Derivation of a function's formulae from its graph

Question

I am given the plot of a signal function. The expectation is to derive the full function given the graph's parameters.

Here's what I've come up with so far for the negative side of the function: $\Pi(\frac{t}{10})+3\Lambda(\frac{(t+2)}{3})$

where

and

Unfortunately I am lost as to how to deal with the plateau in the middle and subsequently the positive side. Any hints or tips please?

Answer 1

Think about what is happening geometrically. There are two or three features of the graph which seem important to capture:

1. There is a triangle on the right which appears to correspond to a scaled and translated copy of $\Lambda$. Let $$ f(x) = 3 \Lambda\left(\frac{x-2}{3}\right). $$ This function has the same graph as $\Lambda$, but translated to the right by $2$ units and scaled both horizontally and vertically by a factor of $3$.

2. There is a triangle on the left which appears to correspond to a scaled and translated copy of $\Lambda$. Let $$ g(x) = 3 \Lambda\left(\frac{x+2}{3}\right). $$ This function has the same graph as $\Lambda$, but translated to the right by $2$ units and scaled both horizontally and vertically by a factor of $3$.

3. It appears that both of these triangles have been translated up by one unit, but only on the interval $[-5,5]$. This corresponds adding a horizontally scaled copy of $\Pi$ which has been left-translated by $5$ units. Let $$ h(x) = \Pi\left( \frac{x+5}{10} \right). $$

Adding these three functions together seems to get the job done: $$ f(x) + g(x) + h(x) = 3\left( \Lambda\left( \frac{x-2}{3} \right) + \Lambda\left( \frac{x+2}{3} \right) \right) + \Pi\left( \frac{x+5}{10} \right). $$

(linky: https://www.geogebra.org/calculator/czywnyrh )

Answer 2

A different solution adding even functions:

$$\Pi(t/10)+5 \Lambda(t/5) -4 \Lambda(t/2)+\Lambda(t)$$