Basic Equations

The LO2 frequency used to track a spectral line at a given frequency Frest centered in the IF band (350 $MHz$) is

$$Flo2 = \frac{ Frest \times Doppler + S \times 350 + M \times L \times Eps } { M \times H + S }$$ (6)

and, from eq. (5), the image rest frequency at the band center is

$$F_{I} = \frac{ (M \times H-S) \times Flo2 - M \times L \times Eps + S \times 350} {Doppler}$$ (7)

A little algebra follows

$F_{I} = \frac{ \frac{M \times H-S}{M \times H+S} \times ( Frest \times Dopple... ...0 + M \times L \times Eps ) - M \times L \times Eps + S \times 350} {Doppler}$

$F_{I} = \frac{ (M \times H-S) \times ( Frest \times Doppler + S \times 350 + ... ...times (S \times 350 - M \times L \times Eps) } {(M \times H+S)\times Doppler}$

$F_{I} = \frac{M \times H-S}{M \times H+S} \times Frest + \frac{ (M\times H-S+... ...- M \times H+S) \times M \times L \times Eps } {(M \times H+S)\times Doppler}$

$F_{I} = \frac{M \times H-S}{M \times H+S} \times Frest + \frac{ 2 \times M \t... ...0 - 2 \times S \times M \times L \times Eps } {(M \times H+S) \times Doppler}$

cm and result in

$$F_{I} = \frac{M \times H-S}{M \times H+S} \times Frest + \... ...H \times 350 - L \times Eps)} {(M \times H+S) \times Doppler}$$ (8)

This result is not independent of the Doppler effect. Accordingly, it the doppler tracking is not exact for the image frequency. and for 2 different values, $D1$ and $D2$, corresponding to different velocities $V1$ and $V2$, we obtain (assuming no change of $H$)

$$\delta F_{I} = \frac{D2-D1}{ D2 \times D1} \times \frac{ ... ...imes M \times ( H \times 350 - L \times Eps) } {M \times H+S}$$ (9)

or, assuming $V1$ and $V2 << c$, and with $H >> 1$, $M > 1$, the frequency shift in MHz is

$$\delta F_{I} = \frac{\delta V}{c} \times 700$$ (10)

or, in velocity (in km.s$^{-1}$)

$$\delta V_{I} = \delta V \frac{700}{F_I}$$ (11)