Contains reports on two research projects.California Institute of Technology (Contract 952568)U. S. Air Force Air Force Systems Command (Contract F33615-72-C-2129

Kebabian, P. L.

Poon, Ronnie K. L.

Staelin, David H.

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II. RADIO ASTRONOMYAcademic and Research StaffProf. A. H. Barrett Prof. D. H. Staelin Dr. J. SanderProf. B. F. Burke Dr. P. L. Kebabian Dr. J. W. WatersProf. Susan G. Kleinmann Dr. K. F. Kunzi J. W. BarrettProf. R. M. Price Dr. P. C. Myers D. C. PapaDr. G. D. PapadopoulosGraduate StudentsB. G. Anderson T. S. Giuffrida R. V. McGahanO. Appiah A. D. Haschick R. L. PettyjohnK. P. Bechis P. T. Ho R. K. L. PoonW. G. Brodsky K-S. Lam R. C. WalkerP. C. Crane S. Lee T. J. WarrenR. W. Freund K-Y. Lo R. W. WilcoxR. N. MartinA. ANOMALOUS OXYGEN ABSORPTION INFERRED FROM NIMBUS-5MICROWAVE EXPERIMENTCalifornia Institute of Technology (Contract 952568)R. K. L. Poon, D. H. StaelinAtmospheric absorption near 5-mm wavelength is dominated by oxygen. We shalldescribe 10 methods for computing its absorption coefficient (see Expressions 1-10 inTable II-1). The pressure-broadening theories of Van Vleck and Weisskopf (VV-W)1 andof Gross and Reber (G-R) 2 lead to different spectral line shapes (Methods 1 and 2). Onemajor uncertainty associated with these theories is the linewidth, whose value has beenexperimentally determined but not well explained.3 For this reason, we have developedExpressions 3-10 in Table II-1.We assume that the absorption coefficient k varies with total air pressure p ask= a p , where a and x are frequency-dependent parameters. Good fit of this expressionto Methods 1 and 2 is obtained when a and x are determined by regressing k againstpressure from 10 to 1000 mb. The regression is performed on a sample of pressure-temperature pairs that occur in the ARDC model. 4  This yields Expressions 3 and 4,respectively, for the two line shapes. As a check, a "random sample" is prepared withpressures and temperatures T independently chosen in the range of atmospheric interest.Regression is then performed on this "random sample" in the form k = a px (Expres-sions 5 and 6) and in the form k = a pX Ty (Expressions 7 and 8).Table II-1 shows the result of different expressions of regression in the generalnormalized form k =ko(p/ 0 )x (T/T o) . The parameters ko, x, and y give best fit toMethod 1 values when we use Expressions 3, 5, or 7 for regression. Expressions 2, 4,6, and 8 are counterparts referred to the Gross-.Reber (G-R) line shape. The parametersQPR No. 111Table II-I. Parameters inferred from different expressions of regression inthe normalized form k = ko(p/po ) x (T/T )YPARAMETE1R CH 1 3 5 7 2 4 6 8 9 10p = 617 mb 3 0. 191 0. 189 0. 192 0. 192 0. 181 0. 178 0. 182 0. 182 0. 186 0. 244ko P = 265 mb 4 0. 203 0. 195 0. 199 0. 198 0. 174 0. 167 0. 170 0. 170 0. 232 0. 247p = 89 mb 5 0. 302 0. 297 0. 293 0.293 0. 239 0. 241 0. 242 0. 242 0. 339 0. 7723 - 1. 26 1. 29 1. 29 1. 28 1. 25 1. 24 1.3 1. 5x 4 - 1.40 1.32 1.32 3 1.47 1.45 1.45 1.54 1.755 1.79 1.57 1.57 1.51 1.48 1.48 2.05 4TF = 262'K 3 - 0 0 -0. 10 0 0 -0.07 0 0y T = 223 K 4 - 0 0 -0. 21 - 0 0 -0. 15 0 0T = 217oK 5 - 0 0 -0.96 - 0 0 -0. 84 0 0NEMS Exact ARDC Random Exact ARDC Random Nimbus-5 NEMSChannel (p, T) (p, T) (p, T) User's DataFrequencies _ Guide3 53. 6 5 GHz VV-W G-R WF4 54.9 GHz Line Shape Line Shape5 58. 8 GHz(II. RADIO ASTRONOMY)30FOR EACH CHANNEL ,EXPRESSION 1k: VAN VLECK -WEISSKOPFLINE SHAPE10 - - EXPRESSION 3k=op0 -- 2 -10 2 5 10 2 5 10  2 5 1ABSORPTION COEFFICIENT (km- I )Fig. II-1. Fitted absorption coefficient for the ARDC model atmosphere.are given for the absorption at 53. 65, 54. 9, and 58. 