Mathematical Representations

Three representations of a cosine with amplitude a > 0, frequency w >= 0, and phase angle q:
Real Trigonometric:a cos(wt + q)
Complex Exponential: ½ a ejqejwt + ½ a e-jqe-jwt
Real Part of a Phasor: Re {aej(wt + q) }

A phasor can be viewed as a rotating vector in the complex plane that has length a and, at any time t, angle wt + q. The real part of a phasor is the projection onto the horizontal (real) axis. However for graphical demonstration it is more convenient to project onto the vertical axis. Therefore we rotate all phasors 90° in in the movie presentations and project on the vertical. In mathematical terms, we are using the relationship:

Im {a ej(wt + q + 90°)} = Re {a ej(wt + q)}

Three representations of a Fourier Series with fundamental frequency w:
Real Trigonometric Sum: Sumk=0,...,oo ak cos(kwt + qk)
Complex Exponential Sum: Sumk=-oo,...,oo ½ ak ejqk ejkwt
Real Part of a Phasor Sum Re { Sumk=0,...,oo ak ej(kwt + qk)}

Because projections on the vertical axis are easier to view than projections on the horizontal (real) axis, all phasors are rotated 90° in the movie presentations.

Time Shift
Time shift of a periodic signal corresponds to a phase shift of each phasor by an amount proportional to the phasor's frequency.

Re { Sumk=0,...,oo ak ej [kw (t - to) + qk] } = Re { Sumk=0,...,oo e-jkwto ak ej [kwt + qk ] }

Fourier Series Approximation of a Signal
The N-harmonic Fourier series approximation xN (t) of a signal x(t) is, in complex form:
xN(t) = Sumk=-N,...+N ½ ak ejkwo t
where the coefficients ak are given by:
ak = ( 1/To) To /|/ x(t) e-jkwo t dt

Convergence of the Fourier Series
The coefficients of the N-harmonic Fourier series approximation minimize the integral squared-error over one period of the signal:
EN = To /|/ | x(t) - xN (t) | ² dt
Indeed, as the number of harmonics N is increased, the value of the integral squared-error converges to 0:
limN->ooEN = 0.

Windowing Techniques
Windowing techniques provide attenuation factors for the kth harmonic in an N-harmonic Fourier series approximation.

Féjer Window:(N-k)/N
Hamming Window:[ 0.54 + 0.46cos (pi × k/N) ]

Note that for the k = N harmonic, the Féjer window yields a zero-multiplier while the Hamming window uses a 0.08 multiplier. For the DC term, both windows use a unity multiplier.