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}], "Text"], Cell[TextData[{ Cell[BoxData[ FormBox[ ButtonBox["TYPO", ButtonData:>"Ed:Change15", Active->True, ButtonStyle->"Hyperlink"], TextForm]]], "\"equations (14), (15) and (16)\" changed to \"equation (14), equation \ (15) and equation (16)\" to accommodate three hyperlinks" }], "Text"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["A PROPOSAL FOR A NEW METHOD OF CALIBRATING RADAR SENSITIVITY", "Title"], Cell["Dr S P Luttrell", "Author"], Cell["\<\ This paper appeared as BS1 Divisional Memo, No. 37, January 1987.\ \>", "Text"], Cell["\<\ Abstract We derive an invariant form for the degree of overlap of a pair of \ probability density functions. The invariance property is such that the \ functional form of the overlap expression is the same for all underlying \ coordinate systems which are related by nonsingular transformations. A \ particular application of these results is in the definition of a new \ calibration procedure for quantifying the ability of a radar system to \ discriminate between signal plus noise and noise alone. The invariance \ property of our definition ensures that the underlying receiver law (e.g. \ linear or logarithmic) does not need to be known in order to conduct the \ calibration. We present a practical means of implementing our new calibration \ procedure, and for ensuring that it is consistent with the old \"tangential \ method\".\ \>", "Abstract"], Cell[CellGroupData[{ Cell["Statement of the problem", "Section 1"], Cell[TextData[{ "There is a long standing interest in the calibration of radar systems to \ provide a quantitative measure of the ability to discriminate between signal \ plus noise and noise alone. Clearly such a figure of merit is an essential \ property of the overall radar system. Hitherto such calibrations have been \ carried out by performing an eyeball comparison of the receiver output both \ with and without a signal present. A specific criterion for discrimination is \ the \"tangential method\" [", ButtonBox["1", ButtonData:>"Ref:Lucas1966", ButtonStyle->"Hyperlink"], "], where a signal is deemed to be discriminated if the minimum voltage \ which is observed when both signal and noise are present is greater than the \ maximum voltage when noise alone is present. The figure of merit is that \ signal to noise ratio which causes the above maximum and minimum to be the \ same: this signal level is the weakest that is deemed to be discriminated." }], "Text"], Cell[TextData[{ "Clearly the tangential method of radar sensitivity calibration is ", StyleBox["ad hoc", FontSlant->"Italic"], ", but it nevertheless produces a useful figure of merit with minimal \ investment of resources (i.e. an oscilloscope). We shall now define a new \ figure of merit which has little of the ", StyleBox["ad hoc", FontSlant->"Italic"], " nature of the tangential method, and which has the added virtue of being \ completely insensitive to the law of the receiver (e.g. linear or \ logarithmic). The only penalty which we have to pay for this elegance is a \ concomitant increase in the complexity of the processing that has to be \ performed on the receiver output. However, as we shall see, the necessary \ operations may easily be performed using only an analogue to digital \ conversion of the receiver output, followed