Z-Transform —
Discrete-Time System Analysis
Digital audio: Your smartphone's equalizer, echo cancellation, and noise reduction are all DT-LTI filters designed using the Z-transform. The output of a digital filter y[n] = b₀x[n] + b₁x[n-1] + b₂x[n-2] − a₁y[n-1] − a₂y[n-2] is completely characterized by H(z) = (b₀ + b₁z⁻¹ + b₂z⁻²)/(1 + a₁z⁻¹ + a₂z⁻²).
Image processing: Blur and sharpen filters applied to pixel arrays are DT-LTI systems analyzed with 2D Z-transforms.
Control systems (digital): A PI controller implemented in a microcontroller uses the bilinear transform z=(1+sT/2)/(1-sT/2) to convert a continuous H(s) to a discrete H(z), which is then run every sample interval T.
The unit circle |z|=1 is the DT analog of the jω-axis. Stability condition: all poles inside unit circle (|pᵢ|<1). The ROC for a causal signal is the region |z|>r₀ (outside a circle). For stability, ROC must include the unit circle.
1. Discrete-Time Signals
A discrete-time (DT) signal x[n] is defined only at integer values of n ∈ ℤ. It is obtained by sampling a CT signal at interval T: x[n] = x_c(nT).
Elementary DT Signals
DT signals are defined only at integer n. Key: u[n] = Σδ[n-k] (superposition of shifts). For aⁿu[n]: if |a|<1 → decays (stable); if |a|>1 → grows (unstable); if a=1 → u[n] (unit step, marginally stable).
2. DTFT — Discrete-Time Fourier Transform
3. Z-Transform — Definition & ROC
ROC in the z-Plane
| Signal Type | ROC | Example |
|---|---|---|
| Right-sided (causal): x[n]=0 for n<N | |z| > r₀ (outside a circle) | aⁿu[n] → ROC: |z|>|a| |
| Left-sided: x[n]=0 for n>N | |z| < r₀ (inside a circle) | −aⁿu[−n−1] → ROC: |z|<|a| |
| Two-sided | r₁ < |z| < r₂ (annular ring) | |z| between two poles |
| Finite-duration (causal) | |z|>0, entire z-plane except z=0 | FIR filters |
| All time (δ[n]) | Entire z-plane | X(z)=1 |
4. Standard Z-Transform Pairs
| x[n] | X(z) | ROC |
|---|---|---|
| δ[n] | 1 | All z |
| u[n] | 1/(1 − z⁻¹) = z/(z−1) | |z| > 1 |
| n·u[n] | z⁻¹/(1−z⁻¹)² = z/(z−1)² | |z| > 1 |
| aⁿu[n] | 1/(1−az⁻¹) = z/(z−a) | |z| > |a| |
| n·aⁿu[n] | az⁻¹/(1−az⁻¹)² = az/(z−a)² | |z| > |a| |
| −aⁿu[−n−1] | 1/(1−az⁻¹) = z/(z−a) | |z| < |a| |
| cos(ω₀n)u[n] | (z²−z·cos ω₀)/(z²−2z·cos ω₀+1) | |z| > 1 |
| sin(ω₀n)u[n] | (z·sin ω₀)/(z²−2z·cos ω₀+1) | |z| > 1 |
| aⁿcos(ω₀n)u[n] | (z²−az·cos ω₀)/(z²−2az·cos ω₀+a²) | |z| > |a| |
| δ[n−k] | z⁻ᵏ | All z (except 0 if k>0) |
X(z) = z/(z−a) with ROC |z|>|a| → x[n] = aⁿu[n] (causal, right-sided)
X(z) = z/(z−a) with ROC |z|<|a| → x[n] = −aⁿu[−n−1] (anti-causal, left-sided)
The ROC determines which x[n] corresponds to X(z). Always state ROC!
5. Properties of Z-Transform
| Property | Sequence x[n] | Z-Transform X(z) |
|---|---|---|
| Linearity | ax[n] + by[n] | aX(z) + bY(z) |
| Time Shifting | x[n−k] | z⁻ᵏ X(z) [delay k → multiply by z⁻ᵏ] |
| z-Domain Scaling | aⁿx[n] | X(z/a) [ROC scaled by |a|] |
| Time Reversal | x[−n] | X(z⁻¹) [replace z by 1/z] |
| Differentiation (z-domain) | n·x[n] | −z·dX(z)/dz |
| Convolution | x[n] * h[n] | X(z) · H(z) [key theorem] |
| Initial Value | x[0] | lim_{z→∞} X(z) [causal x[n]] |
| Final Value | x[∞] | lim_{z→1} (z−1)X(z) [poles inside unit circle] |
| Parseval's | Σ|x[n]|² | (1/2πj)∮ X(z)X*(1/z*)z⁻¹ dz |
6. Inverse Z-Transform
Three methods: (1) Partial fraction expansion + table lookup, (2) Power series (long division), (3) Contour integration (Cauchy's residue theorem).
