Topic 04 Fourier Series.pdf - ELEC3241\/2201 Signals and Linear Systems \u2028 Topic 6 Fourier Series\u2028 Kaibin Huang Dept of Electrical Electronic

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Unformatted text preview: ELEC3241/2201 Signals and Linear Systems   Topic 6: Fourier Series  Kaibin Huang Dept. of Electrical & Electronic Engineering The University of Hong Kong Hong Kong Representing a Signal in Frequency Domain                         History of Fourier series "\$""\$####"%## \$# ##\$!!"%!###!#!!& #\$\$"!\$#!!"#& #!#!"!"  #"#"\$!!  '  # \$#  Fourier Series for Periodic Signals            2π • Fundamental frequency: ω0 = 2πf0 = T0 • nthHarmonic frequency: nω0  Trigonometric Fourier Series             • Fourier spectrum: frequency domain description of x(t) • Frequencies:    • Amplitudes: &                !"# \$       cn =     a2n + b2n Trigonometric Fourier Series • The trigonometric Fourier series is a periodical signal with the period T0. Can you prove this property? • The property shows that any combination of sinusoids of frequencies 0, f0, 2f0, …, kf0 is a periodic signal of period T0 regardless of the amplitudes {ak} and {bk}. • Thus, by varying the values of {ak} and {bk}, we can construct a variety of periodic signals, all of the same period T0.  a0 sin(ω0t) cos(ω0t) b1 a1 sin(2ω0t) cos(2ω0t) b2 a2  sin(3ω0t)        cos(3ω0t) b3 a3             Compute Fourier Series Coefficients  T0 1 x(t)dt     T0 0  T0  T0  T0   1 1 1 a0 dt + an cos(nω0 t)dt + bn sin(nω0 t)dt = T0 0 T T 0 0 0 0 n n =a0 1 a0 = T0  0  T0 x(t)dt Compute Fourier Series Coefficients 2 an = T0  T0 0 x(t) cos(nωn t)dt  T0 1 x(t) cos(nω0 t)dt    T0 0   am  T0  bm  T0 a0 T0 cos(nω0 t)dt + cos(mω0 t) cos(nω0 t)dt + sin(mω0 t) cos(nω0 t)dt = T0 0 T T 0 0 0 0 m m  am  T0 cos(mω0 t) cos(nω0 t)dt = T 0 0 m an = 2     2 bn = T0  T0 0 x(t) sin(nω0 t)dt  Compact Trigonometric Form of Fourier series x(t) = C0 + ∞  Cn cos(nω0 t + θn ) n=1 • Definition: magnitude coefficients cn = • Definition: phase coefficients  a2n + b2n θn = tan−1   −bn an   ;5  /: :5 /< ;  .= >      ? % @ \$   AB: : /8C:23  .  >      • Fundamental frequency            (DE • Computation of trigonometric Fourier series                                                                            2 π − t ej2nπt + e−j2nπt dt an = e 2 π 0 2  π  π 1 1 (−1/2+j2nπ)t = e dt + e(−1/2−j2nπ)t dt π 0 π 0 −π −π 2 −1 e e 2 −1 +  1  =  1 π − 2 + j2n π − 2 − j2n 1.008 ≈ 1 + 16n2            ;5  /: :5 /< ;  .= >      ? % @ \$   AB: : /8C:23  .  >          • Computation of Fourier series in compact form                                                                ;5  /: :5 /< ;  .= >      ? % @ \$   AB: : /8C:23  .  >        • Amplitude spectrum:                                            !    "     #\$% &' ( )  • Fundamental frequency     +    • Computation of trigonometric Fourier series   & (           , + -    , + '   . ' .  /         / ' /                       !    "     #\$% &' ( )  • Computation of trigonometric Fourier series (continue)                                                 8      ! "  9    !&!   1               6 6 #\$  : 6     :6    6 6                        !    "     #\$% &' ( )  • Computation of trigonometric Fourier series (continue)      >  > 8   %  "'      :              &  # (#           2                                      !"   #  \$ %  &            • Fundamental frequency   '(    '( )   • Computation of trigonometric Fourier series  *      '(   '                                       !"   #  \$ %  &               •   Computation of trigonometric Fourier series (continue)         D F GFH I D J    K&KL   E #) M N E O > O     HNEPEQ M E M  EO O O   Q         D GF   T FU  **  M    M M M D FV W D NF D FV X D PF   N  P                                 !"   #  \$ %  &               Spectrum    **  M T FU     M M M D FV W D NF D FV X D PF   N  P              Effect of Symmetry • If x(t) is an even function:   H   H - !  - !        + ,+  + ./,+  H   • If x(t) is an odd function: 0 H  H b+#h\$  - !       + 1+,+ Effect of Symmetry                       Compute Fourier Series Coefficients  1 1 a0 = t2 dt = 3 0  1 an = 2 t2 cos(nπt)dt 0 1  1 2t t = sin(nπt) − 4 sin(nπt)dt nπ nπ 0 0 1  1 4 cos(nπt)t cos(nπt) = − 4 dt 2 (nπ)2 0 (nπ) 0 4 cos(nπ) = (nπ)2 2   Effect of Symmetry             v               Fourier Series  v  2 m n o7 p 3425 m Dqrst m DNrstn u D6342  N 34  # v    8      m  wD 934 2  N 347   9 7        Effect of Symmetry          Fourier Series     > qx > v > Dy3425 m Dy q342 m Dy N3425 m Dy 342   2 34  q N           qz D342: #;  D q!"2 #<  wD N 3425 #; "  D6342#<  {   Fourier Series for Signal Approximation • In a Fourier series, harmonics with lower frequencies determine the large-scale behavior of x(t); • Harmonics with higher frequencies determine the small-scale behavior of x(t) e.g., sharp changes.   The Fourier series for a square pulse train:      K * L L L F   G H I  =2I   J F  >2   * # M N O   Fourier Series for Signal Approximation      (DC term only)           (DC + 1 harmonic)  (DC + 2 harmonics)  (DC + 3 harmonics)     (DC + 4 harmonics)    Exponential Fourier Series • Derive from trigonometric form…  Exponential Fourier Series  ?      • Exponential Fourier spectrum • Frequencies: · · · , −2ω0 , −ω0 , 0, ω0 , 2ω0 , · · · • Amplitudes: · · · , |D−2 |, |D−1 |, |D0 |, |D1 |, |D2 |, · · ·       C0 = D 0 Cn = 2|Dn |  Computing Fourier Series Coefficients 1 Dn = T0  T0 0  x(t)e−jnω0 t dt Exponential Fourier Series           Spectrum                 Exponential Fourier Series • What is a negative frequency?    can create sinusoids: • A pair of    ,  ah 1 jnω0 t (e + e−jnω0 t ) 2 1 sin nω0 t = (ejnω0 t − e−jnω0 t ) 2j cos nω0 t =     • Bandwidth of a signal, B, is the difference between the highest and the lowest frequencies.     ' (           B=9  Exponential Fourier Series    Find the spectrum of an impulse train:.          !        \$   (Exponential Fourier) Spectrum  """         # \$ E  "#           @ABCF GHIDJK, -  L @ABC !  """ %\$   \$ M -  Exponential Fourier Series   """         #  \$ N  E  O -     "#  (Exponential) """ %\$         \$  &'()(''*+)    :6 > > > (Trigonometric)  Parseval’s Theorem • The power of x(t) is given as follows.      ' /   * + )*  , -. / 0 1         V = V x ex + V y ey + V z e z ex = [1, 0, 0] ey = [0, 1, 0]  0  V  (power) V2 = |Vx |2 + |Vy |2 + |Vz |2 ez = [0, 0, 1]    Summary of Coefficient Computation           9: P QRSTURTVT P QRSTURT            3   3  +                3   2 2      3   )   1      1 #  4  5 6 7 !" # \$  *8 9    A  BC , = D EF  G)H B  ) !           2     I  G J ) K %&  . L ) /0123435678 + 9' (   )*+ ,- ./0 45678  :;+  < 2  #  \$ 4 , = > [email protected] 1 , -  ...
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