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==
Das '''Matched-Filter''' – auch Korrelationsfilter genannt dient zum Nachweis der Signalexistenz. Es kann mit größtmöglicher Sicherheit anders ausgedrückt: mit maximalem SNR – entscheiden, ob ein durch additives Rauschen $n(t)$ gestörtes impulsförmiges Nutzsignal $g(t)$ vorhanden ist oder nicht. Zur Herleitung des Matched-Filters wird folgende Anordnung betrachtet.  
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<br>
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{{BlaueBox|TEXT= 
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$\text{Definition:}$&nbsp; The&nbsp; &raquo;'''matched filter'''&laquo;&nbsp; also called&nbsp; "correlation filter"&nbsp; &nbsp; is used to prove the signal existence.
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[[File:EN_Sto_T_5_4_S1_neu2.png |right|frame| Block diagram of the&nbsp; "matched filter receiver"]]
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The&nbsp; &raquo;'''matched filter receiver'''&laquo;&nbsp; can decide with the greatest possible certainty&nbsp; in other words: &nbsp; with maximum SNR –&nbsp; whether or not a pulse&nbsp; $g(t)$&nbsp; disturbed by additive white noise&nbsp; $n(t)$&nbsp; is present.
  
[[File:P_ID568__Sto_T_5_4_S1_neu.png |frame| Blockschaltbild des Matched-Filter-Empfängers]]
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 +
To derive the "matched filter receiver",&nbsp; consider the block diagram on the right. }}
  
Für die einzelnen Komponenten gelten folgende Voraussetzungen:  
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*Der Nutzanteil $g(t)$ des Empfangssignals $r(t)$ sei impulsförmig und somit ''energiebegrenzt''. Das heißt: Das Integral über $g^2(t)$ von $–∞$ bis $+∞$ liefert den endlichen Wert $E_g$.  
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The following conditions apply to the individual components:
*Das Störsignal $n(t)$ sei ''Weißes Gaußsches Rauschen'' mit der Rauschleistungsdichte $N_0$.  
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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".&nbsp;
*Das Filterausgangssignal $d(t)$ setzt sich additiv aus zwei Anteilen zusammen. Der Anteil $d_{\rm S}(t)$ geht auf das „'''S'''ignal” $g(t)$ zurück und der Anteil $d_{\rm N}(t)$ auf das „'''N'''oise” $n(t)$.  
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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; $E_g$.  
*Der Empfänger, bestehend aus einem linearen Filter ⇒  Frequenzgang $H_{\rm MF}(f)$ und dem Entscheider, ist so zu dimensionieren, dass das momentane S/N-Verhältnis am Ausgang maximal wird:  
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*Let the noise signal&nbsp; $n(t)$&nbsp; be&nbsp; "white Gaussian noise"&nbsp; with&nbsp; (one&ndash;sided)&nbsp; noise power density&nbsp; $N_0$.  
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*The 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; $n(t)$.  
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*The receiver,&nbsp; consisting of a linear filter &nbsp; &nbsp; frequency response&nbsp; $H_{\rm MF}(f)$&nbsp; and the&nbsp; "decision",&nbsp; is to be dimensioned <br>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 $σ_d^2$ die ''Varianz'' (Leistung) von $d_{\rm N}(t)$ und $T_{\rm D}$ den (geeignet gewählten) ''Detektionszeitpunkt.''
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*Here, &nbsp;$σ_d^2$&nbsp; denotes the&nbsp; variance&nbsp; ("power")&nbsp; of the signal&nbsp; $d_{\rm N}(t)$,&nbsp; and &nbsp;$T_{\rm D}$&nbsp; denotes the (suitably chosen)&nbsp; "detection time".  
  
