GBT Calibration and Aperture Ef ciency

Transcription

1 Precision Telescope Control System PTCS Project Note 37.1 GBT Calibration and Aperture Ef ciency Dana S. Balser GBT Archive: PR048 File: PROJECTS Keys: PTCS, pointing, calibration 19 July 2004 Abstract Routine pointing and tipping observations are used to explore a method of calibration using total power, cold sky data. At X-band (9 GHz) the results are consistent with alternative methods. At Q-band (43 GHz) the results are not very promising. These data have been compromised by the recent discoveries of nonlinearities in the GBT IF system. Contents 1 Calibration Methodology Y-factor Method Astronomical Method Aperture Ef ciency Results X-band Q-band Aperture Ef ciency 8 4 Conclusions 15 5 References 17 A Matlab Function ta 18 B Matlab Function cal 19 C Matlab Function plotcal 21 History 37.0 Original Draft (Dana S. Balser).

2 PTCS/PN/ Calibration Methodology Radio frequency waves with ux density S ν (ergss 1 cm 2 Hz 1 ) induce currents in the antenna system with a power that is proportioinal to the temperature of a charateristic resistor in the Rayleigh-Jeans limit (hν kt). This temperature is called the antenna temperature, T a, and it is often used as the intensity scale for single-dish, cm-wave astronomy. To calibrate the antenna temperature, power is injected into the signal path using a noise diode with a measured intensity in Kelvins. An alternative method is to bypass this temperature scale and calibrate the intensity relative to astronomical sources. There are a handful of astronomical calibration sources, point sources for many telescopes, that have well measured ux densities. Therefore, observations of these calibrators allow the observer to convert their intensity scale from detected power in counts to Jy/beam, where 1 Jy (Jansky) is ergss 1 cm 2 Hz 1. This intensity scale is independent of the telescope. (For example, see Maddelena et al. 2002). Therefore in principle the observer does not need to know much about the details of the telescope (e.g., aperture ef ciency) to perform their calibration. In practice, however, there are corrections due to gravitational deformation of the primary with elevation (gain corrections), atmospheric amplitude corrections that depend on the weather conditions and frequency, etc., that must measured. Moreover, these details are important in evaluating the performance of the telescope. To measure the aperture ef ciency we have to compare a known input power with the detected output power. Typically, astronomical sources are used for large radio telescopes since we need to know the ux density in the far eld and this distance for the GBT is, well, far ( D 2 /λ). We also require good measurements of the noise diode temperature scale the CAL to calibrate the detected output power. The noise diode values are typically measured either in the lab or on the telescope using the Y-factor method (see below). The data must be corrected for the atmosphere at higher frequencies where the aperture ef ciency measurements are most interesting. 1.1 Y-factor Method This method involves using a known hot and cold load that is coupled into the receiver system. Typcially an air temperature absorber and liquid nitrogen are used as the hot and cold loads, respectively. The CAL is then red ON and OFF. The total power is measured for each of these phases: P on h = (T rx + T h + T cal )G (1) P off h = (T rx + T h )G (2) P on c = (T rx + T c + T cal )G (3) P off c = (T rx + T c )G (4) where P denotes the total detected power in counts, T rx is the receiver temperature, T cal is the noise diode temperature, T h is the hot temperature load, T c is the cold temperature load, and G is the gain. The superscript on and off corrrespond to the CAL turned ON and OFF, respectively. The receiver temperature is then T rx = T h yt c y 1 (5) where

3 PTCS/PN/ The CAL can be determined for the hot and cold measurements and is given by, y = Ph off /Poff c. (6) and T cal (hot) = ( Pon h Ph off 1)(T rx + T h ) (7) T cal (cold) = ( Pon c Pc off 1)(T rx + T c ) (8) 1.2 Astronomical Method During routine pointing observations a measure of the system temperature on cold sky is made at different elevations. So effectively there exists many measurements of total power with different input levels. If we can estimate the input power level on cold sky then these data can be used to measure the receiver and noise diode temperatures. Consider a total power measurement on cold sky. The antenna temperature is given by T cal T a = T rx + η l T atm (1 e τa ) + (1 η l )T spill + T fss + η l T cmb e τa, (9) where T a is the antenna temperature in units of the noise dioide calibration temperature (T cal ), T rx is the receiver temperature, T atm is the mean temperature of the atmosphere, T spill is the effective temperature of the real spillover, T fss is the forward spillover and scattering temperature, T cmb is the cosmic microwave background temperature, η l is the rear spillover ef ciency, τ is the opacity at zenith, and A is the number of air masses relative to zenith (1/sin(E)). Let us assume that we have independently determined values for T atm, T spill, T fss, T cmb, and η l. Then for each polarization (R for RCP and L for LCP), T L a = T L rx + η l T atm (1 e τa ) + (1 η l )T spill + T fss + η l T cmb e τa T L cal (10) and T R a = T rx R + η l T atm (1 e τa ) + (1 η l )T spill + T fss + η l T cmb e τa. (11) We can formulate these equations as a two-dimentional nonlinear least-squares problem with a dependent variable A and independent variables Ta L and Ta R. The parameters we seek are Tcal L, T cal R, T rx, L Trx R, and τ. T R cal 1.3 Aperture Ef ciency The aperture ef ciency (η a ) is given by (e.g., Condon 2003), S ν = 2k T A η a A g, (12)

