Digital mixer implementation for DDC/DUC

I am trying to implement digital up/down converter in which the implementation flow is

DUC

Interpolation->Digital mixer

DDC

Digital mixer->Decimation

In DUC(digital mixer) the input is I AND Q symbols then same I,Q are recovered at DDC digital mixer I want to find out complete equations of Mixer so that i can easily implement it

According to my understanding in DUC digital mixer the equation is Transmit_signal=I*Cos(wt)-Qsin(wt);

And in the receiver side the I and Q is recovered through following sequence

Reciver_I=Transmit_signal*Cos(wt); Reciver_Q=Transmit_signal*sin(wt);

Reciver_I_fil=lowpassfilter(Reciver_I); Reciver_Q_fil=lowpassfilter(Reciver_Q);

but i cannot be able to recover the exact I and Q symobls at the receiver side please help to find the exact equations