1. 08/10/2001 1 High Performance Parallel Implementation of Adaptive Beamforming Using Sinusoidal Dithers High Performance Embedded Computing Workshop Peter Vouras and Trac D. Tran Johns Hopkins University September 19, 2007

2. 08/10/2001 2 Overview n Gradient Estimation Using Sinusoidal Dithers n Application to Adaptive Beamforming n Parallel Implementation n Results n Conclusion

3. 08/10/2001 3 Gradient Estimation Using Dithers n Consider any objective function f(x) where x is an N-by-1 vector. For each component of x, superimpose a sinusoidal dither of different frequency, as in where α is a small scalar. The Taylor series expansion of f(x) yields,

4. 08/10/2001 4 Gradient Estimation Using Dithers - 2 n The components of the gradient vector can be determined exactly after we multiply f(x') by cos(ω j t), for j = 1, 2, …, N. The result after using trigonometric identities is, n Note that the only constant term on the right hand side is the jth component of the gradient vector scaled by the factor α/2, and can be recovered exactly by low-pass filtering f(x')cos(ω j t).

5. 08/10/2001 5 Dither Application n Dithers can be applied sequentially or in parallel n If applied in parallel, care must be taken to avoid crosstalk between adjacent channels which may corrupt the gradient estimate l Requires high sampling rate, and narrow low pass filter with steep transition regions

6. 08/10/2001 6 Advantages of Using Dithers in Adaptive Beamforming PROs n Estimates gradient exactly and results in higher SINR than other recursive algorithms n Very scalable algorithm l Each dither computation is independent of others and can therefore be distributed across parallel processors CONs n Increased computations and FLOP count

7. 08/10/2001 7 Brief Overview of Adaptive Beamforming n Linearly Constrained Minimum Variance (LCMV) Beamformer n Minimize output power of an array, y(n) = w H u(n), subject to a set of linear constraints

8. 08/10/2001 8 Adaptive Array Configuration With Dithers n Separate dither must be applied to real and imaginary part of each array element weight

9. 08/10/2001 9 Adaptive Array Performance Jammer at 44º azimuth

10. 08/10/2001 10 Adaptive Array Performance - 2

11. 08/10/2001 11 Subarray Architecture For this study, techniques to minimize grating lobes were not considered

12. 08/10/2001 12 Software Design MPI Implementation on Cray XD1

13. 08/10/2001 13 Measured Results Scalability Speed-Up Load is perfectly balanced

14. 08/10/2001 14 Conclusions n Directly estimating the components of the gradient vector using sinusoidal dithers may improve the performance of steepest descent algorithms l Adaptive beamforming n Subarray implementation of adaptive beamformer evaluated on Cray XD1 using parallel processors l Convolution (dot product) and mixing operations may be implemented on FPGAs for further performance improvements