7. 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)
9. Gradient Based Algorithms
properties
convergence speed
steady-state performance
computation complexity
method of steepest descent
greatest rate of decrease (negative gradient)
iterative (recursive)
10. LMS Algorithm
statistics of d(n) and x(n) are unknown
estimation of MSE
avoids explicit computation of matrix inversion,
squaring, averaging or differentiating
11. 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
12. Performance Analysis
convergence speed
large μ => fast convergence
λ => relation between stability and speed of
convergence
estimation
13. Performance Analysis
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
14. 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
15. 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
16. Applications
operate in an unknown enviroment
track time variations
identification
inverse modeling
prediction
interference canceling
18. 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
20. 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)
22. 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