alma cross correlation bandpass stability · 2015. 3. 16. · thedraft system veri cation report on...

ALMA Cross Correlation Bandpass Stability

ALMA Technical Note Number: 15

Status: DRAFT

Prepared by: Organization: Date:

Seiji Kameno (ALMA System Verification Scientist)

JAO/NAOJ 17 December 2014

ALMA Cross Correlation Bandpass Stability(Draft)

Seiji Kameno

December 17, 2014

This report addresses the cross-correlation bandpass stability of ALMA at Band3, 4, 6, 7, 8, 9, and 10.

Aims. Verify present performance of cross-correlation bandpass stability whichis critical for observations that require high spectral dynamic range. Then, identifyorigins of the bandpass instability and quest for solutions to mediate them.Methods. Bright continuum point-like sources were observed with 21–35 antennas

for longer than 1.5 hours. Antenna-based bandpass shape were determined in twomethods: real–imaginary and amplitude–phase solutions. The bandpass variationwas evaluated in terms of amplitude and phase by comparing the reference BPtable. Linear regression was applied to find the control parameters to explain thebandpass variation. The bandpass flatness was measured by taking the spectralAllan variance.Results. The real–imaginary solution showed better accuracy than the amplitude–

phase solution. The bandpass stability was partially compliant at Band 3, 4, and6, while the signal-to-noise ratio at Band 7 or higher was not sufficient to achievethe accuracy of 0.1% that is necessary to verify the requirement. We found thatthe bandpass variation was dominated by the random noise component and con-stant intercept, with weak dependence of atmospheric component. Time stabilitywas sufficient for the timescale of ∼ 1 day. The bandpass shape after bandpasscalibration in TDM was flat enough in the frequency scale of ∆ν < 125 MHz. Thebandpass shapes of TDM and FDM were different by ∼ 2 %.

Conclusions. We got practical performance of cross-correlation bandpass stabilityat all working frequency bands. It is possible to predict the standard deviation ofbandpass amplitude variation, ∆|B| as ∆|B| ∼ a0 + a1S, where a0 is the interceptdepending on frequency bands and a1 is the proportional coefficient on the random

noise, S, given as S2 =1

(Nant,BP − 2)|VBP|2∆νTBP+

1

(Nant,tg − 2)|Vtg|2∆νTtg. See

subsection 5.1 for detail values of the parameters.

1 Introduction

The draft System Verification report on the SYS273.2 cross correlation bandpass stability wasissued covering frequency bands of Band 3, 6, 7, and 9 with 15 – 21 antennas. The results didnot verify the required accuracy of −30 dB because of insufficient signal-to-noise ratio.

1

https://wikis.alma.cl/bin/view/SystemEngineering/SYS2732CrossCorrBandpassStabilityDraft

The bandpass stability campaign in 2013 Oct – Nov and the high frequency campaign in 2014May – Sep covered frequency bands of Band 3, 4, 6, 7, 8, 9, and 10 with 21–35 antennas. Weattempted to retake the verification of cross-correlation bandpass stability using the datasetstaken in the campaigns.

1.1 Requirements

The requirement for the bandpass stability is defined as Req#273.2 in the ALMA SystemTechnical Requirements (ALMA-80.04.00.00-005-C-SPE):

Bandpass Stability:spectral gain vrs time(T)

The 1 hr: Temporal change in bandpass gain or shape ofcross correlation < −30 dB over 3600 seconds.

The requirement of 1000:1 is defined for looking for weak lines in presence of strong contin-uum emission. Further description on the requirement states:

Case (b) determines the bandpass stability for interferometric observations. Thestability of the continuum within the passband has to be adequate and over longtimescales. Assume that a bandpass calibration is done every 60 minutes and thatspectra are sufficiently over-sampled that ringing is not a problem and that ther-mal noise is not significant. Then thermal changes which alter reflections and/orbandpass ripple in the analogue system may limit the spectral dynamic range.Therefore, the stability of the passband profile, with respect to the mean signalstrength within the total IF, has to be better than −30 dB for a time of ∼ 3600seconds at a resolution bandwidth of 2 MHz.

1.2 Conditions for the Requirements

1. A strong point-like unpolarized continuum source that has a flat spectrum should beobserved for the verification of bandpass characteristics. An ideal source yields constantvisibilities so that the observed visibilities response only the complex gain of the observingsystem.

2. To evaluate the bandpass gain or shape, random (thermal) noise should be reduced asfar as possible.

3. Complex (amplitude and phase) gain calibration should be employed before time inte-gration of visibilities to avoid coherence loss. This treatment doesn’t affect the bandpassshape because the applied complex gain is uniform across the band. The calibrationincludes residual-delay compensation that yields linear slope of phase across the band.

4. Band edges are excluded from the evaluation. In this report we trimmed 5% bandwidthoff at band edges.

5. To evaluate antenna-based gain stability, solution for them is applied using the baseline-based visibilities. This improves the accuracy in determination of bandpass shape.

6. To evaluate the stability of the passband profile with respect to the mean signal strength,

we employed the standard deviation (SD) defined as√⟨

(Bk − B)2⟩

and the peak excess

2

http://edm.alma.cl/forums/alma/dispatch.cgi/documents/docProfile/100722/

(PE) defined as max(|Bk− B|) as the index of bandpass variation. Here, Bk is the band-pass amplitude at the k−th spectral channel and B is the mean across the bandwidth.PE is used for judgement of compliant/non-compliant referring the system technical re-quirements, while SD is used for further statistical analysis such as p

2 Methods of Measurements

The model to describe complex visibility, V m,n(t, ν), of the baseline consisting of antenna mand n at the frequency, ν, and time, t, is:

V ab(t, ν) = Ga(t)G∗b(t)Ba(ν)B∗

b(ν)S exp(i2πν(τa − τb)). (1)

Here, G(t) is the antenna-based complex gain, B(ν) is the antenna-based bandpass, S is theflux density of a flat-spectrum point source, and τ is the delay.

Since G and B are coupled, we can set B = 1 and ascribe time variation of channel-averagedamplitude and phase to G. Time variation of B indicates the temporal change in bandpassshape that is described in the System Technical Requirements.

2.1 Antenna-Based Solutions

To determine the antenna-based bandpass using the baseline-based visibilities, we tried twosolutions and compared their performances. The following subsections of 2.1.1 and 2.1.2 statesolutions for G and the same arguments are applicable to determine B.

2.1.1 Amplitude-Phase Solutions

This method solved for amplitudes and phases independently. The visibility amplitude isproportional to the multiplication of antenna-based gain amplitudes. Taking logarithm of them,we have a linear equation between them. The visibility phase is the difference of antenna-basedgain phase. Thus, we can employ a linear equation to solve them, too.

Let us formulate the equations with the visibilities V and the antenna-based gain G

log |V | =

log |V 1,0|log |V 2,0|log |V 2,1|log |V 3,0|log |V 3,1|log |V 3,2|

...

, argV =

arg |V 1,0|arg |V 2,0|arg |V 2,1|arg |V 3,0|arg |V 3,1|arg |V 3,2|

...

(2)

log |G| =

log |G0|log |G1|log |G2|log |G3|

...

argG =

argG1

argG2

argG3...

(3)

3

and the matrices

Pamp =

1 1 0 0 0 . . .1 0 1 0 0 . . .0 1 1 0 0 . . .1 0 0 1 0 . . .0 1 0 1 0 . . .0 0 1 1 0 . . ....

......

......

. . .

Pphs =

1 0 0 0 . . .0 1 0 0 . . .−1 1 0 0 . . .0 0 1 0 . . .−1 0 1 0 . . .0 −1 1 0 . . ....

......

.... . .

(4)

as

W log |V | = WPamp log |G|W argV = WPphs argG, (5)

where W = diag(1/σ2vis) is the diagonal weighting matrix. The least-squares solution forequation 5 is given as

log |G| =(P TampWPamp

)−1W log |V |

argG =(P TphsWPphs

)−1W argV . (6)

2.1.2 Real-Imaginary Solutions

< and = stand for real and imaginary parts of a complex variable, respectively. Since equation1 involves multiplication of antenna-based complex gains, it is a non-linear equation. To solveit in real-imaginary form, we employ the iterative approach. Let the initial estimate of gain asG, the difference of the true gain, ∆G, and the visibility residual, ∆V are defined as:

∆V =

<[V 1,0 − G1G∗0]

<[V 2,0 − G2G∗0]

<[V 2,1 − G2G∗1]

<[V 3,0 − G3G∗0]

<[V 3,1 − G3G∗1]

<[V 3,2 − G3G∗2]

...