by some simple calculations which \ are well within the capabilities of a small personal computer system." }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell["The proposed solution", "Section"], Cell[TextData[{ "In order to formulate our proposed solution to the above radar calibration \ problem we must first of all introduce some mathematical notation. Thus we \ shall denote the receiver output voltage by the symbol ", Cell[BoxData[ \(TraditionalForm\`V\)]], ": this is the quantity which is used to control the ", Cell[BoxData[ \(TraditionalForm\`y\)]], " deflection of the oscilloscope, display in the tangential method. The \ properties of ", Cell[BoxData[ \(TraditionalForm\`V\)]], " are described by its probability density function (PDF) ", Cell[BoxData[ \(TraditionalForm\`P(V)\)]], ": this is the function which, when the response of the oscilloscope \ phosphor is taken into account as well, describes the appearance of the \ signal on the oscilloscope display. Typically ", Cell[BoxData[ \(TraditionalForm\`P(V)\)]], " is unimodal and so it creates a well defined horizontal band on the \ oscilloscope." }], "Text"], Cell[TextData[{ "We shall define two possible forms for ", Cell[BoxData[ \(TraditionalForm\`P(V)\)]], ", namely" }], "Text"], Cell[BoxData[ FormBox[GridBox[{ {\(P(V) = N(V)\), " ", " ", " ", \((PDF\ for\ noise\ alone)\)} }], TraditionalForm]], "NumberedEquation"], Cell["and", "Text"], Cell[BoxData[ FormBox[GridBox[{ {\(P(V) = S(V)\), " ", " ", " ", \((PDF\ for\ signal\ plus\ noise)\)} }], TraditionalForm]], "NumberedEquation"], Cell[TextData[{ "where the signal strength which corresponds to ", Cell[BoxData[ \(TraditionalForm\`S(V)\)]], " is predetermined. In practice both ", Cell[BoxData[ \(TraditionalForm\`S(V)\)]], " and ", Cell[BoxData[ \(TraditionalForm\`N(V)\)]], " are unimodal with the peak of ", Cell[BoxData[ \(TraditionalForm\`S(V)\)]], " lying at a higher voltage than the peak of ", Cell[BoxData[ \(TraditionalForm\`N(V)\)]], ". Loosely speaking the signal level which is sought by the tangential \ method is such that the peaks of ", Cell[BoxData[ \(TraditionalForm\`N(V)\)]], " and ", Cell[BoxData[ \(TraditionalForm\`S(V)\)]], " are only just separated, so that the upper tail of ", Cell[BoxData[ \(TraditionalForm\`N(V)\)]], " is hidden under the peak of ", Cell[BoxData[ \(TraditionalForm\`S(V)\)]], " and the lower tail of ", Cell[BoxData[ \(TraditionalForm\`S(V)\)]], " is hidden under the peak of ", Cell[BoxData[ \(TraditionalForm\`N(V)\)]], "." }], "Text"], Cell[TextData[{ "The tangential method is ", StyleBox["ad hoc", FontSlant->"Italic"], " and so it can not be formulated theoretically with any rigour. \ Nevertheless the essence of the tangential method is to discover that signal \ level which separates ", Cell[BoxData[ \(TraditionalForm\`N(V)\)]], " and ", Cell[BoxData[ \(TraditionalForm\`S(V)\)]], " (as above), and so we shall seek a theoretical alternative which \ encapsulates this essence whilst possibly differing in inessential details. \ Furthermore we shall strive to endow the theory with the desirable property \ of insensitivity to details of the receiver law (e.g. linear of logarithmic). \ Such insensitivity in the resulting calibration procedure will lead to \ identical calibration results for radar systems which are identical in all \ respects other than the