10.00 Z-Transform and Discrete-Time Analysis Compact Review

ECE 2107 Master Roadmap | 10.01 Z-Transform Definitions & ROC | 10.02 Z-Transform Properties

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Chapter 10: Z-Transform & Discrete Analysis - Compact Review

10.01 Z-Transform Core Definitions

*(Target: Theory Descriptive / 5-Mark Formula)* [10.01]

• Bilateral (Two-Sided) Z-Transform: Maps a discrete-time sequence $x[n]$ defined over all integers onto the complex z-plane [10.01]: $X(z)=\mathcal{Z}\{x[n]\}={\sum }_{n=-\infty }^{\infty }x[n]z^{-n}ext(derivation)$ [10.01]
• Unilateral (One-Sided) Z-Transform: Maps strictly causal signals ($n\ge 0$) [10.01]. This is uniquely suited for solving difference equations with non-zero initial conditions [10.05]: $X_{+}(z)={\mathcal{Z}}_{+}\{x[n]\}={\sum }_{n=0}^{\infty }x[n]z^{-n}$ [10.01]

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10.02 s-Plane to z-Plane Exponential Mapping

*(Target: Mathematical Proof / 6-Mark Derivation)* [10.01]

• The Mapping Operator: Sampling a continuous-time signal $x_a(t)$ at uniform intervals $T_s$ maps Laplace domain variables to discrete z-plane variables [10.01]: $z=e^{s{T}_s}ext(derivation)$ [10.01]
• Let $s=\sigma +j\Omega$ and $z=r{e}^{j\omega }$ [10.01]: $r{e}^{j\omega }=e^{(\sigma +j\Omega )T_s}=e^{\sigma T_s}{e}^{j\Omega T_s}⟹r=e^{\sigma T_s},\omega =\Omega T_{s}ext(derivation)$ [10.01]

Table 10.1: s-Plane to z-Plane Boundary Mappings [10.01]

s-Plane Domain / Path	z-Plane Equivalent Domain / Path	Physical Significance
Imaginary Axis ($\sigma =0$)	**Unit Circle ($r = 1 \implies \lvert z
vert = 1$)**	Standard DTFT boundary path [10.01]
Left-Half s-Plane ($\sigma <0$)	**Interior of Unit Circle ($r < 1 \implies \lvert z
vert < 1$)**	Stable discrete pole region [10.01]
Right-Half s-Plane ($\sigma >0$)	**Exterior of Unit Circle ($r > 1 \implies \lvert z
vert > 1$)**	Unstable discrete pole region [10.01]
Origin ($s=0$)	DC Unit-Circle Crossing (z = 1 ngle 0^\circ)	Zero-frequency/DC gain point [10.01]

         s-plane (Laplace)                     z-plane (Z-Transform)
            +j\Omega                                    Im(z)
               |                                         ^
      LHP      |  RHP                                _---|___
  (Stable)     | (Unstable)                        /   * *     ~~~~~~~~~*   |   *                              |  * (o) *  | --> Re(z)
  ~~~~~~~~~    +---------> \sigma                 |  Stable   |     |z|=1 (Unit Circle)
  ~~~~~~~~~    |                                   \   * *   /
               |                                     `---|---'
               |                                      (Interior)

[DIAGRAM: Visual mapping showing s-plane boundary lines and shaded areas projecting onto unit circle disk envelopes inside the z-plane] [10.01]

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10.03 Region of Convergence (ROC) Core Symmetries

*(Target: Theory Descriptive / 5-to-13 Mark Properties Mapping)* [10.01]

• The Region of Convergence (ROC) is the vertical annulus (ring) centered at the origin of the complex z-plane where the Laurent power series converges absolutely [10.01]:

ight vert < \infty$$ [10.01]

Table 10.2: ROC Geometries & Signal Classifications [10.01]

