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Beamforming
Beamforming is a technique whereby the receiver (typically at a base-station) adjusts its transmission or more typically reception parameters, so as to concentrate on particular parts of the cell and not in others. This can be static i.e. in a given cell, we know where our users are concentrated and whether they are not or dynamic, in that the focus area is continuously adjusted to match the location of the user. The purpose of beamforming is two fold. One, to maximize the receptivity from the user and two, to minimize receptivity from a noise source. The diagram below shows how this works 
How does it work
Beam-forming antenna is nothing but an electronically controlled adaptive diffractor for electromagnetic waves. The aim of the diffractor is to accept a signal at an angle \phi but create nulls at angles \psi_1,\ \psi_2 etc. The diagram below shows the a picture of an adaptive antenna structure, using 5 elements.
The diagram shows a 5-element adaptive antenna structure. Each element is at a fixed distance from the previous one. The primary source is sufficiently far from the antenna that we can consider the different rays to be parallel to each other; each ray hits the corresponding element at the angle \psi . Each element is multiplied by a certain weight, and the output is consequently summed. There is a feedback element to this structure; the output is processed to generate an error signal, which is then fed back to the antenna which uses it to adjust the weights.
Angle of maximum receptivity
To understand how this works, let us start with the basic diffraction formula for an N-line elements spaced at an equal distance 'd' . Each element gets the data at a phase difference \chi from the previous one. The intensity is given by:
\begin{eqnarray}
I_{\phi} &=& \mathcal{R} \big\{ \sum_{k=0}^{N-1}e^{\{\omega t + k \chi \}} \big\} \\
&=& \mathcal{R} \big\{ e^{\omega t}\sum_{k=0}^{N-1}e^{\{k \chi \}} \big \}
\end{eqnarray}
which implies
I_{\phi} = \mathcal{R}\big\{e^{iwt} \frac{1 - e^{iN\chi}}{1 - e^{i \chi}} \big\}
We know that 1 - e^{ix} = 2i \sin (\frac{x}{2})e^{ix/2} . Substituting in the equation above, we get
I_{\phi} = \mathcal{R}\big\{e^{iwt + \chi(N-1)} \frac{\sin(\frac{\chi N}{2})}{\sin(\frac{\chi}{2})} \big\}
To maximize this, we have to remember that sin(x) reaches its absolute maximum when x = (2k+1)\frac{\pi}{2} and that E(e^{ix}) = 1 . But the phase \chi is related to the inter element distance d, by the formula \chi = \frac{2 \pi d sin(\phi)}{\lambda} . The maximum thus occurs when \phi = \arcsin{\frac{\lambda}{2d}}. This gives the direction in which the receptivity is the maximum.
A similar effect can be brought about by inserting a time-delay between elements \nabla t. The formula for the phase difference then becomes \chi = \nabla t 2 \pi f
. The direction of maximum receptivity thus becomes \phi_{max} = \nabla f
Noise rejection
From the section above, it is clearing that we can make the antenna structure have maximum receptivity in a particular direction, simply by selecting the phase adjustment suitably, either by setting a distance between the antenna elements, or, more practically, by inserting a variable time delay between the different antenna elements, prior to the summing. However, the second problem is to reject noise signals, which come from other directions.
To do this, the signal from each line element is multiplied by a certain weight, which is computed dynamically. The signal thus becomes: I_\phi = \mathcal{R}\big\{e^{i \omega t}\sum_k w_k e^{i k \chi} \big \} where \sum_k w_k = 1 . Very simplistically, the problem can be solved in the following steps:
- Based on the direction of the signal source, compute the required phase difference, \lambda
- Given that we know the directions of arrival for the m noise sources, \psi_i, i = 1,2...m , we can formulate the problem as a simple linear programming problem as follows:
Let \vec{W} = \{ w_0, w_1, \ldots \} and
S_x\{\vec{W}\} = E(I_x\{\vec{W}\} = \mathcal{R}\big\{ \sum_k w_k e^{i k \chi} \big \} = \sum_k w_k cos(k\chi)
.
Choose W, so as to maximize S_\phi\{\vec{W}\} , subject to S_{\chi_i} \le \delta and \sum_k w_k = 1 . When all angles of arrival are fixed, we can solve this as a linear programming problem and compute the optimal weight vector. \delta is the noise rejection threshold.
However, for this scheme to work, we need to know the angle of arrival for the main signal source, as well as all the noise sources. Also, with each change, the weight vectors need to be recomputed. Clearly, this is an impractical solution for a real system.
Adaptive beam-steering
However, a very simple yet clever adaptation strategy was proposed long back by [Widrow] , which provides a very simple mechanism to adaptively compute the weights and thus steer the beam optimally. Since the algorithm uses the criterion of Least Mean squares, it is known as the LMS algorithm. The mechanism is very simple
- Start with an initial weight vector W_0 . At any given point of time j, let the current weight vector be W_j.
- Let the input to the antenna elements at the jth instant be X_j
- Compute the error at the jth instant using the following formula \epsilon_j = d_j - W^TX_j , where d(j) is taken from the list of possible output vectors to have the least mean square error.
- Update the weight vector using W_{j+1} = W_j - 2 \mu \epsilon_j X_j . It can be proven that the weight vector W will move to an optimal equilibrium.
The constant \mu controls the robustness and speed of convergence of this algorithm. There are many variants to this basic formula, claiming to achieve faster convergence or higher rejection, etc. For example, [WidrowHoff] proposes to use W_{j+1} = W_j - 2 \mu K \epsilon_j^{2K-1} X_j to get better suppression for the Kth side-band, etc. However, it is well-proven that even the basic Widrow algorithm and its derivatives are sufficient to get extremely good performance, given suitable separation of the signal and noise sources.
Categories: Wireless
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