Maximum possible CFO that can be estimated

Question

I was trying to model the CFO (Carrier Frequency Offset) estimation in MATLAB. The Frame I was working on is of the following structure, yellow being the information data.

How CFO was introduced?

CFO of deltaF was introduced by up converting (baseband to passband) the frame to Fc+deltaF and then down converting (passband to baseband) to -Fc

What I did in Receiver Side?

Extracted the content from the same position of the Reference data in the received Frame.

Correlated between the known Reference data and the extracted content, CFO value deltaF was successfully estimated.

All is good!!

My question is now what is the max limit of CFO that this frame can estimate, because when i increased the value of introduced CFO, the estimation was going wrong after some value. What is the limit of this max possible CFO that can be estimated. I am assuming that will be dependent on the following and more

• Number of samples in these two Reference Data,
• Separation between them.
• Sampling rate.

Thanks in advance.

Answer 1

The CFO is estimated specifically from the unwrapped phase difference between two reference samples in time as frequency is the derivative of phase and a difference is the estimate of the derivative. Or more simply, a change in phase versus a change in time is the frequency. See this post for more intuitive details on measuring frequency by two samples delayed in time.

That said, the maximum unambiguous phase difference is $\pm \pi$ from which we can determine the maximum frequency offset in radians/sec given by $\pm \pi/T$ where $T$ is the time difference in seconds.

For this we must know the expected phase at each reference symbol under the condition of no offset. For the case of the reference being multiple symbols, we correlate each with the reference pattern and extract the phase from the complex correlation result.

Note too for correlation the magnitude of the correlation vs frequency offset has a Sinc function response with first null at $f=1/T$, where $T$ is the duration of the correlation sequence. We see this with OFDM and demonstrates why subcarriers are orthogonal. See this post for further intuitive details as to the relationship between correlation and frequency offset.

Also see this link further explaining how correlation would decrease vs frequency offset intuitively:

Derivation of the Optimal Matched Filter - Convolution vs. Correlation