Let’s look at another spectral correlation function estimator: the FFT Accumulation Method (FAM). This estimator is in the time-smoothing category, is exhaustive in that it is designed to compute estimates of the spectral correlation function over its entire principal domain, and is efficient, so that it is a competitor to the Strip Spectral Correlation Analyzer (SSCA) method. I implemented my version of the FAM by using the paper by Roberts et al (The Literature [R4]). If you follow the equations closely, you can successfully implement the estimator from that paper. The tricky part, as with the SSCA, is correctly associating the outputs of the coded equations to their proper 
spectral coherence
Computational Costs for Spectral Correlation Estimators
Let’s look at the computational costs for spectral-correlation analysis using the three main estimators I’ve previously described on the CSP Blog: the frequency-smoothing method (FSM), the time-smoothing method (TSM), and the strip spectral correlation analyzer (SSCA).
We’ll see that the FSM and TSM are the low-cost options when estimating the spectral correlation function for a few cycle frequencies and that the SSCA is the low-cost option when estimating the spectral correlation function for many cycle frequencies. That is, the TSM and FSM are good options for directed analysis using prior information (values of cycle frequencies) and the SSCA is a good option for exhaustive blind analysis, for which there is no prior information available.
CSP Estimators: The Strip Spectral Correlation Analyzer
In this post I present a very useful blind cycle-frequency estimator known in the literature as the strip spectral correlation analyzer (SSCA) (The Literature [R3-R5]). We’ve covered the basics of the frequency-smoothing method (FSM) and the time-smoothing method (TSM) of estimating the spectral correlation function (SCF) in previous posts. The TSM and FSM are efficient estimators of the SCF when it is desired to estimate it for one or a few cycle frequencies (CFs). The SSCA, on the other hand, is efficient when we want to estimate the SCF for all CFs.
See also a competing method of exhaustive SCF estimation: The FFT Accumulation Method.
The Spectral Coherence Function
In this post I introduce the spectral coherence function, or just coherence. It deserves its own post because the coherence is a useful detection statistic for blindly determining significant cycle frequencies of arbitrary data records.
Let’s start with reviewing the standard correlation coefficient 
![\rho = \displaystyle \frac{E[(X - m_X)(Y - m_Y)]}{\sigma_X \sigma_Y}, \hfill (1)](https://s0.wp.com/latex.php?latex=%5Crho+%3D+%5Cdisplaystyle+%5Cfrac%7BE%5B%28X+-+m_X%29%28Y+-+m_Y%29%5D%7D%7B%5Csigma_X+%5Csigma_Y%7D%2C+%5Chfill+%281%29&bg=e9e7e3&fg=6a524a&s=0 "\rho = \displaystyle \frac{E[(X - m_X)(Y - m_Y)]}{\sigma_X \sigma_Y}, \hfill (1)")
where 
![m_X = E[X] \hfill (2)](https://s0.wp.com/latex.php?latex=m_X+%3D+E%5BX%5D+%5Chfill+%282%29&bg=e9e7e3&fg=6a524a&s=0 "m_X = E[X] \hfill (2)")
![m_Y = E[Y] \hfill (3)](https://s0.wp.com/latex.php?latex=m_Y+%3D+E%5BY%5D+%5Chfill+%283%29&bg=e9e7e3&fg=6a524a&s=0 "m_Y = E[Y] \hfill (3)")
![\sigma_X^2 = E[(X-m_X)^2] \hfill (4)](https://s0.wp.com/latex.php?latex=%5Csigma_X%5E2+%3D+E%5B%28X-m_X%29%5E2%5D+%5Chfill+%284%29&bg=e9e7e3&fg=6a524a&s=0 "\sigma_X^2 = E[(X-m_X)^2] \hfill (4)")
![\sigma_Y^2 = E[(Y-m_Y)^2] \hfill (5)](https://s0.wp.com/latex.php?latex=%5Csigma_Y%5E2+%3D+E%5B%28Y-m_Y%29%5E2%5D+%5Chfill+%285%29&bg=e9e7e3&fg=6a524a&s=0 "\sigma_Y^2 = E[(Y-m_Y)^2] \hfill (5)")
So the correlation coefficient is the covariance between
