# Correlation of Complex Random Variables

• EngWiPy
In summary, correlation is a measure of how much two variables change together. For an autocorrelation, x and y are the same except for their time displacement, which doesn't affect the variance. So, 0.5 is just a normalization factor.
EngWiPy
Hi,

Why there is a half factor in the definition of the correlation of complex random variables, like:

$$\phi_{zz}(\tau)=\frac{1}{2}\mathbf{E}\left[z^*(t+\tau)z(t)\right]$$?

S_David said:
Hi,

Why there is a half factor in the definition of the correlation of complex random variables, like:

$$\phi_{zz}(\tau)=\frac{1}{2}\mathbf{E}\left[z^*(t+\tau)z(t)\right]$$?

I don't think that's true as a general rule. For the example you give, an autocorrelation, the general formula would be

$$\rho_{zz}(\tau)=\frac{\mathbf{E}\left[z^*(t+\tau)z(t)\right]}{\mathbf{E}\left[z^*(t)z(t)\right]}$$

I'm guessing that in your case, 1/2 is just the normalization factor 1/E[z*z], perhaps because the real and imaginary parts of z are independent with mean square 1.

pmsrw3 said:
I don't think that's true as a general rule. For the example you give, an autocorrelation, the general formula would be

$$\rho_{zz}(\tau)=\frac{\mathbf{E}\left[z^*(t+\tau)z(t)\right]}{\mathbf{E}\left[z^*(t)z(t)\right]}$$

I'm guessing that in your case, 1/2 is just the normalization factor 1/E[z*z], perhaps because the real and imaginary parts of z are independent with mean square 1.

does this general formula apply to the real-valued case, too?

S_David said:
does this general formula apply to the real-valued case, too?
Yes.

The general formula for a correlation is $$\frac{Cov(x,y)}{\sqrt{Var(x)Var(y)}}$$. In the case of an autocorrelation, x, and y are the same (except displaced in time, which doesn't affect the variance), so the denominator reduces to Var(x) = E[x^2].

pmsrw3 said:
Yes.

The general formula for a correlation is $$\frac{Cov(x,y)}{\sqrt{Var(x)Var(y)}}$$. In the case of an autocorrelation, x, and y are the same (except displaced in time, which doesn't affect the variance), so the denominator reduces to Var(x) = E[x^2].

So, 0.5 is just a normalization factor. Ok thanks a lot.

Regards

## What is the definition of correlation of complex random variables?

The correlation of complex random variables is a statistical measure that describes the relationship between two or more complex-valued random variables. It measures the strength and direction of the linear relationship between these variables.

## How is correlation of complex random variables calculated?

The correlation of complex random variables is calculated using the complex-valued covariance and standard deviations of the variables. The formula for correlation of complex random variables is (covariance of X and Y) / (standard deviation of X * standard deviation of Y).

## What does a correlation coefficient of complex random variables indicate?

A correlation coefficient of complex random variables ranges from -1 to 1, with 0 indicating no linear relationship, -1 indicating a perfect negative linear relationship, and 1 indicating a perfect positive linear relationship. A correlation coefficient close to 0 indicates a weak relationship, while a coefficient close to -1 or 1 indicates a strong relationship.

## Can correlation of complex random variables be used to determine causation?

No, correlation of complex random variables only measures the strength and direction of the linear relationship between variables. It cannot determine causation, as there may be other underlying factors or variables that are influencing the relationship.

## What are some limitations of using correlation of complex random variables?

Correlation of complex random variables only measures linear relationships, so it may not capture nonlinear relationships between variables. It also does not account for other factors that may be influencing the relationship between variables. Additionally, it can be influenced by outliers and may not accurately reflect the true relationship between variables in a small sample size.

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