8 GHz, three center frequenciesnear the 5-mm oxygen resonances (Channels 3, 4, 5) which have been selected forthe Nimbus-5 Microwave Experiment (NEMS).5 Figure II-1 shows the good fit ofExpression 3 to the ARDC model atmosphere.The form k = a pX leads to a surprisingly simple description of the temperatureweighting function (WF), defined 3 as k(z) exp[-T(z)], which takes on the simple forms exp -[sz + exp(-sz )] at any height z, where s = x/H, z = z - zm is the height refer-x1 apoenced at the WF peak height z = -n a , and the atmospheric pressure p is assumedm s sto equal po at the surface of the Earth and decrease exponentially with a scale height H.The opacity at z is T(z)= fo k(z) dz = k(z)/s. In particular, at the WF peak, T(z m) = 1,while k(zm) = s, or roughly I dB/km, since x ~ 1. 5, H - 7 km and 1 neper " 4. 343 dB.The peak value of the WF is s/e, which reduces to a fraction f of the peak value whenthe shoulder width of the WF is c/s, where c is a tabulated function of f (for example,c = 2. 45 at f = 0. 5). The exact relationships derived from the form k = a pX are shownin Fig. 11-2. They help us to fit a given WF by a judicious choice of a and x (Method 9).In Table II-1 the normalization constants po and To are the pressures and temperatureswhere the three weighting functions peak. Figure II-3 shows the result of fitting theNEMS weighting function as given in "The Nimbus-5 User's Guide." 5This form of WF, in conjunction with the radiative transfer equation, is used toQPR No. 111It - -- -1 _HEIGHTzW (z) = s exp - sz* + exp (-sz*) s HT (z) = k (z)/s z*=z - zmz2 - z 1 = c/s1 ~ TOTAL I p l2 /p e -c/Xm AREA k2 /kl =T 2 / e-p = 1 -Tz =0SPo - 0 W (Z)k= 0 p k e fsk = k =a P o e WEIGHTINGr= o= ko/s 0<f<l FUNCTIONFig. 11-2. Exact relationships of the analytic weighting function.3224E168W=k e x k =apxp = 1000 e -z/7 km--- WEIGHTING FUNCTIONS IN THE NIMBUS 5 USER'S GUIDEWEIGHTING FUNCTION W (z)x102 km- IFig. II-3. Fitted weighting functions for the NEMS oxygen channels.QPR No. 111_ __za --4(a) (b)Fig. II-4. Brightness-temperature error: (a) Channel 3, (b) Channel 4, (c) Channel 5.C I I I I I I I__ __ ____ -e~ __,___~__ __ _ __ __ _- ~--~-----~ _~____ _-~---(II. RADIO ASTRONOMY)calculate brightness temperatures TB,calc for comparison with NEMS measured valuesTB,NEMS for 100 major frames of data (December 1972) with wide spatial and temporalextent (Method 10). At each NEMS frequency, a and x are varied as free parameters incomputing the absorption for the subsatellite temperature profile as analyzed by theNational Oceanic and Atmospheric Administration (NOAA). We base our compari-son on 4 error criteria: (i) mean error E, which is the average of the discrepancyE = TB,calc T B,NEMS; (ii) tilt t, defined as E = t(T BNEMS-T BNEMS) + E; (iii) rmserror about E; and (iv) rms error about the tilted line. (In these expressions, overbarsdenote averaging over the 100 major frames.) These discrepancies depend much moreon x than on a. In Fig. II-4 the x dependence is shown for the value of a that gives theleast rms about E. Besides uncertainty in the absorption coefficient, there is inherentdiscrepancy associated with instrumental noise, error in the NOAA analysis, and otherapproximations in the radiative transfer computation. Everything considered, however,we recommend the values x = 1. 5, 1.75, and 4 for Channels 3, 4, and 5, respectively,and z = 4, 9. 25, and 17. 5 km. This suggests more absorption than is given by theVV-W line shape, which in turn predicts more absorption than is given by the G-R lineshape, at least near the peak of each WF. Absorption measurements with a Fabry-Perotinterferometer are being made in our laboratory to resolve the present discrepancy.References1. J. H. Van Vleck, "The Absorption of Microwaves by Oxygen," Phys. Rev. 71, 413-424 (1947); J. H. Van Vleck and V. F. Weisskopf, "On the Shape of Collision Broad-ened Lines," Rev. Mod. Phys. 17, 227 (1945).2. E. E. Reber, "Absorption of the 4- to 6-mm Wavelength Band in the Atmosphere,"J. Geophys. Res. 77, 3831-3845 (1972).3. M. L. Meeks and A. E. Lilley, "The Microwave Spectrum of Oxygen in the Earth'sAtmosphere," J. Geophys. Res. 68, 1683-1703 (1963).4. R. A. Minzner, K. S. W. Champion, and H. L. Pond, "The ARDC Model Atmo-sphere, 1959," AFCRC-TR-59-267, Air Force Surveys in Geophysics No. 115,Geophysics Research