14.5 : 畳み込み特性-II

The important convolution properties are width, area, differentiation, and integration properties.

The width property indicates that if the durations of input signals are T₁ and T₂, respectively, then the width of the output response equals the sum of both durations, irrespective of the shapes of the two functions. For instance, convolving two rectangular pulses with durations of 2s and 1s results in a function with a width of 3s.

The area property asserts that the area under the convolution of two functions equals the product of the areas under each function.

The differentiation property states that the derivative of a convolution equals the convolution of the input signal's derivative and impulse response or vice versa.

Combining the equations and generalizing for higher order derivatives gives the general differentiation relation.

The integration property for a given convolution yields the convolution of the impulse response and input response. This can also be stated in the opposite sequence.

If a convolution is performed with a step function, the LTI system behaves like an ideal integrator.