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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 127467, 3994]*) (*NotebookOutlinePosition[ 145511, 4517]*) (* CellTagsIndexPosition[ 141638, 4388]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["Notes", "Section 1"], Cell[CellGroupData[{ Cell["Editorial Changes", "Subsection"], Cell[TextData[{ "Double subscript notation \"", Cell[BoxData[ \(TraditionalForm\`\[SelectionPlaceholder]\_\(\[SelectionPlaceholder]\ \ \[SelectionPlaceholder]\)\)]], "\" changed to \"", Cell[BoxData[ \(TraditionalForm\`\[SelectionPlaceholder]\_\(\[SelectionPlaceholder], \ \[SelectionPlaceholder]\)\)]], "\" throughout the paper." }], "Text"], Cell[TextData[{ "Fraction notation \"", Cell[BoxData[ \(TraditionalForm\`\[SelectionPlaceholder]/\[SelectionPlaceholder]\)]], "\" changed to \"", Cell[BoxData[ \(TraditionalForm\`\[SelectionPlaceholder]\/\[SelectionPlaceholder]\)]],\ "\" throughout the paper." }], "Text"], Cell["\<\ Equations reformatted to remove restrictions due to the original 2-column \ format.\ \>", "Text"], Cell[TextData[{ Cell[BoxData[ FormBox[ ButtonBox["TYPO", ButtonData:>"Ed:Change2", Active->True, ButtonStyle->"Hyperlink"], TextForm]]], "\"", Cell[BoxData[ \(TraditionalForm\`S\_out\)]], "\" changed to \"", Cell[BoxData[ FormBox[ SubscriptBox["S", StyleBox["out", FontSlant->"Italic"]], TraditionalForm]]], "\", and \"Variance\" changed to \"variance\"." }], "Text"], Cell[TextData[{ Cell[BoxData[ FormBox[ ButtonBox["TYPO", ButtonData:>"Ed:Change1", Active->True, ButtonStyle->"Hyperlink"], TextForm]]], "\"'Manpack' SHF ground terminal'\" changed to \"''Manpack' SHF ground \ terminal'\"." }], "Text"], Cell[TextData[{ Cell[BoxData[ FormBox[ ButtonBox["TYPO", ButtonData:>"Ed:Change3", Active->True, ButtonStyle->"Hyperlink"], TextForm]]], "\"section 3\" changed to \"Section 3\"." }], "Text"], Cell[TextData[{ Cell[BoxData[ FormBox[ ButtonBox["TYPO", ButtonData:>"Ed:Change4", Active->True, ButtonStyle->"Hyperlink"], TextForm]]], "\"", Cell[BoxData[ \(TraditionalForm\`n\ \[Pi]/B\)]], "\" changed to \"", Cell[BoxData[ \(TraditionalForm\`\(n\ \[Pi]\)\/B\)]], "\"." }], "Text"], Cell[TextData[{ Cell[BoxData[ FormBox[ ButtonBox["TYPO", ButtonData:>"Ed:Change5", Active->True, ButtonStyle->"Hyperlink"], TextForm]]], Cell[BoxData[ FormBox[ ButtonBox["TYPO", ButtonData:>"Ed:Change37", Active->True, ButtonStyle->"Hyperlink"], TextForm]]], "\"", Cell[BoxData[ \(TraditionalForm\`exp[i\ n\ \[Pi]\ \[Omega]/B]\)]], "\" changed to \"", Cell[BoxData[ \(TraditionalForm\`exp[\(i\ n\ \[Pi]\ \[Omega]\)\/B]\)]], "\"." }], "Text"], Cell[TextData[{ Cell[BoxData[ FormBox[ ButtonBox["TYPO", ButtonData:>"Ed:Change6", Active->True, ButtonStyle->"Hyperlink"], TextForm]]], "\"", Cell[BoxData[ FormBox[GridBox[{ {"0", \(n\^\[Prime] > n\)} }], TraditionalForm]]], "\" changed to \"", Cell[BoxData[ FormBox[GridBox[{ {\(\(0\)\(,\)\), \(n\^\[Prime] > n\)} }], TraditionalForm]]], "\"." }], "Text"], Cell[TextData[{ Cell[BoxData[ FormBox[ ButtonBox["TYPO", ButtonData:>"Ed:Change7", Active->True, ButtonStyle->"Hyperlink"], TextForm]]], "\"", Cell[BoxData[ \(TraditionalForm\`\(c\_\[PlusMinus]\) \[Congruent] \(\[PlusMinus]\({1 \ - 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\ \(\[Theta]\_2\) \[Omega]\_2)\)/B]\/\(1 - A\ exp[\(-i\)\ \[Pi]\ \((\[Omega]\_1 \ - \[Omega]\_2)\)/B]\)\)]], "\" changed to \"", Cell[BoxData[ \(TraditionalForm\`exp[\(-\(\(i\ \[Pi]\ \((\(\[Theta]\_1\) \[Omega]\_1 \ - \(\[Theta]\_2\) \[Omega]\_2)\)\)\/B\)\)]\/\(1 - A\ exp[\(-\(\(i\ \[Pi]\ \((\ \[Omega]\_1 - \[Omega]\_2)\)\)\/B\)\)]\)\)]], "\"." }], "Text"], Cell[TextData[{ Cell[BoxData[ FormBox[ ButtonBox["TYPO", ButtonData:>"Ed:Change12", Active->True, ButtonStyle->"Hyperlink"], TextForm]]], "\"", Cell[BoxData[ \(TraditionalForm\`8/\[Pi]\^2\)]], "\" changed to \"", Cell[BoxData[ \(TraditionalForm\`8\/\[Pi]\^2\)]], "\"." }], "Text"], Cell[TextData[{ Cell[BoxData[ FormBox[ ButtonBox["TYPO", ButtonData:>"Ed:Change13", Active->True, ButtonStyle->"Hyperlink"], TextForm]]], "\"", Cell[BoxData[ \(TraditionalForm\`\[Pi]\ \((\[Omega]\_1 - \[Omega]\_2)\)/B\)]], "\" changed to \"", Cell[BoxData[ \(TraditionalForm\`\(\[Pi]\ \((\[Omega]\_1 - \[Omega]\_2)\)\)\/B\)]], "\"." }], "Text"], Cell[TextData[{ Cell[BoxData[ FormBox[ ButtonBox["TYPO", ButtonData:>"Ed:Change14", Active->True, ButtonStyle->"Hyperlink"], TextForm]]], Cell[BoxData[ FormBox[ ButtonBox["TYPO", ButtonData:>"Ed:Change19", Active->True, ButtonStyle->"Hyperlink"], TextForm]]], "Single line equation reformatted as a multi-line equation." }], "Text"], Cell[TextData[{ Cell[BoxData[ FormBox[ ButtonBox["TYPO", ButtonData:>"Ed:Change15", Active->True, ButtonStyle->"Hyperlink"], TextForm]]], "\"", Cell[BoxData[ \(TraditionalForm\`1/256\)]], "\" changed to \"", Cell[BoxData[ \(TraditionalForm\`1\/256\)]], "\", and \"", Cell[BoxData[ \(TraditionalForm\`256/\[Pi]\)]], "\" changed to \"", Cell[BoxData[ \(TraditionalForm\`256\/\[Pi]\)]], "\"." }], "Text"], Cell[TextData[{ Cell[BoxData[ FormBox[ ButtonBox["TYPO", ButtonData:>"Ed:Change16", Active->True, ButtonStyle->"Hyperlink"], TextForm]]], "\"", Cell[BoxData[ \(TraditionalForm\`sin(\(n\^\[Prime]\) \[Pi]\ \[Omega]\^\[Prime]/ B - \[Theta])\)]], "\" changed to \"", Cell[BoxData[ \(TraditionalForm\`sin(\(\(n\^\[Prime]\) \[Pi]\ \[Omega]\^\[Prime]\)\/B \ - \[Theta])\)]], "\", \"", Cell[BoxData[ \(TraditionalForm\`e\^\(i\ n\ \[Pi]\ \((\[Omega] - \ \[Omega]\^\[Prime])\)/B\)\)]], "\" changed to \"", Cell[BoxData[ \(TraditionalForm\`e\^\(\(i\ n\ \[Pi]\ \((\[Omega] - \ \[Omega]\^\[Prime])\)\)\/B\)\)]], "\", and \"", Cell[BoxData[ \(TraditionalForm\`e\^\(i\ \[Pi]\ \((\[Omega] - \[Omega]\^\[Prime])\)/B\ \)\)]], "\" changed to \"", Cell[BoxData[ \(TraditionalForm\`e\^\(\(i\ \[Pi]\ \((\[Omega] - \[Omega]\^\[Prime])\)\ \)\/B\)\)]], "\"." }], "Text"], Cell[TextData[{ Cell[BoxData[ FormBox[ ButtonBox["TYPO", ButtonData:>"Ed:Change17", Active->True, ButtonStyle->"Hyperlink"], TextForm]]], "Single line equations reformatted as a multi-line equations, and \"", Cell[BoxData[ \(TraditionalForm\`\(sin\^2\)(n\ \[Pi]\ \[Omega]/B - \[Theta])\)]], "\" changed to \"", Cell[BoxData[ \(TraditionalForm\`\(sin\^2\)(\(n\ \[Pi]\ \[Omega]\)\/B - 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1)\)/ 2\)\) exp {\(-D\)\ \((E + \(b\^\[Prime]\)\^2)\)/\(a\^\[Prime]\)\ \^2}\), "\[IndentingNewLine]", \(TraditionalForm\`\(\(\[Times]\)\(I\_\(D - 1\)\) \({2 D\ \(E\_D\%\(1\/2\)\) b\^\[Prime]/\(a\^\[Prime]\)\^2}\)\)\)}]], "\" changed to \"", Cell[BoxData[ \(TraditionalForm\`\(\((E\_D\/\(b\^\[Prime]\)\^2)\)\^\(\(D - 1\)\/2\)\) exp {\(-\(\(D\ \((E + \ \(b\^\[Prime]\)\^2)\)\)\/\(a\^\[Prime]\)\^2\)\)} \(I\_\(D - 1\)\) {\(2 D\ \(E\_D\%\(1\/2\)\) \ b\^\[Prime]\)\/\(a\^\[Prime]\)\^2}\)]], "\"." }], "Text"], Cell[TextData[{ Cell[BoxData[ FormBox[ ButtonBox["TYPO", ButtonData:>"Ed:Change29", Active->True, ButtonStyle->"Hyperlink"], TextForm]]], "\"", Cell[BoxData[ \(TraditionalForm\`1 + 1/D\)]], "\" changed to \"", Cell[BoxData[ \(TraditionalForm\`1 + 1\/D\)]], "\" (2 times)." }], "Text"], Cell[TextData[{ Cell[BoxData[ FormBox[ ButtonBox["TYPO", ButtonData:>"Ed:Change30", Active->True, ButtonStyle->"Hyperlink"], TextForm]]], "\"eqns. 26 and 27\" changed to \"eqn. 26 and eqn. 27\" to accommodate two \ hyperlinks." }], "Text"], Cell[TextData[{ Cell[BoxData[ FormBox[ ButtonBox["TYPO", ButtonData:>"Ed:Change31", Active->True, ButtonStyle->"Hyperlink"], TextForm]]], "\"", Cell[BoxData[ \(TraditionalForm\`P(E\_D)\)]], "\" changed to \"", Cell[BoxData[ \(TraditionalForm\`\(P\_D\)(E\_D)\)]], "\" (twice)." }], "Text"], Cell[TextData[{ Cell[BoxData[ FormBox[ ButtonBox["TYPO", ButtonData:>"Ed:Change32", Active->True, ButtonStyle->"Hyperlink"], TextForm]]], "\"", Cell[BoxData[ \(TraditionalForm\`Var\ \((E\_D)\)\^\(1/2\)\)]], "\" changed to \"", Cell[BoxData[ \(TraditionalForm\`\((Var(E\_D))\)\^\(1\/2\)\)]], "\"." }], "Text"], Cell[TextData[{ Cell[BoxData[ FormBox[ ButtonBox["TYPO", ButtonData:>"Ed:Change35", Active->True, ButtonStyle->"Hyperlink"], TextForm]]], Cell[BoxData[ FormBox[ ButtonBox["TYPO", ButtonData:>"Ed:Change36", Active->True, ButtonStyle->"Hyperlink"], TextForm]]], Cell[BoxData[ FormBox[ ButtonBox["TYPO", ButtonData:>"Ed:Change40", Active->True, ButtonStyle->"Hyperlink"], TextForm]]], Cell[BoxData[ FormBox[ ButtonBox["TYPO", ButtonData:>"Ed:Change42", Active->True, ButtonStyle->"Hyperlink"], TextForm]]], Cell[BoxData[ FormBox[ ButtonBox["TYPO", ButtonData:>"Ed:Change44", Active->True, ButtonStyle->"Hyperlink"], TextForm]]], Cell[BoxData[ FormBox[ ButtonBox["TYPO", ButtonData:>"Ed:Change46", Active->True, ButtonStyle->"Hyperlink"], TextForm]]], Cell[BoxData[ FormBox[ ButtonBox["TYPO", ButtonData:>"Ed:Change47", Active->True, ButtonStyle->"Hyperlink"], TextForm]]], "Figure moved to a more appropriate place in the text." }], "Text"], Cell[TextData[{ Cell[BoxData[ FormBox[ ButtonBox["TYPO", ButtonData:>"Ed:Change33", Active->True, ButtonStyle->"Hyperlink"], TextForm]]], "\"eqns. 29 and 30\" changed to \"eqn. 29 and eqn. 30\" to accommodate two \ hyperlinks." }], "Text"], Cell[TextData[{ Cell[BoxData[ FormBox[ ButtonBox["TYPO", ButtonData:>"Ed:Change34", Active->True, ButtonStyle->"Hyperlink"], TextForm]]], "\"", Cell[BoxData[ \(TraditionalForm\`\(b\^\[Prime]\)\^2/\(a\^\[Prime]\)\^2\)]], "\" changed to \"", Cell[BoxData[ \(TraditionalForm\`\(b\^\[Prime]\)\^2\/\(a\^\[Prime]\)\^2\)]], "\"." }], "Text"], Cell[TextData[{ Cell[BoxData[ FormBox[ ButtonBox["TYPO", ButtonData:>"Ed:Change38", Active->True, ButtonStyle->"Hyperlink"], TextForm]]], "\"", Cell[BoxData[ \(TraditionalForm\`12 kHz/128\)]], "\" changed to \"", Cell[BoxData[ \(TraditionalForm\`\(12 kHz\)\/128\)]], "\"." }], "Text"], Cell[TextData[{ Cell[BoxData[ FormBox[ ButtonBox["TYPO", ButtonData:>"Ed:Change41", Active->True, ButtonStyle->"Hyperlink"], TextForm]]], "\"", Cell[BoxData[ \(TraditionalForm\`1/30\)]], "\" changed to \"", Cell[BoxData[ \(TraditionalForm\`1\/30\)]], "\"." }], "Text"], Cell[TextData[{ Cell[BoxData[ FormBox[ ButtonBox["TYPO", ButtonData:>"Ed:Change43", Active->True, ButtonStyle->"Hyperlink"], TextForm]]], "\"Figs. 5 and 6\" changed to \"Fig. 5 and Fig. 6\" to accommodate two \ hyperlinks." }], "Text"], Cell[TextData[{ Cell[BoxData[ FormBox[ ButtonBox["TYPO", ButtonData:>"Ed:Change45", Active->True, ButtonStyle->"Hyperlink"], TextForm]]], "\"", Cell[BoxData[ \(TraditionalForm\`4/9\)]], "\" changed to \"", Cell[BoxData[ \(TraditionalForm\`4\/9\)]], "\"." }], "Text"], Cell[TextData[{ Cell[BoxData[ FormBox[ ButtonBox["TYPO", ButtonData:>"Ed:Change48", Active->True, ButtonStyle->"Hyperlink"], TextForm]]], "\"\[EmptyUpTriangle]\[EmptyUpTriangle]\[EmptyUpTriangle] experimental\" \ changed to \"", "\[EmptyUpTriangle] experimental results", "\"." }], "Text"], Cell[TextData[{ Cell[BoxData[ FormBox[ ButtonBox["TYPO", ButtonData:>"Ed:Change49", Active->True, ButtonStyle->"Hyperlink"], TextForm]]], "\"Figs. 5, 6 and 7\" changed to \"Fig. 5, Fig. 6 and Fig. 7\" to \ accommodate three hyperlinks." }], "Text"], Cell[TextData[{ Cell[BoxData[ FormBox[ ButtonBox["TYPO", ButtonData:>"Ed:Change50", Active->True, ButtonStyle->"Hyperlink"], TextForm]]], "\"A.C Baynham\" changed to \"A.C. Baynham\"." }], "Text"], Cell[TextData[{ Cell[BoxData[ FormBox[ ButtonBox["TYPO", ButtonData:>"Ed:Change52", Active->True, ButtonStyle->"Hyperlink"], TextForm]]], "\"", Cell[BoxData[ \(TraditionalForm\`\(\(exp(\(-\(b\^\[Prime]\)\^2\)/\(a\^\[Prime]\)\^2)\)\ \/\(1 + i\ k\ \(a\^\[Prime]\)\^2\)\) exp[\(\(b\^\[Prime]\)\^2/\(a\^\[Prime]\)\^2\)\/\(1 + i\ k\ \(a\^\ \[Prime]\)\^2\)]\)]], "\" changed to \"", Cell[BoxData[ \(TraditionalForm\`\(\(exp(\(-\(\(b\^\[Prime]\)\^2\/\(a\^\[Prime]\)\^2\)\ \))\)\/\(1 + i\ k\ \(a\^\[Prime]\)\^2\)\) exp[\(1\/\(1 + i\ k\ \(a\^\[Prime]\)\^2\)\) \(b\^\[Prime]\)\^2\/\(a\^\ \[Prime]\)\^2]\)]], "\"." }], "Text"], Cell[TextData[{ Cell[BoxData[ FormBox[ ButtonBox["TYPO", ButtonData:>"Ed:Change53", Active->True, ButtonStyle->"Hyperlink"], TextForm]]], Cell[BoxData[ FormBox[ ButtonBox["TYPO", ButtonData:>"Ed:Change54", Active->True, ButtonStyle->"Hyperlink"], TextForm]]], Cell[BoxData[ FormBox[ ButtonBox["TYPO", ButtonData:>"Ed:Change60", Active->True, 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\(b\^\[Prime]\)\^2)\)/\(a\^\[Prime]\)\^2}\), "\[IndentingNewLine]", \(TraditionalForm\`\(\(\[Times]\)\(1\/\(2 \[Pi]\ i\)\) \(\(\[Integral]\ \_\(\[Xi]\_-\)\%\(\[Xi]\_+\)d\[Xi]\ \(exp( s\[CenterDot] D\ E\_D)\)\)\[IndentingNewLine]\(\(\[Times]\)\(1\/\[Xi]\^\ D\) \(exp[\(D\ \(b\^\[Prime]\)\^2/\(a\^\[Prime]\)\^4\)\/\[Xi]]\)\)\)\)\)}]], "\" changed to \"", Cell[BoxData[ \(TraditionalForm\`\(D\/\((a\^\[Prime])\)\^\(2 D\)\) exp {\(-\(\(D\ \((E\_D + \ \(b\^\[Prime]\)\^2)\)\)\/\(a\^\[Prime]\)\^2\)\)} \(1\/\(2 \[Pi]\ i\)\) \(\ \[Integral]\_\(\[Xi]\_-\)\%\(\[Xi]\_+\)d\[Xi]\ \(exp( s\ D\ E\_D)\) \(1\/\[Xi]\^D\) exp[\(D\ \(b\^\[Prime]\)\^2\)\/\(\[Xi]\ \ \(a\^\[Prime]\)\^4\)]\)\)]], "\"." }], "Text"], Cell[TextData[{ Cell[BoxData[ FormBox[ ButtonBox["TYPO", ButtonData:>"Ed:Change57", Active->True, ButtonStyle->"Hyperlink"], TextForm]]], "\"", Cell[BoxData[ \(TraditionalForm\`1/\(a\^\[Prime]\)\^2 \[PlusMinus] i\ \[Infinity]\)]], "\" changed to \"", Cell[BoxData[ 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ButtonData:>"Ed:Change63", Active->True, ButtonStyle->"Hyperlink"], TextForm]]], "\"", Cell[BoxData[ \(TraditionalForm\`\(C(k/D)\)\^D\)]], "\" changed to \"", Cell[BoxData[ \(TraditionalForm\`\(C(k\/D)\)\^D\)]], "\" (4 times)." }], "Text"], Cell[TextData[{ Cell[BoxData[ FormBox[ ButtonBox["TYPO", ButtonData:>"Ed:Change64", Active->True, ButtonStyle->"Hyperlink"], TextForm]]], Cell[BoxData[ FormBox[ ButtonBox["TYPO", ButtonData:>"Ed:Change67", Active->True, ButtonStyle->"Hyperlink"], TextForm]]], "\"", Cell[BoxData[ \(TraditionalForm\`\((1 + 1/D)\)\)]], "\" changed to \"", Cell[BoxData[ \(TraditionalForm\`\((1 + 1\/D)\)\)]], "\" (2 times)." }], "Text"], Cell[TextData[{ Cell[BoxData[ FormBox[ ButtonBox["TYPO", ButtonData:>"Ed:Change66", Active->True, ButtonStyle->"Hyperlink"], TextForm]]], "\"", Cell[BoxData[ FormBox[ RowBox[{\(1\/\(2 \[Pi]\)\), RowBox[{\(\[Integral]\_\(-\[Infinity]\)\%\(+\[Infinity]\)\), RowBox[{ StyleBox[ RowBox[{ StyleBox["d", FontSlant->"Plain"], StyleBox["k", FontSlant->"Italic"]}]], " ", \(exp(i\ k\ E\^\((D)\))\), "\[IndentingNewLine]", "\[Times]", \(1\/\((1 + i\ k\ \(a\^\[Prime]\)\^2/D)\)\^r\), \ \(exp[\(D\ \(b\^\[Prime]\)\^2/\(a\^\[Prime]\)\^2\)\/\(1 + i\ k\ \(a\^\[Prime]\ \)\^2/D\)]\)}]}]}], TraditionalForm]]], "\" changed to \"", Cell[BoxData[ FormBox[ RowBox[{\(1\/\(2 \[Pi]\)\), RowBox[{\(\[Integral]\_\(-\[Infinity]\)\%\(+\[Infinity]\)\), RowBox[{ StyleBox[ RowBox[{ StyleBox["d", FontSlant->"Plain"], StyleBox["k", FontSlant->"Italic"]}]], " ", \(exp( i\ k\ E\_D)\), \(1\/\((1 + \(i\ k\ \(a\^\[Prime]\)\^2\)\/D)\)\ \^r\), \(exp[\(1\/\(1 + \(i\ k\ \(a\^\[Prime]\)\^2\)\/D\)\) \(D\ \ \(b\^\[Prime]\)\^2\)\/\(a\^\[Prime]\)\^2]\)}]}]}], TraditionalForm]]], "\", and \"", Cell[BoxData[ \(TraditionalForm\`exp(D\ \(b\^\[Prime]\)\^2/\(a\^\[Prime]\)\^2)\)]], "\" changed to \"", Cell[BoxData[ \(TraditionalForm\`exp(\(D\ \ \(b\^\[Prime]\)\^2\)\/\(a\^\[Prime]\)\^2)\)]], "\"." }], "Text"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Rapid acquisition of low signal-to-noise carriers", "Title"], Cell["\<\ S.P. Luttrell J.A.S. Pritchard Royal Signals and Radar Establishment, St. Andrews Road, Malvern WR14 3PS, \ United Kingdom \ \>", "Author"], Cell[TextData[StyleBox["Indexing terms: Satellite communications, Algorithms, \ Signal Processing", FontSlant->"Italic"]], "Text"], Cell["\<\ This paper appeared in IEE Proceedings, Part I, 1989, vol. 136, no. 1, pp. \ 100-108.\ \>", "Text"], Cell["\<\ Paper 6463I (E9, E16), first received 18th January and in revised form 20th \ October 1988\ \>", "Text"], Cell[TextData[{ StyleBox["Abstract:", FontWeight->"Bold"], " A parallel bank of 1-bit digital filters is proposed as a solution to the \ rapid carrier acquisition problem in satellite communications. The hardware \ is compact and cheap, and using a crude threshold detection criterion, it can \ localise a 30 dBHz carrier to within 100 Hz in a total bandwidth of 12 kHz in \ about 60 mS. A theoretical analysis of the system performance is also \ presented, together with predictions of its statistical behaviour, which will \ assist in the design of more sophisticated signal detection algorithms." }], "Abstract"], Cell[CellGroupData[{ Cell["List of principal symbols", "Section"], Cell[TextData[Cell[BoxData[ FormBox[GridBox[{ {"t", "=", "time"}, {\(n, n\^\[Prime], k, k\^\[Prime]\), "=", \(sampling\ time\ indices\)}, {\(t\_n\), "=", \(sampling\ time\)}, {\(\([\)\(t\_n, t\_\(n + 1\)\)\(]\)\), "=", \(sampling\ time\ interval\)}, {\(V(t)\), "=", \(input\ voltage\)}, {\(V\_n\), "=", \(sampled\ input\ voltage\)}, {\(l, l\_1, l\_2\), "=", \(Fourier\ component\ indices\)}, {"j", "=", \(integer\ resonance\ condition\)}, {"B", "=", "bandwidth"}, {"A", "=", \(filter\ decay\ constant\)}, {"T", "=", \(filter\ memory\ time\)}, {\(\[Omega], \[Omega]\^\[Prime], \[Omega]\_1, \[Omega]\_2\), "=", \(angular\ frequencies\)}, {\(\[Theta], \[Theta]\_1, \[Theta]\_2\), "=", \(phase\ shifts\)}, {\(C(\[Omega]\_1, \[Omega]\_2)\), "=", \(cross\ correlation\ between\ square\ waves\)}, {\(E(\[Omega]\_1, \[Omega]\_2)\), "=", \(modulus\ square\ of\ \(C(\[Omega]\_1, \[Omega]\_2)\)\)}, {\(\(c\_+\), \(c\_-\)\), "=", \(terms\ in\ \(C(\[Omega]\_1, \[Omega]\_2)\)\)}, {\(x\_1, x\_2\), "=", \(scaled\ frequency\ in\ \(C(\[Omega]\_1, \[Omega]\_2)\)\)}, {\(\(\[CapitalTheta]\_n\)(\[Omega])\), "=", \(phase\ table\ for\ frequency\ \[Omega]\ at\ time\ n\)}, {\(\(F\_n\)(\[Omega])\), "=", \(filter\ output\ for\ frequency\ \[Omega]\ at\ time\ n\)}, {\(\(R\_\(n, n\^\[Prime]\)\)(\[Omega])\), "=", \(filter\ impulse\ response\)}, {\(\[LeftAngleBracket]\(V\_n\) V\_\(n\^\[Prime]\)\[RightAngleBracket]\), "=", \(covariance\ of\ V\_n\)}, {\(\[LeftAngleBracket]\(\(F\_n\)(\[Omega])\) \(\(F\_\(n\^\[Prime]\)\ \%*\)(\[Omega])\)\[RightAngleBracket]\), "=", \(covariance\ of\ \(\(F\_n\)(\[Omega])\)\)}, {\(a\^2\), "=", \(input\ noise\ variance\)}, {\(b\^2\), "=", \(input\ signal\ variance\)}, {\(\(a\^\[Prime]\)\^2\), "=", \(output\ noise\ variance\)}, {\(\(b\^\[Prime]\)\^2\), "=", \(output\ signal\ variance\)}, { SubscriptBox["S", StyleBox["in", FontSlant->"Italic"]], "=", \(input\ signal - to - noise\ ratio\)}, { SubscriptBox["S", StyleBox["out", FontSlant->"Italic"]], "=", \(output\ signal - to - noise\ ratio\)}, { SubscriptBox[\(S\^\[Prime]\), StyleBox["in", FontSlant->"Italic"]], "=", RowBox[{ SubscriptBox["S", StyleBox["in", FontSlant->"Italic"]], " ", "in", " ", "dBHz"}]}, {"E", "=", \(\[LeftBracketingBar]\(F\_n\)(\[Omega])\[RightBracketingBar]\ \^2\)}, {"D", "=", \(number\ of\ output\ samples\ averaged\ over\)}, {\(E\_D\), "=", \(average\ of\ D\ samples\ of\ E\)}, {\(\[LeftAngleBracket]E\_D\[RightAngleBracket]\), "=", \(first\ moment\ of\ \(P(E\_D)\)\)}, {\(\[LeftAngleBracket]\((E\_D)\)\^2\[RightAngleBracket]\), "=", \(second\ moment\ of\ \(P(E\_D)\)\)}, {\(Var(E\_D)\), "=", \(variance\ of\ E\_D\)}, {\(P(E)\), "=", \(PDF\ of\ E\ \((Rician)\)\)}, {\(\(P\_D\)(E\_D)\), "=", \(PDF\ of\ E\_D\)}, {\(M\_r\), "=", RowBox[{ StyleBox[ RowBox[{ StyleBox["r", FontSlant->"Italic"], "th"}]], " ", "moment", " ", "of", " ", \(\(P\_D\)(E\_D)\)}]}, {"k", "=", \(Fourier\ conjugate\ of\ E\ \((or\ E\_D)\)\)}, {\(Q(k)\), "=", \(characteristic\ function\ of\ \(P(E)\)\)}, {"\[Xi]", "=", \(Lapace\ conjugate\ of\ D\ E\_D\)}, {\(\(\[Xi]\_+\), \(\[Xi]\_-\)\), "=", \(limits\ of\ Bromwich\ integral\)}, {\(L {\[CenterDot]}\), "=", \(Laplace\ transform\ operator\)}, {\(\(J\_\[Nu]\)(z)\), "=", \(Bessel\ function\)}, {\(\(I\_0\)(z), \(I\_\[Nu]\)(z), \(I\_\(D - 1\)\)(z)\), "=", \(modified\ Bessel\ functions\)}, {"s", "=", \(number\ of\ standard\ deviations\ in\ E\_D - \ \[LeftAngleBracket]E\_D\[RightAngleBracket]\)}, {\(Q(\(b\^\[Prime]\)\^2, \(a\^\[Prime]\)\^2, s)\), "=", \(threshold\ s\ standard\ deviations\ above\ \ \[LeftAngleBracket]E\_D\[RightAngleBracket]\)}, { RowBox[{ RowBox[{ SubscriptBox["P", StyleBox["MD", FontSlant->"Italic"]], "(", \(\(b\^\[Prime]\)\^2, \(a\^\[Prime]\)\^2, s\), ")"}], ",", SubscriptBox["P", StyleBox["MD", FontSlant->"Italic"]]}], "=", \(probability\ of\ missed\ detection\)}, { RowBox[{ RowBox[{ SubscriptBox["P", StyleBox["FA", FontSlant->"Italic"]], "(", \(\(a\^\[Prime]\)\^2, s\), ")"}], ",", SubscriptBox["P", StyleBox["FA", FontSlant->"Italic"]]}], "=", \(probability\ of\ false\ alarm\)}, {"d", "=", \(register\ word\ length\)} }], TraditionalForm]]]], "Text", TextAlignment->AlignmentMarker, GridBoxOptions->{ColumnAlignments->{Left}}, CellTags->"Ed:Change2"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ CounterBox["Section"], " Introduction" }], "Section 1", CounterAssignments->{{"Section", 0}}], Cell["\<\ Geostationary satellites are now widely used for reliable long distance \ communications. In both business and military fields, the need for long \ range, flexible and portable communications is growing. For many users, the \ amount of information which needs to be communicated is relatively small, but \ often it is the portability of the remote terminal and the ease of use which \ is most important. The soldier may need to relay strategic information from \ the battlefield, or the mobile business user might wish to send back \ commercial information to his own country. The frequencies which are usually \ used for this kind of traffic are the military and civilian bands which lie \ between 7 and 13 GHz. These bands are high enough in frequency to support a \ large number of users, yet are not so high as to suffer high atmospheric \ attenuation or to demand prohibitively expensive microwave components.\ \>", "Text"], Cell[TextData[{ "To realise self-contained man-portable tactical terminals working at \ X-band using contemporary technology, a decrease in power with an attendant \ decrease in data rate is necessary and desirable [", ButtonBox["1", ButtonData:>"Ref:Jones1981", ButtonStyle->"Hyperlink"], ", ", ButtonBox["2", ButtonData:>"Ref:SkiltonWestall1982", ButtonStyle->"Hyperlink"], "]." }], "Text"], Cell[TextData[{ "In the case of low data rate telegraph signals at the frequencies \ described, there will be a frequency ambiguity introduced over a microwave \ satellite link which can be orders of magnitude greater than the bandwidth of \ the signal. In a channel where the link is implemented most efficiently, no \ more power than that necessary to support error-free communications is used. \ In the examples which concern us, the signal-to-noise ratio in the band of \ ambiguity can be worse than ", Cell[BoxData[ \(TraditionalForm\`\(-13\)\ dB\)]], ". Consequently, for both the base station and the remote terminal, the \ rapid acquisition of the correct signal is a nontrivial task." }], "Text"], Cell[TextData[{ "An X-band satellite transponder in geostationary orbit will typically \ introduce a maximum Doppler shift of about \[PlusMinus]2 kHz to the signal. \ The reference frequency uncertainty in the remote terminal, using a \ temperature compensated crystal oscillator for a reference, is about \ \[PlusMinus]3 kHz. The total frequency ambiguity of the signal is therefore \ of the order of 10 kHz; the search bandwidth used (with a guard margin) is 12 \ kHz. Where the data rate is such that the modulated signal bandwidth is much \ less than the carrier centre frequency ambiguity, carrier acquisition using a \ conventional swept filter and integrator is no longer satisfactory. For \ example, the data modulation scheme which is used in the man-portable remote \ terminal prototype known as the 'Manpack' [", ButtonBox["1", ButtonData:>"Ref:Jones1981", ButtonStyle->"Hyperlink"], ", ", ButtonBox["2", ButtonData:>"Ref:SkiltonWestall1982", ButtonStyle->"Hyperlink"], "] is 50 bits per second coherent differential phase shift keying (DPSK). \ The time taken to acquire a signal that has a frequency error of \ \[PlusMinus]6 kHz using this swept filter technique is between 1 and 40 \ seconds, depending on the position of the signal in the band. Clearly this is \ not satisfactory for tactical communications when a brief transmission is \ desirable, or in environments where signal fading occurs." }], "Text"], Cell[TextData[{ "Recent studies have shown [", ButtonBox["3", ButtonData:>"Ref:DuddlePowellWarnerGannonAeigus1982", ButtonStyle->"Hyperlink"], "] that rapid acquisition of such signals, using the most advanced known \ techniques, requires methods which are in general complicated, expensive, or \ bulky, and in any case are not satisfactory in terms of acquisition time. \ State of the art methods implement successive approximations using a chirp \ correlator [", ButtonBox["3", ButtonData:>"Ref:DuddlePowellWarnerGannonAeigus1982", ButtonStyle->"Hyperlink"], "]. Phase lock to a 50 baud DPSK signal in a 12 kHz band may be achieved in \ 7 seconds, regardless of the carrier centre frequency within the band. This \ delay in signal acquisition is still not satisfactory for our exacting \ applications of low data rate links." }], "Text"], Cell["\<\ In this paper, we take advantage of the fact that the full apparatus of the \ Fourier transform is not required if we wish only to find a signal carrier \ frequency. We need only an approximation, and so we emulate a parallel bank \ of filters using simple serial hardware which is constructed from cheap and \ readily available digital components. The scheme which we describe will \ enable lock to be achieved at least an order of magnitude more quickly than \ before.