Directorate, Air Force Cambridge Research Center, U. S.Air Force, Bedford, Massachusetts, August 1959.5. R. R. Sabatini (Ed.), "The Nimbus-5 User's Guide," Goddard Space Flight Center,National Aeronautics and Space Administration, Greenbelt, Maryland, November1972.QPR No. 111(II. RADIO ASTRONOMY)B. STELLAR INTERFEROMETER: DIGITAL FILTERSU. S. Air Force Air Force Systems Command (Contract F33615-72-C-2129)P. L. KebabianIn previous studies of digital filters with variable sampling ratel ' 2 the input to thefilter was a time series of numbers. Obviously, all results derived for that case alsoapply to the case of parallel processing of numbers stored in a table. When the impulseresponse of the filter is of finite duration, which is almost always true of filters ofpractical interest (cf. Schafer and Rabiner3), it is possible to build such a parallelprocessor to obtain any given sample of the filter' s output. This is no advantage in thecase of rate-reducing filters, but it provides an attractive way of using a rate-increasingfilter to interpolate between numbers stored in a table, in particular in a table of cosines.Consider the rate-increasing filter in Fig. II-5a. The input sequence is w 0, w i, w2,... , and the output of the rate-increasing sampler is w, 0, . ... , wl, 0, 0, ... . , with r-1zeros added after each input number. The time-invariant filter interpolates betweenthese numbers to produce the output sequence y0 y, y 2, . . . . . Figure II-5b shows theequivalent parallel processor to generate y l . Note that the filter h. must be causal(hj< 0 = 0), but there is no similar restriction on the coefficients in the parallel proces-sor.Figure II-6 shows the spectrum at the input to the filter, after the sampler, and atthe output when the w are samples of a cosine function, w. = A cos (Zrrj/N). H(f) is thefrequency response of the filter h. and, by assumption, M h. = 1 = H(0). Ideally, the onlyJ j 3components of the output would be at I and (r-1)N, and the design of the filter is governedby the tolerance on the components at N± 1, 2N± 1, etc. For a given tolerance, as rincreases the number of coefficients also increases, but because only 1/r of the samplesWO, 0, 0 ... , .,  , 0,...WO' wI .T -YTIM -Y1N ARIT - 7 YO' YA Y2 ... ASAMPLING RATE f(0) AT INPUT0 1 N-1 N fW01 h A2 + AFTER SAMPLER (r =2)yl 0 1 N-I N N+i rN-1 rN f0 h0 A AH(f)(b) IAT OUTPUT(b) 0 1 N-1 N N+1 rN-I rN fFig. 11-5. (a) Interpolation in a time series. Fig. 11-6. Signal spectra.(b) Parallel interpolation, for thecase r = 2.QPR No. 111I _~ II(II. RADIO ASTRONOMY)sent to the filter are nonzero the number of multiplications per second (in Fig. II-5a)stays the same. For very large r, the memory required for the coefficients becomesimpractically large, and this prevents realizing the interpolator as a single unit cellsuch as in Fig. 1I-5.As shown elsewhere, both rate-decreasing and rate-increasing filters are realizableas a cascade of unit cells, the total rate change being the product of the rate changes ofthe separate cells. Realizable, in this case, means only that the filter in each cellis causal, and that the coefficients in the impulse response may be calculated from thecell's rate change, the overall rate change, and the overall impulse response. Ingeneral, the duration of the cell impulse responses is infinite, even when the overallimpulse response has a finite duration. There is, however, one family of functions,f(b), that generate sets of duplicating filters all of which have finite duration. Theseare fn(b)= 1 +[b/n], n > 1, b > 0, together with f0 (b) = 1 (for b<0, f(b) = 0 in both cases),also with convolutions of these with each other. For example, f 0 generates theduplicating filters that comprise a single boxcar integrator. Likewise, f 0 *f 0 (which isalso fl ) generates the duplicating filters that comprise two boxcar integrators in cas-cade. An important remaining theoretical question is, Under what circumstances willthe duplicating filters used to realize a given finite-duration impulse response filter withrate change all have finite-duration impulse responses?SAMPLINGRATE f SAMPLING RATESAMPLE DELAY r1 2(a)Fig. 11-7. (a) Chain with a delay.