1. What is Discrete Convolution?

Discrete convolution is a mathematical operation that combines two sequences to produce a third sequence. It's fundamental in digital signal processing for systems analysis, filtering, and more.

2. How Does the Calculator Work?

The calculator implements the discrete convolution formula:

Where:

• \( x[k] \) — Input signal (array of values)
• \( h[k] \) — Impulse response (array of values)
• \( y[n] \) — Output signal (array of values)
• \( n \) — Current output index
• \( k \) — Summation index

Explanation: For each output point n, the calculation sums the products of x[k] and h[n-k] for all relevant k values.

3. Importance of Convolution

Details: Convolution is essential for understanding linear time-invariant systems, designing digital filters, and processing signals in both time and frequency domains.

4. Using the Calculator

Tips: Enter comma-separated values for both input signal and impulse response. The output length will be (length of x + length of h - 1).

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between discrete and continuous convolution?
A: Discrete convolution operates on sampled data (arrays), while continuous convolution uses integrals over continuous functions.

Q2: What are typical applications of convolution?
A: Audio processing, image filtering, system response calculation, and implementing FIR filters.

Q3: How does zero-padding affect convolution?
A: Zero-padding can be used to make the arrays the same length for circular convolution or to prevent edge effects.

Q4: What's the computational complexity?
A: Direct computation is O(N²). For large arrays, FFT-based convolution (O(N log N)) is more efficient.

Q5: How does this relate to correlation?
A: Correlation is similar but doesn't reverse one of the signals. Convolution flips the impulse response.