--- title: "Lattice coefficients for fractional FIR" subject: "Digital Signal Analysis and Processing" chapter: 4 key: "semester_7/digital_signal_analysis_and_processing:lattice-coeff-fractions" source: https://www.bctengineer.com/q/semester_7/digital_signal_analysis_and_processing/lattice-coeff-fractions site: BCT Engineering --- # Lattice coefficients for fractional FIR Digital Signal Analysis and Processing · Chapter 4: Discrete Filter Structures - Asked in: 2074 Chaitra, 5 marks - Video: [Q6. b. Lattice Structure of FIR Filters | DSP | EnggClasses](https://www.youtube.com/watch?v=-vur1D7iXyM) — EnggClasses - 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/lattice-coeff-fractions - Whole chapter: https://www.bctengineer.com/md/semester_7/digital_signal_analysis_and_processing/chapter-4 ## Question Determine the lattice coefficients corresponding to the FIR filter with the system function H(z) = A₃(z) = 1 + (52/96)z⁻¹ + (25/40)z⁻² + (1/3)z⁻³. ## Short answer Simplify to A₃(z)=1+(13/24)z⁻¹+(5/8)z⁻²+(1/3)z⁻³. Step-down recursion yields K₁=1/4, K₂=1/2, K₃=1/3. All magnitudes are less than 1, indicating a minimum-phase filter. ## Answer Simplify: 52/96 = 13/24 and 25/40 = 5/8, so A₃(z) = 1 + (13/24)z⁻¹ + (5/8)z⁻² + (1/3)z⁻³. #### Method The FIR lattice of order M uses the stage equations ```text f0(n) = g0(n) = x(n) fm(n) = f(m−1)(n) + Km · g(m−1)(n−1) gm(n) = Km · f(m−1)(n) + g(m−1)(n−1) m = 1 … M y(n) = fM(n) ``` The reflection coefficient of the last stage is Km = am(m) (the last coefficient of the polynomial Am(z)). The lower-order polynomials come from the step-down recursion ```text a(m−1)(i) = [ am(i) − Km · am(m−i) ] / (1 − Km²), i = 1 … m−1 ``` #### Computation A3(z) = 1 + 0.5417z⁻¹ + 0.625z⁻² + 0.3333z⁻³ - A3: coefficients {1, 0.5417, 0.625, 0.3333}, so **K3 = a3(3) = 0.3333** - A2: coefficients {1, 0.375, 0.5}, so **K2 = a2(2) = 0.5** - A1: coefficients {1, 0.25}, so **K1 = a1(1) = 0.25** | Stage | Km | Size | |---|---|---| | 1 | 0.25 | magnitude below 1 | | 2 | 0.5 | magnitude below 1 | | 3 | 0.3333 | magnitude below 1 | **Lattice coefficients:** K1 = 0.25, K2 = 0.5, K3 = 0.3333 #### Structure One lattice stage (repeated for m = 1 … 3 with K1 … K3): ```text f(m-1)[n] ──────●────────────────────(+)──► f(m)[n] │ ┌───────────▲ │ Km │ Km └─────────┼───────────┐ │ │ │ ▼ g(m-1)[n] ─[z⁻¹] ─────────●──────────(+)──► g(m)[n] ``` Cascade (f and g pass together from stage to stage): ```text +------+ +------+ +------+ x[n]-+->| |-f1->| |-f2->| |-f3-> y[n] = f3[n] | | K1 | | K2 | | K3 | +->| |-g1->| |-g2->| | +------+ +------+ +------+ ``` (The upper path carries f and the lower path carries g; f0 = g0 = x[n].) Exact values: **K₁ = 1/4, K₂ = 1/2, K₃ = 1/3**. All |K| < 1, so all the zeros are inside the unit circle (minimum-phase FIR). **Check:** build-up: A₁ = 1 + ¼z⁻¹; A₂ = A₁ + ½z⁻²A₁(z⁻¹) = 1 + 0.375z⁻¹ + 0.5z⁻²; A₃ = A₂ + ⅓z⁻³A₂(z⁻¹) = 1 + 0.5417z⁻¹ + 0.625z⁻² + 0.3333z⁻³. ✓ ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What is A3(z) after simplifying 52/96 and 25/40? A. 1 + (13/24)z^-1 + (5/8)z^-2 + (1/3)z^-3 B. 1 + (26/48)z^-1 + (5/4)z^-2 + (1/3)z^-3 C. 1 + (13/12)z^-1 + (5/8)z^-2 + (1/3)z^-3 D. 1 + (13/24)z^-1 + (5/4)z^-2 + (1/3)z^-3 Answer: A. 52/96 = 13/24 and 25/40 = 5/8; the z^-3 coefficient stays 1/3. (Key point: Simplify coefficients) 2. What is K3 for this filter? A. 13/24 B. 5/8 C. 1/3 D. 1/4 Answer: C. K3 = a3(3), the last coefficient of A3(z), which is 1/3. (Key point: Last stage K3) 3. After step-down, which are the lattice coefficients K1, K2, K3? A. 1/3, 1/2, 1/4 B. 1/4, 1/2, 1/3 C. 1/2, 1/4, 1/3 D. 1/4, 3/8, 1/3 Answer: B. A2 = {1, 0.375, 0.5} gives K2 = 0.5 and A1 = {1, 0.25} gives K1 = 0.25, with K3 = 1/3. (Key point: Final coefficients) 4. All |K| < 1 here. What does this imply? A. All zeros are inside the unit circle (minimum phase) B. The filter is IIR C. The filter has poles at the origin only D. All zeros are on the unit circle Answer: A. For an FIR lattice, |Km| < 1 for all stages means all zeros lie inside the unit circle. 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