# fft

Document Sample

```					                                                     Introduction

FFTCalc

FFTCalc can be used to calculate an aggregate loss distribution with correlated frequency distributions.

Running FFTCalc
1. Input the parameters of the negative binomial event count distributions. I have taken the liberty of
parametrizing the event count distribution in terms of the expected event count and the contagion
parameter, c.

2. Input the probabilities of the event intensity distributions. Because the aggregate calculation is done
with the FFT, it is necessary to express the event intensity distributions as a discrete distribution. I have
chosen units of one. If you change the units, be sure to adjust the means, variances and the covariances
of the aggregate loss distribution and the event severity distributions.

3. Input the omega parameter. FFTCalc warns you if your choice of a positive omega is not permissible.
4. Invoke the macro "AggCalc" with the keyboard command "Control-A". Be sure you have the Excel
"Analysis ToolPak" and the "Analysis ToolPak - VBA" add-ins installed in your Excel.

Calculation Steps
1. Calculate the FFT of each event intensity distribution. Put the results below in the same column under
the heading "FFT of Event Intensity Distribution". Note that the FFT is a vector of complex numbers.
Excel treats complex numbers as labels.
2. Calculate the expression (1-beta(j)*(FFT(j) -1))^(alpha(j)*omega) for each variable's timeseries. Put
the results below the FFT's under the heading "Operations on Intensity FFT."
3. Sum the results of Step 2 and place them to the right under the heading "Sum Ops".
4. Calculate the FFT of the aggregate loss distribution = (Step 3 result - # Lines +1)^(-1/omega). Put the
results to the right of the Step 1 results under the heading "Agg FFT"
5. Invert the Aggregate FFT to obtain the aggregate probabilities. Put the results immediately to the left
of Step 4 under the heading "InvAggFFT". The imaginary part of the inverse FFT will be zero. The real
part then "unlabeled" and put to the right of the event intensity distributions under the heading "Aggregate
Probability".

The Cycle Problem
The FFT has a "Cycle" property that can cause incorrect calculated aggregate probabilities. To illustrate,
the highest aggregate probability calculated on this spreadsheet is 7-bits "127". If the parameters you
select indicate a positive probability for higher aggregate losses, Pr{128} will be added to the indicated
Pr{0}, Pr{129} will be added to the indicated Pr{1}, and so on. This spreadsheet offers ways to check if
this is happening. First if Pr{127} = 0, it will usually be the case that the probability of higher losses is
also 0 (unless only even events are permissible). Next it compares the calculated mean and variance of
the aggregate loss distribution with the indicated theoretical mean and variance. If they are the same, it
is unlikely that there is a cycle problem.

Page 1
FFTCalc

Calculate Aggregate Loss Distribution using the Fast Fourier Transform
Line Statistics              Aggregate Statistics        Count Covariance Matrix
Line 1       Line 2      Omega         0.090000               Line 1      Line 2
Alpha          10.00        10.00 OmegaOK?         Yes       Line 1      7.500000    2.250000
Beta      0.500000     0.500000    E[Agg]      21.000000    Line 2      2.250000    7.500000
E[Count]    5.000000     5.000000    V[Agg]      90.895000
V[Count]    7.500000     7.500000 CalcE[Agg]     21.000000        Count Correlation Matrix
c        0.100000     0.100000 CalcV[Agg]     90.894996                Line 1      Line 2
E[Sev]     2.100000     2.100000   LastProb      0.00E+00    Line 1      1.000000    0.300000
V[Sev]     0.490000     0.490000                             Line 2      0.300000    1.000000
V[LAgg]   35.525000 35.525000

Aggregate    Limited
Event Intensity Distribution       Probability Pure Prem.     Aggregate Covariance Matrix
0                 0            0     0.000895    0.000000              Line 1      Line 2
1        0.200000       0.200000     0.000914    0.999105   Line 1    35.525000    9.922500
2        0.500000       0.500000     0.002796    1.997296   Line 2     9.922500 35.525000
3        0.300000       0.300000     0.004136    2.992692
4        0.000000       0.000000     0.006358    3.983951     Aggregate Correlation Matrix
5        0.000000       0.000000     0.009369    4.968851              Line 1        Line 2
6        0.000000       0.000000     0.012290    5.944383   Line 1     1.000000 0.2793103
7        0.000000       0.000000     0.016020    6.907624   Line 2    0.2793103     1.000000
8        0.000000       0.000000     0.019780    7.854846
9        0.000000       0.000000     0.023607    8.782288
10        0.000000       0.000000     0.027501    9.686123
11        0.000000       0.000000     0.031095   10.562457
12        0.000000       0.000000     0.034424   11.407695
13        0.000000       0.000000     0.037334   12.218510
14        0.000000       0.000000     0.039726   12.991991
15        0.000000       0.000000     0.041590   13.725745
16        0.000000       0.000000     0.042870   14.417910
17        0.000000       0.000000     0.043575   15.067205
18        0.000000       0.000000     0.043725   15.672925
19        0.000000       0.000000     0.043354   16.234920
20        0.000000       0.000000     0.042513   16.753561
21        0.000000       0.000000     0.041262   17.229690
22        0.000000       0.000000     0.039664   17.664556
23        0.000000       0.000000     0.037787   18.059758
24        0.000000       0.000000     0.035697   18.417173
25        0.000000       0.000000     0.033456   18.738892
26        0.000000       0.000000     0.031123   19.027154
27        0.000000       0.000000     0.028750   19.284293
28        0.000000       0.000000     0.026381   19.512682
29        0.000000       0.000000     0.024055   19.714691
30        0.000000       0.000000     0.021804   19.892644