\ \>", "Text"], Cell[TextData[{ "In ", ButtonBox["Section", ButtonData:>"Sect:2", ButtonStyle->"Hyperlink"], " ", CounterBox["Section", "Sect:2"], " we present a theoretical analysis of a 1-bit quantised single pole \ recursive filter bank. We derive the response and aliasing properties of such \ filters. We also derive some statistical properties of the filter outputs; \ these may be used to predict performance in a noisy environment. In ", ButtonBox["Section", ButtonData:>"Sect:3", ButtonStyle->"Hyperlink"], " ", CounterBox["Section", "Sect:3"], " we describe how fast serial hardware can be built to emulate the parallel \ bank of filters, and we multiplex many filters into the wide digital \ bandwidth which is available in a single serial circuit. The filter outputs \ (which are a crude representation of the Fourier transform of the input \ signal) are then sampled using a microprocessor. In ", ButtonBox["Section", ButtonData:>"Sect:4", ButtonStyle->"Hyperlink"], " ", CounterBox["Section", "Sect:4"], " we present the results of testing the circuit." }], "Text", CellTags->"Ed:Change3"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ CounterBox["Section"], " Theory" }], "Section", CellTags->"Sect:2"], Cell["\<\ The primary goal of this research is to design hardware which can rapidly \ localise a carrier frequency, but which at the same time is compact and \ inexpensive. We shall assume that the 'noise environment' is benign (i.e. no \ large amplitude spurious coherent signals are present), and that the \ signal-to-noise ratio (SNR) is small (\[LessTilde] 30 dBHz). Under these \ assumptions, 1-bit quantisation of the input signal retains nearly all the \ signal information, and single (complex) pole recursive filtering of the \ 1-bit signal is sufficient to extract the required carrier frequency.\ \>", "Text"], Cell[TextData[{ "We will therefore develop a theory which describes the operation of a bank \ of single pole recursive filters with 1-bit quantised inputs. The analogue \ input signal ", Cell[BoxData[ \(TraditionalForm\`V(t)\)]], " comprises a set of modulated carrier signals plus noise in general, which \ is bandpass filtered to a band ", Cell[BoxData[ \(TraditionalForm\`\(\([\)\(0, B\)\(]\)\)\)]], " and sampled at a sufficient rate to avoid aliasing. We assume throughout \ that Nyquist sampling for the band ", Cell[BoxData[ \(TraditionalForm\`\(\([\)\(\(-B\), B\)\(]\)\)\)]], " is used, so that the sample times ", Cell[BoxData[ \(TraditionalForm\`t\_n\)]], " are given by" }], "Text"], Cell[BoxData[ \(TraditionalForm\`t\_n \[Congruent] \(n\ \[Pi]\)\/B\)], \ "NumberedEquation", CellTags->{"Eq:1", "Ed:Change4"}], Cell[TextData[{ "where the corresponding analogue input signal value is denoted by ", Cell[BoxData[ \(TraditionalForm\`V\_n\)]], ". We also assume that the noise accompanying the signal is uncorrelated \ from sample to sample." }], "Text"], Cell[TextData[{ "1-bit quantisation of ", Cell[BoxData[ \(TraditionalForm\`V\_n\)]], " is defined as follows:" }], "Text"], Cell[BoxData[ FormBox[ RowBox[{\(V\_n\), "\[LongRightArrow]", TagBox[ StyleBox[ RowBox[{"{", GridBox[{ {\(\(1\)\(,\)\), \(V\_n \[GreaterEqual] 0\)}, {\(\(-1\)\(,\)\), \(V\_n \[LessEqual] 0\)} }]}], ShowAutoStyles->False], (#&)]}], TraditionalForm]], "NumberedEquation", SpanMaxSize->Infinity, GridBoxOptions->{ColumnAlignments->{Left}}, CellTags->"Eq:2"], Cell[TextData[{ "The effect of which is to preserve most of the information content of the \ analogue input signal because the noise acts as a 'dither signal'. There is, \ however, an aliasing problem associated with such 1-bit quantisation because \ it turns sine waves into square waves which have out of band harmonics. The \ Fourier decomposition of a square wave ", Cell[BoxData[ \(TraditionalForm\`V(t)\)]], " of frequency ", Cell[BoxData[ \(TraditionalForm\`\[Omega]\)]], " is" }], "Text"], Cell[BoxData[ \(TraditionalForm\`V( t) = \[Sum]\+l\( 4\/\(\[Pi]\ l\)\) sin[l\ \[Omega]\ t]\)], "NumberedEquation", CellTags->"Eq:3"], Cell[TextData[{ "where the summation is over odd integers ", Cell[BoxData[ \(TraditionalForm\`l\)]], ", and the ", Cell[BoxData[ \(TraditionalForm\`l\)]], "th harmonic is suppressed in amplitude by a factor ", Cell[BoxData[ \(TraditionalForm\`l\)]], ". If we use 2-(or more) bit quantisation, the harmonic amplitudes decay \ faster, but we will see later that the effects of aliassing in the 1-bit case \ are acceptable." }], "Text"], Cell[TextData[{ "We now introduce our basic system model by defining a single pole \ recursive filter with input ", Cell[BoxData[ \(TraditionalForm\`V\_n\)]], " as" }], "Text"], Cell[BoxData[ \(TraditionalForm\`\(F\_\(n + 1\)\)(\[Omega]) = A\ \(\(F\_n\)(\[Omega])\) + \(\(\[CapitalTheta]\_n\)(\[Omega])\) V\_n\)], "NumberedEquation", CellTags->"Eq:4"], Cell[TextData[{ "where ", Cell[BoxData[ \(TraditionalForm\`\(F\_n\)(\[Omega])\)]], " is the complex amplitude response of the filter tuned to ", Cell[BoxData[ \(TraditionalForm\`\[Omega]\)]], ", ", Cell[BoxData[ \(TraditionalForm\`A\)]], " is the relaxation factor for the filter, and ", Cell[BoxData[ \(TraditionalForm\`\(\[CapitalTheta]\_n\)(\[Omega])\)]], " is a 1-bit quantised (in each of the real and imaginary parts) version of \ the complex exponential" }], "Text"], Cell[BoxData[ \(TraditionalForm\`\(\[CapitalTheta]\_n\)(\[Omega]) \[Congruent] exp[\(i\ n\ \[Pi]\ \[Omega]\)\/B]\)], "NumberedEquation", CellTags->{"Ed:Change5", "Eq:5"}], Cell[TextData[{ "The characteristic memory time ", Cell[BoxData[ \(TraditionalForm\`T\)]], " of the filter measured in sample intervals is simply related to ", Cell[BoxData[ \(TraditionalForm\`A\)]], " by" }], "Text"], Cell[BoxData[ \(TraditionalForm\`T = 1\/\(1 - A\)\)], "NumberedEquation", CellTags->"Eq:6"], Cell[TextData[{ "The solution to ", ButtonBox["eqn.", ButtonData:>"Eq:4", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedEquation", "Eq:4"], " is the filter response ", Cell[BoxData[ \(TraditionalForm\`\(F\_n\)(\[Omega])\)]], " which is given by" }], "Text"], Cell[BoxData[ \(TraditionalForm\`\(F\_n\)(\[Omega]) = \[Sum]\+\(n\^\[Prime] = \(-\ \[Infinity]\)\)\%n\(\( R\_\(n, n\^\[Prime]\)\)(\[Omega])\) V\_\(n\^\[Prime]\)\)], "NumberedEquation", CellTags->"Eq:7"], Cell[TextData[{ "where the impulse response ", Cell[BoxData[ \(TraditionalForm\`\(R\_\(n, n\^\[Prime]\)\)(\[Omega])\)]], " is given by" }], "Text"], Cell[BoxData[ FormBox[ RowBox[{\(\(R\_\(n, n\^\[Prime]\)\)(\[Omega])\), "=", TagBox[ RowBox[{ StyleBox["{", ShowAutoStyles->False], StyleBox[GridBox[{ {\(\(\(A\^\(n - n\^\[Prime]\)\) \ \(\(\[CapitalTheta]\_\(n\^\[Prime]\)\)(\[Omega])\)\)\(,\)\), \(n\^\[Prime] \ \[LessEqual] n\)}, {\(\(0\)\(,\)\), \(n\^\[Prime] > n\)} }], ShowAutoStyles->True]}], (#&)]}], TraditionalForm]], "NumberedEquation", SpanMaxSize->Infinity, GridBoxOptions->{ColumnAlignments->{Left}}, CellTags->{"Ed:Change6", "Eq:8"}], Cell[TextData[{ "where both ", Cell[BoxData[ \(TraditionalForm\`V\_\(n\^\[Prime]\)\)]], " and ", Cell[BoxData[ \(TraditionalForm\`\(\[CapitalTheta]\_\(n\^\[Prime]\)\)(\[Omega])\)]], " are 1-bit quantised, but ", Cell[BoxData[ \(TraditionalForm\`A\)]], " and ", Cell[BoxData[ \(TraditionalForm\`\(F\_n\)(\[Omega])\)]], " are represented using many bits. ", ButtonBox["Eqn.", ButtonData:>"Eq:7", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedEquation", "Eq:7"], " for ", Cell[BoxData[ \(TraditionalForm\`\(F\_n\)(\[Omega])\)]], " may be interpreted as a weighted cross-correlation between the past input \ signal ", Cell[BoxData[ \(TraditionalForm\`V\_n\)]], " and the ", Cell[BoxData[ \(TraditionalForm\`\(\[CapitalTheta]\_n\)(\[Omega])\)]], ", and the weighting factor ", Cell[BoxData[ \(TraditionalForm\`A\^\(n - n\^\[Prime]\)\)]], " gives rise to the filter memory time ", Cell[BoxData[ \(TraditionalForm\`T\)]], "." }], "Text"], Cell[TextData[{ "We will now verify that the aliasing, which we alluded to earlier, is \ inconsequential in the case of 1-bit quantisation. The response to an input \ sine wave at frequency ", Cell[BoxData[ \(TraditionalForm\`\[Omega]\_1\)]], " of a filter (as defined in ", ButtonBox["eqn.", ButtonData:>"Eq:4", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedEquation", "Eq:4"], ") tuned to ", Cell[BoxData[ \(TraditionalForm\`\[Omega]\_2\)]], " is obtained by calculating the weighted cross-correlation ", Cell[BoxData[ \(TraditionalForm\`C(\[Omega]\_1, \[Omega]\_2)\)]], " between the two corresponding sampled square waves (as defined in ", ButtonBox["eqn.", ButtonData:>"Eq:3", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedEquation", "Eq:3"], "). This is given by" }], "Text"], Cell[BoxData[ \(TraditionalForm\`C(\[Omega]\_1, \[Omega]\_2) = \[Sum]\+\(n = 0\)\%\ \[Infinity]\( A\^n\) \(\(\[Sum]\+\(l\_1\)\[Sum]\+\(l\_2\)\)\+odd\) \(16\/\(\(\ \[Pi]\^2\) \(l\_1\) l\_2\)\) sin[\(\((n + \[Theta]\_1)\) \[Pi]\ \(l\_1\) \[Omega]\_1\)\/B] exp[\(\(i(n + \[Theta]\_2)\) \[Pi]\ \(l\_2\) \[Omega]\_2\)\/B]\)], \ "NumberedEquation", SpanMaxSize->Infinity, CellTags->"Eq:9"], Cell[TextData[{ "where ", Cell[BoxData[ \(TraditionalForm\`\[Theta]\_1\)]], " and ", Cell[BoxData[ \(TraditionalForm\`\[Theta]\_2\)]], " are arbitrary phase shifts. Summing over ", Cell[BoxData[ \(TraditionalForm\`n\)]], " then yields" }], "Text"], Cell[BoxData[ \(TraditionalForm\`C(\[Omega]\_1, \[Omega]\_2) = \(\(\[Sum]\+\(l\_1\)\ \[Sum]\+\(l\_2\)\)\+odd\) \(8\/\(i\ \(\[Pi]\^2\) \(l\_1\) l\_2\)\)[\(c\_+\) + \(c\_-\)]\)], "NumberedEquation", CellTags->"Eq:10"], Cell["where", "Text"], Cell[BoxData[ \(TraditionalForm\`\(c\_\[PlusMinus]\) \[Congruent] \(\[PlusMinus]\(\(1 - A\ exp[\(\[MinusPlus]\(\(i\ \[Pi]\ \((\(l\_1\) \[Omega]\_1 \ \[PlusMinus] \(l\_2\) \[Omega]\_2)\)\)\/B\)\)]\)\/\(1 + A\^2 - 2 A\ cos[\(\[Pi]\ \((\(l\_1\) \[Omega]\_1 \[PlusMinus] \ \(l\_2\) \[Omega]\_2)\)\)\/B]\)\)\) exp[\(\[PlusMinus]\(\(i\ \[Pi]\ \((\(\[Theta]\_1\) \(l\_1\) \ \[Omega]\_1 \[PlusMinus] \(\[Theta]\_2\) \(l\_2\) \ \[Omega]\_2)\)\)\/B\)\)]\)], "NumberedEquation", SpanMaxSize->Infinity, CellTags->{"Ed:Change7", "Eq:11"}], Cell[TextData[{ "This response is large when one or both of the denominator factors in the \ ", Cell[BoxData[ \(TraditionalForm\`\(c\_\[PlusMinus]\)\)]], " are small, which occurs when" }], "Text"], Cell[BoxData[{ \(TraditionalForm\`\(l\_1\) x\_1 \[PlusMinus] \(l\_2\) x\_2 = \[AlignmentMarker]2 j\), "\n", \(TraditionalForm\`\(l\_1\) l\_2 = \[AlignmentMarker]1, 3, 5, \[CenterEllipsis]\), "\n", \(TraditionalForm\`j = \[AlignmentMarker]0, \(\[PlusMinus]1\), \(\ \[PlusMinus]2\), \[CenterEllipsis]\), "\n", \(TraditionalForm\`0 \[LessEqual] \[AlignmentMarker]x\_i \[LessEqual] 1\)}], "NumberedEquation", TextAlignment->AlignmentMarker, CellTags->"Eq:12"], Cell[TextData[{ "where ", Cell[BoxData[ \(TraditionalForm\`x\_i\)]], " (", Cell[BoxData[ \(TraditionalForm\`0 \[LessEqual] x\_i \[LessEqual] 1\)]], ") is the scaled frequency ", Cell[BoxData[ \(TraditionalForm\`\[Omega]\_i\/B\)]], ". There are many solutions to this equation, some of which we will now \ tabulate." }], "Text", CellTags->"Ed:Change8"], Cell[TextData[{ Cell[BoxData[ \(TraditionalForm\`l\_1 = \(l\_2 = 1\)\)]], " solutions:" }], "Text"], Cell[TextData[{ "\t", Cell[BoxData[ FormBox[GridBox[{ {\(j = 0\), " ", " ", " ", \(x\_1 = x\_2\), " ", " ", " ", \(0 \[LessEqual] x\_1, x\_2 \[LessEqual] 1\)} }], TraditionalForm]]] }], "Text"], Cell[TextData[{ Cell[BoxData[ \(TraditionalForm\`l\_1 = 1, \ l\_2 = 3\)]], " solutions:" }], "Text"], Cell[TextData[{ "\t", Cell[BoxData[ FormBox[GridBox[{ {\(j = \(-1\)\), " ", " ", " ", \(x\_1 = 3 x\_2 - 2\), " ", " ", " ", \(0 \[LessEqual] x\_1 \[LessEqual] 1, \ 2\/3 \[LessEqual] x\_2 \[LessEqual] 1\)}, {\(j = 0\), " ", " ", " ", \(x\_1 = 3 x\_2\), " ", " ", " ", \(0 \[LessEqual] x\_1 \[LessEqual] 1, \ 0 \[LessEqual] x\_2 \[LessEqual] 1\/3\)}, {\(j = 1\), " ", " ", " ", \(x\_1 = \(-3\) x\_2 + 2\), " ", " ", " ", \(0 \[LessEqual] x\_1 \[LessEqual] 1, \ 1\/3 \[LessEqual] x\_2 \[LessEqual] 2\/3\)} }], TraditionalForm]]] }], "Text", GridBoxOptions->{ColumnAlignments->{Left}}, CellTags->"Ed:Change9"], Cell[TextData[{ Cell[BoxData[ \(TraditionalForm\`l\_1 = 3, \ l\_2 = 1\)]], " solutions: Obtain by interchanging ", Cell[BoxData[ \(TraditionalForm\`x\_1\)]], " and ", Cell[BoxData[ \(TraditionalForm\`x\_2\)]], " in the solutions for the ", Cell[BoxData[ \(TraditionalForm\`l\_1 = 