refactor: anti-aliasing and reconstruction filter
summarized both topics in one chapter
This commit is contained in:
Matthias Jung
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% Author: Prof. Dr. Matthias Jung, DL9MJ
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% Year: 2026
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After Width: | Height: | Size: 27 KiB |
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% Author: Prof. Dr. Matthias Jung, DL9MJ
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% Year: 2026
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\begin{tikzpicture}
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\draw(0,0)
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node[anchor=east](){E}
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to [short, o-] ++(0.5,0)
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node[inputarrow](){}
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node[lowpassshape, anchor=west, fill=DARCblue!40](anti-alias){};
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\draw(anti-alias.east)
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to [adc, >] ++(2,0)
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node[inputarrow](){}
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node[dspshape, anchor=west](dsp){};
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\draw(dsp.east)
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to [dac, >] ++(2,0)
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node[inputarrow](){}
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node[lowpassshape, anchor=west, fill=DARCred!40](reconstruction){};
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\draw(reconstruction.east)
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to [short, -o] ++(0.5,0)
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node[anchor=west](){A};
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%
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\draw[DARCblue] (anti-alias.north) node[above](){Anti-Aliasing};
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\draw[DARCred] (reconstruction.north) node[above](){Rekonstruktions};
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%
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\draw[DARCred] (reconstruction.south) node[below](){Filter};
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\end{tikzpicture}
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After Width: | Height: | Size: 27 KiB |
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% Author: Prof. Dr. Matthias Jung, DL9MJ
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% Year: 2026
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blocks/scale=\getDarcImageFactor,
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RF/scale=\getDarcImageFactor
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node[antenna](){}
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to [lowpass, fill=DARCblue!40, >] ++(3,0)
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node[adcshape, fill=DARCorange!40, t={3}, anchor=west](drei){};
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node [inputarrow](ende){};
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node [anchor=west, align=left] {Weitere\\Teile\\des\\Empfängers};
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@@ -1,20 +0,0 @@
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%%% TODO diese beiden textblöcke müssen verbunden werden.
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|
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Wie zuvor schon kurz angesprochen, muss vor dem A/D-Umsetzer ein entsprechendes Filter zur Unterdrückung von Signalanteilen höherer Frequenzen eingefügt werden. Man nennt ein solches Filter auch Antialiasingfilter. Als Filtertypen können hier sowohl Tiefpassfilter als auch Bandpassfilter Anwendung finden. Das Filter muss hierbei so beschaffen sein, dass Signalanteile, die sich oberhalb der halben Sampling-Frequenz befinden, wirksam unterdrückt werden. Wir erinnern uns, dass für die Abtastung eines Signals mehr als die doppelte Abtastfrequenz der zu erfassenden Signalfrequenz erforderlich ist.
|
||||
|
||||
[question:AF622]
|
||||
[question:AF623]
|
||||
|
||||
|
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|
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Aus der Lektion zum Abtasttheorem wissen wir, dass ein Signal mit einer ausreichend hohen Abtastrate abgetastet werden muss. Über eine Antenne empfangen wir jedoch in der Regel viele unterschiedliche Signale – auch solche mit Frequenzen oberhalb des Frequenzbereichs, den wir eigentlich verarbeiten wollen.
|
||||
|
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Treffen solche Signale auf den A/D-Umsetzer, obwohl seine Abtastrate für diese Frequenzen nicht ausreichend ist, können sie als andere, nicht tatsächlich vorhandene Frequenzen im digitalen Signal erscheinen. Diese werden als *Aliase* bezeichnet.
|
||||
|
||||
Um dies zu verhindern, wird vor dem Eingang des A/D-Umsetzers ein *Antialiasing-Filter* eingesetzt. Dabei handelt es sich in der Regel um einen Tiefpass- oder Bandpassfilter, der unerwünschte Frequenzen unterdrückt, bevor das Signal den A/D-Umsetzer erreicht.
|
||||
|
||||
Der A/D-Umsetzer benötigt außerdem einen Taktgenerator, den man auch als Abtastratengenerator bezeichnet. Dieser legt fest, zu welchen Zeitpunkten das Eingangssignal abgetastet wird. Der A/D-Umsetzer erzeugt daraus einen digitalen Datenstrom, der anschließend von weiteren Teilen der Schaltung verarbeitet werden kann. Die Taktrate kann fest eingestellt sein oder beispielsweise durch einen Mikrocontroller gesteuert werden.
