Adaptive Filtering.ppt

Real time DSP
Professors:
 Eng. Diego Barral
 Eng. Mariano Llamedo Soria
 Julian Bruno

Filters
 conventional filters
 time-invariant
 fixed coefficients
 adaptive filters
 time varying
 variable coefficients
 adaptive algorithm
 function of incoming signal
 exact filtering operation is unknown or is non-
stationary!

Random Processes
 random != deterministic
 concepts
 realization
 ensemble
 ergodic
 tools
 mean
 variance
 correlation/autocorrelation
 stationary processes & WSS

Adaptive Filters
 parts
 digital filter
 adaptive algorithm
 filter
 FIR
 IIR (stability problems are difficult to handle)

Adaptive Filters
 d(n) desired signal
 y(n) output of the filter
 x(n) input signal
 e(n) error signal

FIR Filter
 wl(n) adaptive filter coefficients

Performance Function
 coefficients are updated to
optimize some predetermined
performance criterion
 mean-square error (MSE)
 for FIR
 R: input autocorrelation matrix
 p: crosscorrelation between d(n)
and x(n)

Performance Function
 MSE surface
 One global minimum
point!

Gradient Based Algorithms
 properties
 convergence speed
 steady-state performance
 computation complexity
 method of steepest descent
 greatest rate of decrease (negative gradient)
 iterative (recursive)

LMS Algorithm
 statistics of d(n) and x(n) are unknown
 estimation of MSE
 avoids explicit computation of matrix inversion,
squaring, averaging or differentiating

Performance Analysis
 stability constraint
 μ controls the size of the incremental correction
 λmax is the largest eigenvalue of the autocorrelation
matrix R
 Px input signal power
 large filters => small μ
 strong signals => small μ

 convergence speed
 large μ => fast convergence
 λ => relation between stability and speed of
convergence
 estimation

 excess mean-square error
 the gradient estimation prevents w from staying at wo
in steady state
 w varies randomly about wo
 trade-off between the excess MSE and the speed of
convergence
 trade-off between real-time tracking and steady-state
performance

Modified LMS Algorithms
 normalized LMS algorithm
 μ varies with input signal power
 optimize the speed of convergence and maintain
steady-state performance
 independent of reference signal power
 c is a small constant
 μ(n) is bounded
 0 < α < 2

Modified LMS Algorithms
 leaky LMS algorithm
 insufficient spectral excitation may result in divergence
of the weights and long term instability
 where v is the leakage factor
 0 < v ≤ 1
 equivalent of adding low-level white noise
 degradetion in performance
 (1 - v) < μ

Applications
 operate in an unknown enviroment
 track time variations
 identification
 inverse modeling
 prediction
 interference canceling

Applications
 adaptive system identification
 experimental modeling of a process or a plant

Applications
 adaptive linear prediction
 provides an estimate of the value of an input
process at a future time
 in y(n) appear the highly correlated components of
x(n)
 i. e. speech coding and separating signals
from noise
 output is e(n) for spread spectrum corrupted
by an additive narrowband interference

Applications
 adaptive linear prediction

Applications
 adaptive noise cancellation (ANC)
 most signal processing techniques are developed
under noise-free assumptions
 the reference sensor is placed close to the noise
source to sense only the noise, because noise from
primary sensor and reference sensor must be
correlated
 the reference sensor can be placed far from the
primary sensor to reduce crosstalk, but it requires a
large-order filter
 P(z) represents the transfer function between the
noise source and the primary sensor
 uses x(n) to estimate x’(n)

Applications
 adaptive noise cancellation (ANC)

Applications
 adaptive channel equalization
 transmission of data is limited by distortion in the
transmission channel
 channel transfer function C(z)
 design of an equalizer in the receiver that counteracts
the channel distortion
 training of an equalizer
 agreed sequence by the transmitter and the receiver
 Decision device

Applications
 adaptive channel equalization

Implementation considerations
 finite-precision effects
 prevent overflow
 scaling of coefficients (or signal)
 quantization & roundoff
 => excess MSE
 => stalling of convergence
 depends on μ
 threshold of e(n) -> LSB

Adaptive Filtering.ppt

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