Signal Representation/Fourier Series - Revision history

Maintenance script: Add German interlanguage link

2026-03-16T12:28:32Z

Add German interlanguage link

← Older revision		Revision as of 14:28, 16 March 2026
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	:$$x(t)=\sum^{+\infty}_{n=-\infty}D_{\it n}\cdot \rm e^{j \it n \omega_{\rm 0} t}$$		:$$x(t)=\sum^{+\infty}_{n=-\infty}D_{\it n}\cdot \rm e^{j \it n \omega_{\rm 0} t}$$

	and the   [[Signal_Representation/~~Fourier_Transform_Theorems~~#Shifting_Theorem\|»shifting theorem«]]  $($for the frequency domain$)$  one gets the following spectrum for the periodic signal  $x(t)$:		and the   [[Signal_Representation/The_Fourier_Transform_Theorems#Shifting_Theorem\|»shifting theorem«]]  $($for the frequency domain$)$  one gets the following spectrum for the periodic signal  $x(t)$:

	:$$X(f)=\sum^{+\infty}_{n=-\infty}D_n\cdot\delta(f-n\cdot f_0).$$		:$$X(f)=\sum^{+\infty}_{n=-\infty}D_n\cdot\delta(f-n\cdot f_0).$$
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	{{Display}}		{{Display}}
			[[de:Signaldarstellung/Fourierreihe]]

Guenter at 16:21, 15 November 2023

2023-11-15T16:21:22Z

← Older revision		Revision as of 18:21, 15 November 2023
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	\|Untermenü=Periodic Signals		\|Untermenü=Periodic Signals
	\|Vorherige Seite=Harmonic Oscillation		\|Vorherige Seite=Harmonic Oscillation
	\|Nächste Seite=Fourier Transform and ~~Its~~ Inverse		\|Nächste Seite=The Fourier Transform and its Inverse
	}}		}}

Guenter at 16:18, 15 November 2023

2023-11-15T16:18:56Z

← Older revision		Revision as of 18:18, 15 November 2023
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	{{Header		{{Header
	\|Untermenü=~~Periodische Signale~~		\|Untermenü=Periodic Signals
	\|Vorherige Seite=Harmonic Oscillation		\|Vorherige Seite=Harmonic Oscillation
	\|Nächste Seite=Fourier Transform and Its Inverse		\|Nächste Seite=Fourier Transform and Its Inverse

Guenter at 15:58, 10 June 2023

2023-06-10T15:58:07Z

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Guenter at 16:45, 9 June 2023

2023-06-09T16:45:03Z

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Guenter at 17:35, 8 February 2023

2023-02-08T17:35:19Z

← Older revision		Revision as of 19:35, 8 February 2023
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	[[File:P_ID528__Sig_T_2_4_S6_neu.png\|right\|frame\|Spectrum of a periodic rectangular signal]]		[[File:P_ID528__Sig_T_2_4_S6_neu.png\|right\|frame\|Spectrum of a periodic rectangular signal]]

	is an even  (German:  "gerade"   ⇒   "$\rm g$")  function, composed of different cosine parts.		is an even  (German:  "gerade"   ⇒   "$\rm g$")  function,  composed of different cosine parts.

			Therefore:
	*The corresponding spectral function  $X_{\rm g}(f)$  is thus purely real.		*The corresponding spectral function  $X_{\rm g}(f)$  is thus purely real.
	::<u>Reason:</u>  As described in the section  [[Signal_Representation/Harmonic_Oscillation#Representation_with_cosine_and_sine_components\|"Spectral Representation of a cosine signal"]]  the basic wave returns two Dirac delta functions at  $\pm f_0$, each weighted with  $2/\pi$.		::<u>Reason:</u>  As described in the section  [[Signal_Representation/Harmonic_Oscillation#Representation_with_cosine_and_sine_components\|"Spectral Representation of a cosine signal"]]  the basic wave returns two Dirac delta functions at  $\pm f_0$, each weighted with  $2/\pi$.
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	*This weighting corresponds to the (generally complex) Fourier coefficients  $D_1 = D_{ - 1}^\ast$, which are only real in the special case of an even function.		*This weighting corresponds to the (generally complex) Fourier coefficients  $D_1 = D_{ - 1}^\ast$, which are only real in the special case of an even function.

