Signal Representation/Fourier Series - Revision history

Guenter at 16:21, 15 November 2023

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Guenter at 17:43, 2 December 2022

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Guenter at 17:39, 2 December 2022

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@@ Line 3: / Line 3: @@
 |Untermenü=Periodic Signals
 |Vorherige Seite=Harmonic Oscillation
-|Nächste Seite=Fourier Transform and Its Inverse
+|Nächste Seite=The Fourier Transform and its Inverse
 }}

@@ Line 1: / Line 1: @@
 {{Header
-|Untermenü=Periodische Signale
+|Untermenü=Periodic Signals
 |Vorherige Seite=Harmonic Oscillation
 |Nächste Seite=Fourier Transform and Its Inverse

@@ Line 238: / Line 238: @@
 [[File:P_ID528__Sig_T_2_4_S6_neu.png|right|frame|Spectrum of a periodic rectangular signal]]
-is an even&nbsp; (German:&nbsp; "gerade" &nbsp; &rArr; &nbsp; "$\rm g$")&nbsp; function, composed of different cosine parts.&nbsp;
+is an even&nbsp; (German:&nbsp; "gerade" &nbsp; &rArr; &nbsp; "$\rm g$")&nbsp; function,&nbsp; composed of different cosine parts.&nbsp;
 *The corresponding spectral function&nbsp; $X_{\rm g}(f)$&nbsp; is thus purely real.
 ::<u>Reason:</u>&nbsp; As described in the section&nbsp; [[Signal_Representation/Harmonic_Oscillation#Representation_with_cosine_and_sine_components|"Spectral Representation of a cosine signal"]]&nbsp; the basic wave returns two Dirac  delta functions at&nbsp; $\pm f_0$, each weighted with&nbsp; $2/\pi$.
@@ Line 244: / Line 246: @@
 *This weighting corresponds to the (generally complex) Fourier coefficients&nbsp; $D_1 = D_{ - 1}^\ast$, which are only real in the special case of an even function.
-*Other Dirac delta functions are available in&nbsp; $\pm 3f_0$ (negative),&nbsp; $\pm 5f_0$ (positive),&nbsp; $\pm 7f_0$ (negative) etc.
+*Other Dirac delta functions are available in&nbsp;
 *All phase values&nbsp; $\varphi_n$&nbsp; are either zero or&nbsp; $\pi$ due to the alternating signs.
@@ Line 253: / Line 260: @@
 *As described in the section&nbsp; [[Signal_Representation/Harmonic_Oscillation#General_spectral_representation|"General Spectral Representation"]]&nbsp;  the basic wave provides two Dirac  delta functions
-**at&nbsp; $+f_0$&nbsp; $($weighted with&nbsp; $-\text{j}\cdot 2/\pi)$&nbsp; resp.
+:*at&nbsp; $+f_0$&nbsp; $($weighted with&nbsp; $-\text{j}\cdot 2/\pi)$&nbsp; resp.
-**at&nbsp; $-f_0$&nbsp; $($weighted with &nbsp; $+\text{j}\cdot 2/\pi)$.
 *All other Dirac delta functions at&nbsp; $\pm 3f_0$,&nbsp; $\pm 5f_0$, ...&nbsp; are also purely imaginary and located in the same direction as the Dirac delta functions at&nbsp; $\pm f_0$.
@@ Line 266: / Line 274: @@
 Not every periodic signal is suitable for the Fourier series.&nbsp; 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.
 *At those places&nbsp; $t=t_i$, where&nbsp; $x(t)$&nbsp; 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.&nbsp; This error is of principle kind, i.e. it could not be avoided too, if infinite summands would be considered.&nbsp; One speaks of the&nbsp; "Gibbs phenomenon", named after the physicist&nbsp; [https://en.wikipedia.org/wiki/Josiah_Willard_Gibbs $\text{Josiah Willard Gibbs}$].
-*An increase of&nbsp; $N$&nbsp; reduces the erroneous range but not the maximum deviation between&nbsp; $x(t)$&nbsp; and the Fourier series representation&nbsp; $x_N(t)$.&nbsp; The maximum error is about&nbsp; $9\%$&nbsp; of the jumping amplitude - independent of&nbsp; $N$.
@@ Line 280: / Line 291: @@
 [[File:P_ID2720__Sig_T_2_4_S7_neu.png|right|frame|On the Gibbs phenomenon]]
 *The basic wave here has the amplitude value&nbsp; $4/\pi \approx 1.27$.
 *Even with&nbsp; $N = 5$&nbsp; $($this means because of&nbsp; $A_2 = A_4 = 0$&nbsp; three "relevant" summands$)$ the Fourier series still differs significantly from the approximated rectangular signal, especially in the area of the edge.
@@ Line 287: / Line 299: @@
 From the right graphic you can see that the flank and the inner area are are well reproduced with&nbsp;
 $N = 100$&nbsp; but due to the Gibbs phenomenon there are still oscillations around&nbsp; $9\%$&nbsp; at the jumping point.
 *Since the jump amplitudes here are equal to&nbsp; $2$&nbsp; the maximum values are approximately&nbsp; $\pm1.18$.
 *With&nbsp; $N = 1000$&nbsp; the oscillations would be exactly the same size, but limited to a narrower space and possibly not recognizable with discrete-time representation.}}

