Signal Processing - January 2016 - 126

EMSE (dB)

-32
-34

le
vab

SE

EM

LMS Achievable EMSE

e

S
RL

-36

hi
Ac

µ0-LMS
β0-RLS
Combination

Combination
Exclusive Area

-38
0

0.2

0.4

α

0.6

0.8

1

[fig3] the tracking performance of a combination of lms and
rls filters when q smoothly changes between R -1 (for a = 0)
and r (for a = 1) . the performance of optimally adjusted filters
is depicted with curves. the blue and light green regions
represent emses that can be obtained with lms and rls filters,
respectively, whereas the dark green area comprises feasible
emse values for both lms and rls filters. finally, the red region
contains emse values that can be obtained with combinations of
lms and rls (but not with these filters individually).

we can reinterpret the adaptation of m (n) as a "second layer"
adaptive filter of length one, so that in principle any adaptive rule
can be used for adjusting the mixing parameter. However, this filtering problem has some particularities, specifically, the strong
and time-varying correlation between y 1 (n) and y 2 (n). This
implies that the power of the difference signal, [y 1 (n) - y 2 (n)], is
also time-varying depending, e.g., on the signal-to-noise conditions and on whether the individual filters are operating in convergence phase or steady state, etc.
Using a stochastic gradient search to minimize the quadratic
error e 2 (n) of the overall filter, defined in (4), [16] proposed the
following adaptation for the mixing parameter in an affine combination (i.e., without imposing restrictions on m (n)),
n m 2e 2 (n)
m (n + 1) = m (n) 2 2m (n)

= m (n) + n m e (n) [y 1 (n) - y 2 (n)] .

(14)

We will refer to this case as aff-LMS adaptation, with n m being a
step-size parameter. As discussed in [16], a large step size should
be used in this rule to ensure an adequate behavior of the affine
combination. However, this can cause instability during the initial
convergence of the algorithm, which was circumvented in [16] by
[table 4] the Power normalized adaPtation of m (n)
in affine combinations and of auxiliary Parameter
a (n) in convex combinations. unnormalized rules
aff-lms and cvx-lms are obtained setting p (n) = 1.
algorithm

uPdate equations

aff-PN-LMS [18]

m (n + 1) = m (n) +

nm
e (n) [y 1 (n) - y 2 (n)]
f + p (n)

p (n) = h p (n - 1) + (1 - h) [y 1 (n) - y 2 (n)] 2
0 % h 1 1; f 2 0 is a sMaLL Constant
cvx-PN-LMS [37]

a (n + 1) = a (n) +

na

p (n)

m (n) [1 - m (n)] e (n) [y 1 (n) - y 2 (n)]

constraining m (n) to be less than or equal to one. Even with this
constraint, the combination scheme may stay away from the optimum EMSE (see, e.g., [18]). This is a direct consequence of the
time-varying power of [y 1 (n) - y 2 (n)], which makes the selection
of n m a difficult task.
To obtain a more robust scheme, it is possible to recur to normalized adaptation schemes. Using a rough (low-pass filtered) estimation of the power of [y 1 (n) - y 2 (n)], a power normalized
version of aff-LMS was proposed in [18], and is summarized in
Table 4. This aff-PN-LMS algorithm does not impose any constraints on m (n), is less sensitive to the filtering scenario, and
converges in the mean-square sense if the step size is selected in
the interval 0 1 n m 1 2.
Rather than directly adjust m (n) as in the affine case, convex
combination schemes recur to activation functions to keep the
mixing parameter in the range of interest. For example, [12] proposed an adaptation scheme for an auxiliary parameter a (n) that
is related to m (n) via the sigmoid function
m (n) = sgm [a (n)] =

(15)

Recurring to this activation function (or similar ones), a (n) can
be adapted without constraints, and m (n) will be automatically
kept inside the interval (0, 1) at all times. Using a gradient descent
method to minimize the quadratic error of the overall filter,
e 2 (n), two algorithms were proposed to update a (n): the cvxLMS algorithm [12] and its power normalized version, cvx-PNLMS [37], whose update equations are given by
a (n + 1) = a (n) + n a m (n)
[1 - m (n)] e (n) [y 1 (n) - y 2 (n)],
a (n + 1) = a (n) +

(cvx-LMS)

(16)

(cvx-PN-LMS)

(17)

na

m (n)
p (n)
[1 - m (n)] e (n) [y 1 (n) - y 2 (n)] .

