Correlation of Complex Random Variables

[EngWiPy]

Hi,

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

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

Thanks in advance

 

[pmsrw3]

Hi,

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

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

Thanks in advance

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.

 

[EngWiPy]

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?

 

[pmsrw3]

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].

 

[EngWiPy]

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