For millimeter wave massive MIMO communication in 5G, we observe that the number of available multipath components (MPCs) is much smaller than the maximum possible connections between the transmitter (TX) and receiver (RX). Only a few MPCs reach the receiver with significant signal strength. For example, let the number of strong MPCs be $L$, while there are $N_t$ antennas at the transmitter and $N_r$ antennas at the receiver.
From the channel matrix of the massive MIMO system, the total number of antenna-to-antenna connections is $N_r \times N_t$. In a beamspace representation, if we use a grid of $N_{tBeams}$ and $N_{rBeams}$, the total possible beam combinations is $N_{tBeams} \times N_{rBeams}$.
The key condition for sparsity is: $L \ll (N_t \times N_r)$
If we look up the massive MIMO channel matrix, then, H=
Because the number of strong MPCs ($L$) is much lower than the maximum connections, the beamspace channel matrix ($h_b$) is sparse. This means most elements in $h_b$ are zero or near-zero. We use the Orthogonal Matching Pursuit (OMP) algorithm, a "Compressive Sensing" method, to identify the indices of these $L$ strong paths and estimate their gains without processing all $N_r \times N_t$ combinations.
Orthogonal Matching Pursuit Algorithm Logic:
Let’s assume we use a beamforming codebook where the number of possible transmit and receive beams are $N_{tBeams}$ and $N_{rBeams}$.
If $N_t = N_r = 32$, the full channel matrix has $32 \times 32 = 1024$ elements. However, if there are only $L=5$ physical paths, only 5 entries in the beamspace matrix $h_b$ will be significantly non-zero.
Mathematically,
$y = Q h_b + n$
Here:
$y$ = received signal vector
$Q$ = Equivalent sensing matrix (containing the steering vectors/codebook)
$h_b$ = Sparse beamspace channel vector (size $N^2 \times 1$)
$n$ = noise
OMP Iteration Steps:
1. Correlation: We find the column of $Q$ that has the maximum projection (correlation) with the received signal $y$.
$i(1) = \text{argmax}_j |(q_j^H / ||q_j||) \cdot y|$
2. Estimation: Use the Least Squares (LS) solution to find the gain ($h_{b1}$) for that specific path:
$h_{b1} = (q_{i1}^H q_{i1})^{-1} q_{i1}^H y$
3. Residual Update: Subtract the estimated signal from the received signal to find the "residue" ($r_1$):
$r_1 = y - q_{i1} h_{b1}$
4. Repeat: In the next iteration, find the column of $Q$ that correlates most with the residue $r_1$. Repeat this until the residue falls below a threshold or we reach $L$ iterations.
Mathematical Example of Orthogonal Matching Pursuit (OMP):
Let's assume, for a MIMO communication system,
The size of the equivalent sensing matrix, Q is 4 X 6
And received signal matrix, y=
Now, y = Qhb
Or,
Where, hb =beamspace matrix =
Let assume, Q = [ q1 q2 q3 q4 q5 q6]
Here, q1 is the first column of Q, q2 is the second column of Q, and so on.
First Iteration of Orthogonal Matching Pursuit:
Now we find the maximum correlation of y by finding out which column in Q generates the maximum value with the projection of y,
Or,
QT*y =
Here, we can see the element in the 5th row is the maximum among all elements. So, we’ll select the 5th column of Q with which y has the maximum correlation value.
Now, hb1 = (q5'*q5)^(-1) * q5'*y
Where q5’ denotes the transpose of q5
Or,
hb1 = 4
Residue Matrix, r1 = y – q5* hb1
Or, r1 =
Here, we observe residual matrix r1 is not a zero matrix. So, we go for 2nd iteration.
Second Iteration of Orthogonal Matching Pursuit
Where we find the maximum correlation of r1 with respect to Q matrix.
Alternatively,
QT*r1=
Now, we see the element in the 2nd row of the above matrix generates the maximum value so r1 has a maximum correlation with the 2nd column of Q Matrix.
Now, we’ll form a new matrix, Qn = [q5 q2]
We find hb2 = (Qn'*Qn)^(-1)*Qn'*y;
Or, hb2 =
Now updated residue matrix, r2 = y – Qn* hb2
Or, r2=
Now we get the desired value in residue matrix r2 where all elements are zeros. So, beamspace matrix, hb will be
Here, we replace the elemental value of hb in that rows which are equal with the number of columns which generates maximum values with projection with y, then r1, and so on.
Now, from the mathematics we can say,
y = Q* hb
Or,
Further Reading
[1] Orthogonal Matching Pursuit (OMP) in Compressive Sensing (Theory)#beamforming