# Carrier and timing offset

Assume my reference is 1 ppm so if my transmit frequency is 400 MHz and symbolrate is 100 kHz.

Then, my carrier will have +/- 400 Hz offset and then my symbol will have +/-0.001 Hz offset. Similarly, the reciever carrier has +/- 300 Hz offset and symbol has +/- 0.002 Hz offset.

So is the carrier recovery is to track 400 + 300 Hz and timing recovery is 0.001 +0.002 Hz ?

• I follow and agree with a +/-400 Hz carrier offset error, worst case given your reference is with +/-1ppm or better, but I get +/-0.1 Hz offset for your symbol clock- how did you compute 0.001 Hz (I could easily be in error)? Also it is not clear from your description where the 300 Hz comes from? Are you saying the transmitter is within +/-1 ppm, and the receiver is as well? If you are transmitting 400 MHz why would you not be receiving that same 400 MHz carrier? – Dan Boschen Dec 18 '19 at 2:10

If you need to guarantee operation within these ranges, then yes. (Assuming the 0.001 Hz is a typo and meant 0.1 Hz.)

# Stochastic Modelling

That "1ppm" is typically a "condensed" number, e.g. the observed maximum over 24h of oscillations, or can be found as integral of phase noise power, which is often given in an oscillator's datasheet. Typically, small frequency errors are more likely than large ones.

It might make sense to model your system probabilistically: Your frequency errors in receiver and transmitter, $$X_{RX}$$ and $$X_{TX}$$ are independent and will have a probability density function $$f_{X_{RX}}(x)$$ and $$f_{X_{TX}}(x)$$, respectively.

So, the probability density function for the difference $$\Delta=X_{TX}-X_{RX}$$ is given by $$f_\Delta(\delta) = f_{X_{TX}}(x)*f_{X_{TX}}(-x)$$, $$*$$ being the convolution operator.

This will allow you to define something like "in x % of cases, it should work, so I only need to account for a range of at most this and that in frequency error".

For example: This makes a lot of sense, since frequency synchronization is not the only thing that can fail in a receiver; if your channel has a 0.1% chance of simply being too noisy, then building a PLL loop filter that is wide enough to work for 99.99996% of cases doesn't make much sense, and you should probably rather reduce your loop filter bandwidth to make the receiver's phase recovery less sensitive to noise.

• Hi Marcus, it is difficult to understand in depth analysis. But i just want to know is my understanding of carrier and timing offset right ? Nancy – nancy Dec 18 '19 at 0:24
• @MarcusMuller interesting perspective, I would be interested in your feedback on how I have approached this (choice of loop bandwidth in regards to phase noise) at this post: dsp.stackexchange.com/questions/31170/… – Dan Boschen Dec 18 '19 at 2:14