# What type of correlation is this equation?

In a book , gives an equation (equation 5.31 in the book) for estimating frequency offset in OFDM systems proposed by Classen. the procedure is:

first, two OFDM symbols, $y_l [n]$ and $y_{l+D} [n]$ , are saved in a memory after synchronization. then, the signals are transformed into ${Y_l [k]}_{k=0}^{N-1}$ and ${Y_{l+D} [k]}_{k=0}^{N-1}$ via FFT, from which pilot tones are extracted. After estimating CFO from pilot tones in the frequency domain, the signal is compensated in the time domain.

Here is the equation:

$\hat \epsilon_{acq} = \dfrac{1}{2\pi \cdot T_{sub}} \underset{\epsilon}{max} \left \{ \left \vert \sum_{j=0}^{L-1} Y_{l+D}[p[j],\epsilon]\cdot Y_{l}^{*}[p[j],\epsilon]\cdot X_{l+D}^{*}[p[j]]\cdot X_l[p[j]] \right \vert \right \}$

where $L$, $p[j]$, and $X_{l}[p[j]]$ are the number of pilot tones, the location of the jth pilot tone, and the pilot tone located at $p[j]$ in the frequency domain at the $l^{th}$ symbol period, respectively.

From the equation, It's supposed to find the normalized CFO, $\epsilon$. but, the parameter we are finding is involved in the equation $[P[j],ε]$. My question is: how this parameter we're supposed to estimate is in the middle of the equation? What type of correlation is this? Or did I misunderstand the equation? Please explain.

•  Yong Soo Cho et al., "MIMO-OFDM wireless communications with MATLAB", John Wiley & Sons (asia), 2010
• Well, the equation says: evaluate the right hand side for every $\epsilon$ and choose the $\epsilon$ that yields the maximum value. Without having the book at hand it's hard to understand the equation. Can you provide some further information? What is $Y,L,D,X,p,\ldots$?
– Deve
Apr 23, 2014 at 11:37
• Please, see now. you may also get the book from here dropbox.com/s/8azwy1vvlfsp4p8/…
– Hamd
Apr 23, 2014 at 13:44

I understand your confusion because the equation is barely understandable from the information given in the book. It becomes more clear from the original paper by Classen and Meyr  from which it has been taken. They propose a two stage frequency offset estimation that consists of an acquistion stage and a tracking stage. The equation you've cited represents the acquisition algorithm. In fact, it is a searching algorithm that "tries out" various values of frequency offset $\epsilon$ and chooses the one value $\hat\epsilon_\mathrm{acq}$ that yields the maximum correlation sum. It can be described as follows

For every $\epsilon$

• Compensate the received time domain signal for the supposed frequency offset $\epsilon$
• Calculate the DFT of the frequency offset compensated RX signal. This yields $Y_{l}[p[j],\epsilon]$
• Calculate the correlation sum $R_\epsilon$. (The term inside $|\cdots|$)

Then, find the maximum $\hat R_\epsilon$ of all $R_\epsilon$ and calculate the estimated frequency offset by $$\hat \epsilon_{acq} = \dfrac{\hat R_\epsilon}{2\pi \cdot T_{sub}}$$

Of course it's not feasible to evaluate the correlation sum for every $\epsilon$ because the number of possbile values is infinite. Instead, you have to restrict the range of values to a reasonable frequency offset that is determined by the hardware you're using. Additionaly, this "search range" has to be partitioned into discrete values. The authors of  note

In practice we found that it is sufficient to space the trial parameters $0.1/T_{sub}$ apart from each other.

Where $T_\mathrm{sub}$ is the length of one OFDM symbol including guard interval.

 Classen, F. and Myer, H. (June 1994) Frequency synchronization algorithm for OFDM systems suitable for communication over frequency selective fading channels. IEEE VTC’94, pp. 1655 1659.