receiver law." }], "Text"], Cell["\<\ We shall call any theoretical expression which is insensitive to receiver law \ an \"invariant\". The fundamental invariants from which all other invariants \ are constructed are\ \>", "Text"], Cell[BoxData[ FormBox[ RowBox[{\(I(V)\), "=", RowBox[{\(N(V)\), " ", StyleBox[ RowBox[{"d", StyleBox["V", FontSlant->"Italic"]}]]}]}], TraditionalForm]], "NumberedEquation", CellTags->"Eq:3"], Cell["and", "Text"], Cell[BoxData[ FormBox[ RowBox[{\(J(V)\), "=", RowBox[{\(S(V)\), " ", StyleBox[ RowBox[{"d", StyleBox["V", FontSlant->"Italic"]}]]}]}], TraditionalForm]], "NumberedEquation", CellTags->"Eq:4"], Cell[TextData[{ "where ", Cell[BoxData[ FormBox[ StyleBox[ RowBox[{"d", StyleBox["V", FontSlant->"Italic"]}]], TraditionalForm]]], " is an infinitesimal voltage change. The invariance of ", Cell[BoxData[ \(TraditionalForm\`I(V)\)]], " and ", Cell[BoxData[ \(TraditionalForm\`J(V)\)]], " with respect to changes in the receiver law is demonstrated as follows. \ Let a new receiver law be defined by the transformation" }], "Text"], Cell[BoxData[ \(TraditionalForm\`V\^\[Prime] = \(V\^\[Prime]\)(V)\)], "NumberedEquation"], Cell["and the inverse transformation", "Text"], Cell[BoxData[ \(TraditionalForm\`V = V(V\^\[Prime])\)], "NumberedEquation"], Cell[TextData[{ "where the output voltage ", Cell[BoxData[ \(TraditionalForm\`V\^\[Prime]\)]], " under the new receiver law is functionally related to the output voltage \ ", Cell[BoxData[ \(TraditionalForm\`V\)]], " under the old receiver law , and vice versa. It is important that both \ the forward and reverse tranformations should both be well behaved within the \ range of voltages under consideration in order that the above invariants be \ truly invariant: this condition is equivalent to the requirement that the \ transformation has no singularities within the range of interest. A \ particular example of a commonly occurring transformation arises when ", Cell[BoxData[ \(TraditionalForm\`V\)]], " corresponds to a linear law and ", Cell[BoxData[ \(TraditionalForm\`V\^\[Prime]\)]], " corresponds to a logarithmic law. This leads to" }], "Text"], Cell[BoxData[ \(TraditionalForm\`V\^\[Prime] = log(V)\)], "NumberedEquation", CellTags->"Eq:7"], Cell["and", "Text"], Cell[BoxData[ \(TraditionalForm\`V = exp(V\^\[Prime])\)], "NumberedEquation"], Cell[TextData[{ "where the logarithmic singularity in ", ButtonBox["equation", ButtonData:>"Eq:7", ButtonStyle->"Hyperlink"], " (", CounterBox["NumberedEquation", "Eq:7"], ") does not lie within the range of practical interest, and is therefore \ acceptable." }], "Text"], Cell[TextData[{ Cell[BoxData[ \(TraditionalForm\`I(V)\)]], " and ", Cell[BoxData[ \(TraditionalForm\`J(V)\)]], " transform to ", Cell[BoxData[ \(TraditionalForm\`I(V\^\[Prime])\)]], " and ", Cell[BoxData[ \(TraditionalForm\`J(V\^\[Prime])\)]], " as follows" }], "Text"], Cell[BoxData[ FormBox[ RowBox[{\(\(I\^\[Prime]\)(V\^\[Prime])\), "=", "\[AlignmentMarker]", RowBox[{ RowBox[{\(I(V)\), FractionBox[ SuperscriptBox[ StyleBox[ RowBox[{"d", StyleBox["V", FontSlant->"Italic"]}]], "\[Prime]"], StyleBox[ RowBox[{"d", StyleBox["V", FontSlant->"Italic"]}]]]}], "\[IndentingNewLine]", "=", "\[AlignmentMarker]", RowBox[{ RowBox[{\(N(V(V\^\[Prime]))\), SuperscriptBox[ StyleBox[ RowBox[{"d", StyleBox["V", FontSlant->"Italic"]}]], "\[Prime]"]}], "\[IndentingNewLine]", "=", "\[AlignmentMarker]", RowBox[{\(\(N\^\[Prime]\)(V\^\[Prime])\), SuperscriptBox[ StyleBox[ RowBox[{"d", StyleBox["V", FontSlant->"Italic"]}]], "\[Prime]"]}]}]}]}], TraditionalForm]], "NumberedEquation", TextAlignment->AlignmentMarker, CellTags->{"Ed:Change1", "Eq:9"}], Cell["and similarly", "Text"], Cell[BoxData[ FormBox[ RowBox[{\(\(J\^\[Prime]\)(V\^\[Prime])\), "=", RowBox[{\(\(S\^\[Prime]\)(V\^\[Prime])\), SuperscriptBox[ StyleBox[ RowBox[{"d", StyleBox["V", FontSlant->"Italic"]}]], "\[Prime]"]}]}], TraditionalForm]], "NumberedEquation", CellTags->"Eq:10"], Cell[TextData[{ "where ", Cell[BoxData[ \(TraditionalForm\`\(N\^\[Prime]\)(V\^\[Prime])\)]], " is the output distribution for noise alone for the new receiver law (and \ similarly ", Cell[BoxData[ \(TraditionalForm\`\(S\^\[Prime]\)(V\^\[Prime])\)]], "). ", ButtonBox["Equation", ButtonData:>"Eq:9", ButtonStyle->"Hyperlink"], " (", CounterBox["NumberedEquation", "Eq:9"], ") and ", ButtonBox["equation", ButtonData:>"Eq:10", ButtonStyle->"Hyperlink"], " (", CounterBox["NumberedEquation", "Eq:10"], ") are manifestly of the same form as ", ButtonBox["equation", ButtonData:>"Eq:3", ButtonStyle->"Hyperlink"], " (", CounterBox["NumberedEquation", "Eq:3"], ") and ", ButtonBox["equation", ButtonData:>"Eq:4", ButtonStyle->"Hyperlink"], " (", CounterBox["NumberedEquation", "Eq:4"], "), so the procedure which is used to evaluate ", Cell[BoxData[ \(TraditionalForm\`I(V)\)]], " and ", Cell[BoxData[ \(TraditionalForm\`J(V)\)]], " is the same whatever the receiver law is." }], "Text", CellTags->"Ed:Change2"], Cell[TextData[{ "Our task now is to formulate a calibration procedure which is expressed \ entirely in terms of the invariants ", Cell[BoxData[ \(TraditionalForm\`I(V)\)]], " and ", Cell[BoxData[ \(TraditionalForm\`J(V)\)]], ", and which is therefore insensitive to the receiver law. As we have \ pointed out we need only preserve the essence of the tangential method in our \ formulation, so we shall construct invariants which express the degree of \ \"overlap\" of the distributions ", Cell[BoxData[ \(TraditionalForm\`N(V)\)]], " and ", Cell[BoxData[ \(TraditionalForm\`S(V)\)]], ". There are two very simple invariants which are relevant to quantifying \ the amount of overlap, namely" }], "Text"], Cell[BoxData[ FormBox[ RowBox[{\(K(A)\), "=", RowBox[{\(\[Integral]\+\(\(N(V)\)\/\(S(V)\) < A\)\), RowBox[{\(N(V)\), " ", StyleBox[ RowBox[{"d", StyleBox["V", FontSlant->"Italic"]}]]}]}]}], TraditionalForm]], "NumberedEquation", CellTags->{"Ed:Change3", "Eq:11"}], Cell["and", "Text"], Cell[BoxData[ FormBox[ RowBox[{\(L(B)\), "=", RowBox[{\(\[Integral]\+\(\(S(V)\)\/\(N(V)\) < B\)\), RowBox[{\(S(V)\), " ", StyleBox[ RowBox[{"d", StyleBox["V", FontSlant->"Italic"]}]]}]}]}], TraditionalForm]], "NumberedEquation", CellTags->{"Ed:Change4", "Eq:12"}], Cell[TextData[{ "The integrands are both constructed directly out of the invariants ", Cell[BoxData[ \(TraditionalForm\`I(V)\)]], " and ", Cell[BoxData[ \(TraditionalForm\`J(V)\)]], ", and the ranges of integration are constructed out of the ratio of \ invariants ", Cell[BoxData[ \(TraditionalForm\`\(I(V)\)\/\(J(V)\)\)]], " together with a pair of constants ", Cell[BoxData[ \(TraditionalForm\`A\)]], " and ", Cell[BoxData[ \(TraditionalForm\`B\)]], " (which are also invariants). ", Cell[BoxData[ \(TraditionalForm\`K\)]], " and ", Cell[BoxData[ \(TraditionalForm\`L\)]], " are therefore manifestly invariant quantities which have the following \ interpretation: ", Cell[BoxData[ \(TraditionalForm\`K\)]], " is the amount of the noise distribution ", Cell[BoxData[ \(TraditionalForm\`N(V)\)]], " which lies \"beneath\" the signal distribution ", Cell[BoxData[ \(TraditionalForm\`S(V)\)]], ", and vice versa for ", Cell[BoxData[ \(TraditionalForm\`L\)]], ". In fact this simple interpretation is only true if ", Cell[BoxData[ \(TraditionalForm\`A = \(B = 1\)\)]], "." }], "Text", CellTags->"Ed:Change5"], Cell[TextData[{ "Our task is to construct a suitable function of ", Cell[BoxData[ \(TraditionalForm\`K\)]], " and ", Cell[BoxData[ \(TraditionalForm\`L\)]], " (which is therefore itself invariant) which quantifies the overlap of the \ distributions ", Cell[BoxData[ \(TraditionalForm\`N(V)\)]], " and ", Cell[BoxData[ \(TraditionalForm\`S(V)\)]], ". There are many candidate solutions to this problem, but we require a \ particularly simple form in order that the resulting calibration procedure is \ practically realisable. We therefore suggest that a class of invariants \ should be defined as" }], "Text"], Cell[BoxData[ \(TraditionalForm\`M(A, B) = \(K(A) + L(B)\)\/2\)], "NumberedEquation", CellTags->"Ed:Change6"], Cell[TextData[{ "A particular member of this class, ", Cell[BoxData[ \(TraditionalForm\`M(1, 1)\)]], ", measures the combined extent to which ", Cell[BoxData[ \(TraditionalForm\`N(V)\)]], " and ", Cell[BoxData[ \(TraditionalForm\`S(V)\)]], " lie under each other, and so we recommend that ", Cell[BoxData[ \(TraditionalForm\`M(1, 1)\)]], " should be adopted as the required quantitative measure of the overlap of \ ", Cell[BoxData[ \(TraditionalForm\`N(V)\)]], " and ", Cell[BoxData[ \(TraditionalForm\`S(V)\)]], ". ", Cell[BoxData[ \(TraditionalForm\`M(1, 1)\)]], " has the following general behaviour: ", Cell[BoxData[ \(TraditionalForm\`\(M(1, 1)\)\[LongRightArrow]1\)]], " as the signal level decreases to zero, and ", Cell[BoxData[ \(TraditionalForm\`\(M(1, 1)\)\[LongRightArrow]0\)]], " as the signal level goes to infinity. Between these two extremes lies the \ signal level which causes ", Cell[BoxData[ \(TraditionalForm\`N(V)\)]], " and ", Cell[BoxData[ \(TraditionalForm\`S(V)\)]], " to be just discriminated according to the tangential method, and we must \ empirically find the corresponding value ", Cell[BoxData[ \(TraditionalForm\`T\)]], " of ", Cell[BoxData[ \(TraditionalForm\`M(1, 1)\)]], " in order to calibrate our new (invariant) method against the old \ (tangential) method. Having done this we can then calibrate the radar in \ future by finding the signal level which satisfies the calibration formula" }], "Text"], Cell[BoxData[ \(TraditionalForm\`\(K(1) + L(1)\)\/2 = T\)], "NumberedEquation", CellTags->{"Ed:Change7", "Eq:14"}], Cell["\<\ and then calculating the corresponding signal to noise ratio.