📘 Solving Difference Equations Using Z-Transform
Difference equations describe discrete-time LTI systems. Z-transform converts them into algebraic equations.
🔹 1. General Form
y[n] + a₁ y[n−1] + a₂ y[n−2] = b₀ x[n] + b₁ x[n−1]
This represents a linear constant coefficient difference equation.🔹 2. Step-by-Step Method
1. Take Z-transform on both sides 2. Use time-shift property 3. Solve for Y(z) 4. Find Transfer Function H(z) 5. Take inverse Z-transform🔹 3. Example 1
Given:y[n] − 0.5 y[n−1] = x[n]
Step 1: Take Z-transform
Y(z) − 0.5 z^{-1} Y(z) = X(z)
Step 2: Factor Y(z)
Y(z) [1 − 0.5 z^{-1}] = X(z)
Step 3: Transfer Function
H(z) = Y(z)/X(z) = 1 / (1 − 0.5 z^{-1})
Step 4: Inverse Z-transform
h[n] = (0.5)^n u[n]
🔹 4. Example 2 (Second Order)
Given:y[n] − 0.7 y[n−1] + 0.1 y[n−2] = x[n]
Step 1: Z-transform
Y(z)[1 − 0.7z^{-1} + 0.1z^{-2}] = X(z)
Step 2: Transfer Function
H(z) = 1 / (1 − 0.7z^{-1} + 0.1z^{-2})
Step 3: Find Poles
Solve denominator = 0 Pole locations determine stability.🔹 5. Stability Condition
System stable if:All poles inside unit circle
|z| < 1🔹 6. Zero Input vs Zero State Response
Total response =- Zero Input Response (due to initial conditions)
- Zero State Response (due to input)
🎯 GATE Important Points
- Transfer function directly from equation
- Poles determine stability
- Repeated poles → n term appears
- Unit circle is key concept
Difference Equation → Algebra → Transfer Function → Time Response
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