Difference between revisions of "Theory of Stochastic Signals/Matched Filter"

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{{Header
 
{{Header
|Untermenü=Filterung stochastischer Signale
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|Untermenü=Filtering of Stochastic Signals
 
|Vorherige Seite=Erzeugung vorgegebener AKF-Eigenschaften
 
|Vorherige Seite=Erzeugung vorgegebener AKF-Eigenschaften
 
|Nächste Seite=Wiener–Kolmogorow–Filter
 
|Nächste Seite=Wiener–Kolmogorow–Filter
 
}}
 
}}
==Optimierungskriterium des Matched–Filters==
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==Optimization criterion of the matched filter==
 
<br>
 
<br>
 
{{BlaueBox|TEXT=   
 
{{BlaueBox|TEXT=   
$\text{Definition:}$&nbsp; Das&nbsp; '''Matched-Filter'''&nbsp; – auch ''Korrelationsfilter''&nbsp; genannt dient zum Nachweis der Signalexistenz.  
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$\text{Definition:}$&nbsp; The&nbsp; '''matched filter'''&nbsp; – also called ''correlation filter''&nbsp; – is used to prove the signal existence.
[[File:EN_Sto_T_5_4_S1.png |right|frame| Blockschaltbild des Matched-Filter-Empfängers]]
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[[File:EN_Sto_T_5_4_S1.png |right|frame| Block diagram of the matched-filter receiver]]
*Der&nbsp; '''Matched-Filter-Empfänger'''&nbsp; kann mit größtmöglicher Sicherheit anders ausgedrückt: &nbsp; mit maximalem SNR – entscheiden, ob ein durch additives Rauschen&nbsp; $n(t)$&nbsp; gestörtes impulsförmiges Nutzsignal&nbsp; $g(t)$&nbsp; vorhanden ist oder nicht.
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*The&nbsp; '''matched filter receiver'''&nbsp; can decide with the greatest possible certainty in other words: &nbsp; with maximum SNR – whether or not a pulsed useful signal&nbsp; $g(t)$&nbsp; distrubed by additive noise&nbsp; $n(t)$&nbsp; is present.
  
 
   
 
   
*Zur Herleitung des Matched-Filter-Empfängers wird die skizzierte Anordnung betrachtet. }}
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*To derive the matched-filter receiver, consider the outlined arrangement. }}
  
  
Für die einzelnen Komponenten gelten folgende Voraussetzungen:  
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The following conditions apply to the individual components:
*Der Nutzanteil&nbsp; $g(t)$&nbsp; des Empfangssignals&nbsp; $r(t)=g(t)+n(t)$&nbsp; sei impulsförmig und somit&nbsp; ''energiebegrenzt''.  
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*Let the useful component&nbsp; $g(t)$&nbsp; of the received signal&nbsp; $r(t)=g(t)+n(t)$&nbsp; be pulse-shaped and thus&nbsp; ''energy-limited''.  
*Das heißt: &nbsp; Das Integral über&nbsp; $\big [g(t)\big ]^2$&nbsp; von&nbsp; $–∞$&nbsp; bis&nbsp; $+∞$&nbsp; liefert den endlichen Wert&nbsp; $E_g$.  
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*That means: &nbsp; The integral over&nbsp; $\big [g(t)\big ]^2$&nbsp; from&nbsp; $–∞$&nbsp; to&nbsp; $+∞$&nbsp; yields the finite value&nbsp; $E_g$.  
*Das Störsignal&nbsp; $n(t)$&nbsp; sei&nbsp; ''Weißes Gaußsches Rauschen''&nbsp; mit der Rauschleistungsdichte&nbsp; $N_0$.  
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*Let the noise signal&nbsp; $n(t)$&nbsp; be&nbsp; ''white Gaussian noise''&nbsp; with noise power density&nbsp; $N_0$.  
*Das Filterausgangssignal&nbsp; $d(t)$&nbsp; setzt sich additiv aus zwei Anteilen zusammen.&nbsp; Der Anteil&nbsp; $d_{\rm S}(t)$&nbsp; geht auf das&nbsp; $\rm S$ignal&nbsp; $g(t)$&nbsp; zurück, der Anteil&nbsp; $d_{\rm N}(t)$&nbsp; auf das&nbsp; $\rm N$oise&nbsp; $n(t)$.  
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*The filter output signal&nbsp; $d(t)$&nbsp; is additively composed of two components.&nbsp; The component&nbsp; $d_{\rm S}(t)$&nbsp; is due to the &nbsp; $\rm S$ignal&nbsp; $g(t)$,&nbsp; the component&nbsp; $d_{\rm N}(t)$&nbsp; is due to the &nbsp; $\rm N$oise&nbsp; $n(t)$.  
*Der Empfänger, bestehend aus einem linearen Filter &nbsp;  ⇒ &nbsp;  Frequenzgang&nbsp; $H_{\rm MF}(f)$&nbsp; und dem Entscheider, ist so zu dimensionieren, dass das momentane S/N-Verhältnis am Ausgang maximal wird:  
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*The receiver, consisting of a linear filter &nbsp;  ⇒ &nbsp;  frequency response&nbsp; $H_{\rm MF}(f)$&nbsp; and the decision maker, is to be dimensioned so that the instantaneous S/N ratio at the output is maximized:
 
:$$\rho _d ( {T_{\rm D} } ) = \frac{ {d_{\rm S} ^2 ( {T_{\rm D} } )} }{ {\sigma _d ^2 } }\mathop  = \limits^{\rm{!} }\hspace{0.1cm} {\rm{Maximum} }.$$
 
:$$\rho _d ( {T_{\rm D} } ) = \frac{ {d_{\rm S} ^2 ( {T_{\rm D} } )} }{ {\sigma _d ^2 } }\mathop  = \limits^{\rm{!} }\hspace{0.1cm} {\rm{Maximum} }.$$
*Hierbei bezeichnen &nbsp;$σ_d^2$&nbsp; die&nbsp; ''Varianz''&nbsp; (Leistung) von $d_{\rm N}(t)$ und &nbsp;$T_{\rm D}$&nbsp; den (geeignet gewählten)&nbsp; ''Detektionszeitpunkt.''  
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*Here, &nbsp;$σ_d^2$&nbsp; denote the&nbsp; ''variance''&nbsp; (power) of $d_{\rm N}(t)$ and &nbsp;$T_{\rm D}$&nbsp; denotes the (suitably chosen)&nbsp; ''detection time.''  
  
==Matched-Filter-Optimierung==
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==Matched filter optimization==
 
<br>
 
<br>
Gegeben sei ein energiebegrenztes Nutzsignal&nbsp; $g(t)$&nbsp; mit dem zugehörigen Spektrum&nbsp; $G(f)$.  
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Let be given an energy-limited useful signal&nbsp; $g(t)$&nbsp; with the corresponding spectrum&nbsp; $G(f)$.  
*Damit kann das Filterausgangssignal zum Detektionszeitpunkt&nbsp; $T_{\rm D}$&nbsp; für jedes beliebige Filter mit der Impulsantwort&nbsp; $h(t)$&nbsp; und dem Frequenzgang&nbsp; $H(f) =\mathcal{ F}\{h(t)\}$ wie folgt geschrieben werden&nbsp; (ohne Berücksichtigung des Rauschens &nbsp; ⇒ &nbsp; Index &nbsp;$\rm S$&nbsp; für „Signal”):  
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*Thus, the filter output signal at detection time&nbsp; $T_{\rm D}$&nbsp; for any filter with impulse response&nbsp; $h(t)$&nbsp; and frequency response&nbsp; $H(f) =\mathcal{ F}\{h(t)\}$ can be written as follows&nbsp; (ignoring noise &nbsp; ⇒ &nbsp; index &nbsp;$\rm S$&nbsp; for "signal"):  
 
:$$d_{\rm S} ( {T_{\rm D} } ) = g(t) * h(t) = \int_{ - \infty }^{ + \infty } {G(f) \cdot H(f) \cdot {\rm{e}}^{ {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} 2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f \hspace{0.05cm}\cdot \hspace{0.05cm}T_{\rm D}  }\hspace{0.1cm} {\rm{d}}f} .$$
 
