This document proposes an individual activation factor proportional normalized least mean square (IAF-PNLMS) algorithm. The standard PNLMS algorithm uses a single activation factor for all coefficients, which does not distribute gains proportionally for inactive coefficients. IAF-PNLMS uses a separate activation factor for each coefficient that converges to the coefficient magnitude, distributing gains more proportionally. Numerical simulations show IAF-PNLMS converges faster than PNLMS and IPNLMS for sparse impulse responses and adapts faster to plant perturbations.

1. Alternative Approach for Computing the Activation
Factor of the PNLMS Algorithm
Authors
Francisco das C. de Souza, Dennis R. Morgan
Orlando José Tobias , and Rui Seara Bell Laboratories, Alcatel-Lucent
LINSE - Circuits and Signal Processing Laboratory drrm@bell-labs.com
Federal University of Santa Catarina
{fsouza, orlando, seara}@linse.ufsc.br

2. INTRODUCTION
Sparse impulse responses are encountered in many real-world applications:
Communications, acoustics, seismic and chemical processes
NLMS algorithm (the same step-size value for all filter coefficients)
⇒ Slow convergence
Algorithms that exploit the sparse nature of the impulse response:
PNLMS (proportionate normalized least-mean-square)
PNLMS ++ (both the NLMS and PNLMS are used
in the coefficient vector update)
IPNLMS (improved PNLMS)
SC-PNLMS (sparseness controlled PNLMS)

3. INTRODUCTION
The standard PNLMS algorithm Formulation
μG (n) e(n)x(n)
Coefficient update ( N × 1) : w (n + 1) = w (n) +
xT (n)G (n)x(n) + ε
Gain distribution matrix ( N × N ) : G (n) = diag [ g1 (n) g 2 (n) g N (n) ]
φi (n)
Individual gain: gi ( n) = N
∑ φi (n)
i =1
Proportionality function: φi (n) = max ⎡ f (n), wi (n) ⎤
⎣ ⎦
Activation factor: f (n) = ρ max ⎡δ, w (n) ∞ ⎤
⎣ ⎦

4. INTRODUCTION
The standard PNLMS algorithm performance depends on predefined parameters:
δ (initialization)
ρ (proportionality or activation)
• These parameters are related to the algorithm variable, termed
ACTIVATION FACTOR f ( n)
• The initialization parameter permits starting the adaptation process at
n = 0, when all filter coefficients are initialized to zero.
⇒ f (0) = ρ max ⎡δ, w (0) ∞ ⎦ = ρδ
⎣
⎤
• The proportionality parameter prevents an individual coefficient from
freezing when its magnitude is much smaller than the largest coefficient
magnitude.
A central point: How to set suitable values for these parameters,
since they impact the algorithm convergence speed?

5. INTRODUCTION
Activation factor in the standard PNLMS algorithm
Common to all coefficients, computed sample-by-sample.
Depends on w (n) ∞ .
Leads to a gain distribution between the adaptive filter coefficients not
entirely in line with the concept of proportionality.
Proposed approach: Individual activation factor PNLMS (IAF-PNLMS)
An individual activation factor is used for each adaptive filter coefficient.
Each individual activation factor is computed in terms of the
corresponding coefficient magnitude.
Consequence
For impulse responses having high sparseness, numerical simulations show that
the proposed approach has faster convergence as well as faster response to
perturbations of the system plant than both the PNLMS and IPNLMS
algorithms.

6. STANDARD PNLMS ALGORITHM DISCUSSION
1
Gain for inactive coefficients g inactive (n) = N
f ( n)
∑ φi (n)
i =1
1
Gain for active coefficients giactive (n) = N
wi (n)
∑ φi (n)
i =1
Total gain distributed over the filter coefficients at each iteration
N−N
tr [G (n) ] = N active f (n) + ∑ giactive (n) = 1
∑ φi (n) i∈A
i =1
The activation factor affects the gains assigned to both
active and inactive coefficients.