==Matched-Filter-Optimierung==
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==Matched filter optimization==
Gegeben sei ein energiebegrenztes Nutzsignal $g(t)$ mit dem zugehörigen Spektrum $G(f)$. Damit kann das Filterausgangssignal zum Detektionszeitpunkt $T_{\rm D}$ für jedes beliebige Filter mit der Impulsantwort $h(t)$ und dem Frequenzgang $H(f) = F${ $h(t)$} wie folgt geschrieben werden (ohne Berücksichtigung des Rauschens  Index S für „Signal”):  
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<br>
:$$d_{\rm S} ( {T_{\rm D} } ) = g(t) * h(t) = \int_{ - \infty }^{ + \infty } {G(f) \cdot H(f) \cdot {\rm{e}}^{{\rm{j}}2{\rm{\pi }}fT_{\rm D} }\hspace{0.1cm} {\rm{d}}f} .$$
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Let be given an energy-limited useful signal&nbsp; $g(t)$&nbsp; with the corresponding spectrum&nbsp; $G(f)$.  
Der „Rauschanteil” $d_{\rm N}(t)$ des Filterausgangssignals rührt allein vom Weißen Rauschen $n(t)$ am Eingang des Empfängers her. Für seine Varianz (Leistung) gilt unabhängig von $T_{\rm D}$:  
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*Thus,&nbsp; 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; $H(f) =\mathcal{ F}\{h(t)\}$&nbsp; can be written as follows&nbsp; <br>(ignoring noise &nbsp; &nbsp; subscript &nbsp;$\rm S$&nbsp; for "signal"):  
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:$$d_{\rm S} ( {T_{\rm D} } ) = g(t) * h(t) = \int_{ - \infty }^{ + \infty } {G(f) \cdot H(f) \cdot {\rm{e}}^{ {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} 2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f \hspace{0.05cm}\cdot \hspace{0.05cm}T_{\rm D} }\hspace{0.1cm} {\rm{d}}f} .$$
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*The&nbsp; "noise component"&nbsp; $d_{\rm N}(t)$&nbsp; of the filter output signal&nbsp; (subscript &nbsp;$\rm N$&nbsp; for "noise")&nbsp; stems solely from the white noise&nbsp; $n(t)$&nbsp; at the input of the receiver.&nbsp; For its variance&nbsp; (power)&nbsp; 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,&nbsp; the optimization problem at hand is:
:$$\rho _d ( {T_{\rm D} } ) = \frac{ {\left| {\int_{ - \infty }^{ + \infty } {G(f) \cdot H(f) \cdot {\rm{e} }^{{\rm{j} }2{\rm{\pi } }fT_{\rm D} }\hspace{0.1cm} {\rm{d} }f} } \right|^2 } }{ {N_0 /2 \cdot \int_{ - \infty }^{ + \infty } {\left| {H(f)} \right|^{\rm{2} }\hspace{0.1cm} {\rm{d} }f} } }  \stackrel{!}{=} {\rm{Maximum} }.$$
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:$$\rho _d ( {T_{\rm D} } ) = \frac{ {\left| {\int_{ - \infty }^{ + \infty } {G(f) \cdot H(f) \cdot {\rm{e} }^{ {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} 2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f \hspace{0.05cm}\cdot \hspace{0.05cm}T_{\rm D} }\hspace{0.1cm} {\rm{d} }f} } \right|^2 } }{ {N_0 /2 \cdot \int_{ - \infty }^{ + \infty } {\left| {H(f)} \right|^{\rm{2} }\hspace{0.1cm} {\rm{d} }f} } }  \stackrel{!}{=} {\rm{Maximum} }.$$
Man kann zeigen, dass der Quotient für den folgenden Frequenzgang am größten wird:  
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$$H(f) = H_{\rm MF} (f) = K_{\rm MF}  \cdot G^{\star}  (f) \cdot {\rm{e}}^{-{\rm{j}}2{\rm{\pi }}fT_{\rm D} } .$$
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{{BlaueBox|TEXT= 
Damit erhält man für das Signal-zu-Rauschleistungsverhältnis am Matched–Filter–Ausgang:  
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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;
:$$\rho _d ( {T_{\rm D} } ) = { {2 \cdot E_g } }/{ {N_0 } },$$
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:$$H(f) = H_{\rm MF} (f) = K_{\rm MF}  \cdot G^{\star}  (f) \cdot {\rm e}^{- {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} 2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f \hspace{0.05cm}\cdot \hspace{0.05cm}T_{\rm D} } . $$
und zwar unabhängig von der dimensionsbehafteten Konstante $K_{\rm MF}$. Zur Erklärung:
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*Thus,&nbsp; for the signal&ndash;to&ndash;noise power ratio at the matched filter output&nbsp; $($independent of the dimensionally constant&nbsp; $K_{\rm MF})$,&nbsp; we obtain:  
* $E_g$ bezeichnet die Energie des Eingangsimpulses, die man nach dem [https://de.wikipedia.org/wiki/Satz_von_Parseval Satz von Parseval] sowohl im Zeit– als auch im Frequenzbereich berechnen kann:
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:$$\rho _d ( {T_{\rm D} } ) = { {2 \cdot E_g } }/{ {N_0 } }.$$
:$$E_g  = \int_{ - \infty }^{ + \infty } {g^2 (t)\hspace{0.1cm}{\rm{d}}t}  = \int_{ - \infty }^{ + \infty } {\left| {G(f)} \right|^{\rm{2}}\hspace{0.1cm} {\rm d}f} .$$
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 +
* $E_g$&nbsp; denotes the energy of the input pulse,&nbsp; which can be calculated using&nbsp; [https://en.wikipedia.org/wiki/Parseval%27s_theorem $\text{Parseval's theorem}$]&nbsp; in both the time and frequency domains:
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:$$E_g  = \int_{ - \infty }^{ + \infty } {g^2 (t)\hspace{0.1cm}{\rm{d} }t}  = \int_{ - \infty }^{ + \infty } {\left \vert {G(f)} \right\vert ^{\rm{2} }\hspace{0.1cm} {\rm d}f} .$$}}
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{{GraueBox|TEXT= 
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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.
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*Thus the pulse energy&nbsp; $E_g = \rm 5 · 10^{–4} \hspace{0.05cm}V^2s$.
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*Let the noise power density be&nbsp; $N_0 = \rm 10^{–6} \hspace{0.05cm}V^2/Hz$.
  
{{Beispiel}}''':'''&nbsp; Ein rechteckförmiger Impuls $g(t)$ mit der Amplitude $\rm 1\hspace{0.05cm}V$ und der Dauer $0.5\hspace{0.05cm} \rm ms$ und unbekannter Lage soll in einer verrauschten Umgebung aufgefunden werden. Somit ist die Impulsenergie $E_g = \rm 5 · 10^{–4} \hspace{0.05cm}V^2s$. Die Rauschleistungsdichte sei $N_0 = \rm 10^{–6} \hspace{0.05cm}V^2/Hz$.
 