:$$d_{\rm S} ( {T_{\rm D} } ) = g(t) * h(t) = \int_{ - \infty }^{ + \infty } {G(f) \cdot H(f) \cdot {\rm{e}}^{ {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} 2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f \hspace{0.05cm}\cdot \hspace{0.05cm}T_{\rm D}  }\hspace{0.1cm} {\rm{d}}f} .$$
*Der&nbsp; „Rauschanteil”&nbsp; $d_{\rm N}(t)$&nbsp; des Filterausgangssignals&nbsp; (Index &nbsp;$\rm N$&nbsp; für „Noise”) rührt allein vom Weißen Rauschen&nbsp; $n(t)$&nbsp; am Eingang des Empfängers her.&nbsp; Für seine Varianz (Leistung) gilt unabhängig vom Detektionszeitpunkt&nbsp; $T_{\rm D}$:  
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*The&nbsp; "noise component"&nbsp; $d_{\rm N}(t)$&nbsp; of the filter output signal&nbsp; (index &nbsp;$\rm N$&nbsp; for "noise") stems solely from the white noise&nbsp; $n(t)$&nbsp; at the input of the receiver.&nbsp; For its variance (power) applies independently of the detection time&nbsp; $T_{\rm D}$:  
 
:$$\sigma _d ^2  = \frac{ {N_0 } }{2} \cdot \int_{ - \infty }^{ + \infty } {\left| {H(f)} \right|^{\rm{2} }\hspace{0.1cm} {\rm{d} }f} .$$
 
:$$\sigma _d ^2  = \frac{ {N_0 } }{2} \cdot \int_{ - \infty }^{ + \infty } {\left| {H(f)} \right|^{\rm{2} }\hspace{0.1cm} {\rm{d} }f} .$$
*Damit lautet das hier vorliegende Optimierungsproblem:
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*Thus, the optimization problem at hand is:
 
:$$\rho _d ( {T_{\rm D} } ) = \frac{ {\left| {\int_{ - \infty }^{ + \infty } {G(f) \cdot H(f) \cdot {\rm{e} }^{ {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} 2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f \hspace{0.05cm}\cdot \hspace{0.05cm}T_{\rm D}  }\hspace{0.1cm} {\rm{d} }f} } \right|^2 } }{ {N_0 /2 \cdot \int_{ - \infty }^{ + \infty } {\left| {H(f)} \right|^{\rm{2} }\hspace{0.1cm} {\rm{d} }f} } }  \stackrel{!}{=} {\rm{Maximum} }.$$
 
:$$\rho _d ( {T_{\rm D} } ) = \frac{ {\left| {\int_{ - \infty }^{ + \infty } {G(f) \cdot H(f) \cdot {\rm{e} }^{ {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} 2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f \hspace{0.05cm}\cdot \hspace{0.05cm}T_{\rm D}  }\hspace{0.1cm} {\rm{d} }f} } \right|^2 } }{ {N_0 /2 \cdot \int_{ - \infty }^{ + \infty } {\left| {H(f)} \right|^{\rm{2} }\hspace{0.1cm} {\rm{d} }f} } }  \stackrel{!}{=} {\rm{Maximum} }.$$
  
 
{{BlaueBox|TEXT=   
 
{{BlaueBox|TEXT=   
$\text{Hier zunächst ohne Beweis:}$&nbsp; &nbsp; Man kann zeigen, dass dieser Quotient für den folgenden Frequenzgang&nbsp; $H(f)$&nbsp; am größten wird:
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$\text{Here first without proof:}$&nbsp; &nbsp; one can show that this quotient becomes largest for the following frequency response&nbsp; $H(f)$:&nbsp;  
 
:$$H(f) = H_{\rm MF} (f) = K_{\rm MF}  \cdot G^{\star}  (f) \cdot {\rm e}^{- {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} 2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f \hspace{0.05cm}\cdot \hspace{0.05cm}T_{\rm D}  } . $$
 
:$$H(f) = H_{\rm MF} (f) = K_{\rm MF}  \cdot G^{\star}  (f) \cdot {\rm e}^{- {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} 2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f \hspace{0.05cm}\cdot \hspace{0.05cm}T_{\rm D}  } . $$
*Damit erhält man für das Signal&ndash;zu&ndash;Rauschleistungsverhältnis am Matched&ndash;Filter&ndash;Ausgang&nbsp; $($unabhängig von der dimensionsbehafteten Konstante&nbsp; $K_{\rm MF})$:  
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*Thus, for the signal&ndash;to&ndash;noise power ratio at the matched filter output&nbsp; $($independent of the dimensionally constant&nbsp; $K_{\rm MF})$, we obtain:  
 
:$$\rho _d ( {T_{\rm D} } ) = { {2 \cdot E_g } }/{ {N_0 } }.$$
 
:$$\rho _d ( {T_{\rm D} } ) = { {2 \cdot E_g } }/{ {N_0 } }.$$
  
* $E_g$ bezeichnet die Energie des Eingangsimpulses, die man nach dem&nbsp; [https://de.wikipedia.org/wiki/Satz_von_Parseval Satz von Parseval]&nbsp; sowohl im Zeit– als auch im Frequenzbereich berechnen kann:
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* $E_g$ denotes the energy of the input pulse, which can be calculated using&nbsp; [https://en.wikipedia.org/wiki/Parseval%27s_theorem Parseval's theorem]&nbsp; in both the time and frequency domains:
 
:$$E_g  = \int_{ - \infty }^{ + \infty } {g^2 (t)\hspace{0.1cm}{\rm{d} }t}  = \int_{ - \infty }^{ + \infty } {\left \vert {G(f)} \right\vert ^{\rm{2} }\hspace{0.1cm} {\rm d}f} .$$}}
 
:$$E_g  = \int_{ - \infty }^{ + \infty } {g^2 (t)\hspace{0.1cm}{\rm{d} }t}  = \int_{ - \infty }^{ + \infty } {\left \vert {G(f)} \right\vert ^{\rm{2} }\hspace{0.1cm} {\rm d}f} .$$}}
  
  
 
{{GraueBox|TEXT=   
 
{{GraueBox|TEXT=   
$\text{Beispiel 1:}$&nbsp; &nbsp;Ein rechteckförmiger Impuls&nbsp; $g(t)$&nbsp; mit Amplitude&nbsp; $\rm 1\hspace{0.05cm}V$,&nbsp; Dauer&nbsp; $0.5\hspace{0.05cm} \rm ms$&nbsp; und unbekannter Lage soll in einer verrauschten Umgebung aufgefunden werden.  
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$\text{Example 1:}$&nbsp; &nbsp;A rectangular pulse&nbsp; $g(t)$&nbsp; with amplitude&nbsp; $\rm 1\hspace{0.05cm}V$,&nbsp; duration&nbsp; $0.5\hspace{0.05cm} \rm ms$&nbsp; and unknown position is to be found in a noisy environment.
*Somit ist die Impulsenergie&nbsp; $E_g = \rm 5 · 10^{–4} \hspace{0.05cm}V^2s$.  
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*Thus the pulse energy&nbsp; $E_g = \rm 5 · 10^{–4} \hspace{0.05cm}V^2s$.  
*Die Rauschleistungsdichte sei&nbsp; $N_0 = \rm 10^{–6} \hspace{0.05cm}V^2/Hz$.  
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*Let the noise power density be&nbsp; $N_0 = \rm 10^{–6} \hspace{0.05cm}V^2/Hz$.  
  
  
Das beste Ergebnis  &nbsp; ⇒  &nbsp; das&nbsp; '''maximale S/N–Verhältnis'''&nbsp; erzielt man mit dem Matched-Filter:  
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The best result &nbsp; ⇒  &nbsp; the&nbsp; '''maximum S/N ratio'''&nbsp; is obtained with the matched filter:
 
:$$\rho _d ( {T_{\rm D} } ) = \frac{ {2 \cdot E_g } }{ {N_0 } } =
 
:$$\rho _d ( {T_{\rm D} } ) = \frac{ {2 \cdot E_g } }{ {N_0 } } =
 
\frac{ {2 \cdot 5 \cdot 10^{-4}\, {\rm V^2\,s} } }{ {10^{-6}\, {\rm V^2/Hz} } } = 1000  
 
\frac{ {2 \cdot 5 \cdot 10^{-4}\, {\rm V^2\,s} } }{ {10^{-6}\, {\rm V^2/Hz} } } = 1000  
Line 58: Line 58:
  
  
Das oben angegebene Matched–Filter–Kriterium wird nun schrittweise hergeleitet.&nbsp; Wenn Sie daran nicht interessiert sind, so springen Sie bitte zur Fortsetzungsseite&nbsp; [[Theory_of_Stochastic_Signals/Matched-Filter#Interpretation_des_Matched-Filters|Interpretation des Matched–Filters]].  
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The matched filter criterion given above is now derived step by step.&nbsp; If you are not interested in this, please skip to the next page&nbsp; [[Theory_of_Stochastic_Signals/Matched-Filter#Interpretation_des_Matched-Filters|Interpretation of the matched filter]].  
  
