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Category:Statistics templates
Category:Sidebar templates
Category:Statistics templates
Category:Sidebar templates by topic
Symmetry properties
[edit]
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
[edit]
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
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Plot
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Frequency domain (sine-cosine form)
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Remarks
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Reference
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Full-wave rectified sine
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[1]: p. 193
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Full-wave rectified sine cut by a phase-fired controller

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Half-wave rectified sine
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[1]: p. 193
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[1]: p. 192
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[1]: p. 192
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[1]: p. 193
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denotes the Dirac delta function.
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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
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Time domain
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Frequency domain (exponential form)
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Remarks
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Reference
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Linearity
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complex numbers
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Time reversal / Frequency reversal
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[2]: p. 610
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Time conjugation
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[2]: p. 610
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Time reversal & conjugation
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Real part in time
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Imaginary part in time
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Real part in frequency
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Imaginary part in frequency
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Shift in time / Modulation in frequency
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real number
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[2]: p. 610
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Shift in frequency / Modulation in time
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integer
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[2]: p. 610
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Differencing in frequency
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Summation in frequency
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Derivative in time
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Derivative in time ( times)
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Integration in time
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Convolution in time / Multiplication in frequency
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denotes continuous circular convolution.
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Multiplication in time / Convolution in frequency
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denotes Discrete convolution.
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Cross correlation
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Parseval's theorem
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[3]: p. 236
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- ^ a b c d e Papula, Lothar (2009). Mathematische Formelsammlung: für Ingenieure und Naturwissenschaftler. Vieweg+Teubner Verlag. ISBN 3834807575.
- ^ a b c d Shmaliy, Y.S. (2007). Continuous-Time Signals. Springer. ISBN 1402062710.
- ^ Cite error: The named reference
ProakisManolakis
was invoked but never defined (see the help page).