Writing in terms of step function

[rock.freak667]

Homework Statement

f(t)= 4t t<1
f(t)= 4 t≥1

I need to write f(t) in terms of H(t)

Homework Equations

H(t-t₀)=0 for t<t₀
= 1 for t≥t₀

The Attempt at a Solution

I am not too sure how it is done, as what I read online, people can just immediately write it down but I don't get the the thought process.

 

[LCKurtz]

Here's the idea. Say you want your function to be f(t) for t < 0, g(t) between 0 and 2, h(t) between 2 and 4, and 0 for t > 4.

So start building your function with f(t). At t=0 you want to get rid of f and start g:

f(t) + (g(t) - f(t))u(t)

At 2 you want to get rid of the g and start h

f(t) + (g(t) - f(t))u(t) + (h(t) - g(t))u(t-2)

At t = 4 you want to get rid of the h.

f(t) + (g(t) - f(t))u(t) + (h(t) - g(t))u(t-2) - h(t)u(t-4)

and you are done. Equivalently you could collect terms on the functions:

f(t)(1 - u(t)) + g(t)(u(t)-u(t-2)) + h(t)(u(t-2)-u(t-4))

Sometimes terms like u(t-b)-u(t-a) are called filter functions because they just pick up whatever they are multiplied by between a and b.

 

[rock.freak667]

Here's the idea. Say you want your function to be f(t) for t < 0, g(t) between 0 and 2, h(t) between 2 and 4, and 0 for t > 4.

So start building your function with f(t). At t=0 you want to get rid of f and start g:

f(t) + (g(t) - f(t))u(t)

At 2 you want to get rid of the g and start h

f(t) + (g(t) - f(t))u(t) + (h(t) - g(t))u(t-2)

At t = 4 you want to get rid of the h.

f(t) + (g(t) - f(t))u(t) + (h(t) - g(t))u(t-2) - h(t)u(t-4)

and you are done. Equivalently you could collect terms on the functions:

f(t)(1 - u(t)) + g(t)(u(t)-u(t-2)) + h(t)(u(t-2)-u(t-4))

Sometimes terms like u(t-b)-u(t-a) are called filter functions because they just pick up whatever they are multiplied by between a and b.

Thank you, I wish my lecturer would have explained it like this