Convolutional integral of square signal and triangle signal.

[ttokoro]

Convolutional integral of square signal and triangle signal is well knowen probrem.
How to find the maximum value of the output signal? Prime's symbolic engine sometimes fail for convolution integral.

7/6 is the answer I want to get. Prime 11 sheet attached.

[Werner_E]

Why would you need a symbolical solution ??  We know that Mathcad's and Prime's symbolic engines are not top notch and state of the art. Wouldn't a numerical result suffice for practical needs?

[LucMeekes]

Mathcad 11 (not Prime 11) is lazy:

Try the laplace route:

Yep, at t=5/2=2.5 that's where the summed length of x(t) and h(t) ends, so y(t) is 0 and y'(t) is 0.

Success!
Luc

[Werner_E]

Here is your approach done with MC15.

While MC11 seems to find 5/2 twice (exact and numerically, a bit strange IMHO), MC15 finds the solution 0.

None of the two finds the solution 7/6 😞

[Werner_E · Accepted solution]

Luc's approach sure has advantages. Not only the much faster numerical evaluation but also significantly better precision as this picture shows:

Decreasing the value of TOL significantly makes it better, but still ...

[ttokoro]

With using range tau=0.0.01..3 plot showes no error.

But without using range tau=0.0.01..3 plot shows the errors for symbolic engine's results. (Red and dotted lines are errors.)

So all errors comes from how to treat the unit step functions, Φ(t). If we use derivative, it shows impulse function Δ(t).

And derivative of convolution integral shows bug.

But we want to use Φ(t) such as if function to use the function for only the separated period.

This convolution error or bug is not fixed for long time.