Linear Convolution Calculator

Example Data Table

Formula Used

Linear convolution combines two finite sequences by shifting one sequence across the other and summing each overlap.

Formula: y[n] = Σ x[k] h[n - k]

For every output index n, multiply each valid pair x[k] and h[n - k]. Add those products. The first output index equals the sum of the two starting indices. The total output length equals Lx + Lh - 1.

This page also lists each valid product for every output term. That makes the result easier to verify during homework, revision, and signal processing practice.

How to Use This Calculator

1. Enter the values of x[n] using commas, spaces, or separate lines.
2. Enter the values of h[n] in the second sequence field.
3. Set the starting index for each sequence if indices do not begin at zero.
4. Choose the decimal precision you want in the output.
5. Select whether you want the detailed term-by-term breakdown.
6. Press the calculation button to display the result above the form.
7. Review the output sequence, graph, table, and optional steps.
8. Use the export buttons to save the results as CSV or PDF.

About Linear Convolution

Linear convolution is a standard operation in discrete mathematics and signal analysis. It measures how one finite sequence overlaps another during shifting. Each new shift creates one output term. Every output term is a sum of products.

This process is useful when studying digital filters, recurrence style models, impulse responses, sequence algebra, and sampled systems. Students often need both the final sequence and the working steps. This calculator gives both.

The first sequence can represent an input signal. The second sequence can represent a system response. Their convolution returns the total response produced by all overlaps. That is why convolution appears in transforms, systems, and communication topics.

Start indices matter. Some sequences begin at zero. Others begin at negative or positive offsets. This calculator handles those offsets and returns the correct output index range. That keeps the table and the graph aligned with the sequence definitions.

The graph helps you compare x[n], h[n], and y[n] on one view. The table lists every output index. The optional step section expands each term so you can audit the arithmetic without doing extra manual work.

Use this page for coursework, practice, checking manual solutions, or building intuition about sequence overlap. Keep the inputs short when you want to inspect every term. Use longer lists when you mainly need the final sequence and exports.

FAQs

1. What is linear convolution?

It is the sum of pairwise products formed while one finite sequence shifts across another. Each shift creates one output value in the final sequence.

2. How is linear convolution different from circular convolution?

Linear convolution uses full sequence overlap and produces Lx + Lh - 1 outputs. Circular convolution wraps indices around a fixed period.

3. Why do start indices matter?

Start indices define the true position of each sample. They control the output index range and help match textbook sequence notation correctly.

4. What separators can I use for inputs?

You can use commas, spaces, or new lines. Brackets are also ignored, so simple list formats work well.

5. Can this calculator handle negative values?

Yes. It accepts positive values, negative values, and decimals as long as every entry is numeric.

6. Why is the output longer than the inputs?

The overlap grows, reaches full coverage, then shrinks. Because of that pattern, the output length becomes the sum of both lengths minus one.

7. What does the step breakdown show?

It lists each valid product contributing to one output term. Then it shows the sum that produces that output value.

8. When should I export CSV or PDF?

Export CSV for spreadsheets and saved data analysis. Export PDF when you need a clean report for revision, printing, or sharing.