Assume that the rotating frame frequency is no and that the signal in the laboratory frame of reference is at n. The input signal to the quadrature detector, and hence the two DBMs, is

Cos(2pnt).

The input reference to the quadrature detector is

Cos(2pnot).

The inputs to the two DBMs within the quadrature detector become

Cos(2pnot) and Sin(2pnot)

since one of the DBMs has a 90o phase shifter before it.

According to the trigonometric identities introduced in Chapter 2, the outputs from the two DBMs in the quadratre detector are

Cos(2pnt) Cos(2pnot) = 1/2 Cos(2pnt - 2pnot) + 1/2 Cos(2pnt + 2pnot)

Sin(2pnt) Cos(2pnot) = 1/2 Sin(2pnt + 2pnot) + 1/2 Sin(2pnt - 2pnot)

The filters remove the high frequency components, leaving

RE = 1/2 Cos(2pnt - 2pnot)

IM = 1/2 Sin(2pnt - 2pnot)

Note that one output of the quadrature detector gives the cosine and the other the sine of the rotating frame frequencies.