Signal Sequence Classification	Time-Domain Constraints	ROC Geometric Bounds in z-Plane	Illustration Placeholder
Finite Causal	$x[n]=0$ for $n<0$, length $N$	Entire z-plane except $z=0$	[GRAPH: ROC covering entire plane except center] [10.01]
Finite Anticausal	$x[n]=0$ for $n\ge 0$, length $N$	Entire z-plane except $z=\infty$	[GRAPH: ROC covering plane except infinite boundary] [10.01]
Finite Two-Sided	$x[n]$ non-zero only for $N_1\le n\le N_2$	Entire z-plane except $z=0$ and $z=\infty$	[GRAPH: ROC plane punctured at origin and outer rim] [10.01]
Infinite Right-Sided (Causal)	$x[n]=0$ for $n<0$	Exterior of a circle: $\lvert z
vert > \lvert p_{ ext{max}}
vert$	[GRAPH: Shaded exterior bounding highest pole outward] [10.01]
Infinite Left-Sided (Anticausal)	$x[n]=0$ for $n\ge 0$	Interior of a circle: $\lvert z
vert < \lvert p_{ ext{min}}
vert$	[GRAPH: Shaded disk bounding lowest pole inward] [10.01]
Infinite Two-Sided	$x[n]$ defined over $(-\infty ,\infty )$	Vertical ring/annulus: $r_1 < \lvert z
vert < r_2$	[GRAPH: Shaded ring bounded between two pole radii] [10.01]

WARNING

The Pole ROC Boundary Principle: The ROC cannot contain any poles. It must always be bounded by poles or extend to infinity [10.01]. If a system is stable, the ROC must include the unit circle ($|zvert=1$) [10.01].

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10.04 Master Z-Transform Properties Matrix

*(Target: Theory Descriptive / Properties Ingestion)* [10.02]

Table 10.3: Bilateral Z-Transform Properties [10.02]

Property Name	Time-Domain Sequence $x[n]$	z-Domain Function $X(z)$	ROC Bound
Linearity	$a_1{x}_1[n]+a_2{x}_2[n]$	$a_1{X}_1(z)+a_2{X}_2(z)$	At least $R_1\cap R_2$ [10.02]
Time Shifting	$x[n-k]$	$z^{-k}X(z)$	Same as $R$ except $z=0$ (if $k>0$) or $z=\infty$ (if $k<0$) [10.02]
Time Reversal	$x[-n]$	$X(z^{-1})$	Inverted boundary: $1/R$ (i.e., $rac{1}{r_2} < \lvert z
vert < rac{1}{r_1}$) [10.02]
Scaling in z	$a^{n}x[n]$	$X(a^{-1}z)$	Scaled boundary: $\lvert a
vert R$(i.e.,$\lvert a
vert r_1 < \lvert z
vert < \lvert a
vert r_2$) [10.02]
Differentiation	$nx[n]$	-z rac{dX(z)}{dz}	Same as $R$ [10.02]
Convolution	$x_1[n]\ast x_2[n]$	$X_1(z)X_2(z)$	At least $R_1\cap R_2$ [10.02]
Correlation	$r_{x_1{x}_2}[l]=\sum x_1[n]x_2[n-l]$	$X_1(z)X_2(z^{-1})$	At least $R_1\cap 1/R_2$ [10.02]
Complex Conjugate	$x^{\ast }[n]$	$X^{\ast }(z^{\ast })$	Unchanged: $R$ [10.02]
Real Part	$extRe\{x[n]\}$	$rac{1}{2}\left[ X(z) + X^(z^)
ight]$	At least $R$ [10.02]
Imaginary Part	$extIm\{x[n]\}$	$rac{1}{2j}\left[ X(z) - X^(z^)
ight]$	At least $R$ [10.02]

• Z-Domain Differentiation Property Proof Outline: ext{Given: } X(z) = \sum_{n=-\infty}^{\infty} x[n] z^{-n} \implies rac{dX(z)}{dz} = \sum_{n=-\infty}^{\infty} -n x[n] z^{-n-1} \implies -zrac{dX(z)}{dz} = \sum_{n=-\infty}^{\infty} n x[n] z^{-n} \quad ext{(derivation)} [10.02]

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10.05 Master Z-Transform Pair Lookup Sheet

*(Target: Numerical Solving / Rapid Lookups)* [10.02]

Table 10.4: High-Yield Discrete Transform Pairs [10.02]

