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Scalable Joint Distributed Transmit Beamforming And Nullforming

Joint distributed transmit beamforming and nullforming is a spatial filtering technique where N cooperative nodes transmit a common message signal to M_1 receivers whiles at the same time forming nulls at M_2 receivers, making a total of M=M_1+M_2 receivers. All the M receiver nodes cooperates with the N transmits by broadcasting a feedback message containing the received signal (RS) strength for the previous epoch. The k^{th} sample at the m^{th} receiver node is as follows:

 

r_m[k] = m[k] \sum\limits_{i=1}^{N} h_{i,m}e^{j \theta_i[k]} +w_m[k]

 

where h_{i,m} is the channel gain from the i^{th} sensor to the m^{th} receiver, \theta_i[k] is the received phase from sensor i at the k^{th} epoch, w_m[k] is a zero mean complex Gaussian additive noise and m[k] is a prearranged signal from all transmitters at epoch k. Precisely, given N transmitters and $M$ receivers, determine in a scalable and distributed manner, appropriate complex weights such that beams are formed at M_1 receivers and nulls are formed at M_2 receivers. The receivers making up the two subsets, M_1 and M_2, 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 s_m,m \in \{1,\cdots,M\} is the target signal strength gain, with s_m =0 for nullforming receivers, and s_m =1 for beamforming receivers, then we can define a minimizing cost function:

 

J(\theta) = m^2[k]\sum\limits_{m=1}^{M} \left |s_m -\sum\limits_{i=1}^{N} h_{i,m} e^{j\theta_i[k]}\right |^2

 

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

 

\theta_i[k+1] = \theta_i[k] +2\mu m[k](\cos{(\theta_i[k])}Im\{X_i[k]\} -\sin{(\theta_i[k])}Re\{X_i[k]\})

 

X_i[k] = \sum\limits_{m=1}^{M} \left (r_m[k]- m[k] s_m \hat{h}_{i,m}\right)

where Im\{.\} and Re\{.\} denote the imaginary and real parts of a complex number respectively and \hat{.} 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 k^{th} sensor and uses the common feedback signal from all receiver nodes.

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