From Real to Complex to Vector Gaussians

Gaussian distributions are probably the most widely used distributions in mathematics and engineering. Though one strange aspect of them is that they have slightly different equations for the real and complex cases. This is an interesting problem, and has a really nice reasoning, which generally is not taught in classes. In an attempt to answer the why here goes…

All of us have definitely at some point of time seen this familiar real Gaussian distribution
p_x(x) = \frac{1}{\sqrt{2\pi \sigma^2}} e^{-\frac{x^2}{2\sigma^2}}

which changes to a real vector form as
p_x(\mathbf{x}) = \frac{1}{\sqrt{(2\pi)^N |\mathbf{C_x}|}} e^{-\frac{1}{2}\mathbf{x}^T \mathbf{C_x}^{-1}\mathbf{x}}

where x is a vector of Gaussian distributions x_1, x_2, \ldots, x_N with 0 mean and covariance matrix \mathbf{C_x}. This can be written directly under the assumption that the individual x_i are uncorrelated, and since uncorrelated gaussians are independent, a multiplication of the individual gaussian distributions can be carried out.

Further, there is a change when referring to complex distributions. This is due to the difference in the definition of the Covariance matrix itself, which can be now written as a covariance matrix of the real and imaginary parts \mathbf{C_z} or the covariance matrix generated by using the complex number directly as \mathbf{C_s} where \mathbf{z} = [\mathbf{x y}]^T and \mathbf{u} =  [\mathbf{u u^*}]^T are both column vectors of 2N size.

This gives a relation between \mathbf{C_z} = \frac{1}{2} \mathbf{T}^H \mathbf{C_s} \mathbf{T} where \mathbf{T} is a 2-D unitary matrix. This half factor is responsible for the disappearance of the 2 in the fraction, and |\mathbf{C_s}| = |\mathbf{C_u}|^2 is responsible for the removal of the square root. Thus the final Complex Guassian Multivariate distribution ends up (different from the real one) as

p_u(\mathbf{u}) = \frac{1}{(\pi)^N |\mathbf{C_u}|} e^{-\mathbf{u}^T \mathbf{C_u}^{-1}\mathbf{u}}

when \mathbf{u} is circulant complex random variable, i.e. the covariance of real and imaginary parts is same, and they are uncorrelated (generally satisfied in applications).

One last point, its important to note that independence implies uncorrelation, but uncorrelation need not imply independence except for the nice and widely used Gaussian case 🙂

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