---
title: "Direct form I and II, y(n)=−0.1y(n−1)+0.2y(n−2)+3x(n)…"
subject: "Digital Signal Analysis and Processing"
chapter: 4
key: "semester_7/digital_signal_analysis_and_processing:direct-form-1-2-3x-3-6x"
source: https://www.bctengineer.com/q/semester_7/digital_signal_analysis_and_processing/direct-form-1-2-3x-3-6x
site: BCT Engineering
---

# Direct form I and II, y(n)=−0.1y(n−1)+0.2y(n−2)+3x(n)…

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

- Asked in: 2072 Chaitra, 5 marks
- Video: [Obtain Direct form I & Direct form II of y(n)=-0.1y(n-1)+0.2y(n-2)+3x(n)+3.6(x(n-1)+0.6x(n-2)](https://www.youtube.com/watch?v=2X6smSrL1ts) — 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/direct-form-1-2-3x-3-6x
- Whole chapter: https://www.bctengineer.com/md/semester_7/digital_signal_analysis_and_processing/chapter-4

## Question

Determine the Direct Form I and Direct Form II realization of the following system: y(n) = −0.1y(n−1) + 0.2y(n−2) + 3x(n) + 3.6x(n−2) + 0.6x(n−2).

## Short answer

Assuming the equation is y(n) = −0.1y(n−1) + 0.2y(n−2) + 3x(n) + 3.6x(n−1) + 0.6x(n−2). DF-I needs 4 delays with coefficients 3, 3.6, 0.6 (feedforward) and −0.1, 0.2 (feedback). DF-II uses 2 delays via w[n] = x[n] −0.1w[n−1] +0.2w[n−2] and y[n]=3w[n]+3.6w[n−1]+0.6w[n−2].

## Answer

The printed equation has x(n−2) twice. I read it as y(n) = −0.1y(n−1) + 0.2y(n−2) + 3x(n) + 3.6x(n−1) + 0.6x(n−2) (the second term should be x(n−1)). If it is taken literally, the two x(n−2) terms combine to 4.2x(n−2) and only b₁ = 0, b₂ = 4.2 change.

H(z) = (3 + 3.6z⁻¹ + 0.6z⁻²) / (1 + 0.1z⁻¹ − 0.2z⁻²)

#### Direct Form I
Separate delays for input and output (2 + 2 = 4 delays).

```text
 x[n]   ─────►[×3]──────────┐
   │z⁻¹                     │
 x[n-1] ─────►[×3.6]────────┤
   │z⁻¹                     ▼
 x[n-2] ─────►[×0.6]─────►( Σ )────┬──► y[n]
                            ▲      │
 y[n-1] ─────►[×−0.1]───────┤      │
   │z⁻¹    (y[n] is delayed to      │
 y[n-2] ─────►[×0.2]────────┘      │ give y[n-1], y[n-2])
```
In words: y[n] = 3x[n] + 3.6x[n−1] + 0.6x[n−2] + (−0.1)y[n−1] + (0.2)y[n−2], with x[n−1], x[n−2] from a two-delay line on the input and y[n−1], y[n−2] from a two-delay line on the output.

#### Direct Form II
Let w[n] = x[n] − 0.1w[n−1] + 0.2w[n−2], and y[n] = 3w[n] + 3.6w[n−1] + 0.6w[n−2].

```text
 x[n]──►(+)────────┬──────►[×3]────►(+)──┬──► y[n]
         ▲   w[n]  ▼ z⁻¹                 ▲  │
         │         ├────────►[×3.6]──────┤
         ├─[×−0.1]─┤                     │
         │         ▼ z⁻¹                 │
         │         └────────►[×0.6]──────┘
         └─[×0.2]──┘
```
(−0.1 and 0.2 are the feedback multipliers on w[n−1] and w[n−2]; 3, 3.6, 0.6 are the feed-forward multipliers.)

Direct Form II needs only **2 delays** (canonic), against 4 for Direct Form I.

**Answer: DF-I: 4 delays; DF-II: 2 delays; feedback −0.1, 0.2; feed-forward 3, 3.6, 0.6.**

## Check yourself

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

1. What is H(z) for y(n) = -0.1y(n-1) + 0.2y(n-2) + 3x(n) + 3.6x(n-1) + 0.6x(n-2)?
   A. (3 + 3.6z^-1 + 0.6z^-2)/(1 - 0.1z^-1 + 0.2z^-2)
   B. (3 + 3.6z^-1 + 0.6z^-2)/(1 + 0.1z^-1 - 0.2z^-2)
   C. (3 + 3.6z^-1 + 0.6z^-2)/(1 + 0.1z^-1 + 0.2z^-2)
   D. (1 + 0.1z^-1 - 0.2z^-2)/(3 + 3.6z^-1 + 0.6z^-2)
   Answer: B. Moving output terms to the left gives denominator 1 + 0.1z^-1 - 0.2z^-2. (Key point: Transfer function)

2. How many delays do DF-I and DF-II need?
   A. 4 and 2
   B. 2 and 4
   C. 4 and 4
   D. 2 and 2
   Answer: A. DF-I has 2 input + 2 output delays; DF-II shares them, needing only 2 (canonic). (Key point: Delay count)

## Close variants set in other exams

- [Direct form II of y(n)=−0.1y(n−1)+0.72y(n−2)…](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Adirect-form-2-0-72) — Digital Signal Analysis and Processing, chapter 4, asked 5×
- [Direct form I and II of 3y[n]+y[n−1]+2y[n−4]](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Adirect-form-1-2-delay4) — Digital Signal Analysis and Processing, chapter 4, asked 4×
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