31        0.000000       0.000000     0.019651   20.048793
32        0.000000       0.000000     0.017616   20.185290
33        0.000000       0.000000     0.015710   20.304172
34        0.000000       0.000000     0.013942   20.407343
35        0.000000       0.000000     0.012315   20.496572
36        0.000000       0.000000     0.010829   20.573486
37        0.000000       0.000000     0.009482   20.639571

Page 2
FFTCalc

38   0.000000   0.000000   0.008269     20.696173
39   0.000000   0.000000   0.007182     20.744507
40   0.000000   0.000000   0.006215     20.785659
41   0.000000   0.000000   0.005358     20.820596
42   0.000000   0.000000   0.004603     20.850176
43   0.000000   0.000000   0.003942     20.875152
44   0.000000   0.000000   0.003364     20.896186
45   0.000000   0.000000   0.002863     20.913856
46   0.000000   0.000000   0.002428     20.928663
47   0.000000   0.000000   0.002054     20.941042
48   0.000000   0.000000   0.001733     20.951367
49   0.000000   0.000000   0.001457     20.959959
50   0.000000   0.000000   0.001223     20.967093
51   0.000000   0.000000   0.001023     20.973005
52   0.000000   0.000000   0.000854     20.977894
53   0.000000   0.000000   0.000712     20.981928
54   0.000000   0.000000   0.000591     20.985251
55   0.000000   0.000000   0.000490     20.987982
56   0.000000   0.000000   0.000406     20.990223
57   0.000000   0.000000   0.000335     20.992059
58   0.000000   0.000000   0.000276     20.993559
59   0.000000   0.000000   0.000227     20.994784
60   0.000000   0.000000   0.000186     20.995782
61   0.000000   0.000000   0.000152     20.996594
62   0.000000   0.000000   0.000125     20.997254
63   0.000000   0.000000   0.000102     20.997788
64   0.000000   0.000000   0.000083     20.998221
65   0.000000   0.000000   0.000067     20.998572
66   0.000000   0.000000   0.000055     20.998854
67   0.000000   0.000000   0.000044     20.999082
68   0.000000   0.000000   0.000036     20.999265
69   0.000000   0.000000   0.000029     20.999413
70   0.000000   0.000000   0.000023     20.999531
71   0.000000   0.000000   0.000019     20.999626
72   0.000000   0.000000   0.000015     20.999702
73   0.000000   0.000000   0.000012     20.999763
74   0.000000   0.000000   0.000010     20.999812
75   0.000000   0.000000   0.000008     20.999851
76   0.000000   0.000000   0.000006     20.999882
77   0.000000   0.000000   0.000005     20.999906
78   0.000000   0.000000   0.000004     20.999926
79   0.000000   0.000000   0.000003     20.999941
80   0.000000   0.000000   0.000003     20.999954
81   0.000000   0.000000   0.000002     20.999963
82   0.000000   0.000000   0.000002     20.999971
83   0.000000   0.000000   0.000001     20.999977
84   0.000000   0.000000   0.000001     20.999982
85   0.000000   0.000000   0.000001     20.999986
86   0.000000   0.000000   0.000001     20.999989
87   0.000000   0.000000   0.000001     20.999991
88   0.000000   0.000000   0.000000     20.999993
89   0.000000   0.000000   0.000000     20.999995

Page 3
FFTCalc

90       0.000000    0.000000    0.000000     20.999996
91       0.000000    0.000000    0.000000     20.999997
92       0.000000    0.000000    0.000000     20.999997
93       0.000000    0.000000    0.000000     20.999998
94       0.000000    0.000000    0.000000     20.999998
95       0.000000    0.000000    0.000000     20.999999
96       0.000000    0.000000    0.000000     20.999999
97       0.000000    0.000000    0.000000     20.999999
98       0.000000    0.000000    0.000000     20.999999
99       0.000000    0.000000    0.000000     21.000000
100       0.000000    0.000000    0.000000     21.000000
101       0.000000    0.000000    0.000000     21.000000
102       0.000000    0.000000    0.000000     21.000000
103       0.000000    0.000000    0.000000     21.000000
104       0.000000    0.000000    0.000000     21.000000
105       0.000000    0.000000    0.000000     21.000000
106       0.000000    0.000000    0.000000     21.000000
107       0.000000    0.000000    0.000000     21.000000
108       0.000000    0.000000    0.000000     21.000000
109       0.000000    0.000000    0.000000     21.000000
110       0.000000    0.000000    0.000000     21.000000
111       0.000000    0.000000    0.000000     21.000000
112       0.000000    0.000000    0.000000     21.000000
113       0.000000    0.000000    0.000000     21.000000
114       0.000000    0.000000    0.000000     21.000000
115       0.000000    0.000000    0.000000     21.000000
116       0.000000    0.000000    0.000000     21.000000
117       0.000000    0.000000    0.000000     21.000000
118       0.000000    0.000000    0.000000     21.000000
119       0.000000    0.000000    0.000000     21.000000
120       0.000000    0.000000    0.000000     21.000000
121       0.000000    0.000000    0.000000     21.000000
122       0.000000    0.000000    0.000000     21.000000
123       0.000000    0.000000    0.000000     21.000000
124       0.000000    0.000000    0.000000     21.000000
125       0.000000    0.000000    0.000000     21.000000
126       0.000000    0.000000    0.000000     21.000000
127       0.000000    0.000000    0.000000     21.000000

FFT of Claim Severity Distribution   Agg FFT    InvAggFFT
0     1            1            1          8.9488120911458E-004
1                  0.994104407566567-0.102841247366872i
0.470034360025417-0.76445925141148i 0.200000
0.994104407566567-0.102841247366872i 9.13787348166209E-004                0.9987955
2                  0.976511686255717-0.204233992250315i
-0.258445333404849-0.604989801342938i0.500000
0.976511686255717-0.204233992250315i 2.79604994043846E-003                0.9951847
3                  0.947502257796094-0.302754961547388i
-0.38742858092028-0.131148042891966i 0.300000
0.947502257796094-0.302754961547388i 4.13630358885264E-003                0.9891765
4                  0.907537706027053-0.397030850491651i
-0.210857238781355+0.0823093263605581i