1, \ l\_2 = 3\)]], " case." }], "Text"], Cell[TextData[{ "The ", Cell[BoxData[ \(TraditionalForm\`l\_1 = 1, \ l\_2 = 1\)]], " case is ideally the only solution, and we will define its amplitude as \ unity. The contribution of particular ", Cell[BoxData[ \(TraditionalForm\`\((l\_1, l\_2)\)\)]], " solution to ", Cell[BoxData[ \(TraditionalForm\`C(\[Omega]\_1, \[Omega]\_2)\)]], " is suppressed by factor ", Cell[BoxData[ \(TraditionalForm\`\(l\_1\) l\_2\)]], ", and so the dominant cases are those in which the fundamental frequency \ cross correlates with the third harmonic, and which therefore have a typical \ amplitude contribution of ", Cell[BoxData[ \(TraditionalForm\`1\/3\)]], ". The worst case occurs when ", Cell[BoxData[ \(TraditionalForm\`\((x\_1, x\_2)\)\)]], ", coincidentally satisfies two of the above equations, for then the \ interference between the crosscorrelation terms can lead to an amplitude \ contribution of ", Cell[BoxData[ \(TraditionalForm\`2\/3\)]], ". However, there are limited number of such coincidental solutions which \ are ", Cell[BoxData[ \(TraditionalForm\`\((x\_1, x\_2)\) = \((1\/5, 3\/5)\), \((1\/4, 3\/4)\), \((2\/5, 4\/5)\), \((3\/5, 1\/5)\), \((3\/4, 1\/4)\), \((4\/5, 2\/5)\)\)]], ". We have ignored the solutions with ", Cell[BoxData[ \(TraditionalForm\`x\_1 = 0\)]], " or ", Cell[BoxData[ \(TraditionalForm\`x\_2 = 0\)]], " because we are not concerned with DC behaviour; and we have ignored the \ solution ", Cell[BoxData[ \(TraditionalForm\`x\_1 = \(x\_2 = 1\/2\)\)]], " because the sum over contributions from all ", Cell[BoxData[ \(TraditionalForm\`\((l\_1, l\_2)\)\)]], " yields no adverse interference (this is most easily seen in terms of \ crosscorrelated square waves directly). In general, higher order harmonics \ will produce further spurious contributions to the amplitude response which \ are suppressed by larger factors than is the third harmonic." }], "Text", CellTags->"Ed:Change10"], Cell["\<\ In summary, for 1-bit quantisation, provided that the existence of spurious \ responses (i.e. aliasing) is acknowledged, the outputs of a bank of single \ pole recursive filters may be easily interpreted if the number of signals \ present is not too large. Usually there is only a single signal present, and \ so the interpretation problem is trivial.\ \>", "Text"], Cell[TextData[{ "With these provisos in mind we will now derive approximations to the \ squared filter response and its half-width by approximating the weighted \ crosscorrelation ", Cell[BoxData[ \(TraditionalForm\`C(\[Omega]\_1, \[Omega]\_2)\)]], " in the vicinity of ", Cell[BoxData[ \(TraditionalForm\`\[Omega]\_1 = \[Omega]\_2\)]], " using only the first term of the ", Cell[BoxData[ \(TraditionalForm\`\((l\_1, l\_2)\)\)]], " sum in ", ButtonBox["eqn.", ButtonData:>"Eq:9", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedEquation", "Eq:9"], ". Thus" }], "Text"], Cell[BoxData[{ \(TraditionalForm\`C(\[Omega]\_1, \[Omega]\_2)\[AlignmentMarker] \ \[TildeEqual] \(8\/\(i\ \[Pi]\^2\)\) \(\(c\_-\)( l\_1 = \(l\_2 = 1\))\)\), "\n", \(TraditionalForm\`\(\(=\)\(\(\(8 i\)\/\[Pi]\^2\) exp[\(-\(\(i\ \[Pi]\ \((\(\[Theta]\_1\) \[Omega]\_1 - \(\[Theta]\_2\) \ \[Omega]\_2)\)\)\/B\)\)]\/\(1 - A\ exp[\(-\(\(i\ \[Pi]\ \((\[Omega]\_1 - \ \[Omega]\_2)\)\)\/B\)\)]\)\)\)\)}], "NumberedEquation", TextAlignment->AlignmentMarker, CellTags->{"Eq:13", "Ed:Change11"}], Cell[TextData[{ "The numerical factor ", Cell[BoxData[ \(TraditionalForm\`8\/\[Pi]\^2\)]], " is unimportant, and so (up to a constant factor) this result is what \ would be obtained if we were to ignore the 1-bit quantisation altogether; we \ shall henceforth use this approximate equivalence to simplify our \ calculations. We may thus approximate the squared response ", Cell[BoxData[ \(TraditionalForm\`E(\[Omega]\_1, \[Omega]\_2)\)]], " by" }], "Text", CellTags->"Ed:Change12"], Cell[TextData[Cell[BoxData[{ \(TraditionalForm\`E(\[Omega]\_1, \[Omega]\_2) \[Congruent] \ \[AlignmentMarker]\[LeftBracketingBar]C(\[Omega]\_1, \[Omega]\_2)\ \[RightBracketingBar]\^2\), "\n", \(TraditionalForm\`\(\(\[TildeEqual]\)\(\[AlignmentMarker]\)\(1\/\(\((1 - \ A)\)\^2 + A\ [\(\[Pi]\ \((\[Omega]\_1 - \[Omega]\_2)\)\)\/B]\^2\)\)\)\)}], TextAlignment->AlignmentMarker]], "NumberedEquation", TextAlignment->AlignmentMarker, CellTags->{"Ed:Change13", "Eq:14"}], Cell["which is a Lorentzian with a scaled half-width given by", "Text"], Cell[TextData[Cell[BoxData[{ \(TraditionalForm\`\[LeftBracketingBar]\[Omega]\_1 - \[Omega]\_2\ \[RightBracketingBar]\/B \[TildeEqual] \[AlignmentMarker]\(1 - A\)\/\[Pi]\), \ "\n", \(TraditionalForm\`\(\(=\)\(\[AlignmentMarker]\)\(1\/\(\[Pi]\ T\)\)\)\)}], TextAlignment->AlignmentMarker]], "NumberedEquation", TextAlignment->AlignmentMarker, CellTags->{"Ed:Change14", "Eq:15"}], Cell[TextData[{ "In this paper, we will deploy 128 filters across the band ", Cell[BoxData[ \(TraditionalForm\`\(\([\)\(0, B\)\(]\)\)\)]], ", and so the scaled half-width of each filter must be ", Cell[BoxData[ \(TraditionalForm\`1\/256\)]], ", thus ", Cell[BoxData[ \(TraditionalForm\`T = 256\/\[Pi] \[TildeEqual] 82\)]], " samples." }], "Text", CellTags->"Ed:Change15"], Cell[TextData[{ "To analyse the performance of our system in more detail we now derive some \ of its higher order statistical properties. From ", ButtonBox["eqn.", ButtonData:>"Eq:7", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedEquation", "Eq:7"], " we obtain" }], "Text"], Cell[BoxData[ \(TraditionalForm\`\[LeftAngleBracket]\(\(F\_n\)(\[Omega])\) \(\(F\_\(n\^\ \[Prime]\)\%*\)(\[Omega])\)\[RightAngleBracket] = \[Sum]\+\(k = \ \(-\[Infinity]\)\)\%n\(\[Sum]\+\(k\^\[Prime] = \(-\[Infinity]\)\)\%\(n\^\ \[Prime]\)\(\(R\_\(n, k\)\)(\[Omega])\) \(\(R\_\(n\^\[Prime], k\^\[Prime]\)\%*\)(\[Omega])\) \[LeftAngleBracket]\(V\_k\) V\_\(k\^\[Prime]\)\[RightAngleBracket]\)\)], \ "NumberedEquation", CellTags->"Eq:16"], Cell[TextData[{ "where ", Cell[BoxData[ \(TraditionalForm\`\(R\_\(n, k\)\)(\[Omega])\)]], " is given in ", ButtonBox["eqn.", ButtonData:>"Eq:8", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedEquation", "Eq:8"], " and the angle brackets ", Cell[BoxData[ \(TraditionalForm\`\[LeftAngleBracket]\[CenterEllipsis]\ \[RightAngleBracket]\)]], " denote time averaging. When ", Cell[BoxData[ \(TraditionalForm\`V(t)\)]], " is zero mean white Gaussian noise with covariance ", Cell[BoxData[ \(TraditionalForm\`\[LeftAngleBracket]\(V\_n\) V\_\(n\^\[Prime]\)\[RightAngleBracket] = \(a\^2\) \[Delta]\_\(n, \ n\^\[Prime]\)\)]], " we obtain" }], "Text"], Cell[BoxData[ \(TraditionalForm\`\[LeftAngleBracket]\(\(F\_n\)(\[Omega])\) \(\(F\_\(n\^\ \[Prime]\)\%*\)(\[Omega])\)\[RightAngleBracket] = \ \[AlignmentMarker]\(\(a\^2\) \(\[Sum]\+\(k = \(-\[Infinity]\)\)\%\(min(n, n\^\ \[Prime])\)\(\(R\_\(n, k\)\)(\[Omega])\) \(\(R\_\(n\^\[Prime], k\)\%*\)(\[Omega])\)\)\)\[IndentingNewLine]\(\(=\)\(\ \[AlignmentMarker]\)\(\(a\^2\) A\^\[LeftBracketingBar]n - n\^\[Prime]\[RightBracketingBar]\/\(1 \ - A\^2\)\)\)\)], "NumberedEquation", TextAlignment->AlignmentMarker, CellTags->"Eq:17"], Cell[TextData[{ Cell[BoxData[ \(TraditionalForm\`\(F\_n\)(\[Omega])\)]], " is zero mean, and so ", Cell[BoxData[ \(TraditionalForm\`\[LeftAngleBracket]\(\(F\_n\)(\[Omega])\) \(\(F\_\(n\ \^\[Prime]\)\%*\)(\[Omega])\)\[RightAngleBracket]\)]], " measures its covariance, thus we see that ", Cell[BoxData[ \(TraditionalForm\`\(F\_n\)(\[Omega])\)]], " decorrelates on a time scale ", Cell[BoxData[ \(TraditionalForm\`\[LeftBracketingBar]n - n\^\[Prime]\[RightBracketingBar]\)]], " such that ", Cell[BoxData[ \(TraditionalForm\`A\^\[LeftBracketingBar]n - n\^\[Prime]\ \[RightBracketingBar] \[LessTilde] 1\)]], ". Using ", ButtonBox["eqn.", ButtonData:>"Eq:6", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedEquation", "Eq:6"], ", this reduces to ", Cell[BoxData[ \(TraditionalForm\`\[LeftBracketingBar]n - n\^\[Prime]\[RightBracketingBar] \[GreaterTilde] T\)]], ". We henceforth assume that the filter output is sampled at time intervals \ ", Cell[BoxData[ \(TraditionalForm\`T\)]], " (", Cell[BoxData[ \(TraditionalForm\`\(\(=\)\(\ \)\(82\)\)\)]], " samples), and call this scheme 'sparse sampling'." }], "Text"], Cell[TextData[{ "We now compare the squared response to both signal and noise. First, for \ zero mean white Gaussian noise with input variance ", Cell[BoxData[ \(TraditionalForm\`a\^2\)]], ", we obtain the output variance ", Cell[BoxData[ \(TraditionalForm\`\(a\^\[Prime]\)\^2\)]], " from ", ButtonBox["eqn.", ButtonData:>"Eq:17", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedEquation", "Eq:17"], " (the ", Cell[BoxData[ \(TraditionalForm\`n = n\^\[Prime]\)]], " case) as" }], "Text"], Cell[BoxData[ \(TraditionalForm\`\(a\^\[Prime]\)\^2 = a\^2\/\(1 - A\^2\)\)], "NumberedEquation", CellTags->"Eq:18"], Cell[TextData[{ "Secondly, for a unit amplitude input sine wave with frequency ", Cell[BoxData[ \(TraditionalForm\`\[Omega]\^\[Prime]\)]], ", we obtain" }], "Text"], Cell[TextData[Cell[BoxData[{ \(TraditionalForm\`\(F\_n\)(\[Omega]) = \[AlignmentMarker]\[Sum]\+\(n\^\ \[Prime] = \(-\[Infinity]\)\)\%n\( A\^\(n - n\^\[Prime]\)\) \(\(\[CapitalTheta]\_\(n\^\[Prime]\)\)(\ \[Omega])\) \(sin(\(\(n\^\[Prime]\) \[Pi]\ \[Omega]\^\[Prime]\)\/B - \ \[Theta])\)\), "\n", \(TraditionalForm\`\(\(\[TildeEqual]\)\(\[AlignmentMarker]\)\(\(\(i\ \ e\^\(i\ \[Theta]\)\)\/2\) e\^\(\(i\ n\ \[Pi]\ \((\[Omega] - \[Omega]\^\[Prime])\)\)\/B\)\/\(1 - \ A\ e\^\(\(i\ \[Pi]\ \((\[Omega] - \[Omega]\^\[Prime])\)\)\/B\)\)\)\)\)}], TextAlignment->AlignmentMarker]], "NumberedEquation", TextAlignment->AlignmentMarker, CellTags->{"Eq:19", "Ed:Change16"}], Cell[TextData[{ "This result should be compared with ", ButtonBox["eqn.", ButtonData:>"Eq:13", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedEquation", "Eq:13"], ". By modulus squaring these amplitudes we may relate the sine wave results \ to the Gaussian noise results. Thus we obtain the input and output variances \ ", Cell[BoxData[ \(TraditionalForm\`b\^2\)]], " and ", Cell[BoxData[ \(TraditionalForm\`\(b\^\[Prime]\)\^2\)]], ", respectively (these are analogous to ", Cell[BoxData[ \(TraditionalForm\`a\^2\)]], " and ", Cell[BoxData[ \(TraditionalForm\`\(a\^\[Prime]\)\^2\)]], ") as" }], "Text"], Cell[BoxData[{ FormBox[\(\(b\^2 \[Congruent] \ \[AlignmentMarker]\[LeftAngleBracket]\(sin\^2\)(\(n\ \[Pi]\ \[Omega]\)\/B - \ \[Theta])\[RightAngleBracket]\)\[IndentingNewLine]\(\(=\)\(\[AlignmentMarker]\ \)\(1\/2\)\)\), TraditionalForm], "\n", FormBox[ RowBox[{\(\(b\^\[Prime]\)\^2 \[Congruent] \[AlignmentMarker]\ \[LeftAngleBracket]\[LeftBracketingBar]\(F\_n\) \((\[Omega])\)\ \[RightBracketingBar]\^2\[RightAngleBracket]\), "\[IndentingNewLine]", "=", "\[AlignmentMarker]", TagBox[ StyleBox[ RowBox[{"{", GridBox[{ {\(1\/\(4 \((1 - A)\)\^2\)\), \((best\ case)\)}, {\(1\/\(8 \((1 - A)\)\^2\)\), \((worst\ case)\)} }]}], ShowAutoStyles->False], (#&)]}], TraditionalForm]}], "NumberedEquation", TextAlignment->AlignmentMarker, SpanMaxSize->Infinity, GridBoxOptions->{ColumnAlignments->{Left}}, CellTags->{"Eq:20", "Ed:Change17"}], Cell[TextData[{ "The best case occurs when ", Cell[BoxData[ \(TraditionalForm\`\[Omega] = \[Omega]\^\[Prime]\)]], " and the worst case occurs when ", Cell[BoxData[ \(TraditionalForm\`\[LeftBracketingBar]\[Omega]\_1 - \[Omega]\_2\ \[RightBracketingBar]\)]], " is a Lorentzian half-width (see ", ButtonBox["eqn.", ButtonData:>"Eq:15", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedEquation", "Eq:15"], "); in the worst case the sine wave causes two adjacent filters to respond \ each with half the best case result." }], "Text"], Cell[TextData[{ "From ", ButtonBox["eqn.", ButtonData:>"Eq:18", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedEquation", "Eq:18"], " and ", ButtonBox["eqn.", ButtonData:>"Eq:20", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedEquation", "Eq:20"], " we may derive the relationship between the input and output \ signal-to-noise ratios (", Cell[BoxData[ FormBox[ SubscriptBox["S", StyleBox["in", FontSlant->"Italic"]], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ SubscriptBox["S", StyleBox["out", FontSlant->"Italic"]], TraditionalForm]]], " respectively). Thus" }], "Text", CellTags->"Ed:Change18"], Cell[BoxData[{ FormBox[ RowBox[{ SubscriptBox["S", StyleBox["out", FontSlant->"Italic"]], "\[Congruent]", "\[AlignmentMarker]", \(\(b\^\[Prime]\)\^2\/\(a\^\[Prime]\)\^2\)}], TraditionalForm], "\n", FormBox[ RowBox[{ SubscriptBox["S", StyleBox["in", FontSlant->"Italic"]], "\[Congruent]", "\[AlignmentMarker]", \(b\^2\/a\^2\)}], TraditionalForm], "\n", FormBox[ RowBox[{ SubscriptBox["S", StyleBox["out", FontSlant->"Italic"]], "=", "\[AlignmentMarker]", RowBox[{\(1\/2\), \(\(1 + A\)\/\(1 - A\)\), SubscriptBox["S", StyleBox["in", FontSlant->"Italic"]]}]}], TraditionalForm]}], "NumberedEquation", TextAlignment->AlignmentMarker, CellTags->{"Eq:21", "Ed:Change19"}], Cell[TextData[{ "For the particular hardware which we have constructed, the relevant \ figures are ", Cell[BoxData[ \(TraditionalForm\`T \[TildeEqual] 81.5\)]], " samples, and so ", Cell[BoxData[ \(TraditionalForm\`A \[TildeEqual] 0.98773\)]], " and ", Cell[BoxData[ FormBox[ RowBox[{ FractionBox[ SubscriptBox["S", StyleBox["out", FontSlant->"Italic"]], SubscriptBox["S", StyleBox["in", FontSlant->"Italic"]]], "\[TildeEqual]", "81"}], TraditionalForm]]], ". Note that ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["S", StyleBox["out", FontSlant->"Italic"]], "\[TildeEqual]", RowBox[{"T", " ", SubscriptBox["S", StyleBox["in", FontSlant->"Italic"]]}]}], TraditionalForm]]], " because the memory ", Cell[BoxData[ \(TraditionalForm\`T\)]], " of the filter coherently combines approximately ", Cell[BoxData[ \(TraditionalForm\`T\)]], " sine wave input samples, whereas it only incoherently combines the same \ number of Gaussian noise input samples." }], "Text", CellTags->"Ed:Change20"], Cell[TextData[{ "We may re-express these results in an alternative form where ", Cell[BoxData[ FormBox[ SubscriptBox["S", StyleBox["in", FontSlant->"Italic"]], TraditionalForm]]], " is expressed in dBHz (which we denote as ", Cell[BoxData[ FormBox[ SubscriptBox[\(S\^\[Prime]\), StyleBox["in", FontSlant->"Italic"]], TraditionalForm]]], "). ", Cell[BoxData[ FormBox[ SubscriptBox["S", StyleBox["in", FontSlant->"Italic"]], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ SubscriptBox[\(S\^\[Prime]\), StyleBox["in", FontSlant->"Italic"]], TraditionalForm]]], " are then related by" }], "Text"], Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox[\(S\^\[Prime]\), StyleBox["in", FontSlant->"Italic"]], "=", RowBox[{"10", RowBox[{\(log\_10\), "(", RowBox[{"B", " ", SubscriptBox["S", StyleBox["in", FontSlant->"Italic"]]}], ")"}]}]}], TraditionalForm]], "NumberedEquation", CellTags->{"Eq:22", "Ed:Change21"}], Cell[TextData[{ "whence from ", ButtonBox["eqn.", ButtonData:>"Eq:21", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedEquation", "Eq:21"], " and ", ButtonBox["eqn.", ButtonData:>"Eq:22", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedEquation", "Eq:22"] }], "Text", CellTags->"Ed:Change22"], Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["S", StyleBox["out", FontSlant->"Italic"]], "=", "\[AlignmentMarker]", RowBox[{\(1\/2\), \(\(1 + A\)\/\(1 - A\)\), FractionBox[ SuperscriptBox["10", FractionBox[ SubscriptBox[\(S\^\[Prime]\), StyleBox["in", FontSlant->"Italic"]], "10"]], "B"]}]}], TraditionalForm]], "NumberedEquation", CellTags->{"Eq:23", "Ed:Change23"}], Cell[TextData[{ "For our hardware, ", Cell[BoxData[ \(TraditionalForm\`B = 12\ kHz\)]], ", ", Cell[BoxData[ \(TraditionalForm\`T = 81.5\)]], ", ", Cell[BoxData[ \(TraditionalForm\`A = 0.98773\)]], ", and so ", ButtonBox["eqn.", ButtonData:>"Eq:23", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedEquation", "Eq:23"], " reduces to ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["S", StyleBox["out", FontSlant->"Italic"]], "=", RowBox[{"0.0067", "\[Times]", SuperscriptBox["10", FractionBox[ SubscriptBox[\(S\^\[Prime]\), StyleBox["in", FontSlant->"Italic"]], "10"]]}]}], TraditionalForm]]], "." }], "Text", CellTags->"Ed:Change24"], Cell["\<\ We have presented results for the statistical properties of the filter \ response to a sine wave and to Gaussian noise, considered separately. We will \ now extend these results to the (more realistic) case of a sum of a sine wave \ plus Gaussian noise. In this case, the statistical properties of the squared \ response are not completely described by means and covariances, and so we \ resort to a treatment using a full probability density function (PDF).\ \>", "Text"], Cell[TextData[{ "The probability density function of the squared filter response ", Cell[BoxData[ \(TraditionalForm\`E \[Congruent] \ \[LeftBracketingBar]\(F\_n\)(\[Omega])\[RightBracketingBar]\^2\)]], " is given by a Rician distribution" }], "Text"], Cell[BoxData[ \(TraditionalForm\`P(E) = \(1\/\(a\^\[Prime]\)\^2\) exp {\(-\(\((E + \(b\^\[Prime]\)\^2)\)\/\(a\^\[Prime]\)\^2\)\)} \ \(I\_0\) {\(2 \( E\^\(1\/2\)\) b\^\[Prime]\)\/\(a\^\[Prime]\)\^2}\)], \ "NumberedEquation", SpanMaxSize->Infinity, CellTags->{"Eq:24", "Ed:Change25"}], Cell[TextData[{ "where ", Cell[BoxData[ \(TraditionalForm\`\(I\_0\)(z)\)]], " is the zeroth order modified Bessel function of the first kind. This \ result is strictly correct only for complex Gaussian noise, but the \ approximation is good for real-valued noise when ", Cell[BoxData[ \(TraditionalForm\`T \[GreaterGreater] \(2 \[Pi]\)\/\[Omega]\)]], " (i.e. for all filters other than those near DC). Independent observations \ of ", Cell[BoxData[ \(TraditionalForm\`E\)]], " are obtained by using the sparse sampling scheme which was designed for \ independent observations of ", Cell[BoxData[ \(TraditionalForm\`\(F\_n\)(\[Omega])\)]], ". Strictly, ", Cell[BoxData[ \(TraditionalForm\`E\)]], " decorrelates twice as rapidly as ", Cell[BoxData[ \(TraditionalForm\`\(F\_n\)(\[Omega])\)]], ", and so the sparse sampling scheme errs on the side of caution for the \ purposes of an average squared response." }], "Text", CellTags->"Ed:Change26"], Cell[TextData[{ "We implement an averaging procedure in the hardware, and so we must also \ derive the PDF of the averaged squared response. Thus, defining ", Cell[BoxData[ \(TraditionalForm\`E\_D\)]], " as the average of ", Cell[BoxData[ \(TraditionalForm\`D\)]], " independent observations of ", Cell[BoxData[ \(TraditionalForm\`E\)]], ", we obtain (see ", ButtonBox["Appendix", ButtonData:>"Sect:8", ButtonStyle->"Hyperlink"], " ", CounterBox["Section", "Sect:8"], " for a detailed derivation of ", ButtonBox["eqn.", ButtonData:>"Eq:25", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedEquation", "Eq:25"], " and ", ButtonBox["eqn.", ButtonData:>"Eq:26", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedEquation", "Eq:26"], ")" }], "Text", CellTags->"Ed:Change27"], Cell[BoxData[ \(TraditionalForm\`\(P\_D\)( E\_D) = \(D\/\(a\^\[Prime]\)\^2\) \ \(\((E\_D\/\(b\^\[Prime]\)\^2)\)\^\(\(D - 1\)\/2\)\) exp {\(-\(\(D\ \((E + \ \(b\^\[Prime]\)\^2)\)\)\/\(a\^\[Prime]\)\^2\)\)} \(I\_\(D - 1\)\) {\(2 D\ \(E\_D\%\(1\/2\)\) \ b\^\[Prime]\)\/\(a\^\[Prime]\)\^2}\)], "NumberedEquation", CellTags->{"Eq:25", "Ed:Change28"}], Cell[TextData[{ "whence the first two moments of ", Cell[BoxData[ \(TraditionalForm\`\(P\_D\)(E\_D)\)]], " are given by" }], "Text"], Cell[BoxData[{ \(TraditionalForm\`\[LeftAngleBracket]E\_D\[RightAngleBracket] = \ \[AlignmentMarker]\(a\^\[Prime]\)\^2 + \(b\^\[Prime]\)\^2\), "\n", \(TraditionalForm\`\[LeftAngleBracket]\((E\_D)\)\^2\[RightAngleBracket] = \ \[AlignmentMarker]\((1 + 1\/D)\) \(a\^\[Prime]\)\^4 + 2 \((1 + 1\/D)\) \(\(a\^\[Prime]\)\^2\) \(b\^\[Prime]\)\^2 + \(b\^\ \[Prime]\)\^4\)}], "NumberedEquation", TextAlignment->AlignmentMarker, CellTags->{"Eq:26", "Ed:Change29"}], Cell[TextData[{ "thus the normalised variance of ", Cell[BoxData[ \(TraditionalForm\`E\_D\)]], " is given by" }], "Text"], Cell[BoxData[ \(TraditionalForm\`\(Var(E\_D)\)\/\[LeftAngleBracket]E\_D\ \[RightAngleBracket]\^2 = \(\(a\^\[Prime]\)\^2\ \((\(a\^\[Prime]\)\^2 + 2 \( \ b\^\[Prime]\)\^2)\)\)\/\(D\ \((\(a\^\[Prime]\)\^2 + \ \(b\^\[Prime]\)\^2)\)\^2\)\)], "NumberedEquation", CellTags->"Eq:27"], Cell[TextData[{ "The statistics in ", ButtonBox["eqn.", ButtonData:>"Eq:26", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedEquation", "Eq:26"], " and ", ButtonBox["eqn.", ButtonData:>"Eq:27", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedEquation", "Eq:27"], " provide a simpler description of the properties of ", Cell[BoxData[ \(TraditionalForm\`E\_D\)]], " than does the full PDF ", Cell[BoxData[ \(TraditionalForm\`\(P\_D\)(E\_D)\)]], ". This simplicity is obtained at the cost of sacrificing knowledge of the \ higher order statistical properties of the nonGaussian ", Cell[BoxData[ \(TraditionalForm\`\(P\_D\)(E\_D)\)]], "." }], "Text", CellTags->{"Ed:Change30", "Ed:Change31"}], Cell[TextData[{ "For practical applications we must define a suitable threshold for signal \ detection in the presence of noise. The average squared output ", Cell[BoxData[ \(TraditionalForm\`E\_D\)]], " follows a distribution ", Cell[BoxData[ \(TraditionalForm\`\(P\_D\)(E\_D)\)]], " of values which depend on both the signal level ", Cell[BoxData[ \(TraditionalForm\`a\^2\)]], " and the noise level ", Cell[BoxData[ \(TraditionalForm\`b\^2\)]], ". Loosely speaking, a signal is deemed to be present if very little of the \ upper tail of ", Cell[BoxData[ \(TraditionalForm\`\(P\_D\)(E\_D)\)]], " (without signal present) lies above the value of ", Cell[BoxData[ \(TraditionalForm\`E\_D\)]], " which is actually observed. We will quantify this by defining a threshold \ function" }], "Text"], Cell[BoxData[ \(TraditionalForm\`Q(\(b\^\[Prime]\)\^2, \(a\^\[Prime]\)\^2, s) \[Congruent] \[LeftAngleBracket]E\_D\[RightAngleBracket] {1 + \(s\ \ \((Var(E\_D))\)\^\(1\/2\)\)\/\[LeftAngleBracket]E\_D\[RightAngleBracket]}\)], \ "NumberedEquation", SpanMaxSize->Infinity, CellTags->{"Eq:28", "Ed:Change32"}], Cell[TextData[{ "which is the value that ", Cell[BoxData[ \(TraditionalForm\`E\_D\)]], " would have if it were displaced by ", Cell[BoxData[ \(TraditionalForm\`s\)]], " standard deviations from its mean ", Cell[BoxData[ \(TraditionalForm\`\[LeftAngleBracket]E\_D\[RightAngleBracket]\)]], ". Assuming that the effect of the signal appears in a single filter \ response, the probability ", Cell[BoxData[ FormBox[ SubscriptBox["P", StyleBox["MD", FontSlant->"Italic"]], TraditionalForm]]], " of a missed detection is given by" }], "Text"], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SubscriptBox["P", StyleBox["MD", FontSlant->"Italic"]], "(", \(\(b\^\[Prime]\)\^2, \(a\^\[Prime]\)\^2, s\), ")"}], "=", RowBox[{\(\[Integral]\_0\%\(Q(\(b\^\[Prime]\)\^2, \(a\^\[Prime]\)\^2, s)\)\), RowBox[{ SubscriptBox[ StyleBox[ RowBox[{ StyleBox["d", FontSlant->"Plain"], StyleBox["E", FontSlant->"Italic"]}]], "D"], \(\(P\_D\)(E\_D)\)}]}]}], TraditionalForm]], "NumberedEquation", CellTags->"Eq:29"], Cell[TextData[{ "Assuming that the averaged squared responses of all 128 filters are \ independent in the presence of pure noise, the probability ", Cell[BoxData[ FormBox[ SubscriptBox["P", StyleBox["FA", FontSlant->"Italic"]], TraditionalForm]]], " of a false alarm in one or more of the 128 filters is given by" }], "Text"], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SubscriptBox["P", StyleBox["FA", FontSlant->"Italic"]], "(", \(\(a\^\[Prime]\)\^2, s\), ")"}], "=", RowBox[{"1", "-", SuperscriptBox[ RowBox[{"[", RowBox[{\(\[Integral]\_0\%\(Q(0, \(a\^\[Prime]\)\^2, s)\)\), RowBox[{ SubscriptBox[ StyleBox[ RowBox[{ StyleBox["d", FontSlant->"Plain"], StyleBox["E", FontSlant->"Italic"]}]], "D"], \(\(P\_D\)( E\_D)\)}]}], "]"}], "128"]}]}], TraditionalForm]], "NumberedEquation", SpanMaxSize->Infinity, CellTags->"Eq:30"], Cell[TextData[{ "These assumptions about the filter responses are not strictly true in \ practice: the signal usually affects more than one filter response, and the \ responses of the filters to pure noise are not independent when they are \ sparse sampled according to the decorrelation time ", Cell[BoxData[ \(TraditionalForm\`T\)]], " of a single filter (i.e. the filter responses overlap)." }], "Text"], Cell[TextData[Cell[TextData[{ " ", ButtonBox["OPEN", ButtonData:>{ URL[ "http://www.luttrell.org.uk/papers/rapidlok/fig1.gif"], None}, Active->True, ButtonStyle->"Hyperlink"], " " }]]], "NumberedFigure", TextAlignment->Center, CellTags->{"Fig:1", "Ed:Change35"}], Cell[TextData[{ "Plot of thresholds against ", Cell[BoxData[ \(TraditionalForm\`D\)]], " for various ", Cell[BoxData[ FormBox[ SubscriptBox[\(S\^\[Prime]\), StyleBox["in", FontSlant->"Italic"]], TraditionalForm]]], " for a 1% missed detection and 1% false alarm probability" }], "Caption"], Cell[TextData[{ "We may obtain a useful approximation to the above signal detection results \ by assuming that ", Cell[BoxData[ \(TraditionalForm\`D\)]], " is large enough to enable us to use a Gaussian approximation for ", Cell[BoxData[ \(TraditionalForm\`\(P\_D\)(E\_D)\)]], ". The integrals in ", ButtonBox["eqn.", ButtonData:>"Eq:29", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedEquation", "Eq:29"], " and ", ButtonBox["eqn.", ButtonData:>"Eq:30", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedEquation", "Eq:30"], " may then be looked up in a table of error functions. Thus we may select \ ", Cell[BoxData[ FormBox[ SubscriptBox["P", StyleBox["MD", FontSlant->"Italic"]], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ SubscriptBox["P", StyleBox["FA", FontSlant->"Italic"]], TraditionalForm]]], " and deduce suitable values of ", Cell[BoxData[ \(TraditionalForm\`s\)]], " for various ", Cell[BoxData[ \(TraditionalForm\`\(b\^\[Prime]\)\^2\/\(a\^\[Prime]\)\^2\)]], " and ", Cell[BoxData[ \(TraditionalForm\`D\)]], ". In ", ButtonBox["Fig.", ButtonData:>"Fig:1", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedFigure", "Fig:1"], " we show plots of the threshold function for ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["P", StyleBox["MD", FontSlant->"Italic"]], "=", RowBox[{ SubscriptBox["P", StyleBox["FA", FontSlant->"Italic"]], "=", "0.01"}]}], TraditionalForm]]], ". Note that we have subtracted ", Cell[BoxData[ \(TraditionalForm\`\(a\^\[Prime]\)\^2\)]], " from all the threshold values to make the plots clearer. The false alarm \ line marks the lower bound of permitted thresholds (as a function of ", Cell[BoxData[ \(TraditionalForm\`D\)]], ") which guarantee ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["P", StyleBox["FA", FontSlant->"Italic"]], "\[LessEqual]", "0.01"}], TraditionalForm]]], ", and each missed detection curve marks the upper bound of permitted \ thresholds which guarantee ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["P", StyleBox["MD", FontSlant->"Italic"]], "\[LessEqual]", "0.01"}], TraditionalForm]]], " for a given ", Cell[BoxData[ FormBox[ SubscriptBox[\(S\^\[Prime]\), StyleBox["in", FontSlant->"Italic"]], TraditionalForm]]], ". Combining these inequalities reveals that for each signal-to-noise ratio \ there is a minimum value of ", Cell[BoxData[ \(TraditionalForm\`D\)]], " for which the missed detection and false alarm criteria are \ simultaneously satisfied." }], "Text", CellTags->{"Ed:Change33", "Ed:Change34"}] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ CounterBox["Section"], " Hardware" }], "Section", CellTags->"Sect:3"], Cell[TextData[{ "We wish to design the hardware in such a way that it most simply reflects \ the processes required to emulate a bank of single (complex) pole recursive \ filters, as defined in ", ButtonBox["eqn.", ButtonData:>"Eq:4", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedEquation", "Eq:4"], ". We envisage that this hardware device will be used as a front-end \ processor for a microprocessor as detailed in Reference ", ButtonBox["4", ButtonData:>"Ref:LuttrellPritchard1987", ButtonStyle->"Hyperlink"], "." }], "Text"], Cell[TextData[Cell[TextData[{ " ", ButtonBox["OPEN", ButtonData:>{ URL[ "http://www.luttrell.org.uk/papers/rapidlok/fig2.gif"], None}, Active->True, ButtonStyle->"Hyperlink"], " " }]]], "NumberedFigure", TextAlignment->Center, CellTags->{"Fig:2", "Ed:Change36"}], Cell["Block diagram of the complex pole recursive filter hardware", "Caption"], Cell[TextData[{ ButtonBox["Fig.", ButtonData:>"Fig:2", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedFigure", "Fig:2"], " shows the block diagram of a simple realisation of the circuit. Because \ we are using 1-bit quantisation of the analogue input voltage ", Cell[BoxData[ \(TraditionalForm\`V(t)\)]], ", only the sign of sample ", Cell[BoxData[ \(TraditionalForm\`V\_n\)]], " is retained; this may be achieved using a comparator and a latch. The \ latch is triggered each time a new sample is required for processing." }], "Text"], Cell[TextData[{ "The ", Cell[BoxData[ \(TraditionalForm\`\(\[CapitalTheta]\_n\)(\[Omega])\)]], " have the general form ", Cell[BoxData[ \(TraditionalForm\`exp[\(i\ n\ \[Pi]\ \[Omega]\)\/B]\)]], " (see ", ButtonBox["eqn.", ButtonData:>"Eq:5", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedEquation", "Eq:5"], ") and hence may be generated by sine and cosine functions. One bit \ quantisation suffices, and so look-up tables are chosen. This has the added \ advantage of retaining complete flexibility of choice in the single bit \ functions which may be used. Care must be taken to eliminate the DC component \ by alternating the sign of the 'zero' crossing points of each of the \ functions. The single bit representations are stored in an 8 K by 8 EPROM, \ where the eight outputs are multiplexed to use it as a 32 K \[Times] 2 \ memory. The 32 K is configured as 128 sets of 256 tables which can be \ accessed sequentially. We call this look-up table the 'phase table'." }], "Text", CellTags->"Ed:Change37"], Cell[TextData[{ "Each ", Cell[BoxData[ \(TraditionalForm\`\(F\_n\)(\[Omega])\)]], " is represented by ", Cell[BoxData[ \(TraditionalForm\`2 d\)]], " bits (", Cell[BoxData[ \(TraditionalForm\`d\)]], " for each of the real and imaginary parts), where ", Cell[BoxData[ \(TraditionalForm\`d\)]], " is determined by the value of ", Cell[BoxData[ \(TraditionalForm\`T\)]], " which is used. A suitable value of ", Cell[BoxData[ \(TraditionalForm\`d\)]], " is obtained by calculating ", Cell[BoxData[ \(TraditionalForm\`\(a\^\[Prime]\)\^2\)]], " from ", ButtonBox["eqn.", ButtonData:>"Eq:18", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedEquation", "Eq:18"], " using appropriate values of ", Cell[BoxData[ \(TraditionalForm\`a\^2\)]], " and ", Cell[BoxData[ \(TraditionalForm\`A\^2\)]], " to estimate the mean of the squared noise response. For a 1-bit quantised \ input, ", Cell[BoxData[ \(TraditionalForm\`a\^2\)]], " is forced to satisfy ", Cell[BoxData[ \(TraditionalForm\`a\^2 \[TildeEqual] 1\)]], ", and so from ", ButtonBox["eqn.", ButtonData:>"Eq:18", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedEquation", "Eq:18"], " we obtain ", Cell[BoxData[ \(TraditionalForm\`\(a\^\[Prime]\)\^2 \[TildeEqual] 8\)]], ". A suitable value of ", Cell[BoxData[ \(TraditionalForm\`d\)]], " should accomodate this squared response, and so we choose ", Cell[BoxData[ \(TraditionalForm\`d = 4\)]], ". The simplest practical method of implementing the transformation ", Cell[BoxData[ \(TraditionalForm\`\(\(F\_n\)(\[Omega])\)\[LongRightArrow] A\ \(\(F\_n\)(\[Omega])\)\)]], " for ", Cell[BoxData[ \(TraditionalForm\`d < 16\)]], " is by using tables, which we call 'decrement tables'. These tables are \ realised as a ", Cell[BoxData[ \(TraditionalForm\`2\^d\[Times]d\)]], " EPROM; this would also permit us to use a more complicated memory term \ than ", Cell[BoxData[ \(TraditionalForm\`A\ \(\(F\_n\)(\[Omega])\)\)]], ", if required. The 4-bit quantisation of ", Cell[BoxData[ \(TraditionalForm\`A\ \(\(F\_n\)(\[Omega])\)\)]], " is sufficient to accomodate ", Cell[BoxData[ \(TraditionalForm\`\(a\^\[Prime]\)\^2\)]], ", but is nevertheless rather coarse and leads to difficulties in \ implementing the required decrement factor ", Cell[BoxData[ \(TraditionalForm\`A\)]], ". We therefore introduce a further 4 bits to accommodate a binary fraction \ correction to the leading 4 bits. This may be viewed equivalently as an 8-bit \ representation, where the upper 4 bits are used to record the (1-bit) output \ of the multiplier, and the lower 4 bits are used to attain the required \ finesse for the decrement operation." }], "Text"], Cell["\<\ The decrement tables may be generated by calculating a decaying exponential \ using real numbers, which is then rounded to produce 8-bit entries for the \ table. Care must be taken to hard limit the extrema of the tables to avoid \ overflow or underflow in the single pole recursive filters. It should be \ noted that the 8-bit approximation used in these tables is a very coarse \ representation which divides an exponential decay into five regions: there is \ a band around zero where there is no decay, there are two hard limiting \ regions at the positive and negative extremes, and there are two bands in \ between where the decrement is one least significant bit.\ \>", "Text"], Cell[TextData[{ "As the input is a time series ", Cell[BoxData[ \(TraditionalForm\`V\_n\)]], ", consisting only of 1 and 0 (or equivalently 1 and -1), a complex \ multiplication to generate ", Cell[BoxData[ \(TraditionalForm\`\(\(\[CapitalTheta]\_n\)(\[Omega])\) V\_n\)]], " in ", ButtonBox["eqn.", ButtonData:>"Eq:4", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedEquation", "Eq:4"], " is not necessary. Two real multiplications are used, each of which \ reduces to an exclusive NOR with 1-bit quantisation. These outputs are added \ to ", Cell[BoxData[ \(TraditionalForm\`A\ \(\(F\_n\)(\[Omega])\)\)]], " as in ", ButtonBox["eqn.", ButtonData:>"Eq:4", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedEquation", "Eq:4"], ". In practical realisations of the circuit, an exclusive OR gate is used \ together with two's complement addition at the adder. This inverts ", Cell[BoxData[ \(TraditionalForm\`\(F\_n\)(\[Omega])\)]], " in the complex plane twice (leaving it unchanged) and simplifies the \ hardware. If the output of the multiplier is 0, then ", Cell[BoxData[ \(TraditionalForm\`A\ \(\(F\_n\)(\[Omega])\)\)]], " is incremented, and if the output is 1, then ", Cell[BoxData[ \(TraditionalForm\`A\ \(\(F\_n\)(\[Omega])\)\)]], " is decremented; the new value ", Cell[BoxData[ \(TraditionalForm\`\(F\_\(n + 1\)\)(\[Omega])\)]], " is stored in the register. The circuit shown in ", ButtonBox["Fig.", ButtonData:>"Fig:2", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedFigure", "Fig:2"], " is designed to operate in a 'lock step' mode. The value ", Cell[BoxData[ \(TraditionalForm\`\(F\_n\)(\[Omega])\)]], " is latched at the output of its register, and the decrement table is then \ allowed to settle. Simultaneously, the input ", Cell[BoxData[ \(TraditionalForm\`V\_n\)]], " is multiplied with the corresponding output of the phase table and this \ too is allowed to settle." }], "Text"], Cell[TextData[{ "If the digital bandwidth available exceeds the analogue bandwidth which is \ being analysed, then it is advantageous to use the same circuit in parallel \ to calculate ", Cell[BoxData[ \(TraditionalForm\`\(F\_n\)(\[Omega])\)]], " for several frequencies. The elements of the block diagram which are \ represented as three diemensional shapes are blocks of memory where the extra \ dimension represents ", Cell[BoxData[ \(TraditionalForm\`\[Omega]\)]], ". As a single input sample ", Cell[BoxData[ \(TraditionalForm\`V\_n\)]], " is held in the input latch, a counter scans through the registers and the \ look-up tables. In any particular application we would seek to accommodate \ the update of all ", Cell[BoxData[ \(TraditionalForm\`\(F\_n\)(\[Omega])\)]], " within the input sample time interval ", Cell[BoxData[ \(TraditionalForm\`\(\([\)\(t\_n, t\_\(n + 1\)\)\(]\)\)\)]], " (see ", ButtonBox["eqn.", ButtonData:>"Eq:1", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedEquation", "Eq:1"], ")." }], "Text"], Cell[TextData[{ "We arrange 128 filters to be evenly spaced in frequency so that they span \ the range ", Cell[BoxData[ \(TraditionalForm\`\(\([\)\(0, B\)\(]\)\)\)]], "; the phase table entries then repeat with a period equal to that of the \ tables in the lowest frequency filter. In the example, the filter widths are \ 94 Hz (", Cell[BoxData[ \(TraditionalForm\`\(\(=\)\(\(12 kHz\)\/128\)\)\)]], "), and so the lowest frequency filter is centred on 47 Hz, and the repeat \ period is 21 ms. This determines the length of the phase table whose elements \ are accessed cyclically as the input samples arrive." }], "Text", CellTags->"Ed:Change38"], Cell[TextData[{ "The real and imaginary parts of each ", Cell[BoxData[ \(TraditionalForm\`\(F\_n\)(\[Omega])\)]], " may be multiplexed into the same memory to save space and hardware, but \ this results in a speed and timing complexity penalty. For the breadboard \ circuit, the real and imaginary parts were separated for simplicity." }], "Text"], Cell[TextData[{ "When an external processor requires a sample ", Cell[BoxData[ \(TraditionalForm\`\(F\_n\)(\[Omega])\)]], ", the address of the register is placed on the channel request lines shown \ in ", ButtonBox["Fig.", ButtonData:>"Fig:2", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedFigure", "Fig:2"], ". When the counter next reaches this address, a pulse is output from the \ comparator which loads the current value of ", Cell[BoxData[ \(TraditionalForm\`\(F\_n\)(\[Omega])\)]], " into a second set of latches. The contents of these may be examined at \ any time before the next value of ", Cell[BoxData[ \(TraditionalForm\`\(F\_n\)(\[Omega])\)]], " is loaded." }], "Text"], Cell[TextData[{ "In the prototype circuit, we use a switch on the channel request lines and \ direct the real and imaginary outputs to 8-bit digital-to-analogue \ convertors. The complex value of any one of the ", Cell[BoxData[ \(TraditionalForm\`\(F\_n\)(\[Omega])\)]], " can then be viewed by using it to drive the ", Cell[BoxData[ \(TraditionalForm\`x\)]], " and ", Cell[BoxData[ \(TraditionalForm\`y\)]], " inputs of an oscilloscope to create an Argand diagram (or Lissajous \ figure). This proves to be a very useful representation for diagnostic \ purposes." }], "Text"], Cell["\<\ The prototype circuit is designed to reduce the centre frequency ambiguity of \ a low data rate signal in a satellite band. The width of the signal in the \ band is about 100 Hz (which is commensurate with the width 94 Hz of each \ digital filter), and the width of the band is 12 kHz. The communications band \ is translated down to baseband (0-12 kHz) and Nyquist sampled, followed by \ input to each of the 128 filters. This requires a clock frequency of 3.072 \ MHz (128 \[Times] 24 kHz) which is well within the scope of TTL logic. The \ maximum propagation delay around the recursive filter loop is therefore 325 \ ns, as is the propagation delay from the counters through the phase tables, \ exclusive OR gates and adders to the registers. This allows 150 ns 8K \ \[Times] 8 CMOS EPROMS to be used for the phase and decrement tables, \ although faster (50 ns) random access memories have to be used for the \ registers.