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Q1: Fourier Cosine Series - Coggle Diagram
Q1: Fourier Cosine Series
Properties
Applied to even functions
Symmetric across 0.
This property can be exploited to simplify the integrals!
The area from -π to 0 will be the same as from 0 to π, and so an integral from -π to π can be written as 2 times the integral from 0 to π
C(x) = C(-x)
All elements of these series are even, and so the
Fourier Cosine Series are also even!
Unlike sine series, cosine series can be started at n = 0
This happens because the first term won't be 0
In the form:
The coefficients can be acquired from formulas:
a0 is the average value of C(x):
ak (each an) comes from the orthogonality of cosines:
The smoother the function, faster is ak's rate of decay:
Delta functions
Step functions (integral of delta functions)
Ramp functions (integral of step functions)
1 more item...
1/k decay
No decay
Applications
Compression
Compression of audio, for example, is done by taking only the first few terms of a fourier series instead of the original wave since they converge so fast
Doesn't necessarily reduce the quality since the human ear has a limit of the frequencies it can hear
Noise cancellation
Known the behavior of a sound wave, it can be removed in post-audio editing
Active noise cancellation can be used to remove periodic waves (i.e. static, hisses) with minimal change to the user's voice
Control theory
Fourier series can be used to find out the dynamics for the solution of a differential equation
Solve PDEs through separation of variables
