Joint distributed transmit beamforming and nullforming is a spatial filtering technique where cooperative nodes transmit a common message signal to receivers whiles at the same time forming nulls at receivers, making a total of receivers. All the receiver nodes cooperates with the transmits by broadcasting a feedback message containing the received signal (RS) strength for the previous epoch. The sample at the receiver node is as follows:
where is the channel gain from the sensor to the receiver, is the received phase from sensor at the epoch, is a zero mean complex Gaussian additive noise and is a prearranged signal from all transmitters at epoch . Precisely, given transmitters and $M$ receivers, determine in a scalable and distributed manner, appropriate complex weights such that beams are formed at receivers and nulls are formed at receivers. The receivers making up the two subsets, and , receivers are known apriori and each receiver send a broadcast feedback signal which contains information on the RS received at the previous epoch. The setup is as illustrated in Figure 1 below:
Figure 1: Joint distributed transmit beamforming and nullforming.
Suppose is the target signal strength gain, with for nullforming receivers, and for beamforming receivers, then we can define a minimizing cost function:
The algorithmic solution to minimizing the above cost function is a distributed implementation where each transmit sensor implements the following:
where and denote the imaginary and real parts of a complex number respectively and denotes estimated value. The scalability of the algorithm is evident since each node independently implements the algorithm by estimating its channel impulse response to the sensor and uses the common feedback signal from all receiver nodes.