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(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.• ak= 2/T 0

T x(t) cos(k0t) dt, k=1,2,… (3.5)

• bk= 2/T 0T x(t) sin(k0t) 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(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.