Method 1 — Partial Fraction Expansion (Most Used in GATE)
Method 2 — Long Division (Power Series)
7. DT-LTI Systems & Transfer Function H(z)
3-tap moving average filter: y[n] = (x[n] + x[n-1] + x[n-2])/3
H(z) = (1 + z⁻¹ + z⁻²)/3
All poles at z=0 (inside unit circle) → always stable! (This is why FIR filters are always BIBO stable.)
Zeros: 1+z⁻¹+z⁻² = 0 → z = e^(±j2π/3), at 120° on unit circle → attenuates those frequencies.
This filter smooths noisy sensor data — used in accelerometers, ECG preprocessing, and stock price averaging.
8. Stability in the z-Plane
s-plane: stable ↔ Re(pᵢ) < 0 (poles in left half-plane)
z-plane: stable ↔ |pᵢ| < 1 (poles inside unit circle)
Mapping: z = e^(sT). LHP (σ<0) → inside unit circle (|z|=|e^σT|=e^(σT)<1 since σ<0).
GATE Questions — Z-Transform & DT Signals (10)
Standard pair: aⁿu[n] ↔ z/(z−a) = 1/(1−az⁻¹), ROC: |z| > |a|
With a = 0.5: X(z) = z/(z−0.5), ROC: |z| > 0.5
Answer: (A) z/(z−0.5), |z| > 0.5
H(z) = 1/(1−0.8z⁻¹) = z/(z−0.8). Pole at z = 0.8.
|0.8| = 0.8 < 1 → pole is inside unit circle → system is BIBO stable.
h[n] = (0.8)ⁿu[n] → Σ(0.8)ⁿ = 1/(1−0.8) = 5 < ∞ ✓
Answer: (A) BIBO stable
Time shift: x[n−k] ↔ z⁻ᵏ·X(z) [delay by k samples = multiply by z⁻ᵏ]
Analog: in s-domain, delay by T → e^(−sT)·X(s). In z-domain, delay by k → z⁻ᵏ·X(z).
z⁻¹ is often called the "unit delay" operator — fundamental building block of digital filters.
Answer: (B) z⁻ᵏ·X(z)
X(z) = 1/(1−2z⁻¹) = z/(z−2) has a pole at z=2.
ROC |z|<2 → left-sided (anti-causal) signal.
Using the pair: −aⁿu[−n−1] ↔ z/(z−a) with ROC |z|<|a|
With a=2: x[n] = −2ⁿu[−n−1]
Contrast: if ROC were |z|>2 → x[n] = 2ⁿu[n] (unstable, causal).
Answer: (B) −2ⁿu[−n−1]
Taking Z-transform of y[n] − 0.5y[n−1] = x[n]:
Pole at z=0.5 (inside unit circle) → stable causal system.
Answer: (A) z/(z−0.5)
X(e^jω) = Z{x[n]}|_{z=e^jω} = z/(z−0.5)|_{z=e^jω}
At ω=0: z = e^j0 = 1
Alternatively: X(e^j0) = Σₙ (0.5)ⁿu[n] = Σₙ₌₀^∞ (0.5)ⁿ = 1/(1−0.5) = 2 [geometric series]
Answer: (C) 2
Z-domain: Z{u[n]} = z/(z−1). Convolution → multiplication:
Inverse Z-transform: z²/(z−1)² → (n+1)u[n]
Verify: y[n] = Σₖ₌₀ⁿ 1·1 = (n+1) for n≥0, 0 for n<0 = (n+1)u[n] ✓
Answer: (C) (n+1)u[n]
X(e^jω) = Σₙ x[n]e^(-jωn). Since e^(-j(ω+2π)n) = e^(-jωn)·e^(-j2πn) = e^(-jωn) for integer n:
X(e^j(ω+2π)) = X(e^jω) → DTFT is periodic with period 2π.
ω is a continuous variable → DTFT is a continuous, periodic function of ω.
Answer: (B)
Using the z-differentiation property: n·x[n] ↔ −z·dX(z)/dz
X(z) = Z{u[n]} = z/(z−1)
Alternatively, direct table: n·u[n] ↔ z/(z−1)² [standard pair for ramp]
Answer: (B) z/(z−1)²
H(z) = Z{δ[n]−δ[n-1]} = 1 − z⁻¹ = (z−1)/z
X(z) = Z{u[n]} = z/(z−1)
Y(z) = 1 → y[n] = δ[n]
Intuition: h[n] = difference filter. Applying it to u[n] gives u[n]−u[n-1] = δ[n]. The difference filter is the DT analog of differentiation (which converts a step to an impulse).
Answer: (A) δ[n]