--- title: "Difference equation need; pole-zero, magnitude response" subject: "Digital Signal Analysis and Processing" chapter: 3 key: "semester_7/digital_signal_analysis_and_processing:why-difference-eq-magnitude-response" source: https://www.bctengineer.com/q/semester_7/digital_signal_analysis_and_processing/why-difference-eq-magnitude-response site: BCT Engineering --- # Difference equation need; pole-zero, magnitude response Digital Signal Analysis and Processing · Chapter 3: Analysis of LTI System in Frequency Domain - Asked in: 2069 Chaitra, 2+2+6 marks - Video: [Z transformation - pole zero plot, transfer function, difference equation, stable/unstable](https://www.youtube.com/watch?v=CE_h0gqgGV4) — Shrenik Jain - 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/why-difference-eq-magnitude-response - Whole chapter: https://www.bctengineer.com/md/semester_7/digital_signal_analysis_and_processing/chapter-3 ## Question Why do we need difference equation? Draw pole-zero in z-plane and plot magnitude response (not to the scale) of the system described by difference equation y[n] − 0.4y[n−1] + 0.2y[n−2] = x[n] + 0.1x[n−1] − 0.06x[n−2]. ## Short answer Difference equations serve as time-domain implementations and provide H(z) for pole‑zero analysis. For y[n]−0.4y[n−1]+0.2y[n−2]=x[n]+0.1x[n−1]−0.06x[n−2], H(z) has poles at 0.2±0.4j (r=0.447), zeros at −0.3 and0.2; magnitude is low‑pass with a peak of1.51 around ω≈0.30π, |H(0)|=1.3, |H(π)|=0.525. ## Answer #### Why we need a difference equation - It is the time-domain description of a discrete system, giving a direct recipe for implementation: each output is a weighted combination of present/past inputs and past outputs, needing only adders, multipliers and unit delays. - It represents recursive (IIR) systems, whose impulse response is infinite, with a finite number of coefficients. - Taking its z-transform gives the system function H(z), from which poles, zeros, stability and the frequency response are obtained. - It is easy to program on a DSP processor and is the basis of realisation structures (direct form, cascade, parallel, lattice). - It is the discrete counterpart of the differential equation describing analog systems. #### System function Taking the z-transform of y[n] − 0.4y[n−1] + 0.2y[n−2] = x[n] + 0.1x[n−1] − 0.06x[n−2] with zero initial conditions (x[n−k] ↔ z⁻ᵏ X(z)): H(z) = Y(z)/X(z) = (1 + 0.1z⁻¹ − 0.06z⁻²) / (1 − 0.4z⁻¹ + 0.2z⁻²) Multiplying numerator and denominator by z^2: H(z) = N(z)/D(z) with - **Poles** (roots of D(z)): 0.200 + 0.400j and its conjugate (radius 0.447) - **Zeros** (roots of N(z)): -0.300 (real); 0.200 (real) All poles are inside the unit circle (largest radius 0.447), so the causal system is stable. #### Pole-zero plot ```text | | | ....... ...... | ...... ... | ... ... | ... .. | x .. .. | .. . | . --------.---------o----|--o-----------.-------- . | . .. | .. .. | x .. ... | ... ... | ... ...... | ...... ....... | | | ``` (x = pole, o = zero, dotted circle = unit circle; Re axis horizontal, Im axis vertical; range ±1.5) #### Magnitude response Put z = e^(jω): |H(e^(jω))| = (product of distances from e^(jω) to the zeros) / (product of distances to the poles). The response is large where e^(jω) passes close to a pole and small near a zero. | w | 0 | pi/6 | pi/3 | pi/2 | 2pi/3 | 5pi/6 | pi | |---|---|---|---|---|---|---|---| | \|H\| | 1.300 | 1.401 | 1.498 | 1.190 | 0.814 | 0.595 | 0.525 | ```text 1.51 | ******** 1.34 |****************** 1.17 |******************** 1.00 |*********************** 0.84 |************************** 0.67 |******************************* 0.50 |************************************* 0.33 |************************************* 0.17 |************************************* +------------------------------------- 0 pi/2 pi (w) ``` **Shape:** low-pass with a small peak. |H(0)| = 1.3, a maximum of 1.51 near ω ≈ 0.30π (pole pair 0.2 ± 0.4j at angle 63.4°, radius 0.447), then a smooth fall to |H(π)| = 0.525. The zeros (−0.3, 0.2) are near the origin and hardly shape the response. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Why is a difference equation needed? A. It avoids the need for any z-transform B. It only describes FIR systems C. It is used only for analog systems D. It gives a direct time-domain recipe to implement the system with adders, multipliers and delays Answer: D. It describes how each output is computed from present/past inputs and past outputs. (Key point: Purpose) 2. What advantage does it give for IIR systems? A. An infinite impulse response is described with finitely many coefficients B. It makes the impulse response finite C. It guarantees stability D. It removes poles Answer: A. Recursion (feedback of past outputs) gives infinite h[n] from a few coefficients. (Key point: IIR representation) 3. Where are the poles of y[n] - 0.4y[n-1] + 0.2y[n-2] = ...? A. -0.3 and 0.2 B. 0.2 +/- 0.4j, radius 0.447, so stable C. 0.4 +/- 0.2j, radius 1 D. 0.1 and -0.06 Answer: B. Roots of z^2 - 0.4z + 0.2 are 0.2 +/- 0.4j. (Key point: Poles) 4. Describe the magnitude response. A. High-pass with a deep notch at 0 B. Sharp resonance of about 7 C. Low-pass with a small peak (about 1.5) near 0.30 pi, falling to 0.525 at pi D. All-pass Answer: C. The zeros near the origin hardly shape the response; the pole pair gives a small peak. 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