13.1.2. Definition of Convolution

Let 𝑓(𝑡) and 𝑔(𝑡) be two piecewise continuous functions defined for 𝑡 ≥ 0. The convolution of 𝑓(𝑡) and 𝑔(𝑡), denoted by (𝑓∗𝑔)(𝑡), is defined as:

(𝑓∗𝑔)(𝑡) = ∫ 𝑓(𝜏)𝑔(𝑡−𝜏) 𝑑𝜏

This operation produces a new function by integrating the product of one function and a time-reversed version of another.

Convolution is an operation that combines two functions to create a third function. In our case, we have two functions, f(t) and g(t), which are both defined for time t greater than or equal to zero. The resulting convolution, denoted (f*g)(t), is calculated using an integral that integrates the product of f(τ) (where τ is a dummy variable representing time) and g(t - τ). This effectively involves multiplying f(τ) by g(t) after shifting it by an amount τ, and then summing this product over all values of τ from 0 to t. The time-reversed nature of the g(t - τ) term is what allows convolution to mix the two functions in a meaningful way. It's often used in mathematics to analyze signals and systems, especially in engineering.

Imagine you are baking cookies (representing g(t)), and you have various ingredients (representing f(t)) that you must combine in specific amounts at different times. Here, convolution would represent how all these ingredients are mixed in the bowl over time, taking into account the order and timing when each ingredient is added. The resulting cookie dough (the convolution result) is unique to the specific way you combined both the process of adding ingredients (f(t)) and the cookie recipe (g(t)).

In this context, the convolution is defined mathematically as:

(𝑓∗𝑔)(𝑡) = ∫ 𝑓(𝜏)𝑔(𝑡−𝜏) 𝑑𝜏

for 0 ≤ 𝜏 ≤ 𝑡.

The integral shown here is a specific mathematical representation of the convolution operation. The integral runs from 0 to t, meaning it takes into account all values of τ from the start of the function up to a specific time t. When we say 𝑔(𝑡−𝜏), we are shifting the function g(t) backwards in time based on the current value of τ. This shift is crucial, as it allows us to see how the function g(t) interacts with the function f(t) at different delayed times. By integrating this product, we sum up all these interactions over time to yield a single new function that captures the cumulative effect.

Think of a concert where multiple musical instruments are playing. Each instrument (representing f(τ)) contributes sound over time, while the overall sound that people hear is affected by factors like reverb and delay (representing g(t - τ)). The convolution represents the combined sound that the audience experiences at each moment, resulting from the layering of sounds produced by each instrument adjusted by the effects applied.

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