# What's the Intuitive Description of Circular Symmetric Complex Zero Mean White Gaussian Noise?

Suppose we have an added discrete noise signal defined as:

$$y[n]=x[n]+w[n],$$ where $w$ is zero mean white Gaussian noise.

Question:

1. When we say white noise, is it sufficient to say that it is always be uncorrelated with signal $x$, no matter what the distribution of signal is? (I think yes)

2. Can we conclude that both $x$ and $w$ are independent IF noise is white Gaussian BUT the signal have unknown distribution ? (I am Confused about it)

3. Lastly, What the information we glean-out of the model (intuitively) if we are informed about noise that besides being zero mean, white and Gaussian, it is circular symmetric complex as well? (I do know white noise has impulse auto-correlation but confused with circular-symmetric term).

• 1. No. 2. No/Yes 3. Depends on whether $x$ and $w$ are complex or real or _complex-baseband representations of narrowband bandpass signals. You might want to read this answer and the link contained therein) and then edit your question. – Dilip Sarwate Dec 7 '14 at 4:23

3. What is meant by circularly symmetric Gaussian noise is that the noise looks the same in all directions. Imagine a certain Gaussian complex noise that (for some physical reason) produces larger noise in specific directions; for example, along a line at an angle of $\pi$/4 on the complex plane. You could then design a QAM constellation (and detector) that takes advantage of this behavior to get better than expected performance given the noise power in the channel. When a problem states that the noise is circularly symmetric, it's saying that you can't play this kind of game.