What is Linear Convolution of Two Sequences?

Linear convolution of two sequences is a mathematical operation that produces a third sequence expressing how the shape of one sequence modifies the other. Specifically, if you convolve a sequence of length P with another sequence of length Q, the resulting sequence will have a length of P + Q - 1.

Here's a breakdown:

• Core Idea: Linear convolution essentially slides one sequence across the other, performing a weighted sum at each position. The weights are determined by the values of the second sequence.

• Resultant Sequence Length: As stated, the output sequence has a length equal to the sum of the lengths of the input sequences minus 1. This is key to understanding the extent of the convolution.

• Mathematical Representation: If we denote two sequences as x[n] and h[n], their linear convolution, denoted as y[n] = x[n] * h[n], is defined as:

  y[n] = Σ x[k] * h[n - k]

  where the summation is taken over all possible values of k.

• Zero Index Location: The location of the zero index is important, especially when implementing convolution. Be mindful of how your sequences are indexed and how the h[n-k] term affects the alignment.

Example:

Let's say we have two sequences:

• x[n] = [1, 2, 3] (length P = 3)
• h[n] = [4, 5, 6] (length Q = 3)

The linear convolution y[n] will have a length of P + Q - 1 = 3 + 3 - 1 = 5.

The process involves flipping h[n] to get h[-n] and then sliding it across x[n], performing a sum of products at each shift. This results in:

y[n] = [4, 13, 28, 27, 18]

Key Points:

• Linear convolution is fundamental in signal processing, image processing, and many other areas of engineering and science.
• It's used for tasks like filtering, blurring, edge detection, and system analysis.
• It differs from circular convolution, which is relevant in the context of the Discrete Fourier Transform (DFT). Circular convolution assumes the sequences are periodic.

In summary, linear convolution is a process of combining two sequences to produce a third sequence that reflects how one sequence modifies the other. The resulting sequence's length is the sum of the lengths of the original sequences, minus one.