=[V 1,0 − G1G∗0]

=[V 2,0 − G2G∗0]

=[V 2,1 − G2G∗1]

=[V 3,0 − G3G∗0]

=[V 3,1 − G3G∗1]

=[V 3,2 − G3G∗2]

, ∆G =

<[G0 − G0]

<[G1 − G1]

<[G2 − G2]

<]G3 − G3]...

=[G1 − G1]

=[G2 − G2]

=]G3 − G3]

. (7)

4

Then, set the derivative matrix as:

P =

<G1 <G0 0 0 0 . . .

<G2 0 <G0 0 0 . . .

0 <G2 <G1 0 0 . . . =G2 =G1 0 . . .

<G3 0 0 <G0 0 . . .

0 <G3 0 <G1 0 . . . =G3 0 =G1 0 . . .

0 0 <G3 <G2 0 . . . 0 =G3 =G2 0 . . ....

=G1 0 0 0 0 . . . <G0 0 . . .

=G2 0 0 0 0 . . . 0 <G0 0 . . .

0 =G2 −=G1 0 0 . . . −<G2 <G1 0 . . .

=G3 0 0 0 0 . . . 0 0 <G0 0 . . .

0 =G3 0 −=G1 0 . . . −<G3 0 <G1 0 . . .

0 0 =G3 −=G2 0 . . . 0 −<G3 <G2 0 . . .

. (8)

The equation of difference values will be

W∆V = WP∆G. (9)

The solution of equation 9 is given as

∆G =(P TWP

)−1W∆V (10)

Then, the initial estimate of the gain will be modified as

G ← G + ∆G. (11)

By applying the iteration for a couple of cycles, the solution will converge to the neighborhoodof the true value.

2.2 Evaluation of Time Stability

The time stability of bandpass shape is evaluated by comparing the antenna-based bandpassshapes between those at the reference epoch, B0(ν) and those at different epoch, B(t, ν), interms of amplitude and phase. Figure 1 shows the example. Upper and lower panels showbandpass amplitudes and phases, respectively. Blue lines and dots indicate the instantaneousantenna-based bandpass shape, while green lines and dots indicate the difference from the

reference BP table. The differences of amplitude and phase are calculated as

∣∣∣∣B(t, ν)

B0(ν)

∣∣∣∣−1 and

argB(t, ν)

B0(ν), respectively.

2.3 Spectral Allan Variance—Evaluation of Bandpass Flatness

The flatness of the bandpass shape was evaluated by the spectral Allan variance (SAV), σ2y(∆ν),defined as

σ2y(∆ν) =

⟨[B(ν + ∆ν)− 2B(ν) +B(ν −∆ν)]2

⟩2∆ν2

. (12)

5

97.0 97.5 98.0

0.0

0.2

0.4

0.6

0.8

1.0

Am

plit

ude

DV04 Pol-X Amplitude

RawBP calibed

97.0 97.5 98.00.20

0.15

0.10

0.05

0.00

0.05

0.10

0.15

0.20

Phase

[ra

d]

DV04 Pol-X Phase

RawBP calibed

97.0 97.5 98.0

0.0

0.2

0.4

0.6

0.8

1.0

Am

plit

ude

DV04 Pol-Y Amplitude

RawBP calibed

97.0 97.5 98.00.20

0.15

0.10

0.05

0.00

0.05

0.10

0.15

0.20

Phase

[ra

d]

DV04 Pol-Y Phase

RawBP calibed

uid___A002_X70c186_X4d3 Cross Power Spectrum

Frequency [GHz]

Figure 1: Cross power spectra of DV04 at Band-3. Top and bottom panels show the ampli-tudes and phases. Left and right panels stand for X and Y polarizations. Blue andgreen colors indicate the cross power spectra before and after bandpass calibration,respectively. The mean values of amplitude and phase are scaled to be unity andzero, respectively.

When bandpass variation is dominated by random thermal noise, we expect σ2y ∝ ∆ν−2 becausethe numerator of equation 12 is independent on ∆ν. If the bandpass distortion is greater thanthe random thermal noise, we will have significant excess of SAV from the power law with thepower index of −2.

Figure 2 shows an example. After bandpass calibration was applied, the BP residual becameflat and thus the SAV follows the power law with the power index of −2. Note that the SAVat δν = 1 ch was underestimated because of Hanning-window apodization in the correlator.

3 Observations

Table 1 lists the observation log used in this report. Observations consisted of 1.5-hour SBEXs.Each SBEX consisted of 60 – 480 sec tracking scans toward a bright continuum source. Ob-servations at Band 3, 6, 7, and 9 spread longer than one day inserted changes of frequencysetups, while Band 4, 8, and 10 covers time range shorter than one day. Number of antennasare listed in the fourth column with the number of antennas unflagged in the parentheses. Theasterisks marked with the source name indicates the bandpass calibration scan.

6

Table 1: List of observations

Band uid://A002/ Date and Time (UT) # Ants Sources

3 X70c186/X168 2013/10/29 02:33 - 04:03 29 (27) J042315-012034, J0522-364∗, J0538-4403 X70c186/X4d3 2013/10/29 04:10 - 05:40 29 (28) J042315-012034, J0522-364, J0538-4403 X70c186/X8b8 2013/10/29 05:45 - 07:16 29 (28) J042315-012034, J0522-364, J0538-4403 X70c186/X1091 2013/10/29 09:00 - 10:31 29 (28) J042315-012034, J0522-364, J0538-4403 X70f4b6/X39 2013/10/30 04:22 - 05:52 21 (20) J042315-012034, J0522-364, J0538-4403 X70f4b6/X4e4 2013/10/30 05:56 - 07:27 21 (20) J042315-012034, J0522-364, J0538-4403 X70f4b6/X9da 2013/10/30 07:31 - 08:34 21 (20) J042315-012034, J0522-364, J0538-4403 X73d9ce/X1ba 2013/11/24 04:44 - 06:15 32 (32) J042315-012034, J0522-364, J0538-440

4 X89d540/X2ee 2014/08/22 05:13 - 06:48 30 (29) 3C 454.3∗, J2232+117, J1924-292

6 X73a7bb/X3b9 2013/11/23 01:23 - 02:29 33 (33) J042315-012034, J0522-364, J0538-4406 X740d04/X4e5 2013/11/26 00:14 - 01:42 30 (29) J042315-012034, J0522-364, J0538-4406 X740d04/X7f7 2013/11/26 01:48 - 03:16 30 (30) J042315-012034, J0522-364, J0538-4406 X740d04/Xa57 2013/11/26 03:19 - 04:47 30 (30) J042315-012034, J0522-364∗, J0538-440

7 X812f8d/X1057 2014/05/09 10:51 - 11:29 21 (21) J1924-292, 3C454.3∗

7 X812f8d/X11bb 2014/05/09 11:37 - 12:15 21 (21) J1924-292, 3C454.37 X8161ed/X1936 2014/05/11 10:26 - 11:58 21 (21) 3C 454.3, J2232+117, J1924-292

8 X8867ff/X1fbf 2014/08/02 12:23 - 13:55 26 (24) J0522-364∗, J0538-4405, J0510+1800

9 X8666c7/X1fa 2014/07/10 07:45 - 09:15 32 (32) 3C 454.4∗

9 X8a2c58/Xb0 2014/08/26 06:42 - 08:16 35 (34) 3C 454.3

10 X8b8415/X5d7 2014/09/11 03:33 - 05:06 21 (21) 3C454.3∗, J1924-292

7

10210-10

10-9

10-8

10-7

10-6

Pol-X Amplitude

RawBP calibed

10210-10

10-9

10-8

10-7

10-6

Pol-Y Amplitude

RawBP calibed

10210-9

10-8

10-7

10-6

10-5

10-4

Pol-X Phase

RawBP calibed

10210-9

10-8

10-7

10-6

10-5

10-4

Pol-Y Phase

RawBP calibed

uid___A002_X70c186_X4d3 Spectral Allan Variance

Spect

ral A

llan V

ari

ance

Frequency Separation [MHz]

Figure 2: The spectral Allan variances (SAVs) of amplitude (top panel) and phase (bottompanel) as a function of frequency separation. Median values of the SAVs among 27antennas are plotted.