4 PTCS/PN/ where k is Boltzmann s contstant, A g is the geometric primary area, and T A is the antenna temperature corrected for atmospheric attenuation given by For the GBT, T A = T A e τa (13) η a = 0.351T A [K] S ν [Jy] (14) 2 Results Both pointing data and tipping scans were used to produce system temperatures in units of the CAL (i.e., T cal = 1) as a function of elevation. The data were then ported into MatLab where the function lsqnonlin was used to solve the nonlinear least-squares problem as outlined above. The user de ned functions are listed in the appendix. The function ta calculates the difference between the observed antenna temperature and the model estimate and is used by lsqnonlin directly. The function cal imports the data, calls lsqnonlin, and then returns both the input and output data. The function plotcal plot the results. Since we are using cold sky as our known load we have to estimate the various contributions to the input power. We assume η l = 0.99, T atm = 260K, T spill = 300K, T fss = 1K, and T cmb = 2.7K. Except for the rear spillover ef ciency the solution is not very sensitive to variations in these values. For example, varing the mean atmospheric temperature from K only changes the results by 2. The solution is more sensitive to η l = 0.99, however. Changing η l from 0.99 to 0.98 increases the CAL values by X-band Three data sets were processed. Projects TPTCSDSB and TPTCSRMP were all-sky pointing observations during excellent weather conditions. Project TPTCSDSB consisted of tipping scan 4. Inital values of [Tcal L, T cal R, T rx, L Trx R, τ] were [2.8, 2.7, 19.7, 16.9, 0.01]. Small perturbations of the initial values alter the results by about 5 in some cases. Larger perturbations of the initial values converges on other local minima. For example, initial values of [1.5, 1.5, 10, 10, 0.01] yields a solution of [1.9, 1.9, 9.7, 9.4, ]. The data were restricted to elevations between Srikanth (1989) predicts that the forward spillover will signi cantly increase below 20 and we have assumed that this term is constant. Also, for X-band the system temperature begins to increase above It it not clear why this occurs but it is visible in all data sets. It seems unlikely given the optical design that the system temperautre would increase for X-band but not other frequency bands. Discussions with Roger Norrod provided some possible explainations. Maybe the active surface increases the spillover at high elevations since we do not observe this effect at lower frequencies, where the active surface is turned off, and the variations are too small to detect at Q-band where the system temperature is higher. Another possibility is that the cryogenic system is degraded at high elevations; this seems unlikely since we are not at the high elevations for very long periods during the all-sky pointing observations. Nonetheless, the results only vary by a few percent if the entire elevation range is used. The results are summarized in Table 1. The values of σ correspond to ( 1 N ta2 (x)) 1/2, where ta 2 is the sum of the squared residuals returned by lsqnonlin. Overall the results agree very well. Figures 1 3 plot the observations (symbols) and the MatLab ts (solid curve). 2.2 Q-band Four data sets were processed from projects TPTCSRMP and TPTCSRMP040418, each consisting of pointing data and a tipping scan. Inital values of [Tcal L, T cal R, T rx, L Trx R, τ] were [9.9, 9.6, 22.3, 28.7, 0.1]. Overall the solutions at Q-band are more sensitive to perturbations in the initial conditions.

