How to decode a MIMO stream using V-BLAST?

Question

I am attempting to implement the V-BLAST algorithm for MIMO streams as described by Wolniansky et al. in the following paper: https://ieeexplore.ieee.org/document/738086 (I am not commenting on its useful-ness or how efficient it is, I am simply trying to implement it using the paper as written.)

The algorithm assumes the following:

• The channel, H, is known perfectly (or already assumed).
• It is a rich scattering environment with some number of Transmitters and some number of Receivers, but for simplicity, I myself will assume that the number of Transmitters and the number of Receivers are equal, N.
• The vector of received symbols follows the form r = Ha + n, where H is the channel, a is the encoded transmitted symbols, and n is Additive White Gaussian Noise (unit-mean).

The algorithm is described as follows in the paper:

$$ \begin{align*} &\textrm{initialization:}\\ &(1)i\leftarrow 1\\ &(2)G_1 = H^+\\ &(3)k_1 = \textrm{argmin}_j||(G_1)_j||^2\\\\ &\textrm{recursion:}\\ &(4)w_{k_i} = (G_i)_{k_i}\\ &(5)y_{k_i} = w_{k_i}^Tr_i\\ &(6)\hat{a}_{k_i} = Q(y_{k_i})\\ &(7)r_{i+1} = r_i - \hat{a}_{k_i}(H)_{k_i}\\ &(8)G_{i+1} = H^+_{\overline{k_i}}\\ &(9)k_{i+1} = \textrm{argmin}_{j\not\in\{k_1,...,k_i\}}||(G_{i+1})_j||^2\\ &(10)i\leftarrow i+1 \end{align*} $$

(An image from the paper can also be found here.)

Some definitions:

• $H^+$ is the pseudoinverse of $H$. (As in a zero-forcing decoder)
• The algorithm confusingly mixes notations. On (3), $(G_1)_j$ is the j-th row of $G_1$, but on (7), $H_{k_i}$ is the $k_i$th column of H.
• $Q(y)$ is the quantization (slicing) operation, according to the constellation used.

With that out of the way, I'm getting lost in the implementation. I think I understand it on a surface level, but I get tripped up in the recursion. So then, I have a few questions:

1. In my implementation with QPSK symbols and N = 4, $r$ is a vector of size 4x1, and $G$ is 4x4. So in line (5), what would the expected size of $y_{k_i}$ be? My MATLAB implementation gives me a 1x4 array (row vector), but I'm not 100% certain that's right.
2. What exactly is the Quantization (slicing) operation on line (6)? Is it simply demodulating whatever it is (using qpskdemod in MATLAB), or am I approximating the received points to its nearest points on the ideal constellation? What would the expected size of $\hat{a}$ be, a 4x1 row vector?

I think with this I'll be able to fully implement the algorithm in MATLAB. I originally wrote this question going line-by-line with my MATLAB implementation at each line of the algorithm, but the question got really long. I can provide my MATLAB implementation if need be. I'd also greatly appreciate anyone's attempts at a MATLAB implementation so I can fully understand the workings of this algorithm, but that isn't necessarily necessary.

Thank you so much in advance!

Answer 1

In V-BLAST, $y_{ki}$ is a scalar (that is, a complex number). It is the (equalized) symbol transmitted by antenna $k_i$, with noise.

Keep in mind that all vectors are column vectors ($n \times 1$). Then, $w_{ki}$ is $4 \times 1$, and $w^T_{ki}$ is $1 \times 4$. Then, $w^T_{ki} r_i$ is $1 \times 1$, i.e. a scalar.

The "quantization" operation $Q(y_{ki})$ consists in finding the constellation element that is closest (in Euclidean distance) to $y_{ki}$. So, $Q(y_{ki})$ is the estimate of the symbol transmitted by antenna $k_i$.