ESTIMATING UNCERTAINTIES in ANTENNA MEASUREMENTS
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1 ESTIMATING UNCERTAINTIES i ANTENNA MEASUREMENTS EuCAP April 013 Gotheburg, Swede Michael H. Fracis Natioal Istitute of Stadards ad Techology 35 Broadway Boulder, CO USA fracis@boulder/ist.gov M. H. Fracis U.S. Natl. Ist. Stad. Tech. 1
2 OUTLINE 1. Itroductio. Sources of Ucertaity 3. Distributig Errors 4. Type A & Type B Ucertaities 5. Systematic ad Radom Errors 6. Methods of Estimatig Ucertaities 7. Example: ULSA Array Patter Ucertaities 8. Coclusio M. H. Fracis U.S. Natl. Ist. Stad. Tech.
3 INTRODUCTION Why do ucertaity aalysis? M. H. Fracis U.S. Natl. Ist. Stad. Tech. 3
4 I geeral, the result of a measuremet is oly a estimate of the value of the quatity subject to measuremet (the measurad) ad the result is complete oly whe it is accompaied by a quatitative statemet of its ucertaity. A A A 0, where A 0 is the best estimate ad ΔA is the ucertaity. M. H. Fracis U.S. Natl. Ist. Stad. Tech. 4
5 The error is the differece betwee the true value ad the measured value. The true value is ukowable so the error is ukowable. Error A A 0 t The ucertaity is the spread of values i which the true value may reasoably be expected to lie. M. H. Fracis U.S. Natl. Ist. Stad. Tech. 5
6 SOURCES of UNCERTAINTY i ANTENNA A good ucertaity aalysis requires that all major sources of ucertaity be idetified. M. H. Fracis U.S. Natl. Ist. Stad. Tech. 6
7 Probe/referece atea properties Aligmet/positioig Receive system Drift Impedace mismatch Leakage ad crosstalk Flexig cables Multipath Normalizatio No-uiform illumiatio M. H. Fracis U.S. Natl. Ist. Stad. Tech. 7
8 Aliasig Measuremet area trucatio Atea-atea multiple reflectios Radom errors (e.g., oise) M. H. Fracis U.S. Natl. Ist. Stad. Tech. 8
9 DISTRIBUTING ERRORS (ISO) The ISO approach is to assume that by some meas we have kowledge of how the errors are distributed. Let us ote some properties of distributio fuctios. 1. The variables x 1, x,..., x are idepedet oly if the distributio fuctio is separable, i.e., f ( x) f ( x, x,..., x ) f ( x ) f ( x )... f ( x ) f ( x ) i i i1. The expectatio value E of ay fuctio g(x) = g(x 1,x,...,x ) is E( g( x))... g( x) f ( x ) f ( x )... f ( x ) dxdx... dx M. H. Fracis U.S. Natl. Ist. Stad. Tech. 9
10 Mea ad Variace The mea ad variace of a liear fuctio y xi are ad y i i1 y i i1 i1 (1) () M. H. Fracis U.S. Natl. Ist. Stad. Tech. 10
11 To make use of (1) ad () for cases where y caot be writte as a sum of other fuctios, we must liearize the problem by usig the first two terms of the Taylor series expasio. This will give us the familiar form for g y xi i1 xi x (3) where the subscript x implies the derivative is evaluated at x, x,..., x. 1 x1 x x M. H. Fracis U.S. Natl. Ist. Stad. Tech. 11
12 If the errors are large, oe might be tempted to use the ext term i the Taylor series. This is ofte problematic sice the ext term i the Taylor series requires higher momets (see Appedix C), which may ot be well kow, especially type B ucertaities. I such cases, it usually makes sese to work directly with y rather tha its Taylor series expasio. M. H. Fracis U.S. Natl. Ist. Stad. Tech. 1