4 Results

The performances of amplitude-phase and real-imaginary solutions are compared in subsection4.1. While time stability was evaluated in TDM for all of frequency bands in subsection 4.2.The SAV was obtained to check spectral flatness of bandpass shape in subsection 4.3. FDMtime stability was investigated in only at Band 4 in subsection 4.4 to test the performance ofthe TDM-to-FDM bandpass transfer method.

4.1 Amplitude-Phase versus Real-Imaginary Solutions

The comparison test of two solution methods, amplitude-phase and real-imaginary, was takenusing the Band-7 observation toward J2232+117 using the narrow-band SPW in the FDMmode (234.375 MHz / 3840 ch, 61-kHz channel spacing). Figure 3 compares the solutionsof the two methods, indicating better accuracy by the real-imaginary method than by theamplitude-phase. The standard deviation of amplitude and phase were 15.03% and 0.1820 radby real-imaginary solution, while amplitude-phase method resulted in 15.08% and 0.1822 rad.We employed the real-imaginary method after this.

8

+3.433e22001000100200 DA41 PE=1.731e-01 SD=6.767e-02

+3.433e22001000100200 DA41 PE=1.986e-01 SD=5.956e-02

+3.433e22001000100200 DA42 PE=4.352e-01 SD=1.499e-01

+3.433e22001000100200 DA42 PE=3.297e-01 SD=1.185e-01

+3.433e22001000100200 DA44 PE=5.433e-01 SD=1.829e-01

+3.433e22001000100200 DA44 PE=3.936e-01 SD=1.307e-01

+3.433e22001000100200 DA46 PE=4.285e-01 SD=1.577e-01

+3.433e22001000100200 DA46 PE=5.068e-01 SD=1.377e-01

+3.433e22001000100200 DA47 PE=4.132e-01 SD=1.354e-01

+3.433e22001000100200 DA47 PE=5.213e-01 SD=1.544e-01

+3.433e22001000100200 DA48 PE=5.870e-01 SD=1.747e-01

+3.433e22001000100200 DA48 PE=4.181e-01 SD=1.355e-01

+3.433e22001000100200 DA50 PE=5.399e-01 SD=1.661e-01

+3.433e22001000100200 DA50 PE=5.135e-01 SD=1.619e-01

+3.433e22001000100200 DA51 PE=3.937e-01 SD=1.362e-01

+3.433e22001000100200 DA51 PE=4.131e-01 SD=1.575e-01

+3.433e22001000100200 DA54 PE=4.638e-01 SD=1.113e-01

+3.433e22001000100200 DA54 PE=3.640e-01 SD=1.268e-01

+3.433e22001000100200 DA55 PE=4.075e-01 SD=1.593e-01

+3.433e22001000100200 DA55 PE=4.171e-01 SD=1.399e-01

+3.433e22001000100200 DA57 PE=4.480e-01 SD=1.634e-01

+3.433e22001000100200 DA57 PE=5.028e-01 SD=1.456e-01

+3.433e22001000100200 DA60 PE=5.034e-01 SD=1.836e-01

+3.433e22001000100200 DA60 PE=4.384e-01 SD=1.517e-01

+3.433e22001000100200 DA61 PE=3.663e-01 SD=1.408e-01

+3.433e22001000100200 DA61 PE=3.555e-01 SD=1.361e-01

+3.433e22001000100200 DA63 PE=6.334e-01 SD=1.855e-01

+3.433e22001000100200 DA63 PE=4.011e-01 SD=1.175e-01

+3.433e22001000100200 DV01 PE=4.470e-01 SD=1.193e-01

+3.433e22001000100200 DV01 PE=3.796e-01 SD=1.473e-01

+3.433e22001000100200 DV02 PE=5.487e-01 SD=1.624e-01

+3.433e22001000100200 DV02 PE=3.484e-01 SD=1.290e-01

+3.433e22001000100200 DV10 PE=3.276e-01 SD=1.240e-01

+3.433e22001000100200 DV10 PE=5.448e-01 SD=1.534e-01

+3.433e22001000100200 DV11 PE=4.602e-01 SD=1.532e-01

+3.433e22001000100200 DV11 PE=6.090e-01 SD=1.442e-01

+3.433e22001000100200 DV13 PE=4.337e-01 SD=1.375e-01

+3.433e22001000100200 DV13 PE=4.693e-01 SD=1.470e-01

+3.433e22001000100200 DV14 PE=5.992e-01 SD=1.680e-01

+3.433e22001000100200 DV14 PE=4.572e-01 SD=1.623e-01

+3.433e22001000100200 DV17 PE=5.376e-01 SD=1.578e-01

+3.433e22001000100200 DV17 PE=3.930e-01 SD=1.612e-01

+3.433e22001000100200 DV18 PE=6.972e-01 SD=1.578e-01

+3.433e22001000100200 DV18 PE=4.520e-01 SD=1.761e-01

+3.433e22001000100200 DV20 PE=3.913e-01 SD=1.368e-01

+3.433e22001000100200 DV20 PE=5.847e-01 SD=1.723e-01

+3.433e22001000100200 DV21 PE=5.023e-01 SD=1.645e-01

+3.433e22001000100200 DV21 PE=3.786e-01 SD=1.536e-01

+3.433e22001000100200 DV22 PE=4.817e-01 SD=1.442e-01

+3.433e22001000100200 DV22 PE=4.662e-01 SD=1.382e-01

+3.433e22001000100200 DV23 PE=4.818e-01 SD=1.811e-01

+3.433e22001000100200 DV23 PE=3.287e-01 SD=1.089e-01

+3.433e22001000100200 DV24 PE=7.038e-01 SD=1.588e-01

+3.433e22001000100200 DV24 PE=8.055e-01 SD=1.814e-01

0.10 0.15 0.20 0.25 0.30+3.433e2

2001000100200 DV25 PE=4.554e-01 SD=1.682e-01

0.10 0.15 0.20 0.25 0.30+3.433e2

2001000100200 DV25 PE=4.579e-01 SD=1.446e-01

uid___A002_X8161ed_X1936 Bandpass Variation

Am

plit

ude V

ari

ati

on [

%]

Frequency [GHz]

Amp PE=4.537e-01(median) SD=1.508e-01 (median)