|
||||
|
||||
[question:AF620]
|
||||
@@ -0,0 +1,57 @@
|
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Bei der A/D- und D/A-Umsetzung werden analoge und digitale Signalverarbeitung miteinander verbunden. Dabei werden sowohl vor dem A/D-Umsetzer als auch nach dem D/A-Umsetzer analoge Filter benötigt. Die Abbildung [ref:a_adc_dac_filter] zeigt die gesamte Signalkette. Auf der Eingangsseite befindet sich vor dem A/D-Umsetzer ein *Anti-Aliasing-Filter*. Es begrenzt den Frequenzbereich des analogen Eingangssignals, bevor dieses abgetastet wird. Nach der digitalen Signalverarbeitung erzeugt der D/A-Umsetzer wieder ein analoges Signal. Ein nachgeschaltetes *Rekonstruktionsfilter* entfernt dabei unerwünschte hochfrequente Signalanteile. Warum beide Filter benötigt werden, betrachten wir in dem folgenden Abschnitt.
|
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|
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<margin>
|
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[picture:1131:a_adc_dac_filter:A/D- und D/A-Umsetzung mit Anti-Aliasing- und Rekonstruktionsfilter]
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</margin>
|
||||
|
||||
---
|
||||
|
||||
Aus der Lektion zum Abtasttheorem wissen wir, dass ein Signal mit einer ausreichend hohen Abtastrate abgetastet werden muss. Für ein Signal mit der höchsten zu erfassenden Frequenz $f_\mathrm{max}$ muss die Abtastrate größer als $2\cdot f_\mathrm{max}$ sein.
|
||||
|
||||
Über eine Antenne empfangen wir jedoch in der Regel viele unterschiedliche Signale – auch solche mit Frequenzen oberhalb des Frequenzbereichs, den wir eigentlich verarbeiten wollen. Treffen solche Signalanteile auf den A/D-Umsetzer, obwohl dessen Abtastrate für diese Frequenzen nicht ausreichend ist, können sie im digitalen Signal als andere, tatsächlich nicht vorhandene Frequenzen erscheinen. Diese werden als *Aliase* bezeichnet.
|
||||
|
||||
Um dies zu verhindern, wird vor dem Eingang des A/D-Umsetzers ein *Anti-Aliasing-Filter* eingesetzt. Dabei handelt es sich je nach Anwendung beispielsweise um einen Tiefpass- oder Bandpassfilter. Ein Bandpassfilter könnte z. B. für Sprache verwendet werden. Das Filter muss unerwünschte Signalanteile, die beim Sampling zu Aliasing führen könnten, ausreichend unterdrücken. Insbesondere dürfen Frequenzanteile oberhalb der halben Samplingfrequenz nicht ungehindert zum A/D-Umsetzer gelangen.
|
||||
|
||||
[question:AF622]
|
||||
[question:AF623]
|
||||
|
||||
<indepth>
|
||||
Ein anschauliches Beispiel für *Aliasing* begegnet uns auch im Alltag bei digitalen Bildern. Fotografiert man mit einer Kamera sehr feine, regelmäßig wiederkehrende Strukturen, beispielsweise ein engmaschiges Gitter, einen Stoff mit feinen Streifen oder ein Fliegengitter, können im Bild plötzlich größere Muster entstehen, die im Original gar nicht vorhanden sind. Diese werden als *Moiré-Muster* bezeichnet.
|
||||
|
||||
Die Ursache ist ähnlich wie beim Abtasten eines elektrischen Signals. Ein Kamerasensor kann ein Bild nicht an beliebig vielen Stellen erfassen, sondern besitzt nur eine endliche Anzahl von Bildpunkten. Ist eine Struktur feiner als die räumliche Auflösung des Sensors, wird sie nicht mehr eindeutig abgetastet. Aus der eigentlich vorhandenen feinen Struktur kann dadurch scheinbar eine andere, gröbere Struktur entstehen.
