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 , Brasília - DF Homepage:

Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos 2 Introduction to Array & Adaptive Signal Processing (1)   Filter Sensors Filter Noisy data Extracted Information   Example of application: Single Input Single Output (SISO) communication system TXRX   The extracted information: the transmitted signal s(n)   Given x(n), we desire s(n).

Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos 3 Introduction to Array & Adaptive Signal Processing (2)   Information processing tasks   Filtering: given x(n), x(n - 1), …, x(n – L), we desire s(n)   Smoothing: given x(n+L), …, x(n + 1), x(n), x(n - 1), …, x(n – L), we desire s(n)   Prediction: given x(n), x(n - 1), …, x(n – L), we desire s(n+  )   Types of filters (besides being Filtering, Smoothing and Prediction):   Linear:   Nonlinear:   For linear filters:   Availability of statistical parameters (e.g. mean or covariance) of the signal or of the unwanted noise   Minimize the noise effects according to some statistical criterion

Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos 4 Introduction to Array & Adaptive Signal Processing (3)   Example of approach (statistical criterion) for linear filters:   Minimize the mean-squared value of the error signal e(t), where   For stationary input, i.e. if the statistical parameters of the input are constant, the Wiener filter is the optimum in the mean-square sense.   If the signal or noise is nonstationary   The optimum filter has to be time varying. The Kalman filter is a solution.   Continuous vs. Discrete   Only discrete is considered due to the digital signal-processing devices   No loss of information since the sampling theorem is respected.

Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos 5 Introduction to Array & Adaptive Signal Processing (4)   Adaptive Filter   Wiener filter: requires a priori information of the statistics of the data   Statistics of the filter equal to statistics of the data to be processed: optimal solution   If not the equal, then two-step solution (non-recursive): 1 st step – estimate the signals and their parameters 2 nd step – with the previous step, estimate the parameters of the filter   The two-step solution: high computational complexity!   Therefore, the adaptive filter (recursive approach) is used. It converges to the optimum Wiener solution (after iterations). In nonstationary case, it has the tracking capability.

Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos 6 Introduction to Array & Adaptive Signal Processing (5)   Several recursive algorithms in the literature and the choice depends on the following factors   Convergence rate: Number of iterations to converge to the Wiener solution   Misadjustment: Mean-squared error of the algorithm is compared to the mean-squared of the Wiener solution.   Tracking: For nonstationary, the algorithms tracks the statistical properties.   Robustness: For disturbances implies on small estimation errors.   Computational requirements: Number of arithmetic operations and the memory requirements for the algorithm.   Numerical properties: inaccuracies due to the quantization errors (Analog-to-Digital Conversion and internal digital representation).

Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos 7 Introduction to Array & Adaptive Signal Processing (6)   Linear filter structures   Tapped-delay line filter or transversal filter   The asterisk stands for complex conjugation.   is the unit-delay operator.

Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos 8 Introduction to Array & Adaptive Signal Processing (7)   Linear filter structures   Tapped-delay line filter or transversal filter Finite convolution sum: convolves the finite duration impulse response of the filter with x(n).   L – 1 is the filter order.   Other types of filter structures: Lattice predictor Systolic array

Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos 9 Introduction to Array & Adaptive Signal Processing (8)   Adaptive linear filter   Tapped-delay line filter or transversal filter   The weights are unknown   For the stationary case: The mean-squared error gives a minimum, which corresponds to the values of the weights and to the Wiener solution.   For the nonstationary case: Incorporation of the steepest descent method on the Wiener equation. The gradient vector is defined. results to the least-mean-squares (LMS) algorithm

Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos 10 Introduction to Array & Adaptive Signal Processing (9)   Adaptive linear filter   The LMS algorithm Low convergence rate Sensitivity to the condition number Condition number: Ratio of the largest and smallest eigenvalue of a Hermitian matrix Most popular algorithm of the stochastic gradient family of the linear adaptive filters

Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos 11 Introduction to Array & Adaptive Signal Processing (10)   Adaptive linear filter   Tapped-delay line filter or transversal filter   Another option for the nonstationary case: Least squares methods Recursive least squares: special case of Kalman filter For being recursive, it requires less storage than a block estimator. There are three categories of RLS algorithms:   Standard RLS   Square-root RLS   Fast RLS

Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos 12 Introduction to Array & Adaptive Signal Processing (11)   Nonlinear adaptive filters   Volterra-based nonlinear adaptive filters Generalization of the Taylor series of a function in 1880 First used by Norbert Wiener in 1958   Neural networks Nonlinearity Continuous input-output mapping Weak statistical assumptions Learning capability Fault tolerance (failure of some part of the neurons) Generalization

Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos 13 Introduction to Array & Adaptive Signal Processing (12)   Identification: channel estimation Plant Adaptive filter   Plant, e.g. channel in communications   When e(n) = 0, then the adaptive filter is equal to the plant. Therefore, the plant is identified!

Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos 14 Introduction to Array & Adaptive Signal Processing (13)   Inverse modeling: adaptive equalization Plant Adaptive filter   Plant, e.g. channel in communications   When e(n) = 0, then the adaptive filter inverts the plant. Therefore, the symbols s(n) are correctly estimated! Delay

Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos 15 Introduction to Array & Adaptive Signal Processing (14)   Prediction Delay Adaptive filter   When e(n) = 0, then y(n) = s(n). Therefore, the symbols s(n) are correctly predicted!

Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos 16 Introduction to Array & Adaptive Signal Processing (15)   Interference canceling: adaptive noise cancelation Adaptive filter   When e(n) = s(n), then the output is equal to the signal. Therefore, the noise is removed!

Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos 17 Mathematical Background: Z Transform (1)   Z Transform   also seen as the discrete form of the Laplace transform   The Laplace transform is defined as h(t) is continuous.   In digital processing devices, samples are used. Therefore, the sampled version of h(t) is given by where T s is sampling period.   By replacing h[n] in the Laplace transform

Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos 18 Mathematical Background: Z Transform (2)   Z Transform   defining:   Therefore:

Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos 19 Mathematical Background: Z Transform (3)   Z Transform: the z plane and the unit circle Reference: Saeed Vaseghi, Communication and Multimedia Signal Processing group, Brunel University

Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos 20 Mathematical Background: Z Transform (4)   Z Transform: Frequency to angle mapping Reference: Saeed Vaseghi, Communication and Multimedia Signal Processing group, Brunel University

Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos 21 Mathematical Background: Z Transform (5)   Z Transform: Transfer function   Discrete-time linear system:   Computing the z transform:   Transfer function:

Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos 22 Mathematical Background: Z Transform (6)   Z Transform:   Transfer function: Poles and zeros representation Reference: Saeed Vaseghi, Communication and Multimedia Signal Processing group, Brunel University

Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos 23 Mathematical Background: Z Transform (7)   Z Transform: Example 1 – Find the z transform and pole-zero diagram

Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos 24 Mathematical Background: Z Transform (8)   Z Transform: Example 2 – First order system with a single zero - - Zero on the middle: All pass filter - - Zero on the right: High pass filter - - Zero on the left: Low pass filter

Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Mathematical Background: Z Transform (9)   Z Transform: Example 2 – First order system with a single zero

Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos 26 Mathematical Background: Z Transform (10)   Z Transform: Example 2

Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Mathematical Background: Z Transform (11)   Z Transform: Example 3 – First order filter with single pole - - Pole on the middle: All pass filter - - Pole on the right: Low pass filter - - Pole on the left: High pass filter 27

Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Mathematical Background: Z Transform (12)   Z Transform: Example 3 – First order system with a single pole 28

Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Mathematical Background: Z Transform (13)   Z Transform: Example 3 29