A dual of wireless distributed beamforming is nullforming, a spatial filtering technique used to limit the energy of a beam from a given direction. In distributed nullforming, the problem is as follows: a set of cooperating transmitter nodes transmitting a common signal, spatially align their individual transmit signals in such a manner as to cancel each other out at a designated receiver node. The receiver node cooperates with the transmit array by sending a feedback signal concerning the received signal (RS) strength for the previous epoch. The k^{th} sample at the receiver node follows the following:

r[k] = \sum\limits_{i=1}^{N} h[i,k]e^{j(\theta_i[k]-\phi_i[k])} +w[k]

where h[i,k] is the channel gain for the i^{th} sensor at the k^{th} epoch, \phi_i[k] is the channel phase estimation error at the k^{th} epoch, \theta_i[k] is the received phase from sensor i at the k^{th} epoch and w[k] is a zero mean additive noise. The setup is as illustrated in Figure 1.


Figure 1: Distributed transmit nullforming.

Without loss of generality, assume that the channel is stationary, then h[i,k] = h[i] and \phi_i[k] = \phi_i \forall k. To form a null at the receiver node is equivalent to minimizing the cost function:

J(\theta) = \sum\limits_{i=1}^{N} \sum\limits_{m=1}^{n} h[i]h[m] e^{j((\theta_i-\theta_m) -(\phi_i +\phi_m))}

The algorithmic solution to minimizing the above cost function is a distributed implementation where each transmit sensor implements the algorithm:

\theta_i[k+1] = \theta_i[k] -2\mu \hat{h}_i (\cos{(\theta_i[k])}Im\{r[k]\} -\sin{(\theta_i[k])}Re\{r[k]\})

where Im\{.\} and Re\{.\} denote the imaginary and real parts of the received signal at the receiver node respectively and \hat{.} denotes estimated value. The receiver sends a feedback signal containing the RS for the k^{th} epoch. The scalability of the algorithm is evident since each node independently implements the algorithm by estimating its channel characteristics and using the common feedback signal from the receiver node.

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