13 Cetral Limit Theorem It is ofte mistakely asserted that if the umber of error sources the the distributio fuctio approaches a ormal distributio. The followig examples show that this is ot a sufficiet coditio. First, cosider the fuctio y 101 x i1 i with distributio fuctios f 1 (x 1 )...f 101 (x 101 ), with f 1 =...=f 100 = the ormal distributio with μ 1 =...=μ 100 =0 ad σ 1 =...=σ 100 =σ a. f 101 = the rectagular distributio with μ 101 =0, ad σ 101 =10σ a. As expected, 101 y i 00 a i1 but the resultig distributio (see graph) is ot a ormal distributio. Secod, cosider σ 101 =0σ a with 500. We have plotted these curves plus a ormal distributio y a agaist y/σ y so they have the same ormalized area ad variace. M. H. Fracis U.S. Natl. Ist. Stad. Tech. 13
14 1 0.8 Relative Probability ormal + rectagular ormal ormal + wide rect. M. H. Fracis U.S. Natl. Ist. Stad. Tech. 14
15 What is the problem? We have ot satisfied the full set of criteria required for the cetral limit theorem to be valid! These criteria are 1. y x, i1 i. x i are radom idepedet variables with distributios f i (x i ), 3. for all x i, var(x i ) is bouded, 4. 64, 5. limvar(y)64, M. H. Fracis U.S. Natl. Ist. Stad. Tech. 15
16 y y the the limitig distributio of f is the stadard ormal y distributio. Criteria 3, 4, ad 5 imply that var(y) >> var(x i ) for all x i. It is this requiremet that is ot satisfied i the above examples. Aother example that does ot satisfy the above criteria is: Why? 0 y = x, ad var(x i ) =, i1 i i The secod order Taylor series also does ot satisfy the requiremets. Why? M. H. Fracis U.S. Natl. Ist. Stad. Tech. 16
17 Oe cosequece of these requiremets: If there is a domiat source of ucertaity that is ot ormally distributed, the it is difficult to associate a cofidece level with the value of σ. M. H. Fracis U.S. Natl. Ist. Stad. Tech. 17
18 TYPE A ad TYPE B UNCERTAINTIES Type A ucertaities are those that ca be estimated usig statistics (large umber of measuremets). Type B ucertaities are those that are estimated i ay other way. The ISO guide idicates that cofidece levels should ot be associated with the overall ucertaity i the presece of type B ucertaities. M. H. Fracis U.S. Natl. Ist. Stad. Tech. 18
19 ISO Method of Combiig Ucertaities We combie ucertaities as follows. For type A ucertaities the overall ucertaity is u A 1 where u i is the stadard deviatio of the i th compoet. i1 u i, (4) For type B ucertaities u B where u j is a estimate of the stadard deviatio for the distributio fuctio for the j th compoet. j1 u j, (5) M. H. Fracis U.S. Natl. Ist. Stad. Tech. 19
20 The overall ucertaity is u u u. c A B (6) The expaded ucertaity is U Ku, c (7) where K is a coverage factor typically with a value of or 3. K = is the value ormally used i iteratioal itercompariso measuremets. M. H. Fracis U.S. Natl. Ist. Stad. Tech. 0
21 Geometric Motivatio for Combiig Ucertaities I the real world, we sometimes have large errors but poor iformatio about the distributio of the errors ad we ca sometimes oly estimate a approximate upperboud error. If the errors are idepedet, the we ca visualize that they are orthogoal vectors i -space. The resultat vector is the root sum square (RSS) of the idividual compoets. We would ot associate a cofidece level with this result (or ay type B ucertaity). M. H. Fracis U.S. Natl. Ist. Stad. Tech. 1
22 Systematic ad Radom Errors Radom error ISO defiitio: The result of a measuremet mius the mea that would result from a ifiite umber of measuremets of the same measurad carried out uder repeatability coditios. [Compare this to a commo defiitio of: ucorrelated from measuremet to measuremet. ] Systematic error The mea that would result from a ifiite umber of measuremets of the same measurad carried out uder repeatability coditios mius the true value of the measurad. M. H. Fracis U.S. Natl. Ist. Stad. Tech.
