What Is the PDF of a Circularly Symmetric Complex Gaussian Vector?

[EngWiPy]

Hi,

Suppose that an n-dimensional vector $$\mathbf{z}=\begin{pmatrix}z_1&z_2&\cdots & z_n\end{pmatrix}^T$$ is characterized as a zero-mean circularly symmetric complex Gaussian random vector. What is the distribution (the probability distribution function PDF) of this vector in both: complex and real representations?

Thanks in advance

 

[Stephen Tashi]

What is the meaning of "circularly symmetric" in n dimensions ? Do you mean "spherically symmetric" in the sense that $|z_1|^2 + |z_2|^2 + ...|z_n|^2 = c$ is a surface where the PDF is constant? And can we also assume that $$|RE(z_1)|^2 + |IM( z_1)|^2 + |RE(z_2)|^ + |IM( z_2)|^2 + ... |RE( z_n)|^2 + |IM (z_n)|^2 = c$$ is a surface where the PDF is constant?

Is $|z|$ the quantity that has a gaussian distribution?

 

[EngWiPy]

What is the meaning of "circularly symmetric" in n dimensions ? Do you mean "spherically symmetric" in the sense that $|z_1|^2 + |z_2|^2 + ...|z_n|^2 = c$ is a surface where the PDF is constant? And can we also assume that $$|RE(z_1)|^2 + |IM( z_1)|^2 + |RE(z_2)|^ + |IM( z_2)|^2 + ... |RE( z_n)|^2 + |IM (z_n)|^2 = c$$ is a surface where the PDF is constant?

Is $|z|$ the quantity that has a gaussian distribution?

A complex random vector $$\mathbf{x}\in C^n$$ is said to be Gaussian, if the real vector $$\mathbf{\hat{x}}\in R^{2n}$$ consisting of the real and imaginary parts of $$\mathbf{x}$$ as $$\mathbf{\hat{x}}=\begin{pmatrix}\text{Re}\{\mathbf{x}\}&\text{Im}\{\mathbf{x}\}\end{pmatrix}^T$$ is Gaussian.

A complex Gaussian random vector $$\mathbf{x}$$ is said to be circularly symmetric if the covariance of the corresponding $$\mathbf{\hat{x}}$$ has the following structure:

$$E(\left(\mathbf{\hat{x}-\mu}\right)\left(\mathbf{\hat{x}-\mu}\right)^H)=\frac{1}{2}\begin{pmatrix}\text{Re}(Q)&-\text{Im}(Q)\\\text{Im}(Q)&\text{Re}(Q)\end{pmatrix}$$

where $$E(\mathbf{\hat{x}})=\mu$$ and $$Q$$ is some non negative matrix.

 

[Stephen Tashi]

If I take a Gaussian distribution of the components of a vector to mean a multivariate Gaussian distribution and take the covariance matrix as given, is the question "What is the PDF of a multivariate Gaussian distribution?". I guess I still don't understand the question.

 

[EngWiPy]

If I take a Gaussian distribution of the components of a vector to mean a multivariate Gaussian distribution and take the covariance matrix as given, is the question "What is the PDF of a multivariate Gaussian distribution?". I guess I still don't understand the question.

That is right, but for complex Gaussian. Actually, I got the result, which is:

$$f_{\mathbf{z}}(\mathbf{z})=\frac{1}{\pi^n\text{det}(\mathbf{R}_z)}\text{exp}\left(-(\mathbf{z}-\overline{\mathbf{z}})^H\mathbf{R}_z^{-1}(\mathbf{z}-\overline{\mathbf{z}})\right)$$

Now the problem with me was that, I read in some paper that the distribution is given by:

$$f_{\mathbf{z}}(\mathbf{z})=\frac{1}{\text{det}(\pi\mathbf{R}_z)}\text{exp}\left(-(\mathbf{z}-\overline{\mathbf{z}})^H\mathbf{R}_z^{-1}(\mathbf{z}-\overline{\mathbf{z}})\right)$$

But knowing that:

$$\text{det}(cA)=c^n\text{det}(A)$$

solved the confusion.

Thanks

 

[EngWiPy]

Ok, now what if Z is a circularly symmetric complex Gaussian matrix not vector, what then the PDF of Z?

 

[EngWiPy]

Any suggestion?

 

[Stephen Tashi]

From the appropriate point of view, matrices are vectors. What property would an nxn "circularly symmetric" matrix have that an n^2 dimensional circularly symmetric vector wouldn't?

 

[EngWiPy]

From the appropriate point of view, matrices are vectors. What property would an nxn "circularly symmetric" matrix have that an n^2 dimensional circularly symmetric vector wouldn't?

So, you are saying it is just like the vector case. But I have a formula in matrix form, and I am not sure how the authors got there. I mean it is like the following:

$$\frac{1}{\pi^{2NK}\text{det}^KQ}\text{exp}\left\{-\|Q^{-1/2}(Y-HX)\|_F^2\right\}$$

where Y is 2N-by-K matrix, Q is 2N-by-2N, H is 2N-by-M, and X is M-by-K. Any hint in this?