 
{{BlaueBox|TEXT=   
 
{{BlaueBox|TEXT=   
$\text{Herleitung des Matched–Filter–Kriteriums:}$&nbsp;  
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$\text{Derivation of the matched filter criterion:}$&nbsp;  
  
$(1)$&nbsp; Die Schwarzsche Ungleichung lautet mit den beiden (im allgemeinen komplexen) Funktionen&nbsp; $A(f)$&nbsp; und&nbsp; $B(f)$:
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$(1)$&nbsp; The Schwarz inequality with the two (generally complex) functions&nbsp; $A(f)$&nbsp; and&nbsp; $B(f)$:
 
:$$\left \vert  {\int_a^b {A(f) \cdot B(f)\hspace{0.1cm}{\rm{d} }f} } \right  \vert ^2  \le \int_a^b {\left \vert {A(f)} \right \vert^{\rm{2} } \hspace{0.1cm}{\rm{d} }f}  \cdot \int_a^b {\left\vert {B(f)} \right \vert^{\rm{2} } \hspace{0.1cm}{\rm{d} }f} .$$
 
:$$\left \vert  {\int_a^b {A(f) \cdot B(f)\hspace{0.1cm}{\rm{d} }f} } \right  \vert ^2  \le \int_a^b {\left \vert {A(f)} \right \vert^{\rm{2} } \hspace{0.1cm}{\rm{d} }f}  \cdot \int_a^b {\left\vert {B(f)} \right \vert^{\rm{2} } \hspace{0.1cm}{\rm{d} }f} .$$
$(2)$&nbsp; Wir wenden nun diese Gleichung auf das Signal&ndash;zu&ndash;Rauschverhältnis an:  
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$(2)$&nbsp; We now apply this equation to the signal&ndash;to&ndash;noise ratio:  
 
:$$\rho _d ( {T_{\rm D} } ) = \frac{ {\left  \vert {\int_{ - \infty }^{ + \infty } {G(f) \cdot H(f) \cdot {\rm{e} }^{ {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} 2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f \hspace{0.05cm}\cdot \hspace{0.05cm}T_{\rm D}  } \hspace{0.1cm}{\rm{d} }f} } \right  \vert^2 } }{ {N_0 /2 \cdot \int_{ - \infty }^{ + \infty } {\left  \vert {H(f)} \right  \vert^{\rm{2} }\hspace{0.1cm} {\rm{d} }f} } }.$$
 
:$$\rho _d ( {T_{\rm D} } ) = \frac{ {\left  \vert {\int_{ - \infty }^{ + \infty } {G(f) \cdot H(f) \cdot {\rm{e} }^{ {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} 2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f \hspace{0.05cm}\cdot \hspace{0.05cm}T_{\rm D}  } \hspace{0.1cm}{\rm{d} }f} } \right  \vert^2 } }{ {N_0 /2 \cdot \int_{ - \infty }^{ + \infty } {\left  \vert {H(f)} \right  \vert^{\rm{2} }\hspace{0.1cm} {\rm{d} }f} } }.$$
$(3)$&nbsp; Mit&nbsp; $A(f) = G(f)$&nbsp; und&nbsp; $B(f) = H(f) · {\rm e}^{ {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} 2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f \hspace{0.05cm}\cdot \hspace{0.05cm}T_{\rm D}  }$&nbsp; ergibt sich somit die folgende Schranke:
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$(3)$&nbsp; Thus, with&nbsp; $A(f) = G(f)$&nbsp; and&nbsp; $B(f) = H(f) · {\rm e}^{ {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} 2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f \hspace{0.05cm}\cdot \hspace{0.05cm}T_{\rm D}  }$&nbsp; the following bound is obtained:
 
:$$\rho_d ( {T_{\rm D} } ) \le \frac{1}{ {N_0 /2} } \cdot \int_{ - \infty }^{ + \infty } {\left \vert  {G(f)} \right \vert^{\rm{2} } }\hspace{0.1cm}{\rm{d} }f .$$
 
:$$\rho_d ( {T_{\rm D} } ) \le \frac{1}{ {N_0 /2} } \cdot \int_{ - \infty }^{ + \infty } {\left \vert  {G(f)} \right \vert^{\rm{2} } }\hspace{0.1cm}{\rm{d} }f .$$
$(4)$&nbsp; Wir setzen für den Filterfrequenzgang nun versuchsweise ein:  
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$(4)$&nbsp; We now tentatively set for the filter frequency response:
 
:$$H(f) = H_{\rm MF} (f) = K_{\rm MF}  \cdot G^{\star}  (f) \cdot {\rm{e} }^{- {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} 2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f \hspace{0.05cm}\cdot \hspace{0.05cm}T_{\rm D}  }.$$
 
:$$H(f) = H_{\rm MF} (f) = K_{\rm MF}  \cdot G^{\star}  (f) \cdot {\rm{e} }^{- {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} 2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f \hspace{0.05cm}\cdot \hspace{0.05cm}T_{\rm D}  }.$$
$(5)$&nbsp; Dann erhält man aus der obigen Gleichung&nbsp; $(2)$&nbsp; folgendes Ergebnis:
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$(5)$&nbsp; Then, from the above equation&nbsp; $(2)$,&nbsp; we obtain the following result:
 
:$$\rho _d ( {T_{\rm D} } ) = \frac{ {\left \vert  K_{\rm MF}\cdot {\int_{ - \infty }^{ + \infty } {\left \vert  {G(f)} \right \vert ^{\rm{2} }\hspace{0.1cm} {\rm{d} }f} } \right \vert ^2 } }{ {N_0 /2 \cdot K_{\rm MF} ^2  \cdot \int_{ - \infty }^{ + \infty } {\left \vert {G(f)} \right \vert ^{\rm{2} }\hspace{0.1cm} {\rm{d} }f} } } = \frac{1}{ {N_0 /2} } \cdot \int_{ - \infty }^{ + \infty } {\left \vert {G(f)} \right \vert ^{\rm{2} }\hspace{0.1cm} {\rm{d} }f} .$$
 
:$$\rho _d ( {T_{\rm D} } ) = \frac{ {\left \vert  K_{\rm MF}\cdot {\int_{ - \infty }^{ + \infty } {\left \vert  {G(f)} \right \vert ^{\rm{2} }\hspace{0.1cm} {\rm{d} }f} } \right \vert ^2 } }{ {N_0 /2 \cdot K_{\rm MF} ^2  \cdot \int_{ - \infty }^{ + \infty } {\left \vert {G(f)} \right \vert ^{\rm{2} }\hspace{0.1cm} {\rm{d} }f} } } = \frac{1}{ {N_0 /2} } \cdot \int_{ - \infty }^{ + \infty } {\left \vert {G(f)} \right \vert ^{\rm{2} }\hspace{0.1cm} {\rm{d} }f} .$$
  
$\text{Das heißt:}$  
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$\text{This means:}$  
*Mit dem Ansatz&nbsp; $(4)$&nbsp; für das Matched&ndash;Filter $H_{\rm MF}(f)$ wird in obiger Abschätzung tatsächlich der maximal mögliche Wert erreicht.  
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*With the approach&nbsp; $(4)$&nbsp; for the matched filter $H_{\rm MF}(f)$, the maximum possible value is indeed obtained in the above estimation.
*Mit keinem anderen Filter&nbsp; $H(f) ≠ H_{\rm MF}(f)$&nbsp; kann man ein höheres Signal&ndash;zu&ndash;Rauschleistungsverhältnis erzielen.
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*No other filter&nbsp; $H(f) ≠ H_{\rm MF}(f)$&nbsp; can achieve a higher signal&ndash;to&ndash;noise power ratio.
*Das Matched–Filter ist in Bezug auf das ihm zugrunde gelegte Maximierungskriterium optimal.
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*The matched filter is optimal with respect to the maximization criterion on which it is based.
 
