---
title: "Lattice coefficients of fourth order FIR"
subject: "Digital Signal Analysis and Processing"
chapter: 4
key: "semester_7/digital_signal_analysis_and_processing:lattice-coeff-fir-4th"
source: https://www.bctengineer.com/q/semester_7/digital_signal_analysis_and_processing/lattice-coeff-fir-4th
site: BCT Engineering
---

# Lattice coefficients of fourth order FIR

Digital Signal Analysis and Processing · Chapter 4: Discrete Filter Structures

- Asked in: 2070 Chaitra, 6 marks
- Video: [A FIR filter is given by, 𝒚(𝒏)=𝒙(𝒏)+𝟐/𝟓 𝒙(𝒏−𝟏)+𝟑/𝟒 𝒙(𝒏−𝟐)+𝟏/𝟑 𝒙(𝒏−𝟑) Draw the Lattice structure.](https://www.youtube.com/watch?v=bjcf2ilEWHo) — Engg-Course-Made-Easy
- 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-fir-4th
- Whole chapter: https://www.bctengineer.com/md/semester_7/digital_signal_analysis_and_processing/chapter-4

## Question

Compute the lattice coefficients and draw the lattice structure of the following FIR system H(z) = 1 + 3.1z⁻¹ + 5.5z⁻² + 4.2z⁻³ + 2.3z⁻⁴.

## Short answer

Using step-down recursion: K₄ = 2.3, K₃ = 0.683, K₂ = 1.1664, K₁ = 0.3381. The lattice structure cascades four stages with these reflection coefficients in reverse order, forming an FIR filter.

## Answer

#### 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
A4(z) = 1 + 3.1z⁻¹ + 5.5z⁻² + 4.2z⁻³ + 2.3z⁻⁴

- A4: coefficients {1, 3.1, 5.5, 4.2, 2.3}, so **K4 = a4(4) = 2.3**
- A3: coefficients {1, 1.5291, 1.6667, 0.683}, so **K3 = a3(3) = 0.683**
- A2: coefficients {1, 0.7325, 1.1664}, so **K2 = a2(2) = 1.1664**
- A1: coefficients {1, 0.3381}, so **K1 = a1(1) = 0.3381**

| Stage | Km | Size |
|---|---|---|
| 1 | 0.3381 | magnitude below 1 |
| 2 | 1.1664 | magnitude above 1 |
| 3 | 0.683 | magnitude below 1 |
| 4 | 2.3 | magnitude above 1 |

**Lattice coefficients:** K1 = 0.3381, K2 = 1.1664, K3 = 0.683, K4 = 2.3

#### Structure
One lattice stage (repeated for m = 1 … 4 with K1 … K4):

```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->|      |-f4-> y[n] = f4[n]
     |  |  K1  |     |  K2  |     |  K3  |     |  K4  |
     +->|      |-g1->|      |-g2->|      |-g3->|      |
        +------+     +------+     +------+     +------+
```
(The upper path carries f and the lower path carries g; f0 = g0 = x[n].)

**Check:** simulating the 4-stage lattice with a unit impulse gives {1, 3.1, 5.5, 4.2, 2.3}. ✓ (Since |K2| and |K4| exceed 1, some zeros lie outside the unit circle.)

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. For the FIR polynomial A4(z) = 1 + 3.1z^-1 + 5.5z^-2 + 4.2z^-3 + 2.3z^-4, which value is the stage-4 reflection coefficient K4?
   A. 3.1
   B. 2.3
   C. 1
   D. 5.5
   Answer: B. The reflection coefficient of the last stage is Km = am(m), the last coefficient of Am(z), so K4 = a4(4) = 2.3. (Key point: Last coefficient is K)

2. How are the lower-order polynomials A3, A2, A1 obtained from A4?
   A. a(m-1)(i) = am(i) + Km am(m-i), no division
   B. a(m-1)(i) = [am(i) - Km am(m-i)] / (1 - Km^2)
   C. a(m-1)(i) = am(i) / Km
   D. By dropping the highest-order term only
   Answer: B. The step-down recursion subtracts Km times the reversed coefficient and divides by (1 - Km^2). (Key point: Step-down recursion)

3. Which set gives the lattice coefficients K1, K2, K3, K4 for this filter?
   A. 0.3381, 1.1664, 0.683, 2.3
   B. 2.3, 0.683, 1.1664, 0.3381
   C. 0.3381, 0.7325, 1.5291, 3.1
   D. 1, 3.1, 5.5, 4.2
   Answer: A. Each Km is the last coefficient of Am(z): K1 = 0.3381, K2 = 1.1664, K3 = 0.683, K4 = 2.3. (Key point: Lattice coefficients result)

## Close variants set in other exams

- [Lattice structure of 1+2z⁻¹−3z⁻²+4z⁻³](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Alattice-coeff-1-2-minus3-4) — Digital Signal Analysis and Processing, chapter 4, asked 4×
- [Lattice coefficients of 1+2z⁻¹+z⁻²](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Alattice-coeff-1-2-1) — Digital Signal Analysis and Processing, chapter 4, asked 4×

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