# Channel direction

I 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 $$\frac{\mathbf{h}}{\sqrt{\text{E}\left\{\|\mathbf{h}\|^2\right\}}}$$

• are you sure about the numerator? Could you cite the book and where to find the defintion? – AlexTP Jan 1 '19 at 16:26
• @AlexTP I added a snapshot from the book to my original post. – BlackMath Jan 2 '19 at 4:31
• 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. – AlexTP Jan 2 '19 at 10:43
• 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. – AlexTP Jan 2 '19 at 10:44
• en.wikipedia.org/wiki/Euclidean_vector – AlexTP Jan 2 '19 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.

• Thanks. I quoted the book I am reading in my original post. – BlackMath Jan 2 '19 at 4:32
• 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. – BlackMath Jan 2 '19 at 9:46
• 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 – Engineer Sep 30 '19 at 20:03