5 PTCS/PN/ Table 1: X-band (9 GHz) Calibration Tcal L Tcal R Trx L Trx R σ Project [K] [K] [K] [K] τ [K] Comment Laboratory Y-factor TPTCSDSB Pointing TPTCSRMP Pointing TPTCSDSB Tipping 12.5 TPTCSDSB (X band) El T cal (LCP) = T cal (RCP) = T rx (LCP) = 19.2 T rx (RCP) = τ = T a /T cal Elevation [deg] Figure 1: Calibration results from project TPTCSDSB all-sky pointing run. The LCP and RCP observations are denoted by red crosses and blue stars, respectively. The green solid curve is the t to the data. Temperature units are in Kelvins.

6 PTCS/PN/ TPTCSRMP (X band) El T a /T cal T (LCP) = cal T (RCP) = cal T rx (LCP) = T (RCP) = rx τ = Elevation [deg] Figure 2: Calibration results from project TPTCSRMP all-sky pointing run. The LCP and RCP observations are denoted by red crosses and blue stars, respectively. The green solid curve is the t to the data. Temperature units are in Kelvins.

7 PTCS/PN/ TPTCSDSB (X band) El T a /T cal T cal (LCP) = T cal (RCP) = T (LCP) = rx T (RCP) = rx τ = Elevation [deg] Figure 3: Calibration results from project TPTCSDSB tipping scan 4. The LCP and RCP observations are denoted by red crosses and blue stars, respectively. The green solid curve is the t to the data. Temperature units are in Kelvins.

8 PTCS/PN/ Table 2: Q-band ( GHz) Calibration (beam 1) Tcal L Tcal R Trx L Trx R σ Project [K] [K] [K] [K] τ [K] Comment Laboratory Y-factor TPTCSRMP Pointing TPTCSRMP Tipping TPTCSRMP Pointing TPTCSRMP Tipping Only elevations between were used. As discussed earlier below 20 the forward spillover increases. The Q-band data at higher elevations looks good for the pointing data; the increase in system temperature above 60 at X-band is not present. But there are some uctuations in the Q-band tipping scan due to instabilities in one of the channels. Therefore the data were limted to elevations below 70. Using the entire range of elevations can change the results by as much as 20, even for the pointing data which appears good at all elevations. The results are summarized in Table 2 and in Figures 4 7. There are some signi cant differences in the calibration values listed in Table 2. Note that the RCP receiver temperatures are much higher for the tipping results. To better understand the Q-band calibration, which is critical in determining the Q-band ef ciency, additional observations were made using the Y-factor method. The same procedures were used except the GBT IF system was used instead of the antenna test range IF system. The hot load was a room temperature chopper wheel that is under computer control. The cold load was a styrofoam container of liquid nitrogen. Both the odd and even optical driver modules (ODMs) were used where ODMs 1, 3, 5, 7 correspond to channels L1, R1, L2, R2 and 2, 4, 6, 8 correspond to channels L1, R1, L2, R2. The results are in Table 3 for the odd ODMs and in Table 4 for the even ODMs. There are differences in T cal larger than 10 between the hot and cold load data, larger than the measured rms. The measurements are quite repeatable, however. Comparison of the odd and even ODMs reveals differences in the receiver temperature as large as 50, although the T cal values are similar. Also, the results in Tables 3 4 are not consistent with the laboratory data in Table 2 that used the same method. Minter et al. (2004) have revealed nonlinearities in the GBT IF system as large as 5. Some optical ber paths are worse than others. During the Q-band measurements an investigation of the nonlinearities revealed that ODM 3 had the largest gain compression and ODM s 5 and 8 had the largest gain expansion. Indeed these ODMs have the largest discrepancy in Tables 3 4. For astronomical data the X-band system is con gured using typically ODM s 2 and 4 for LCP and RCP, respectively. While at Q-band the rst beam is typically connected to ODMs 1 and 3 for LCP and RCP, respectively. We were aware of this problem during project TPTCSRMP and carefully balanced the IF system ON source for each elevation. Although we should have the same power on source the system temperature values will be at different power levels depending on the relative strengths of the calibrator sources. 3 Aperture Ef ciency The aperture ef ciency is calculated using the equations in 1.3. The X-band data consisted of all-sky pointing observations. Therefore the sources of interest were not necessarily ux calibrators but bright pointing sources. However, 3C147 was used in the pointing catalog for each observing epoch. Typically 3C147 was only observed at a few different elevations and here we measure the ef ciency near an elevation of 25 for each epoch. The results are listed in Table 5 for X-band where both the LCP and RCP results are shown. These ef ciencies are consistent with expectations (Norrod 1995). For Q-band we concentrated on the data for project TPTCSRMP where the weather was reasonable good. The results are summarized in Figure 8. The calibrator 3C286 was observed over a range of elevations. The active surface was adjusted based on OOF holography results at a high elevation and were dialed in using Zernike polynomials. This model was applied for all of the points in Figure 8. Three different sets of CAL values are

