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Units, Constants and Useful Formulas
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Last modified: November 18, 2021 ✓
Fourier Series and Transforms
-----------------------------
The Fourier Series
------------------
The Fourier Series breaks down a periodic function
with period T into the sum of sinusoidal functions.
N
f(x) = ao + Σ(cos(2πnt/T) + bnsin(2πnt/T))
n=1
T
ao = (1/T)∫f(x)dx
0
T
an = (2/T)∫cos(2πnt/T)dx
0
L
bn = (2/T)∫sin(2πnt/T)dx
0
These equations can also be written in terms of
the repeat distance, L, by replacing t with x
and T with L.
The Fourier Transform
---------------------
The Fourier Transform decomposes a non-periodic
waveform into into sinusoidal functions. A
non-periodic waveform is one where
T (or L) -> ∞.
∞
F(f) = ∫f(t)exp(-2πift)dt
-∞
This represents the waveform in the frequency
domain.
The inverse transform is:
∞
f(t) = ∫F(f)exp(2πift)df;
-∞
This represents the waveform in the time domain.
F(f) contains information about the amplitude
and phase of the sinusoidal components that make
up the function f(t).
The FT pairs allow a mapping between the time
and frequency domains.