  
Das beste Ergebnis  ⇒  das '''maximale S/N–Verhältnis''' erzielt man mit dem Matched-Filter:  
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The best result &nbsp; ⇒  &nbsp; the&nbsp; &raquo;'''maximum S/N ratio'''&laquo;&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  
 
\hspace{0.3cm}\Rightarrow\hspace{0.3cm}
 
\hspace{0.3cm}\Rightarrow\hspace{0.3cm}
  10 \cdot {\rm lg}\hspace{0.15cm}\rho _d ( {T_{\rm D} } ) = 30\,{\rm dB}.$$
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  10 \cdot {\rm lg}\hspace{0.15cm}\rho _d ( {T_{\rm D} } ) = 30\,{\rm dB}.$$}}
{{end}}
 
  
  
Das oben angegebene Matched–Filter–Kriterium wird nun schrittweise hergeleitet. Wenn Sie daran nicht interessiert sind, so springen Sie bitte zur Fortsetzungsseite [[Stochastische_Signaltheorie/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 section&nbsp; [[Theory_of_Stochastic_Signals/Matched_Filter#Interpretation_of_the_matched_filter|"Interpretation of the matched filter"]].  
  
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{{BlaueBox|TEXT= 
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$\text{Derivation of the matched filter criterion:}$&nbsp;
  
{{Box}}
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$(1)$&nbsp; The Schwarz inequality with the two&nbsp; (generally complex)&nbsp; functions&nbsp; $A(f)$&nbsp; and&nbsp; $B(f)$:
'''Herleitung des Matched–Filter–Kriteriums:'''&nbsp; Die Schwarzsche Ungleichung lautet mit den beiden (im allgemeinen komplexen) Funktionen $A(f)$ und $B(f)$:
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:$$\left \vert  {\int_a^b {A(f) \cdot B(f)\hspace{0.1cm}{\rm{d} }f} } \right \vert ^2  \le \int_a^b {\left \vert {A(f)} \right \vert^{\rm{2} } \hspace{0.1cm}{\rm{d} }f}  \cdot \int_a^b {\left\vert {B(f)} \right \vert^{\rm{2} } \hspace{0.1cm}{\rm{d} }f} .$$
:$$\left| {\int_a^b {A(f) \cdot B(f)\hspace{0.1cm}{\rm{d} }f} } \right|^2  \le \int_a^b {\left| {A(f)} \right|^{\rm{2} } \hspace{0.1cm}{\rm{d}}f}  \cdot \int_a^b {\left| {B(f)} \right|^{\rm{2} } \hspace{0.1cm}{\rm{d} }f} .$$
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$(2)$&nbsp; We now apply this equation to the signal&ndash;to&ndash;noise ratio:  
Wir wenden nun diese Gleichung auf das Signal–zu–Rauschverhältnis an:  
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:$$\rho _d ( {T_{\rm D} } ) = \frac{ {\left \vert {\int_{ - \infty }^{ + \infty } {G(f) \cdot H(f) \cdot {\rm{e} }^{ {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} 2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f \hspace{0.05cm}\cdot \hspace{0.05cm}T_{\rm D} } \hspace{0.1cm}{\rm{d} }f} } \right \vert^2 } }{ {N_0 /2 \cdot \int_{ - \infty }^{ + \infty } {\left \vert {H(f)} \right \vert^{\rm{2} }\hspace{0.1cm} {\rm{d} }f} } }.$$
:$$\rho _d ( {T_{\rm D} } ) = \frac{ {\left| {\int_{ - \infty }^{ + \infty } {G(f) \cdot H(f) \cdot {\rm{e}}^{ {\rm{j} }2{\rm{\pi } }fT_{\rm D} } \hspace{0.1cm}{\rm{d} }f} } \right|^2 } }{ {N_0 /2 \cdot \int_{ - \infty }^{ + \infty } {\left| {H(f)} \right|^{\rm{2} }\hspace{0.1cm} {\rm{d} }f} } }.$$
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$(3)$&nbsp; Thus,&nbsp; 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:
Mit $A(f) = G(f)$ und $B(f) = H(f) · {\rm exp}({\rm j2}πfT_{\rm D})$ ergibt sich somit die folgende Schranke:  
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:$$\rho_d ( {T_{\rm D} } ) \le \frac{1}{ {N_0 /2} } \cdot \int_{ - \infty }^{ + \infty } {\left \vert  {G(f)} \right \vert^{\rm{2} } }\hspace{0.1cm}{\rm{d} }f .$$
$$\rho_d ( {T_{\rm D} } ) \le \frac{1}{ {N_0 /2} } \cdot \int_{ - \infty }^{ + \infty } {\left| {G(f)} \right|^{\rm{2} } }\hspace{0.1cm}{\rm{d} }f .$$
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$(4)$&nbsp; We now tentatively set for the filter frequency response:
Setzt man für den Filterfrequenzgang versuchsweise
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:$$H(f) = H_{\rm MF} (f) = K_{\rm MF}  \cdot G^{\star}  (f) \cdot {\rm{e} }^{- {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} 2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f \hspace{0.05cm}\cdot \hspace{0.05cm}T_{\rm D} }.$$
:$$H(f) = H_{\rm MF} (f) = K_{\rm MF}  \cdot G^{\star}  (f) \cdot {\rm{e} }^{ {\rm{ - j} }2{\rm{\pi } }fT_{\rm D} }$$
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$(5)$&nbsp; Then,&nbsp; from the above equation&nbsp; $(2)$,&nbsp; we obtain the following result:
ein, so erhält man aus der obigen Gleichung:  
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:$$\rho _d ( {T_{\rm D} } ) = \frac{ {\left \vert  K_{\rm MF}\cdot {\int_{ - \infty }^{ + \infty } {\left \vert  {G(f)} \right \vert ^{\rm{2} }\hspace{0.1cm} {\rm{d} }f} } \right \vert ^2 } }{ {N_0 /2 \cdot K_{\rm MF} ^2  \cdot \int_{ - \infty }^{ + \infty } {\left \vert {G(f)} \right \vert ^{\rm{2} }\hspace{0.1cm} {\rm{d} }f} } } = \frac{1}{ {N_0 /2} } \cdot \int_{ - \infty }^{ + \infty } {\left \vert {G(f)} \right \vert ^{\rm{2} }\hspace{0.1cm} {\rm{d} }f} .$$
:$$\rho _d ( {T_{\rm D} } ) = \frac{ {\left| K_{\rm MF}\cdot {\int_{ - \infty }^{ + \infty } {\left{G(f)} \right|^{\rm{2} }\hspace{0.1cm} {\rm{d} }f} } \right|^2 } }{ {N_0 /2 \cdot K_{\rm MF} ^2  \cdot \int_{ - \infty }^{ + \infty } {\left| {G(f)} \right|^{\rm{2} }\hspace{0.1cm} {\rm{d} }f} } } = \frac{1}{ {N_0 /2} } \cdot \int_{ - \infty }^{ + \infty } {\left| {G(f)} \right|^{\rm{2} }\hspace{0.1cm} {\rm{d} }f} .$$
 