<div align="right">'''q.e.d.'''</div>
 
<div align="right">'''q.e.d.'''</div>
 
}}
 
}}
  
==Interpretation des Matched-Filters==
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==Interpretation of the matched filter==
 
<br>
 
<br>
Auf der letzten Seite wurde der Frequenzgang des Matched-Filters wie folgt hergeleitet:  
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On the last page, the frequency response of the matched filter was derived as follows:
 
:$$H_{\rm MF} (f) = K_{\rm MF}  \cdot G^{\star}  (f) \cdot {\rm{e} }^{- {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} 2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f \hspace{0.05cm}\cdot \hspace{0.05cm}T_{\rm D}  } .$$
 
:$$H_{\rm MF} (f) = K_{\rm MF}  \cdot G^{\star}  (f) \cdot {\rm{e} }^{- {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} 2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f \hspace{0.05cm}\cdot \hspace{0.05cm}T_{\rm D}  } .$$
Durch&nbsp; [[Signal_Representation/Fourier_Transform_and_Its_Inverse#Das_zweite_Fourierintegral|Fourierrücktransformation]]&nbsp; erhält man die dazugehörige Impulsantwort:  
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By&nbsp; [[Signal_Representation/Fourier_Transform_and_its_Inverse#The_second_Fourier_integral|Fourier inverse transformation]]&nbsp; the corresponding impulse response is obtained:
 
:$$h_{\rm MF} (t) = K_{\rm MF}  \cdot g(T_{\rm D}  - t).$$
 
:$$h_{\rm MF} (t) = K_{\rm MF}  \cdot g(T_{\rm D}  - t).$$
  
Diese beiden Funktionen lassen sich wie folgt interpretieren:  
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These two functions can be interpreted as follows:
*Das&nbsp; ''Matched-Filter''&nbsp; ist durch den Term &nbsp;$G^{\star}(f)$&nbsp; an das Spektrum des aufzufindenden Impulses &nbsp;$g(t)$&nbsp; angepasst daher sein Name (englisch: ''to match'' ≡ anpassen).  
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*The&nbsp; ''matched filter''&nbsp; is matched by the term &nbsp;$G^{\star}(f)$&nbsp; to the spectrum of the pulse &nbsp;$g(t)$&nbsp; which is to be found hence its name.  
*Die&nbsp; ''Konstante'' &nbsp;$K_{\rm MF}$&nbsp; ist aus Dimensionsgründen notwendig.  
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*The&nbsp; ''constant'' &nbsp;$K_{\rm MF}$&nbsp; is necessary for dimensional reasons.
*Ist&nbsp; $g(t)$&nbsp; ein Spannungsimpuls, so hat diese Konstante die Einheit „Hz/V”.&nbsp; Der Frequenzgang ist somit dimensionslos.  
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*If&nbsp; $g(t)$&nbsp; is a voltage pulse, this constant has the unit "Hz/V".&nbsp; The frequency response is therefore dimensionless.
*Die&nbsp; ''Impulsantwort'' &nbsp;$h_{\rm MF}(t)$&nbsp; ergibt sich aus dem Nutzsignal &nbsp;$g(t)$&nbsp; durch Spiegelung &nbsp; ⇒ &nbsp; aus $g(t)$ wird $g(–t)$  &nbsp; &nbsp; sowie einer Verschiebung um&nbsp; $T_{\rm D}$&nbsp; nach rechts.  
+
*The&nbsp; ''impulse response'' &nbsp;$h_{\rm MF}(t)$&nbsp; results from the useful signal &nbsp;$g(t)$&nbsp; by mirroring &nbsp; ⇒ &nbsp; from $g(t)$ becomes $g(–t)$  &nbsp; &nbsp; as well as a shift by&nbsp; $T_{\rm D}$&nbsp; to the right.
*Der&nbsp; ''früheste Detektionszeitpunkt'' &nbsp;$T_{\rm D}$&nbsp; folgt für realisierbare Systeme aus der Bedingung&nbsp; $h_{\rm MF}(t < 0)\equiv 0$ &nbsp; $($„Kausalität”,&nbsp; siehe Buch [[Lineare_zeitinvariante_Systeme|Lineare zeitinvariante Systeme]]$)$.  
+
*The&nbsp; ''earliest detection time'' &nbsp;$T_{\rm D}$&nbsp; follows for realizable systems from the condition&nbsp; $h_{\rm MF}(t < 0)\equiv 0$ &nbsp; $($"causality",&nbsp; see book [[Linear_and_Time_Invariant_Systems|Linear and Time-Invariant Systems]]$)$.  
*Der&nbsp; ''Nutzanteil'' &nbsp;$d_{\rm S} (t)$&nbsp; des Filterausgangssignals ist formgleich mit der&nbsp; [[Digital_Signal_Transmission/Grundlagen_der_codierten_Übertragung#AKF.E2.80.93Berechnung_eines_Digitalsignals|Energie-AKF]] &nbsp; $\varphi^{^{\bullet} }_{g} (t )$&nbsp; und gegenüber dieser um &nbsp;$T_{\rm D}$&nbsp; verschoben. Es gilt:  
+
*The&nbsp; ''useful component'' &nbsp;$d_{\rm S} (t)$&nbsp; of the filter output signal is equal in shape to the&nbsp; [[Digital_Signal_Transmission/Grundlagen_der_codierten_Übertragung#AKF.E2.80.93Berechnung_eines_Digitalsignals|energy ACF]] &nbsp; $\varphi^{^{\bullet} }_{g} (t )$&nbsp; and shifted with respect to it by &nbsp;$T_{\rm D}$.&nbsp; It holds:  
 
:$$d_{\rm S} (t) = g(t) * h_{\rm MF} (t) = K_{\rm MF}  \cdot g(t) * g(T_{\rm D}  - t) = K_{\rm MF}  \cdot \varphi^{^{\bullet} }_{g} (t - T_{\rm D} ).$$
 
:$$d_{\rm S} (t) = g(t) * h_{\rm MF} (t) = K_{\rm MF}  \cdot g(t) * g(T_{\rm D}  - t) = K_{\rm MF}  \cdot \varphi^{^{\bullet} }_{g} (t - T_{\rm D} ).$$
  
 
{{BlaueBox|TEXT=   
 
{{BlaueBox|TEXT=   
$\text{Bitte beachten Sie:}$&nbsp;  
+
$\text{Please note:}$&nbsp;  
Bei einem energiebegrenzten Signal&nbsp; $g(t)$&nbsp; kann man nur die&nbsp; ''Energie–AKF''&nbsp; angeben:
+
For an energy-limited signal&nbsp; $g(t)$,&nbsp; one can only specify the&nbsp; ''energy ACF'':&nbsp;  
 
:$$\varphi^{^{\bullet} }_g (\tau ) = \int_{ - \infty }^{ + \infty } {g(t) \cdot g(t + \tau )\,{\rm{d} }t} .$$
 
:$$\varphi^{^{\bullet} }_g (\tau ) = \int_{ - \infty }^{ + \infty } {g(t) \cdot g(t + \tau )\,{\rm{d} }t} .$$
Gegenüber der AKF-Definition eines leistungsbegrenzten Signals&nbsp; $x(t)$, nämlich
+
Compared to the ACF definition of a power-limited signal&nbsp; $x(t)$, viz.
 
:$$\varphi _x (\tau ) = \mathop {\lim }_{T_{\rm M}  \to \infty } \frac{1}{ {T_{\rm M} } }\int_{ - T_{\rm M} /2}^{+T_{\rm M} /2} {x(t) \cdot x(t + \tau )\hspace{0.1cm}\,{\rm{d} }t} ,$$
 
:$$\varphi _x (\tau ) = \mathop {\lim }_{T_{\rm M}  \to \infty } \frac{1}{ {T_{\rm M} } }\int_{ - T_{\rm M} /2}^{+T_{\rm M} /2} {x(t) \cdot x(t + \tau )\hspace{0.1cm}\,{\rm{d} }t} ,$$
wird bei der Berechnung der Energie-AKF auf die Division durch die Messdauer&nbsp; $T_{\rm M}$&nbsp; sowie auf den Grenzübergang&nbsp; $T_{\rm M} → ∞$&nbsp; verzichtet.}}  
+
the division by the measurement duration&nbsp; $T_{\rm M}$&nbsp; and the boundary transition&nbsp; $T_{\rm M} → ∞$&nbsp; are omitted in the calculation of the energy ACF.}}  
  
  
 
{{GraueBox|TEXT=   
 
{{GraueBox|TEXT=   
$\text{Beispiel 2:}$&nbsp; Wir gehen davon aus, dass der Rechteckimpuls zwischen &nbsp;$\rm 2\hspace{0.08cm}ms$&nbsp; und &nbsp;$\rm 2.5\hspace{0.08cm}ms$&nbsp; liegt und der Detektionszeitpunkt &nbsp;$T_{\rm D} =\rm 2\hspace{0.08cm}ms$&nbsp; gewünscht wird.  
+
$\text{Example 2:}$&nbsp; We assume that the square pulse is between &nbsp;$\rm 2\hspace{0.08cm}ms$&nbsp; and &nbsp;$\rm 2.5\hspace{0.08cm}ms$&nbsp; and the detection time &nbsp;$T_{\rm D} =\rm 2\hspace{0.08cm}ms$&nbsp; is desired.
  
