Consideration to the Correction Factor for Single-DishObservations with the ACA Correlator
ALMA Technical Note Number: 13
Status: DRAFT
Prepared by:

Organization:

Date:

Seiji Kameno

JAO/NAOJ

26 October 2014

Consideration to the Correction Factor forSingle-Dish Observations with the ACA
Correlator (draft)
Seiji Kameno
October 26, 2014
This report theoretically addresses the non-linearity of total power (TP) measurementsof single-dish observations using the ACA correlator (ACAC). We review the digital signal
processing in the digitizers and the ACA correlator. The non-linear relation between input
signal power and total power measurements with the correlator will be discussed. For continuum sources, the theoretically derived correction factor is 1.256 that meets the observational
implication of 1.28 ± 0.04. In the case of spectral lines, the correction factor depends on theline width relative to the bandwidth. For narrow spectral lines, the theoretical correction
factor is 1.14. The empirical correction factor of 1.206 ± 0.04 is slightly different from thetheoretical implication.

1 Introduction
The single-dish observations with the TP (total power) array in ALMA is carried out using the ACAcorrelator. Accuracy of flux measurements in the TP array is crucial to obtain high-quality images
combined with the interferometric observations of the Morita array and the main array. However, itis acknowledged that the single-dish flux measurements using the ACA correlator is affected by nonlinearity, which sometimes achieve at ∼ 50% error (CSV-2880), and has been the blocker of single-dishSBs of Cycle-1/2 observations.
The Total Power Non-Linearity Tiger Team has worked to investigate the characteristics of the nonlinearity and have quested for a remedy to correct the non-linearity. Verification observations toward
continuum sources and emission line sources have carried out, as logged in JIRA CSV-2880 and CSV3079. Statistical analyses of these tests concluded the correction factor of 1.28 and 1.2 to be multiplied
to continuum and spectral line observations, respectively. The conclusion of the TP Tiger Teem was toapply a single correction factor of 1.25 to meet the tentative requirement of 10% accuracy for Cycle 1and 2.
Besides empirical determination of the correction factor, theoretical reasoning was desired (1) toexplain the necessity of the correction factor to the community, (2) to identify the scope and theapplication range of of the correction factor, and (3) to quest further accuracy targeting 1%.
In this report we attempt to derive the non linearity and the correction factor inductively.

1V̂jvhq
wq
Pq
C(τ )
Ĉjs(ν)S(ν)
ŜkTable 1: Symbols used in this report
digitized voltage at t = j∆t where ∆t is the sampling period
threshold voltages (h = −3, −2, . . . , +3)
3-bit quantized levels (q = 0, 1, . . . , 7)weight at the q-th quantization levelprobability at the q-th quantization levelanalog correlation function of the time lag, τdigitized correlation function at τ = j∆tcomplex voltage spectrum as a function of the baseband frequency, νanalog power spectrum
digitized power spectrum at the frequency of ν = k∆ν where ∆ν is the channel
spacing

Symbol
vth8
vth16

Value
0.586
0.335

Table 2: Constants used in this report
Remarks
The optimal vth /σ value to maximize the 3-bit quantization efficiency
The optimal vth /σ value to maximize the 4-bit quantization efficiency

F80.7964

The 3-bit quantized power sensitivity at the optimal level, derived by

F16
η80.9123
0.9625

The 4-bit re-quantized spectral sensitivity at the optimal level
The 3-bitquantization
efficiency for cross correlation at the optimal level, de∂ ρ̂
rived by∂ρ ρ=0

∂ Ĉ0
∂C(0)

1.1 Basic Theory on Spectrum and Correlation Function
Let V (t) the analog baseband signal to input to the digitizer, and s(ν) is the complex voltage spectrum
as the Fourier transform of V (t) as
1T →∞ TZ T /2V (t)e−i2πνt dt.

s(ν) = lim(1)

t=−T /2(2)
Hereafter, expression of the Fourier transform is simplified as s(ν) = FT [V (t)]. The power spectrum isgiven as S(ν) = |s(ν)|2 .
According to the Wiener–Khinchin theorem, the power spectrum relates to the correlation function,C(τ ) = hV (t)V (t − τ )i, as1 .
S(ν)
C(τ )

=FT [C(τ )] ,(3)

=−1

(4)

FT[S(ν)] .

