**NATIONAL RADIO ASTRONOMY OBSERVATORY**

**MEMORANDUM**

DATE:

TO: Operators and Observers

FROM: Ron Maddalena

This document outlines the calibration of 140-ft data. It is divided into three sections covering what the Mark IV Autocorrelator (AC) does to the data and the special calibration techniques used for total-power and switched-power observing.

The AC at the 140-ft takes two spectra for each AC bank, one spectra is called signal (Sig) and one called reference (Ref). The AC cycles between accumulating data in Sig and Ref--usually, half a second on Sig followed by half a second on Ref (with some blanking time in between). This cycling continues through out the course of an AC integration cycle (20 or 60 secs, usually).

For position switching, Sig and Ref should be almost identical. For frequency or beam switching, Sig and Ref correspond to data taken when the Local Oscillator (LO) or subreflector are at their signal and reference frequencies or positions. For each Sig and Ref cycle, the noise tube is on for half the time and of for the other half. The total power counts (TPC) for the four phases (sig_on, sig_off, ref_on, and ref_off) are accumulated separately and sent along with the data to the control computer.

If T_sig(i) and T_ref(i) are the actual intensity distribution of the sky in Sig and Ref, a_sig(i) and a_ref(i) the frequency response of the instrument, Tnt = noise tube temperature, and Tsys_sig and Tsys_refthe contribution to the system temperature from the instrument, receiver, sky, continuum, ground pick up, etc. (i.e. everything BUT the noise tube and the line) then:

TPC(sig_on) = integral over i { a_sig(i)*[Tsys_sig + Tnt + T_sig(i)] }

TPC(sig_off) = integral over i { a_ref(i)*[Tsys_sig + T_sig(i)] }

TPC(ref_on) = integral over i { a_ref(i)*[Tsys_ref + Tnt + T_ref(i)] }

TPC(ref_off) = integral over i { a_ref(i)*[Tsys_ref + T_ref(i)] }

and

Sig(i) = a_sig(i) * [Tsys_sig + Tnt/2 + T_sig(i)]

Ref(i) = a_ref(i) * [Tsys_ref + Tnt/2 + T_ref(i)]

[The 2 in the Tnt/2 is because the noise tube is only on half of the time. I'll use Tsys from now on for this uncomplete definition of system temperature and TSYS for the definition of the complete system temperature (= Tsys + any contribution from the noise tube + any from the line). ]

The AC normalizes the two spectra (Sig and Ref) by the integral over i of Sig and Ref), before transferring them to the control computer. That is, if S and R are what is being sent to the control computer:

S(i) = Sig(i) / integral over i { Sig(i) }

R(i) = Ref(i) / integral over i { Ref(i) }

Thus, S(i) and R(i) should have a median value of 1 and the integrals should
be proportional to the total reference and signal system temperatures. That is:

integral over i { Sig(i) } = [ TPC(sig_on) + TPC(sig_off) ] / 2

integral over i { Ref(i) } = [ TPC(ref_on) + TPC(ref_off) ] / 2

or:

S(i) = 2 * Sig(i) / [ TPC(sig_on) + TPC(sig_off) ]

R(i) = 2 * Ref(i) / [ TPC(ref_on) + TPC(ref_off) ]

The control computer calculates the TOTAL system temperature (TSYS) for SIG
and REF from the total power counter numbers sent with the data:

TPC(sig_on) + TPC(sig_off)

TSYS(SIG) = Tnt * -------------------------

TPC(sig_on) - TPC(sig_off)

TPC(ref_on) + TPC(ref_off)

TSYS(REF) = Tnt * -------------------------

TPC(ref_on) - TPC(ref_off)

(These TSYS are higher than the usual definition of system temperature by 1/2 the noise tube value since the noise tube are on half the time.) As is obvious:

S(i) is proportional to Sig(i) / TSYS(sig)

R(i) is proportional to Ref(i) / TSYS(ref)

The control computer, for total power observing, sums R(i) and S(i) together
and passes the data [ SUM(i) ] to the analysis system. Thus, the data the
analysis system should have a median value of 2. The computer also averages the
Tsys(SIG) and Tsys(REF) values together into Tsys(SUM).

