Basic Equations

For the Plateau de Bure interferometer, the conversion from sky frequencies down to the IF band (100-600 MHz) is done using two local oscillators. The first frequency conversion (with the LO1 oscillator) is double sideband from sky frequency down to 1.5 GHz. The second frequency is a lower sideband conversion with a second oscillator LO2 around 1.875 GHz. In this document, I will use the following frequencies :

• Fsyn the synthesizer frequency
• Flo2 the second LO frequency
• Flo1 the first LO frequency
• Fsky the sky frequency corresponding to the IF band center
• Fcent a frequency in the IF band (100-600 $MHz$)
• Frest the corresponding rest frequency at the source

All frequencies are derived from Fsyn, with frequency multipliers to generate the relevant harmonic numbers, according to the following equations (with frequencies in MHz)

$$Flo2 = Fsyn - 0.5$$ (1)

$$Flo1 = M \times (H \times Flo2 - L \times Eps)$$ (2)

$$Fsky = Flo1 + S \times (Flo2-350)$$ (3)

$$Frest = \frac{Fsky - S \times (Fcent-350)}{Doppler}$$ (4)

$$Frest = \frac{ ( (M \times H+S) \times Flo2 - M \times L \times Eps ) - S \times Fcent}{Doppler}$$ (5)

Five parameters introduced in these equations, M, H, L, S, and Eps, depend on the phase lock and receiver system, while Doppler depends on the source:

• H is the harmonic number used to generate the LO1 from the LO2. With Flo2 near 1.875 GHz, H is roughly 50 at 95 GHz.
• M is the multiplier order: 1 for the 3mm system, 3 for the 1.3mm system.
• L is +1 if the LO1 is in ``HIGH lock'', -1 in ``LOW lock''
• Eps is equal to (100 + 50/512)
• S is +1 for the upper sideband of the receiver (USB), -1 for the lower sideband (LSB).
• Doppler is the total Doppler correction $(1-v/c)$ for the current source.

The phase lock system nominal performances are obtained for Flo2 in the range 1.850 - 1.900 GHz.