Discrete Sequence $x[n]$	z-Domain Transform $X(z)$	Discrete ROC Boundary
Unit Impulse $\delta [n]$	$1$	Entire z-plane [10.02]
Shifted Impulse $\delta [n-k]$	$z^{-k}$	Entire plane except $z=0$ (if $k>0$) [10.02]
Unit Step $u[n]$	rac{1}{1-z^{-1}} = rac{z}{z-1}	$\lvert z
vert > 1$ [10.02]
Negative Unit Step $-u[-n-1]$	rac{1}{1-z^{-1}} = rac{z}{z-1}	$\lvert z
vert < 1$ [10.02]
Causal Exponential $a^{n}u[n]$	rac{1}{1-a z^{-1}} = rac{z}{z-a}	$\lvert z
vert > \lvert a
vert$ [10.02]
Anticausal Exponential $-a^{n}u[-n-1]$	rac{1}{1-a z^{-1}} = rac{z}{z-a}	$\lvert z
vert < \lvert a
vert$ [10.02]
Ramp Function $nu[n]$	rac{z^{-1}}{(1-z^{-1})^2} = rac{z}{(z-1)^2}	$\lvert z
vert > 1$ [10.02]
Scaled Ramp $n{a}^{n}u[n]$	rac{a z^{-1}}{(1-a z^{-1})^2} = rac{a z}{(z-a)^2}	$\lvert z
vert > \lvert a
vert$ [10.02]
Discrete Cosine $\cos ({\omega }_{0}n)u[n]$	rac{1 - z^{-1}\cos(\omega_0)}{1 - 2z^{-1}\cos(\omega_0) + z^{-2}} = rac{z(z - \cos(\omega_0))}{z^2 - 2z\cos(\omega_0) + 1}	$\lvert z
vert > 1$ [10.02]
Discrete Sine $\sin ({\omega }_{0}n)u[n]$	rac{z^{-1}\sin(\omega_0)}{1 - 2z^{-1}\cos(\omega_0) + z^{-2}} = rac{z\sin(\omega_0)}{z^2 - 2z\cos(\omega_0) + 1}	$\lvert z
vert > 1$ [10.02]
Exponential Cosine $a^n\cos ({\omega }_{0}n)u[n]$	rac{z(z - a\cos(\omega_0))}{z^2 - 2az\cos(\omega_0) + a^2}	$\lvert z
vert > \lvert a
vert$ [10.02]
Exponential Sine $a^n\sin ({\omega }_{0}n)u[n]$	rac{az\sin(\omega_0)}{z^2 - 2az\cos(\omega_0) + a^2}	$\lvert z
vert > \lvert a
vert$ [10.02]

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10.06 Initial & Final Value Theorems

*(Target: Mathematical Proof & Numerical Solving)* [10.03]

• Initial Value Theorem (IVT): Determines the starting value $x[0]$ of a causal sequence directly from its Z-transform without requiring inverse operations [10.03]: $x[0]={\lim }_{zo\infty }X(z)ext(derivation)$ [10.03]
• Final Value Theorem (FVT): Finds the steady-state DC limit $x[\infty ]$ of a stable sequence [10.03]:

ight) X(z) \quad ext{(derivation)}$$ [10.03]

WARNING

The Critical FVT Stability Constraint: The final value theorem yields a correct steady-state limit if and only if all poles of $(1-z^{-1})X(z)$ lie strictly inside the unit circle ($|p_{i}vert<1$) [10.03]. If the system has multiple poles on the unit circle or poles outside the unit circle (e.g., an oscillatory system like $\cos ({\omega }_{0}n)$ or an unstable system like $2^n$), the FVT mathematically fails and yields invalid values [10.03].

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10.07 Unilateral Z-Transform Shift properties

*(Target: Numerical Solving / Recursive Difference Equations)* [10.05] When solving linear constant-coefficient difference equations (LCCDE) with non-zero initial conditions (e.g., $y[-1],y[-2]$), the standard bilateral shift theorem fails. We must apply the Unilateral Shift properties [10.05]:

• First-Order Delay Shift ($k=1$): ${\mathcal{Z}}_{+}\{y[n-1]\}=z^{-1}{Y}_{+}(z)+y[-1]ext(derivation)$ [10.05]
• Second-Order Delay Shift ($k=2$): ${\mathcal{Z}}_{+}\{y[n-2]\}=z^{-2}{Y}_{+}(z)+z^{-1}y[-1]+y[-2]ext(derivation)$ [10.05]
• General Delay Shift ($k>0$):

ight] \quad ext{(derivation)}$$ [10.05]

