Convolution Result
| n | y[n] | Raw sum | Contributing products |
|---|
Calculator Input
Example Data Table
The table below uses x[n] = [1, 2, 1] and h[n] = [1, -1, 2]. Both sequences start at n = 0.
| n | x[n] | h[n] | Full convolution y[n] |
|---|---|---|---|
| 0 | 1 | 1 | 1 |
| 1 | 2 | -1 | 1 |
| 2 | 1 | 2 | 1 |
| 3 | - | - | 3 |
| 4 | - | - | 2 |
Formula Used
Discrete time convolution:
y[n] = Σ x[k] h[n - k]
The summation is taken over all integer values of k where both samples exist.
Finite sequence index range:
n starts at xStart + hStart
n ends at xEnd + hEnd
Optional output scaling:
displayed y[n] = normalized(gain × raw y[n]) + offset
How to Use This Calculator
- Enter the samples of x[n] in the first sequence box.
- Enter the samples of h[n] in the second sequence box.
- Set the starting index for each sequence.
- Select full, same, or valid output mode.
- Adjust gain, offset, decimals, or normalization if needed.
- Press the calculate button to view the output above the form.
- Download the result as CSV or PDF for reports.
Discrete Time Convolution Guide
What Convolution Means
Discrete time convolution combines two indexed sequences. One sequence is usually the input. The other sequence is often an impulse response. The result shows how a system responds to every shifted sample of the input. This is why convolution is central in digital signal processing.
Why Indexing Matters
Each sample has a position. That position is called n. When a sequence starts at zero, the first value is easy to read. Yet many real problems start at negative or positive indexes. This calculator lets you set those starts. The output range then follows the true support of both sequences.
How the Sum Is Built
For each output index, the calculator finds matching pairs. It multiplies each x sample by the matching shifted h sample. Then it adds those products. The contribution column makes this process visible. It is helpful when checking homework, filter designs, and system models.
Output Modes
Full mode displays every sample created by the two finite sequences. Same mode keeps only the index span of x[n]. Valid mode keeps samples where the shorter sequence fully overlaps the longer sequence. These views are useful for different analysis tasks.
Practical Uses
Convolution can model filters, echoes, smoothing, difference systems, and weighted averages. It also appears in probability, image processing, control systems, and communications. A short impulse response can transform a long input signal. A symmetric response can smooth noise. A difference response can highlight sudden changes.
Reading the Results
The result table lists n, the displayed value, the raw convolution sum, and the products that formed it. The plot gives a fast visual check. Scaling, offset, and normalization help prepare values for comparison. CSV and PDF exports make the result easy to save, share, and include in notes.
FAQs
1. What is discrete time convolution?
It is an operation that combines two indexed sequences. Each output sample is made by shifting one sequence, multiplying matching samples, and adding the products.
2. What does x[n] represent?
x[n] usually represents the input signal. It can also be any finite sequence used in a digital system, probability problem, or numerical model.
3. What does h[n] represent?
h[n] often represents the impulse response of a system. It describes how the system reacts to a single impulse at a specific index.
4. What is full convolution mode?
Full mode returns every output sample created by the two finite sequences. Its length is usually length x plus length h minus one.
5. What is same mode?
Same mode keeps the output samples that match the index span of x[n]. It is useful when you want an output aligned with the input.
6. What is valid mode?
Valid mode keeps only positions where the shorter sequence fully overlaps the longer sequence. It removes partial edge-overlap results.
7. Can I use negative indexes?
Yes. Enter a negative start index for x[n] or h[n]. The calculator uses those starts when building the output index range.
8. Why use normalization?
Normalization makes results easier to compare. Peak normalization scales by the largest magnitude. Sum normalization can help with averaging and probability sequences.