  
Das heißt: Mit diesem Ansatz für das Matched-Filter $H_{\rm MF}(f)$ wird in obiger Abschätzung tatsächlich der maximal mögliche Wert erreicht. Mit keinem anderen Filter $H(f) ≠ H_{\rm MF}(f)$ kann man ein höheres Signal–zu–Rauschleistungsverhältnis erzielen  &nbsp; ⇒  &nbsp; Das Matched–Filter ist in Bezug auf das ihm zugrunde gelegte Maximierungskriterium optimal.
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$\text{This means:}$
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*With the approach&nbsp; $(4)$&nbsp; for the matched filter $H_{\rm MF}(f)$,&nbsp; the maximum possible value is indeed obtained in the above estimation.
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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.
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*The matched filter is optimal with respect to the maximization criterion on which it is based.
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<div align="right">'''q.e.d.'''</div>
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}}
  
q.e.d.
 
{{end}}
 
  
==Interpretation des Matched-Filters==
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We refer here to the HTML5/JavaScript applet &nbsp; [[Applets:Matched_Filter_Properties|"Matched Filter Properties"]].
Auf der letzten Seite wurde der Frequenzgang des Matched-Filters wie folgt abgeleitet:  
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:$$H_{\rm MF} (f) = K_{\rm MF}  \cdot G^{\star}  (f) \cdot {\rm{e} }^{ {\rm{ - j} }2{\rm{\pi } }fT_{\rm D} } .$$
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==Interpretation of the matched filter==
Durch [[Signaldarstellung/Fouriertransformation_und_-rücktransformation#Das_zweite_Fourierintegral|Fourierrücktransformation]] erhält man die dazugehörige Impulsantwort:  
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<br>
 +
In the last section,&nbsp; 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&nbsp; [[Signal_Representation/Fourier_Transform_and_its_Inverse#The_second_Fourier_integral|$\text{Fourier inverse transformation}$]]&nbsp; the corresponding impulse response is obtained:
 
:$$h_{\rm MF} (t) = K_{\rm MF}  \cdot g(T_{\rm D}  - t).$$
 
:$$h_{\rm MF} (t) = K_{\rm MF}  \cdot g(T_{\rm D}  - t).$$
  
 
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These two functions can be interpreted as follows:
Diese beiden Funktionen lassen sich wie folgt interpretieren:  
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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&nbsp; &nbsp; hence its name.  
*Das '''Matched-Filter''' ist durch den Term $G^{\star}(f)$ an das Spektrum des aufzufindenden Impulses $g(t)$ angepasst daher sein Name (englisch: ''to match'' ≡ anpassen).  
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*The&nbsp; "constant" &nbsp;$K_{\rm MF}$&nbsp; is necessary for dimensional reasons.
*Die '''Konstante''' $K_{\rm MF}$ ist aus Dimensionsgründen notwendig. Ist $g(t)$ ein Spannungsimpuls, so hat diese Konstante die Einheit „Hz/V”. Der Frequenzgang ist somit dimensionslos.  
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*If&nbsp; $g(t)$&nbsp; is a voltage pulse,&nbsp; this constant has the unit "Hz/V".&nbsp; The frequency response&nbsp; $H_{\rm MF} (f)$&nbsp; is therefore dimensionless.
*Die '''Impulsantwort''' $h_{\rm MF}(t)$ ergibt sich aus dem Nutzsignal $g(t)$ durch Spiegelung  aus $g(t)$ wird $g(–t)$ –  sowie einer Verschiebung um $T_{\rm D}$ nach rechts.  
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*The&nbsp; "impulse response" &nbsp;$h_{\rm MF}(t)$&nbsp; results from the useful signal &nbsp;$g(t)$&nbsp; by mirroring &nbsp; &nbsp; from&nbsp; $g(t)$&nbsp; becomes $g(–t)$&nbsp;$]$ &nbsp; as well as a shift by&nbsp; $T_{\rm D}$&nbsp; to the right.
*Der '''früheste Detektionszeitpunkt''' $T_{\rm D}$ folgt für realisierbare Systeme aus der Bedingung $h_{\rm MF}(t < 0)$ ≡ 0  ⇒  „Kausalität” (siehe Buch [[Lineare_zeitinvariante_Systeme|Lineare zeitinvariante Systeme]] ).  
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*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&nbsp; [[Linear_and_Time_Invariant_Systems/System_Description_in_Time_Domain#Causal_systems|"Linear and Time-Invariant Systems"]]$)$.  
*Für den '''Nutzanteil''' des Filterausgangssignals gilt:  
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*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/Basics_of_Coded_Transmission#ACF_calculation_of_a_digital_signal|$\text{energy auto-correlation function}$]] &nbsp; $\varphi^{^{\bullet} }_{g} (t )$&nbsp; and shifted with respect to it by &nbsp;$T_{\rm D}$.&nbsp; It holds:  
 
