Piecewise Convolution Notebook

The linear convolution of f (t) and g(t) is defined as

#math254#

f (t)#tex2html_wrap_indisplay3331#g(t) = #tex2html_wrap_indisplay3332# f (u) g(t - u) du .

Here, f (t) and g(t) are piecewise-continuous functions. Piecewise convolution is a time-domain approach that finds the convolution by computing the integral over each interval where the two functions overlap as g(t - u) ``slides'' through f (u). This is usually the first approach for solving convolution problems that engineers learn.

The piecewise convolution Notebook shows the student how to solve one-dimensional continuous-time convolution problems using the piecewise convolution package. In the introduction, the Notebook informs the user that a piecewise function can be represented as a list of intervals. Each interval is itself a list containing the start of the interval, the end of the interval, and the function defined on that interval. For example, the function #math255##tex2html_wrap_inline3338#(t), a unit triangle defined on #math256#t∈[- 1, 1], can be represented as an expression

<#666#>(1 + t) CPulse[1, 1 + t] + (1 - t) CPulse[1, t]<#666#>

or in list form as

<#669#>{{1 + t, -1, 0}, {1 - t, 0, 1}}.<#669#>

The Notebook gives several examples of piecewise convolution using the newly defined object <#671#>PiecewiseConvolution<#671#>, which requires three arguments--- f (t), g(t), and t. For example, the <#1184#>Mathematica<#1184#> dialogue

verbatim196#

illustrates the shifting property of the Dirac delta function. We could have used the piecewise form of <#674#>Delta[t - a]<#674#>, which is <#675#>{Area[1], a, a}<#675#>, in this example. The Notebook also gives an example of a convolution that produces an unstable function:

104
<#676#>Integrate::divg: Integral does not converge<#676#>
105

Part of computing a piecewise convolution requires identifying the overlapping intervals by drawing pictures of f (u) and g(t - u). So, for two simple functions, the Notebook shows an <#677#>animation<#677#> of g(t - u) sliding through f (u). Each frame shows an important interval over which to integrate, and the reader can control the frame rate.