Transcript Fourier Series

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(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.
T
• ak= 2/T 0 x(t) cos(k0t) dt, k=1,2,… (3.5)
T
• bk= 2/T 0 x(t) sin(k0t) 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(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.