:$$d_{\rm S} (t) = g(t) * h_{\rm MF} (t) = K_{\rm MF}  \cdot g(t) * g(T_{\rm D}  - t) = K_{\rm MF}  \cdot \varphi^{^{\bullet} }_{g} (t - T_{\rm D} ).$$
 
:$$d_{\rm S} (t) = g(t) * h_{\rm MF} (t) = K_{\rm MF}  \cdot g(t) * g(T_{\rm D}  - t) = K_{\rm MF}  \cdot \varphi^{^{\bullet} }_{g} (t - T_{\rm D} ).$$
:Das bedeutet: Das Ausgangssignal ist formgleich mit der [[Digitalsignalübertragung/Grundlagen_der_codierten_Übertragung&action=submit#AKF.E2.80.93Berechnung_eines_Digitalsignals|Energie-AKF]] (in diesem Tutorial durch einen Punkt gekennzeichnet) und gegenüber dieser um $T_{\rm D}$ verschoben.
 
  
 +
{{BlaueBox|TEXT= 
 +
$\text{Please note:}$&nbsp;
 +
For an energy-limited signal&nbsp; $g(t)$,&nbsp; one can only specify the&nbsp; &raquo;'''energy ACF'''&laquo;:&nbsp;
 +
:$$\varphi^{^{\bullet} }_g (\tau ) = \int_{ - \infty }^{ + \infty } {g(t) \cdot g(t + \tau )\,{\rm{d} }t} .$$
 +
Compared to the ACF definition of a power-limited signal&nbsp; $x(t)$, viz.
 +
:$$\varphi _x (\tau ) = \mathop {\lim }_{T_{\rm M}  \to \infty } \frac{1}{ {T_{\rm M} } }\int_{ - T_{\rm M} /2}^{+T_{\rm M} /2} {x(t) \cdot x(t + \tau )\hspace{0.1cm}\,{\rm{d} }t} ,$$
 +
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.}}
  
''Anmerkung:'' Bei einem energiebegrenzten Signal $g(t)$ kann man nur die ''Energie–AKF'' angeben:
 
:$$\varphi^{^{\bullet}}_g (\tau ) = \int_{ - \infty }^{ + \infty } {g(t) \cdot g(t + \tau )\,{\rm{d}}t} .$$
 
Gegenüber der AKF-Definition eines leistungsbegrenzten Signals $x(t)$, nämlich
 
:$$\varphi _x (\tau ) = \mathop {\lim }_{T_{\rm M}  \to \infty } \frac{1}{ {T_{\rm M} } }\int_{ - T_{\rm M} /2}^{+T_{\rm M} /2} {x(t) \cdot x(t + \tau )\hspace{0.1cm}\,{\rm{d} }t} ,$$
 
wird bei der Berechnung der Energie-AKF auf die Division durch die Messdauer $T_{\rm M}$ sowie auf den Grenzübergang $T_{\rm M} → ∞$ verzichtet.
 
  
{{Beispiel}}''':'''&nbsp; Wir gehen wieder davon aus, dass der Rechteckimpuls zwischen $\rm 2\hspace{0.05cm}ms$ und $\rm 2.5\hspace{0.05cm}ms$ liegt und der Detektionszeitpunkt $T_{\rm D} =\rm 2\hspace{0.05cm}ms$ gewünscht wird. Dann gilt:
+
{{GraueBox|TEXT= 
*Die Matched–Filter–Impulsantwort $h_{\rm MF}(t)$ muss im Bereich von $t_1 (= 4 – 2.5) =\rm 1.5\hspace{0.05cm}ms$ bis $t_2 (= 4 – 2) =\rm 2\hspace{0.05cm}ms$ konstant sein. Für $t < t_1$ sowie für $t > t_2$ darf sie keine Anteile besitzen.
+
$\text{Example 2:}$&nbsp; We assume that the rectangular 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.
*Der Betragsfrequenzgang $|H_{\rm MF}(f)|$ ist hier si–förmig. Die Höhe der Impulsantwort $h_{\rm MF}(t)$ spielt für das S/N–Verhältnis keine Rolle, da dieses unabhängig von $K_{\rm MF}$ ist.
 
