Convolution - Definition

Definition

The convolution of f and g is written f∗g, using an asterisk or star. It is defined as the integral of the product of the two functions after one is reversed and shifted. As such, it is a particular kind of integral transform:


	(commutativity)

While the symbol t is used above, it need not represent the time domain. But in that context, the convolution formula can be described as a weighted average of the function f(τ) at the moment t where the weighting is given by g(−τ) simply shifted by amount t. As t changes, the weighting function emphasizes different parts of the input function.

More generally, if f and g are complex-valued functions on Rd, then their convolution may be defined as the integral:

Visual explanation of convolution.
Express each function in terms of a dummy variable Reflect one of the functions: → Add a time-offset, t, which allows to slide along the -axis. Start t at -∞ and slide it all the way to +∞. Wherever the two functions intersect, find the integral of their product. In other words, compute a sliding, weighted-average of function, where the weighting function is The resulting waveform (not shown here) is the convolution of functions f and g. If f(t) is a unit impulse, the result of this process is simply g(t), which is therefore called the impulse response.

Visual explanation of convolution.

Express each function in terms of a dummy variable
Reflect one of the functions: →
Add a time-offset, t, which allows to slide along the -axis.
Start t at -∞ and slide it all the way to +∞. Wherever the two functions intersect, find the integral of their product. In other words, compute a sliding, weighted-average of function, where the weighting function is

The resulting waveform (not shown here) is the convolution of functions f and g. If f(t) is a unit impulse, the result of this process is simply g(t), which is therefore called the impulse response.

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