Unter diesen Voraussetzungen gilt:  
+
Under these conditions:  
*Die Matched–Filter–Impulsantwort &nbsp;$h_{\rm MF}(t)$&nbsp; muss im Bereich von &nbsp;$t_1 (= 4 - 2.5) =\rm 1.5\hspace{0.08cm}ms$&nbsp; bis&nbsp; $t_2 (= 4 - 2) =\rm 2\hspace{0.08cm}ms$&nbsp; konstant sein.
+
*The matched filter impulse response &nbsp;$h_{\rm MF}(t)$&nbsp; must be constant in the range from &nbsp;$t_1 (= 4 - 2.5) =\rm 1.5\hspace{0.08cm}ms$&nbsp; to&nbsp; $t_2 (= 4 - 2) =\rm 2\hspace{0.08cm}ms$.&nbsp;  
*Für &nbsp;$t < t_1$&nbsp; sowie für &nbsp;$t > t_2$&nbsp; darf sie keine Anteile besitzen.  
+
*For &nbsp;$t < t_1$&nbsp; as well as for &nbsp;$t > t_2$&nbsp; it must not have any components.
*Der Betragsfrequenzgang &nbsp;$\vert H_{\rm MF}(f)\vert$&nbsp; ist hier&nbsp; $\rm si$–förmig.  
+
*The magnitude frequency response &nbsp;$\vert H_{\rm MF}(f)\vert$&nbsp; is&nbsp; $\rm si$–shaped here.  
*Die Höhe der Impulsantwort &nbsp;$h_{\rm MF}(t)$&nbsp; spielt für das S/N–Verhältnis keine Rolle, da dieses unabhängig von &nbsp;$K_{\rm MF}$&nbsp; ist.}}
+
*The magnitude of the impulse response &nbsp;$h_{\rm MF}(t)$&nbsp; is not important for the S/N ratio, because it is independent of &nbsp;$K_{\rm MF}$.&nbsp;}}
  
  
==Verallgemeinertes Matched-Filter für den Fall farbiger Störungen==
+
==Generalized matched filter for the case of colored disturbances==
 
<br>
 
<br>
Bei den Herleitungen dieses Abschnittes wurde bisher stets von Weißem Rauschen ausgegangen.&nbsp; Nun soll die folgende Frage geklärt werden:  
+
In the derivations of this section, white noise has always been assumed so far.&nbsp; Now the following question shall be clarified:
  
Wie ist das Empfangsfilter&nbsp; $H(f) = H_{\rm MF}(f)$&nbsp; '''bei farbiger Störung'''&nbsp; $n(t)$&nbsp; zu gestalten, damit das Signal&nbsp;zu&nbsp;Rauschleistungsverhältnis maximal wird?  
+
How should the receive filter&nbsp; $H(f) = H_{\rm MF}(f)$&nbsp; be designed in the presence of '''colored interference'''&nbsp; $n(t)$&nbsp; so that the signal&nbsp;to&nbsp;noise power ratio is maximized?
  
 
{{BlaueBox|TEXT=   
 
{{BlaueBox|TEXT=   
$\text{Zur Erläuterung einiger Begrifflichkeiten:}$&nbsp; Der Begriff „Störung” ist etwas allgemeiner als „Rauschen”.  
+
$\text{To explain some terminology:}$&nbsp; The term "interference" is somewhat more general than "noise."
*Vielmehr ist Rauschen eine Teilmenge aller Störungen, zu denen zum Beispiel auch das Nebensprechen von benachbarten Leitungen zählt.  
+
*Rather, noise is a subset of all interference, which includes, for example, crosstalk from adjacent lines.
*Wir sprechen nur dann von (weißem) Rauschen&nbsp; $n(t)$, wenn das Leistungsdichtespektrum&nbsp; ${\it Φ}_n(f)$&nbsp; für alle Frequenzen gleich ist.  
+
*We speak of (white) noise&nbsp; $n(t)$ only if the power-spectral density&nbsp; ${\it Φ}_n(f)$&nbsp; is the same for all frequencies.
*Ist dies nicht erfüllt, so bezeichnen wir&nbsp; $n(t)$&nbsp; als farbige Störung.}}  
+
*If this is not satisfied, we refer to&nbsp; $n(t)$&nbsp; as colored noise.}}  
  
  
Die obere Grafik zeigt das Blockschaltbild zur Herleitung des Matched–Filters&nbsp; $H_{\rm MF}(f)$&nbsp; bei farbiger Störung&nbsp; $n(t)$, gekennzeichnet durch das Leistungsdichtespektrum&nbsp; ${\it Φ}_n(f) ≠\text{ const}$.&nbsp; Alle weiteren bisher für diesen Abschnitt genannten Voraussetzungen gelten weiterhin.  
+
The top diagram shows the block diagram for deriving the matched filter&nbsp; $H_{\rm MF}(f)$&nbsp; in the presence of colored interference&nbsp; $n(t)$, denoted by the power-spectral density&nbsp; ${\it Φ}_n(f) ≠\text{ const}$.&nbsp; All other conditions stated so far for this section still apply.
  
[[File:EN_Sto_T_5_4_S4.png |center|frame| Zum Matched-Filter bei farbiger Störung]]   
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[[File:EN_Sto_T_5_4_S4.png |center|frame| Matched filter with colored interference]]   
Zum modifizierten Modell gemäß der unteren Grafik ist anzumerken:
+
Regarding the modified model according to the diagram below, note:
*Das farbige Störsignal&nbsp; $n(t)$&nbsp; mit dem Leistungsdichtespektrum&nbsp; ${\it Φ}_n(f)$&nbsp; kann man zumindest gedanklich durch eine „weiße” Rauschquelle&nbsp; $n_{\rm WR}(t)$&nbsp; mit der konstanten (zweiseitigen) Rauschleistungsdichte&nbsp; $N_0/2$&nbsp; und ein Formfilter mit dem Frequenzgang&nbsp; $H_{\rm N}(f)$&nbsp; modellieren:
+
*The colored interference signal&nbsp; $n(t)$&nbsp; with the power-spectral density&nbsp; ${\it Φ}_n(f)$&nbsp; can be modeled at least mentally by a "white" noise source&nbsp; $n_{\rm WR}(t)$&nbsp; with the constant (two-sided) noise power density&nbsp; $N_0/2$&nbsp; and a shape filter with the frequency response&nbsp; $H_{\rm N}(f)$:&nbsp;  
 
:$${\it{\Phi} }_n \left( f \right) = { {N_{\rm 0} } }/{\rm 2} \cdot \left| {H_{\rm N} \left( f \right)} \right|^{\rm 2} .$$
 
:$${\it{\Phi} }_n \left( f \right) = { {N_{\rm 0} } }/{\rm 2} \cdot \left| {H_{\rm N} \left( f \right)} \right|^{\rm 2} .$$
  
*Da Realisierungsaspekte hier nicht betrachtet werden, wird&nbsp; $H_{\rm N}(f)$&nbsp; (stark vereinfachend) als reell angenommen.&nbsp; Der Phasengang von&nbsp; $H_{\rm N}(f)$&nbsp; spielt für das Folgende keine Rolle.&nbsp; In dieser Darstellung ist zudem das Formfilter&nbsp; $H_{\rm N}(f)$&nbsp; auf die rechte Seite der Störaddition verschoben.&nbsp; Um ein auch bezüglich des Nutzsignals&nbsp; $d_{\rm S}(t)$&nbsp; äquivalentes Modell zu erhalten, wird das Formfilter im Nutzsignalzweig durch das inverse Filter&nbsp; $H_{\rm N}(f)^{–1}$&nbsp; kompensiert.
+
*Since implementation aspects are not considered here, &nbsp; $H_{\rm N}(f)$&nbsp; is assumed to be real (for simplicity).&nbsp; The phase response of&nbsp; $H_{\rm N}(f)$&nbsp; is not important for what follows.&nbsp; In this representation the shape filter&nbsp; $H_{\rm N}(f)$&nbsp; is shifted to the right side of the spurious addition.&nbsp; To obtain a model which is also equivalent with respect to the useful signal&nbsp; $d_{\rm S}(t)$,&nbsp; the shape filter in the useful signal branch is compensated by the inverse filter&nbsp; $H_{\rm N}(f)^{–1}$.&nbsp;
  
  
Anhand dieses modifizierten Modells wird nun das Matched&ndash;Filter für den Fall farbiger Störungen hergeleitet.&nbsp; Besitzt&nbsp; $H_{\rm N}(f)$&nbsp; keine Nullstelle, was für das Folgende vorausgesetzt werden soll, so ist diese Anordnungen mit dem obigen Blockschaltbild identisch.  
+
Using this modified model, the matched filter is now derived for the case of colored interference.&nbsp; If&nbsp; $H_{\rm N}(f)$&nbsp; has no zero, which shall be assumed for the following, this arrangement is identical to the block diagram above.
  
