NOTE THIS IS A TEXT FILE FROM PDF- MUCH BETTER TO READ THE PDF FILE, ctstx.pdf, IF AT ALL POSSIBLE. CTS: Frequency Response as a Function of Temperature C. Backus and S. Gulkis Jet Propulsion Laboratory, Pasadena, CA 91109-8099, Email: backus@jpl.nasa.gov Abstract. This is an investigation report concerning the use of ground-based thermal vacuum test results to calibrate IF frequency as a function of SAW temperature, and also channel number. 1. Introduction The MIRO instrument utilizes a Chirp Transform Spectrometer (CTS) to provide a very high spectral resolution capability for the submillimeter band. The CTS provides 4096 channels, each approximately 44 kHz wide, for a total bandwidth of 180 MHz. The input bandwidth and center frequency are 178 MHz and 1350 MHz respectively. The Surface Acoustic Wave (SAW) devices used in the CTS are constructed of LiNiO3. The SAW devices are temperature controlled at discrete setpoints from 0 to 70. C in increments of 10. The input frequencies to the CTS are preprocessed by an analog Intermediate Frequency processor (IFP) that accepts certain frequencies in the 6-16 GHz range and mixes them down to the 1350 MHz center frequency range of the CTS. IFP frequency translation is described in Table 1. Frequency stability in the IFP is provided by an ultra-stable oscillator (USO). This technical note addresses the susceptibility of the CTS frequency response to changes in temperature. During the course of the MIRO instrument testing program, we placed the entire MIRO instrument inside a thermal vacuum (TV) chamber at JPL and performed a variety of tests. One of these tests was to insert a set of .xed narrow-band frequency signals into the input of the IFP and measure the response of the CTS. These tests were performed with the SAW .lters stabilized at di.erent thermal regimes chosen to sample the range of temperatures expected in .ight. In the following analysis an expression is derived for converting a measured channel number and SAW temperature into a calibrated frequency, along with a procedure that is needed for shifting spectra taken at di.erent temperatures so that they can be coadded and calibrated. 2. Test Con.guration and Data The entire MIRO .ight instrument was placed inside a Thermal Vacuum (TV) chamber at JPL and tested prior to delivery to ESA. In May and June, 2001, tests were carried out to determine the temperature susceptibility of the CTS. For these tests a broadband signal coupler was inserted into the .rst IF ahead of the IFP as shown in Fig 1 below. The signal generator was outside the thermal vacuum chamber; its output signal was passed through a vacuum isolation port. The frequency uncertainty of the signal generator was estimated to be less than 1.5 Khz. The SAW devices were temperature controlled by the MIRO instrument control software. These 1 Table 1. LO Mapping for IFP LO 1 : 140703.25 4xLO 1 = 562813 LO 2 : 2182 LO3 : 7147 LO4 : 7728 Spectrometer Bandwidth: 1260 -1440 MHz RF IF1 IF2 IF3 IF4 IF5 IF6 IF7 IF8 IF9IF10Output Lines smm mixer M1 M2 M3 M4 M5 M6 M7 M8 M9 to (MHz) 4xLO1 2xLO2 LO4 L04 LO4 LO3 LO2 LO3 2xLO2 LO4 cts H2 18O 547677 15136 . . . . . . . . . . . . 7989 5807 1340 . . . . . . 1340 H2 17O 552022 10791 6427 . . . . . . 1301 . . . . . . . . . . . . . . . 1301 CH3OH 553146 9667 14031 6303 1425 . . . . . . . . . . . . . . . . . . 1425 H2 16O 556936 5877 . . . . . . . . . . . . 1270 . . . . . . . . . . . . 1270 CH3OH 568566 5753 1389 . . . . . . . . . . . . . . . . . . . . . . . . 1389 NH3 572498 9685 14049 6321 1407 . . . . . . . . . . . . . . . . . . 1407 CO 576268 13455 9091 1363 . . . . . . . . . . . . . . . . . . . . . 1363 CH3OH 579151 16338 . . . . . . . . . . . . 9191 . . . 