# Low data rate with high frequency offset

I am trying to implement a basic digital modulation scheme (BPSK) with a low data rate of 5 kbps. The requirement is to support a frequency offset of +/- 10 KHz. I would like to know what are the criteria to decide how much offset can be corrected? My burst data header is only 50 bits. I want to know if it is possible to track the frequency offset and also do the timing recovery (gardner) in the header bits.

The answer is yes it is possible and any amount of frequency offset can be corrected to the extent your sampling rate is sufficiently high enough for unambiguous frequency resolution over ±10 KHz. A very simple frequency discriminator that can be used in a carrier tracking loop is further described at this post:Carrier frequency offset estimation for QPSK and asymmetrical spectrum and results in determining the frequency offset simply from the product of two successive (complex) samples given as $$I_1Q_2-I_2Q_1$$. For BPSK you can also remove the modulation from the frequency error result by first squaring the signal, in which case the result will be twice the actual frequency offset; which is of no consequence since the loop will drive the offset to 0 in any event).