--- title: "Fibonacci difference equation: limiting ratio 1.6180" subject: "Digital Control System" chapter: 2 key: "semester_6/digital_control_system:fibonacci-limit-ratio" source: https://www.bctengineer.com/q/semester_6/digital_control_system/fibonacci-limit-ratio site: BCT Engineering --- # Fibonacci difference equation: limiting ratio 1.6180 Digital Control System · Chapter 2: The Z-Transform - Asked in: 2073 Magh, 8 marks - Video: [Fibonacci Difference Equation | Z transform | DSP | MATLAB](https://www.youtube.com/watch?v=p9f9oRCT5ME) — Knowledge Amplifier - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_6/digital_control_system/fibonacci-limit-ratio - Whole chapter: https://www.bctengineer.com/md/semester_6/digital_control_system/chapter-2 ## Question Find X(k) when x(k+2)=x(k+1)+x(k), x(0)=0 and x(1)=1, and show that the limiting value of x(k+1)/x(k)=1.6180 when k tends to infinity. ## Short answer Solve via z-transform: X(z)=z/(z^2–z–1). Partial fractions give x(k)=(1/√5)[((1+√5)/2)^k – ((1–√5)/2)^k]. The dominant root φ=(1+√5)/2, so lim x(k+1)/x(k)=φ=1.6180. ## Answer This is the **Fibonacci** equation $x(k+2) - x(k+1) - x(k) = 0$. Solve it by z-transform, then take the limit of the ratio. #### Step 1: Take the z-transform Use $\mathcal{Z}[x(k+2)] = z^2X(z) - z^2x(0) - zx(1)$ and $\mathcal{Z}[x(k+1)] = zX(z) - zx(0)$, with $x(0) = 0$ and $x(1) = 1$: $$ \left[z^2X(z) - z\right] - zX(z) - X(z) = 0 $$ $$ X(z) = \frac{z}{z^2 - z - 1} $$ #### Step 2: Factorise the denominator $$ z^2 - z - 1 = 0 \;\Rightarrow\; z_{1,2} = \frac{1 \pm \sqrt{5}}{2}, \quad z_1 = 1.6180,\; z_2 = -0.6180 $$ #### Step 3: Partial fractions of $X(z)/z$ $$ \frac{X(z)}{z} = \frac{1}{(z-z_1)(z-z_2)} = \frac{1}{z_1 - z_2}\left[\frac{1}{z-z_1} - \frac{1}{z-z_2}\right], \quad z_1 - z_2 = \sqrt{5} $$ $$ X(z) = \frac{1}{\sqrt5}\left[\frac{z}{z - z_1} - \frac{z}{z - z_2}\right] $$ #### Step 4: General (closed-form) solution $$ x(k) = \frac{1}{\sqrt5}\left[\left(\frac{1+\sqrt5}{2}\right)^k - \left(\frac{1-\sqrt5}{2}\right)^k\right], \quad k = 0, 1, 2, \dots $$ Here $1/\sqrt5 = 0.4472$. Check: $x(0) = 0$, and $x(1) = \frac{1}{\sqrt5}\cdot\sqrt5 = 1$. First terms: $0, 1, 1, 2, 3, 5, 8, 13, 21, 34, \dots$ #### Step 5: Limiting value of $x(k+1)/x(k)$ Let $z_1 = 1.6180$ and $z_2 = -0.6180$: $$ \frac{x(k+1)}{x(k)} = \frac{z_1^{k+1} - z_2^{k+1}}{z_1^{k} - z_2^{k}} = \frac{z_1 - z_2\left(\dfrac{z_2}{z_1}\right)^{k}}{1 - \left(\dfrac{z_2}{z_1}\right)^{k}} $$ Since $\left|\dfrac{z_2}{z_1}\right| = \dfrac{0.618}{1.618} = 0.382 < 1$, we get $\left(\dfrac{z_2}{z_1}\right)^k \to 0$ as $k \to \infty$: $$ \lim_{k\to\infty}\frac{x(k+1)}{x(k)} = z_1 = \frac{1+\sqrt5}{2} = 1.6180 $$ This is the golden ratio. #### Numerical check | $k$ | 5 | 6 | 7 | 8 | 9 | |---|---|---|---|---|---| | $x(k+1)/x(k)$ | 8/5 = 1.600 | 13/8 = 1.625 | 21/13 = 1.6154 | 34/21 = 1.6190 | 55/34 = 1.6176 | The ratio oscillates about 1.6180 and settles there. **Answer:** $x(k) = \dfrac{1}{\sqrt5}\left[(1.6180)^k - (-0.6180)^k\right]$, and $\lim_{k\to\infty} x(k+1)/x(k) = \dfrac{1+\sqrt5}{2} = 1.6180$. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. With x(0)=0, x(1)=1, the z-transform gives X(z) = ? A. z^2/(z^2 - z - 1) B. z/(z^2 + z - 1) C. z/(z^2 - z - 1) D. 1/(z^2 - z - 1) Answer: C. z^2X - z - zX - X = 0. (Key point: Transform of difference equation) 2. What are the roots of z^2 - z - 1 = 0? A. 1.618 and 0.618 B. -1.618 and 0.618 C. 2 and -1 D. 1.618 and -0.618 Answer: D. z = (1 +- sqrt5)/2. (Key point: Roots) 3. Which is the closed form? A. [(1+sqrt5)/2]^k - [(1-sqrt5)/2]^k all over sqrt5 B. The same with a plus sign C. The same without 1/sqrt5 D. [(1+sqrt5)/2]^k only Answer: A. Partial fractions give 1/sqrt5 [z/(z-z1) - z/(z-z2)]. (Key point: Closed form) 4. Why does x(k+1)/x(k) tend to 1.618? A. Initial conditions force it B. (z2/z1)^k -> 0 since |z2/z1| = 0.382 < 1, leaving z1 C. z2 dominates D. Both roots equal Answer: B. The smaller root's term dies out, leaving ratio z1 = golden ratio. (Key point: Limit) ## Related in Digital Control System - [Fibonacci difference equation: closed form and limit](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Afibonacci-general-solution) — Digital Control System, chapter 2, asked 2× - [Fibonacci difference equation: series to 8 terms](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Afibonacci-series-8-terms) — Digital Control System, chapter 2 - [Third-order difference equation: limit x(k+2)/x(k)](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Athird-order-difference-limit-k) — Digital Control System, chapter 2, asked 2× - [Third-order difference equation: limit x(k+2)/x(k+1)](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Athird-order-difference-limit-k1) — Digital Control System, chapter 2 - [Difference equation x(k+2)+3x(k+1)+2x(k)=0](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Adifference-equation-x0-0-x1-1) — Digital Control System, chapter 2, asked 5× - [Difference equation with step input](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Adifference-equation-step-0-25) — Digital Control System, chapter 2, asked 2× ## Related in other subjects - [Short vs long column; limiting slenderness ratio](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Ashort-long-column-slenderness) — Strength of Materials, chapter 8 - [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× - [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× - [Difference equation by Z-transform](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Adifference-equation-by-z-transform-two) — Mathematics III, chapter 3, asked 2×