Hi Fuzzification,

Below please find a detailed response. I tried to include both a high school level response and a more advanced response. I also included a simple example. I also responded to your side extra question.

If you liked the effort and are satisfied, don't forget to tip me. Thanks.

**Simple explanation (since you mentioned you want a more digestible explanation for someone with high school math knowledge):**

Filtering a signal means removing certain parts of it while keeping others. In your case, you want to "flatten" the peaks of a signal that looks like a sine wave as shown in your example. Now, to better visualize and understand this, imagine you have a rope that you shake up and down to make waves (like the rope exercise in gyms 😉). The rope represents the signal, and the waves represent the peaks and valleys. If you shake the rope faster, the waves get closer together and the peaks and valleys become more frequent. If you shake the rope slower, the waves get farther apart and the peaks and valleys become less frequent.

Similarly, a sine wave signal has peaks and valleys that happen at a certain frequency. If the frequency is high, the peaks and valleys are closer together and the signal looks more like a wave (like your picture number 1). If the frequency is low, the peaks and valleys are farther apart and the signal looks flatter (like your picture number 2). Now, you want to make (1) more like (2), so you need to flatten the first sine wave. In other words, you need to lower the frequency, or in the rope case, you would want to shake it slower.

To filter out the high-frequency parts of the signal and make it flatter, you can use a special filter called a low-pass filter. Think of this filter like a strainer that lets some things pass through while blocking others. The low-pass filter lets the low-frequency parts of the signal (the flatter parts) pass through while blocking the high-frequency parts (the wavy parts). It is getting a bit complex if you want to dig deeper. But, you can use math to make this filter. You can use a special formula called a “Fourier transform” to break down the signal into different parts, like how a song is broken down into different notes. Then, you can remove the high-frequency parts of the signal using a special equation, and put it all back together to get the filtered signal. This whole thing relates to digital signal processing and can get complicated for you. To give you some general instructions, see below.

In the context of your data, this means that the filter will reduce the amplitude of the peaks while preserving the overall shape of the curve. Here are some general steps you could take to apply a low-pass filter to your data:

- Choose a filter type: There are several types of low-pass filters you could choose from, such as Butterworth, Chebyshev, or Bessel filters. Each filter has its own characteristics in terms of roll-off rate and passband ripple, so you should choose a filter type that is appropriate for your specific application.
- Choose a cutoff frequency: The cutoff frequency is the frequency below which the filter will allow signals to pass through unchanged. In the context of your data, you should choose a cutoff frequency that is lower than the frequency of the peaks you want to flatten.
- Apply the filter: Once you have chosen a filter type and cutoff frequency, you can apply the filter to your data using a software package or programming language that supports digital signal processing. For example, you could use Python coding language and use the scipy.signal module in to apply a Butterworth filter with a specified cutoff frequency.

After performing these steps, you can check and see if the peaks have been sufficiently flattened. You could plot the filtered data and compare it to the original data to visually inspect the changes, or you could calculate statistics such as the peak-to-peak amplitude to quantify the degree of flattening to check/review what has been produced.

**To explain some important details from above and a simple example:**The mathematical concept behind a low-pass filter is called a Fourier transform. A Fourier transform is a mathematical tool that allows you to represent a signal as a sum of sine waves of different frequencies. Each sine wave in the sum is called a component or frequency component. To filter a signal with a low-pass filter, you first take the Fourier transform of the signal. Then, you remove the frequency components that are above a certain frequency cutoff. This is done by setting the amplitude of those frequency components to zero. Finally, you take the inverse Fourier transform of the modified signal to get the filtered signal.

**See this example to illustrate this process:**

Suppose you have a signal that is a sine wave with a frequency of 100 Hz. You want to filter this signal with a low-pass filter that has a cutoff frequency of 10 Hz. Here are the steps we would follow:

- Take the Fourier transform of the signal. The Fourier transform gives you a representation of the signal in terms of its frequency components.
- Remove the frequency components that are above the cutoff frequency. In this example, you would remove all frequency components above 10 Hz. This means setting the amplitude of those frequency components to zero.
- Take the inverse Fourier transform of the modified signal. This gives you the filtered signal.

The result of filtering the signal with a low-pass filter is that the high-frequency components of the sine wave are attenuated or removed, leaving only the lower-frequency components. This results in a flattened signal with less pronounced peaks.

**Regarding your extra side question:**The field of math that deals with the study of curves and their transformations is called "Differential Geometry." Differential geometry is the branch of mathematics that deals with the geometry of curves and surfaces, and it has many applications in physics, engineering, computer graphics, and other fields. But, first you need to be good at the basics and foundations. If you want to learn how to manipulate curves and transform their shape, you could start by studying calculus and linear algebra, as they are the foundation for much of differential geometry. You could also study differential equations and topology, which are other important fields within differential geometry. For a more specific focus on curve transformations, you could also look into the field of "Geometric Modeling," which is a subfield of computer graphics that deals with creating and manipulating geometric shapes, including curves. Some popular textbooks on this topic include "Geometric Modeling" by Michael E. Mortenson and "Mathematical Methods for Curves and Surfaces" by Alain Fournier. You can also Google these key words to find more resources online. I am sure there are many resources available and you could also take free courses from Coursera or LinkedIn.

Best,

Kav10

Low bounty!

What do you reckon should it be? I'm not trying to be stingy, I just don't know how hard of a problem it actually is to be honest :)

To prepare a page or two response to explain details for you, it can take about an hour or so. So, a fair bounty is $20-$30.