|
||||
|
||||
Beim A/D-Umsetzer geschieht das gleiche Prinzip auf der Zeitachse: Wird eine zu hohe Signalfrequenz mit einer zu niedrigen Samplingrate abgetastet, erscheint im digitalisierten Signal eine andere, niedrigere Frequenz, die ursprünglich gar nicht vorhanden war.
|
||||
|
||||
Ein Moiré-Muster kann daher als sichtbares Beispiel dafür betrachtet werden, wie durch eine unzureichende Abtastung neue, scheinbare Strukturen entstehen.
|
||||
|
||||
% TODO: Bild besorgen
|
||||
%<margin>
|
||||
%[picture:XXXX:a_moire:Moiré-Muster als Beispiel für räumliches Aliasing]
|
||||
%</margin>
|
||||
</indepth>
|
||||
|
||||
---
|
||||
|
||||
Der A/D-Umsetzer benötigt außerdem einen Taktgenerator, den man auch als Abtastratengenerator bezeichnet. Dieser legt fest, zu welchen Zeitpunkten das Eingangssignal abgetastet wird und bestimmt damit die Samplingrate. Die Taktrate kann fest eingestellt sein oder beispielsweise durch einen Mikrocontroller gesteuert werden.
|
||||
|
||||
<margin>
|
||||
[picture:1132:a_anit_alias:Anti-Aliasing-Filter, A/D-Umsetzer und Taktgenerator]
|
||||
</margin>
|
||||
|
||||
[question:AF620]
|
||||
|
||||
---
|
||||
|
||||
Auf der anderen Seite der digitalen Signalverarbeitung übernimmt der D/A-Umsetzer den umgekehrten Vorgang. Er setzt die digitalen Samples wieder in analoge Spannungswerte um. Da die einzelnen Werte nur in festen zeitlichen Abständen ausgegeben werden, entsteht am Ausgang des D/A-Umsetzers zunächst kein ideal glatter Signalverlauf.
|
||||
|
||||
Durch die zeitdiskrete Ausgabe entstehen neben dem gewünschten Nutzsignal auch unerwünschte höherfrequente Signalanteile (Z.B. in Abbildung [ref:a_adc_4bit], die scharfen Kanten des Signals enthalten die hohen Frequenzanteile). Um diese zu unterdrücken, wird hinter dem D/A-Umsetzer ein *Rekonstruktionsfilter* eingesetzt. Auch hierbei kann je nach Anwendung beispielsweise ein Tiefpass- oder Bandpassfilter verwendet werden.
|
||||
|
||||
Das Rekonstruktionsfilter lässt den gewünschten Nutzfrequenzbereich passieren und unterdrückt die unerwünschten höherfrequenten Signalanteile des D/A-Umsetzers. Dadurch entsteht am Ausgang wieder ein möglichst sauberes analoges Signal (vgl. Abbildung [ref:a_adc_12bit], das Rekonstruktionsfilter glättet das Signal).
|
||||
|
||||
[question:AF624]
|
||||
[question:AF625]
|
||||
|
||||
<margin>
|
||||
[picture:300:a_adc_4bit:Signal vor dem Rekonstruktionsfilter]
|
||||
[picture:299:a_adc_12bit:Singnal nach dem Rekonstruktionsfilter]
|
||||
</margin>
|
||||
@@ -1,7 +0,0 @@
|
||||
|
||||
Sehen wir uns nun die Umsetzung von digitalen Signalen (Samples) zurück in analoge Signale durch einen D/A-Umsetzer etwas genauer an. Hierbei werden die digitalen Daten im D/A-Umsetzer in Spannungsstufen zurück übersetzt. Dies geschieht in einem festen zeitlichen Abstand der Samples zueinander. Man nennt diesen Vorgang auch Rekonstruktion.
|
||||
|
||||
Aufgrund der zeitdiskreten Auflösung durch die begrenzte Sampling-Frequenz eines D/A-Umsetzers entsteht ein stufenförmiges Signal, das unerwünschte, höhere Frequenzen enthält. Um diese unerwünschten Anteile zu entfernen und somit das ursprüngliche Signal wiederherzustellen, benötigen wir ebenso wie am Eingang zum A/D-Umsetzer ein Tiefpass- oder Bandpassfilter. Dieses Rekonstruktionsfilter muss hierbei alle Signalanteile, die sich oberhalb der halben Abtastrate (Samplingfrequenz) befinden, wirksam unterdrücken.
