(HP-67/97) RADAR - Speckle Target Calculations

[Thomas Klemm]

From page 13.

Eq. (11) becomes

\(P_D=\frac{1}{\sqrt{\pi}} \int_{-\infty}^{\infty} \: d\tau \: e^{-\tau^2} \: P_F \: ^{\frac{1}{1 \, + \, CNR_s \, e^{2\sigma(\tau\sqrt{2}-\sigma)}}}\)

\((21)\)

Eq. (21) has a form suitable for Gaussian-Hermite quadrature. A 16-point quadrature provided sufficient accuracy for \(\sigma^2 \leq 0.2\). For larger \(\sigma^2\) values the number of quadrature points was increased to 32.

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From APPENDIX, page 87.

PROGRAMMABLE HAND CALCULATOR PROGRAMS

Evaluation of the speckle target detection probability Eq. (21 ) is straightforward and yields excellent accuracy for quadratures with a relatively small number of points. This, coupled with the fact that most objects exhibit a significant diffuse reflection component at optical and infrared wavelengths, has prompted us to develop a program suitable for evaluation on a programmable hand calculator. In Table 21 we detail a program usable with either a Hewlett-Packard HP-67 or HP-97 calculator. This program calculates \(P_D(CUR_s;P_F,\sigma^2)\) for \(P_F\) stored in register A, \(\sigma^2\) stored in register B, and \(CNR_s\) entered as \(x\) using an 18-point Gaussian-Hermite quadrature. The quadrature points and weighing coefficients are entered via a data card. Computation time of this program is roughly 60 seconds and the relative accuracy in \(P_D\) is better than about 0.3%, for \(\sigma^2 = 2.0\) and improves rapidly as \(\sigma^2\) decreases.

…

The preceding programs were also modified for use with a Hewlett-Packard HP-29C calculator. The modified programs are detailed in Table 22. Because of the larger number of storage registers available, a 24—point Gaussian-Hermite quadrature was used. Computation time of this program is roughly 75 seconds and the relative accuracy in \(P_D\) is approximately 0.03% for \(\sigma^2 = 2.0\) and improves rapidly as \(\sigma^2\) decreases.

Nodes and Weights

These tables were calculated with the Nodes and Weights of Gauss-Hermite Calculator.

n = 18

Code:

 1    0.2582677505190967592581  0.4834956947254555528764
 2    0.7766829192674116613167  0.284807285669979578596
 3    1.300920858389617365666   0.0973017476413154293309
 4    1.835531604261628892254   0.01864004238754465192193
 5    2.386299089166686000265   0.00188852263026841789438
 6    2.961377505531606844779   9.18112686792940352915E-5
 7    3.573769068486266079501   1.810654481093430409597E-6
 8    4.248117873568126463023   1.046720579579208244436E-8
 9    5.048364008874466768372   7.8281997721158910293E-12

n = 24

Code:

 1    0.2244145474725155851511  0.4269311638686992496532
 2    0.6741711070372122360003  0.2861795353464430179019
 3    1.126760817611245072133   0.1277396217845591606473
 4    1.584250010961694148506   0.03744547050323074601333
 5    2.04900357366169891179    0.00704835581007267097
 6    2.523881017011426974199   8.23692482688417457918E-4
 7    3.012546137565564825655   5.68869163640437976904E-5
 8    3.52000681303452471129    2.15824570490233363224E-6
 9    4.053664402448149503948   4.01897117494142968454E-8
10    4.625662756423787265049   3.04625426998756390389E-10
11    5.259382927668044367431   6.58462024307817006456E-13
12    6.01592556142573971735    1.66436849648910887377E-16

It's interesting to note that these values were arranged differently in the registers of the HP-67 and the HP-29.
In case of the HP-67 the primary and secondary banks were swapped using the P<>S command.
But since these registers were only addressed indirectly the same approach as for the HP-29 could have been used.

Efficiency

Both program could be improved, though I consider the 2nd one for the HP-29 slightly better.
At least they knew that they could swap \(x\) and \(y\):

HP-67

Code:

050 STO D
051 RCL A
052 RCL D
053 h yˣ

HP-29

Code:

047 RCL 3
048 x↔y
049 f yˣ

It appears they were not aware of the LAST X command.
Both programs use:

Code:

RCL B
√x
-
RCL B
√x
×

The register B contains the parameter \(\sigma^2\) which is constant during the calculation.
They could have stored \(\sigma\) instead of calculating the square root twice in the loop again and again.

Cheers
Thomas