7. Standard PNLMS algorithm performance with respect f (n)
Scenario for all numerical simulations
• Sparse impulse response p with N =100 coefficients,
Active coefficient values:
{0.1, 1.0, 0.5, 0.1} located at positions {1, 30, 35, 85}, respectively.
S (p) = 0.9435
• Input signal: Correlated unity-variance
μ = 0.5, δ = 0.01
AR(2) process with χ = 74
Normalized misalignment measure:
2
p − w ( n)
κ(n) = 10log10 2
2
p 2

8. Total gain distribution over L iterations
L −1
θi = ∑ gi ( n)
n =0
Average of θi over the inactive coefficients
1
θinactive =
mean
N − N active
∑ θi , A = {1, 30, 35,85}
i∉A

9. At the beginning of
the learning phase
(0 ≤ n < 30)
w30 ( n) < w1 (n)
g30 (n) < g1 (n)
Desired condition
p1 = 0.1, p30 = 1.0
g30 (n) > g1 (n)

10. MODIFIED PNLMS ALGORITHM
Features of the standard PNLMS algorithm:
1) When wi (n) is an active coefficient, its gain is always proportional to wi (n) .
2) When wi (n) is inactive, the gain is not proportional to wi (n) .
Objective:
To overcome the drawback (2) by making the gain gi (n)
tend towards being proportional to wi (n) even when
wi (n) is inactive.
⎧⇒ φi (n) = max[ fi (n), wi (n) ]
⎪
⎪ 1
f (n) is replaced by fi (n) ⎨⇒ gi ( n) = N
inactive
f i ( n)
⎪
⎪ ∑ φi (n)
⎩ i =1

11. MODIFIED PNLMS ALGORITHM
Conditions Required for the New Activation Factor fi (n)
C1) fi (n) must converge to the corresponding coefficient magnitude wi (n)
lim [ fi (n) − wi (n) ] = 0 , i = 1, 2, …, N
n→∞
C2) fi (n) must always be greater than zero, i.e.,
f i ( n) > 0 , i = 1, 2, …, N
inactive
If C1 is fulfilled, gi (n) tends to be proportional to wi (n) as n → ∞
C2 ensures that gi (n) > 0 when wi (n) = 0,
avoiding the freezing of wi (n)

12. MODIFIED PNLMS ALGORITHM
Proposed Approach for Computing fi (n)
fi (n) = γ wi (n) + (1 − γ )φi (n − 1)
INTENDED C2
AIM (C1)
where 0 < γ < 1
By considering that no knowledge of the
system plant is available a priori, it is reasonable to choose γ = 1/ 2
The activation factors are initialized
with a small positive constant (typically, fi (0) = 10−2 / N )

13. MODIFIED PNLMS ALGORITHM
For proper algorithm operation, it is required that the instantaneous
magnitude of the estimated coefficients be proportional to the magnitude
of the corresponding plant coefficients.
wi (n) may not be proportional to pi (n)
at the beginning of the adaptation process
IAF-PNLMS
⎧1 1
⎪ wi (n) + φi ( n − 1), n = mN
f i ( n) = ⎨ 2 2
⎪ fi (n − 1),
⎩ otherwise
N : adaptive filter length
m = 1, 2, 3, …

14. NUMERICAL SIMULATIONS
Example 1
A perturbation in the plant takes place at n = 2500, whereby the plant
vector p is changed to −p
Parameter values:
μ = 0.5
ρ = 0.05
δ = 0.01
α=0
fi (0) = 10−4

15. NUMERICAL SIMULATIONS
Example 2
A perturbation in the plant takes place at n = 2500, whereby the plant vector
p is shifted to the right by 12 samples, changing the position of all
active coefficients
{1, 30, 35, 85}
⇓
{13, 42, 47, 97}
Parameter values:
μ = 0.5
ρ = 0.05
δ = 0.01
α=0
fi (0) = 10−4

16. CONCLUSIONS
The IAF-PNLMS algorithm uses an individual activation factor for each
adaptive filter coefficient.
The IAF-PNLMS algorithm presents better gain distribution than the
PNLMS and IPNLMS algorithms.
The IAF-PNLMS algorithm provides an improvement in convergence
speed for plant impulse responses having high sparseness.