Sideband Separation

The PdBI is able to detect and separate signals from both the Upper and Lower sidebands of the receivers. These two bands can be considered to define two different sets of k sky frequencies. Alternatively, we can also assume that k is an index to intermediate frequency. In this representation, Equation 4 is split in two similar equations for the Signal and Image band of the receiver, and becomes after adding superscript b (for Signal or Image band)

V^b_ijk = A^b_i (A^b_j)^* S^b_ik (S^b_jk)^* C_ijk R^b_ijk + O^b_ijk + N^b_ijk (9)

A simplification occurs in this representation because the correlator based error C_ijk is independent of b. For a small IF frequency (1.5 GHz) as compared to the observing frequency, the effects of pointing and focus on the antenna gain terms do not depend on the sideband b. The atmospheric transmission does depend on the sideband (specially at the edge of strong telluric lines like the Oxygen band), but the phase fluctuations are not very chromatic. For this purpose of atmospheric transmission, it is convenient to extract the receiver sideband gain ratio I_i(image over signal, averaged over the receiver bandwidth) from the antenna gain:

V^S_ijk = A_i A_j^* S^S_ik (S^S_jk)^* C_ijk R^S_ijk + N^S_ijk (10)

V^I_ijk = I_i I_j^* A_i A_j^* S^I_ik (S^I_jk)^* C_ijk R^I_ijk + N^I_ijk (11)

where superscripts S and I stand for Signal and Image bands respectively.

Provided the receiver is not retuned, and in the (very good) approximation of non chromatic atmospheric phase fluctuations, I_i is a constant in time. Small variations in the amplitude may happen because of imperfect atmospheric calibration.