\ \>", "Text"], Cell[TextData[{ "The memory which we use for the decrement tables is only 256 \[Times] 8 in \ size. We note that the use of a larger memory could absorb the adder and \ multiplier functions by connecting extra input lines on the EPROM to the \ sample input ", Cell[BoxData[ \(TraditionalForm\`V\_n\)]], " and the phase tables ", Cell[BoxData[ \(TraditionalForm\`\(\[CapitalTheta]\_n\)(\[Omega])\)]], ", and then adjusting the decrement table appropriately." }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ CounterBox["Section"], " Results" }], "Section", CellTags->"Sect:4"], Cell["\<\ To help understand the behaviour of the filter which we have constructed, it \ is instructive to view the Argand plane representation of their state spaces \ during operation. The display produced shows which states of the filter have \ been accessed. The more frequently accessed states are brighter, although the \ transition sequence is not exactly enumerable from the display.\ \>", "Text"], Cell[TextData[Cell[TextData[{ " ", ButtonBox["OPEN", ButtonData:>{ URL[ "http://www.luttrell.org.uk/papers/rapidlok/fig3.gif"], None}, Active->True, ButtonStyle->"Hyperlink"], " " }]]], "NumberedFigure", TextAlignment->Center, CellTags->{"Ed:Change40", "Fig:3"}], Cell[TextData[{ "Response of a filter to various noise levels at centre frequency\n\n", Cell[BoxData[ FormBox[GridBox[{ { RowBox[{ StyleBox["a", FontSlant->"Italic"], " ", "50", " ", "dBHz"}], " ", " ", " ", \(d\ 20\ dBHz\)}, {\(b\ 40\ dBHz\), " ", " ", " ", \(e\ noise\)}, {\(c\ 30\ dBHz\), " ", " ", " ", RowBox[{ "f", " ", "noise", " ", "averaged", " ", "for", " ", "4", " ", StyleBox["s", FontSlant->"Plain"], " ", RowBox[{"(", RowBox[{"4", " ", StyleBox["s", FontSlant->"Plain"], " ", "exposure"}], ")"}]}]} }], TraditionalForm]]] }], "Caption", GridBoxOptions->{ColumnAlignments->{Left}}], Cell[TextData[{ ButtonBox["Fig.", ButtonData:>"Fig:3", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedFigure", "Fig:3"], " shows photographs of the response of a typical filter to the frequency at \ the centre of its band in varying signal-to-noise conditions. The noise is \ Gaussian and covers the input band which is 0-12 kHz. At 50 dBHz the \ registers are fully saturated and there is barely enough noise to create a \ 'dither' signal. As the signal-to-noise ratio is decreased to 30 dBHz, the \ profile of the noise response becomes apparent. At 20 dBHz the display \ structure is similar in appearance to that generated by pure noise; the \ signal perturbs the probability distribution of the states by much less than \ the width of the noise response. For the processor to work well, we require \ that the noise response is contained mostly within the allowed states within \ the Argand plane at low signal-to-noise ratios. This condition is manifestly \ satisfied, and so the choice ", Cell[BoxData[ \(TraditionalForm\`d = 4\)]], " is adequate. All exposures in ", ButtonBox["Fig.", ButtonData:>"Fig:3", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedFigure", "Fig:3"], " are for ", Cell[BoxData[ \(TraditionalForm\`1\/30\)]], " second except for ", ButtonBox["Fig.", ButtonData:>"Fig:3", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedFigure", "Fig:3"], "f, which shows noise averaged (by the film) for 4 seconds. In ", ButtonBox["Fig.", ButtonData:>"Fig:3", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedFigure", "Fig:3"], "f, the striations apparent near the edges of the photograph are due to the \ modulo-16 resynchronisation, which occurs when a state transition across the \ hard limiting boundaries imposed by the registers is attempted." }], "Text", CellTags->"Ed:Change41"], Cell[TextData[Cell[TextData[{ " ", ButtonBox["OPEN", ButtonData:>{ URL[ "http://www.luttrell.org.uk/papers/rapidlok/fig4.gif"], None}, Active->True, ButtonStyle->"Hyperlink"], " " }]]], "NumberedFigure", TextAlignment->Center, CellTags->{"Fig:4", "Ed:Change42"}], Cell[TextData[{ "Response of a filter to a noiseless sine wave at various frequencies", "\n\n", Cell[BoxData[ FormBox[GridBox[{ { RowBox[{ StyleBox["a", FontSlant->"Italic"], " ", "Input", " ", "signal", " ", "at", " ", "centre", " ", "frequency"}], " ", " ", " ", \(d\ + 188\ Hz\ \((2\ BINS)\)\)}, {\(b\ CF + 47 Hz\ \((1\/2\ BIN)\)\), " ", " ", " ", \(e\ + 470\ Hz\ \((5\ BINS)\)\)}, {\(c\ + 97\ Hz\ \((1\ BIN)\)\), " ", " ", " ", \(f\ + 2\ kHz\)} }], TraditionalForm]]] }], "Caption", GridBoxOptions->{ColumnAlignments->{Left}}], Cell[TextData[{ ButtonBox["Fig.", ButtonData:>"Fig:4", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedFigure", "Fig:4"], " shows the response of a filter to various signals with no noise. In ", ButtonBox["Fig.", ButtonData:>"Fig:4", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedFigure", "Fig:4"], "a, the signal frequency is at the centre frequency of the filter; there is \ little difference between this and the 50 dBHz example in ", ButtonBox["Fig.", ButtonData:>"Fig:3", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedFigure", "Fig:3"], "a. As the frequency is shifted, the small cluster of occupied states \ begins to move around the periphery of the Argand plane as expected. At first \ it makes rapid transitions from corner to corner, and then begins to occupy \ more states on the edges. At a frequency difference of about half the width \ of the response of the filter, the occupied states begin to collapse towards \ the origin, as required. An interesting case occurs when the input frequency \ is far away from the filter centre frequency, illustrated by ", ButtonBox["Fig.", ButtonData:>"Fig:4", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedFigure", "Fig:4"], "f. Here, the amplitude of the filter must still be incremented or \ decremented by one during each clock cycle due to the 1-bit quantisation. \ However these increments and decrements are not random, as is the case when \ the input is uncorrelated noise. The result is a state distribution in the \ Argand plane which is smaller than the noise response depicted in ", ButtonBox["Fig.", ButtonData:>"Fig:3", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedFigure", "Fig:3"], "e", ". This is because a greater proportion of the energy of the input signal \ lies outside the pass band of the filter than in the case where the input is \ noise alone." }], "Text"], Cell[TextData[{ "The filter bank is connected to a TMS320-10 microprocessor, so that \ various processing algorithms, such as averaging, may be tested in \ conjunction with this system. The nature of these algorithms and the extra \ hardware are outside the scope of this paper, which seeks only to describe \ the filter system. We therefore confine ourselves to a few simple, empirical \ results which will illustrate the behaviour of the circuit and its \ suitability for our stated application. The power spectra shown in ", ButtonBox["Fig.", ButtonData:>"Fig:5", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedFigure", "Fig:5"], " and ", ButtonBox["Fig.", ButtonData:>"Fig:6", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedFigure", "Fig:6"], " are generated by the microprocessor and are output to an oscilloscope \ using digital-to-analogue convertors." }], "Text", CellTags->"Ed:Change43"], Cell[TextData[Cell[TextData[{ " ", ButtonBox["OPEN", ButtonData:>{ URL[ "http://www.luttrell.org.uk/papers/rapidlok/fig5.gif"], None}, Active->True, ButtonStyle->"Hyperlink"], " " }]]], "NumberedFigure", TextAlignment->Center, CellTags->{"Fig:5", "Ed:Change44"}], Cell["Display of averaged filter output energies showing aliassing", "Caption", GridBoxOptions->{ColumnAlignments->{Left}}], Cell[TextData[{ ButtonBox["Fig.", ButtonData:>"Fig:5", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedFigure", "Fig:5"], " shows an example of the aliasing behaviour. The horizontal axis on the \ display represents frequency, and the vertical axis represents the averaged \ filter response ", Cell[BoxData[ \(TraditionalForm\`E\_D\)]], " which is calculated over a large number ", Cell[BoxData[ \(TraditionalForm\`D\)]], " of sparse samples by the microprocessor. The input is a pure sine wave at \ 3 kHz. The strong response at 3 kHz is accompanied by the large peak at 9 kHz \ with ", Cell[BoxData[ \(TraditionalForm\`4\/9\)]], " of the power. This second large peak is predicted by theory, and is an \ example of the worst case where there are coincident solutions of ", ButtonBox["eqn.", ButtonData:>"Eq:12", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedEquation", "Eq:12"], " (i.e. ", Cell[BoxData[ \(TraditionalForm\`i\_1 = 1, i\_2 = 3, j = \(-1\)\)]], " and ", Cell[BoxData[ \(TraditionalForm\`i\_1 = 3, i\_2 = 1, j = 0\)]], "). The smaller peaks are also predicted by other solutions; for example, \ two of the next largest solutions are when ", Cell[BoxData[ \(TraditionalForm\`i\_1 = 1, i\_2 = 3, j = 0\)]], " and when ", Cell[BoxData[ \(TraditionalForm\`i\_1 = 1, i\_2 = 3, j = 1\)]], ". These give rise to aliases which are about 0.08 and 0.58 of the way up \ the band, respectively. Where there are minor coincident solutions, the \ relative phase of the contributions from each source is not predictable, but \ this does not concern us here, as it will not materially affect the \ performance of the circuit." }], "Text", CellTags->"Ed:Change45"], Cell[TextData[Cell[TextData[{ " ", ButtonBox["OPEN", ButtonData:>{ URL[ "http://www.luttrell.org.uk/papers/rapidlok/fig6.gif"], None}, Active->True, ButtonStyle->"Hyperlink"], " " }]]], "NumberedFigure", TextAlignment->Center, CellTags->{"Fig:6", "Ed:Change46"}], Cell[TextData[{ "Display of filter output energies with various amounts of averaging\n\n", StyleBox["a", FontSlant->"Italic"], " Averaged response after 7 sparse samples\n", StyleBox["b", FontSlant->"Italic"], " Averaged response after 27 sparse samples\n", StyleBox["c", FontSlant->"Italic"], " Averaged response after 46 sparse samples" }], "Caption", GridBoxOptions->{ColumnAlignments->{Left}}], Cell[TextData[{ ButtonBox["Fig.", ButtonData:>"Fig:6", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedFigure", "Fig:6"], " shows the power spectrum produced by a sine wave added to Gaussian noise \ where the signal-to-noise ratio is 30 dBHz. This is the lowest \ signal-to-noise ratio which will normally be present in our application. The \ averaged response is shown after 7, 27 and 46 sparse samples in ", ButtonBox["Fig.", ButtonData:>"Fig:6", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedFigure", "Fig:6"], "a, b and c, respectively. The response to a signal which has 50 bit per \ second phase shift keying (the modulation scheme which we use) is similar in \ appearance but slightly degraded due to spreading of the peak." }], "Text"], Cell[TextData[Cell[TextData[{ " ", ButtonBox["OPEN", ButtonData:>{ URL[ "http://www.luttrell.org.uk/papers/rapidlok/fig7.gif"], None}, Active->True, ButtonStyle->"Hyperlink"], " " }]]], "NumberedFigure", TextAlignment->Center, CellTags->{"Fig:7", "Ed:Change47"}], Cell["\<\ Graph of time to detect a noisy sine wave with 95% confidence against input \ signal-to-noise ratio (using a crude thresholding