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Consider a wireless communication system having $t$ transmitting antennas and $r$ receiving antennas. Then, the received signal is given by

$y = \mathbb{H}x+n \tag{1}$

where $\mathbb{H}$ is a $r \times t$ complex matrix, $x$ is a transmitted vector such that $x \in \mathcal{C}^t$, $y$ is a received vector such that $y \in \mathcal{C}^r$ and $n$ is a zero-mean complex Gaussian noise with independent, equal variance real and imaginary parts.

My question is that in many papers, it is written that $E[nn^{\dagger}] = I_r$, where $I$ is identity matrix. I am not getting clearly where this identity matrix is coming from.

Any help in this regard will be highly appreciated.

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    $\begingroup$ $E[nn^{\dagger}]$ gives the auto-covariance matrix of the noise $n$. Now for gaussian noise, this equals $\sigma^2 I$, where $\sigma^2$ is the variance of the noise. Why the \sigma^2 dis-appeared is probably due to the independent, equal variance real and imaginary parts condition, but I'm not familiar enough with this to tell you. I'd try writing out the generic expression for $n$ and go from there. Or look in your texts if they mention unity variance, that would explain $\sigma^2 = 1$ $\endgroup$
    – Jdip
    Aug 18, 2022 at 12:34
  • $\begingroup$ Ok...thanks a lot sir... $\endgroup$
    – chaaru
    Aug 18, 2022 at 13:05

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The result $E[nn^\dagger] = I_r$ comes from writing out explicitly the diagonal, and the off-diagonal terms in the $r\times r$ matrix $nn^\dagger$, paying special attention to that $\ \dagger$ superscript which means Hermitian transpose, and then evaluating the expected value of each matrix entry in light of the words "independent" and "equal variance" that you have used already.

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  • $\begingroup$ Ok ...Thank u sir... $\endgroup$
    – chaaru
    Aug 18, 2022 at 13:04
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My question is that in many papers, it is written that $E[nn^{\dagger}] = I_r$, where $I$ is identity matrix.

That's basically a definition of the noise characteristics: for $\mathbf n = \begin{bmatrix}n_1 & n_2 & n_3 & \cdots\end{bmatrix}^T$, the individual $n_k$ are independent and with $\sigma = 1$.

This is something that would be true if the significant part of the noise came from each receiver channel's circuitry, and said circuitry was well-designed. If you had some external noise source(s) with a significant noise contribution then this may not be true.

Presumably (A) it's felt that the independent noise source representation is good enough for now, or (B) there's still plenty to be learned about MIMO communications systems without worrying about noise that has channel-channel correlation, or (C) there's some mathematically easy canonical way of dealing with an $\mathbf I_r$ that's not actually an identity matrix.

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