	*Other Dirac delta functions are available in  $\pm 3f_0$ (negative),  $\pm 5f_0$ (positive),  $\pm 7f_0$ (negative) etc.		*Other Dirac delta functions are available in
			:*$\pm 3f_0$ (negative),

			:*$\pm 5f_0$ (positive),

			:*$\pm 7f_0$ (negative) etc.

	*All phase values  $\varphi_n$  are either zero or  $\pi$ due to the alternating signs.		*All phase values  $\varphi_n$  are either zero or  $\pi$ due to the alternating signs.
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	*As described in the section  [[Signal_Representation/Harmonic_Oscillation#General_spectral_representation\|"General Spectral Representation"]]  the basic wave provides two Dirac delta functions		*As described in the section  [[Signal_Representation/Harmonic_Oscillation#General_spectral_representation\|"General Spectral Representation"]]  the basic wave provides two Dirac delta functions
	**at  $+f_0$  $($weighted with  $-\text{j}\cdot 2/\pi)$  resp.		:*at  $+f_0$  $($weighted with  $-\text{j}\cdot 2/\pi)$  resp.
	**at  $-f_0$  $($weighted with   $+\text{j}\cdot 2/\pi)$.
			:*at  $-f_0$  $($weighted with   $+\text{j}\cdot 2/\pi)$.

	*All other Dirac delta functions at  $\pm 3f_0$,  $\pm 5f_0$, ...  are also purely imaginary and located in the same direction as the Dirac delta functions at  $\pm f_0$.		*All other Dirac delta functions at  $\pm 3f_0$,  $\pm 5f_0$, ...  are also purely imaginary and located in the same direction as the Dirac delta functions at  $\pm f_0$.
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	Not every periodic signal is suitable for the Fourier series.  Some restrictions below:		Not every periodic signal is suitable for the Fourier series.  Some restrictions below:
	*An important condition for the convergence of the Fourier series is that the signal may only have a finite number of discontinuities per period.		*An important condition for the convergence of the Fourier series is that the signal may only have a finite number of discontinuities per period.

	*At those places  $t=t_i$, where  $x(t)$  has jumps, the series converges to the arithmetic mean value formed by the respective left and right boundary value.		*At those places  $t=t_i$, where  $x(t)$  has jumps, the series converges to the arithmetic mean value formed by the respective left and right boundary value.

	*In the surrounding area of such discontinuities, high-frequency oscillations usually occur in the series representation.  This error is of principle kind, i.e. it could not be avoided too, if infinite summands would be considered.  One speaks of the  "Gibbs phenomenon", named after the physicist  [https://en.wikipedia.org/wiki/Josiah_Willard_Gibbs $\text{Josiah Willard Gibbs}$].		*In the surrounding area of such discontinuities, high-frequency oscillations usually occur in the series representation.  This error is of principle kind, i.e. it could not be avoided too, if infinite summands would be considered.  One speaks of the  "Gibbs phenomenon", named after the physicist  [https://en.wikipedia.org/wiki/Josiah_Willard_Gibbs $\text{Josiah Willard Gibbs}$].
	*An increase of  $N$  reduces the erroneous range but not the maximum deviation between  $x(t)$  and the Fourier series representation  $x_N(t)$.  The maximum error is about  $9\%$  of the jumping amplitude - independent of  $N$.
			*An increase of  $N$  reduces the erroneous range but not the maximum deviation between  $x(t)$  and the Fourier series representation  $x_N(t)$.  The maximum error is about  $9\%$  of the jumping amplitude – independent of  $N$.


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	[[File:P_ID2720__Sig_T_2_4_S7_neu.png\|right\|frame\|On the Gibbs phenomenon]]		[[File:P_ID2720__Sig_T_2_4_S7_neu.png\|right\|frame\|On the Gibbs phenomenon]]

	*The basic wave here has the amplitude value  $4/\pi \approx 1.27$.		*The basic wave here has the amplitude value  $4/\pi \approx 1.27$.

	*Even with  $N = 5$  $($this means because of  $A_2 = A_4 = 0$  three "relevant" summands$)$ the Fourier series still differs significantly from the approximated rectangular signal, especially in the area of the edge.		*Even with  $N = 5$  $($this means because of  $A_2 = A_4 = 0$  three "relevant" summands$)$ the Fourier series still differs significantly from the approximated rectangular signal, especially in the area of the edge.

