I saw a couple papers by Fred Harris on band edge detectors for carrier frequency correction, in all of them it is said the band-edge detector can be derived from the maximum likelihood principle but none of the papers actually go about showing it.
I was wondering if anyone here could give further insight or link to a paper which could give a closer mathematical examination of what's going on with regards to Maximum Liklihood?
I think you posted this around the time dsprelated.com suffered a host outage. It may have gone unnoticed by some.
What do you recommend? Should I repost it? Also, I'm a bit confused on how you can use the spectral line appearing at the symbol frequency for timing synchronization.
No need to repost. You might want to look at Michael Rice's book, Chapter 8.
Thanks, that answered some general question I had about maximum likelihood but the questions on the band edge detector still persist.
As I understand, the idea is to use two band edge filters (one mirroring the other's frequency response and each centred on band edge), compute power output from each and use that to converge on dc centre.
It might be useful for initial coarse frequency estimate before a fine stage to correctly track frequency and phase.
My view is that for all practical purposes it is too expensive for its purpose and not needed as most current carrier tracking methods do not require coarse initial adjustment anyway since prior knowledge of tx frequency is available.
In a few steps, it is derived as follows.
1. Maximum likelihood implies taking the derivative w.r.t. the CFO (Carrier Frequency Offset).
2. Through some simplifications, it comes down to differentiating the Nyquist pulse frequency response which is constant in the middle and has a quarter cycle of a cosine at band edges.
3. The result of the differentiation implies discontinuous sine lobes at both edges.
4. Continue the sine shape to avoid the discontinuity and that leads to band edge filters.
If you are looking for a reference, I have derived in quite some detail the band edge detectors in chapters 6 (carrier frequency synchronization) and 7 (timing synchronization) of my book below. These are not exactly from maximum likelihood principle but from a correlation principle, but the ideas are almost the same.
Hi, sorry I didn't respond before as classes were sort of a problem.
I think I understand now, I re-read the paper by harris, as you pointed out ML = maximizing correlator though his paper implied a non-overlapping pulse shape, a proper derivation would average over the symbols.
I don't really understand why the need to continue the sine shape - you should be able to apply IDTFT and generate an impulse response which approximates it. The sine continuation probably requires fewer coefficients but why is it needed?
That is correct but the frequency domain derivative is discontinuous at the edge and taking an iDFT would produce an impracticably long impulse response.
Thanks, that makes sense. As a second question - is there an easy explanation of why the same band edge detector works for timing synchronization when you look at the imaginary output? I am reading this paper:
The frequency synchronization procedure is pretty clear now but he also points out that the same system can be used for timing recovery. I understand the square-law timing recovery procedure but I don't immediately see if this is an extension of that.
For this, you have to differentiate between the mechanisms of positive/negative band edge filters versus even/odd band edge filters. If you consider even and odd band edge filters, then their conjugate product is an analog of squaring the signal for spectral line generation. But going further, the odd band edge filter is specifically designed to help in timing estimation.
I think I understand the algebraic operations to get the correct result with the expectation operator but I wish the paper would have done it. I think it follows the same formulation as the standard square-law derivation after you have passed the results through the LTI band edge filters and used the fact the symbols are uncorrelated with mean 0. You get a periodic summation of the pulse squared which you expand into a fourier series, but since it's a periodic summation it's actually the fourier transform and you can show only the first two frequencies expand(DC and symbol interval). I haven't done all the math so maybe this isn't how it goes.
Also the band-edge carrier frequency recovery method I think only really works for pulses whos frequencies occur at the same time or sums of pulses where this is the case. You could pretty easily break the system by having a pulse whos negative frequency component was substantially delayed from the positive frequency component leading to an error term which continually oscillates even if the systems are operating at the same carrier frequency. I guess this could be dampened by the loop filter but it still seems like a problem.