An der Störadditionsstelle liegt nun weißes Rauschen&nbsp; $n_{\rm WR}(t)$&nbsp; an.&nbsp; Die Herleitung der&nbsp; [[Theory_of_Stochastic_Signals/Matched-Filter#Matched-Filter-Optimierung|Matched–Filter–Optimierung bei weißem Rauschen]]&nbsp; lässt sich in einfacher Weise auf das aktuelle Problem anpassen, wenn man Folgendes berücksichtigt:  
+
White noise&nbsp; $n_{\rm WR}(t)$&nbsp; is now present at the interference addition point.&nbsp; The derivation of the&nbsp; [[Theory_of_Stochastic_Signals/Matched-Filter#Matched-Filter-Optimierung|matched filter optimization in the presence of white noise]]&nbsp; can be easily adapted to the current problem by considering the following:
*Anstelle des tatsächlichen Nutzsignals&nbsp; $g(t)$&nbsp; ist das Signal&nbsp; $g_{\rm WR}(t)$&nbsp; vor der Störaddition zu berücksichtigen.  
+
*Instead of the actual useful signal&nbsp; $g(t)$,&nbsp; consider the signal&nbsp; $g_{\rm WR}(t)$&nbsp; before the interference addition.
*Die dazugehörige Spektralfunktion lautet: &nbsp; $G_{\rm WR}(f) = G(f)/H_{\rm N}(f)$.  
+
*The corresponding spectral function is: &nbsp; $G_{\rm WR}(f) = G(f)/H_{\rm N}(f)$.  
*Anstelle von&nbsp; $H_{\rm MF}(f)$&nbsp; ist nun der resultierende Frequenzgang&nbsp; ${H_{\rm MF} }' (f) = H_{\rm N}(f) · H_{\rm MF}$&nbsp; rechts von der Störadditionsstelle einzusetzen.  
+
*Instead of&nbsp; $H_{\rm MF}(f)$,&nbsp; the resulting frequency response&nbsp; ${H_{\rm MF} }' (f) = H_{\rm N}(f) · H_{\rm MF}$&nbsp; is now to be substituted to the right of the interference addition point.
  
  
 
{{BlaueBox|TEXT=   
 
{{BlaueBox|TEXT=   
$\text{Fazit:}$
+
$\text{Conclusion:}$
  
 
'''(1)''' &nbsp;  
 
'''(1)''' &nbsp;  
Für das&nbsp; '''Matched-Filter bei farbigen Störungen'''&nbsp; ergibt sich:
+
For the&nbsp; '''matched filter in the presence of colored interference'''&nbsp; we get:
 
:$${H_{\rm MF} }\hspace{0.01cm}' (f)  = H_{\rm N} (f) \cdot H_{\rm MF} (f) = K_{\rm MF}  \cdot G_{\rm WR} ^ {\star}  (f) \cdot {\rm{e} }^{- {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} 2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f \hspace{0.05cm}\cdot \hspace{0.05cm}T_{\rm D}  } \hspace{0.3cm}\Rightarrow \hspace{0.3cm}H_{\rm MF} (f) = K_{\rm MF} \cdot  \frac{ {G^{\star}  (f)} }{ {\left\vert {H_{\rm N} (f)} \right\vert^2 } } \cdot {\rm{e} }^{- {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} 2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f \hspace{0.05cm}\cdot \hspace{0.05cm}T_{\rm D}  } .$$
 
:$${H_{\rm MF} }\hspace{0.01cm}' (f)  = H_{\rm N} (f) \cdot H_{\rm MF} (f) = K_{\rm MF}  \cdot G_{\rm WR} ^ {\star}  (f) \cdot {\rm{e} }^{- {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} 2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f \hspace{0.05cm}\cdot \hspace{0.05cm}T_{\rm D}  } \hspace{0.3cm}\Rightarrow \hspace{0.3cm}H_{\rm MF} (f) = K_{\rm MF} \cdot  \frac{ {G^{\star}  (f)} }{ {\left\vert {H_{\rm N} (f)} \right\vert^2 } } \cdot {\rm{e} }^{- {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} 2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f \hspace{0.05cm}\cdot \hspace{0.05cm}T_{\rm D}  } .$$
'''(2)''' &nbsp; Das&nbsp; '''Signal-zu-Störleistungsverhältnis'''&nbsp; vor dem Entscheider ist somit maximal:  
+
'''(2)''' &nbsp; Thus, the&nbsp; '''signal-to-interference power ratio'''&nbsp; before the decider is maximum:
 
:$$\rho _{d,\ \max } ( {T_{\rm D} } ) = \frac{1}{ {N_0 /2} }\int_{ - \infty }^{ + \infty } {\left\vert{G_{\rm WR} (f)} \right\vert^2 }\, {\rm{d} }f = \int_{ - \infty }^{ + \infty } \frac{\left \vert  G(f) \right\vert^2 }{ {\it{\Phi _n {\rm (f)} } } } \,{\rm{d} }f.$$
 
:$$\rho _{d,\ \max } ( {T_{\rm D} } ) = \frac{1}{ {N_0 /2} }\int_{ - \infty }^{ + \infty } {\left\vert{G_{\rm WR} (f)} \right\vert^2 }\, {\rm{d} }f = \int_{ - \infty }^{ + \infty } \frac{\left \vert  G(f) \right\vert^2 }{ {\it{\Phi _n {\rm (f)} } } } \,{\rm{d} }f.$$
'''(3)''' &nbsp; Der Fall „Weißes Rauschen” ist in dieser allgemeineren Gleichung für&nbsp; ${\it Φ}_n(f) = N_0/2$&nbsp; mitenthalten.
+
'''(3)''' &nbsp; The "white noise" case is included in this more general equation for&nbsp; ${\it Φ}_n(f) = N_0/2$.&nbsp;  
  
'''(4)''' &nbsp; Alle hier angegebenen Gleichungen führen bei farbiger Störung allerdings nur dann zu sinnvollen, auch für die Praxis verwertbaren Ergebnissen, wenn das Energiespektrum&nbsp; $\vert G(f)\vert ^2$&nbsp; des Nutzsignals asymptotisch schneller abklingt als das Störleistungsdichtespektrum&nbsp; ${\it Φ}_n(f)$.}}
+
'''(4)''' &nbsp; However, all equations given here lead to meaningful results, which can also be used in practice, in case of colored interference only if the energy spectrum&nbsp; $\vert G(f)\vert ^2$&nbsp; of the useful signal decays asymptotically faster than the interference power density spectrum&nbsp; ${\it Φ}_n(f)$.}}
  
==Aufgaben zum Kapitel==
+
==Exercise for the chapter==
 
<br>
 
<br>
[[Aufgaben:5.7 Rechteck-Matched-Filter|Aufgabe 5.7: Rechteck-Matched-Filter]]
+
[[Aufgaben:Exercise_5.7:_Rectangular_Matched_Filter|Exercise 5.7: Rectangular Matched Filter]]
  
[[Aufgaben:5.7Z Matched-Filter - alles gaußisch|Aufgabe 5.7Z: Matched-Filter - alles gaußisch]]
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[[Aufgaben:Exercise_5.7Z:_Matched_Filter_-_All_Gaussian|Exercise 5.7Z: Matched Filter - All Gaussian]]
  
[[Aufgaben:5.8 Matched-Filter für farbige Störung|Aufgabe 5.8: Matched-Filter für farbige Störung]]
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[[Aufgaben:Exercise_5.8:_Matched_Filter_for_Colored_Noise|Exercise 5.8: Matched Filter for Colored Noise]]
  
[[Aufgaben:5.8Z Matched-Filter bei Rechteck-LDS|Aufgabe 5.8Z: Matched-Filter bei Rechteck-LDS]]
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[[Aufgaben:Exercise_5.8Z:_Matched_Filter_for_Rectangular_PSD|Exercise 5.8Z: Matched Filter for Rectangular PSD]]
  
  
 
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Revision as of 11:20, 26 January 2022

Optimization criterion of the matched filter


$\text{Definition:}$  The  matched filter  – also called correlation filter  – is used to prove the signal existence.

Block diagram of the matched-filter receiver
  • The  matched filter receiver  can decide with the greatest possible certainty – in other words:   with maximum SNR – whether or not a pulsed useful signal  $g(t)$  distrubed by additive noise  $n(t)$  is present.


  • To derive the matched-filter receiver, consider the outlined arrangement.