SUM(i) = R(i) + S(I)

TSYS(SUM) = [ TSYS(SIG) + TSYS(REF) ] / 2

Since not much differs between the signal and reference cycles for
total-power observing, R and S and other quantities can be assumed to be
identical. Thus, I can define:

TPC(on) = TPC(ref_on) ~= TPC(sig_on)

TPC(off) = TPC(ref_off) ~= TPC(sig_off)

a(i) = a_sig(i) ~= a_ref(i)

TSYS = TSYS(SIG) ~= TSYS(REF)

Tsys = Tsys(SIG) ~= Tsys(REF)

T(i) = Tsig(i) ~= Tref(i)

RS(i) = Ref(i) ~= Sig(i)

For total-power observing, one takes on-source and off-source spectra. Thus,
there will be an on_source and off_source versions of TPC, TSYS, Tsys, T, and
RS. One combines the spectra with the analysis system using the TEMP verb in the
hope of recreating the difference between T_on_source(i) and T_off_source(i).
TEMP performs the following and produces the final data D(i):

SUM_on_source(i) - SUM_off_source(i)

D(i) = ------------------------------------ * TSYS_on_source

SUM_off_source(i)

Using the above definitions, this becomes, after some substitutions:

2*RS_on_source(i) 2*RS_off_source(i)

----------------- - -----------------

TSYS_on_source TSYS_off_sourceD(i) = -------------------------------------------- * TSYS_on_source

2*RS_off_source(i)

-----------------

TSYS_off_source

or

RS_on_source

D(i) = ------------ * TSYS_off_source - TSYS_on_source

RS_off_source

or

RS_on_source - RS_off_source

D(i) = ---------------------------- * TSYS_off_source + TSYS_off_source - TSYS_on_source.

RS_off_source

Except for the last two terms (which amount to a DC offset to the data), the last equation is the one typically found in the text books for filter-bank spectrometers. That is, because of the normalization performed by the AC, the DC offset one would expect for the data when there is a difference in Tsys between on and off source doesn't exist like it would for a filter-bank spectrometer.

If there is a line only in the on-source measurement (i.e., T_off_source(i)
is zero), the above equation reduces to: D(i) = T_on_source(i) + constant, the
quantity we are after.

For switched-power observing, either the LO or subreflector are switched
between a signal and reference frequency or position synchronously with the
switching between the signal and reference phases of the AC.

The control computer passes S and R, TSYS(SIG) and TSYS(REF) to them analysis
computer. S and R are also stored separately on the telescope (archive) tape.
Before the analysis computer stores the data (D) to disk, it performs:

S(i) - R(i)

D(i) = ----------- * TSYS(Sig)

R(i)

TSYS = sqrt { [ TSYS(sig)**2 + TSYS(ref)**2 ] / 2 }

With a little rearranging, D(i) becomes:

Sig(i) - Ref(i)

D(i) = --------------- * TSYS(Ref) + TSYS(Ref) - TSYS(Sig)

Ref(i)

As in total-power observing, this is the classical filter-bank calibration equation except for a DC-offset. similarly, if T_ref(i) is zero, then the above equation becomes: D(i) = T_sig(i) + constant, the quantity we are after.

For in-band frequency switching, T_ref(i) is NOT zero but is: T_ref(i) =
T_sig(i+amount_of_switch) where amount_of_switch is the number of channels that
the LO is switching between. Usually, one picks the amount_of_switch to be
somewhat larger than the line width so at no I is both T_ref and T_sig non zero.
The RAP verb in the analysis system allows you to properly average T_ref(i) with
T_sig(i+amount_of_switch) so that you can increase a spectrum's signal to noise.
I'll only describe the Green Bank version of RAP since the 12-m version is
mathematically incorrect in a great number of ways.

First, RAP uses the data in the spectrum's header to come up with the
amount_of_switch. RAP rounds the amount of switch to an integer number of
channels (something it probably should NOT do and which I intend to fix
someday). It then calculates for each channel a quantity I'll call d(j):

d(j) = d(i+amount_of_switch)) = -TSYS(ref)*D(i) / [ TSYS(sig) + D(i) ]

In principle, d(j) is what D(i) would be if the roles of Sig and Ref are
reversed. RAP then averages D(j) with d(j).