(b) Equivalent chain.-' 1 '2 SAMPLE DELAY(b)One of the elementary properties of a cascade of unit cells (Fig. II-7a) is that a timedelay associated with the filter may be moved to the high sampling rate end of the chain,as in Fig. II-7b.Assume that the chain has n stages, each having a rate increase of 2, and a filterwith a symmetrical, finite-duration impulse response with 0 time delay. Now let a timedelay of d < 2n samples be inserted after the output of the chain; d/ 2 n = d 2-1 + d2 2-  +-n.+ d 2 n, d. = 0 or 1. This is equivalent to a chain having d1 samples delay in serieswith the first filter, d2 in series with the second filter, and so forth. Suppose that theinput is a sequence of samples from a cosine function, as before, with N = 2m, and thatA and the filter impulse responses are such that the output is a suitably accuratesequence of samples, at 2n higher sampling rate, from a cosine function. By selectingQPR No. 111(II. RADIO ASTRONOMY)the initial phase of the input sequence and d 1 through dn, the output at index 0 is thecosine of any angle, represented as an n+m bit binary fraction of a circle. Since thefilter impulse responses have finite duration, only a finite number of input samples isneeded, usually <<N.This is the central idea of the present interpolation method: a 2m word read-only-n+mmemory (ROM) and n unit cells simulates a 2 word ROM, which might be far toolarge to be built directly.Assume that each of the filters in the unit cells has an odd number of points, k+1, inits impulse response. As noted by Schafer and Rabiner,3 this should be true of the filterto which the chain is equivalent because otherwise there will be a 1/2 sample phaseshift; that is, the interpolator would not generate cos (0). Let the input to the chaintransmit k+l samples to the rate-increasing sampler, which interleaves k+1 zeros andthus transmits 2k+2 samples to the filter, the indices of the samples being from -kthrough (k+l). When the coefficient d of the unit cell is zero, the filter uses sampleswith indices -k/2 through k/2 in computing the output sample with index 0; -k through 0for the output of index -k/2; and 0 through k for the output of index k/2. If d = 1, thehighest index input sample used is k+1. Thus, the k+1 input samples are all that areneeded to find k+1 output samples, and therefore the parallel processor may have anynumber of identical unit cells. Figure II-8 shows an example for k = 2.DECODING BITS2m WORDS MATRIX ADDRESSO INPUTMINPUTS M OUTPUTS OUTPUT(a)M_ _ -._ _-_-- DECODING( ) MATRIX - mBITSADDRESSINPUT2 + P-1WORDSINPUTS h 0 1 OUTPUTP OUTPUTS(P- 3)(b) (b)Fig. 11-8. (a) Parallel unit cell. Fig. 11-9. (a) Conventional read-only-(b) Network M. memory (ROM).(b) Modified ROM.QPR No. 111(II. RADIO ASTRONOMY)The ROM may store samples from the entire cosine function, or from just onequadrant. In that case, means must be provided to convert the address to that quad-rant, and to multiply the outputs by ± 1 as needed. Because the input to the interpolatoruses several consecutive entries from this memory, it differs from the conventionalROM, as illustrated in Fig. 11-9.Duplicating filters, as used for time-series filtering, are inherently discrete-timefilters, but not necessarily digital, except to the extent that analog delay elements suchas charge-coupled devices are not generally available. This limitation does not apply tothe parallel processors described above in which it is perfectly possible for all the sig-nals to be analog. In that case, this system is a form of digital-to-analog converter, inwhich the output is the cosine of the input number. The practical constraints deter-mining the best structure of the unit cell are very different from those that apply to thecase of digital signals, and it remains to be seen if this kind of D/A converter would becompetetive with more