0.907537706027053-0.397030850491651i 6.35834240247215E-003                0.9941044
5                  0.857252220419574-0.485762090847757i
-0.0706825419931899+0.0931023917467658i
0.857252220419574-0.485762090847757i 9.3690924122339E-003
6                  0.797440858546808-0.567745187969515i
-0.0140514764677561+0.0568720863705315i
0.797440858546808-0.567745187969515i 1.22900989136087E-002
7                  0.729044862975939-0.641893195760349i
0.00110164563031686+0.0293077709134086i
0.729044862975939-0.641893195760349i 1.60196840632767E-002
8                  0.653134326805058-0.707253936819678i
0.00308156324564211+0.0146384717751505i
0.653134326805058-0.707253936819678i 1.97795691243878E-002
9                  0.57088855467749-0.763025621325788i
0.00221100999794183+0.00755432858701816i
0.57088855467749-0.763025621325788i  2.36073715186389E-002
10                  0.48357451147834-0.808569571518131i
0.00120960755110871+0.00413033618729614i
0.48357451147834-0.808569571518131i  2.75011926962547E-002
11                  0.392523788114827-0.843419817874374i
0.000539164796145812+0.00239476380579609i
0.392523788114827-0.843419817874374i 3.10951043079941E-002

Page 4
FFTCalc

12                0.299108542038215-0.867289396980533i
0.000154445493881774+0.0014545440985185i
0.299108542038215-0.867289396980533i 3.44240686105582E-002
13                0.204716888905693-0.880073248319497i
-0.0000478821157364069+0.000909509353122559i
0.204716888905693-0.880073248319497i 3.73335655599371E-002
14                0.110728230632811-0.881847676338851i
-0.000144084061182723+0.000574371993769288i
0.110728230632811-0.881847676338851i 3.97263986278266E-002
15                1.84890038880413E-002-0.872866413749696i
-0.000180856201766898+0.000358548351120316i
4.15896726693671E-002
1.84890038880413E-002-0.872866413749696i
16                -7.07106781186556E-002-0.853553390593274i
-0.000185154485788765+0.000214880959160925i
4.28698336755567E-002
-7.07106781186556E-002-0.853553390593274i
17                -0.155659038639571-0.82449237975482i
-0.000172258047010098+0.000117225999027558i
-0.155659038639571-0.82449237975482i 4.35745709217358E-002
18                -0.235242883479842-0.786413751921962i
-0.00015079520274934+0.0000502269233023047i
-0.235242883479842-0.786413751921962i4.37254819061597E-002
19                -0.308465697283652-0.740178630179899i
-0.000125636492543831+4.39882587985466E-06i
-0.308465697283652-0.740178630179899i4.33537111649119E-002
20                -0.374463253699594-0.68676078532099i
-0.0000995215967023145-0.0000263680935257123i
-0.374463253699594-0.68676078532099i 4.25131608286338E-002
21                -0.432516456435907-0.627226656472457i
-0.0000739732958595412-0.0000461465842219164i
-0.432516456435907-0.627226656472457i4.12617513939567E-002
22                -0.482061187146261-0.562713916922075i
-0.0000498153053107157-0.0000577262123162181i
-0.482061187146261-0.562713916922075i3.96639249888821E-002
23                -0.52269499935413-0.494409031335077i
-0.0000274693161374289-0.0000630573845820041i
-0.52269499935413-0.494409031335077i 3.77871431374422E-002
24                -0.554180563873642-0.423524267386003i
-0.0000071276690763467-0.0000635229869218115i
-0.554180563873642-0.423524267386003i3.56969010412638E-002
25                -0.576445839003006-0.351274631860459i
0.000011144974478283-0.0000601097199965242i
-0.576445839003006-0.351274631860459i3.34562196449693E-002
26                -0.589581006709201-0.278855198407148i
0.0000273510903945807-0.0000535185234657764i
-0.589581006709201-0.278855198407148i3.1123157865236E-002
27                -0.59383228264763-0.207419281445419i
0.0000415239584050574-0.0000442363681527539i
-0.59383228264763-0.207419281445419i 2.8749637141085E-002
28                -0.589592771758298-0.138057888572426i
0.0000537059068633749-0.0000325823139595442i
-0.589592771758298-0.138057888572426i2.6380966843053E-002
29                -0.577390601004116-7.17808526831532E-002i
0.0000639357605673232-0.0000187353303690395i
2.40553500510147E-002
-0.577390601004116-7.17808526831532E-002i
30                -0.557874615312041-9.5000056228398E-003i
0.0000722427936511659-2.74809811677191E-06i
2.18040008121326E-002
-0.557874615312041-9.5000056228398E-003i
31                -0.531797970807223+4.79852915836205E-002i
0.00007864644691204+0.0000154510771183915i
1.96514413424103E-002
-0.531797970807223+4.79852915836205E-002i
32                                        1.76159655722903E-002
-0.5+1E-001i -0.5+1E-001i 0.0000831628770957949+0.0000360674301041809i
33                -0.463386755864973+0.14600243191318i
0.0000858215044696276+0.0000594597938234314i
-0.463386755864973+0.14600243191318i 1.57102729645562E-002
34                -0.422910665091189+0.185590316393287i
0.0000866974653278413+0.000086170983497093i
-0.422910665091189+0.185590316393287i1.39421474414293E-002
35                -0.379549734728092+0.218503824571308i
0.0000859694131731954+0.000116965776608425i
-0.379549734728092+0.218503824571308i1.2315171362589E-002
36                -0.334286760752988+0.244625543792662i
0.0000840160483354993+0.000152868408253571i
-0.334286760752988+0.244625543792662i1.08294344689634E-002
37                -0.288088981700725+0.263977455380578i
0.0000815673850212385+0.00019518100221748i
-0.288088981700725+0.263977455380578i9.4822050661545E-003
38                -0.241888605593344+0.276715034612453i
0.000079923876257727+0.000245446952563044i
-0.241888605593344+0.276715034612453i8.26855182658715E-003
39                -0.196564614359731+0.283118652303185i
0.0000812402274803727+0.000305299756537953i
-0.196564614359731+0.283118652303185i7.18190164507862E-003
40                -0.152926217312906+0.283582513800543i
0.0000888301031162764+0.000376116304501911i
-0.152926217312906+0.283582513800543i6.21452799391428E-003
41                -0.111698284809516+0.278601422027659i
0.000107375031938083+0.000458395902450279i
-0.111698284809516+0.278601422027659i5.35796882014252E-003
42                -7.35090458733416E-002+0.26875569538047i
