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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).