algorithm) \[EmptyUpTriangle] experimental results\ \>", "Caption", GridBoxOptions->{ColumnAlignments->{Left}}, CellTags->"Ed:Change48"], Cell[TextData[{ "Simple tests can be made to determine the time required to find which \ filter has its centre frequency nearest to the signal frequency. We do not \ implement the full false alarm and missed detection apparatus of ", ButtonBox["Fig.", ButtonData:>"Fig:1", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedFigure", "Fig:1"], ". Instead we choose a crude 'first past the post' thresholding technique. \ In this scheme, we sum the sparse sample values ", Cell[BoxData[ \(TraditionalForm\`E\)]], " and detect which filter sum first passes some preset threshold criterion. \ We choose this criterion so that (for a given signal-to-noise ratio) the \ probability of finding the correct filter approximately exceeds 95%. We show \ these results in ", ButtonBox["Fig.", ButtonData:>"Fig:7", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedFigure", "Fig:7"], ", where we see that at 30 dBHz we correctly localise the signal in 60 ms. \ This is two orders of magnitude better than the best state of the art method \ [", ButtonBox["3", ButtonData:>"Ref:DuddlePowellWarnerGannonAeigus1982", ButtonStyle->"Hyperlink"], "]. However, the results which we obtain are not as good as the theoretical \ predictions which are plotted in ", ButtonBox["Fig.", ButtonData:>"Fig:1", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedFigure", "Fig:1"], " (these predictions are relatively insensitive to missed detection and \ false alarm levels). This is because of the extremely crude thresholding \ scheme which we use." }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ CounterBox["Section"], " Conclusions" }], "Section"], Cell["\<\ The hardware which we have developed primarily to reduce the frequency \ ambiguity of low data rate communications signals in satellite channels shows \ considerable improvement over state of the art techniques. Using the \ averaging and crude thresholding technique which we have described, followed \ by a short serial search using a swept filter (not described in this paper), \ we have achieved phase lock to a 30 dBHz signal in a 12 kHz band of ambiguity \ more than an order of magnitude faster than the seven seconds required \ before. Using a full false alarm and missed detection analysis, and an \ improved phase lock loop rather than a final sweep, we expect a further \ substantial improvement.\ \>", "Text"], Cell["\<\ There are implications for military and business users of low data rate \ satellite links. The probability of hostile exploitation of the link will be \ reduced, especially for short communications where signal acquisition \ accounts for a large proportion of the time 'on the air'. Simplex or 'over, \ over' communications become feasible with an attendant saving in transmission \ time and battery power. In cases where signal fading is encountered, the \ carrier may be rapidly reacquired.\ \>", "Text"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ CounterBox["Section"], " Acknowledgement" }], "Section"], Cell[TextData[{ "We thank A.C. Baynham for supplying the data that are shown in ", ButtonBox["Fig.", ButtonData:>"Fig:5", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedFigure", "Fig:5"], ", ", ButtonBox["Fig.", ButtonData:>"Fig:6", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedFigure", "Fig:6"], " and ", ButtonBox["Fig.", ButtonData:>"Fig:7", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedFigure", "Fig:7"], "." }], "Text", CellTags->{"Ed:Change49", "Ed:Change50"}] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ CounterBox["Section"], " References" }], "Section"], Cell[TextData[{ "1 JONES, C.H.: 'A manpack satellite communications earth station', ", StyleBox["Radio & Electron. Eng.", FontSlant->"Italic"], ", 1981, ", StyleBox["51", FontWeight->"Bold"], ", (6), pp. 259-271" }], "Reference", CellTags->"Ref:Jones1981"], Cell["\<\ 2 SKILTON, P.J., and WESTALL, I.L.: ''Manpack' SHF ground terminal'. Military \ Microwaves, London, 1982\ \>", "Reference", CellTags->{"Ed:Change1", "Ref:SkiltonWestall1982"}], Cell["\<\ 3 DUDDLE, A.R., POWELL, R.G., WARNER, M.G., GANNON, K.M., and AEIGUS, M.: \ 'Rapid synchronisation of CPSK modems used for tactical satellite \ communications'. Marconi Space and Defence Systems Ltd., Report No. BL 4105, \ Study Contract No. A57A/1059, 1982\ \>", "Reference", CellTags->"Ref:DuddlePowellWarnerGannonAeigus1982"], Cell[TextData[{ Cell[TextData[{ " ", ButtonBox["OPEN", ButtonData:>{ URL[ "http://www.luttrell.org.uk/papers/2190221/2190221.nb"], None}, Active->True, ButtonStyle->"Hyperlink"], " " }]], "4 LUTTRELL, S.P., and PRITCHARD, J.A.S.: 'Signal transforming device'. \ Patent application 8707301, 1987" }], "Reference", CellTags->"Ref:LuttrellPritchard1987"], Cell["\<\ 5 ABRAMOWITZ, M., and STEGUN, I.A.: 'Handbook of mathematical functions' \ (Dover Publications, New York, 1972), pp. 375, 377 & 1026\ \>", "Reference", CellTags->"Ref:AbramowitzStegun1972"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ CounterBox["Section"], " Appendix" }], "Section", CellTags->"Sect:8"], Cell["\<\ In this Appendix we present a detailed derivation of the response PDF of an \ average of squared values of sparse samples.\ \>", "Text"], Cell[TextData[{ "The basic PDF is the Rician ", Cell[BoxData[ \(TraditionalForm\`P(E)\)]] }], "Text"], Cell[BoxData[ \(TraditionalForm\`P(E) = \(1\/\(a\^\[Prime]\)\^2\) exp {\(-\(\((E + \(b\^\[Prime]\)\^2)\)\/\(a\^\[Prime]\)\^2\)\)} \ \(I\_0\) {\(2 \( E\^\(1\/2\)\) b\^\[Prime]\)\/\(a\^\[Prime]\)\^2}\)], \ "NumberedEquation", CellTags->{"Ed:Change51", "Eq:31"}], Cell[TextData[{ "The characteristic function ", Cell[BoxData[ \(TraditionalForm\`C(k)\)]], " is defined as" }], "Text"], Cell[BoxData[ FormBox[ RowBox[{\(C(k)\), "\[Congruent]", RowBox[{\(\[Integral]\_0\%\[Infinity]\), RowBox[{ StyleBox[ RowBox[{ StyleBox["d", FontSlant->"Plain"], StyleBox["E", FontSlant->"Italic"]}]], " ", \(e\^\(\(-i\)\ k\ E\)\), \(P(E)\)}]}]}], TraditionalForm]], "NumberedEquation", CellTags->"Eq:32"], Cell["then", "Text"], Cell[BoxData[ \(TraditionalForm\`C( k) = \(\(exp(\(-\(\(b\^\[Prime]\)\^2\/\(a\^\[Prime]\)\^2\)\))\)\/\(1 \ + i\ k\ \(a\^\[Prime]\)\^2\)\) exp[\(1\/\(1 + i\ k\ \(a\^\[Prime]\)\^2\)\) \(b\^\[Prime]\)\^2\/\(a\^\ \[Prime]\)\^2]\)], "NumberedEquation", SpanMaxSize->Infinity, CellTags->{"Eq:33", "Ed:Change52"}], Cell[TextData[{ "The characteristic function of the PDF of an average ", Cell[BoxData[ \(TraditionalForm\`E\_D\)]], " of ", Cell[BoxData[ \(TraditionalForm\`D\)]], " independently distributed random variables ", Cell[BoxData[ \(TraditionalForm\`E\)]], " is then ", Cell[BoxData[ \(TraditionalForm\`\(C(k\/D)\)\^D\)]], ", and so" }], "Text", CellTags->"Ed:Change53"], Cell[BoxData[ FormBox[ RowBox[{\(\(P\_D\)(E\_D)\), "=", RowBox[{\(1\/\(2 \[Pi]\)\), RowBox[{\(\[Integral]\_\(-\[Infinity]\)\%\(+\[Infinity]\)\), RowBox[{ StyleBox[ RowBox[{ StyleBox["d", FontSlant->"Plain"], StyleBox["k", FontSlant->"Italic"]}]], " ", \(exp(i\ k\ E\_D)\), \(\(C(k\/D)\)\^D\)}]}]}]}], TraditionalForm]], "NumberedEquation", CellTags->{"Eq:34", "Ed:Change54"}], Cell[TextData[{ "Substituting ", Cell[BoxData[ \(TraditionalForm\`\[Xi] = 1 + \(i\ k\ \(a\^\[Prime]\)\^2\)\/D\)]], " yields" }], "Text", CellTags->"Ed:Change55"], Cell[BoxData[ \(TraditionalForm\`\(P\_D\)(E\_D) = \(D\/\((a\^\[Prime])\)\^\(2 D\)\) exp {\(-\(\(D\ \((E\_D + \ \(b\^\[Prime]\)\^2)\)\)\/\(a\^\[Prime]\)\^2\)\)} \(1\/\(2 \[Pi]\ i\)\) \(\ \[Integral]\_\(\[Xi]\_-\)\%\(\[Xi]\_+\)d\[Xi]\ \(exp( s\ D\ E\_D)\) \(1\/\[Xi]\^D\) exp[\(D\ \(b\^\[Prime]\)\^2\)\/\(\[Xi]\ \ \(a\^\[Prime]\)\^4\)]\)\)], "NumberedEquation", CellTags->{"Eq:35", "Ed:Change56"}], Cell[TextData[{ "where ", Cell[BoxData[ \(TraditionalForm\`\(\[Xi]\_\[PlusMinus]\) \[Congruent] 1\/\(a\^\[Prime]\)\^2 \[PlusMinus] i\ \[Infinity]\)]], ". The Bromwich integral in ", ButtonBox["eqn.", ButtonData:>"Eq:35", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedEquation", "Eq:35"], " may be evaluated by noting that (eqn. 29.3.80 of Reference ", ButtonBox["5", ButtonData:>"Ref:AbramowitzStegun1972", ButtonStyle->"Hyperlink"], ")" }], "Text", CellTags->"Ed:Change57"], Cell[BoxData[ FormBox[GridBox[{ {\(\(L {\(\((t\/a)\)\^\(\[Nu]\/2\)\) \(\(J\_\[Nu]\)( 2 \((a\ t)\)\^\(1\/2\))\)} = \(exp(\(-\(a\/\[Xi]\)\))\)\ \/\[Xi]\^\(\[Nu] + 1\)\)\(,\)\), " ", " ", " ", \(\[Nu] > \(-1\)\)} }], TraditionalForm]], "NumberedEquation", CellTags->{"Eq:36", "Ed:Change58"}], Cell[TextData[{ "where ", Cell[BoxData[ \(TraditionalForm\`L {f(t)}\)]], " is the Laplace transform of ", Cell[BoxData[ \(TraditionalForm\`f(t)\)]], ". Now using (eqn. 9.6.3 of Reference ", ButtonBox["5", ButtonData:>"Ref:AbramowitzStegun1972", ButtonStyle->"Hyperlink"], ")" }], "Text"], Cell[BoxData[ \(TraditionalForm\`\(J\_\[Nu]\)( i\ t) = \(i\^\[Nu]\) \(\(I\_\[Nu]\)(t)\)\)], "NumberedEquation", CellTags->"Eq:37"], Cell["which leads to", "Text"], Cell[BoxData[ \(TraditionalForm\`\(P\_D\)( E\_D) = \(D\/\(a\^\[Prime]\)\^2\) \ \(\((E\_D\/\(b\^\[Prime]\)\^2)\)\^\(\(D - 1\)\/2\)\) exp {\(-\(\(D\ \((E + \ \(b\^\[Prime]\)\^2)\)\)\/\(a\^\[Prime]\)\^2\)\)} \(I\_\(D - 1\)\) {\(2 D\ \(E\_D\%\(1\/2\)\) \ b\^\[Prime]\)\/\(a\^\[Prime]\)\^2}\)], "NumberedEquation", CellTags->{"Eq:38", "Ed:Change59"}], Cell[TextData[{ "The moments ", Cell[BoxData[ \(TraditionalForm\`M\_r\)]], " of ", Cell[BoxData[ \(TraditionalForm\`\(P\_D\)(E\_D)\)]], " may be derived by making use of the characteristic function ", Cell[BoxData[ \(TraditionalForm\`\(C(k\/D)\)\^D\)]], ". Thus" }], "Text", CellTags->"Ed:Change60"], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{\(M\_r\), "\[Congruent]", "\[AlignmentMarker]", RowBox[{\(\[Integral]\_0\%\[Infinity]\), RowBox[{ SubscriptBox[ StyleBox[ RowBox[{ StyleBox["d", FontSlant->"Plain"], StyleBox["E", FontSlant->"Italic"]}]], "D"], " ", \(\((E\_D)\)\^r\), \(\(P\_D\)(E\_D)\)}]}]}], "\[IndentingNewLine]", "=", "\[AlignmentMarker]", RowBox[{\(i\^r\/\(2 \[Pi]\)\), RowBox[{\(\[Integral]\_0\%\[Infinity]\), RowBox[{ SubscriptBox[ StyleBox[ RowBox[{ StyleBox["d", FontSlant->"Plain"], StyleBox["E", FontSlant->"Italic"]}]], "D"], RowBox[{\(\[Integral]\_\(-\[Infinity]\)\%\(+\[Infinity]\)\), RowBox[{ StyleBox[ RowBox[{ StyleBox["d", FontSlant->"Plain"], StyleBox["k", FontSlant->"Italic"]}]], " ", \(exp( i\ k\ E\_D)\), \(\[PartialD]\^r\(\([\)\(\(C( k\/D)\)\^D\)\(]\)\)\/\[PartialD]k\^r\)}]}]}]}]}]\ }], TraditionalForm]], "NumberedEquation", TextAlignment->AlignmentMarker, CellTags->{"Eq:39", "Ed:Change61"}], Cell[TextData[{ "We wish to calculate ", Cell[BoxData[ \(TraditionalForm\`M\_1\)]], " and ", Cell[BoxData[ \(TraditionalForm\`M\_2\)]], ", and so we need the following results" }], "Text"], Cell[BoxData[{ \(TraditionalForm\`\[PartialD]\(\([\)\(\(C(k\/D)\)\^D\)\(]\)\)\/\ \[PartialD]k = \[AlignmentMarker]\(-\(i\ [\(a\^\[Prime]\)\^2\/\(1 + \(i\ k\ \ \(a\^\[Prime]\)\^2\)\/D\) + \(b\^\[Prime]\)\^2\/\((1 + \(i\ k\ \ \(a\^\[Prime]\)\^2\)\/D)\)\^2]\)\ [\(C(k\/D)\)\^D]\)\), "\n", \(TraditionalForm\`\[PartialD]\^2\(\([\)\(\(C(k\/D)\)\^D\)\(]\)\)\/\ \[PartialD]k\^2 = \[AlignmentMarker]\[AlignmentMarker]\(-\(\([\)\(\(\((1 + \ 1\/D)\) \(a\^\[Prime]\)\^4\)\/\((1 + \(i\ k\ \(a\^\[Prime]\)\^2\)\/D)\)\^2 + \ \(2 \((1 + 1\/D)\) \(\(a\^\[Prime]\)\^2\) \(b\^\[Prime]\)\^2\)\/\((1 + \(i\ k\ \ \(a\^\[Prime]\)\^2\)\/D)\)\^3 + \(b\^\[Prime]\)\^4\/\((1 + \(i\ k\ \(a\^\ \[Prime]\)\^2\)\/D)\)\^4\)\(]\)\)\ [\(C(k\/D)\)\^D]\)\)}], "NumberedEquation",\ TextAlignment->AlignmentMarker, SpanMaxSize->Infinity, CellTags->{"Eq:40", "Ed:Change62", "Ed:Change63", "Ed:Change64"}], Cell[TextData[{ "which are easily obtained by differentiating ", Cell[BoxData[ \(TraditionalForm\`\(C(k\/D)\)\^D\)]], ". To perform the integration over ", Cell[BoxData[ \(TraditionalForm\`k\)]], " we need to use the result" }], "Text", CellTags->"Ed:Change65"], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{\(K\_r\), "\[Congruent]", "\[AlignmentMarker]", RowBox[{\(1\/\(2 \[Pi]\)\), RowBox[{\(\[Integral]\_\(-\[Infinity]\)\%\(+\[Infinity]\)\), RowBox[{ StyleBox[ RowBox[{ StyleBox["d", FontSlant->"Plain"], StyleBox["k", FontSlant->"Italic"]}]], " ", \(exp( i\ k\ E\_D)\), \(1\/\((1 + \(i\ k\ \(a\^\[Prime]\)\^2\)\/D)\ \)\^r\), \(exp[\(1\/\(1 + \(i\ k\ \(a\^\[Prime]\)\^2\)\/D\)\) \(D\ \(b\^\ \[Prime]\)\^2\)\/\(a\^\[Prime]\)\^2]\)}]}]}]}], "\[IndentingNewLine]", "=", "\[AlignmentMarker]", \(D\ \(exp(\(D\ \(b\^\[Prime]\)\^2\)\/\(a\^\ \[Prime]\)\^2)\) \(\(P\_r\)(D\ E\_D)\)\)}], TraditionalForm]], "NumberedEquation", TextAlignment->AlignmentMarker, CellTags->{"Eq:41", "Ed:Change66"}], Cell[TextData[{ "C