The following conditions apply to the individual components:

  • Let the useful component  $g(t)$  of the received signal  $r(t)=g(t)+n(t)$  be pulse-shaped and thus  energy-limited.
  • That means:   The integral over  $\big [g(t)\big ]^2$  from  $–∞$  to  $+∞$  yields the finite value  $E_g$.
  • Let the noise signal  $n(t)$  be  white Gaussian noise  with noise power density  $N_0$.
  • The filter output signal  $d(t)$  is additively composed of two components.  The component  $d_{\rm S}(t)$  is due to the   $\rm S$ignal  $g(t)$,  the component  $d_{\rm N}(t)$  is due to the   $\rm N$oise  $n(t)$.
  • The receiver, consisting of a linear filter   ⇒   frequency response  $H_{\rm MF}(f)$  and the decision maker, is to be dimensioned so that the instantaneous S/N ratio at the output is maximized:
$$\rho _d ( {T_{\rm D} } ) = \frac{ {d_{\rm S} ^2 ( {T_{\rm D} } )} }{ {\sigma _d ^2 } }\mathop = \limits^{\rm{!} }\hspace{0.1cm} {\rm{Maximum} }.$$
  • Here,  $σ_d^2$  denote the  variance  (power) of $d_{\rm N}(t)$ and  $T_{\rm D}$  denotes the (suitably chosen)  detection time.

Matched filter optimization


Let be given an energy-limited useful signal  $g(t)$  with the corresponding spectrum  $G(f)$.

  • Thus, the filter output signal at detection time  $T_{\rm D}$  for any filter with impulse response  $h(t)$  and frequency response  $H(f) =\mathcal{ F}\{h(t)\}$ can be written as follows  (ignoring noise   ⇒   index  $\rm S$  for "signal"):
$$d_{\rm S} ( {T_{\rm D} } ) = g(t) * h(t) = \int_{ - \infty }^{ + \infty } {G(f) \cdot H(f) \cdot {\rm{e}}^{ {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} 2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f \hspace{0.05cm}\cdot \hspace{0.05cm}T_{\rm D} }\hspace{0.1cm} {\rm{d}}f} .$$
  • The  "noise component"  $d_{\rm N}(t)$  of the filter output signal  (index  $\rm N$  for "noise") stems solely from the white noise  $n(t)$  at the input of the receiver.  For its variance (power) applies independently of the detection time  $T_{\rm D}$:
$$\sigma _d ^2 = \frac{ {N_0 } }{2} \cdot \int_{ - \infty }^{ + \infty } {\left| {H(f)} \right|^{\rm{2} }\hspace{0.1cm} {\rm{d} }f} .$$
  • Thus, the optimization problem at hand is:
$$\rho _d ( {T_{\rm D} } ) = \frac{ {\left| {\int_{ - \infty }^{ + \infty } {G(f) \cdot H(f) \cdot {\rm{e} }^{ {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} 2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f \hspace{0.05cm}\cdot \hspace{0.05cm}T_{\rm D} }\hspace{0.1cm} {\rm{d} }f} } \right|^2 } }{ {N_0 /2 \cdot \int_{ - \infty }^{ + \infty } {\left| {H(f)} \right|^{\rm{2} }\hspace{0.1cm} {\rm{d} }f} } } \stackrel{!}{=} {\rm{Maximum} }.$$

$\text{Here first without proof:}$    one can show that this quotient becomes largest for the following frequency response  $H(f)$: 

$$H(f) = H_{\rm MF} (f) = K_{\rm MF} \cdot G^{\star} (f) \cdot {\rm e}^{- {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} 2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f \hspace{0.05cm}\cdot \hspace{0.05cm}T_{\rm D} } . $$
  • Thus, for the signal–to–noise power ratio at the matched filter output  $($independent of the dimensionally constant  $K_{\rm MF})$, we obtain:
$$\rho _d ( {T_{\rm D} } ) = { {2 \cdot E_g } }/{ {N_0 } }.$$
  • $E_g$ denotes the energy of the input pulse, which can be calculated using  Parseval's theorem  in both the time and frequency domains:
$$E_g = \int_{ - \infty }^{ + \infty } {g^2 (t)\hspace{0.1cm}{\rm{d} }t} = \int_{ - \infty }^{ + \infty } {\left \vert {G(f)} \right\vert ^{\rm{2} }\hspace{0.1cm} {\rm d}f} .$$


$\text{Example 1:}$   A rectangular pulse  $g(t)$  with amplitude  $\rm 1\hspace{0.05cm}V$,  duration  $0.5\hspace{0.05cm} \rm ms$  and unknown position is to be found in a noisy environment.

  • Thus the pulse energy  $E_g = \rm 5 · 10^{–4} \hspace{0.05cm}V^2s$.
  • Let the noise power density be  $N_0 = \rm 10^{–6} \hspace{0.05cm}V^2/Hz$.


The best result   ⇒   the  maximum S/N ratio  is obtained with the matched filter:

$$\rho _d ( {T_{\rm D} } ) = \frac{ {2 \cdot E_g } }{ {N_0 } } = \frac{ {2 \cdot 5 \cdot 10^{-4}\, {\rm V^2\,s} } }{ {10^{-6}\, {\rm V^2/Hz} } } = 1000 \hspace{0.3cm}\Rightarrow\hspace{0.3cm} 10 \cdot {\rm lg}\hspace{0.15cm}\rho _d ( {T_{\rm D} } ) = 30\,{\rm dB}.$$


The matched filter criterion given above is now derived step by step.  If you are not interested in this, please skip to the next page  Interpretation of the matched filter.

$\text{Derivation of the matched filter criterion:}$ 

$(1)$  The Schwarz inequality with the two (generally complex) functions  $A(f)$  and  $B(f)$:

$$\left \vert {\int_a^b {A(f) \cdot B(f)\hspace{0.1cm}{\rm{d} }f} } \right \vert ^2 \le \int_a^b {\left \vert {A(f)} \right \vert^{\rm{2} } \hspace{0.1cm}{\rm{d} }f} \cdot \int_a^b {\left\vert {B(f)} \right \vert^{\rm{2} } \hspace{0.1cm}{\rm{d} }f} .$$

$(2)$  We now apply this equation to the signal–to–noise ratio:

$$\rho _d ( {T_{\rm D} } ) = \frac{ {\left \vert {\int_{ - \infty }^{ + \infty } {G(f) \cdot H(f) \cdot {\rm{e} }^{ {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} 2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f \hspace{0.05cm}\cdot \hspace{0.05cm}T_{\rm D} } \hspace{0.1cm}{\rm{d} }f} } \right \vert^2 } }{ {N_0 /2 \cdot \int_{ - \infty }^{ + \infty } {\left \vert {H(f)} \right \vert^{\rm{2} }\hspace{0.1cm} {\rm{d} }f} } }.$$

$(3)$  Thus, with  $A(f) = G(f)$  and  $B(f) = H(f) · {\rm e}^{ {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} 2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f \hspace{0.05cm}\cdot \hspace{0.05cm}T_{\rm D} }$  the following bound is obtained:

$$\rho_d ( {T_{\rm D} } ) \le \frac{1}{ {N_0 /2} } \cdot \int_{ - \infty }^{ + \infty } {\left \vert {G(f)} \right \vert^{\rm{2} } }\hspace{0.1cm}{\rm{d} }f .$$

$(4)$  We now tentatively set for the filter frequency response:

$$H(f) = H_{\rm MF} (f) = K_{\rm MF} \cdot G^{\star} (f) \cdot {\rm{e} }^{- {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} 2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f \hspace{0.05cm}\cdot \hspace{0.05cm}T_{\rm D} }.$$

$(5)$  Then, from the above equation  $(2)$,  we obtain the following result:

$$\rho _d ( {T_{\rm D} } ) = \frac{ {\left \vert K_{\rm MF}\cdot {\int_{ - \infty }^{ + \infty } {\left \vert {G(f)} \right \vert ^{\rm{2} }\hspace{0.1cm} {\rm{d} }f} } \right \vert ^2 } }{ {N_0 /2 \cdot K_{\rm MF} ^2 \cdot \int_{ - \infty }^{ + \infty } {\left \vert {G(f)} \right \vert ^{\rm{2} }\hspace{0.1cm} {\rm{d} }f} } } = \frac{1}{ {N_0 /2} } \cdot \int_{ - \infty }^{ + \infty } {\left \vert {G(f)} \right \vert ^{\rm{2} }\hspace{0.1cm} {\rm{d} }f} .$$

$\text{This means:}$

  • With the approach  $(4)$  for the matched filter $H_{\rm MF}(f)$, the maximum possible value is indeed obtained in the above estimation.
  • No other filter  $H(f) ≠ H_{\rm MF}(f)$  can achieve a higher signal–to–noise power ratio.
  • The matched filter is optimal with respect to the maximization criterion on which it is based.
q.e.d.