23 M. H. Fracis U.S. Natl. Ist. Stad. Tech. 3
24 Corrected Measuremet Result The corrected (for systematic error) result of the measuremet gives the followig estimate of the measurad V M V M e s, (8) where e s is the best estimate of the systematic error. M. H. Fracis U.S. Natl. Ist. Stad. Tech. 4
25 METHODS OF ESTIMATING UNCERTAINTIES 1. Aalytical Method. Simulatio Method 3. Self-Compariso Method (Measuremet) M. H. Fracis U.S. Natl. Ist. Stad. Tech. 5
26 We wat to estimate the idividual u i ad u j i (4) ad (5). We do ot ecessarily eed to kow or i (3) to do this. Istead, we may be able to estimate directly usig oe of the methods above. y x j x j y xi x i M. H. Fracis U.S. Natl. Ist. Stad. Tech. 6
27 ANALYTICAL METHOD We use our kowledge of the theory to develop equatios to provide estimates of ucertaity. Advatages: More geerally apply; Valid for may situatios; Ca idetify upper-boud ucertaities. Limitatio: Some estimates, particularly worst-case estimates, of ucertaity are urealistically high. Example: I plaar ear-field measuremets, we assume a periodic probepositio error to obtai a equatio to estimate the ucertaity due to probepositio errors. This provides a upper-boud error. Periodic positio errors cause larger (tha o-periodic) errors i the patter but are cocetrated i two directios. Other types of positio errors give smaller patter errors but spread them out over more of the patter (see Yaghjia). M. H. Fracis U.S. Natl. Ist. Stad. Tech. 7
28 Fe (, ) F(, ) db z 13.5 cos B g(, ), where F e ad F are the patter with ad without errors respectively, g(θ,φ) is the ratio of the peak far-field patter to the patter value i the θ,φ directio ad is greater tha 1. Θ B is the directio of the peak far-field patter, ad δ z is the maximum z-positio error. M. H. Fracis U.S. Natl. Ist. Stad. Tech. 8
29 SIMULATION METHOD We use a simulatio to model a error ad estimate its effect o the result. Advatage: Study idividual errors; Use measured or ideal data; Ca study may cases. Limitatio: Must assume the form of the error; Results are ot geeral. Example: We model a receiver o-liearity effect by assumig the receiver output has additioal terms tha the liear term such as Output = c 1 *iput + c *(iput), ad the see how this affects the result. M. H. Fracis U.S. Natl. Ist. Stad. Tech. 9
30 M. H. Fracis U.S. Natl. Ist. Stad. Tech. 30
31 SELF-COMPARISON METHOD We chage the measuremet system i a cotrolled maer to chage the error i some kow way. Advatages: Directly measure some ucertaities; Ca verify simulatio ad theory; Applies to actual measuremet system. Limitatio: Time cosumig; Results are ot geeral. Example: Multiple reflectios betwee the atea uder test ad the probe have period of λ/ i the separatio distace. By chagig the separatio distace by λ/4 we ca chage the multiple reflectios from i phase to out of phase. By comparig the patter ad gai results from both measuremets we ca estimate the ucertaity due to multiple reflectios. M. H. Fracis U.S. Natl. Ist. Stad. Tech. 31
32 M. H. Fracis U.S. Natl. Ist. Stad. Tech. 3
33 M. H. Fracis U.S. Natl. Ist. Stad. Tech. 33