2044 6408 1320 1320 Figure 1. Test Con.guration for CTS Frequency Response. were measured using the permanently installed temperature sensors on the SAW devices. They are estimated to be sensitive at a digital quantization level of about 34 mK. Tests were performed by setting the SAW devices in the CTS to a speci.ed temperature and then inserting a series of 8 di.erent frequencies (5.753, 5.877, 9.667, 9.685, 10.791, 13.455, 15.137, 16.338 Ghz) into the signal coupler. The IFP translated these signal generator tones into intermediate frequencies (1389, 1270, 14.25, 1407, 1301, 1363, 1339, 1320 MHz) at the input to the CTS. After cycling through the 8 frequencies, the SAW temperature was changed and the frequencies were recycled. At each temperature and frequency setting, the bin number of the signal was recorded. Table 2 gives the measured channel number for each frequency and temperature setting. At any given temperature, the test data can be represented approximately by a linear equation of the form of eq (1), below, where the slope is a temperature dependent bin width, along with an o.set, which is also temperature dependent. The approximate linearity of the CTS at a given frequency can be seen by plotting the input frequency as a function of the channel number. Figure 2. shows these data for the minimum (9.6.C) and maximum temperature (68.2.C) settings. A systematic o.set, as well as a discernible change in slope, can be seen with these two curves. This is the temperature dependence under consideration. Figure 2. Frequency vs Bin Number. The diamonds are data taken at 9.C, and the triangles are data at 68.C. 3. The Model Function The assumption that the spectral bin width is linearly dependent upon temperature leads directly to an expression linear in temperature, spectral position, along with a cross term. Let F : IFP Frequency (MHz) T : SAW temperature (.C) N: Bin (or channel) number (0-4095) where N is the bin number showing the response to the input frequency, F. Consider a model F = a(T )N + b(T ) (1) .a .b where a = a0 + .T T0 (T - T0) and b = b0 + .T T0 (T - T0) give the bin width and intercept as || first order Taylor expansions around some arbitrary center temperature T0. Substituting these expressions for a and b into eq (1) results in .b .a F =(T - T0)+ a0N + N(T - T0)+ b0. (2) .T .T |T0 |T0 If it is assumed that a and b are indeed linear functions of T then the partials become constants, and the products of the partials with T0 can be ignored since we are free to choose T0 to be Table 2. Bin Numbers in Response to Injected Frequencies IFP GHz: 5.753 5.877 9.667 9.685 10.791 13.455 15.137 16.338 CTS MHz: 1389 1270 1425 1407 1301 1363 1339 1320 2001 DOY T .Ca 158 19:37 68.2 1084 3802 262 673 3094 1678 2227 2661 154 13:55 58.5 1104 3818 284 694 3111 1697 2245 2678 158 22:10 58.4 1104 3818 284 694 3111 1698 2245 2678 159 09:07 48.7 1125 3834 305 715 3128 1717 2263 2696 154 23:04 39.4 1140b 3846b 324b 735b 3144b 1736 2281 2713 159 12:02 38.9 1145 3849 327 736 3145 1736 2281 2713 155 03:23 29.3 1161b 3863b 347b 757 3161 1755 159 18:19 19.37 1185 3880 370 778 3178 1774 2318 2748 159 21:12 9.6 1205 3896 391 798 3195 1793 2336 2765 aOne of the four SAW temperature sensors, spect t1. b Omitted from .t, poor CTS thermal controller stability. .b .a zero. To simplify notation set a =, b = a0, c = , and d = b0. A predicted F is then .T .T written as F = aT + bN + cNT + d, (3) which provides a model in which F is expressed as a linear function of the parameters to be .tted, a, b, c, and d.1 4. Physical Motivation of the Model The development of a theoretical argument to justify eq (3) is a work in progress. If we assume that the sound velocity of the propagating medium in the SAW is linearly dependent upon temperature, then the constant sampling and integration times at the output will be multiplying segments of the chirp of varying width, leading to the conjectured change in channel width, a, as a function of temperature. An alternative view of the situation is that the distance traversed by the chirp in the medium varies with temperature as the SAW thermally dilates or contracts. In either case the timing of the arrival of the chirp at the SAW output is thermally modulated. Similar arguments can be made for the intercept term, b. 5. Results of the Regression Table 2 tabulates the test data from the thermal vacuum testing at JPL in the Summer of 2001. The CTS thermal control was set above the ambient temperature in each thermal regime. The SAW temperature for each test2 is shown in the temperature column of Table 2. The estimated parameters for the .t are shown in Table 3, along with their standard errors. The residuals 1 The linearity assumption was tested using eq (2) as the model, where the partials, a0, and the constant, b0, were estimated linearly for various trial values of T0. It was found that changes in the estimates as T0 was varied were insignificant. 