+3.433e22001000100200 DA41 PE=1.731e-01 SD=6.747e-02

+3.433e22001000100200 DA41 PE=1.978e-01 SD=5.931e-02

+3.433e22001000100200 DA42 PE=4.346e-01 SD=1.494e-01

+3.433e22001000100200 DA42 PE=3.254e-01 SD=1.184e-01

+3.433e22001000100200 DA44 PE=5.413e-01 SD=1.817e-01

+3.433e22001000100200 DA44 PE=3.913e-01 SD=1.305e-01

+3.433e22001000100200 DA46 PE=4.279e-01 SD=1.575e-01

+3.433e22001000100200 DA46 PE=5.045e-01 SD=1.371e-01

+3.433e22001000100200 DA47 PE=4.117e-01 SD=1.347e-01

+3.433e22001000100200 DA47 PE=5.218e-01 SD=1.546e-01

+3.433e22001000100200 DA48 PE=5.887e-01 SD=1.745e-01

+3.433e22001000100200 DA48 PE=4.187e-01 SD=1.354e-01

+3.433e22001000100200 DA50 PE=5.441e-01 SD=1.660e-01

+3.433e22001000100200 DA50 PE=5.145e-01 SD=1.615e-01

+3.433e22001000100200 DA51 PE=3.887e-01 SD=1.365e-01

+3.433e22001000100200 DA51 PE=4.142e-01 SD=1.580e-01

+3.433e22001000100200 DA54 PE=4.613e-01 SD=1.108e-01

+3.433e22001000100200 DA54 PE=3.626e-01 SD=1.271e-01

+3.433e22001000100200 DA55 PE=4.052e-01 SD=1.588e-01

+3.433e22001000100200 DA55 PE=4.169e-01 SD=1.401e-01

+3.433e22001000100200 DA57 PE=4.464e-01 SD=1.636e-01

+3.433e22001000100200 DA57 PE=4.970e-01 SD=1.453e-01

+3.433e22001000100200 DA60 PE=5.083e-01 SD=1.829e-01

+3.433e22001000100200 DA60 PE=4.399e-01 SD=1.512e-01

+3.433e22001000100200 DA61 PE=3.719e-01 SD=1.410e-01

+3.433e22001000100200 DA61 PE=3.530e-01 SD=1.362e-01

+3.433e22001000100200 DA63 PE=6.252e-01 SD=1.843e-01

+3.433e22001000100200 DA63 PE=3.986e-01 SD=1.170e-01

+3.433e22001000100200 DV01 PE=4.502e-01 SD=1.198e-01

+3.433e22001000100200 DV01 PE=3.842e-01 SD=1.472e-01

+3.433e22001000100200 DV02 PE=5.600e-01 SD=1.635e-01

+3.433e22001000100200 DV02 PE=3.461e-01 SD=1.293e-01

+3.433e22001000100200 DV10 PE=3.286e-01 SD=1.234e-01

+3.433e22001000100200 DV10 PE=5.391e-01 SD=1.525e-01

+3.433e22001000100200 DV11 PE=4.592e-01 SD=1.528e-01

+3.433e22001000100200 DV11 PE=6.106e-01 SD=1.447e-01

+3.433e22001000100200 DV13 PE=4.354e-01 SD=1.370e-01

+3.433e22001000100200 DV13 PE=4.667e-01 SD=1.465e-01

+3.433e22001000100200 DV14 PE=6.006e-01 SD=1.676e-01

+3.433e22001000100200 DV14 PE=4.619e-01 SD=1.631e-01

+3.433e22001000100200 DV17 PE=5.345e-01 SD=1.572e-01

+3.433e22001000100200 DV17 PE=3.833e-01 SD=1.608e-01

+3.433e22001000100200 DV18 PE=6.935e-01 SD=1.578e-01

+3.433e22001000100200 DV18 PE=4.440e-01 SD=1.761e-01

+3.433e22001000100200 DV20 PE=3.941e-01 SD=1.369e-01

+3.433e22001000100200 DV20 PE=5.837e-01 SD=1.722e-01

+3.433e22001000100200 DV21 PE=5.056e-01 SD=1.646e-01

+3.433e22001000100200 DV21 PE=3.761e-01 SD=1.533e-01

+3.433e22001000100200 DV22 PE=4.824e-01 SD=1.438e-01

+3.433e22001000100200 DV22 PE=4.641e-01 SD=1.375e-01

+3.433e22001000100200 DV23 PE=4.836e-01 SD=1.803e-01

+3.433e22001000100200 DV23 PE=3.282e-01 SD=1.098e-01

+3.433e22001000100200 DV24 PE=6.962e-01 SD=1.579e-01

+3.433e22001000100200 DV24 PE=8.139e-01 SD=1.819e-01

0.10 0.15 0.20 0.25 0.30+3.433e2

2001000100200 DV25 PE=4.483e-01 SD=1.675e-01

0.10 0.15 0.20 0.25 0.30+3.433e2

2001000100200 DV25 PE=4.561e-01 SD=1.440e-01

uid___A002_X8161ed_X1936 Bandpass Variation

Am

plit

ude V

ari

ati

on [

%]

Frequency [GHz]

Amp PE=4.493e-01(median) SD=1.503e-01 (median)

Figure 3: Comparison of amplitude-phase (left) and real-imaginary (right) solutions forantenna-based bandpass. The test was carried out using the Band-7 observationstoward J2232 + 117 with 28 12-m antennas. The cross power spectra over 234.375MHz / 3840 ch (61-kHz channel spacing) was smoothed with Rbf(smooth=0.1) forthe reference bandpass and with Rbf(smooth=0.05) for the target spectrum. Therelative amplitude variations of the cross power spectrum with respect to the refer-ence bandpass are shown in each antenna and polarization panel.

9

4.2 TDM Bandpass Variations

The time serias plots of antenna-based bandpass variations are archived in JIRA CSV-2964.Figures 4 – 10 shows the bandpass variations as a function of elapsed time since BP calibrationscan. Four panels in each band stand for SD(amp), PE(amp), SD(phase), and PE(phase) asan indicator of bandpass variation as described in subsection 2.2.

Band 3

5e+02 5e+03 5e+04 5e+05

5e−0

41e−0

32e−0

35e−0

31e−0

22e−0

25e−0

2

Band3 Bandpass Stablitity

Time Difference from BP Cal. [sec]

SD o

f Am

plitu

de V

aria

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(a) SD (left) and PE (right) of bandpass amplitude variation at band 3 as a function of time differencebetween bandpass-calibration scans.

5e+02 5e+03 5e+04 5e+05

5e−0

41e−0

32e−0

35e−0

31e−0

22e−0

25e−0

2



SD o

f Pha

se V

aria

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(b) SD (left) and PE (right) of bandpass phase variation at band 3

Figure 4: Band-3 bandpass variation. Small colored dots stand for antennas and polarizations.The mint filled area indicates the range of dots and the blue circles with solid linesindicate the median values. Two gray vertical lines point the timescales of 1 hourand 1 day, respectively. The red horizontal line indicates the required stability of−30 dB.

10

http://jira.alma.cl/browse/CSV-2964?focusedCommentId=469539&page=com.atlassian.jira.plugin.system.issuetabpanels:comment-tabpanel#comment-469539

Band 4

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5e−0

41e−0

32e−0

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31e−0

2



SD o

f Am

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200 500 1000 2000 5000

5e−0

41e−0

32e−0

35e−0

31e−0

22e−0

25e−0

2



PE o

f Am

plitu

de V

aria

tion

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(a) SD (left) and PE (right) of bandpass amplitude variation at band 4.

200 500 1000 2000 5000

5e−0

41e−0

32e−0

35e−0

31e−0

22e−0

2



SD o

f Pha

se V

aria

tion

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200 500 1000 2000 5000

5e−0

41e−0

32e−0

35e−0

31e−0

22e−0

25e−0

2Band4 Bandpass Stablitity


PE o

f Pha

se V

aria

tion

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(b) SD (left) and PE (right) of bandpass phase variation at band 4

Figure 5: Band-4 bandpass variation. See figure 4 caption for the legend.

11

Band 6

5e+02 2e+03 5e+03 2e+04 5e+04 2e+05

5e−0

41e−0

32e−0

35e−0

31e−0

22e−0

2



SD o

f Am

plitu

de V

aria

tion

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(b) SD (left) and PE (right) of bandpass phase variation at band 6.


12

Band 7

1e+02 5e+02 5e+03 5e+04

5e−0

41e−0

32e−0

35e−0

31e−0

22e−0

25e−0

2



SD o

f Am

plitu

de V

aria

tion

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2



SD o

f Pha

se V

aria

tion

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22e−0



PE o

f Pha

se V

aria

tion

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SD o

f Pha

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25e−0

15e

+00



PE o

f Pha

se V

aria

tion

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PE o

f Am

plitu

de V

aria

tion

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PE o

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tion

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15

Band 10

200 500 1000 2000 5000

1e−0

31e−0

21e−0

11e

+00

1e+0

1



SD o

f Am

plitu

de V

aria

tion

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1e−0

31e−0

11e

+01



PE o

f Am

plitu

de V

aria

tion

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5e−0

45e−0

35e−0

25e−0

1



SD o

f Pha

se V

aria

tion

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200 500 1000 2000 5000

5e−0

45e−0

35e−0

25e−0

15e

+00



PE o

f Pha

se V

aria

tion

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16

4.3 Spectral Allan Variance

SAVs are evaluated for the scans 1-hour later than the bandpass calibration scans towardthe same source with the bandpass calibrator. We calculated each SAV of the antenna-basedbandpass shape and took median values of SAVs among all antennas. The results are plotted infigures 11 – 17. The blue and green lines indicate SAVs before and after bandpass calibration.The red line guides the power-law with the power index of −2, which indicates the ideal casewhere the bandpass shape were flat and the variation were dominated by random noise. Notethat the SAV at ∆ν = 15.6 MHz (1 ch) is underestimated in every case because the spectralresolution is coarser than the channel separation due to the Hanning window in lag domain.

Band 3

10210-10

10-9

10-8

10-7

10-6

10-5

10-4 Pol-X Amplitude

RawBP calibed

10210-10

10-9

10-8

10-7

10-6

10-5

10-4 Pol-Y Amplitude

RawBP calibed

10210-10

10-9

10-8

10-7

10-6

10-5

10-4

Pol-X Phase

RawBP calibed

10210-10

10-9

10-8

10-7

10-6

10-5

10-4

Pol-Y Phase

RawBP calibed

uid___A002_X70c186_X168 Spectral Allan Variance

Spect

ral A

llan V

ari

ance


Figure 11: Band-3 spectral Allan variance

The non-flat spectral undulation significantly decreased through bandpass calibration. Bothamplitude and phase SAVs after bandpass calibration follows the power raw with the index of−2 in the frequency separation range of 32 MHz < ∆ν < 125 MHz. The excess of amplitudeSAV at ∆ν > 250 indicates bandpass shape variation at broad range of bandwidth.