|
||||
|
||||
[question:AF624]
|
||||
[question:AF625]
|
||||
@@ -1,21 +0,0 @@
|
||||
## Antialiasingfilter vor dem A/D-Umsetzer
|
||||
|
||||
* Vor dem A/D-Umsetzer wird ein Filter eingesetzt, um Signalanteile höherer Frequenzen zu unterdrücken
|
||||
* Solche Filter werden als Antialiasingfilter bezeichnet
|
||||
* Ziel: Vermeidung von Alias-Effekten durch Abschwächung unerwünschter Frequenzen
|
||||
|
||||
---
|
||||
|
||||
### Filtertypen und Anforderungen
|
||||
|
||||
* Tiefpass- oder Bandpassfilter können eingesetzt werden
|
||||
* Das Filter muss Signalanteile oberhalb der halben Sampling-Frequenz wirksam dämpfen
|
||||
* Voraussetzung: Die Abtastfrequenz muss mehr als das Doppelte der zu erfassenden Signalfrequenz betragen
|
||||
|
||||
---
|
||||
|
||||
[question:AF622]
|
||||
|
||||
---
|
||||
|
||||
[question:AF623]
|
||||
@@ -0,0 +1,143 @@
|
||||
--- style="font-size: smaller;"
|
||||
|
||||
## Filter bei der A/D- und D/A-Umsetzung
|
||||
|
||||
[picture:1131:a_adc_dac_filter:A/D- und D/A-Umsetzung mit Anti-Aliasing- und Rekonstruktionsfilter]
|
||||
|
||||
* Vor dem A/D-Umsetzer: *Anti-Aliasing-Filter*
|
||||
* Nach dem D/A-Umsetzer: *Rekonstruktionsfilter*
|
||||
* Beide Filter unterdrücken unerwünschte Frequenzanteile
|
||||
|
||||
<note>
|
||||
Die Abbildung zeigt die gesamte Signalkette von einem analogen Eingangssignal über die digitale Signalverarbeitung bis zurück zu einem analogen Ausgangssignal.
|
||||
|
||||
Vor dem A/D-Umsetzer befindet sich das Anti-Aliasing-Filter. Hinter dem D/A-Umsetzer befindet sich das Rekonstruktionsfilter.
|
||||
|
||||
Warum beide Filter benötigt werden, schauen wir uns nun genauer an.
|
||||
</note>
|
||||
|
||||
---
|
||||
|
||||
## Aliasing am A/D-Umsetzer
|
||||
|
||||
* Für die Abtastung gilt: $f_\mathrm{S}>2\cdot f_\mathrm{max}$
|
||||
* Am Eingang können auch unerwünschte höhere Frequenzen vorhanden sein
|
||||
* Ist die Samplingrate dafür zu gering, entstehen scheinbare Frequenzen
|
||||
* Diese werden als *Aliase* bezeichnet
|
||||
|
||||
<note>
|
||||
Über eine Antenne empfangen wir in der Regel nicht nur das gewünschte Signal, sondern viele weitere Frequenzanteile.
|
||||
|
||||
Ist die Samplingrate für einen dieser Frequenzanteile zu gering, kann er nach der Digitalisierung als eine andere Frequenz erscheinen, die im ursprünglichen Signal gar nicht vorhanden war.
|
||||
|
||||
Diesen Effekt haben wir bereits beim Abtasttheorem kennengelernt.
|
||||
</note>
|
||||
|
||||
--- style="font-size: smaller;"
|
||||
|
||||
## Anti-Aliasing-Filter
|
||||
|
||||
[picture:1131:a_adc_dac_filter_aa:A/D- und D/A-Umsetzung mit Anti-Aliasing- und Rekonstruktionsfilter]
|
||||
|
||||
* Das Filter befindet sich *vor* dem A/D-Umsetzer
|
||||
* Es begrenzt den Frequenzbereich des Eingangssignals
|
||||
* Tiefpass- oder Bandpassfilter können eingesetzt werden
|
||||
* Kritische Frequenzanteile müssen ausreichend unterdrückt werden
|
||||
|
||||
<fragment>
|
||||
Insbesondere dürfen Frequenzen oberhalb von $\frac{f_\mathrm{S}}{2}$ nicht ungehindert zum A/D-Umsetzer gelangen.
