Can a Fourier series be adjusted to model a decreasing period function?

In summary, the conversation discusses modeling a periodic function that is "squished" for larger values of x. The proposed function to be modeled is y = SUM{aSin(nx^2)} + SUM{bCos(nx^2)}, which replaces x with x^2 to achieve the desired effect. The application for this function is bladder level as a function of beers consumed, with a basic function of an increasing quadratic or exponential function followed by a linear, steeply sloped drop to zero. The use of a wavelet transform is suggested for this type of function.
  • #1
flatmaster
501
2
I have a function I want to model. It is periodic, but the period keeps decreasing. Basically, it'll be a periodic function "squished" for larger values of x.

The typical Fourier series is...
y = SUM{aSin(nx)} + SUM{bCos(nx)}

I think I will attempt

y = SUM{aSin(nx^2)} + SUM{bCos(nx^2)}

replacing x -->x^2 should give me the "smushing" that I want.

The application is bladder level as a function of beers consumed. The basic function is an increasing (quadratic, exponential) function followed by a linear, steeply sloped drop to zero.
 
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  • #2
I guess you should worry the amplitude as well. Your general function is ##f(t) = A(t)\sin(p(t)+p_0)##. Now you can try some functions for ##A(t)## and ##p(t)##. I would let WolframAlpha do the graphics until I'm satisfied.
 
  • #3
sounds like you need a wavelet transform, not a Fourier transform
 

1. What is an Adjusted Fourier series?

An Adjusted Fourier series is a mathematical representation of a periodic function using a combination of sine and cosine functions. It is a way to break down a complex function into simpler components, making it easier to analyze and manipulate.

2. How is an Adjusted Fourier series different from a regular Fourier series?

An Adjusted Fourier series takes into account any discontinuities or jumps in the function, while a regular Fourier series assumes that the function is continuous. This allows for a more accurate representation of the function and can be useful in real-life applications.

3. What is the process for finding an Adjusted Fourier series?

The process involves finding the Fourier coefficients, which are the weights for each sine and cosine function in the series. This is done by integrating the function over one period and solving for the coefficients using the orthogonality property of sine and cosine functions.

4. Can an Adjusted Fourier series be used for any type of function?

Yes, an Adjusted Fourier series can be used for any periodic function, regardless of its shape or complexity. However, the accuracy of the series may vary depending on the smoothness of the function.

5. What are the practical applications of an Adjusted Fourier series?

An Adjusted Fourier series has many practical applications in fields such as engineering, physics, and signal processing. It can be used to analyze and manipulate signals, model physical phenomena, and solve differential equations.

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