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.3.13.4) 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(k0t)+bksin(k0t) < 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. T • ak= 2/T 0 x(t) cos(k0t) dt, k=1,2,… (3.5) T • bk= 2/T 0 x(t) sin(k0t) dt, k=1,2,... (3.6) T • a0= 1/T 0 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(j0t), -< 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.