@@ Line 236: / Line 236: @@
 :$$x_{\rm u}(t)={4}/{\pi} \cdot \big[\sin(\omega_0 t)+{1}/{3} \cdot \sin(3\omega_0 t)+{1}/{5} \cdot \sin(5\omega_0 t)+ \, \text{...}\big].$$
 [[File:P_ID528__Sig_T_2_4_S6_neu.png|right|frame|Spectrum of a periodic rectangular signal]]
 *As described in the section&nbsp; [[Signal_Representation/Harmonic_Oscillation#General_spectral_representation|"General Spectral Representation"]]&nbsp;  the basic wave provides two Dirac  delta functions

@@ Line 240: / Line 240: @@
 **at&nbsp; $+f_0$&nbsp; $($weighted with&nbsp; $-\text{j}\cdot 2/\pi)$&nbsp; resp.
 **at&nbsp; $-f_0$&nbsp; $($weighted with &nbsp; $+\text{j}\cdot 2/\pi)$.
 *All other Dirac delta functions at&nbsp; $\pm 3f_0$,&nbsp; $\pm 5f_0$, ...&nbsp; are also purely imaginary and located in the same direction as the Dirac delta functions at&nbsp; $\pm f_0$.
 *The two magnitude spectra are equal: &nbsp; $\vert X_{\rm u}(f)\vert = \vert X_{\rm g}(f) \vert$.}}

← Older revision		Revision as of 17:40, 2 December 2022
Line 266:		Line 266:

	*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.