|
||||
</fragment>
|
||||
|
||||
<note>
|
||||
Das Anti-Aliasing-Filter verhindert, dass Frequenzen zum A/D-Umsetzer gelangen, für die die Samplingrate nicht ausreichend hoch ist.
|
||||
|
||||
Je nach Anwendung kann dazu beispielsweise ein Tiefpass- oder Bandpassfilter eingesetzt werden. Für Sprachsignale kann beispielsweise ein Bandpass sinnvoll sein.
|
||||
|
||||
Entscheidend ist, dass Frequenzanteile, die zu Aliasing führen könnten, vor der Digitalisierung ausreichend unterdrückt werden.
|
||||
</note>
|
||||
|
||||
---
|
||||
|
||||
[question:AF622]
|
||||
|
||||
---
|
||||
|
||||
[question:AF623]
|
||||
|
||||
---
|
||||
## Abtastratengenerator
|
||||
|
||||
<left>
|
||||
[picture:1132:a_anit_alias:Anti-Aliasing-Filter, A/D-Umsetzer und Taktgenerator]
|
||||
</left>
|
||||
<right>
|
||||
* Der A/D-Umsetzer benötigt einen Takt für die Abtastung
|
||||
* Der Takt legt die Zeitpunkte der einzelnen Samples fest
|
||||
* Seine Frequenz bestimmt die Samplingrate
|
||||
* Die Taktrate kann fest oder steuerbar sein
|
||||
</right>
|
||||
|
||||
<note>
|
||||
Der Abtastratengenerator gibt vor, wann der A/D-Umsetzer jeweils ein neues Sample aufnimmt.
|
||||
|
||||
Seine Frequenz bestimmt damit unmittelbar die Samplingrate. Die Taktrate kann fest vorgegeben oder beispielsweise durch einen Mikrocontroller gesteuert werden.
|
||||
</note>
|
||||
|
||||
---
|
||||
|
||||
[question:AF620]
|
||||
|
||||
---
|
||||
|
||||
## Zurück zum analogen Signal
|
||||
|
||||
* Der D/A-Umsetzer erzeugt aus digitalen Samples wieder analoge Spannungswerte
|
||||
* Die Werte werden in festen zeitlichen Abständen ausgegeben
|
||||
* Am Ausgang entsteht zunächst kein ideal glatter Signalverlauf
|
||||
* Scharfe Übergänge enthalten hohe Frequenzanteile
|
||||
|
||||
<note>
|
||||
Auf der Ausgangsseite findet der umgekehrte Vorgang statt.
|
||||
|
||||
Der D/A-Umsetzer gibt die einzelnen digitalen Werte als analoge Spannungswerte aus. Durch die zeitdiskrete Ausgabe entsteht zunächst kein ideal glatter Verlauf.
|
||||
|
||||
Insbesondere scharfe Kanten und Übergänge enthalten zusätzliche hochfrequente Signalanteile.
|
||||
</note>
|
||||
|
||||
--- style="font-size: smaller;"
|
||||
|
||||
## Rekonstruktionsfilter
|
||||
|
||||
<left>
|
||||
[picture:300:a_adc_4bit:Signal vor dem Rekonstruktionsfilter]
|
||||
</left>
|
||||
<right>
|
||||
[picture:299:a_adc_12bit:Signal nach dem Rekonstruktionsfilter]
|
||||
</right>
|
||||
|
||||
* Das Rekonstruktionsfilter befindet sich *hinter* dem D/A-Umsetzer
|
||||
* Es lässt den gewünschten Nutzfrequenzbereich passieren
|
||||
* Unerwünschte höherfrequente Signalanteile werden unterdrückt
|
||||
* Tiefpass- oder Bandpassfilter können eingesetzt werden
|
||||
|
||||
<fragment>
|
||||
Dadurch entsteht wieder ein möglichst sauberes analoges Ausgangssignal.