34 EXAMPLE ULSA ARRAY PATTERN UNCERTAINTIES UNCERTAINTY SOURCE UNCERTAINTY LEVEL (Relative to peak) Atea-atea multiple reflectios.8e-04 Multipath (room scatterig) 1.6e-04 Flexig cables 1.6e-04 Aligmet / positioig errors 9.0e-05 Leakage 9.0e-05 Noise / Radom errors 9.0e-05 Measuremet area trucatio 1.6e-05 Aliasig 1.6e-05 Receive system o-liearity 5.0e-06 ROOT SUM SQUARE Expaded ucertaity (K=) 4.0e e-04 M. H. Fracis U.S. Natl. Ist. Stad. Tech. 34
35 SIDELOBE LEVEL Expaded Ucertaity i db -30 db ±. -45 db db db db M. H. Fracis U.S. Natl. Ist. Stad. Tech. 35
36 Measuremet Itercompariso Sice there ca be much subjectivity i evaluatig type B ucertaities, some itercompariso measuremets are ecessary as a reality check! Measuremets from differet rages should agree to withi the combied ucertaities. That is, the ucertaity bars should overlap. M. H. Fracis U.S. Natl. Ist. Stad. Tech. 36
37 M. H. Fracis U.S. Natl. Ist. Stad. Tech. 37
38 CONCLUSION C Rigorous ucertaity aalysis is a oxymoro! If it were rigorous, it would ot be ucertai. C A measuremet is icomplete without a ucertaity aalysis. M. H. Fracis U.S. Natl. Ist. Stad. Tech. 38
39 Refereces 1. Iteratioal Orgaizatio for Stadards, "Guide to the Expressio of Ucertaity i Measuremet," revised Taylor, B.N. ad Kuyatt, C.E., Guidelies for Evaluatig ad Expressig the Ucertaity of NIST Measuremet Results, NIST Tech. Note 197, Miller, I ad Miller, M. Joh E. Freud s Mathematical Statistics, 6 th ed., Pretice Hall, Newell, A.C., "Error Aalysis Techiques for Plaar Near-Field Measuremets," IEEE Tras. Atea Propagat., AP-36, pp , Jue M. H. Fracis U.S. Natl. Ist. Stad. Tech. 39
40 5. IEEE Stadard , Recommeded Practice for Near- Field Atea Measuremets, Clause 9, December Fracis, M.H.; Newell, A.C.; Grimm, K.R.; Hoffma, J.; ad Schrak, H.E., Compariso of Ultralow Sidelobe Atea Far- Field Patters Usig the Plaar Near-Field Method ad the Far- Field Method, IEEE Atea Propagat. Mag., vol. 37, pp. 7-15, Dec Wittma, R.C.; Alpert, B.K. ad Fracis, M.H., Near-Field Atea Measuremets Usig Noideal Measuremet Locatios, IEEE Tras. Atea Propagat., vol. AP-46, pp.716-7, May M. H. Fracis U.S. Natl. Ist. Stad. Tech. 40
41 8. Yaghjia, A.D., Upper-Boud Errors i Far-Field Atea Parameters Determied from Plaar Near-Field Measuremets Part 1: Aalysis, Natl. Bur. Stad. Tech. Note 667, October Joy, E.B., Near-Field Rage Qualificatio Methodology, IEEE Tras. Atea Propagat., vol. AP-36, pp , Jue Wittma, R.C.; Fracis, M.H.; Muth, L.A. ad Lewis, R.L., Proposed Ucertaity Aalysis for RCS Measuremets, US Natl. Ist. Stad. Tech. It. Rep. NISTIR 5019, Ja M. H. Fracis U.S. Natl. Ist. Stad. Tech. 41
42 APPENDICES M. H. Fracis U.S. Natl. Ist. Stad. Tech. 4
43 APPENDIX A Liear Sums of Radom Variables y a x 1 1 If with distributio fuctios f ( x ), f ( x ),..., f ( x ) the the i1 i i mea ad the variace. y i i1 y i i1 M. H. Fracis U.S. Natl. Ist. Stad. Tech. 43
44 axf( x) dx y i i i i i i1 i1 ax 1 1f1( x1) dx1 fi( xi) dxi i1 axf ( x) dx fi( xi) dxi... i y i i1 y ax i i x f( ) i i xi dxi i1 i1 i1 M. H. Fracis U.S. Natl. Ist. Stad. Tech. 44