17

Band 4

10210-10

10-9

10-8

10-7

10-6

10-5


RawBP calibed

10210-10

10-9

10-8

10-7

10-6

10-5


RawBP calibed

10210-10

10-9

10-8

10-7

10-6

10-5

10-4

Pol-X Phase

RawBP calibed

10210-10

10-9

10-8

10-7

10-6

10-5

10-4

Pol-Y Phase

RawBP calibed

uid___A002_X89d540_X2ee Spectral Allan Variance

Spect

ral A

llan V

ari

ance



Behavior of the SAVs is similar to those at Band 3. The SAVs in 32 MHz < ∆ν < 125 MHzfollowed the power raw with the index of −2, which indicated that the bandpass variation isdominated by random white noise. Systematic variation of bandpass shape was significant atbroad band range of ∆ν > 125 MHz.

Band 6

10210-10

10-9

10-8

10-7

10-6

10-5


RawBP calibed

10210-10

10-9

10-8

10-7

10-6

10-5


RawBP calibed

10210-10

10-9

10-8

10-7

10-6

10-5

10-4

Pol-X Phase

RawBP calibed

10210-10

10-9

10-8

10-7

10-6

10-5

10-4

Pol-Y Phase

RawBP calibed

uid___A002_X740d04_Xa57 Spectral Allan Variance

Spect

ral A

llan V

ari

ance



Bandpass calibration efficiently reduced the SAVs at all range of frequency separation. TheSAVs after bandpass calibration showed a slight excess at ∆ν > 125 MHz with respect to thewhite noise component.

18

Band 7

10210-10

10-9

10-8

10-7

10-6

10-5


RawBP calibed

10210-10

10-9

10-8

10-7

10-6

10-5


RawBP calibed

10210-10

10-9

10-8

10-7

10-6

10-5

10-4 Pol-X Phase

RawBP calibed

10210-10

10-9

10-8

10-7

10-6

10-5

10-4 Pol-Y Phase

RawBP calibed

uid___A002_X812f8d_X11bb Spectral Allan Variance

Spect

ral A

llan V

ari

ance



Bandpass calibration significantly reduced the SAV at all range of frequency separation.The SAVs after bandpass calibration followed the power-law with the power index of −2. Thisindicates that the systematic variation of bandpass shape is hidden behind the white noise.

Band 8

10210-10

10-9

10-8

10-7

10-6

10-5


RawBP calibed

10210-10

10-9

10-8

10-7

10-6

10-5


RawBP calibed

10210-10

10-9

10-8

10-7

10-6

10-5

10-4

Pol-X Phase

RawBP calibed

10210-10

10-9

10-8

10-7

10-6

10-5

10-4

Pol-Y Phase

RawBP calibed

uid___A002_X8867ff_X1fbf Spectral Allan Variance

Spect

ral A

llan V

ari

ance



The effect of bandpass calibration was significant in bandpass phase at whole frequencyseparation range, and in amplitude at ∆ν > 170 MHz. This indicates that the amplitudeundulation of the raw bandpass shape was comparable to the white noise.

19

Band 9

10210-10

10-9

10-8

10-7

10-6

10-5


RawBP calibed

10210-10

10-9

10-8

10-7

10-6

10-5


RawBP calibed

10210-10

10-9

10-8

10-7

10-6

10-5

10-4

Pol-X Phase

RawBP calibed

10210-10

10-9

10-8

10-7

10-6

10-5

10-4

Pol-Y Phase

RawBP calibed

uid___A002_X8867ff_X1fbf Spectral Allan Variance

Spect

ral A

llan V

ari

ance



Similar behavior to that at Band 8. The effect of bandpass calibration in amplitude waslimited at the frequency separation range of ∆ν > 170 MHz.

Band 10

10210-7

10-6

10-5

10-4

10-3

10-2


RawBP calibed

10210-7

10-6

10-5

10-4

10-3

10-2

10-1 Pol-X Phase

RawBP calibed

uid___A002_X8b8415_X5d7 Spectral Allan Variance

Spect

ral A

llan V

ari

ance



No significant effect through bandpass calibration. There was no significant undulation inamplitude of raw bandpass shape, compared with the random noise level. The SAVs afterbandpass calibration were greater than those of raw bandpass shape in the whole frequencyseparation range. This indicates that the bandpass calibration at Band 10 was harmful ratherthan efficient for accuracy in spectral measurements.

20

4.4 FDM Bandpass Stability

Figure 18: Comparison between FDM-FDM and TDM-FDM bandpass transfer. Bandpassvariations at band 4 are plotted. (Left): Bandpass table was determined using thesame FDM SPW. (Right): Bandpass table was produced in the TDM SPW andinterpolated for FDM spectral channels.

The bandpass stability in the FDM mode was verified at Band 4 where the best performanceof bandpass stability in the TDM mode was obtained. To evaluate the FDM bandpass stability,we tested two different BP tables generated from TDM and FDM SPWs. The TDM and FDMSPWs were tuned at the same center frequency, with the different bandwidths and the channelseparations of (2000 MHz bandwidth / 128 ch) in TDM and (1875 MHz / 3840 ch) in FDM,respectively.

To realize the idea that TDM-based BP table (2 GHz bandwidth / 128 ch) to apply FDMSPWs, the TDM BP table was interpolated into the spectral resolution of FDM. The TDMBP table was smoothed and interpolated to accommodate the FDM SPW.

Figure 18 compares variations of spectral shape through FDM-BP and TDM-BP calibrations.While FDM-BP calibration resulted in flat spectral shape, significant systematic undulationappeared in TDM-BP calibration.

Figure 19 plots the SD values of amplitude as a function of elapsed time since the scanfor the BP table. Although the residual in the FDM-to-FDM BP calibration was consistentwith the TDM-to-FDM results (see figure 5), that in the TDM-to-FDM BP calibration wasdominated by the systematic undulation and was less dependent on the signal-to-noise ratio.

Figure 20 show the SAVs of bandpass shape. The FDM-to-FDM bandpass calibration re-sulted in flat spectrum whose SAV is dominated by white noise. On the other hand, significantsystematic undulation remained in TDM-to-FDM bandpass calibration that appeared the localpeaks of SAV at ∆ν = 125 MHz and 7.8 MHz for amplitude and phase, respectively.

21

200 500 1000 2000 5000

5e−0

42e−0

35e−0

32e−0

25e−0



SD o

f Am

plitu

de V

aria

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Figure 19: Time series of bandpass variation at band 4. Left and right panels stand for band-pass tables determined using FDM and TDM SPWs, respectively.See figure 4 cap-tion for the legend.

100 101 10210-11

10-10

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10-6


RawBP calibed

100 101 10210-11

10-10

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10-8

10-7

10-6


RawBP calibed

100 101 102

10-10

10-9

10-8

10-7

10-6

10-5

Pol-X Phase

RawBP calibed

100 101 102

10-10

10-9

10-8

10-7

10-6

10-5

Pol-Y Phase

RawBP calibed


Spect

ral A

llan V

ari

ance


100 101 10210-11

10-10

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10-8

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10-6


RawBP calibed

100 101 10210-11

10-10

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RawBP calibed

100 101 102

10-10

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10-5

Pol-X Phase

RawBP calibed

100 101 102

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10-5

Pol-Y Phase

RawBP calibed


Spect

ral A

llan V

ari

ance


Figure 20: Spectral Allan variances of bandpass shape. Green and blue lines stand for the SAVbefore and after bandpass calibration, respectively. (Left): BP table was generatedusing the same FDM SPW. (Right): BP table was generated using the TDM SPWthat covers the bandwidth of the target FDM SPW.

22

5 Discussion

The results in section 4 showed that the antenna-based bandpass stability performance waspartially compliant at Band 3, 4, and 6 for limited cases and non-compliant for others. Wequest for the origins of bandpass instability in this section and attempt to find the remedy forthem.

5.1 Origins of bandpass variation

Finding origins of bandpass variation is crucial to control the bandpass stability. We givefour candidates of control variables, random noise, elapsed time, secZ difference, and azimuthdifference to explain the bandpass variation as the dependent variable.

