Fourier Series 3.1-3.3 Kamen and Heck

3.1 Representation of Signals in Terms of Frequency Components x(t) = k=1,N Ak cos(kt + k), -< t <  Fourier Series Representation Amplitudes Frequencies Phases Example 31. Sum of Sinusoids 3 sinusoids—fixed frequency and phase Different values for amplitudes (see Fig.3.1-3.4)

3.2 Trigonometric Fourier Series Let T be a fixed positive real number. Let x(t) be a periodic continuous-time signal with period T. Then x(t) can be expressed, in general, as an infinite sum of sinusoids. x(t)=a0+k=1, akcos(k0t)+bksin(k0t) < t <  (Eq. 3.4)

3.2 Trig Fourier Series (p.2) a0, ak, bk are real numbers. 0 is the fundamental frequency (rad/sec) 0 = 2/T, where T is the fundamental period. ak= 2/T 0T x(t) cos(k0t) dt, k=1,2,… (3.5) bk= 2/T 0T x(t) sin(k0t) dt, k=1,2,... (3.6) a0= 1/T 0T x(t) dt, k = 1,2,… (3.7) The “with phase form”—3.8,3.9,3.10.

3.2 Trig Fourier Series (p.3) Conditions for Existence (Dirichlet) 1. x(t) is absolutely integrable over any period. 2. x(t) has only a finite number of maxima and minima over any period. 3. x(t) has only a finite number of discontinuities over any period.

3.2 Trig Fourier Series (p.4) Example 3.2 Rectangular Pulse Train 3.2.1 Even or Odd Symmetry Equations become 3.13-3.18. Example 3.3 Use of Symmetry (Pulse Train) 3.2.2 Gibbs Phenomenon As terms are added (to improve the approximation) the overshoot remains approximately 9%. Fig. 3.6, 3.7, 3.8

3.3 Complex Exponential Series x(t) = k=-, ck exp(j0t), -< t <  (3.19) Equations for complex valued coefficients —3.20 – 3.24. Example 3.4 Rectangular Pulse Train 3.3.1Line Spectra The magnitude and phase angle of the complex valued coefficients can be plotted vs.the frequency. Examples 3.5 and 3.6

3.3 Complex Exponential Series (p.2) 3.3.2 Truncated Complex Fourier Series As with trigonometric Fourier Series, a truncated version of the complex Fourier Series can be computed. 3.3.3 Parseval’s Theorem The average power P, of a signal x(t), can be computed as the sum of the magnitude squared of the coefficients.