enter image description hereI am reading a book called Massive MIMO Networks: Spectral, Energy, and Hardware Efficiency, and it implies that $$\frac{\mathbf{h}}{\sqrt{\text{E}\left\{\|\mathbf{h}\|^2\right\}}}$$ is the channel direction, where $\mathbf{h}\in \mathbb{C}^{M}$, where $M$ is a positive integer (see the picture for the snapshot). I don't understand what it means, and how. Could any explain this to me. Thanks in advance

EDIT: It I originally defined the direction as $$\frac{\|\mathbf{h}\|^2}{\sqrt{\text{E}\left\{\|\mathbf{h}\|^2\right\}}}$$ but it should be $$\frac{\mathbf{h}}{\sqrt{\text{E}\left\{\|\mathbf{h}\|^2\right\}}}$$

  • $\begingroup$ are you sure about the numerator? Could you cite the book and where to find the defintion? $\endgroup$ – AlexTP Jan 1 at 16:26
  • $\begingroup$ @AlexTP I added a snapshot from the book to my original post. $\endgroup$ – BlackMath Jan 2 at 4:31
  • $\begingroup$ Well, these are just definitions. I mean the author defined a term $\mathbf{h}/\sqrt{E |\mathbf{h}|^2}$ which can be interpreted as statistical direction (whose its norm can be different from 1, not like $\mathbf{h}/||\mathbf{h}||$). You can see that the denominator is a constant (by expected value) while the numerator is a random variable. $\endgroup$ – AlexTP Jan 2 at 10:43
  • $\begingroup$ This definition leads to a nice property that when the dimension $M$ approaches infinity, some directions (in your example, $l \to i$ and $j \to k$) can be almost surely orthogonal whatever their channel realizations (impulses) $\mathbf{h}$. In brief, if a model (channel direction) is accepted then its analysis can follow. Just to see how the author use his defintions. $\endgroup$ – AlexTP Jan 2 at 10:44
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    $\begingroup$ en.wikipedia.org/wiki/Euclidean_vector $\endgroup$ – AlexTP Jan 2 at 11:53

You should double check the formula.

The classic single input multiple output (SIMO) equation with $N_T$ receive antennas is: $\mathbf{y}=\mathbf{h}x+\mathbf{w}$.

Where $x$ is the transmitted symbol (usually complex valued), $\mathbf{w} \sim CN(0,N_0\mathbf{I})$ is the complex noise, and $\mathbf{y}$ is the received vector, which is a vector of size $N_{R}\times1$.

You are asking about the channel vector $\mathbf{h}$ which contains the channel gains between the transmitter and each of the receive antennas, ie. $\mathbf{h}=[h_1,...,h_{N_R}]^T$. Since $\mathbf{h}$ is just a vector we can talk about its magnitude and direction. To look at a vector's direction we make the vector in question of unit length so we divide by the norm $||\mathbf{h}||$. That is, the direction of $\mathbf{h}$ is given as: $\frac{\mathbf{h}}{||\mathbf{h}||}$.

As far as what does the direction mean? Well, it can be thought of as what is responsible for the rotation of the transmitted symbol. The magnitude of $\mathbf{h}$ scales the transmitted symbol and the direction rotates the symbol in the complex plane.

  • $\begingroup$ Thanks. I quoted the book I am reading in my original post. $\endgroup$ – BlackMath Jan 2 at 4:32
  • $\begingroup$ Why the direction of $\mathbf{h}$ is given by $\frac{\mathbf{h}}{\|\mathbf{h}\|}$? It's close to what was mentioned in the book if you take the square root of the denominator in the direction in the original post, with the difference of expectation. $\endgroup$ – BlackMath Jan 2 at 9:46
  • $\begingroup$ There is an expectation just because the channel coefficients are usually thought of as random variables. So taking expectation, norm squared, then square root is same as what I put down although I admit a bit sloppy on my part just because technically $\mathbf{h}$ is a random vector $\endgroup$ – Engineer Sep 30 at 20:03

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