The random noise, S, is inversely proportional to the signal-to-noise ratio of visibilities.Since we apply bandpass calibration to the cross power spectra toward target sources, S isgiven by the RSS (root sum square) of random noise in the bandpass table and that in thetarget spectrum as

S2 =1

(Nant,BP − 2)|VBP|2∆νTBP+

1

(Nant,tg − 2)|Vtg|2∆νTtg. (13)

Here, Nant is the number of antennas, VBP and Vtg are visibilities of BP calibrator and targetsources, and TBP and Ttg are integration periods, respectively. The elapsed time, T , is thetime difference between BP calibration and target scans. It relates to time variability of thebandpass shape. ∆ secZ and ∆ AZ are the differences of airmass and azimuth angle betweenBP calibration and target scan, respectively.

We applied linear regression of the bandpass variation, ∆B, using multiple control variablesof S, T , ∆ secZ, and ∆ AZ, as

∆B ∼ a0 + a1S + bT + c∆ secZ + d∆AZ, (14)

where a0, a1, b, c, and d are coefficients for each control variable. We tried four indicators ofthe bandpass variation; SD and PE of amplitudes and phases.

Tables 2, 3, 4, and 5 lists the coefficients determined by the liner regressions to the indicatorsof the bandpass variation, SD of amplitude, PE of amplitude, SD of phase, and PE of phase,respectively. The numbers colored black, green, and yellow stand for the statistical significancewith P ≤ 0.01 (significant), 0.01 < P ≤ 0.05 (possible), and P > 0.05 (non-significant),respectively.

The random noise, related with a0, is always the primary control parameter for amplitudeof bandpass variation, and also phase except Band 6. It is remarkable that the coefficienta0 in SD amplitude lies in a0 = 0.39 ± 0.06 for TDM. This suggests that the bandpass erroris predictable by giving the magnitude of random error, and allows us to design an optimalbandpass calibration procedure that fulfills desired bandpass accuracy. The coefficient forFDM is ×1.4 as great as that for TDM. This possibly relates to the re-quantization loss in theFDM [Comoretto (2008)].

The secondary major component to explain the bandpass variation is the intercept, a0, whichis independent of control parameters listed above. The magnitude of a0 becomes greater athigher frequency bands.

Elapsed time, T , significantly relates to amplitude variation at Band 6 and phase variationat Band 3, 6, and 7. Since we did not measure long-term (> 1 day) stability at Band 4, 8, and

23

Table 2: Linear Regression of SD amplitude

Band a0 (×10−3) a1 b (10−9 s−1) c (10−4) d (10−4) R2

3 0.47± 0.04 0.36± 0.01 −0.02± 0.01 8.04± 0.34 0.04± 0.01 0.924 0.79± 0.38 0.48± 0.04 −104± 115 23.1± 15.5 −7.02± 6.19 0.824 (FDM) 0.83± 0.18 0.56± 0.01 −61.8± 53.5 4.65± 7.19 −0.64± 2.87 1.006 0.96± 0.13 0.30± 0.05 3.07± 0.17 14.5± 0.58 0.80± 0.28 0.997 2.20± 0.35 0.38± 0.07 −2.38± 1.99 87.1± 38.2 −5.04± 2.75 0.998 3.74± 0.36 0.38± 0.02 377± 184 164± 127 −14.8± 16.2 1.009 5.81± 0.85 0.45± 0.03 −0.01± 0.24 −3.99± 20.7 −24.5± 39.9 0.9910 6.44± 9.87 0.41± 0.02 11800± 31200 −690± 5320 −329± 1260 0.95

Table 3: Linear Regression of PE amplitude

Band a0 (×10−3) a1 b (10−9 s−1) c (10−4) d (10−4) R2

3 1.19± 0.12 1.03± 0.03 −0.10± 0.03 19.52± 0.10 1.06± 0.35 0.914 2.08± 1.10 1.34± 0.13 −252± 334 49.6± 45.0 −16.4± 18.0 0.814 (FDM) 3.44± 0.81 2.02± 0.02 −247± 246 29.2± 33.1 −7.42± 13.2 1.006 1.78± 0.45 1.11± 0.17 7.60± 0.60 35.0± 0.20 2.17± 0.96 0.987 5.90± 0.96 1.04± 0.02 −3.39± 5.40 197± 104 −13.9± 7.45 0.998 9.21± 0.86 1.07± 0.04 1270± 433 500± 300 −54.9± 38.2 1.009 19.9± 3.51 1.06± 0.13 −0.18± 1.00 62.1± 85.3 −45.9± 164 0.9810 39.7± 31.0 1.12± 0.06 −32500± 98000 9600± 16700 −1530± 3960 0.95

Table 4: Linear Regression of SD Phase

Band a0 (×10−3) a1 b (10−9 s−1) c (10−4) d (10−4) R2

3 1.31± 0.12 0.32± 0.03 0.70± 0.03 4.46± 1.01 4.37± 0.35 0.874 1.09± 0.51 0.62± 0.06 −87.2± 157 16.0± 21.1 −7.24± 8.34 0.804 (FDM) 12.7± 0.27 0.54± 0.01 156± 83 −9.52± 11.1 −8.39± 4.44 1.006 2.35± 0.35 0.11± 0.13 2.96± 0.47 15.4± 1.57 −0.00± 0.07 0.907 2.94± 0.44 0.51± 0.01 −7.77± 2.51 78.9± 48.0 −4.09± 3.46 0.998 5.28± 0.48 0.51± 0.02 187± 244 43.4± 16.9 10.5± 21.5 1.009 10.2± 2.02 0.55± 0.08 1.09± 0.58 43.3± 49.2 −181± 94.7 0.9710 371± 117 0.20± 0.02 −67700± 37000 9180± 6320 890± 1490 0.78

24

Table 5: Linear Regression of PE Phase

Band a0 (×10−3) a1 b (10−9 s−1) c (10−4) d (10−4) R2

3 3.10± 0.32 1.00± 0.07 1.17± 0.7 13.90± 0.27 10.0± 0.92 0.834 3.41± 1.68 1.69± 0.20 −365± 517 60.3± 68.9 −27.3± 27.5 0.734 (FDM) 46.6± 0.87 1.98± 0.02 282± 265 2.88± 35.7 −38.8± 14.3 0.736 5.53± 0.93 0.56± 0.34 −6.29± 0.24 37.4± 0.41 −0.02± 1.97 1.007 8.07± 1.27 1.37± 0.03 −22.5± 7.2 251± 138 −16.1± 9.93 0.998 15.0± 1.22 1.33± 0.05 1180± 613 814± 424 −86.1± 54.0 1.009 33.8± 4.67 1.22± 0.17 2.17± 1.33 156± 113 −277± 218 0.9810 1207± 344 0.43± 0.07 −248000± 10900 26200± 18600 5570± 4390 0.72

10, we could not obtain enough accuracy to determine the time dependence, b, at these bands.Although we had > 1 day span at Band 9, dependence on b was hidden behind random noise.Dependence on ∆ secZ was significant at only Band 3 and 6.

The adjusted R2 values are greater than 0.9 for most of regressions. This means that thecontrol parameters of S, T , ∆ secZ, and ∆AZ explain the majority of bandpass variation. Inother words, we can predict expected bandpass variations giving these control parameters.

5.2 Prediction of bandpass variation

Using the linear regression coefficients and control parameters, we attempt to predict thebandpass variation. Figures 21 – 28 plots the bandpass variations with four different indicators(SD of amplitude, PE of amplitude, SD of phase, and PE of phase) as a function of thecombination of control parameters weighted by coefficients. The red horizontal line indicatesthe required stability level of −30 dB.

The intercept corresponds to the minimal bandpass variation where control parameters arezero (i.e., infinite signal-to-noise ratio without T , ∆ secZ and ∆ AZ). Since the intercept valuesof SD amplitude At Band 3, 4, and 6 are smaller than the requirement, relevant strategy ofbandpass calibration allows us to fulfill the requirement. However, we have no chance to achievethe required stability at higher frequency than Band 7 where the intercept is greater than thegoal.

25

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0.0000 0.0005 0.0010 0.0015 0.0020

0.00

000.

0005

0.00

100.

0015

0.00

200.

0025

Band3 bandpass stability

aS + b∆T + c∆sec Z + d∆Az

SD o

f Am

plitu

de V

aria

tion

a = 3.63e−01 (t= 40.3)

b = −1.85e−11 (t= −2.0)

c = 8.04e−04 (t= 23.5)

d = 3.60e−05 (t= 3.0)

intercept = 4.70e−04 ± 4.16e−05

R^2 = 0.92

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0.000 0.001 0.002 0.003 0.004 0.005 0.006

0.00

00.

002

0.00

40.

