---
title: "LTI output, h={1,1,1}, x={1,1,1,1}"
subject: "Digital Signal Analysis and Processing"
chapter: 1
key: "semester_7/digital_signal_analysis_and_processing:lti-output-ones-sequences"
source: https://www.bctengineer.com/q/semester_7/digital_signal_analysis_and_processing/lti-output-ones-sequences
site: BCT Engineering
---

# LTI output, h={1,1,1}, x={1,1,1,1}

Digital Signal Analysis and Processing · Chapter 1: Discrete Time Signals and Systems

- Asked in: 2073 Shrawan, 6 marks
- 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/lti-output-ones-sequences
- Whole chapter: https://www.bctengineer.com/md/semester_7/digital_signal_analysis_and_processing/chapter-1

## Question

Determine the system output y(n) of the following signals: h(n) = {1,1,1} and x(n) = {1,1,1,1}.

## Short answer

Convolving h(n)={1,1,1} and x(n)={1,1,1,1} both starting at n=0 gives y(n)={1,2,3,3,2,1} for n=0 to 5.

## Answer

For an LTI system y(n) = Σ x(k) h(n − k). With no origin marked, take both sequences to start at n = 0: h = {1, 1, 1} (3 samples), x = {1, 1, 1, 1} (4 samples). The output has 3 + 4 − 1 = 6 samples, n = 0 … 5.

#### Table method (sum along diagonals)
| | x0=1 | x1=1 | x2=1 | x3=1 |
|---|---|---|---|---|
| h0=1 | 1 | 1 | 1 | 1 |
| h1=1 | 1 | 1 | 1 | 1 |
| h2=1 | 1 | 1 | 1 | 1 |

Adding the entries with the same i + j:

- y(0) = 1
- y(1) = 1 + 1 = 2
- y(2) = 1 + 1 + 1 = 3
- y(3) = 1 + 1 + 1 = 3
- y(4) = 1 + 1 = 2
- y(5) = 1

**Answer:** y(n) = {1, 2, 3, 3, 2, 1}, n = 0 … 5.

**Check:** Σy = 12 = (Σx)(Σh) = 4 × 3. ✓
Also X(z)H(z) = (1 + z⁻¹ + z⁻² + z⁻³)(1 + z⁻¹ + z⁻²) = 1 + 2z⁻¹ + 3z⁻² + 3z⁻³ + 2z⁻⁴ + z⁻⁵. ✓

## Check yourself

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

1. Convolving 3 samples with 4 samples gives how many samples?
   A. 7
   B. 12
   C. 6
   D. 4
   Answer: C. Length = 3 + 4 - 1 = 6. (Key point: Output length)

2. In the table method, which entries are added to get y(n)?
   A. Entries in the same row
   B. Entries along diagonals with the same i + j
   C. Entries in the same column
   D. All entries
   Answer: B. Product x(j) h(i) contributes to y(i + j), so same-index diagonals are summed. (Key point: Table method)

3. What is y(2) for h = {1,1,1} and x = {1,1,1,1}?
   A. 3
   B. 2
   C. 1
   D. 4
   Answer: A. Three products (i + j = 2) each equal 1, so y(2) = 3. (Key point: Peak output)

4. The output is
   A. {1, 2, 3, 4, 3, 2, 1}
   B. {1, 1, 1, 1, 1, 1}
   C. {3, 3, 3, 3}
   D. {1, 2, 3, 3, 2, 1}
   Answer: D. The diagonal sums give 1,2,3,3,2,1 with sum 12 = 4 x 3. (Key point: Full result)

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