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Problem

Why, when assembling an I/Q modulator or demodulator using MIXER_B/MIXER_F, is a phase shift of +90 degrees applied to the LO signal rather than -90 degrees?

Solution

In an ideal I/Q modulator or demodulator, the I channel is mixed with cos(2*π*fLO*t), while the Q channel is mixed with -sin(2*π* fLO*t):

Using the identity:

cos(fLO) = cos(s)·cos(t) - sin(s)·sin(t)

We can relate -sin(2*π*fLO*t) to cos(2*π*fLO*t) by:

cos(fLO + π/2) = cos(fLO)·cos(π/2) - sin(fLO)·sin(π/2) = -sin(fLO)

Therefore, applying a 90 degree phase shift to cos(2*π*fLO*t) is identical to -sin(2*π*fLO*t).