006

0.00

8



PE o

f Am

plitu

de V

aria

tion

a = 1.03e+00 (t= 39.1)

b = −1.03e−10 (t= −3.8)

c = 1.95e−03 (t= 19.5)

d = 1.06e−04 (t= 3.1)

intercept = 1.19e−03 ± 1.21e−04

R^2 = 0.91

(a) SD (left) and PE (right) of bandpass amplitude variation at Band 3

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0.000 0.001 0.002 0.003 0.004

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00.

001

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003

0.00

40.

005



SD o

f Pha

se V

aria

tion

a = 3.21e−01 (t= 12.1)

b = 7.05e−10 (t= 25.6)

c = 4.46e−04 (t= 4.4)

d = 4.37e−04 (t= 12.5)

intercept = 1.31e−03 ± 1.23e−04

R^2 = 0.87

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0.000 0.002 0.004 0.006 0.008 0.010

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006

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010

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20.

014



PE o

f Pha

se V

aria

tion

a = 1.00e+00 (t= 14.3)

b = 1.17e−09 (t= 16.2)

c = 1.39e−03 (t= 5.2)

d = 1.00e−03 (t= 10.8)

intercept = 3.10e−03 ± 3.24e−04

R^2 = 0.83

(b) SD (left) and PE (right) of bandpass phase variation at Band 3

Figure 21: Predicted and measured bandpass variations in Band 3. The red horizontal lineindicates the required stability of −30 dB.

26

Band 4

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0.000 0.001 0.002 0.003 0.004 0.005 0.006 0.007

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00.

002

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40.

006



SD o

f Am

plitu

de V

aria

tion

a = 4.80e−01 (t= 10.7)

b = −1.04e−07 (t= −0.9)

c = 2.31e−03 (t= 1.5)

d = −7.02e−04 (t= −1.1)

intercept = 7.93e−04 ± 3.78e−04

R^2 = 0.82

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0.000 0.005 0.010 0.015 0.020

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005

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015

0.02

0



PE o

f Am

plitu

de V

aria

tion

a = 1.34e+00 (t= 10.3)

b = −2.52e−07 (t= −0.8)

c = 4.96e−03 (t= 1.1)

d = −1.64e−03 (t= −0.9)

intercept = 2.08e−03 ± 1.10e−03

R^2 = 0.81


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0.000 0.002 0.004 0.006 0.008

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002

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006

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010



SD o

f Pha

se V

aria

tion

a = 6.19e−01 (t= 10.1)

b = −8.72e−08 (t= −0.6)

c = 1.60e−03 (t= 0.8)

d = −7.24e−04 (t= −0.9)

intercept = 1.09e−03 ± 5.14e−04

R^2 = 0.80

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0.000 0.005 0.010 0.015 0.020 0.025

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025



PE o

f Pha

se V

aria

tion

a = 1.69e+00 (t= 8.5)

b = −3.65e−07 (t= −0.7)

c = 6.03e−03 (t= 0.9)

d = −2.73e−03 (t= −1.0)

intercept = 3.41e−03 ± 1.68e−03

R^2 = 0.73


Figure 22: Predicted and measured bandpass variations in Band 4

27

Band 4 (FDM)

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SD o

f Am

plitu

de V

aria

tion

a = 5.57e−01 (t= 111.0)

b = −6.18e−08 (t= −1.2)

c = 4.65e−04 (t= 0.6)

d = −6.36e−05 (t= −0.2)

intercept = 8.34e−04 ± 1.75e−04

R^2 = 1.00

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PE o

f Am

plitu

de V

aria

tion

a = 2.02e+00 (t= 87.2)

b = −2.47e−07 (t= −1.0)

c = 2.92e−03 (t= 0.9)

d = −7.42e−04 (t= −0.6)

intercept = 3.44e−03 ± 8.07e−04

R^2 = 1.00

(a) SD (left) and PE (right) of bandpass amplitude variation at Band 4 FDM

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SD o

f Pha

se V

aria

tion

a = 5.42e−01 (t= 69.8)

b = 1.55e−07 (t= 1.9)

c = −9.52e−04 (t= −0.9)

d = −8.39e−04 (t= −1.9)

intercept = 1.27e−02 ± 2.71e−04

R^2 = 1.00

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PE o

f Pha

se V

aria

tion

a = 1.98e+00 (t= 79.3)

b = 2.82e−07 (t= 1.1)

c = 2.88e−04 (t= 0.1)

d = −3.88e−03 (t= −2.7)

intercept = 4.66e−02 ± 8.70e−04

R^2 = 1.00

(b) SD (left) and PE (right) of bandpass phase variation at Band 4 FDM

Figure 23: Predicted and measured bandpass variations in Band 4 FDM SPW

28

Band 6

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003

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4



SD o

f Am

plitu

de V

aria

tion

a = 3.02e−01 (t= 6.3)

b = 3.07e−09 (t= 17.6)

c = 1.45e−03 (t= 24.9)

d = 8.01e−05 (t= 2.9)

intercept = 9.65e−04 ± 1.30e−04

R^2 = 0.99

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006

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010

0.01

2



PE o

f Am

plitu

de V

aria

tion

a = 1.11e+00 (t= 6.7)

b = 7.60e−09 (t= 12.6)

c = 3.50e−03 (t= 17.3)

d = 2.17e−04 (t= 2.3)

intercept = 1.78e−03 ± 4.52e−04

R^2 = 0.98


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0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030

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003

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005



SD o

f Pha

se V

aria

tion

a = 1.15e−01 (t= 0.9)

b = 2.96e−09 (t= 6.3)

c = 1.54e−03 (t= 9.8)

d = −6.47e−06 (t= −0.1)

intercept = 2.34e−03 ± 3.53e−04

R^2 = 0.90

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006

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010

0.01

20.

014



PE o

f Pha

se V

aria

tion

a = 5.59e−01 (t= 1.7)

b = 6.29e−09 (t= 5.1)

c = 3.74e−03 (t= 9.0)

d = −1.96e−05 (t= −0.1)

intercept = 5.53e−03 ± 9.26e−04

R^2 = 0.90



29

Band 7

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SD o

f Am

plitu

de V

aria

tion

a = 3.83e−01 (t= 51.0)

b = −2.38e−09 (t= −1.2)

c = 8.71e−03 (t= 2.3)

d = −5.04e−04 (t= −1.8)

intercept = 2.20e−03 ± 3.53e−04

R^2 = 0.99

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0.14



PE o

f Am

plitu

de V

aria

tion

a = 1.04e+00 (t= 51.0)

b = −3.39e−09 (t= −0.6)

c = 1.97e−02 (t= 1.9)

d = −1.39e−03 (t= −1.9)

intercept = 5.90e−03 ± 9.57e−04

R^2 = 0.99


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0.07



SD o

f Pha

se V

aria

tion

a = 5.12e−01 (t= 54.1)

b = −7.77e−09 (t= −3.1)

c = 7.88e−03 (t= 1.6)

d = −4.09e−04 (t= −1.2)

intercept = 2.94e−03 ± 4.44e−04

R^2 = 0.99

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0.15



PE o

f Pha

se V

aria

tion

a = 1.37e+00 (t= 50.5)

b = −2.25e−08 (t= −3.1)

c = 2.51e−02 (t= 1.8)

d = −1.61e−03 (t= −1.6)

intercept = 8.07e−03 ± 1.27e−03

R^2 = 0.99



30

Band 8

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0.00 0.01 0.02 0.03 0.04

0.00

0.01

0.02

0.03

0.04



SD o

f Am

plitu

de V

aria

tion

a = 3.77e−01 (t= 21.9)

b = 3.77e−07 (t= 2.1)

c = 1.64e−02 (t= 1.3)

d = −1.48e−03 (t= −0.9)

intercept = 3.74e−03 ± 3.64e−04

R^2 = 1.00

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0.00 0.02 0.04 0.06 0.08 0.10

0.00

0.02

0.04

0.06

0.08

0.10

0.12



PE o

f Am

plitu

de V

aria

tion

a = 1.07e+00 (t= 26.4)

b = 1.27e−06 (t= 2.9)

c = 5.00e−02 (t= 1.7)

d = −5.49e−03 (t= −1.4)

intercept = 9.21e−03 ± 8.59e−04

R^2 = 1.00


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0.00 0.01 0.02 0.03 0.04 0.05

0.00

0.01

0.02

0.03

0.04

0.05

0.06



SD o

f Pha

se V

aria

tion

a = 5.14e−01 (t= 22.5)

b = 1.87e−07 (t= 0.8)

c = 4.34e−03 (t= 0.3)

d = 1.05e−03 (t= 0.5)

intercept = 5.28e−03 ± 4.84e−04

R^2 = 1.00

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0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14