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10.08 Methods for Inverse Z-Transform Solving

*(Target: Theory Descriptive & Numerical Solving)* [10.04]

1. Cauchy’s Residue Integration Method

• Concept: Resolves the inverse transform by evaluating contour integrals over a closed counter-clockwise path $C$ in the ROC [10.04]:

ight] \quad ext{(derivation)}$$ [10.04]

• Residue Formula at a Simple Pole ($p_i$):

ight} = \lim_{z o p_i} (z - p_i) X(z) z^{n-1}$$ [10.04]

• Residue Formula at a Pole of Multiplicity $m$:

ight} = rac{1}{(m-1)!} \lim_{z o p_i} rac{d^{m-1}}{dz^{m-1}} \left[ (z - p_i)^m X(z) z^{n-1} ight]$$ [10.04]

2. Partial Fraction Expansion Method ($X(z)/z$)

• Concept: Rather than expanding $X(z)$ directly (which creates $z$ factors in numerators), expand the ratio rac{X(z)}{z}. Once decomposed, multiply back by $z$ to recover standard $a^{n}u[n]$ forms [10.04]:

ight] u[n] \quad ext{(derivation)}$$ [10.04]

3. Power Series Expansion Method (Long Division)

• Causal Case ($|zvert>a$): Divide the numerator by the denominator arranged in descending powers of $z$ to obtain a power series in negative powers of $z$ ($z^{-1},z^{-2}$) [10.04]: $X(z)=x[0]+x[1]z^{-1}+x[2]z^{-2}+\dots ⟹x[n]=\{x[0],x[1],x[2],\dots \}$ [10.04]
• Anticausal Case ($|zvert<a$): Divide the numerator by the denominator arranged in ascending powers of $z$ to obtain a power series in positive powers of $z$ [10.04]: $X(z)=\cdots +x[-2]z^2+x[-1]z^1⟹x[n]=\{\dots ,x[-2],x[-1],0\}$ [10.04]

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10.09 System Realization Architectures

*(Target: Theory Descriptive & Graphical Diagramming)* [10.05] Discrete transfer functions H(z) = rac{Y(z)}{X(z)} = rac{b_0 + b_1 z^{-1} + b_2 z^{-2}}{1 + a_1 z^{-1} + a_2 z^{-2}} are mapped to hardware block elements (Adders, Multipliers, and Unit Delay Registers $z^{-1}$) using two principal topological structures [10.05]:

• Direct Form I realization: Implements poles and zeros separately [10.05]. It requires $2N$ delay registers (non-canonic) and is highly immune to coefficient quantization noise [10.05].
• Direct Form II canonic realization: Merges feedback and feedforward delay registers [10.05]. It minimizes total register count to exactly $N$ delays (canonic form) but is more sensitive to internal register overflow [10.05].

       Direct Form I (Non-Canonic)                Direct Form II (Canonic)
      x[n] --->(+)----------->(+)---> y[n]       x[n] --->(+)------------->(+)---> y[n]
                |              ^                           |     |          ^
              [z-1]          [z-1]                       [z-1]   |        [z-1]
                |              |                           |<-a1 | b1->     |
               ( )*-a1        ( )*b1                       v     |          v
                |              |                         [z-1]   v        [z-1]
              [z-1]          [z-1]                         |<-a2          |
                |              |                           v              v
               ( )*-a2        ( )*b2

[DIAGRAM: Non-canonic Direct Form I vs canonic Direct Form II delay element merging pathways] [10.05]

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10.10 Common Mistakes That Cost Marks