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	From the right graphic you can see that the flank and the inner area are are well reproduced with		From the right graphic you can see that the flank and the inner area are are well reproduced with
	$N = 100$  but due to the Gibbs phenomenon there are still oscillations around  $9\%$  at the jumping point.		$N = 100$  but due to the Gibbs phenomenon there are still oscillations around  $9\%$  at the jumping point.
	*Since the jump amplitudes here are equal to  $2$  the maximum values are approximately  $\pm1.18$.		*Since the jump amplitudes here are equal to  $2$  the maximum values are approximately  $\pm1.18$.

	*With  $N = 1000$  the oscillations would be exactly the same size, but limited to a narrower space and possibly not recognizable with discrete-time representation.}}		*With  $N = 1000$  the oscillations would be exactly the same size, but limited to a narrower space and possibly not recognizable with discrete-time representation.}}

Guenter at 17:22, 8 February 2023

2023-02-08T17:22:44Z

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Guenter at 17:43, 2 December 2022

2022-12-02T17:43:14Z

← Older revision		Revision as of 19:43, 2 December 2022
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	:$$x_{\rm u}(t)={4}/{\pi} \cdot \big[\sin(\omega_0 t)+{1}/{3} \cdot \sin(3\omega_0 t)+{1}/{5} \cdot \sin(5\omega_0 t)+ \, \text{...}\big].$$		:$$x_{\rm u}(t)={4}/{\pi} \cdot \big[\sin(\omega_0 t)+{1}/{3} \cdot \sin(3\omega_0 t)+{1}/{5} \cdot \sin(5\omega_0 t)+ \, \text{...}\big].$$
	[[File:P_ID528__Sig_T_2_4_S6_neu.png\|right\|frame\|Spectrum of a periodic rectangular signal]]		[[File:P_ID528__Sig_T_2_4_S6_neu.png\|right\|frame\|Spectrum of a periodic rectangular signal]]



	*As described in the section  [[Signal_Representation/Harmonic_Oscillation#General_spectral_representation\|"General Spectral Representation"]]  the basic wave provides two Dirac delta functions		*As described in the section  [[Signal_Representation/Harmonic_Oscillation#General_spectral_representation\|"General Spectral Representation"]]  the basic wave provides two Dirac delta functions

Guenter at 17:42, 2 December 2022

2022-12-02T17:42:04Z

← Older revision		Revision as of 19:42, 2 December 2022
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	**at  $+f_0$  $($weighted with  $-\text{j}\cdot 2/\pi)$  resp.		**at  $+f_0$  $($weighted with  $-\text{j}\cdot 2/\pi)$  resp.
	**at  $-f_0$  $($weighted with   $+\text{j}\cdot 2/\pi)$.		**at  $-f_0$  $($weighted with   $+\text{j}\cdot 2/\pi)$.



	*All other Dirac delta functions at  $\pm 3f_0$,  $\pm 5f_0$, ...  are also purely imaginary and located in the same direction as the Dirac delta functions at  $\pm f_0$.		*All other Dirac delta functions at  $\pm 3f_0$,  $\pm 5f_0$, ...  are also purely imaginary and located in the same direction as the Dirac delta functions at  $\pm f_0$.



	*The two magnitude spectra are equal:   $\vert X_{\rm u}(f)\vert = \vert X_{\rm g}(f) \vert$.}}		*The two magnitude spectra are equal:   $\vert X_{\rm u}(f)\vert = \vert X_{\rm g}(f) \vert$.}}

Guenter at 17:40, 2 December 2022

2022-12-02T17:40:38Z

← Older revision		Revision as of 19:40, 2 December 2022
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	*The basic wave here has the amplitude value  $4/\pi \approx 1.27$.		*The basic wave here has the amplitude value  $4/\pi \approx 1.27$.
	*Even with  $N = 5$  $($this means because of  $A_2 = A_4 = 0$  three "relevant" summands$)$ the Fourier series still differs significantly from the approximated ~~square wave '''KORREKTUR:~~ rectangular~~'''~~ signal, especially in the area of the edge.		*Even with  $N = 5$  $($this means because of  $A_2 = A_4 = 0$  three "relevant" summands$)$ the Fourier series still differs significantly from the approximated rectangular signal, especially in the area of the edge.