0.00

0.05

0.10

0.15



PE o

f Pha

se V

aria

tion

a = 1.33e+00 (t= 23.2)

b = 1.18e−06 (t= 1.9)

c = 8.14e−02 (t= 1.9)

d = −8.61e−03 (t= −1.6)

intercept = 1.50e−02 ± 1.22e−03

R^2 = 1.00



31

Band 9

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0.000 0.005 0.010 0.015 0.020 0.025 0.030

0.00

0.01

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0.03

0.04



SD o

f Am

plitu

de V

aria

tion

a = 4.45e−01 (t= 14.0)

b = −1.13e−11 (t= −0.0)

c = −3.99e−04 (t= −0.2)

d = −2.45e−03 (t= −0.6)

intercept = 5.81e−03 ± 8.53e−04

R^2 = 0.99

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0.00 0.02 0.04 0.06 0.08

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0.02

0.04

0.06

0.08

0.10



PE o

f Am

plitu

de V

aria

tion

a = 1.06e+00 (t= 8.1)

b = −1.76e−10 (t= −0.2)

c = 6.21e−03 (t= 0.7)

d = −4.59e−03 (t= −0.3)

intercept = 1.99e−02 ± 3.51e−03

R^2 = 0.98


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0.00 0.01 0.02 0.03 0.04

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0.01

0.02

0.03

0.04

0.05



SD o

f Pha

se V

aria

tion

a = 5.51e−01 (t= 7.3)

b = 1.09e−09 (t= 1.9)

c = 4.33e−03 (t= 0.9)

d = −1.81e−02 (t= −1.9)

intercept = 1.02e−02 ± 2.02e−03

R^2 = 0.97

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0.00 0.02 0.04 0.06 0.08 0.10

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0.02

0.04

0.06

0.08

0.10

0.12



PE o

f Pha

se V

aria

tion

a = 1.22e+00 (t= 7.0)

b = 2.17e−09 (t= 1.6)

c = 1.56e−02 (t= 1.4)

d = −2.77e−02 (t= −1.3)

intercept = 3.38e−02 ± 4.67e−03

R^2 = 0.98

(b) Predicted and measured bandpass variations in Band 9


32

Band 10

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0 1 2 3 4

01

23

4



SD o

f Am

plitu

de V

aria

tion

a = 4.11e−01 (t= 20.1)

b = 1.18e−05 (t= 0.4)

c = −6.91e−02 (t= −0.1)

d = −3.29e−02 (t= −0.3)

intercept = 6.44e−02 ± 9.87e−02

R^2 = 0.95

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0 2 4 6 8 10 12

02

46

810



PE o

f Am

plitu

de V

aria

tion

a = 1.12e+00 (t= 17.5)

b = −3.25e−05 (t= −0.3)

c = 9.60e−01 (t= 0.6)

d = −1.53e−01 (t= −0.4)

intercept = 3.97e−01 ± 3.10e−01

R^2 = 0.93


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0.0 0.5 1.0 1.5

0.0

0.5

1.0

1.5



SD o

f Pha

se V

aria

tion

a = 1.95e−01 (t= 8.0)

b = −6.77e−05 (t= −1.8)

c = 9.18e−01 (t= 1.5)

d = 8.88e−02 (t= 0.6)

intercept = 3.71e−01 ± 1.17e−01

R^2 = 0.78

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−1 0 1 2 3

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5



PE o

f Pha

se V

aria

tion

a = 4.28e−01 (t= 6.0)

b = −2.48e−04 (t= −2.3)

c = 2.62e+00 (t= 1.4)

d = 5.57e−01 (t= 1.3)

intercept = 1.21e+00 ± 3.44e−01

R^2 = 0.72

(b) Predicted and measured bandpass variations in Band 10


33

5.3 Bandpass flatness

As shown in the SAVs, the bandpass shape after bandpass calibration is flat for the frequencyseparation of ∆ν < 100 MHz, compared with the random noise. This indicates that SmoothedBandpass Calibration [Yamaki et al.(2012)] will work to reduce the random noise in the band-pass table by smoothing with the frequency scale < 100 MHz. At higher frequency than Band7, the bandpass undulation becomes less significant compared with the random noise and thusSBC will be more efficient to reduce the random noise that is the major component of bandpassvariation.

Note that SBC should be applied only for the BP table and keep the spectral resolutiontoward target sources as desired.

5.4 Time instability of bandpass shape

The dependence of bandpass variation on T , denoted as b in the linear regression, is significantat Band 3 and 6. The amplitude dependence at Band 8 seems ridiculous because the shorttime span of observations (∼ 1.5 hour) and non-significant dependence in SD amplitude. Atother frequency bands, the detectability for b is not enough because of insufficient time spanor signal-to-noise ratio.

The amplitude and phase dependence on elapsed time is less than b < 10−8 s−1 which willintroduce amplitude variation of < −30 dB at T = 1 day. Thus, the bandpass shape is time-stable so that we can employ the of BP table obtained the day before the observation of targetsources.

5.5 Az and El dependence of bandpass stability

Bandpass variation significantly relate to ∆ secZ at Band 3 and 6, with the coefficient ofc 4 × 10−3 at Band 6 and c 2 × 10−3 at Band 3. These results indicate that the bandpasscalibration should taken near the target source (∆ secZ < 0.25 and ∆ secZ < 0.5 for Band 6and 3, respectively) to suppress the bandpass variation less than −30 dB.

Azimuth dependence is significant at only Band 3. This is probably relate to the influenceof geomagnetism (FENCR-582).

We did not find significant ∆ AZ and ∆ secZ dependences of the bandpass variation at otherfrequency bands because of narrow azimuth or elevation coverage of the observations, or dueto insufficient signal-to-noise ratio to detect them.

5.6 TDM-to-FDM bandpass transfer

The results presented in subsection 4.4 clearly suggest that a BP table generated in the TDMmode is significantly different (∼ 2%) from that in the FDM mode. The TDM-to-FDM band-pass transfer will bring serious systematic errors in bandpass shape.

The different bandpass characteristics between TDM and FDM can caused by quantizationloss and non-linearity through signal processing in the FDM.

Instead of employing TDM bandpass for FDM, SBC is an alternative to improve accuracyin the BP table. The SAV in the raw FDM BP table showed that the amplitude is flat at thefrequency separation of ∆ν < 31 MHz. Thus, SBC with the smoothing window of ∼ 30 MHzwill work to reduce the random noise keeping undulation of the bandpass shape.

34

http://jira.alma.cl/browse/FENCR-582

6 Summary

We verified the cross-correlation bandpass stability at Band 3, 4, 6, 7, 8, 9, and 10 using arrayswith 21–35 antennas toward bright and compact continuum radio sources. Antenna-basedbandpass tables with real-imaginary solutions showed slightly better accuracy than amplitude-phase solutions. The required bandpass stability of −30 dB in 1 hour was partially compliantat Band 3, 4, and 6, while bandpass variation exceeded the requirement.

The primary component of bandpass variation is the random (thermal) noise that relatesto the signal-to-noise ratio. The secondary component is the intercept that becomes greaterat higher frequency bands. The origin of the intercept is unknown. Elapsed time from BPcalibration scan is a minor part to explain the bandpass variation. The BP table taken 1 daybefore will not bring systematic error greater than −30 dB. The atmospheric effect becomescomparable to the requirement when ∆ secZ exceeds 0.25. Azimuth difference responses tothe bandpass variation in only Band 3, probably due to geomagnetism.

The bandpass shapes in TDM and FDM are significantly different by ∼ 2 %. Thus, TDM-to-FDM bandpass transfer will cause serious systematic error in bandpass calibrations. TheFDM bandpass shape is flat enough in the frequency separation of ∆ν < 31 MHz and SBCwill work to improve the accuracy of FDM BP table.

References

[Baart et al.(1986)] Baart, E. E., Cohen, R. J., Davies, R. D., Rowland, P. R., & Norris, R. P.1986, MNRAS, 219, 145

[Comoretto (2008)] Comoretto G. 2008, ALMA memo 583

[Yamaki et al.(2012)] Yamaki, H., Kameno, S., Beppu, H., Mizuno, I., & Imai, H. 2012, PASJ,64, 118

35

alma cross correlation bandpass stability · 2015. 3. 16. · thedraft system veri cation report on...

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