Fourier series of a step function

In summary, the conversation discusses the possibility of making a Fourier series expansion of a step function and the challenges that arise with infinite and finite step functions. It also explores the idea of using other methods to calculate Fourier series for increasing or decreasing functions. The conversation then delves into the purpose of Fourier transforming the function and potential ways to extract a mathematical function for a given graph. The suggestion is made to write the function as a sum of a linear or parabolic function with a periodic term, but it is noted that a piecewise definition may be more effective.
  • #1
chikou24i
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Hello, can we make a Fourier series expansion of a (increasing or decreasing) step function ? like the one that I attached here. I just want to know the idea of that if it is possible.
shNoy.png
 
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  • #2
You can write down the integral, but with an infinite step function like this the integrals will diverge. With a finite step function it works.
 
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  • #3
mfb said:
You can write down the integral, but with an infinite step function like this the integrals will diverge. With a finite step function it works.
So you mean that I should take each portion appart (where f(x) is constant) and calculate the integral. Then, I add them all together ?
Is there any other method where I can calculate Fourier series normally and then I multiply them by another factor which will make them increase or decrease ?
 
  • #4
The sum will diverge as well for most frequencies, and the approach is at best questionable for frequencies where it does not diverge.

Why do you want to Fourier transform that function?
 
  • #5
I want to extract the mathematical function for this graph of voltage response. Could you please help me or guide me in doing that ?
https://fb-s-d-a.akamaihd.net/h-ak-xlp1/v/t1.0-0/p526x296/17457874_1911500359125961_8184758272904545979_n.jpg?oh=50117debf8fd6047caf1f0795e0a9254&oe=59500784&__gda__=1498980987_b71e34fea90ba330474f88d4a58f82d9
 
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  • #6
You could write it as sum of a linear or parabolic function and some periodic term.

I'm not sure if that is really useful, I'm quite sure a piecewise definition of the function will work better.
 
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What is a step function?

A step function, also known as a Heaviside step function, is a mathematical function that is defined as 0 for negative values and 1 for positive values.

What is a Fourier series?

A Fourier series is a mathematical representation of a function as an infinite sum of sines and cosines.

Why is the Fourier series of a step function important?

The Fourier series of a step function is important because it allows us to approximate complex functions by using simpler trigonometric functions. It also has many applications in signal processing and engineering.

What is the formula for the Fourier series of a step function?

The Fourier series of a step function can be represented by the following formula: f(x) = 1/2 + ∑(n=1)∞ (sin(nx)/nπ), where n is an integer.

How is the convergence of the Fourier series of a step function determined?

The convergence of the Fourier series of a step function can be determined by using the Dirichlet conditions, which state that the function must be periodic, piecewise continuous, and have a finite number of discontinuities within a given period. If these conditions are met, the Fourier series will converge to the original function.

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