• The Unilateral Delay Shift Trap: Accidentally writing ${\mathcal{Z}}_{+}\{y[n-1]\}=z^{-1}{Y}_{+}(z)$ when initial conditions are non-zero. You must include the initial condition step $y[-1]$ [10.05]!
• The FVT Pole-Location Oversight: Applying the Final Value Theorem to an oscillatory signal like $\sin ({\omega }_{0}n)$ or an unstable sequence. You must check that the poles of $(1-z^{-1})X(z)$ lie strictly inside the unit circle ($|zvert<1$) before using the theorem [10.03].
• Long Division Order Confusion: Sorting polynomials incorrectly during long division. If seeking a causal sequence, write polynomials in descending powers of $z$ (or ascending powers of $z^{-1}$) [10.04]. If anticausal, reverse the order [10.04].
• Failing to scale the partial fraction expansion: Trying to expand $X(z)$ directly rather than rac{X(z)}{z}, which leads to algebraically intensive terms that do not match standard causal pairs [10.04].

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10.11 Verbatim PYQ Bank & Answer Plans

[PYQ 2025, Question 5a - 5 Marks]

• Question: Find the z-transform of the sequence $x(n)=[u(n)-u(n-5)]$ and its ROC.
• Answer Plan:
  1. Express the window sequence as a sum of impulses or a finite geometric series: $x[n]=\{1,1,1,1,1\}$ for $n=0,1,2,3,4$.
  2. Apply the definition: $X(z)={\sum }_{n=0}^4{z}^{-n}=1+z^{-1}+z^{-2}+z^{-3}+z^{-4}$.
  3. Express in closed ratio form: X(z) = rac{1 - z^{-5}}{1 - z^{-1}} = rac{z^5 - 1}{z^4(z-1)}.
  4. State the ROC: Since it is a finite-duration causal sequence, the ROC is the entire z-plane except $z=0$.

[PYQ 2024, Question 5b - 10 Marks]

• Question: Prove that the final values of $x(n)$ for X(z) = rac{z^2}{(z-1)(z-0.2)} is 1.25 and its initial value is unity.
• Answer Plan:
  1. Initial Value Proof: Apply the IVT: x[0] = \lim_{z o \infty} X(z) = \lim_{z o \infty} rac{z^2}{z^2 - 1.2z + 0.2} = 1.
  2. FVT Prerequisite Check: Locate the poles of (1-z^{-1})X(z) = rac{z-1}{z} rac{z^2}{(z-1)(z-0.2)} = rac{z}{z-0.2}. The only pole is at $z=0.2$, which lies strictly inside the unit circle ($|0.2vert<1$). Thus, FVT is valid.
  3. Final Value Proof: Apply the FVT: x[\infty] = \lim_{z o 1} (1-z^{-1}) X(z) = \lim_{z o 1} rac{z}{z-0.2} = rac{1}{1-0.2} = rac{1}{0.8} = 1.25.

[PYQ 2023, Question 5b - 5 Marks]

• Question: If $x(n)$ is causal, find the inverse z-transform of X(z) = rac{1}{z(z-0.8)(z+0.4)}.
• Answer Plan:
  1. Rewrite X(z) = z^{-1} \left[ rac{1}{(z-0.8)(z+0.4)} ight] or expand rac{X(z)}{z} = rac{1}{z^2(z-0.8)(z+0.4)} using multiple pole residues.
  2. Alternatively, perform partial fraction expansion on the core bracket first: F(z) = rac{z}{(z-0.8)(z+0.4)} = rac{2/3 z}{z-0.8} + rac{1/3 z}{z+0.4} \implies f[n] = \left[ rac{2}{3}(0.8)^n + rac{1}{3}(-0.4)^n ight] u[n].
  3. Apply the time-shifting property: Since $X(z)=z^{-2}F(z)$, we delay by 2 samples: x[n] = f[n-2] = \left[ rac{2}{3}(0.8)^{n-2} + rac{1}{3}(-0.4)^{n-2} ight] u[n-2].

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10.12 Interactive Self-Check Revision List

• Do you know the exact unit-circle boundary coordinates of the s-plane imaginary axis projection ($r=1$)?
• Can you state the 3 distinct finite-duration ROC cases (causal, anticausal, double-sided)?
• Do you remember the unilateral shifting terms for ${\mathcal{Z}}_{+}\{y[n-2]\}$ without looking?
• Have you verified the pole locations of $(1-z^{-1})X(z)$ before solving any FVT?
• Do you know why Direct Form II canonic structures use exactly half the delay elements of Direct Form I?