```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
de Brasília
http://www.pgea.unb.br/~lasp
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Introduction to
Array & Adaptive Signal Processing (1)

Filter
Extracted Information
Noisy data
Filter
Sensors

Example of application: Single Input Single Output (SISO)
communication system
TX


RX
The extracted information: the transmitted signal s(n)
Given x(n), we desire s(n).
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Introduction to
Array & Adaptive Signal Processing (2)

 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
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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.
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Introduction to
Array & Adaptive Signal Processing (4)

 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):
1st step – estimate the signals and their parameters
2nd 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.
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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).
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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.
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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
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Introduction to
Array & Adaptive Signal Processing (8)

 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
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Introduction to
Array & Adaptive Signal Processing (9)

 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
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Introduction to
Array & Adaptive Signal Processing (10)

 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
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Introduction to
Array & Adaptive Signal Processing (11)

 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
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Introduction to
Array & Adaptive Signal Processing (12)

Identification: channel estimation
Plant


Plant, e.g. channel in communications
When e(n) = 0, then the adaptive filter is equal to the plant.
 Therefore, the plant is identified!
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Introduction to
Array & Adaptive Signal Processing (13)

Plant
Delay


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!
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Introduction to
Array & Adaptive Signal Processing (14)

Prediction
Delay

When e(n) = 0, then y(n) = s(n).
 Therefore, the symbols s(n) are correctly predicted!
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Introduction to
Array & Adaptive Signal Processing (15)


When e(n) = s(n), then the output is equal to the signal.
 Therefore, the noise is removed!
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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 Ts is sampling period.
By replacing h[n] in the Laplace transform
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Mathematical Background: Z Transform (2)

Z Transform
 defining:

Therefore:
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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
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Mathematical Background: Z Transform (4)

Z Transform: Frequency to angle mapping
Reference: Saeed Vaseghi, Communication and Multimedia
Signal Processing group, Brunel University
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Mathematical Background: Z Transform (5)

Z Transform: Transfer function
 Discrete-time linear system:

Computing the z transform:

Transfer function:
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Mathematical Background: Z Transform (6)

Z Transform:

Transfer function: Poles and zeros representation
Reference: Saeed Vaseghi, Communication and Multimedia
Signal Processing group, Brunel University
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Mathematical Background: Z Transform (7)

Z Transform: Example 1 – Find the z transform and pole-zero diagram
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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
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Mathematical Background: Z Transform (9)

Z Transform: Example 2 – First order system with a single zero
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Mathematical Background: Z Transform (10)

Z Transform: Example 2
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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
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Mathematical Background: Z Transform (12)

Z Transform: Example 3 – First order system with a single pole
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Mathematical Background: Z Transform (13)

Z Transform: Example 3
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