9 PTCS/PN/ TPTCSRMP (Q band) El 20 70) T (LCP) = cal T cal (RCP) = T rx (LCP) = T (RCP) = rx τ = T a /T cal Elevation [deg] Figure 4: Calibration results from project TPTCSRMP pointing data. The LCP and RCP observations are denoted by red crosses and blue stars, respectively. The green solid curve is the t to the data. Temperature units are in Kelvins.

10 PTCS/PN/ TPTCSRMP (Q band) El 20 70) 20 T a /T cal T (LCP) = cal T cal (RCP) = T (LCP) = rx T rx (RCP) = τ = Elevation [deg] Figure 5: Calibration results from project TPTCSRMP tipping scan 663. The LCP and RCP observations are denoted by red crosses and blue stars, respectively. The green solid curve is the t to the data. Temperature units are in Kelvins.

11 PTCS/PN/ TPTCSRMP (Q band) El 20 70) 18 T cal (LCP) = T cal (RCP) = T rx (LCP) = T rx (RCP) = τ = T a /T cal Elevation [deg] Figure 6: Calibration results from project TPTCSRMP pointing data. The LCP and RCP observations are denoted by red crosses and blue stars, respectively. The green solid curve is the t to the data. Temperature units are in Kelvins.

12 PTCS/PN/ TPTCSRMP (Q band) El 20 70) 16 T cal (LCP) = T cal (RCP) = T rx (LCP) = T rx (RCP) = τ = T a /T cal Elevation [deg] Figure 7: Calibration results from project TPTCSRMP tipping scan 9. The LCP and RCP observations are denoted by red crosses and blue stars, respectively. The green solid curve is the t to the data. Temperature units are in Kelvins.

13 PTCS/PN/ GHz Hot Scan = 12 Cold Scan = 13 Thot = Tcold = 80 Table 3: Q-band Y Factor Method odd ODMs Chan Trx RMS TcalCold RMS TcalHot RMS Yfactor RMS GHz Hot Scan = 15 Cold Scan = 16 Thot = Tcold = 80 Chan Trx RMS TcalCold RMS TcalHot RMS Yfactor RMS GHz Hot Scan = 17 Cold Scan = 18 Thot = 294 Tcold = 80 Chan Trx RMS TcalCold RMS TcalHot RMS Yfactor RMS

14 PTCS/PN/ GHz, Even ODMs Hot Scan = 20 Cold Scan = 19 Thot = Tcold = 80 Table 4: Q-band Y Factor Method even ODMs Chan Trx RMS TcalCold RMS TcalHot RMS Yfactor RMS GHz, Even ODMs Hot Scan = 21 Cold Scan = 22 Thot = Tcold = 80 Chan Trx RMS TcalCold RMS TcalHot RMS Yfactor RMS GHz, Even ODMs Hot Scan = 26 Cold Scan = 27 Thot = Tcold = 80 Chan Trx RMS TcalCold RMS TcalHot RMS Yfactor RMS GHz, Even ODMs Hot Scan = 28 Cold Scan = 29 Thot = Tcold = 80 Chan Trx RMS TcalCold RMS TcalHot RMS Yfactor RMS

15 PTCS/PN/ Table 5: X-band (9 GHz) Ef ciency El η a Project [deg] LCP RCP Comment TPTCSDSB Pointing TPTCSRMP Pointing TPTCSDSB Tipping shown. The typical rms standard deviation of each data point is 5 based on the four cross scans in a peak procedure. Although the data for elevation 44.2 had uncertainties near 20. Condon (2003) determines the Q-band ef ciency from data taken in 2003 November 22 (see Figure 2). During these observations the active surface was turned on but no holography corrections were made. Also, these data were taken with beam 2 using ODM s 5 and 7 for LCP and RCP, respectively. The peak ef ciency is about the same for both periods; although the ef ciency falls off above and below an elevation of 52 degrees for the 2003 data. Based on these comparisons the peak performance does not appear to have been improved by the holography measurements. Nevertheless, direct comparison of pointing data with the OOF holography corrections ON and OFF revealed a improvement in gain and smaller sidelobes. 4 Conclusions All-sky pointing data and tipping scans are used to calibrate the receiver system noise temperature and the noise diode intensity. At X-band (9 GHz) the results are consistent with the traditional Y-factor method. At Q-band there are variations as high as 15 in the CAL and 50 in the receiver temperature using different methods and different data sets. These data have been compromised by the recent discoveries of nonlinearities in the GBT IF system. Jim Condon suggested the method of calibration. Kim Constantikes helped formulate and optimize the nonlinear least-squares problem using the matlab function lsqnonlin. Richard Prestage performed many of the astronomical observations. Roger Norrod, Bill Bennett, and Jonah Bauserman performed the Q-band calibration using the Y-factor method.

