Welcome to our exploration of convolution! Convolution is a fundamental mathematical operation that describes how one function modifies the shape of another function. It's widely used in signal processing, image processing, probability theory, and system analysis. The mathematical definition shows that convolution involves flipping, shifting, multiplying, and summing operations.
The first step in discrete convolution is to flip one of the sequences horizontally. Here we take the sequence g of k and create its flipped version g of negative k. This flipping operation creates a mirror image of the original sequence around k equals zero. The red bars show the original sequence, and the blue bars show the flipped sequence.
The second step is to shift the flipped sequence g of negative k by n positions to create g of n minus k. This shifting operation moves the flipped sequence along the k-axis. For example, when n equals 2, we get g of 2 minus k. As we change the value of n, the sequence slides to different positions, which is essential for computing convolution at different output indices.
The third step involves multiplying f of k by g of n minus k element by element, then summing all the products. For n equals 2, we multiply f of 0 times g of 2, plus f of 1 times g of 1, plus f of 2 times g of 0. This gives us 1 times 0.5 plus 2 times 1 plus 1 times 0.5, which equals 0.5 plus 2 plus 0.5, totaling 3. The green bars show the products, and their sum gives us the convolution output at n equals 2.
To complete the convolution, we repeat this process for all values of n. The final convolved sequence shows the result at each output index. For our example, we get values 0.5, 2.5, 3.0, 1.5, and 0.5 for indices 0 through 4. This convolution operation is fundamental in many applications including digital filters, image processing for blurring and edge detection, and neural networks for feature extraction.