1. Lms algorithm FOR NON-STATIONARY INPUTS FOR THE PIPELINED IMPLEMENTATION OF ADAPTIVE ANTENNAS Prof.Yu Hen Hu Arjun Arunachalam Department of Electrical and Computer Engineering University Of Wisconsin-Madison E-mail: arunacha@cae.wisc.edu ECE-734 Spring 2002

2. ABSTRACT In this project, a modified LMS algorithm has been proposed which is capable of handling non-stationary inputs. The algorithm is then modified for the pipelined implementation of adaptive antennas which have higher throughput.A number of recently proposed Approximation techniques when applied to standard LMS algorithm can result in instability due to a change in the functionality of the algorithm and thus careful calibration of step size and depth of pipelining is needed. This project also studies the issues involved with the various approximations techniques used to reduce the overhead involved when the look-ahead technique is applied to a standard LMS algorithm.

3. Structure of an Adaptive Filter U(N) Y(N) D(N) Transversal Filter With weights W (n) Adaptive Weight Control Mechanism 

4. Equations governing the Behavior of a pipelined Adaptive Filter Pipelined LMS: Computation: For n=0,1,2,3………………n E (n)=D (n)-W (n-M) h *U (n) W (n+1)=W (n-M)+   *E (n-I)* U (n-I) : This is a look ahead only if the error and input are wide sense stationary else this cannot be look-ahead. Note: This approximation Applicable for Stationary Inputs only: Different LMS has been proposed for non- stationary inputs in the report.

5. Approximation Technique for Stationary LMS Replace the summation :  * E (n-I)* U (n-I) I=1,2…M … With the Expression :  *M *E (n)*U (n)……………. This Stabilizes the pipelined implementation of an adaptive filter.approximation techniques such as RLA technique change the functionality of the LMS algorithm which may cause extreme instability necessitating the need for proper selection of step size and pipeline depth etc.

6. The aim of the simulations above was : 1.To show that the approximations introduced were just as effective as RLA if not better in pipelining an adaptive LMS filter. 2.To show that without the additional hardware overhead, we can still obtain the same accuracy of an RLA approximation and also ensure a greater range of pipeline depths. 3.I did not want to make drastic changes in the functionality of the algorithm and yet obtain significant pipelining depths. Results: 1.I achieved the same accuracy of an RLA approximation and in some cases the accuracy is better. 2.The depth of pipeline can be more for this approximation 3.We can see that for a pipeline depth of 20, instability seems to be creeping in but the algorithm still is very accurate. 4.The additional hardware overhead is not involved and the change is functionality is not as drastic as in RLA.

7. Non Stationary LMS-Changes introducedto the original algorithm 1.Forgetting Factor 2.Epoch Length for staggered Update of the weights.

8. Results and analysis: the aim of the NON-LMS simulations was: 1.To pipeline a non-stationary input LMS algorithm and then test its stability 2.To change the epoch length and test for various pipeline depths. Results: The non-stationary algorithm looks a lot more stable at greater pipeline depths than the stationary algorithm. Also, for greater pipeline depths for a fixed epoch length, the algorithm takes a long time to converge. Therefore, we can vary the epoch length or the pipeline depth in order to ensure faster convergence. Therefore, a balance between the throughput and convergence time should be they’re depending on the type of application for which we use the filter. When we reduce the epoch length, the incoming data is broken up into smaller blocks and takes a longer time to converge. This can be seen by comparing the plots of the learning curve for pipelining depth of 5 but with epoch lengths of 10 and 7 respectively. Therefore, we need to reduce the pipeline depth in order to reduce the convergence time. Therefore, for a pipeline depth of 3 as shown above, the convergence time is much lower. The conclusion is that depending on the application we need to strike a balance between the depth of pipelining and the epoch length.

9. Conclusion and Discussion: These courses provided me with the knowledge base to pursue this work. I believe a lot more work can be done with adaptive filters to ensure faster computation for a variety of different applications. For example, changing epoch lengths for different can ensure faster computation for a much higher depth of pipeline. I believe that the work that needs to be done with adaptive filters needs to be more application specific. This project is an implementation type project undertaken by one person. ARJUN ARUNACHALAM