If you have any questions, please ask. I love yapping.
yap
Let's say we have some periodic function f. The goal is to represent it as a bunch of sine waves. This is called a Fourier series (FS).
The FS coefficients are given by
⌠2⋅π
1 ⎮ -i⋅k⋅x
f̂(k) = ─── ⎮ f(x)⋅e dx
2⋅π ⌡0
for whole numbers k. We can then approximate the original function with
n
___
╲ -i⋅k⋅x
╱ f̂(k)⋅e
‾‾‾
k = -n
and make the approximation better by making n larger.
For non-periodic functions, we can still get an approximation! The Fourier Transform (FT) uses a continuous range of frequencies instead of whole numbers, and the sum is replaced by an integral (the inverse FT).
FT's are used everywhere, for example in signal processing and spectroscopy. If a recording contains an annoying high-pitched noise, we can apply the FT and filter the unwanted frequencies. Spectroscopy separates light into its different wavelengths. By comparing the resulting spectral lines with known measurements, we can determine which elements and molecules are present in stars and planets, for example.
idk what more to add ._.
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spoiler
::: spoiler small win
I went to the store. My sister got on the phone with me to help talk me through it. But. I went to the store. I bought food. I bought good food. Like, to cook. Not instant or frozen, but real, good, food. I didn't have spoons to make it today. But. Tomorrow I will make myself a dinner worth eating. And it will be good. I'm proud of myself. The anxiety was strong. But. I did it. I went to the fucking store. Success. I hate that this is a big deal for me but this is the first time ive gone shopping shopping in a very long time. Even if its a little pathetic, I'm really proud. I'm debating going to the park again today. Ive been doing that as well, and have been proud of my ability to go out and be around people. But maybe not today. I'm tired. But proud. Prouds.
I've been having a lot of similar experiences and feelings lately, shit is fucking tough. I can't seem to give myself the love and grace I can extend to others, so to you I say: I'm proud of you too! Good for you, and I hope you enjoy the tasty dinner you worked so hard for
Same. I can't give myself the compassion. I'm trying really hard to tho! I did it today a little bit. I'm just, idk. I internalized the voices of others and now they drown out my own. But I'm learning to shout.
I'm so glad to hear someone else say it so I can recognize it as valid and helpful instead of letting my brain convince me otherwise, thank you!
You deserve the compassion of your own heart. It is one of the most meaningful and most difficult gifts we can give to ourselves.
Real af. I need to internalise this myself.
Well done, I'm so happy for you! It's all about progress, no matter where you start, it's good that you managed to do something you needed to
Thankieeeessssssss I'm trying. Its hard but I'm doing it!
Trying and doing is always good lily!
spoiler
Good for you; I hope the meal tomorrow turns out well.