Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos 1 Adaptive & Array Signal Processing AASP Prof. Dr.-Ing. João Paulo C. Lustosa.

Apresentação em tema: "Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos 1 Adaptive & Array Signal Processing AASP Prof. Dr.-Ing. João Paulo C. Lustosa."— Transcrição da apresentação:

1 Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos 1 Adaptive & Array Signal Processing AASP Prof. Dr.-Ing. João Paulo C. Lustosa da Costa University of Brasília (UnB) Department of Electrical Engineering (ENE) Laboratory of Array Signal Processing PO Box 4386 Zip Code 70.919-970, Brasília - DF Homepage: http://www.pgea.unb.br/~lasp

2 Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Mathematical Background: Z Transform (14)   Z Transform: Example 4 – Second-order feed forward filter Roots: real-valued or complex conjugated 2

3 Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Mathematical Background: Z Transform (15)   Z Transform: Example 4 – Second-order feed forward filter 3

4 Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Mathematical Background: Z Transform (16)   Z Transform: Example 4 4

5 Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Mathematical Background: Z Transform (17)   Discrete Fourier Transform (DFT) from the Z transform 5   By replacing   The inverse DFT is given by   The DFT is the Z transform only over the unitary circle

6 Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Mathematical Background: Complex variables (1)   The complex variables   Example of application: complex variable z of the z transform   Consider where x = Re[z] and y = Im[z].   Assuming a certain function f, such that   Therefore:   The derivative of f(z) is defined as Derivative: independent the way  z approaches to z 0. Writing in terms of u, v, x and y, we have that: 6

7 Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Mathematical Background: Complex variables (2)   The complex variables   The derivative of f(z) can rewritten as   Case 1: 7

8 Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Mathematical Background: Complex variables (3)   The complex variables   Case 2:   To approach in a independent way, both cases should be satisfied   Therefore: These equations are known as Cauchy-Riemann equations. 8 Cauchy-Riemann conditions a necessary and sufficient for the existence of the derivative.

9 Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Mathematical Background: Vector Differentiation (1)   Optimization problem: Differentiate a cost function with respect to a complex-valued vector w   Each element of the vector: 9   The real part of each element:   The imaginary part of each element:   Complex derivatives in terms of real derivatives and   Properties of the complex vector differentiation

10 Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Mathematical Background: Vector Differentiation (2) 10   Complex derivatives of vectors in terms of real derivatives and   Properties of the complex vector differentiation

11 Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Mathematical Background: Vector Differentiation (3) 11   Example 1:   Example 2:   Example 3: Wiener-Hopf equations Zero matrix Zero vector

12 Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Mathematical Background: Vector Differentiation (4) 12   Example 3: Wiener-Hopf equations - Finding the minimum Matrix form of the Wiener- Hopf equations for a transversal filter   Example 4: Least squares method   where F is constant and   Computing the derivative with respect to the complex conjugated w

13 Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Mathematical Background: Vector Differentiation (5) 13   Computing the derivative with respect to the complex conjugated w   To find the value that maximize, where w 0 maximizes the function.

14 Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Mathematical Background: Vector Differentiation (6) 14   Relation between gradient and vector differentiation and Gradient Vector differentiation

15 Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Mathematical Background: Lagrange Multipliers (1) 15   Method of Lagrange Multipliers solves constrained optimization problems Optimization with a single equality constraint f(w) is a quadratic function of a vector w. Subject to the constraint: Rewriting the problem as known as primal equation. Physical interpretation: w is the vector with complex weights, s are the steering vectors, and f(w) is the mean-squared value of the beamformer output.   Method of Lagrange Multipliers transforms the constrained problem above into an unconstrained problem by applying the Lagrange multipliers!

16 Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Mathematical Background: Lagrange Multipliers (2) 16   Method of Lagrange Multipliers: Using f(w) and c(w) to build a new function where We can rewrite our new function as To minimize the function h(w), we apply its derivative with respect to w* equal to zero. known as adjoint equation

17 Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Mathematical Background: Lagrange Multipliers (3) 17   Method of Lagrange Multipliers: Observing the adjoint equation The solution: when the two curves are parallel to each other (tangent of contour lines), i.e., when the derivatives are equal!

18 Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Mathematical Background: Lagrange Multipliers (4) 18   Method of Lagrange Multipliers: Example Real valued case: Minimize the dimensions of a 2000 m 3 oil reservoir (tank) without top: The area to be minimized is given by: The restriction:

19 Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Mathematical Background: Lagrange Multipliers (5) 19   Method of Lagrange Multipliers: Example Using the restriction to find the solution:

20 Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Mathematical Background: Lagrange Multipliers (6) 20   Example: Find the vector that minimizes the function: given the following constraint:   The adjoint equation is given by   Applying the previous vector differentiation rules

21 Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Mathematical Background: Lagrange Multipliers (7) 21   Replacing the optimal w H in the primal equation:   Replacing back to the optimal w equation:

22 Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Mathematical Background: Lagrange Multipliers (8) 22   Method of Lagrange Multipliers with several constraints: Constraints We can write our function as To minimize the function h(w), we apply its derivative with respect to w* equal to zero. known as adjoint equation