Interpretation of the matched filter


On the last page, the frequency response of the matched filter was derived as follows:

$$H_{\rm MF} (f) = K_{\rm MF} \cdot G^{\star} (f) \cdot {\rm{e} }^{- {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} 2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f \hspace{0.05cm}\cdot \hspace{0.05cm}T_{\rm D} } .$$

By  Fourier inverse transformation  the corresponding impulse response is obtained:

$$h_{\rm MF} (t) = K_{\rm MF} \cdot g(T_{\rm D} - t).$$

These two functions can be interpreted as follows:

  • The  matched filter  is matched by the term  $G^{\star}(f)$  to the spectrum of the pulse  $g(t)$  which is to be found – hence its name.
  • The  constant  $K_{\rm MF}$  is necessary for dimensional reasons.
  • If  $g(t)$  is a voltage pulse, this constant has the unit "Hz/V".  The frequency response is therefore dimensionless.
  • The  impulse response  $h_{\rm MF}(t)$  results from the useful signal  $g(t)$  by mirroring   ⇒   from $g(t)$ becomes $g(–t)$     as well as a shift by  $T_{\rm D}$  to the right.
  • The  earliest detection time  $T_{\rm D}$  follows for realizable systems from the condition  $h_{\rm MF}(t < 0)\equiv 0$   $($"causality",  see book Linear and Time-Invariant Systems$)$.
  • The  useful component  $d_{\rm S} (t)$  of the filter output signal is equal in shape to the  energy ACF   $\varphi^{^{\bullet} }_{g} (t )$  and shifted with respect to it by  $T_{\rm D}$.  It holds:
$$d_{\rm S} (t) = g(t) * h_{\rm MF} (t) = K_{\rm MF} \cdot g(t) * g(T_{\rm D} - t) = K_{\rm MF} \cdot \varphi^{^{\bullet} }_{g} (t - T_{\rm D} ).$$

$\text{Please note:}$  For an energy-limited signal  $g(t)$,  one can only specify the  energy ACF

$$\varphi^{^{\bullet} }_g (\tau ) = \int_{ - \infty }^{ + \infty } {g(t) \cdot g(t + \tau )\,{\rm{d} }t} .$$

Compared to the ACF definition of a power-limited signal  $x(t)$, viz.

$$\varphi _x (\tau ) = \mathop {\lim }_{T_{\rm M} \to \infty } \frac{1}{ {T_{\rm M} } }\int_{ - T_{\rm M} /2}^{+T_{\rm M} /2} {x(t) \cdot x(t + \tau )\hspace{0.1cm}\,{\rm{d} }t} ,$$

the division by the measurement duration  $T_{\rm M}$  and the boundary transition  $T_{\rm M} → ∞$  are omitted in the calculation of the energy ACF.


$\text{Example 2:}$  We assume that the square pulse is between  $\rm 2\hspace{0.08cm}ms$  and  $\rm 2.5\hspace{0.08cm}ms$  and the detection time  $T_{\rm D} =\rm 2\hspace{0.08cm}ms$  is desired.

Under these conditions:

  • The matched filter impulse response  $h_{\rm MF}(t)$  must be constant in the range from  $t_1 (= 4 - 2.5) =\rm 1.5\hspace{0.08cm}ms$  to  $t_2 (= 4 - 2) =\rm 2\hspace{0.08cm}ms$. 
  • For  $t < t_1$  as well as for  $t > t_2$  it must not have any components.
  • The magnitude frequency response  $\vert H_{\rm MF}(f)\vert$  is  $\rm si$–shaped here.
  • The magnitude of the impulse response  $h_{\rm MF}(t)$  is not important for the S/N ratio, because it is independent of  $K_{\rm MF}$. 


Generalized matched filter for the case of colored disturbances


In the derivations of this section, white noise has always been assumed so far.  Now the following question shall be clarified:

How should the receive filter  $H(f) = H_{\rm MF}(f)$  be designed in the presence of colored interference  $n(t)$  so that the signal to noise power ratio is maximized?

$\text{To explain some terminology:}$  The term "interference" is somewhat more general than "noise."

  • Rather, noise is a subset of all interference, which includes, for example, crosstalk from adjacent lines.
  • We speak of (white) noise  $n(t)$ only if the power-spectral density  ${\it Φ}_n(f)$  is the same for all frequencies.
  • If this is not satisfied, we refer to  $n(t)$  as colored noise.


The top diagram shows the block diagram for deriving the matched filter  $H_{\rm MF}(f)$  in the presence of colored interference  $n(t)$, denoted by the power-spectral density  ${\it Φ}_n(f) ≠\text{ const}$.  All other conditions stated so far for this section still apply.

Matched filter with colored interference

Regarding the modified model according to the diagram below, note:

  • The colored interference signal  $n(t)$  with the power-spectral density  ${\it Φ}_n(f)$  can be modeled – at least mentally – by a "white" noise source  $n_{\rm WR}(t)$  with the constant (two-sided) noise power density  $N_0/2$  and a shape filter with the frequency response  $H_{\rm N}(f)$: 
$${\it{\Phi} }_n \left( f \right) = { {N_{\rm 0} } }/{\rm 2} \cdot \left| {H_{\rm N} \left( f \right)} \right|^{\rm 2} .$$
  • Since implementation aspects are not considered here,   $H_{\rm N}(f)$  is assumed to be real (for simplicity).  The phase response of  $H_{\rm N}(f)$  is not important for what follows.  In this representation the shape filter  $H_{\rm N}(f)$  is shifted to the right side of the spurious addition.  To obtain a model which is also equivalent with respect to the useful signal  $d_{\rm S}(t)$,  the shape filter in the useful signal branch is compensated by the inverse filter  $H_{\rm N}(f)^{–1}$. 


Using this modified model, the matched filter is now derived for the case of colored interference.  If  $H_{\rm N}(f)$  has no zero, which shall be assumed for the following, this arrangement is identical to the block diagram above.

White noise  $n_{\rm WR}(t)$  is now present at the interference addition point.  The derivation of the  matched filter optimization in the presence of white noise  can be easily adapted to the current problem by considering the following:

  • Instead of the actual useful signal  $g(t)$,  consider the signal  $g_{\rm WR}(t)$  before the interference addition.
  • The corresponding spectral function is:   $G_{\rm WR}(f) = G(f)/H_{\rm N}(f)$.
  • Instead of  $H_{\rm MF}(f)$,  the resulting frequency response  ${H_{\rm MF} }' (f) = H_{\rm N}(f) · H_{\rm MF}$  is now to be substituted to the right of the interference addition point.


$\text{Conclusion:}$

(1)   For the  matched filter in the presence of colored interference  we get:

$${H_{\rm MF} }\hspace{0.01cm}' (f) = H_{\rm N} (f) \cdot H_{\rm MF} (f) = K_{\rm MF} \cdot G_{\rm WR} ^ {\star} (f) \cdot {\rm{e} }^{- {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} 2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f \hspace{0.05cm}\cdot \hspace{0.05cm}T_{\rm D} } \hspace{0.3cm}\Rightarrow \hspace{0.3cm}H_{\rm MF} (f) = K_{\rm MF} \cdot \frac{ {G^{\star} (f)} }{ {\left\vert {H_{\rm N} (f)} \right\vert^2 } } \cdot {\rm{e} }^{- {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} 2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f \hspace{0.05cm}\cdot \hspace{0.05cm}T_{\rm D} } .$$

(2)   Thus, the  signal-to-interference power ratio  before the decider is maximum:

$$\rho _{d,\ \max } ( {T_{\rm D} } ) = \frac{1}{ {N_0 /2} }\int_{ - \infty }^{ + \infty } {\left\vert{G_{\rm WR} (f)} \right\vert^2 }\, {\rm{d} }f = \int_{ - \infty }^{ + \infty } \frac{\left \vert G(f) \right\vert^2 }{ {\it{\Phi _n {\rm (f)} } } } \,{\rm{d} }f.$$

(3)   The "white noise" case is included in this more general equation for  ${\it Φ}_n(f) = N_0/2$. 

(4)   However, all equations given here lead to meaningful results, which can also be used in practice, in case of colored interference only if the energy spectrum  $\vert G(f)\vert ^2$  of the useful signal decays asymptotically faster than the interference power density spectrum  ${\it Φ}_n(f)$.

Exercise for the chapter


Exercise 5.7: Rectangular Matched Filter

Exercise 5.7Z: Matched Filter - All Gaussian

Exercise 5.8: Matched Filter for Colored Noise

Exercise 5.8Z: Matched Filter for Rectangular PSD