---
title: "State LCCDE and system function"
subject: "Digital Signal Analysis and Processing"
chapter: 1
key: "semester_7/digital_signal_analysis_and_processing:lccde-system-function"
source: https://www.bctengineer.com/q/semester_7/digital_signal_analysis_and_processing/lccde-system-function
site: BCT Engineering
---

# State LCCDE and system function

Digital Signal Analysis and Processing · Chapter 1: Discrete Time Signals and Systems

- Asked in: 2073 Shrawan, 3 marks
- Video: [3.4 : Linear Constant Coefficient Difference Equation](https://www.youtube.com/watch?v=z0_h_uJppTk) — IIT Madras - BS in Electronic Systems
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_7/digital_signal_analysis_and_processing/lccde-system-function
- Whole chapter: https://www.bctengineer.com/md/semester_7/digital_signal_analysis_and_processing/chapter-1

## Question

State linear constant coefficient difference equation and corresponding system function.

## Short answer

An LCCDE is sum_{k=0}^N a_k y[n-k] = sum_{k=0}^M b_k x[n-k], a0=1, or y[n]= -sum a_k y[n-k]+sum b_k x[n-k]. Its z-transform gives H(z)=Y(z)/X(z)= (sum_{k=0}^M b_k z^{-k})/(1+sum_{k=1}^N a_k z^{-k}); numerator zeros, denominator poles.

## Answer

#### Linear constant-coefficient difference equation (LCCDE)
An LTI discrete-time system of order N is described by

Σ(k=0 to N) aₖ y[n − k] = Σ(k=0 to M) bₖ x[n − k],  a₀ = 1

or y[n] = −Σ(k=1 to N) aₖ y[n − k] + Σ(k=0 to M) bₖ x[n − k].

The aₖ and bₖ are constants (independent of n). The first sum is the feedback (recursive) part, the second the feed-forward part. The equation is solved with initial rest conditions for an LTI, causal system.

#### System function
Take the z-transform of both sides. Since x[n − k] ↔ z⁻ᵏ X(z):

Y(z) [1 + a₁z⁻¹ + … + a_N z⁻ᴺ] = X(z) [b₀ + b₁z⁻¹ + … + b_M z⁻ᴹ]

H(z) = Y(z)/X(z) = (Σ bₖ z⁻ᵏ) / (1 + Σ aₖ z⁻ᵏ)

The roots of the numerator are the zeros, and the roots of the denominator are the poles.

**Example:** y[n] = 0.5y[n−1] + x[n] + 2x[n−1] gives H(z) = (1 + 2z⁻¹)/(1 − 0.5z⁻¹), pole at z = 0.5, zero at z = −2.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. The LCCDE with a0 = 1 is
   A. sum a_k y[n-k] = sum b_k x[n-k]
   B. sum a_k x[n-k] = sum b_k y[n-k]
   C. y[n] = x[n]^2
   D. y[n] = sum x[k]
   Answer: A. Feedback side on y, feed-forward side on x. (Key point: LCCDE form)

2. Delay by k samples becomes
   A. multiplication by z^k
   B. addition of k
   C. division by k
   D. multiplication by z^-k
   Answer: D. x[n-k] <-> z^-k X(z). (Key point: Delay in z-domain)

3. The poles are roots of
   A. the numerator
   B. the input
   C. the denominator
   D. a_0
   Answer: C. Numerator roots are zeros. (Key point: Poles and zeros)

## Close variants set in other exams

- [Need for difference equation; LCCDE and system function](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Awhy-difference-equation-lccde) — Digital Signal Analysis and Processing, chapter 1, asked 2×