- Claus Benjaminsen and Shyam Bharat ECE 734 Project Presentation

1. A Recursive Method for the Solution of the Linear Least Squares Formulation Algorithm and Performance in Connection with the PLX Instruction Set - Claus Benjaminsen and Shyam Bharat ECE 734 Project Presentation

2. Aim • Linear Least Squares Estimation (LLSE) for Digital Signal Processing (DSP) applications • Updating of weight vector in time, based on value at previous instant

3. Background • Least Squares Estimation (LSE) – minimization of squared error between observed data sequence and assumed signal model • Linear LSE (LLSE) – signal model is linear function of parameter to be estimated • Computation of inverse of autocorrelation matrix of assumed signal model (P x P) • For DSP applications, P  ∞ !!

4. Motivation • Practically impossible to compute inverse of matrix with infinite entries ! • Need for alternate method not involving matrix inverse computations • Answer: Recursive Linear Least Squares!

5. Preliminaries Function to be minimized w.r.t parameter: , 0 < λ < 1 where en(k) = d(k) – wH(n)u(k) In matrix form, J(w(n)) = [d* - uHw(n)]HΛ[d – uHw(n)] where Λ = diag{λn-1, λn-2, … , λ, 1} Solution to this equation: w(n) = (uΛuH)-1uΛd

6. Alternate Representation w(n) = Φ(n)-1ө(n) where: Recursively, Φ(n) = λΦ(n-1) + u(n)uH(n) ө(n) = λө(n-1) + u(n)d*(n)

7. Recursive LLSE Update the weight vector: w(n) = w(n-1) + k(n)[d*(n) – uH(n)w(n-1)] where: and P(n) = Φ-1(n) where P(n) has dimensions M x M The algorithm is ….

8. Recursive LLSE Algorithm Initialization: P(0) = δ-1I where δ-1 is small and positive w(0) = 0 For n = 1,2,3,…. x(n) = λ-1P(n-1)u(n) k(n) = [1 + uH(n)x(n)]-1x(n) α(n) = d(n) – wH(n-1)u(n) w(n) = w(n-1) + k(n)α*(n) P(n) = λ-1P(n-1) – k(n)xH(n)

9. Dimensions • Simplifications: • Real values for all quantities • d is a scalar

10. Data Flow Graph

11. Analysis of the DFG The DFG has 6 loops: A – B – A t1: tm+ta = tm+ta A – C – D – E – B – A t2: tm+tm+tt+tm+ta = 3tm+ta+tt A – C – J – E – B – A t3: tm+tm+tm+tm+ta = 4tm+ta A – C – G – H – I – J – E – B – A t4: tm+tm+tm+ta+td+tm+tm+ta = 5tm+2ta+td K – L – M – O – N – K t5: tm+ta+tm+ta+tt = 2tm+ta+tt O – O t6: ta = ta

12. Iteration Bound (T) • Definition: T is the maximum, over all loops, of the total loop computational time divided by the number of delays in that loop • T = max{t1, t2, t3, t4, t5, t6} = t4 = 5tm+2ta+td

13. Dependence Graph

14. Implementation Analysis • Numerical analysis - Finding upper/lower bounds of various quantities used in analysis - Generalization of bounds using variables is a complex task - Recursion leads to unboundedness for some variables - Depending on availability of resources, appropriate bounds may be placed on the concerned variables