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Category:Statistics templates

Category:Sidebar templates Category:Statistics templates Category:Sidebar templates by topic


Symmetry properties

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Symmetry poperties of the Fourier series.
  • If is a real function, then (Hermitian symmetric) which implies:
    • (real part is even symmetric)
    • (imaginary part is odd symmetric)
    • (absolut value is even symmetric)
    • (argument is odd symmetric)
  • If is a real and even function (), then all coefficients are real and (even symmetric) which implies:
    • for all
  • If is a real and odd function (), then all coefficients are purely imaginary and (odd symmetric) which implies:
    • for all
  • If is a purely imaginary function, then which implies:
    • (real part is odd symmetric)
    • (imaginary part is even symmetric)
    • (absolut value is even symmetric)
    • (argument is odd symmetric)
  • If is a purely imaginary and even function (), then all coefficients are purely imaginary and (even symmetric).
  • If is a purely imaginary and odd function (), then all coefficients are real and (odd symmetric).

Table of Fourier Series coefficients

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Some common pairsof periodic functions and their Fourier Series coefficients are shown in the table below. The following notation applies:

  •   designates a periodic function defined on  .
  •   designates a ...
  •   designates a ...
Time domain
 
Plot Frequency domain (sine-cosine form)
 
Remarks Reference
 
 
  Full-wave rectified sine [1]: p. 193 
 
 
  Full-wave rectified sine cut by a phase-fired controller
 
 
 
  Half-wave rectified sine [1]: p. 193 
 
 
   
 
 
  [1]: p. 192 
 
 
  [1]: p. 192 
 
 
  [1]: p. 193 
 
 
   
 
 
   
 
  denotes the Dirac delta function.

Properties

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This table shows some mathematical operations in the time domain and the corresponding effects in the frequency domain.

  •   is the complex conjugate of  .
  •   designate a  -periodic functions defined on  .
  •   designates the Fourier series coefficients (exponential form) of   and   as defined in equation TODO!!!
Property Time domain Frequency domain (exponential form) Remarks Reference
Linearity     complex numbers  
Time reversal / Frequency reversal     [2]: p. 610 
Time conjugation     [2]: p. 610 
Time reversal & conjugation    
Real part in time    
Imaginary part in time    
Real part in frequency    
Imaginary part in frequency    
Shift in time / Modulation in frequency     real number   [2]: p. 610 
Shift in frequency / Modulation in time     integer   [2]: p. 610 
Differencing in frequency
Summation in frequency
Derivative in time    
Derivative in time (  times)
Integration in time
Convolution in time / Multiplication in frequency       denotes continuous circular convolution.
Multiplication in time / Convolution in frequency       denotes Discrete convolution.
Cross correlation    
Parseval's theorem     [3]: p. 236 
  1. ^ a b c d e Papula, Lothar (2009). Mathematische Formelsammlung: für Ingenieure und Naturwissenschaftler. Vieweg+Teubner Verlag. ISBN 3834807575.
  2. ^ a b c d Shmaliy, Y.S. (2007). Continuous-Time Signals. Springer. ISBN 1402062710.
  3. ^ Cite error: The named reference ProakisManolakis was invoked but never defined (see the help page).