45 y ax i i i ax i i i ajxj j fi( xi) dxi i1 i1 ji i1 The secod term i the square brackets is zero whe x i ad x j are idepedet. Thus, ( ) y ax i i i fi xi dxi i i1 i1 i1 M. H. Fracis U.S. Natl. Ist. Stad. Tech. 45
46 APPENDIX B Taylor Series to First Order If y g( x1,..., x ), x,..., 1 xare idepedet radom variables with distributio fuctios f ( x ),..., f ( x ) ad we expad g i a Taylor series to 1 1 g first order, the y g( x,..., ) ad. 1 x y xi 1 x i i x First, we expad g i a Taylor series about the mea values of x 1,..., x, the g yx ( 1,..., x) g( x,..., ) 1 x x i x i 1 x i i x M. H. Fracis U.S. Natl. Ist. Stad. Tech. 46
47 The g y g( ) xi x f ( ) i i xi dxi i1 x i x i1 Sice Ex ( ) 0, it follows i x i g( ) y g y E g x y x i x f ( ) i i xi dxi i1 x i x i1 M. H. Fracis U.S. Natl. Ist. Stad. Tech. 47
48 g y xi i1 xi x M. H. Fracis U.S. Natl. Ist. Stad. Tech. 48
49 APPENDIX C Taylor Series to Secod Order Cosider the case of Appedix B but expadig to secod order i the Taylor series. Fid ad. y y 1 g y g( ) xi x i x g i1 j1 i j i1 i x x i x xj x i j x x x. M. H. Fracis U.S. Natl. Ist. Stad. Tech. 49
50 y i x i x j x i i i i xi i j xi x x j i1 x. g 1 g g( ) x x x f ( x ) dx i i j Sice Ex ( ) 0 ad E(( x ) ) the the secod term i the i x i i x x i square brackets is zero ad oly those terms with j=i cotribute i the third term. Thus, 1 g y g( ) xi 1 x y i i x g 1 g 1 g xi x x ( ) i i x x i j x j x f i i xi dxi i xi i j xi x j x i 1 x i x i 1 x i M. H. Fracis U.S. Natl. Ist. Stad. Tech. 50
51 g g y xi x x... i j x j i j x i x x j x g g g g... xi x x... i j x x j k x k x x i j x j i j k x k x x i x j i j x x i x j 1 g g... xi x x... i j x x j k x x k x 4 i j k xi x j x x kx x 1 g g x x k i x x i j x j x i j k ix j x x k x 1 g g... ( ) x i x f j i xi dxi 4 x i j i x j i1 x x M. H. Fracis U.S. Natl. Ist. Stad. Tech. 51
52 Sice Ex ( ) 0 ad E(( x ) ) the the first term is zero i x i i x x i uless i=j, the secod term is zero uless i=j=k, the third term is zero, the fourth term is zero uless k=i (j), R=j (i) ad the fifth term is zero uless i=j. Thus i g g g 3 1 g y x Exi i x x i1 xi i xi x i x x i 1 j i xi x j x i i j 1 4 g x i i x E x 4 4 i x x i i M. H. Fracis U.S. Natl. Ist. Stad. Tech. 5
53 Note that there are third order Taylor series terms of the form 3 g g which are of the same order as terms i the x i x j x i x x i x j above equatio. x M. H. Fracis U.S. Natl. Ist. Stad. Tech. 53
54 Normal distributio APPENDIX D Mea ad Variace for Selected Distributios f( x) e x x 1 x M. H. Fracis U.S. Natl. Ist. Stad. Tech. 54
55 Expoetial distributio 1 f ( x) x x x e M. H. Fracis U.S. Natl. Ist. Stad. Tech. 55
56 Biomial distributio f( x;, ) x (1 ) x x! (1 ) x!( x)! x M. H. Fracis U.S. Natl. Ist. Stad. Tech. 56
57 M. H. Fracis U.S. Natl. Ist. Stad. Tech. 57 Chi-squared distributio with ν degrees of freedom ( ) x x x x e f x
58 Rectagular distributio 1 f x a a xa 0 otherwise x x 0 a 3 M. H. Fracis U.S. Natl. Ist. Stad. Tech. 58
59 U distributio 1 x f x a xa; a iteger greater tha 0 a a 0 otherwise x 0 1 a 3 x M. H. Fracis U.S. Natl. Ist. Stad. Tech. 59
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