|
||||
</fragment>
|
||||
|
||||
<note>
|
||||
Das Rekonstruktionsfilter entfernt unerwünschte höherfrequente Signalanteile am Ausgang des D/A-Umsetzers.
|
||||
|
||||
Links ist ein Signal vor der Filterung dargestellt. Die deutlich sichtbaren Kanten enthalten hohe Frequenzanteile.
|
||||
|
||||
Nach der Filterung entsteht ein glatterer Signalverlauf, der dem gewünschten analogen Signal näherkommt.
|
||||
|
||||
Je nach Anwendung kann dafür beispielsweise ein Tiefpass- oder Bandpassfilter verwendet werden.
|
||||
</note>
|
||||
|
||||
---
|
||||
|
||||
[question:AF624]
|
||||
|
||||
---
|
||||
|
||||
[question:AF625]
|
||||
@@ -1,22 +0,0 @@
|
||||
## Rekonstruktion durch den D/A-Umsetzer
|
||||
|
||||
* Wandelt digitale Samples zurück in analoge Spannungsstufen
|
||||
* Samples werden in festgelegten zeitlichen Abständen wieder ausgegeben
|
||||
* Vorgang: Rekonstruktion des ursprünglichen Signals
|
||||
|
||||
---
|
||||
|
||||
### Stufenförmiges Signal und Filterung
|
||||
|
||||
* Zeitdiskrete Abtastung führt zu einem stufenförmigen Signal
|
||||
* Unerwünschte höhere Frequenzen (Artefakte) können auftreten
|
||||
* Einsatz eines Rekonstruktionsfilters (Tiefpass- oder Bandpassfilter)
|
||||
* Filter unterdrückt alle Signalanteile oberhalb der halben Abtastrate
|
||||
|
||||
---
|
||||
|
||||
[question:AF624]
|
||||
|
||||
---
|
||||
|
||||
[question:AF625]
|
||||
+2
-9
@@ -852,15 +852,8 @@
|
||||
"video_url": null
|
||||
},
|
||||
{
|
||||
"title": "Anti-Alias-Filter",
|
||||
"ident": "anti_alias_filter",
|
||||
"status": "alpha",
|
||||
"class": "A",
|
||||
"video_url": null
|
||||
},
|
||||
{
|
||||
"title": "Rekonstruktionsfilter",
|
||||
"ident": "rekonstruktionsfilter",
|
||||
"title": "Anti-Alias- und Rekonstruktionsfilter",
|
||||
"ident": "anti_alias_rekonstruktionsfilter",
|
||||
"status": "alpha",
|
||||
"class": "A",
|
||||
"video_url": null
|
||||
|
||||
+2
-9
@@ -1400,15 +1400,8 @@
|
||||
"video_url": null
|
||||
},
|
||||
{
|
||||
"title": "Anti-Alias-Filter",
|
||||
"ident": "anti_alias_filter",
|
||||
"status": "alpha",
|
||||
"class": "A",
|
||||
"video_url": null
|
||||
},
|
||||
{
|
||||
"title": "Rekonstruktionsfilter",
|
||||
"ident": "rekonstruktionsfilter",
|
||||
"title": "Anti-Alias- und Rekonstruktionsfilter",
|
||||
"ident": "anti_alias_rekonstruktionsfilter",
|
||||
"status": "alpha",
|
||||
"class": "A",
|
||||
"video_url": null
|
||||
|
||||
+2
-9
@@ -2041,15 +2041,8 @@
|
||||
"video_url": null
|
||||
},
|
||||
{
|
||||
"title": "Anti-Alias-Filter",
|
||||
"ident": "anti_alias_filter",
|
||||
"status": "alpha",
|
||||
"class": "A",
|
||||
"video_url": null
|
||||
},
|
||||
{
|
||||
"title": "Rekonstruktionsfilter",
|
||||
"ident": "rekonstruktionsfilter",
|
||||
"title": "Anti-Alias- und Rekonstruktionsfilter",
|
||||
"ident": "anti_alias_rekonstruktionsfilter",
|
||||
"status": "alpha",
|
||||
"class": "A",
|
||||
"video_url": null
|
||||
|
||||
Reference in New Issue
Block a user