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 defined for two random variables X and Y as

\rho = \displaystyle \frac{E[(X - m_X)(Y - m_Y)]}{\sigma_X \sigma_Y}, \hfill (1)

where m_X and m_Y are the mean values of X and Y, and \sigma_X and \sigma_Y are the standard deviations of X and Y. That is,

m_X = E[X] \hfill (2)

m_Y = E[Y] \hfill (3)

\sigma_X^2 = E[(X-m_X)^2] \hfill (4)

\sigma_Y^2 = E[(Y-m_Y)^2] \hfill (5)

So the correlation coefficient is the covariance between X and Y divided by the geometric mean of the variances of X and Y.

Now consider the spectral correlation function,

S_x^\alpha(f) = \displaystyle \lim_{T\rightarrow\infty} \frac{1}{T} E[ X_T(t, f+\alpha/2) X_T^* (t, f-\alpha/2)], \hfill (6)

where the expectation operator E[\cdot] can be either a time average or the ensemble average used in a stochastic-process formulation. For each finite T, we can consider the quantity

\displaystyle \frac{1}{T} E[X_T(t, f+\alpha/2) X_T^* (t, f-\alpha/2)], \hfill (7)

which is the correlation between the two random variables \displaystyle \frac{1}{T^{1/2}} X_T(t, f+\alpha/2) and \displaystyle \frac{1}{T^{1/2}} X_T(t, f-\alpha/2). To form the correlation coefficient, we need the variances for these two random variables,

\sigma_1^2(T) = E[\displaystyle\frac{1}{T} \left| X_T(t, f+\alpha/2) \right|^2] \hfill (8)

and

\sigma_2^2(T) = E[\displaystyle\frac{1}{T} \left| X_T(t, f-\alpha/2) \right|^2]. \hfill (9)

Here we are assuming that the two variables have zero means, which is true when there is no additive sine-wave component in the data with frequency f\pm \alpha/2.

For finite T, then, we can form the correlation coefficient

\rho(T) = \displaystyle \frac{\frac{1}{T} E[X_T(t, f+\alpha/2) X_T^* (t, f-\alpha/2)]}{[\sigma_1^2(T) \sigma_2^2(T)]^{1/2}}. \hfill (10)

Now, as T\rightarrow \infty, the numerator of \rho(T) converges to the spectral correlation function S_x^\alpha(f) and

\displaystyle \lim_{T\rightarrow\infty} \sigma_1^2(T) = S_x^0(f+\alpha/2) \hfill (11)

and

\displaystyle \lim_{T\rightarrow\infty} \sigma_2^2(T) = S_x^0(f-\alpha/2). \hfill (12)

If the limit of the quotient exists, then the correlation coefficient is given by

\rho = C_x^\alpha(f) = \displaystyle\frac{S_x^\alpha(f)}{[S_x^0(f+\alpha/2)S_x^0(f-\alpha/2)]^{1/2}}. \hfill (13)

We call this function the spectral coherence. A similar argument holds for the conjugate spectral coherence,

C_{x^*}^\alpha(f) = \displaystyle\frac{S_{x^*}^\alpha(f)}{[S_x^0(f+\alpha/2)S_x^0(\alpha/2-f)]^{1/2}}. \hfill (14)

Since the coherence is a valid correlation coefficient, its magnitude will be less than or equal to one, and since the involved random variables are complex-valued, in general, so is the spectral correlation function and the coherence. Therefore, the coherence lies in the closed unit disk in the complex plane.

Here is an FSM-based estimate of the coherence for our rectangular-pulse BPSK signal:

ww_coh

The data-block length is 65536 samples and the frequency resolution (width of g(f) in the FSM) is set to 0.01 (one percent of the sampling rate, which here is 1.0) for both the numerator spectral correlation function and the denominator PSDs.

In practice, of course, the numerator and denominator of the coherence are estimates corresponding to finite-duration data blocks. The tricky part of estimating the coherence involves good selection of the estimator for the denominator PSDs S_x^0(f\pm\alpha/2). If the PSDs are not well resolved, or contain zeros, the coherence quotient can be numerically unstable or erroneous.

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