--- title: "Direct form and lattice of FIR, 1+0.7z⁻¹+1.2z⁻²−z⁻³" subject: "Digital Signal Analysis and Processing" chapter: 4 key: "semester_7/digital_signal_analysis_and_processing:lattice-fir-1-0-7-1-2" source: https://www.bctengineer.com/q/semester_7/digital_signal_analysis_and_processing/lattice-fir-1-0-7-1-2 site: BCT Engineering --- # Direct form and lattice of FIR, 1+0.7z⁻¹+1.2z⁻²−z⁻³ Digital Signal Analysis and Processing · Chapter 4: Discrete Filter Structures - Asked in: 2074 Asoj, 3+7 marks - Video: [Q6. a. Lattice Structure of FIR Filters | DSP | EnggClasses](https://www.youtube.com/watch?v=AcMcntjNpM8) — 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-fir-1-0-7-1-2 - Whole chapter: https://www.bctengineer.com/md/semester_7/digital_signal_analysis_and_processing/chapter-4 ## Question Draw the direct form and lattice structure of a filter with system function H(z) = 1 + 0.7z⁻¹ + 1.2z⁻² − z⁻³. ## Short answer Direct form: y[n]=x[n]+0.7x[n-1]+1.2x[n-2]-x[n-3] with 3 delays. Lattice: K3=-1, but step-down fails because 1-K3^2=0 and coefficients are not antisymmetric, so no finite K1,K2 exist; only direct form implementation is possible. ## Answer #### Direct form H(z) = 1 + 0.7z⁻¹ + 1.2z⁻² − z⁻³, so y[n] = x[n] + 0.7x[n−1] + 1.2x[n−2] − x[n−3]. ```text x[n] ─┬──►[z⁻¹]──┬──►[z⁻¹]──┬──►[z⁻¹]──┐ │ │ │ │ ×b0 ×b1 ×b2 ×b3 │ │ │ │ └──►(+)◄───┘ │ │ └───►(+)◄────────┘ │ └────►(+)◄─────────┘ │ ▼ y[n] ``` with b0 = 1, b1 = 0.7, b2 = 1.2, b3 = -1. Output: y[n] = b0x[n] + b1x[n−1] + b2x[n−2] + b3x[n−3]; 3 delays, 4 multipliers, 3 adders. #### Lattice structure Order M = 3, a₃(1) = 0.7, a₃(2) = 1.2, a₃(3) = −1. The last reflection coefficient is K₃ = a₃(3) = **−1** The step-down formula divides by 1 − K₃² = 0, and the numerator for i = 1 is a₃(1) − K₃a₃(2) = 0.7 + 1.2 = 1.9 ≠ 0, so K₂ would be infinite. Check by building the filter up from a lattice: A₃(z) = A₂(z) + K₃z⁻³A₂(z⁻¹) with A₂ = 1 + a₁z⁻¹ + a₂z⁻² gives the coefficients {1, a₁ + K₃a₂, a₂ + K₃a₁, K₃}. With K₃ = −1 this is {1, a₁ − a₂, a₂ − a₁, −1}, which needs the z⁻¹ and z⁻² coefficients to be negatives of each other; here they are 0.7 and 1.2. **Conclusion:** because |K₃| = 1 and the coefficients are not (anti)symmetric, this H(z) has no finite lattice coefficients K₁, K₂ (the FIR lattice requires |K| ≠ 1 at the last stage). K₃ = −1 is the only coefficient that can be stated; the lattice stage for m = 3 is ```text f3(n) = f2(n) − g2(n−1) g3(n) = −f2(n) + g2(n−1) ``` and the filter must be implemented in the direct form drawn above. (If the last coefficient were not ±1, say a₃(3) = k with |k| < 1, the usual step-down procedure and the stage diagram of the lattice below would apply.) One stage of the lattice (for reference, repeated for m = 1, 2, 3): ```text f(m-1)[n] ──────●────────────────────(+)──► f(m)[n] │ ┌───────────▲ │ Km │ Km └─────────┼───────────┐ │ │ │ ▼ g(m-1)[n] ─[z⁻¹] ─────────●──────────(+)──► g(m)[n] ``` ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. How many delays, multipliers and adders does the direct form need? A. 3 delays, 4 multipliers, 3 adders B. 4 delays, 3 multipliers, 3 adders C. 3 delays, 3 multipliers, 4 adders D. 2 delays, 4 multipliers, 3 adders Answer: A. Order 3 with 4 coefficients. (Key point: Direct form) 2. What is K3 for this filter? A. 1 B. -1 C. 0.7 D. 1.2 Answer: B. K3 = a3(3) = -1. (Key point: Last coefficient) 3. Why can K2 not be computed with the usual formula? A. a3(2) is zero B. The filter is unstable C. 1 - K3^2 = 0, so the step-down divides by zero D. The coefficients are complex Answer: C. With |K3| = 1 the denominator vanishes and the numerator is nonzero. (Key point: Step-down failure) 4. What is the conclusion? A. The lattice has K1 = K2 = 0 B. The lattice has K1 = 0.7, K2 = 1.2 C. The filter is unstable D. No finite K1, K2 exist, so implement it in direct form Answer: D. With |K3| = 1 and non-antisymmetric coefficients, no finite lattice exists. 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