2 Thermal stability at the various setpoints is important. By re-examining the acquired data corresponding to outlier residuals, many tests were found to have been run during unstable regimes. Figure 3. Residuals plotted against temperature, channel number, and cross term. are plotted in Figure 3, and can be seen generally to be within one bin width, nominally about 44KHz. Measurements that were excluded from the .t owing to lack of thermal stability are marked. Because the residuals are small, within the precision limited by the bin width, and because they exhibit no noticeable patterns, we conclude that higher order terms in the Taylor expansions used in eq (2) are not required. Table 3. Fitted Model Parameters Units Value Std error a MHz per deg C -0.0996154 0.0002 b MHz per Channel -0.0443027 0.0000042 c d MHz per [chnl*deg C] MHz 7.66259 x 10-6 1443.27 8.9 x 10-8 6. Independent Model Veri.cation using an Anomalous Birdie The CTS exhibits an intermittent, weak, but sharp, spike feature, whose conjectured origin in the IFP circuitry is under investigation. An example is plotted in Figure 4. The conditions for its appearance or disappearance are not yet known. The feature does not appear in the di.erenced spectra, one indication that it originates in the IFP, downstream from the .rst mixing stage, where a 10MHz frequency shift occurs. Remarkable constancy is demonstrated using the thermal correction parameter estimates to predict the CTS frequency needed to Table 4. Selected IFP Birdie Appearances 2001 SAW Bin Predicted DOY T .C Number Input MHz 153 21:05 48.69 1694 1364.00 154 04:44 54.06 1682 1364.06 154 07:38 52.9 1686 1363.99 154 13:54 58.47 1674 1364.03 154 18:56 48.71 1694 1364.00 154 23:09 39.6 1711 1364.04 155 04:06 29.2 1732 1364.02 156 02:30 68.2 1655 1364.02 158 22:10 58.4 1675 1363.99 159 09:07 48.6 1694 1364.01 159 15:08 29.14 1732 1364.02 159 18:19 19.4 1752 1363.98 159 21:12 9.54 1771 1363.99 Figure 4. Example of the IFP birdie. produce the responses shown in Table 4, where the calculated frequencies are listed in the righthand column. The average value is 1364.01 +/- 0.007 MHz. With reference to the LO mapping shown in Table 1, it is indicated that the signal originates somewhere along the path for 13456 MHz, in the penultimate row of the table. 7. Frequency Corrections for Calculating with Spectra To perform calculations with spectra acquired at di.erent CTS SAW temperatures, it is necessary to calculate with power levels from the same frequency instead of the same bin number. It is therefore required to rebin a set of spectra to re.ect what would have been acquired at a standard operating temperature. For T0 as the standard temperature, data taken at T1 can be rewritten as a T0 spectrum using N0 = a(T1 - T0)+ bN1 + cT1N1 , (4)cT0 + b where N1 runs over all the bins from 0 to 4095. Eq (4) is derived from eq (3) by equating the input frequencies giving (N1,T1) to those for a set of (N0,T0), in other words by writing aT0 + bN0 + cT0N0 + d = aT1 + bN1 + cT1N1 + d (5) and solving for N0. The value from each N1 is placed at the corresponding N0 of the temperature standardized spectrum. The quantization error introduced could be mitigated by re.ning this type of calculation with a suitable linear interpolation, using the same techniques employed to adjust spectra for Doppler shifts. 