Digital Mixer implementation for DUC/DDC

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