What is the Fourier series in simple terms?

Fourier series find many applications in engineering fields like telecommunication, electrical, and other applied sciences. However the important contribution is in analysis of simple and even complex waveforms. In geophysics this mathematical model is highly applied in the field of geophysical signal processing such as in seismic signals and others. The model can be applied over a wide range of waveforms. However, today I have written to you this piece of post so that you can grasp the basics of Fourier series on the range of Periodic function. However in order to adapt easy I assume you know some bit knowledge of integral calculus and trigonometric waveforms, Let's move on,

The fundamental basics laid behind Fourier series is that all functions which can be defined in the interval −π ≤ x ≤ π can be expressed as convergent trigonometric series of the form:

f (x) = a0 + a1cosx + a2cos2x + a3cos3x +···+ b1sinx + b2sin2x + b3sin3x +···
where a0, a1, a2, ... b1, b2, ... are real constants, such that
f(x) = a0 + Σn=1 to ∞(ancosnx + bnsinnx)..............(1)

For the range  -π to π

a0 = 1/2πf(x)dx, 

an = 1/πf(x)cosnxdx, 

bn = 1/πf(x)sinnxdx, 

Where n = 1,2,3....

From expression 1 above a0, an and bn are known as the Fourier coefficients of the series and if these can be determined, the series of equation (1) is the Fourier series corresponding to f(x).

Fourier cosine series

The Fourier series of periodic functions f(x) having period 2π is regarded as Fourier cosine series if the f(x) is an even function. This series will contain cosine terms only and may contain a constant term.

Example of even function, y = sinx,  y = x3 such that f(-x) = - f(x)
Hence f(x) = a0 + Σn=1 to ∞(ancosnx)
where a0 = 1/2πf(x)dx and an = 1/πf(x)cosnxdx

an = 2/πf(x)cosnxdx

For the range 0 to π


Fourier sine series
The Fourier series of a periodic function f(x) having period 2π is regarded as Fourier sine series if the f(x) is an odd function.This series will contain sine terms only and with no constant term.

Example of odd function, y = cosx,  y = x2 such that f(-x) = f(x).
Hence f(x) = Σn=1 to ∞(bnsinnx) 

where bn = 1/πf(x)sinnxdx
bn = 2/πf(x)sinnxdx

For the range 0 to π
Since from equation 1 the series having cosines and sines, we can write them by using trigonometric form acosx + bsinx = csin(x + α), as
f(x) = a0 + c1sin(x + α1) + c2sin(2x + α2)+···+ cnsin(nx + αn),
where a0 is a constant,
c1 = √(a12+ b12) and cn = √(an2 + bn2)
Where cn is the amplitude of the various components, and Phasor angle αn = tan-1(an/bn). This is essential when you want to compute the amplitudes and phasor angle of the resulting waveform.

N.B: [a1cosx + b1sinx] or c1sin(x + α1) is known as first harmonic while
[a2cos2x + b2sin2x] or c2sin(2x + α2) is known as second harmonic.

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