16 PTCS/PN/ Figure 8: Aperture ef ciency versus elevation. The data are from project TPTCSRMP consisting of peak measurements towards 3C286 using the OOF holography results for high elevation (python funtion setchigh()). Three different calibration values are used from Table 2: laboratory (red circles), pointing (green squares), and tipping (blue triangles). The LCP and RCP channels correspond to the lled and open symbols, respectively. The antenna temperature values used to determine the aperture ef ciency were calculated using the mean value of the four peak scans in (Az, El). Typically the rms of these numbers was 5; except for the data at elevation 44.2 were the rms was 20.

17 PTCS/PN/ References Condon, J. J. 2003, GBT Ef ciency at 43 GHz, PTCSPN Maddalena, R., Fisher, R., Jewell, P., Braatz, J., Kemball, A., Langston, G., & McMullin, J. 2002, Speci cation of GBT Astronomical Intensity Calibration. Minter, T., Maddalena, R., & Mason, B. 2004, GBT IF System Non-Linearity April 4 and 8, 2004 Tests Results. Norrod, R. D. 1995, GBT Surface Accuracy, GBT Memo 119. Srikanth, S. 1989, Spillover Noise Temperature Calculations for the Green Bank Clear Aperture Antenna, GBT Memo 16.

18 PTCS/PN/ A Matlab Function ta function F = ta(x, p1, p2) TA function for use by lsqnonlin in GBT calibration x - parameters p1 - independent variable p2 - dependent variable usage: x0 = [9.8, 9.5, 30.0, 30.0, 0.1] [x,resnorm] = lsqnonlin(@ta, x0, xl, xu, options, p1, p2 ) parameters in x: tcallcp - LCP noise diode value (9.8 K) tcalrcp - RCP noise diode value (9.5 K) trxlcp - LCP receiver temperature (30 K) trxlcp - RCP receiver temperature (30 K) tau - zenith opacity (0.1) independent values in p1: TLCP - system temperature (units of CAL) first half of array TRCP - system temperature (units of CAL) second half of array dependent values in p2: A - 1/sin(E) dumplicated in array constants eta - rear spillover efficiency (0.99) tatm - temperature in atmosphere (260.0 K) tspill - rear spillover temperature (300.0 K) tfss - forward spillover and scattering temperature (1 K) tcmb - cosmic microwave background temperature (2.7 K) n - number of LCP measurements m - number of RCP measurements eta = 0.99; tatm = 260.0; tspill = 300.0; tfss = 1.0; tcmb = 2.7; n = length(p1)/2; m = n; F1 = ( x(3) + eta*tatm*(1.0 - exp(-x(5)*p2(1:n))) + (1.0 - eta)*tspill... + tfss + eta*tcmb*exp(-x(5)*p2(1:n)) ) / x(1); F2 = ( x(4) + eta*tatm*(1.0 - exp(-x(5)*p2(n+1:n+m))) + (1.0 - eta)*tspill... + tfss + eta*tcmb*exp(-x(5)*p2(n+1:n+m)) ) / x(2); Ft = [F1,F2]; F = p1 - Ft;

