--- title: "Difference equation example; poles/zeros response" subject: "Digital Signal Analysis and Processing" chapter: 3 key: "semester_7/digital_signal_analysis_and_processing:pole-zero-define-difference-eq" source: https://www.bctengineer.com/q/semester_7/digital_signal_analysis_and_processing/pole-zero-define-difference-eq site: BCT Engineering --- # Difference equation example; poles/zeros response Digital Signal Analysis and Processing · Chapter 3: Analysis of LTI System in Frequency Domain - Asked in: 2073 Chaitra, 2+8 marks - Video: [Difference Equation Descriptions for Systems](https://www.youtube.com/watch?v=MtHpbGUIGaA) — Barry Van Veen - 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/pole-zero-define-difference-eq - Whole chapter: https://www.bctengineer.com/md/semester_7/digital_signal_analysis_and_processing/chapter-3 ## Question Define the difference equation with example. The poles of a system are located at 0.45 + 0.77i and 2 ± 0.7i and zeros at 1.2 ± 0.43i. Plot the magnitude response of this system. ## Short answer A difference equation relates present output to past outputs and inputs, e.g., y[n]=0.5y[n-1]+x[n]. For the given poles and zeros, the magnitude response has a sharp resonance peak of about 2.09 at ω≈π/3 from the pole pair 0.45±0.77i near the unit circle; the system is unstable because poles 2±0.7i lie outside the unit circle. ## Answer #### Difference equation A difference equation relates the present output to present and past inputs and past outputs. A linear constant-coefficient difference equation of order N is Σ(k=0 to N) aₖ y[n − k] = Σ(k=0 to M) bₖ x[n − k] **Example:** y[n] = 0.5y[n − 1] + x[n]. For the input x[n] = δ[n], y[0] = 1, y[1] = 0.5, y[2] = 0.25, …, so h[n] = (0.5)ⁿ u[n] and H(z) = 1/(1 − 0.5z⁻¹). It is the standard way to describe and implement a digital filter. #### Magnitude response from the given poles and zeros **Reading:** real filters have conjugate pairs, so I take poles at 0.45 ± 0.77i and 2 ± 0.7i, and zeros at 1.2 ± 0.43i. No gain is given, so K = 1: H(z) = (z − z₁)(z − z₁*) / [(z − p₁)(z − p₁*)(z − p₂)(z − p₂*)] | Point | Radius | Angle | |---|---|---| | Poles 0.45 ± 0.77i | 0.892 | ±59.7° | | Poles 2 ± 0.7i | 2.119 | ±19.3° | | Zeros 1.2 ± 0.43i | 1.275 | ±19.7° | ```text | | | | | | ........... .... | x.... x .. | .. o . | .. -------------.---------|---------.------------- . | .. .. | .. o .... | x.... x ........... | | | | | | ``` (x = pole, o = zero, dotted circle = unit circle; Re axis horizontal, Im axis vertical; range ±2.4) |H(e^(jω))| = (distances to the two zeros) / (distances to the four poles), computed with K = 1: | w | 0 | pi/6 | pi/3 | pi/2 | 2pi/3 | 5pi/6 | pi | |---|---|---|---|---|---|---|---| | \|H\| | 0.169 | 0.268 | 2.095 | 0.506 | 0.281 | 0.214 | 0.196 | ```text 2.09 | * 1.86 | * 1.63 | *** 1.40 | *** 1.16 | ***** 0.93 | ****** 0.70 | ********* 0.47 | *************** 0.23 |************************************* +------------------------------------- 0 pi/2 pi (w) ``` **Observations:** - A **sharp resonance peak (about 2.09) at ω ≈ π/3** (60°) because the pole pair 0.45 ± 0.77i (radius 0.89) lies close to the unit circle at that angle. - A small dip (about 0.16) near ω ≈ 0.08π, and the response then settles to a flat, low value (about 0.2) at high frequencies. - The zeros at angle 19.7° almost coincide with the poles at angle 19.3° (outside the circle), so their effects nearly cancel and no extra peak appears near ω ≈ 0.1π. - The poles at radius 2.12 are outside the unit circle, so a causal realisation is unstable; the response above is for the ROC containing the unit circle (non-causal system). The pole pair at radius 0.89 alone would be stable. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What does a difference equation relate? A. Only present and past inputs, never outputs B. The present output to present and past inputs and past outputs C. Output and input in the frequency domain only D. Derivatives of the output to the input Answer: B. It is a constant-coefficient linear recursion in discrete time. (Key point: Definition) 2. For y[n] = 0.5y[n-1] + x[n] with x[n] = delta[n], the impulse response is: A. (0.5)^n for all n B. 0.5 delta[n] C. (0.5)^n u[n] D. u[n] Answer: C. y[0] = 1, y[1] = 0.5, y[2] = 0.25 and so on. (Key point: Example) 3. Why is there a sharp peak at about w = pi/3? A. The zeros lie on the unit circle at 60 degrees B. The poles at radius 2.12 are at 60 degrees C. K = 1 causes it D. The pole pair 0.45 +/- 0.77i (radius 0.89) is close to the unit circle at 60 degrees Answer: D. A pole near the circle at that angle makes the denominator distance small. (Key point: Resonance) ## Related in Digital Signal Analysis and Processing - [Map given poles and zeros, magnitude response](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Apole-zero-map-given-poles) — Digital Signal Analysis and Processing, chapter 3, asked 2× - [Difference equation need; pole-zero, magnitude response](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Awhy-difference-eq-magnitude-response) — Digital Signal Analysis and Processing, chapter 3, asked 3× - [Pole-zero, magnitude response of y=x+x(n−2)](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Asystem-y-x-plus-x-delay-2) — Digital Signal Analysis and Processing, chapter 3 - [Magnitude response of second order system](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Amagnitude-response-0-5-0-3) — Digital Signal Analysis and Processing, chapter 3, asked 2× - [Pole-zero and magnitude response, third order](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Amagnitude-response-0-4-0-2-0-5) — Digital Signal Analysis and Processing, chapter 3, asked 3× - [Pole-zero and magnitude response, y−0.4y+0.1y](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Amagnitude-response-0-4-0-1) — Digital Signal Analysis and Processing, chapter 3, asked 3× ## Related in other subjects - [H(e^(jω)), magnitude plot and h[n] for y[n] = 0.5y[n−1] + x[n]](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Afirst-order-difference-plot) — Signals and Systems, chapter 6, asked 9× - [Inverse Chebyshev characteristics, order, poles and zeros](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Ainverse-chebyshev-poles-zeros) — Filter Design, chapter 2, asked 2× - [Poles and zeros of a network function](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_circuit_theory%3Apoles-and-zeros-significance) — Electrical Circuit Theory, chapter 5 - [Transfer function, poles and zeros from 3×3 state model](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Ass-tf-poles-zeros-3x3) — Control Systems, chapter 8, asked 3×