\ \>", "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Practical implementation of the proposed solution", "Section"], Cell[TextData[{ ButtonBox["Equation", ButtonData:>"Eq:14", ButtonStyle->"Hyperlink"], " (", CounterBox["NumberedEquation", "Eq:14"], ") together with ", ButtonBox["equation", ButtonData:>"Eq:11", ButtonStyle->"Hyperlink"], " (", CounterBox["NumberedEquation", "Eq:11"], ") and ", ButtonBox["equation", ButtonData:>"Eq:12", ButtonStyle->"Hyperlink"], " (", CounterBox["NumberedEquation", "Eq:12"], ") define the calibration procedure, where we assume that ", Cell[BoxData[ \(TraditionalForm\`T\)]], " has been determined by an independent means. The only numerically hard \ parts are the two integrals which need to be evaluated. However we may \ approximate these integrals by using corresponding summation formulae over \ histograms" }], "Text", CellTags->"Ed:Change8"], Cell[BoxData[ \(TraditionalForm\`K(1) = h \(\[Sum]\+\(N\_i\/S\_i < 1\)N\_i\)\)], "NumberedEquation", CellTags->{"Ed:Change9", "Eq:15"}], Cell["and", "Text"], Cell[BoxData[ \(TraditionalForm\`L(1) = h \(\[Sum]\+\(S\_i\/N\_i < 1\)S\_i\)\)], "NumberedEquation", CellTags->{"Ed:Change10", "Eq:16"}], Cell[TextData[{ "where ", Cell[BoxData[ \(TraditionalForm\`h\)]], " is the histogram bin width, ", Cell[BoxData[ \(TraditionalForm\`i\)]], " indexes the ", Cell[BoxData[ \(TraditionalForm\`i\)]], "th histogram bin, and ", Cell[BoxData[ \(TraditionalForm\`N\_i\)]], " is the number of counts in the ", Cell[BoxData[ \(TraditionalForm\`i\)]], "th bin of the histogram which represents ", Cell[BoxData[ \(TraditionalForm\`N(V)\)]], " normalised so that the total number of counts in all bins is 1 (and \ similarly for ", Cell[BoxData[ \(TraditionalForm\`S\_i\)]], " and ", Cell[BoxData[ \(TraditionalForm\`S(V)\)]], "). 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The \ theoretical treatment has two principal advantages over the tangential \ method: theoretical rigour and insensitivity to receiver law (which we have \ called invariance). These advantages have been secured without an inordinate \ computational cost. The principal formulae for radar calibration are \ contained in ", ButtonBox["equation", ButtonData:>"Eq:14", ButtonStyle->"Hyperlink"], " (", CounterBox["NumberedEquation", "Eq:14"], "), ", ButtonBox["equation", ButtonData:>"Eq:15", ButtonStyle->"Hyperlink"], " (", CounterBox["NumberedEquation", "Eq:15"], ") and ", ButtonBox["equation", ButtonData:>"Eq:16", ButtonStyle->"Hyperlink"], " (", CounterBox["NumberedEquation", "Eq:16"], "), where the threshold parameter ", Cell[BoxData[ \(TraditionalForm\`T\)]], " is determined so that the new (invariant) method is consistent with the \ old (tangential) method as explained in the text." }], "Text", CellTags->"Ed:Change15"] }, Closed]], Cell[CellGroupData[{ Cell["Acknowledgement", "Section"], Cell["\<\ I am grateful to T B Nichols and B A Wyndham for bringing my attention to the \ problem of radar sensitivity calibration.\ \>", "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Reference", "Section"], Cell[TextData[{ "[1] Lucas W J, 1966, \"The tangential sensitivity of a detector video \ system with RF preamplification\", ", StyleBox["Proc IRE", FontSlant->"Italic"], ", ", StyleBox["113", FontWeight->"Bold"], "(8), 1321-1330" }], "Reference", CellTags->"Ref:Lucas1966"] }, Closed]] }, Open ]] }, FrontEndVersion->"5.0 for Microsoft Windows", ScreenRectangle->{{0, 1280}, {0, 941}}, WindowToolbars->{}, WindowSize->{665.375, 641}, WindowMargins->{{307.25, Automatic}, {Automatic, 50}}, Magnification->1, StyleDefinitions -> "Report.nb" ] (******************************************************************* Cached data follows. 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