conventional methods.In the digital case, errors in the output result from errors in the numbers in theROM, rounding errors in the unit cell, and errors inherent in the quantized angle, all inaddition to the unwanted frequencies transmitted by the filter. Thus, all parts of thesystem must use a word length comparable to that of the output, and this, as well as thefact that each unit cell has several parallel data paths, means that bit-parallel repre-sentation of the numbers would involve an excessive amount of circuitry. Using bit-serial numbers simplifies the structure of the ROM, since a simple shift register willdrive the bit-select lines, instead of a complete decoder.The filters in the unit cells contain a cascade of k boxcar integrators. This struc-ture, as shown in Fig. II-10 for the case k = 4, has the advantage of being built from aregular array of serial adders, and thus is suitable for large-scale integration. Forthis kind of filter, the only significant unwanted components in the spectrum are atINPUTS T L OUTPUTSI +d 1 oFig. II-10. Unit cell built from 4 cascaded boxcar integrators.QPR No. 111(II. RADIO ASTRONOMY)N+ 1, with images at (r-1)N± 1. These have amplitude =(1/N)k , =2 - m k for N = 2mThey must produce negligible error compared with the quantization error, and thus2 -mk+1+t < 2 -(m + n) , or k(m) > (m+n+t+l)/m, where t is the tolerance in bits (typically2). Also, k must be even so that the number of points in the impulse response is odd.Each word in the ROM has length m+n+k+t' bits, where m+n is the length of the out-put, k is the increase in length within one unit cell before the number is rounded to formthe cell's output, and t' is the tolerance to insure that the rounding error in the filter isnegligible compared with the quantization error (typically 2 bits). Assume the ROM canhave, at most, a total of 2s bits, and that only one quadrant of the function is stored;s = 13 is reasonable, given the current state of the art, and the greater complexity ofthis kind of ROM. Then 2m(m+n+k+t ' ) < 2 s is a limit that must be satisfied by k(m), andm should be as large as possible, since for fixed n+m the computation time is propor-tional to n.Each unit cell rounds the outputs from the adder array, and resynchronizes them byadding a delay of at least one bit. It is possible to use a real chain of separate unitcells, but a better way to get the same result is to store the outputs in shift registers,and at the end of each interpolation return them to the input of the same unit cell forthe next interpolation. While this is happening, the ROM may send a new set of numbersto another unit cell, thereby effectively doubling the computation speed in the ordinarycase when both the sine and cosine of an angle are used.Table II-2 summarizes the performance of the system. The times are in units ofthe clock period of the serial data. If this is conservatively taken to be 1 Ls, the timescompare favorably with the 1-2 ms required for a typical 16-bit general-purpose compu-ter (NOVA 1200) with hardware multiply/divide to compute a 24-bit answer.Table 11-2. System performance.m+n Bits m k Number of Adders Time=k2 +k/216 8 4 18 17624 8 4 18 48032 7 6 39 100064 6 12 150 4500The advantage of this kind of cosine generator is that it requires relatively littlecircuitry and is comparable in speed to a small general-purpose computer with hard-ware multiply/divide. In computers without such a multiplier, it will greatly increaseQPR No. 111(II. RADIO ASTRONOMY)the speed of jobs such as spectral analysis, and in computers with one it will increasethe speed somewhat, by permitting a measure of parallelism in operation of the program.References1. P. L. Kebabian, Quarterly Progress Report No. 101, Research Laboratory of Elec-tronics, M.I.T., April 15, 1971, pp. 1-11.2. P. L. Kebabian, Ph. D. Thesis, Department of Electrical Engineering, M. I. T.,1972.3. R. Schafer and L. Rabiner, Proc. IEEE 61, 692 (1973).QPR No. 111

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