# Writing in terms of step function

• rock.freak667
In summary, when writing a function in terms of H(t), you can use the formula f(t) + (g(t) - f(t))u(t) + (h(t) - g(t))u(t-2) - h(t)u(t-4) where f(t), g(t), and h(t) represent different parts of the function and u(t) is a filter function that picks up whatever it is multiplied by between certain values. This method can be used to write a function in terms of H(t) for different ranges of t.
rock.freak667
Homework Helper

## 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.

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.

LCKurtz said:
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

## 1. What is a step function in writing?

A step function in writing is a literary technique where the story or plot progresses in a series of distinct and defined steps or stages, rather than a smooth and continuous flow. This can add structure and create a sense of anticipation and resolution in the story.

## 2. How can step functions enhance my writing?

Step functions can enhance your writing by creating a clear and organized structure for your story. By breaking the plot into distinct steps, the reader can follow along more easily and the story can build to a climax or resolution. This technique can also create a sense of tension and keep the reader engaged.

## 3. Are there any common examples of step functions in literature?

Yes, there are many examples of step functions in literature. One common example is the hero's journey, where the protagonist goes through a series of steps or challenges before reaching their ultimate goal. Other examples include the classic three-act structure, where the story is divided into three distinct parts, and the five-act structure, commonly used in plays and screenplays.

## 4. Can I use step functions in any genre of writing?

Yes, step functions can be used in any genre of writing. While they may be more commonly associated with certain genres, such as fantasy or adventure, step functions can be applied to any type of story. It is up to the writer to determine how to best use this technique to enhance their particular genre and story.

## 5. Are there any tips for effectively using step functions in my writing?

Some tips for effectively using step functions in writing include planning out the steps or stages in advance, making sure each step builds upon the previous one, and using transitions or cliffhangers between steps to create a sense of anticipation and keep the reader engaged. It can also be helpful to study and analyze how other writers use step functions in their works to gain inspiration and insights.

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