19 PTCS/PN/ B Matlab Function cal function F = cal(filein, x0, xl, xu, options, elmin, elmax) CAL performs GBT calibration filein - input filename csv file: Elevation, Ta/Tcal, 1/Sin(Elevation) x0 - parameters initial value LCP noise diode value (K) RCP noise diode value (K) LCP receiver temperature (K) RCP receiver temperature (K) Zenith opacity x0 = [3, 3, 20.0, 20.0, 0.01]; xl - lower bounds on x ( [] for default ) xu - upper bounds on x ( [] for default ) options - options for lsqnonlin options = optimset( lsqnonlin ); optnew = optimset(options, MaxFunEvals, ); optnew = optimset(optnew, MaxIter, ); optnew = optimset(optnew, TolFun, 1.0e-6); optnew = optimset(optnew, TolX, 1.0e-4); optnew = optimset(optnew, Diagnostics, on ); optnew = optimset(optnew, Display, iter ); elmin - minimum elevation to use elmax - maximum elevation to use Usage: f = cal( data/tptcsrmp031120_xband_tsys.dat, x0, xl, xu, optnew, 20, 65); read the data from the input file d = dlmread(filein,, ); el = d(:,1); elevation p1 = d(:,3); Ta/Tcal p2 = d(:,2); 1/Sin(E) return input F.filein = filein; F.x0 = x0; F.xl = xl; F.xu = xu; F.options = options; F.elmin=elmin; F.elmax=elmax;

20 PTCS/PN/ F.el = el; F.p1 = p1; F.p2 = p2; only process data within el range j = 0; for i = 1:length(el) if(el(i) > elmin && el(i) < elmax) j = j+1; lsq.el(j) = el(i); lsq.p1(j) = p1(i); lsq.p2(j) = p2(i); end end run lsqnonlin [x, resnorm, residual, exitflag, output, lambda, jacobian] =... lsqnonlin(@ta, x0, xl, xu, options, lsq.p1, lsq.p2); return results F.x = x; F.resnorm = resnorm; F.residual = residual; F.exitflag = exitflag; F.output = output; F.lambda = lambda; F.jacobian = jacobian;

21 PTCS/PN/ C Matlab Function plotcal function F = plotcal(el, ta, fit, com1) plotcal Plot results of lsqnonlin el - elevation ta - antenna temperature in units of the CAL fit - results of the lsqnonlin fit com1 - Comment for plots constants eta - rear spillover efficiency (0.99) tatm - temperature in atmosphere (260.0 K) tspill - rear spillover temperature (300.0 K) tfss - forward spillover and scattering temperature (1 K) tcmb - cosmic microwave background temperature (2.7 K) n - number of LCP measurements m - number of RCP measurements Usage: plotcal(f.el, f.p1, f.x, TPTCSDSB (X-band) 0-90 ) eta = 0.99; tatm = 260.0; tspill = 300.0; tfss = 1.0; tcmb = 2.7; n = length(el)/2; m = n; sort by elevation elf_lcp = sort(el(1:n)); elf_rcp = sort(el(n+1:n+m)); af_lcp = 1.0./sin(elf_lcp*pi/180.0); af_rcp = 1.0./sin(elf_rcp*pi/180.0); determine the model curve taf_lcp = ( fit(3) + eta*tatm*(1.0 - exp(-fit(5)*af_lcp)) + (1.0 - eta)*tspill... + tfss + eta*tcmb*exp(-fit(5)*af_lcp) ) / fit(1); taf_rcp = ( fit(4) + eta*tatm*(1.0 - exp(-fit(5)*af_rcp)) + (1.0 - eta)*tspill... + tfss + eta*tcmb*exp(-fit(5)*af_rcp) ) / fit(2); set the data points el_lcp = el(1:n); el_rcp = el(n+1:n+m); ta_lcp = ta(1:n); ta_rcp = ta(n+1:n+m); plot the data and model figure; plot(el_lcp, ta_lcp, xr ); xlim([5 90]); xlabel( Elevation [deg] ); ylabel( T_{a}/T_{cal} ) title(com1); hold on;

22 PTCS/PN/ plot(elf_lcp, taf_lcp, g ); plot(el_rcp, ta_rcp, *b ); plot(elf_rcp, taf_rcp, g ); hold off; ymax=max(max(ta_lcp), max(ta_rcp)); ymin=min(min(ta_lcp), min(ta_rcp)); delta=(ymax-ymin)/13.0; y1=ymax-delta; x1=55.0; dum = sprintf( T_{cal}(LCP) = 0.4g, fit(1)); text(x1, y1, dum); dum = sprintf( T_{cal}(RCP) = 0.4g, fit(2)); text(x1, y1-delta, dum); dum = sprintf( T_{rx}(LCP) = 0.4g, fit(3)); text(x1, y1-2.0*delta, dum); dum = sprintf( T_{rx}(RCP) = 0.4g, fit(4)); text(x1, y1-3.0*delta, dum); dum = sprintf( 0.5g, fit(5)); text(x1, y1-4.0*delta, [ \tau =, dum]);