0.000142826197671894+0.00055084958166647i
4.60337513214599E-003
-7.35090458733416E-002+0.26875569538047i
43                -3.88802803900915E-002+0.254694607875897i
0.000201722900677205+0.000649345612216713i
3.94179460909213E-003
-3.88802803900915E-002+0.254694607875897i
44                -8.22017866549626E-003+0.237118747190296i
0.000289717097779813+0.000746107551748576i
3.36439583409704E-003
-8.22017866549626E-003+0.237118747190296i
45                1.81810200291883E-002+0.216761705552309i
0.000409393427551737+0.000829783473723437i
2.86263961655612E-003
1.81810200291883E-002+0.216761705552309i
46                4.01525614637131E-002+0.194371528481268i
0.000557991956313243+0.000886966353333678i
2.42840434765314E-003
4.01525614637131E-002+0.194371528481268i
47                5.76418983100093E-002+0.170692346917376i
0.00072611413243743+0.000905239540211048i
2.05407224930567E-003
5.76418983100093E-002+0.170692346917376i
48                7.07106781186545E-002+0.146446609406726i
0.000898474582996207+0.000876938332329067i
1.73258310016317E-003
7.07106781186545E-002+0.146446609406726i
49                7.95281364415182E-002+0.122318312922501i
0.00105698181096666+0.000802084103465779i
1.4574615370613E-003
7.95281364415182E-002+0.122318312922501i
50                8.43620913833157E-002+9.89376040643795E-002i
0.00118520011299038+0.000689016819308311i
1.22282341794009E-003
8.43620913833157E-002+9.89376040643795E-002i
51                8.55677883487678E-002+7.68670874127114E-002i
0.00127238843304145+0.000552286415648863i
1.02336607682832E-003
8.55677883487678E-002+7.68670874127114E-002i
52                8.35748903268739E-002+5.65901355307538E-002i
0.00131549944755918+0.000408735136626959i
8.54346627103993E-004
8.35748903268739E-002+5.65901355307538E-002i
53                7.88729487111694E-002+3.85014464739815E-002i
0.00131862871996591+0.000273475409472414i
7.11551816078907E-004
7.88729487111694E-002+3.85014464739815E-002i
54                7.19957215412614E-002+2.29000407844141E-002i
0.00129062249139141+0.000157211213530991i
5.91262323754533E-004
7.19957215412614E-002+2.29000407844141E-002i
55                6.35047294861544E-002+9.98483203694905E-003i
0.00124213266516086+0.0000654275686936312i
4.90213842986407E-004
6.35047294861544E-002+9.98483203694905E-003i
56                5.3972454381489E-002-1.47155633130003E-004i
0.00118320818012259-9.13256931758336E-07i
4.05556786015658E-004
5.3972454381489E-002-1.47155633130003E-004i
57                4.39655903867973E-002-7.4999115967026E-003i
0.00112191285808894-0.0000439335482022705i
3.34816035139968E-004
4.39655903867973E-002-7.4999115967026E-003i
58                3.40287537557368E-002-1.21749549499121E-002i
0.00106391414012077-0.000067350581280255i
2.75851791982633E-004
3.40287537557368E-002-1.21749549499121E-002i
59                2.46690439287806E-002-1.4365354021759E-002i
0.00101271408512141-0.0000753279897026411i
2.26822276920489E-004
2.46690439287806E-002-1.4365354021759E-002i
60                1.63418264842334E-002-1.4347418126561E-002i
0.000970170555896829-0.0000717738753207594i
1.86148782491642E-004
1.63418264842334E-002-1.4347418126561E-002i
61                9.43807793611495E-003-1.24702842929253E-002i
0.00093705887697316-0.0000600212634055284i
1.52483386091928E-004
9.43807793611495E-003-1.24702842929253E-002i
62                4.27359414751305E-003-9.14367023418611E-003i
0.000913545387018841-0.000042769105887041i
1.24679471699739E-004
4.27359414751305E-003-9.14367023418611E-003i
63                1.08031910562967E-003-4.82410703731122E-003i
0.000899532480772942-0.0000221718508796514i
1.01765091513922E-004
1.08031910562967E-003-4.82410703731122E-003i

Page 5
FFTCalc

64   0            0                       8.2919110348688E-005
0.000894881192129819
65                1.08031910562967E-003+4.82410703731183E-003i
0.000899532480772942+0.0000221718508796543i
6.74500129834906E-005
1.08031910562967E-003+4.82410703731183E-003i
66                4.27359414751316E-003+9.1436702341867E-003i
0.000913545387018841+0.0000427691058870438i
5.47772125548658E-005
4.27359414751316E-003+9.1436702341867E-003i
67                9.43807793611523E-003+1.24702842929258E-002i
0.00093705887697316+0.0000600212634055306i
4.44146722608065E-005
9.43807793611523E-003+1.24702842929258E-002i
68                1.63418264842337E-002+1.43474181265614E-002i
0.000970170555896829+0.0000717738753207611i
3.59566394754255E-005
1.63418264842337E-002+1.43474181265614E-002i
69                2.4669043928781E-002+1.43653540217595E-002i
0.00101271408512141+0.0000753279897026436i
2.9065287775641E-005
2.4669043928781E-002+1.43653540217595E-002i
70                3.40287537557372E-002+1.21749549499124E-002i
0.00106391414012077+0.0000673505812802563i
2.34600658174282E-005
3.40287537557372E-002+1.21749549499124E-002i
71                4.39655903867978E-002+7.49991159670288E-003i
0.00112191285808894+0.0000439335482022723i
1.89085604001184E-005
4.39655903867978E-002+7.49991159670288E-003i
72                5.39724543814896E-002+1.47155633130169E-004i
0.00118320818012259+9.13256931759361E-07i
1.52186927665776E-005
5.39724543814896E-002+1.47155633130169E-004i
73                6.35047294861549E-002-9.98483203694889E-003i
0.00124213266516086-0.0000654275686936301i
1.22320809129925E-005
6.35047294861549E-002-9.98483203694889E-003i
74                7.1995721541262E-002-2.29000407844141E-002i
0.00129062249139141-0.000157211213530991i
9.81841542387757E-006
7.1995721541262E-002-2.29000407844141E-002i
75                7.88729487111698E-002-3.85014464739816E-002i
0.00131862871996591-0.000273475409472414i
7.87071136607032E-006
7.88729487111698E-002-3.85014464739816E-002i
76                8.35748903268744E-002-5.6590135530754E-002i
0.00131549944755918-0.00040873513662696i
6.30131352557378E-006
8.35748903268744E-002-5.6590135530754E-002i
77                8.55677883487683E-002-7.68670874127118E-002i
0.00127238843304145-0.000552286415648866i
5.03854637875015E-006
8.55677883487683E-002-7.68670874127118E-002i