The correlation function at zero-lag is proportional to the power, W , as C(0) = V 2 = ZW ,where Z is the impedance of the signal transmission. This is also identical to the integral of theZ B
power spectrum as C(0) =S(ν) dν, where B is the bandwidth. When the probability density
ν=−Bfunction of V (t) follows the zero-bias normal distribution, N (µ = 0, σ 2 ) with the variance of σ 2 , wehave σ 2 = V 2 = C(0).
1 Since V (t) is real, s(ν) is an Hermite function. Thus, S(ν) is an even real function. In some physical description wherefrequency is limited in positive values, the power spectrum is folded as S̃(|ν|) = 2S(|ν|).

2

2 Digital Spectral Measurements
In this section, we formularize the power spectral measurements in 3-bit quantized digital signals. We
employ the zero-bias normal distribution of input analog voltage. The sampling period is at the Nyquist
rate of ∆t = 1/(2B).

2.1 3-bit Quantization of Voltages
We assume that the threshold voltages are uniformly spaced as vh = hvth , h = −3, −2, . . . , +3. In theACAC, the quantized voltages, wq , at 8 levels (q = 0, 1, . . . , 7) are coded as wq = q − 3.5.
The power spectrum of digitized signal is calculated via FFT (Fast Fourier Transform) in the ACAC
The time-series data, V̂j = wq (t − j∆t) will be converted to spectrum via FFT and then complex
multiplication will be take to produce the digitized power spectrum, Ŝk , where k stands for the spectral
channel. The digitized correlation function, Ĉj relates to the digitized power spectrum via FFT as,

N −1
2πkj
1 X
Sk exp i,Ĉj =NN(5)k=0where j stands for the time lag and N is the number of spectral channels.

2.2 Relation between Analog and Digital Correlation Coefficient
Since quantization is a non-linear operation to reduce amount of information in the received signal, thedigitized correlation function is distorted from the original analog correlation function. The digitized
power spectrum is indeed modified from the original one, too.

0.150.100.00

0.05

Probability Density

0.20

0.25

3−bit Optimal Level−4

W=−3.5W=−2.5W=−1.5

W=−0.5

W=0.5

W=1.5

W=2.5

W=3.5

P= 3.9%P= 8.1%P=15.8%P=22.1%

P=22.1%P=15.8%P= 8.1%

P= 3.9%−202

4Threshold Voltage

Figure 1: The probability density of input analog voltages at the optimal with respect to the thresholdvoltages. The vertical dashed lines indicate the threshold voltages. The standard deviation, σis set at σ = 1/vth8 = 1.706. Digitized code (weight) and probability at each quantized level
are also shown onset.

32.2.1 Zero-Lag Correlation Coefficient
The zero-lag value of the digitized correlation function, Ĉ0 is given byĈ0 = wq2 =7
X

Pq wq2 .

(6)

q=0Here, Pq is the probability where a sample states at the q-th quantization level.
For the normal distribution of analog voltages, the probabilities of quantized levels are given as


vq−4
vq−3
√
√
− erf,(7)
Pq = erf
σ 2σ 2where v±4 = ±∞. Figure 1 shows the probability of quantized level, Pq , for the case of vth /σ = vth8 =0.586.
Figure 2a shows the relation between analog power and digital power measurements. Both zero-lag
values, C(0) and Ĉ0 are scaled to be unity at the optimal level of vth /σ = vth8 .
For position-switching observations toward a continuum source, the antenna temperature, Ta is calculated asTa

=T̂a=C(0)on − C(0)off
Tsys ,C(0)offĈ0,on − Ĉ0,offĈ0,off

Tsys ,(8)(9)

