UnilateralConvolve

UnilateralConvolve[f,g,u,t]

gives the unilateral convolution with respect to u of the expressions f and g.

UnilateralConvolve[f,g,{u₁,…,u_n},{t₁,…,t_n}]

gives the multidimensional unilateral convolution.

Details and Options

UnilateralConvolve is also known as causal convolution.
The concept of unilateral convolution arises naturally when examining causal systems. The output of such systems at any time depends only on values of the input at the present time and in the past.
The unilateral convolution of two functions and is given by .
The lower limit of the integral is effectively taken to be , so that where is the DiracDelta function.
The following demonstrates the unilateral convolution of exponential and unit box functions. represents an exponential function and represents the reflected and shifted unit box function. The convolution is the area under the product from to .

The multidimensional convolution is given by .
The following options can be given:

Assumptions	$Assumptions	assumptions to make about parameters
GenerateConditions	False	whether to generate conditions on parameters
Method	Automatic	method to use
PrincipalValue	False	whether to use principal value integrals

Examples

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Basic Examples (3)

Unilateral convolution of a function with DiracDelta:

Convolve two unit pulses and plot the result:

Convolve two exponential functions and plot the result:

Scope (5)

Univariate Convolution (3)

The convolution yields the integrated product of translated functions:

Elementary functions:

A convolution typically smooths the function:

Multivariate Convolution (2)

Unilateral convolution with a multivariate delta function acts as a point operator:

Unilateral convolution with a function of bounded support acts as a filter:

Plot the result:

Options (2)

Assumptions (1)

Specify assumptions on a variable or parameter:

GenerateConditions (1)

Generate conditions for the range of a parameter:

Applications (3)

Obtain a particular solution for a linear ordinary differential equation using convolution:

Obtain the step response of a linear, time-invariant system given its impulse response h:

The step response of the system:

Evaluate the inverse Laplace transform for the following example, using the convolution theorem:

Multiplication in the domain corresponds to a convolution operation in the domain. Begin by computing the inverse transforms for each individual term of the product:

Then convolve them in domain:

Properties & Relations (7)

UnilateralConvolve computes an integral over a finite interval:

Convolution with DiracDelta gives the function itself:

Scaling:

Commutativity:

Distributivity:

Convolve coincides with UnilateralConvolve, when the input functions are causal:

The Laplace transform of a causal convolution is a product of the individual transforms:

Verify the convolution theorem for Laplace transforms on the following example:

Interactive Examples (2)

This demonstrates the unilateral convolution operation between exponential and unit box functions:

This demonstrates the unilateral convolution operation between exponential and unit step functions:

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UnilateralConvolve

Details and Options

Examples

Basic Examples (3)