78                8.4362091383316E-002-9.89376040643798E-002i
0.00118520011299038-0.000689016819308313i
4.02391340696695E-006
8.4362091383316E-002-9.89376040643798E-002i
79                7.95281364415184E-002-0.122318312922502i
0.00105698181096666-0.000802084103465786i
3.20976254518379E-006
7.95281364415184E-002-0.122318312922502i
80                7.07106781186546E-002-0.146446609406726i
0.000898474582996207-0.000876938332329067i
2.55734562865789E-006
7.07106781186546E-002-0.146446609406726i
81                5.76418983100093E-002-0.170692346917377i
0.000726114132437425-0.000905239540211052i
2.03520965451343E-006
5.76418983100093E-002-0.170692346917377i
82                4.0152561463713E-002-0.194371528481269i
0.000557991956313238-0.000886966353333682i
1.61786653064822E-006
4.0152561463713E-002-0.194371528481269i
83                1.81810200291881E-002-0.21676170555231i
0.000409393427551737-0.000829783473723437i
1.28469579754875E-006
1.81810200291881E-002-0.21676170555231i
84                -8.22017866549654E-003-0.237118747190296i
0.000289717097779813-0.000746107551748576i
1.01904164810998E-006
-8.22017866549654E-003-0.237118747190296i
85                -3.88802803900919E-002-0.254694607875898i
0.000201722900677197-0.000649345612216714i
8.0747151787533E-007
-3.88802803900919E-002-0.254694607875898i
86                -7.35090458733421E-002-0.26875569538047i
0.000142826197671894-0.00055084958166647i
6.39168657480538E-007
-7.35090458733421E-002-0.26875569538047i
87                -0.111698284809517-0.278601422027659i
0.000107375031938083-0.000458395902450279i
-0.111698284809517-0.278601422027659i5.05435514962238E-007
88                -0.152926217312906-0.283582513800543i
0.0000888301031162764-0.000376116304501911i
-0.152926217312906-0.283582513800543i3.99288531696684E-007
89                -0.196564614359732-0.283118652303185i
0.0000812402274803727-0.000305299756537953i
-0.196564614359732-0.283118652303185i3.15128167890227E-007
90                -0.241888605593345-0.276715034612453i
0.000079923876257727-0.000245446952563044i
-0.241888605593345-0.276715034612453i2.48470694682451E-007
91                -0.288088981700726-0.263977455380578i
0.0000815673850212385-0.00019518100221748i
-0.288088981700726-0.263977455380578i1.95730587033288E-007
92                -0.334286760752989-0.244625543792662i
0.0000840160483354993-0.000152868408253571i
-0.334286760752989-0.244625543792662i1.54044280298651E-007
93                -0.379549734728093-0.218503824571307i
0.0000859694131731965-0.000116965776608424i
-0.379549734728093-0.218503824571307i1.21127670844878E-007
94                -0.422910665091189-0.185590316393287i
0.0000866974653278413-0.000086170983497093i
-0.422910665091189-0.185590316393287i9.51610893435567E-008
95                -0.463386755864974-0.146002431913179i
0.0000858215044696279-0.000059459793823431i
-0.463386755864974-0.146002431913179i7.4696598536747E-008
96                                        5.85833984621809E-008
-0.5-1E-001i -0.5-1E-001i 0.0000831628770957949-0.0000360674301041809i
97                -0.531797970807223-4.79852915836194E-002i
0.0000786464469120401-0.0000154510771183911i
4.59078922307454E-008
-0.531797970807223-4.79852915836194E-002i
98                -0.557874615312042+9.50000562284103E-003i
0.0000722427936511658+2.74809811677226E-06i
3.59456015968165E-008
-0.557874615312042+9.50000562284103E-003i
99                -0.577390601004117+7.17808526831547E-002i
0.0000639357605673231+0.0000187353303690399i
2.8122643938415E-008
-0.577390601004117+7.17808526831547E-002i
100                -0.589592771758298+0.138057888572428i
0.0000537059068633745+0.0000325823139595443i
-0.589592771758298+0.138057888572428i2.19849121225676E-008
101                -0.59383228264763+0.20741928144542i
0.000041523958405057+0.0000442363681527543i
-0.59383228264763+0.20741928144542i  1.71734501567405E-008
102                -0.589581006709201+0.27885519840715i
0.0000273510903945803+0.0000535185234657767i
-0.589581006709201+0.27885519840715i 1.3404804601394E-008
103                -0.576445839003006+0.351274631860461i
0.0000111449744782825+0.0000601097199965242i
-0.576445839003006+0.351274631860461i1.04553665222079E-008
104                -0.554180563873642+0.423524267386005i
-7.12766907634726E-06+0.0000635229869218114i
-0.554180563873642+0.423524267386005i8.14890873486115E-009
105                -0.522694999354129+0.494409031335079i
-0.0000274693161374294+0.0000630573845820038i
-0.522694999354129+0.494409031335079i6.34667846229064E-009
106                -0.48206118714626+0.562713916922076i
-0.0000498153053107162+0.0000577262123162177i
-0.48206118714626+0.562713916922076i 4.93953066166633E-009
107                -0.432516456435906+0.627226656472458i
-0.0000739732958595412+0.0000461465842219164i
-0.432516456435906+0.627226656472458i3.84168879477217E-009
108                -0.374463253699593+0.686760785320992i
-0.0000995215967023146+0.0000263680935257114i
-0.374463253699593+0.686760785320992i2.98580207541872E-009
109                -0.30846569728365+0.7401786301799i
-0.000125636492543831-4.39882587985466E-06i
-0.30846569728365+0.7401786301799i 2.31903387725579E-009
110                -0.235242883479841+0.786413751921962i
-0.00015079520274934-0.0000502269233023047i
-0.235242883479841+0.786413751921962i1.79996971841345E-009
111                -0.155659038639569+0.824492379754821i
-0.000172258047010095-0.000117225999027559i
-0.155659038639569+0.824492379754821i1.39617544847419E-009
112                -7.07106781186535E-002+0.853553390593274i
-0.000185154485788765-0.000214880959160925i
1.08227106118165E-009
-7.07106781186535E-002+0.853553390593274i
113                1.84890038880435E-002+0.872866413749696i
-0.000180856201766898-0.000358548351120316i
8.38412565285288E-010
1.84890038880435E-002+0.872866413749696i
114                0.110728230632814+0.881847676338851i
-0.000144084061182723-0.000574371993769288i
0.110728230632814+0.881847676338851i 6.49096892338039E-010
115                0.204716888905696+0.880073248319497i