  
 +
Under these conditions:
 +
*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;
 +
*For &nbsp;$t < t_1$&nbsp; as well as for &nbsp;$t > t_2$&nbsp; it must not have any components.
 +
*The magnitude frequency response &nbsp;$\vert H_{\rm MF}(f)\vert$&nbsp; is&nbsp; $\rm sinc$–shaped here.
 +
*The magnitude of the impulse response &nbsp;$h_{\rm MF}(t)$&nbsp; is not important for the S/N ratio, because&nbsp; $\rho _d ( {T_{\rm D} } )$&nbsp; is independent of &nbsp;$K_{\rm MF}$.}}
  
{{end}}
 
  
==Matched-Filter bei farbigen Störungen==
+
We refer here to the HTML5/JavaScript applet &nbsp; [[Applets:Matched_Filter_Properties|"Matched Filter Properties"]].
Bei den Herleitungen dieses Abschnittes wurde bisher stets von Weißem Rauschen ausgegangen. Nun soll die Frage geklärt werden, wie das '''Empfangsfilter''' $H(f) = H_{\rm MF}(f)$ '''bei farbiger Störung''' $n(t)$ zu gestalten ist, damit das Signal–zu–Rauschleistungsverhältnis maximal wird.  
 
  
''Hinweis:'' Der Begriff „Störung” ist etwas allgemeiner als „Rauschen”. Vielmehr ist Rauschen eine Teilmenge aller Störungen, zu denen z. B. auch das Nebensprechen von benachbarten Leitungen zählt. Wir sprechen nur dann von (weißem) Rauschen $n(t)$, wenn das Leistungsdichtespektrum ${\it Φ}_n(f)$ für alle Frequenzen gleich ist. Ist dies nicht erfüllt, so bezeichnen wir $n(t)$ als farbige Störung.
 
  
[[File:P_ID644__Sto_T_5_4_S4ab_neu.png |frame| Zum Matched-Filter bei farbiger Störung]]
+
==Generalized matched filter for the case of colored interference==
 +
<br>
 +
In the derivations of this section,&nbsp; white noise has always been assumed so far.&nbsp; Now the following question shall be clarified:<br> &nbsp; &nbsp; How should the receiver  filter&nbsp; $H(f) = H_{\rm MF}(f)$&nbsp; be designed in the presence of&nbsp; &raquo;'''colored interference'''&laquo;&nbsp; $n(t)$&nbsp; so that the signal&nbsp;to&nbsp;noise power ratio is maximized?
  
Zum hier betrachteten Modell ist anzumerken:
+
{{BlaueBox|TEXT= 
*Die obere Grafik zeigt das Blockschaltbild zur Herleitung des Matched–Filters $H_{\rm MF}(f)$ bei farbiger Störung $n(t)$, gekennzeichnet durch das Leistungsdichtespektrum ${\it Φ}_n(f) ≠$ const. Alle bisher für diesen Abschnitt genannten Voraussetzungen gelten weiterhin.  
+
$\text{To explain some terminology:}$&nbsp; The term&nbsp; "interference"&nbsp; is somewhat more general than&nbsp; "noise."
*Das farbige Störsignal $n(t)$ mit dem Leistungsdichtespektrum ${\it Φ}_n(f)$ kann man – zumindest gedanklich – durch eine „weiße” Rauschquelle $n_{\rm WR}(t)$ mit der konstanten (zweiseitigen) Rauschleistungsdichte $N_0/2$ und ein Formfilter mit dem Frequenzgang $H_{\rm N}(f)$ modellieren:
+
*Rather,&nbsp; noise is a subset of all interference,&nbsp; which includes,&nbsp; for example,&nbsp; crosstalk from adjacent lines.
:$${\it{\Phi} }_n \left( f \right) = { {N_{\rm 0} } }/{\rm 2} \cdot \left| {H_{\rm N} \left( f \right)} \right|^{\rm 2} .$$
+
*We speak of&nbsp; (white)&nbsp; noise&nbsp; $n(t)$&nbsp; only if the power-spectral density&nbsp; ${\it Φ}_n(f)$&nbsp; is the same for all frequencies.
*Diese Modifikation ist in der unteren Grafik berücksichtigt. Da Realisierungsaspekte hier nicht betrachtet werden, wird $H_{\rm N}(f)$ vereinfacht als reell angenommen. Der Phasengang von $H_{\rm N}(f)$ spielt für das Folgende keine Rolle.
+
*If this is not satisfied,&nbsp; we refer to&nbsp; $n(t)$&nbsp; as&nbsp; "colored interference".}}  
*In der unteren Darstellung ist das Formfilter $H_{\rm N}(f)$ auf die rechte Seite der Störaddition verschoben. Um ein auch bezüglich des Nutzsignals $d_{\rm S}(t)$ äquivalentes Modell zu erhalten, wird das Formfilter im Nutzsignalzweig durch das inverse Filter $H_{\rm N}(f)^{–1}$ kompensiert.
 
