]\} = 3 z^0 = 3$ * $\mathcal{Z}\{7\delta[n-3]\} = 7 z^{-3}$ * $\mathcal{Z}\{-3\delta[n-4]\} = -3 z^{-4}$ #### Step 2: Assemble the final expression By linearity: $$X(z) = z + 3 + 7z^{-3} - 3z^{-4} \quad [168]$$ #### Step 3: Establish the ROC * The term $z$ has a pole at $z = \infty$ {magnitude blows up as $z o \infty$}. * The terms $z^{-3}$ and $z^{-4}$ have poles of order 3 and 4 at the origin $z = 0$ {magnitude blows up as $z o 0$}. * **Final ROC:** **The entire $z$-plane except $z = 0$ and $z = \infty$**. This matches **Property 5** for two-sided finite sequences. --- ## 5. Common Mistakes That Cost Marks > [!WARNING] **Critical Exam Pitfalls** > > 1. **Omitting the ROC Statement:** In discrete-time systems, a Z-domain expression $X(z)$ without an ROC does **not** represent a unique time-domain sequence! For example, $X(z) = rac{z}{z-a}$ can represent the causal sequence $a^n u[n]$ (if $|z| > |a|$) or the anticausal sequence $-a^n u[-n-1]$ (if $|z| < |a|$). **Always state the ROC explicitly, or you will lose 2 to 3 marks immediately!** > 2. **Algebraic Sign Errors in Anticausal Substitutions:** When evaluating anticausal geometric series like $\sum_{n=-\infty}^{0} a^n z^{-n}$, students often forget that replacing $n$ with $-k$ flips the exponent of both $a$ and $z$ {yielding $(a^{-1}z)^k$, not $(az)^k$}. Work through the substitution steps explicitly. > 3. **Confusing the Stability Boundary with Laplace:** Do not mix up the stability regions of the $s$-plane and the $z$-plane! In Laplace, the imaginary axis $\sigma = 0$ separates stable and unstable territories. In the Z-transform, the **unit circle $|z| = 1$** acts as this boundary. A discrete-time LTI system is BIBO stable if and only if the ROC of its transfer function **includes the unit circle $|z| = 1$**. --- ## 6. PYQ Bank — Verbatim Questions & Answer Plans ### Q1: Relationship between Laplace and Z-Transform [PYQ 2024, 2019, 2017, 2015] * **Question:** How is the Z-transform obtained from the Laplace transform? Describe the relationship between Laplace transform and Z-transform. **(08 Marks)** * **Answer Plan:** 1. Model the discrete sequence $x[n]$ as a sampled continuous signal $x^*(t)$ convolved with an impulse train. 2. Take the Laplace transform of $x^*(t)$ and prove that $X^*(s) = \sum x(nT_s) e^{-n s T_s}$. 3. Compare this to the Z-transform definition $\sum x[n] z^{-n}$ and prove the mapping equation $z = e^{s T_s}$. 4. Draw the $s$-plane to $z$-plane mapping diagram, detailing why the imaginary axis maps to $|z|=1$, the LHP maps to $|z|<1$, and the RHP maps to $|z|>1$. ### Q2: Bilateral vs. Unilateral Z-Transforms [PYQ 2020] * **Question:** Briefly discuss the properties of two-sided Z-transform and compare them with one-sided Z-transform. **(06 Marks)** * **Answer Plan:** 1. Write the mathematical equations for both bilateral ($\sum_{-\infty}^{\infty}$) and unilateral ($\sum_{0}^{\infty}$) Z-transforms. 2. Provide a structured comparison table analyzing limits, ROC behavior, causality assumptions, and their continuous Laplace analogues. --- ## 7. Self-Check Before Moving On - [ ] Can you define the mathematical differences between bilateral and unilateral Z-transforms? - [ ] Can you derive the mapping relationship $z = e^{s T_s}$ and sketch the mapped LHP and imaginary axis territories? - [ ] Do you know why a rational Z-transform cannot have poles inside its Region of Convergence? - [ ] Can you calculate the Z-transform and ROC of a causal decaying exponential sequence? - [ ] Can you calculate the Z-transform and ROC of an anticausal left-sided sequence using geometric series convergence rules? --- ## 8. Source Citations * *(k.Deergha Rao) signals and systems.pdf, Springer Nature, 2018* * *04 Z-transform.pdf, Digital Signal Processing lecture slides* * *Signals, Systems, and Networks MOC & Checklist*