-0.0000478821157364069-0.000909509353122559i
0.204716888905696+0.880073248319497i 5.02221750948648E-010

Page 6
FFTCalc

116              0.299108542038217+0.867289396980533i
0.000154445493881774-0.0014545440985185i
3.88346937940653E-010
0.299108542038217+0.867289396980533i
117              0.392523788114829+0.843419817874373i
0.000539164796146006-0.00239476380579634i
3.00114246334134E-010
0.392523788114829+0.843419817874373i
118              0.483574511478342+0.80856957151813i
0.00120960755110867-0.00413033618729618i
2.31792653322162E-010
0.483574511478342+0.80856957151813i
119              0.570888554677492+0.763025621325787i
0.0022110099979426-0.007554328587019i
1.78921595638329E-010
0.570888554677492+0.763025621325787i
120              0.65313432680506+0.707253936819677i
0.00308156324564194-0.0146384717751506i
1.38031994336004E-010
0.65313432680506+0.707253936819677i
121              0.729044862975941+0.641893195760347i
0.00110164563031642-0.0293077709134088i
1.06427548496107E-010
0.729044862975941+0.641893195760347i
122              0.797440858546809+0.567745187969513i
-0.0140514764677571-0.0568720863705316i
0
0.797440858546809+0.567745187969513i
123              0.857252220419575+0.485762090847755i
-0.0706825419931918-0.0931023917467651i
0
0.857252220419575+0.485762090847755i
124              0.907537706027054+0.397030850491649i
-0.210857238781358-0.0823093263605546i
0
0.907537706027054+0.397030850491649i
125              0.947502257796095+0.302754961547386i
-0.38742858092028+0.131148042891974i
0
0.947502257796095+0.302754961547386i
126              0.976511686255718+0.204233992250313i
-0.258445333404838+0.604989801342947i
0
0.976511686255718+0.204233992250313i
127              0.994104407566567+0.10284124736687i
0.470034360025434+0.764459251411473i
0
0.994104407566567+0.10284124736687i

Operations on Severity FFT    Sum Ops
0     1           1           2
1                 1.00277116504672+0.0462627137948894i
2.00554233009344+0.0925254275897788i
1.00277116504672+0.0462627137948894i
2                 1.01102592292843+0.0917809648712576i
2.02205184585686+0.183561929742515i
1.01102592292843+0.0917809648712576i
3                 1.02459132003821+0.135833363696474i
2.04918264007642+0.271666727392948i
1.02459132003821+0.135833363696474i
4                 1.04318929657289+0.177742202125682i
2.08637859314578+0.355484404251364i
1.04318929657289+0.177742202125682i
5                 1.06644994889008+0.216889895893815i
2.13289989778016+0.43377979178763i
1.06644994889008+0.216889895893815i
6                 1.09392688865389+0.252730676095738i
2.18785377730778+0.505461352191476i
1.09392688865389+0.252730676095738i
7                 1.12511293939903+0.284797903558378i
2.25022587879806+0.569595807116756i
1.12511293939903+0.284797903558378i
8                 1.15945504078925+0.31270789857543i
2.3189100815785+0.62541579715086i
1.15945504078925+0.31270789857543i
9                 1.19636785730525+0.336161264964838i
2.3927357146105+0.672322529929676i
1.19636785730525+0.336161264964838i
10                 1.23524602562161+0.354942512111622i
2.47049205124322+0.709885024223244i
1.23524602562161+0.354942512111622i
11                 1.27547519551311+0.368918519517458i
2.55095039102622+0.737837039034916i
1.27547519551311+0.368918519517458i
12                 1.31644207893772+0.37803615616093i
2.63288415787544+0.75607231232186i
1.31644207893772+0.37803615616093i
13                 1.35754369310979+0.382319203428755i
2.71508738621958+0.76463840685751i
1.35754369310979+0.382319203428755i
14                 1.39819592080877+0.381864635295783i
2.79639184161754+0.763729270591566i
1.39819592080877+0.381864635295783i
15                 1.43784144651284+0.376838266099868i
2.87568289302568+0.753676532199736i
1.43784144651284+0.376838266099868i
16                 1.47595707420644+0.367469766146669i
2.95191414841288+0.734939532293338i
1.47595707420644+0.367469766146669i
17                 1.51206039563607+0.354047053861229i
3.02412079127214+0.708094107722458i
1.51206039563607+0.354047053861229i
18                 1.54571575532452+0.336910090454504i
3.09143151064904+0.673820180909008i
1.54571575532452+0.336910090454504i
19                 1.57653944808679+0.316444123150951i
3.15307889617358+0.632888246301902i
1.57653944808679+0.316444123150951i
20                 1.60420408324451+0.293072442620482i
3.20840816648902+0.586144885240964i
1.60420408324451+0.293072442620482i
21                 1.62844205473649+0.267248737683339i
3.25688410947298+0.534497475366678i
1.62844205473649+0.267248737683339i
22                 1.64904806592594+0.239449144841068i
3.29809613185188+0.478898289682136i
1.64904806592594+0.239449144841068i
23                 1.66588067063098+0.210164101459675i
3.33176134126196+0.42032820291935i
1.66588067063098+0.210164101459675i
24                 1.67886280664099+0.179890119477095i
3.35772561328198+0.35978023895419i
1.67886280664099+0.179890119477095i
25                 1.68798131388215+0.149121601439452i
3.3759626277643+0.298243202878904i
1.68798131388215+0.149121601439452i
26                 1.69328544579823+0.118342822666733i
3.38657089159646+0.236685645333466i
1.69328544579823+0.118342822666733i
27                 1.69488439889602+0.088020202623411i
3.38976879779204+0.176040405246822i
1.69488439889602+0.088020202623411i
28                 1.69294390133988+0.0585949853643533i
3.38588780267976+0.117189970728707i
1.69294390133988+0.0585949853643533i
29                 1.68768191662478+0.0304764435060691i
3.37536383324956+0.0609528870121382i
1.68768191662478+0.0304764435060691i
30                 1.67936353243881+0.00403571282573506i
3.35872706487762+0.00807142565147012i
1.67936353243881+0.00403571282573506i
31                 1.66829511764944-0.0203996443774867i
3.33659023529888-0.0407992887549734i
1.66829511764944-0.0203996443774867i
32                 1.65481784179491-0.0425502592934322i
3.30963568358982-0.0851005185868644i
1.65481784179491-0.0425502592934322i
33                 1.6393006615072-0.0621886833732544i