where Tsys is the system noise temperature. The ratio of T̂a to Ta is given by the derivative of Ĉ0 , i.e.,∂ Ĉ0∂ Ĉ0, if Ta  Tsys . Figure 2b shows the derivative,. The derivative is F8 = 0.7964 at the∂C(0)
∂C(0)
!−1
∂ Ĉ0
optimal level, which corresponds to the correction factor of= 1.256.
∂C(0)2.2.2 Correlation Coefficients at Non-Zero Lag
Correlation coefficients at lags other than 0, C(τ 6= 0) are usually much smaller than C(0). The relation
between the digitized correlation coefficient and analog one is known as the Van Vleck relation. SinceC(τ )the Van Vleck relation is established in the normalized correlation coefficient, ρ(τ ) =, we argue
C(0)about the normalized correlation coefficient and then multiply the zero-lag correlation coefficient ifnecessary.
Figure 3(a) shows the Van Vleck relation of 3-bit quantization at the optimal level. By definition,
the digitized normalized correlation  coefficient, ρ̂ takes 0 and 1 at ρ =0 and 1, respectively.
∂ ρ̂The slope of ρ̂ at ρ = 0 is= 0.9626, which is also known as the quantization coefficient. For∂ρ ρ=0small correlation coefficients, ρ̂ is approximately proportional to ρ with the slope of the quantization
coefficient. Figure 3(b) shows the departure of ρ̂ from the linear regression. The departure is negligiblefor ρ < 0.2 and less than 1% for ρ < 0.9.
In the case of white noise, the correlation function will be the delta function where C(τ 6= 0) =0. For bandwidth-limited white noise, the correlation function will be the sinc function as C(τ ) =sin 2πBτ, and the discrete sampling of the correlation function with the sampling period of ∆τ =C(0)2πBτ
1/(2B) will result in C(j∆τ ) = 0 (j 6= 0).
Non-flat bandpass shape or presence of line spectrum causes non-zero values of C(τ 6= 0). Nevertheless, these values are usually small, compared with C(0). Figure 4 shows the real power spectra andthe correlation function. The correlation coefficient at τ 6= 0 is indeed much smaller than the unity.

4

1.0
0.8
0.6
0.4
0.0

0.2

Channel−Averaged Power (scaned at the optimal level)

2.52.01.5
1.0
0.5

Channel−Averaged Power (scaned at the optimal level)

0.0

Sensitivity = 0.7964

0123450Analog Power (scaled at the optimal level)

12345Analog Power (scaled at the optimal level)

Figure 2: (a):Relation between analog power, C(0) and corresponding digital power, Ĉ0 . Both values
are scaled at the optimal input level. The digital power measurements become significant
underestimation than the ideal linear relation (dashed diagonal line) at greater power input∂ Ĉ0. At the optimal level, thethan the optimal level. (b):The derivative of digital power,∂C(0)derivative is F8 = 0.7964 that yields the correction factor of 1/F8 = 1.256.
In summary, the correlation coefficient at non-zero lag is approximately given as Ĉj6=0 = Ĉ0 ρ̂ =Ĉ0
η8 Ĉ0 ρ = η8C(τ 6= 0) = η8 C(τ 6= 0).
C(0)2.2.3 4-bit Re-Quantization in Spectra
In the ACAC, 4-bit re-quantization is took place for both real and imaginary parts of complex voltage
spectrum, s(ν). This introduces additional non-linearity, especially in autocorrelation.
We assume that the complex voltage spectrum follows the normal distribution. Then, we can recall
similar arguments of §2.2.1 for the relation between power spectra before and after the re-quantization.
Figure 5 shows the relation between non-re-quantized and re-quantized power spectra at a spectralchannel. The derivative of re-quantized power spectra at the optimal power level is F16 = 0.9123, whichcorresponds to the correction factor of 1/F16 = 1.0961.
Note that the correction factor is spectral-channel based unlike the zero-lag value with 3-bit quantization. The channel-by-channel power was scaled into a flat-spectrum at optimal power level. The scaling
factor is determined through calibration scans before every sub scans. Thus, the 4-bit non-linearity
correction factor should be applied only for difference from the calibration scan, i.e., line features.
Alternatively, it is identical to apply the correction factor to the correlation function except zero lag.

3 Correction Factor for Single-Dish Observations
Now we argue the correction factor to be applied to the autocorrelation power spectrum generated in
the ACAC.

5Departure from linear Regression

0.02

3−bit corr. − 0.96256* analog corr

0.01

0.6
0.4
0.0

0.00

0.2

3−bit Correlation Coefficient

0.8

0.03

1.0

Optimal Threshold Level

0.0

0.2

0.4

0.6

0.8

1.0

0.0

Analog Correlation Coefficient

0.2

0.4

0.6

0.8

1.0

Analog Correlation Coefficient

Figure 3: Relation between correlation coefficient in analog (horizontal axis) and 3-bit quantized (vertical axis) measurements.

3.1 ContinuumRecall the discussion in §2.2.1, the zero-lag correlation coefficient of 3-bit digitized signals is affectedby non linearity and its sensitivity at the optimal level is F8 = 0.7964. Thus, we should apply thecorrection factor of 1/F8 = 1.256 to the on-off power. This value is consistent with the empirical factorof 1.28 for continuum, reported in CSV-2880/CSV-3079.

