# 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.