3.2786013230144-0.124377366746509i
1.6393006615072-0.0621886833732544i
34                 1.62213288700733-0.0791421572031492i
3.24426577401466-0.158284314406298i
1.62213288700733-0.0791421572031492i
35                 1.60371644936934-0.0932943692410936i
3.20743289873868-0.186588738482187i
1.60371644936934-0.0932943692410936i
36                 1.58445799590938-0.104586164953121i
3.16891599181876-0.209172329906242i
1.58445799590938-0.104586164953121i
37                 1.56476094714165-0.113015180060734i
3.1295218942833-0.226030360121468i
1.56476094714165-0.113015180060734i

Page 7
FFTCalc

38              1.54501765457124-0.11863438545063i
3.09003530914248-0.23726877090126i
1.54501765457124-0.11863438545063i
39              1.52560180436783-0.121549546320285i
3.05120360873566-0.24309909264057i
1.52560180436783-0.121549546320285i
40              1.50686121763639-0.121915615102296i
3.01372243527278-0.243831230204592i
1.50686121763639-0.121915615102296i
41              1.48911120309168-0.119932097551836i
2.97822240618336-0.239864195103672i
1.48911120309168-0.119932097551836i
42              1.47262862139294-0.115837454988753i
2.94525724278588-0.231674909977506i
1.47262862139294-0.115837454988753i
43              1.45764682048495-0.109902633610902i
2.9152936409699-0.219805267221804i
1.45764682048495-0.109902633610902i
44              1.44435159573415-0.102423843854216i
2.8887031914683-0.204847687708432i
1.44435159573415-0.102423843854216i
45              1.43287831491947-0.0937147476622236i
2.86575662983894-0.187429495324447i
1.43287831491947-0.0937147476622236i
46              1.42331032404947-0.0840982465926944i
2.84662064809894-0.168196493185389i
1.42331032404947-0.0840982465926944i
47              1.41567871442607-0.0738980950416925i
2.83135742885214-0.147796190083385i
1.41567871442607-0.0738980950416925i
48              1.40996348501911-0.0634305859186938i
2.81992697003822-0.126861171837388i
1.40996348501911-0.0634305859186938i
49              1.40609607984538-0.0529965665120843i
2.81219215969076-0.105993133024169i
1.40609607984538-0.0529965665120843i
50              1.40396322237624-0.0428740370308435i
2.80792644475248-0.085748074061687i
1.40396322237624-0.0428740370308435i
51              1.40341191387036-0.0333115626194081i
2.80682382774072-0.0666231252388162i
1.40341191387036-0.0333115626194081i
52              1.40425541568425-0.0245226933289438i
2.8085108313685-0.0490453866578876i
1.40425541568425-0.0245226933289438i
53              1.40628000146226-0.0166815396301155i
2.81256000292452-0.033363079260231i
1.40628000146226-0.0166815396301155i
54              1.40925224586581-0.00991959889234133i
2.81850449173162-0.0198391977846827i
1.40925224586581-0.00991959889234133i
55              1.41292661199154-0.0043238762348233i
2.82585322398308-0.0086477524696466i
1.41292661199154-0.0043238762348233i
56              1.41705310768286+0.0000637042965756998i
2.83410621536572+0.0001274085931514i
1.41705310768286+0.0000637042965756998i
57              1.42138479822648+0.00324564232263607i
2.84276959645296+0.00649128464527214i
1.42138479822648+0.00324564232263607i
58              1.42568498583117+0.00526703310977874i
2.85136997166234+0.0105340662195575i
1.42568498583117+0.00526703310977874i
59              1.42973389162511+0.00621266781473049i
2.85946778325022+0.012425335629461i
1.42973389162511+0.00621266781473049i
60              1.43333470131957+0.00620317706207908i
2.86666940263914+0.0124063541241582i
1.43333470131957+0.00620317706207908i
61              1.43631885972558+0.00539034359848474i
2.87263771945116+0.0107806871969695i
1.43631885972558+0.00539034359848474i
62              1.43855052132558+0.00395171649054208i
2.87710104265116+0.00790343298108416i
1.43855052132558+0.00395171649054208i
63              1.43993008401382+0.00208466341451024i
2.87986016802764+0.00416932682902048i
1.43993008401382+0.00208466341451024i
64              1.44039675118833
1.44039675118833       2.88079350237666
65              1.43993008401382-0.00208466341451051i
2.87986016802764-0.00416932682902102i
1.43993008401382-0.00208466341451051i
66              1.43855052132558-0.00395171649054234i
2.87710104265116-0.00790343298108468i
1.43855052132558-0.00395171649054234i
67              1.43631885972558-0.00539034359848496i
2.87263771945116-0.0107806871969699i
1.43631885972558-0.00539034359848496i
68              1.43333470131957-0.00620317706207925i
2.86666940263914-0.0124063541241585i
1.43333470131957-0.00620317706207925i
69              1.42973389162511-0.00621266781473071i
2.85946778325022-0.0124253356294614i
1.42973389162511-0.00621266781473071i
70              1.42568498583117-0.00526703310977887i
2.85136997166234-0.0105340662195577i
1.42568498583117-0.00526703310977887i
71              1.42138479822648-0.0032456423226362i
2.84276959645296-0.0064912846452724i
1.42138479822648-0.0032456423226362i
72              1.41705310768286-0.0000637042965757717i
2.83410621536572-0.000127408593151543i
1.41705310768286-0.0000637042965757717i
73              1.41292661199154+0.00432387623482323i
2.82585322398308+0.00864775246964646i
1.41292661199154+0.00432387623482323i
74              1.40925224586581+0.00991959889234133i
2.81850449173162+0.0198391977846827i
1.40925224586581+0.00991959889234133i
75              1.40628000146226+0.0166815396301155i
2.81256000292452+0.033363079260231i
1.40628000146226+0.0166815396301155i
76              1.40425541568425+0.0245226933289439i
2.8085108313685+0.0490453866578878i
1.40425541568425+0.0245226933289439i
77              1.40341191387036+0.0333115626194083i
2.80682382774072+0.0666231252388166i
1.40341191387036+0.0333115626194083i
78              1.40396322237624+0.0428740370308436i
2.80792644475248+0.0857480740616872i
1.40396322237624+0.0428740370308436i
79              1.40609607984538+0.0529965665120848i
2.81219215969076+0.10599313302417i
1.40609607984538+0.0529965665120848i
80              1.40996348501911+0.0634305859186938i
2.81992697003822+0.126861171837388i
1.40996348501911+0.0634305859186938i
