[Discuss-gnuradio] Basic question about carrier synchronize in PM demodulation

[adream]

Hello everyone, let me explain my question:
Let the input signal,Fc=cos(2*pi*fc*t+a(t)),fc is the carrier's frequency,and a(t) is the information which modulated into the carrier.
But the SDR(software define radio)doesn't processes the input signal Fc directly, it just deal with the I/Q signal.
Multiplying the input signal Fc by cosine and sine wave, and putting result into low pass filter, the output of this low pass filter are I/Q signals. The frequency of the cosine and sine wave is fs. Usually,fs should equal to carrier's frequency fc.
So the I/Q signals can be expressed as follows:
I signal SI=1/2*cos(2*pi*(fc-fs)*t+a(t));
Q signal SQ=-1/2*sin(2*pi*(fc-fs)*t+a(t));
If the fs is equal to fc, then fc-fs is equal to 0, and SI and SQ just contain the modulated information a(t).
But in the real time project, fs isn't equal to fc, there may be little difference.
Older oscillator and doppler effective will cause the fs having little difference with fc.
so my question is that, how to remove the errors in SI and SQ which called by fs-fc not equal to zeros, and get SI and SQ as follow:
I signal SI=1/2*cos(a(t));
Q signal SQ=-1/2*sin(a(t));
thanks.