8. CTS Pulse Position The CTS digital logic accepts a command to determine the timing of the pulse to be converted to a chirp. This a.ects the bin number corresponding to a given test frequency. The nominal .ight value for this parameter is 410, which is the value at which all of the frequency response testing in this analysis was run. During ground testing and also as part of the standard in .ight payload checkout procedure, the e.ect of this parameter was tested. The e.ect is measured by a shift in the bin number of the birdie discussed above. Table 5. Bin Numbers of Birdie at Test CTS Pulse Positions Year/ Pulse Birdie Bin No. Bins per DOY T .Ca Position Bin @ 68.C Pulse Pos. 2001 164 1103 45.95b 412c 1698 1654.4 2001 164 1113 44.07 510 2102 2056 4.10 2001 164 1122 42.53 310 1305 1253 4.02 2001 164 1128 41.34 410 1707 1654.3 4.02 2004 091 0258 23.88 410 1743 1656.2 2004 091 0304 24.20 510 2142 2059.0 4.03 2004 091 0314 25.01 410 1740 1655.4 4.04 2005 088 0656 67.84d 410 1654 1654 2005 088 0704 67.94 510 2054 2054 4.00 2005 088 0730 67.94 410 1654 1654 4.00 2005 276 1353 67.90 410 1654 1654 2005 276 1358 67.90 510 2054 2054 4.00 2005 276 1411 67.90 410 1654 1654 4.00 aOne of the four SAW temperature sensors, spect t1. b Thermal vacuum testing. Not thermally stable. c Questionable, from test log entry, probably 410. d Only this later .ight data was thermally stable. Table 5 shows results from pulse position testing to date. Using the technique described in the preceding section, the birdie position is transformed to a standard temperature, in this case 68.C. The rate of change in bin numbers per single digit of pulse position parameter setting is shown in the right hand column. Because the CTS SAW .lter was thermally stable for testing after 2005, it is believed that 4.0 bins per parameter unit is the best estimate. This rate may well be independent of SAW temperature, but subsequent tests with adequate thermal stability will be required to verify that. Figure 5 portrays the situation in Figure 5. Plot showing relative e.ects of changes in temperature and pulse position. schematic form, where the spectra are shown oriented with frequency increasing from left to right, and bin number going the opposite way, with time. The spectrum emerges from the CTS as a time series which is sequentially digitized at .xed sampling intervals. Changing the pulse position parameter changes a delay, by .P , from the CTS cycle start time for the pulse transmitted to the SAW to create the chirp. Increasing that delay advances the bin number at which a given .xed spectral feature, in this case the birdie, is seen. Temperature change, on the other hand, has a di.erent e.ect, in which the spectrum is dilated or contracted with respect to time, and consequently with respect to bin number. In the highly unlikely event that the CTS needs to be run at a di.erent pulse position, the procedure for frequency calibration would involve an initial step to adjust the measured bin number to the number that would have been seen using the nominal value of 410, B. = B - 4(P - 410), (6) where P is the new pulse position parameter value. After that the corresponding frequency would again be calculated with eq (3). 9. Concluding Remarks We have shown that a simple linear model accounts for enough variation in frequency response to restrict the great majority of the residuals to a band comparable to the measurement sensitivity, the bin width, of the CTS, namely to a range within 44 kHz. With more sophisticated physics in support of a rami.ed model it might be possible to use our test data to measure a second order temperature e.ect. Because we would be moving within our levels of measurement quantization error, more demanding statistical analysis would be required, in which the variances implied by the quantization errors, 1/12 of the measurement intervals3, would be added as weights to the regression problem. Furthermore, in our measurement situation the independent variables were in fact SAW temperature and injected IF frequency, with the the peak bin number as the response or dependent variable. For a truly rigorous statistical analysis the model should be written that way, in order to make signi.cance tests meaningful. The desired results, calibrated frequency as a function of T and B, and a rebinning procedure, could then be constructed by solving the model equation, with its marginally more rigorously estimated parameters, for F . 3This also applies to the digitzed SAW temperature sensor data.