81              1.41567871442607+0.0738980950416929i
2.83135742885214+0.147796190083386i
1.41567871442607+0.0738980950416929i
82              1.42331032404947+0.0840982465926948i
2.84662064809894+0.16819649318539i
1.42331032404947+0.0840982465926948i
83              1.43287831491947+0.0937147476622236i
2.86575662983894+0.187429495324447i
1.43287831491947+0.0937147476622236i
84              1.44435159573415+0.102423843854216i
2.8887031914683+0.204847687708432i
1.44435159573415+0.102423843854216i
85              1.45764682048495+0.109902633610903i
2.9152936409699+0.219805267221806i
1.45764682048495+0.109902633610903i
86              1.47262862139294+0.115837454988753i
2.94525724278588+0.231674909977506i
1.47262862139294+0.115837454988753i
87              1.48911120309168+0.119932097551836i
2.97822240618336+0.239864195103672i
1.48911120309168+0.119932097551836i
88              1.50686121763639+0.121915615102296i
3.01372243527278+0.243831230204592i
1.50686121763639+0.121915615102296i
89              1.52560180436783+0.121549546320285i
3.05120360873566+0.24309909264057i
1.52560180436783+0.121549546320285i

Page 8
FFTCalc

90              1.54501765457124+0.11863438545063i
3.09003530914248+0.23726877090126i
1.54501765457124+0.11863438545063i
91              1.56476094714165+0.113015180060734i
3.1295218942833+0.226030360121468i
1.56476094714165+0.113015180060734i
92              1.58445799590938+0.104586164953121i
3.16891599181876+0.209172329906242i
1.58445799590938+0.104586164953121i
93              1.60371644936934+0.0932943692410927i
3.20743289873868+0.186588738482185i
1.60371644936934+0.0932943692410927i
94              1.62213288700733+0.0791421572031492i
3.24426577401466+0.158284314406298i
1.62213288700733+0.0791421572031492i
95              1.6393006615072+0.062188683373254i
3.2786013230144+0.124377366746508i
1.6393006615072+0.062188683373254i
96              1.65481784179491+0.0425502592934322i
3.30963568358982+0.0851005185868644i
1.65481784179491+0.0425502592934322i
97              1.66829511764944+0.0203996443774862i
3.33659023529888+0.0407992887549724i
1.66829511764944+0.0203996443774862i
98              1.67936353243881-0.00403571282573558i
3.35872706487762-0.00807142565147116i
1.67936353243881-0.00403571282573558i
99              1.68768191662478-0.0304764435060698i
3.37536383324956-0.0609528870121396i
1.68768191662478-0.0304764435060698i
100              1.69294390133988-0.0585949853643542i
3.38588780267976-0.117189970728708i
1.69294390133988-0.0585949853643542i
101              1.69488439889602-0.0880202026234119i
3.38976879779204-0.176040405246824i
1.69488439889602-0.0880202026234119i
102              1.69328544579823-0.118342822666734i
3.38657089159646-0.236685645333468i
1.69328544579823-0.118342822666734i
103              1.68798131388215-0.149121601439453i
3.3759626277643-0.298243202878906i
1.68798131388215-0.149121601439453i
104              1.67886280664099-0.179890119477096i
3.35772561328198-0.359780238954192i
1.67886280664099-0.179890119477096i
105              1.66588067063098-0.210164101459676i
3.33176134126196-0.420328202919352i
1.66588067063098-0.210164101459676i
106              1.64904806592594-0.239449144841069i
3.29809613185188-0.478898289682138i
1.64904806592594-0.239449144841069i
107              1.62844205473649-0.267248737683339i
3.25688410947298-0.534497475366678i
1.62844205473649-0.267248737683339i
108              1.60420408324451-0.293072442620483i
3.20840816648902-0.586144885240966i
1.60420408324451-0.293072442620483i
109              1.57653944808679-0.316444123150951i
3.15307889617358-0.632888246301902i
1.57653944808679-0.316444123150951i
110              1.54571575532452-0.336910090454504i
3.09143151064904-0.673820180909008i
1.54571575532452-0.336910090454504i
111              1.51206039563607-0.35404705386123i
3.02412079127214-0.70809410772246i
1.51206039563607-0.35404705386123i
112              1.47595707420644-0.367469766146669i
2.95191414841288-0.734939532293338i
1.47595707420644-0.367469766146669i
113              1.43784144651284-0.376838266099868i
2.87568289302568-0.753676532199736i
1.43784144651284-0.376838266099868i
114              1.39819592080877-0.381864635295783i
2.79639184161754-0.763729270591566i
1.39819592080877-0.381864635295783i
115              1.35754369310979-0.382319203428755i
2.71508738621958-0.76463840685751i
1.35754369310979-0.382319203428755i
116              1.31644207893772-0.37803615616093i
2.63288415787544-0.75607231232186i
1.31644207893772-0.37803615616093i
117              1.2754751955131-0.368918519517458i
2.5509503910262-0.737837039034916i
1.2754751955131-0.368918519517458i
118              1.23524602562161-0.354942512111621i
2.47049205124322-0.709885024223242i
1.23524602562161-0.354942512111621i
119              1.19636785730524-0.336161264964838i
2.39273571461048-0.672322529929676i
1.19636785730524-0.336161264964838i
120              1.15945504078925-0.312707898575429i
2.3189100815785-0.625415797150858i
1.15945504078925-0.312707898575429i
121              1.12511293939903-0.284797903558377i
2.25022587879806-0.569595807116754i
1.12511293939903-0.284797903558377i
122              1.09392688865389-0.252730676095737i
2.18785377730778-0.505461352191474i
1.09392688865389-0.252730676095737i
123              1.06644994889008-0.216889895893814i
2.13289989778016-0.433779791787628i
1.06644994889008-0.216889895893814i
124              1.04318929657289-0.177742202125681i
2.08637859314578-0.355484404251362i
1.04318929657289-0.177742202125681i
125              1.02459132003821-0.135833363696473i
2.04918264007642-0.271666727392946i
1.02459132003821-0.135833363696473i
126              1.01102592292843-0.0917809648712567i
2.02205184585686-0.183561929742513i
1.01102592292843-0.0917809648712567i
127              1.00277116504672-0.0462627137948885i
2.00554233009344-0.092525427589777i
1.00277116504672-0.0462627137948885i

Page 9
FFTCalc

Page 10
FFTCalc

Page 11
FFTCalc

0.049068
0.098017
0.14673
0.102841

Page 12

```
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