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:
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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. | |
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
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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.
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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. |