  
 +
<br>
 +
[[File:EN_Sto_T_5_4_S4.png |right|frame| Matched filter with colored resp. white noise]]
  
Anhand dieses modifizierten Modells wird nun das verallgemeinerte Matched-Filter für den Fall farbiger Störungen hergeleitet. Besitzt $H_{\rm N}(f)$ keine Nullstelle, was für das Folgende vorausgesetzt werden soll, so ist diese Anordnungen mit dem obigen Blockschaltbild identisch.  
+
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; $n(t)$,&nbsp; 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.
 +
 
 +
Regarding the modified model according to the diagram below,&nbsp; note:
 +
*The colored interference signal&nbsp; $n(t)$&nbsp; with power-spectral density&nbsp; ${\it Φ}_n(f)$&nbsp; can be modeled – at least mentally – by a&nbsp; "white"&nbsp; noise source&nbsp; $n_{\rm WN}(t)$&nbsp; with the constant&nbsp; (two-sided)&nbsp; noise power density&nbsp; $N_0/2$&nbsp; and a shape filter with frequency response&nbsp; $H_{\rm N}(f)$:&nbsp;
 +
:$${\it{\Phi} }_n \left( f \right) = { {N_{\rm 0} } }/{\rm 2} \cdot \left| {H_{\rm N} \left( f \right)} \right|^{\rm 2} .$$
  
[[File:P_ID645__Sto_T_5_4_S4b_neu.png |frame| Äquivalentes Matched-Filter bei farbigen Störungen]]
+
*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 interference 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;
  
An der Störadditionsstelle liegt nun weißes Rauschen $n_{\rm WR}(t)$ an. Die Herleitung der [[Stochastische_Signaltheorie/Matched-Filter#Matched-Filter-Optimierung_.281.29|Matched–Filter–Optimierung]] bei weißem Rauschen lässt sich in einfacher Weise auf das aktuelle Problem anpassen, wenn man Folgendes berücksichtigt:
 
*Anstelle des tatsächlichen Sendesignals $g(t)$ ist das Signal $g_{\rm WR}(t)$ vor der Störaddition zu berücksichtigen. Die dazugehörige Spektralfunktion lautet: $G_{\rm WR}(f) = G(f)/H_{\rm N}(f)$.
 
*Anstelle von $H_{\rm MF}(f)$ ist nun der resultierende Frequenzgang ${H_{\rm MF} }' (f) = H_{\rm N}(f) · H_{\rm MF}$ rechts von der Störadditionsstelle einzusetzen.
 
  
 +
Using this modified model,&nbsp; the matched filter is now derived for the case of colored interference.&nbsp; If&nbsp; $H_{\rm N}(f)$&nbsp; has no zero,&nbsp; which shall be assumed for the following,&nbsp; this arrangement is identical to the block diagram above.
  
 +
White noise&nbsp; $n_{\rm WN}(t)$&nbsp; is now present at the interference addition point.&nbsp; The derivation of the&nbsp; [[Theory_of_Stochastic_Signals/Matched_Filter#Matched_filter_optimization|$\text{matched filter optimization in the presence of white noise}$]]&nbsp; can be easily adapted to the current problem by considering the following:
 +
*Instead of the actual useful signal&nbsp; $g(t)$,&nbsp; consider the signal&nbsp; $g_{\rm WN}(t)$&nbsp; before the interference addition.
 +
*The corresponding spectral function is: &nbsp; $G_{\rm WN}(f) = G(f)/H_{\rm N}(f)$.
 +
*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.
  
{{Box}} '''Endergebnis:'''
 
Damit ergibt sich für das '''Matched-Filter bei farbigen Störungen''':
 
$${H_{\rm MF} }' (f)  = H_{\rm N} (f) \cdot H_{\rm MF} (f) = K_{\rm MF}  \cdot G_{\rm WR} ^ {\star}  (f) \cdot {\rm{e} }^{ - {\rm{j} }2{\rm{\pi } }fT_{\rm D} } \hspace{0.3cm}\Rightarrow \hspace{0.3cm}H_{\rm MF} (f) = K_{\rm MF} \cdot  \frac{ {G^{\star}  (f)} }{ {\left| {H_{\rm N} (f)} \right|^2 } } \cdot {\rm{e} }^{ - {\rm{j} }2{\rm{\pi } }fT_{\rm D} } .$$
 
Das '''Signal-zu-Störleistungsverhältnis''' vor dem Entscheider ist somit maximal:
 
$$\rho _{d,\max } ( {T_{\rm D} } ) = \frac{1}{ {N_0 /2} }\int_{ - \infty }^{ + \infty } {\left| {G_{\rm WR} (f)} \right|^2 }\, {\rm{d} }f = \int_{ - \infty }^{ + \infty } \frac{\left| G(f) \right|^2 }{ {\it{\Phi _n {\rm (f)} } } } \,{\rm{d} }f.$$
 
Der Fall „Weißes Rauschen” ist in dieser allgemeineren Gleichung für ${\it Φ}_n(f) = N_0/2$ mitenthalten.
 