3.2 Line
In the case of spectral line sources, we have to apply different correction factors considering its spectral
shape. We discuss in the correlation function in lag domain because of different quantum efficiencies
influenced on the zero-lag and non-zero-lag values.
The increment of zero-lag correlation coefficient in 3-bit quantization is F8 times the un-quantizedcase. On the oter hand, the correlation coefficients at non-zero-lag values are multiplied by the quantum
efficiency, η8 for 3-bit quantization. Furthermore, the 4-bit re-quantization reduces the on–off spectral
power by 1 − F16 Thus, the correlation function will be(F8 C on (0) − C off (0)(j = 0)
onoffĈj − Ĉj=(10)onoffη8 F16 C (j∆τ ) − C (j∆τ ) (j 6= 0)
Figure 6 shows the effect of 3-bit and 4-bit quantizations to a simulated emission-line spectrum. The
correlation function of quantized signals (red) is affected from the un-quantized one (black) as expressedin equation 10. The intensity of corresponding spectrum is reduced by the factor of η8 F16 . On the other
hand, the efficiency in the zero-lag value affects reduction in power spectrum in the band uniformly.
Therefore, the ratio of quantized line intensity to un-quantized is given as η = η8 F16 (1 − b) + F8 b, whereb is the ratio of effective line width to the bandwidth. Thus, the line correction factor, c is dependent on
1. Figure 7 plots the line correction factor as a function
the line width as c = 1/η =η8 F16 (1 − b) + F8 bof line width.
The correction factor at the narrowest end of the line profile is 1/(η8 F16 ) = 1.139 and approaches
to 1/F8 = 1.256 at the widest line width (i.e. continuum). There exist a small difference from thedeductive implication of 1.2 for line emission reported in CSV-2880. The line width is less than 3% of6

uid___A002_X82e118_X51c.ms IRC+10216 CO(3-2)

Autocorrelation Function
1.0
0.8
Corrlation CoefficientPower Spectrum

15

105

0.6
0.4
0.2

00.0
345.0

345.5

346.0
Frequency [GHz]

346.5

05

1015
Lag

202530Figure 4: (a): Power spectra of on-source (blue) and off-source (green) toward IRC+10216. CO (3-2)
line emission can be seen together with the bandpass shape. (b): Normalized autocorrelation
functions of on-source (blue) and off-source (green). The correlation coefficient at τ 6= 0 is
less than 0.1.
the bandwidth and the correction factor of 1.14 was expected. At this moment it is unclear to explain
the difference between theoretical and observed correction factors was 5.7%.

4 Conclusions
We reviewed the digital signal processing in the digitizers and the ACA correlator, and argued thenon-linearity of quantized signals to evaluate the correction factor for single-dish observations.
For continuum observations, the theoretical correction factor is 1.256. The verification results of
1.28 ± 0.04 was consistent with the theoretical implication.
In the case of spectral line observations, the correction factor depends on the line width relative tothe bandwidth. For narrow spectral line, the theoretical correction factor is 1.14. The observationalcorrection factor of 1.206 ± 0.04 was slightly different from the theoretical implication, hence, it isnecessary to clarify the cause of the difference to quest better accuracy of 1%.

7

4−bit Re−Quantization

1.00.40.60.8Sensitivity = 0.9123

0.00.2Channel−Averaged Power (scaled at the optimal level)

432
1
0

Channel−Averaged Power (scaled at the optimal level)54−bit Re−Quantization

01

2

3

4

501

Analog Power (scaled at the optimal level)

2

3

4

5

Analog Power (scaled at the optimal level)

0.4
0.3

−0.06

0.0

−0.04

0.1

0.2

Intensity (relative to Tsys)

0.02
0.00
−0.02

Correlation Coefficient

0.04

0.5

0.06

Figure 5: (a):Re-quantized power versus un-re-quantized power at a spectral channel. Both powers are
scaled at the optimal level. (b):The derivative of the re-quantized power spectra. At theoptimal level, the derivative is F16 = 0.9123.

01020304050600Lag

20406080100120

Frequency [ch]Figure 6: (a): On - Off correlation function of simulated line emission. Solid black line indicates correlation function of unquantized signal. The red line shows after 3-bit quantization + 4-bit
re-quantization; quantum efficiencies for Ĉ0 and Ĉj6=0 are F8 and η8 × F16 = 0.8781, respectively. (b): Simulated line profile (black) and quantized profile (red).

8

1.22
1.20
1.18
1.14

1.16

Line Correction Factor

1.24

ACAC line correction factor

0.0

0.2

0.4

0.6

0.8

1.0

Line Width / Bandwidth

Figure 7: Correction factor as a function of line width relative to the bandwidth.

9