How to Calculate Convolution

Definition of the convolution of f(t) and g(t) along with various properties.

A Laplace transform is a method engineers use to make difficult problems easier to solve. This transform replaces a calculus problem with one that only requires algebraic manipulation to solve. The engineer solves the transformed equation and then performs an inverse transform on the result so that it matches the original problem. A convolution is the inverse transform of a function (H), which is the product of two other transformed functions (F, G). The calculation for convolution is commutative, distributive and associative, in much the same way as multiplication.

Instructions

1. 

   Determine the two transformed function (F, G). Sometimes, these will be given values; other times, you may have to factor the product function (H).

2. 

   Perform an inverse Laplace transform on the functions F and G. Inverse transforms of common functions are available in engineering reference books or textbooks. Some special cases can be calculated directly, but you will most often either have the inverse transform memorized or you must look it up.

3. 

   Use the inverse transforms to build the integral to calculate the convolution and simplify. In this example, the product-to-sum identity converts a multiplication problem involving trigonometric functions into an addition problem.

4. 

   Solve the resulting integral or integrals.

Tips & Warnings

• To calculate convolution, you must have a firm grasp of derivatives, integrals and Laplace transforms.

References

• Advanced Engineering Mathematics, 7th ed.; Erwin Kreyszig; 1993