{{end}}
 
  
 +
{{BlaueBox|TEXT= 
 +
$\text{Conclusion:}$
  
''Hinweis:'' Alle hier angegebenen Gleichungen führen bei farbiger Störung allerdings nur dann zu sinnvollen, auch in der Praxis verwertbaren Ergebnissen, wenn das Energiespektrum $|G(f)|^2$ des Nutzsignals asymptotisch schneller abklingt als das Störleistungsdichtespektrum ${\it Φ}_n(f)$.
+
'''(1)''' &nbsp;
 +
For the&nbsp; &raquo;'''matched filter in the presence of colored interference'''&laquo;&nbsp; we get:
 +
:$${H_{\rm MF} }\hspace{0.01cm}' (f)  = H_{\rm N} (f) \cdot H_{\rm MF} (f) = K_{\rm MF}  \cdot G_{\rm WN} ^ {\star}  (f) \cdot {\rm{e} }^{- {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} 2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f \hspace{0.05cm}\cdot \hspace{0.05cm}T_{\rm D}  } \hspace{0.3cm}\Rightarrow \hspace{0.3cm}H_{\rm MF} (f) = K_{\rm MF} \cdot  \frac{ {G^{\star}  (f)} }{ {\left\vert {H_{\rm N} (f)} \right\vert^2 } } \cdot {\rm{e} }^{- {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} 2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f \hspace{0.05cm}\cdot \hspace{0.05cm}T_{\rm D}  } .$$
 +
'''(2)''' &nbsp; Thus,&nbsp; the&nbsp; &raquo;'''signal-to-interference power ratio'''&laquo;&nbsp; before the decider is maximum:
 +
:$$\rho _{d,\ \max } ( {T_{\rm D} } ) = \frac{1}{ {N_0 /2} }\int_{ - \infty }^{ + \infty } {\left\vert{G_{\rm WN} (f)} \right\vert^2 }\, {\rm{d} }f = \int_{ - \infty }^{ + \infty } \frac{\left \vert  G(f) \right\vert^2 }{ {\it{\Phi _n {\rm (f)} } } } \,{\rm{d} }f.$$
 +
'''(3)''' &nbsp; The&nbsp; "white noise"&nbsp; case is included in this more general equation for&nbsp; ${\it Φ}_n(f) = N_0/2$.&nbsp;
  
==Aufgaben zum Kapitel==
+
'''(4)''' &nbsp; However,&nbsp; all equations given here lead to meaningful results,&nbsp; which can also be used in practice,&nbsp; '''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-spectral density&nbsp; ${\it Φ}_n(f)$'''.}}
  
[[Aufgaben:5.7 Rechteck-Matched-Filter|Aufgabe 5.7: &nbsp; Rechteck-Matched-Filter]]
+
==Exercises for the chapter==
 +
<br>
 +
[[Aufgaben:Exercise_5.7:_Rectangular_Matched_Filter|Exercise 5.7: Rectangular Matched Filter]]
  
[[Aufgaben:5.7Z Matched-Filter - alles gaußisch|Zusatzaufgabe 5.7Z: &nbsp; Matched-Filter - alles gaußisch]]
+
[[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: &nbsp; Matched-Filter für farbige Störung]]
+
[[Aufgaben:Exercise_5.8:_Matched_Filter_for_Colored_Interference|Exercise 5.8: Matched Filter for Colored Interference]]
  
[[Aufgaben:5.8Z Matched-Filter bei Rechteck-LDS|Zusatzaufgabe 5.8Z: &nbsp; Matched-Filter bei Rechteck-LDS]]
+
[[Aufgaben:Exercise_5.8Z:_Matched_Filter_for_Rectangular_PSD|Exercise 5.8Z: Matched Filter for Rectangular PSD]]
  
  
 
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Latest revision as of 10:22, 22 December 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 pulse  $g(t)$  disturbed by additive white noise  $n(t)$  is present.


To derive the "matched filter receiver",  consider the block diagram on the right.


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  (one–sided)  noise power density  $N_0$.
  • The 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",  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,  $σ_d^2$  denotes the  variance  ("power")  of the signal  $d_{\rm N}(t)$,  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   ⇒   subscript  $\rm S$  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  (subscript  $\rm N$  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  $\text{Parseval's theorem}$  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 section  "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.


We refer here to the HTML5/JavaScript applet   "Matched Filter Properties".

Interpretation of the matched filter


In the last section,  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  $\text{Fourier inverse transformation}$  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  $H_{\rm MF} (f)$  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  $\text{energy auto-correlation function}$   $\varphi^{^{\bullet} }_{g} (t )$  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 rectangular 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  $\rm sinc$–shaped here.
  • The magnitude of the impulse response  $h_{\rm MF}(t)$  is not important for the S/N ratio, because  $\rho _d ( {T_{\rm D} } )$  is independent of  $K_{\rm MF}$.


We refer here to the HTML5/JavaScript applet   "Matched Filter Properties".


Generalized matched filter for the case of colored interference


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


Matched filter with colored resp. white 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.

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

  • The colored interference signal  $n(t)$  with power-spectral density  ${\it Φ}_n(f)$  can be modeled – at least mentally – by a  "white"  noise source  $n_{\rm WN}(t)$  with the constant  (two-sided)  noise power density  $N_0/2$  and a shape filter with 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 interference 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  $n_{\rm WN}(t)$  is now present at the interference addition point.  The derivation of the  $\text{matched filter optimization in the presence of white noise}$  can be easily adapted to the current problem by considering the following:

  • Instead of the actual useful signal  $g(t)$,  consider the signal  $g_{\rm WN}(t)$  before the interference addition.
  • The corresponding spectral function is:   $G_{\rm WN}(f) = G(f)/H_{\rm N}(f)$.
  • 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:

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

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

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

(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-spectral density  ${\it Φ}_n(f)$.

Exercises for the chapter


Exercise 5.7: Rectangular Matched Filter

Exercise 5.7Z: Matched Filter - All Gaussian

Exercise 5.8: Matched Filter for Colored Interference

Exercise 5.8Z: Matched Filter for Rectangular PSD