@@ Line 208: / Line 208: @@
 This means:
 *The (amplitude) spectrum of a periodic signal with&nbsp; $T_0$&nbsp; is a&nbsp; &raquo;'''line spectrum'''&laquo;&nbsp; for integer multiples of the basic frequency&nbsp; $f_0 = 1/T_0$.
-*The&nbsp; &raquo;'''DC signal component'''&laquo; &nbsp; returns a Dirac '''KORREKTUR: delta?''' function at&nbsp; $f=0$&nbsp; with the impulse weight&nbsp; $A_0$.
+*The&nbsp; &raquo;'''DC signal component'''&laquo; &nbsp; returns a &nbsp; &raquo;Dirac delta function&laquo;&nbsp; at&nbsp; $f=0$&nbsp; with the impulse weight&nbsp; $A_0$.
-*There are also &nbsp; &raquo;'''Dirac KORREKTUR: delta? functions'''&laquo;&nbsp; $\delta(f \pm n \cdot f_0)$&nbsp; at the multiples of&nbsp;  $f_0$,
+*There are also Dirac delta functions&nbsp; $\delta(f \pm n \cdot f_0)$&nbsp; at the multiples of&nbsp;  $f_0$,
-**where&nbsp; $\delta(f - n \cdot f_0)$&nbsp; denotes a Dirac '''KORREKTUR: delta?''' function at &nbsp; $f= n \cdot f_0$&nbsp; (namely in the positive frequency domain)
+**where&nbsp; $\delta(f - n \cdot f_0)$&nbsp; denotes a Dirac delta function at &nbsp; $f= n \cdot f_0$&nbsp; (namely in the positive frequency domain)
 **and&nbsp; $\delta(f + n \cdot f_0)$&nbsp; denotes a Dirac at the frequency&nbsp;  $f= -n \cdot  f_0$&nbsp; (in the negative frequency domain).
 *The&nbsp; &raquo;'''pulse weights''' &laquo;&nbsp; are generally complex.
@@ Line 219: / Line 219: @@
 {{GraueBox|TEXT=
 $\text{Example 4:}$&nbsp;
-We consider as in&nbsp; [[Signal_Representation/Harmonic_Oscillation#Time_domain_representation|$\text{Example 1}$]]&nbsp; two periodic square wave '''KORREKTUR: rectangular''' signals, each with period duration&nbsp; $T_0$&nbsp; and basic frequency&nbsp; $f_0=1/T_0$.&nbsp;
+We consider as in&nbsp; [[Signal_Representation/Harmonic_Oscillation#Time_domain_representation|$\text{Example 1}$]]&nbsp; two periodic   rectangular signals, each with period duration&nbsp; $T_0$&nbsp; and basic frequency&nbsp; $f_0=1/T_0$.&nbsp;
 The upper signal
@@ Line 226: / Line 226: @@
 is an even&nbsp; (German:&nbsp; "gerade" &nbsp; &rArr; &nbsp; $\rm g$)&nbsp; function, composed of different cosine parts.&nbsp; The corresponding spectral function&nbsp; $X_{\rm g}(f)$&nbsp; is thus purely real.
-''Reason:''&nbsp; As described in the section&nbsp; [[Signal_Representation/Harmonic_Oscillation#Representation_with_cosine_and_sine_components|"Spectral Representation of a cosine signal"]]&nbsp; the basic wave returns two Dirac '''KORREKTUR: delta?''' functions at&nbsp; $\pm f_0$, each weighted with&nbsp; $2/\pi$.
+''Reason:''&nbsp; As described in the section&nbsp; [[Signal_Representation/Harmonic_Oscillation#Representation_with_cosine_and_sine_components|"Spectral Representation of a cosine signal"]]&nbsp; the basic wave returns two Dirac  delta functions at&nbsp; $\pm f_0$, each weighted with&nbsp; $2/\pi$.
 *This weighting corresponds to the (generally complex) Fourier coefficients&nbsp; $D_1 = D_{ - 1}^\ast$, which are only real in the special case of an even function.
-*Other Dirac '''KORREKTUR: delta?''' functions are available in&nbsp; $\pm 3f_0$ (negative),&nbsp; $\pm 5f_0$ (positive),&nbsp; $\pm 7f_0$ (negative) etc.
+*Other Dirac delta functions are available in&nbsp; $\pm 3f_0$ (negative),&nbsp; $\pm 5f_0$ (positive),&nbsp; $\pm 7f_0$ (negative) etc.
 *All phase values&nbsp; $\varphi_n$&nbsp; are either zero or&nbsp; $\pi$ due to the alternating signs.
 The function&nbsp; $x_{\rm u}(t)$&nbsp; shown below is odd&nbsp; (German:&nbsp; "ungerade" &nbsp; &rArr; &nbsp; $\rm u$):
 :$$x_{\rm u}(t)={4}/{\pi} \cdot \big[\sin(\omega_0 t)+{1}/{3} \cdot \sin(3\omega_0 t)+{1}/{5} \cdot \sin(5\omega_0 t)+ \, \text{...}\big].$$
-*As described in the section&nbsp; [[Signal_Representation/Harmonic_Oscillation#General_spectral_representation|"General Spectral Representation"]]&nbsp;  the basic wave provides two Dirac '''KORREKTUR: delta?''' functions
+*As described in the section&nbsp; [[Signal_Representation/Harmonic_Oscillation#General_spectral_representation|"General Spectral Representation"]]&nbsp;  the basic wave provides two Dirac  delta functions
 **at&nbsp; $+f_0$&nbsp; $($weighted with&nbsp; $-\text{j}\cdot 2/\pi)$&nbsp; resp.
 **at&nbsp; $-f_0$&nbsp; $($weighted with &nbsp; $+\text{j}\cdot 2/\pi)$.
-*All other Dirac '''KORREKTUR: delta?''' functions at&nbsp; $\pm 3f_0$,&nbsp; $\pm 5f_0$, ...&nbsp; are also purely imaginary and located in the same direction as the Dirac '''KORREKTUR: delta?''' functions at&nbsp; $\pm f_0$.
 *The two magnitude spectra are equal: &nbsp; $\vert X_{\rm u}(f)\vert = \vert X_{\rm g}(f) \vert$.}}