Here, p (n) is a low-pass filtered estimation of the power of
[y 1 (n) - y 2 (n)] . These algorithms are also shown in Table 4 for
further reference. Compared to the cvx-LMS scheme, cvx-PNLMS is more robust to signal-to-noise ratio (SNR) changes, and
simplifies the selection of step size n a [37]. [According to the
linear regression model (1), the SNR is defined as SNR =
[w 


http://www.1.pn
Table of Contents for the Digital Edition of Signal Processing - January 2016

Signal Processing - January 2016 - Cover1

Signal Processing - January 2016 - Cover2

Signal Processing - January 2016 - 1

Signal Processing - January 2016 - 2

Signal Processing - January 2016 - 3

Signal Processing - January 2016 - 4

Signal Processing - January 2016 - 5

Signal Processing - January 2016 - 6

Signal Processing - January 2016 - 7

Signal Processing - January 2016 - 8

Signal Processing - January 2016 - 9

Signal Processing - January 2016 - 10

Signal Processing - January 2016 - 11

Signal Processing - January 2016 - 12

Signal Processing - January 2016 - 13

Signal Processing - January 2016 - 14

Signal Processing - January 2016 - 15

Signal Processing - January 2016 - 16

Signal Processing - January 2016 - 17

Signal Processing - January 2016 - 18

Signal Processing - January 2016 - 19

Signal Processing - January 2016 - 20

Signal Processing - January 2016 - 21

Signal Processing - January 2016 - 22

Signal Processing - January 2016 - 23

Signal Processing - January 2016 - 24

Signal Processing - January 2016 - 25

Signal Processing - January 2016 - 26

Signal Processing - January 2016 - 27

Signal Processing - January 2016 - 28

Signal Processing - January 2016 - 29

Signal Processing - January 2016 - 30

Signal Processing - January 2016 - 31

Signal Processing - January 2016 - 32

Signal Processing - January 2016 - 33

Signal Processing - January 2016 - 34

Signal Processing - January 2016 - 35

Signal Processing - January 2016 - 36

Signal Processing - January 2016 - 37

Signal Processing - January 2016 - 38

Signal Processing - January 2016 - 39

Signal Processing - January 2016 - 40

Signal Processing - January 2016 - 41

Signal Processing - January 2016 - 42

Signal Processing - January 2016 - 43

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Signal Processing - January 2016 - 45

Signal Processing - January 2016 - 46

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Signal Processing - January 2016 - 64

Signal Processing - January 2016 - 65

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Signal Processing - January 2016 - 67

Signal Processing - January 2016 - 68

Signal Processing - January 2016 - 69

Signal Processing - January 2016 - 70

Signal Processing - January 2016 - 71

Signal Processing - January 2016 - 72

Signal Processing - January 2016 - 73

Signal Processing - January 2016 - 74

Signal Processing - January 2016 - 75

Signal Processing - January 2016 - 76

Signal Processing - January 2016 - 77

Signal Processing - January 2016 - 78

Signal Processing - January 2016 - 79

Signal Processing - January 2016 - 80

Signal Processing - January 2016 - 81

Signal Processing - January 2016 - 82

Signal Processing - January 2016 - 83

Signal Processing - January 2016 - 84

Signal Processing - January 2016 - 85

Signal Processing - January 2016 - 86

Signal Processing - January 2016 - 87

Signal Processing - January 2016 - 88

Signal Processing - January 2016 - 89

Signal Processing - January 2016 - 90

Signal Processing - January 2016 - 91

Signal Processing - January 2016 - 92

Signal Processing - January 2016 - 93

Signal Processing - January 2016 - 94

Signal Processing - January 2016 - 95

Signal Processing - January 2016 - 96

Signal Processing - January 2016 - 97

Signal Processing - January 2016 - 98

Signal Processing - January 2016 - 99

Signal Processing - January 2016 - 100

Signal Processing - January 2016 - 101

Signal Processing - January 2016 - 102

Signal Processing - January 2016 - 103

Signal Processing - January 2016 - 104

Signal Processing - January 2016 - 105

Signal Processing - January 2016 - 106

Signal Processing - January 2016 - 107

Signal Processing - January 2016 - 108

Signal Processing - January 2016 - 109

Signal Processing - January 2016 - 110

Signal Processing - January 2016 - 111

Signal Processing - January 2016 - 112

Signal Processing - January 2016 - 113

Signal Processing - January 2016 - 114

Signal Processing - January 2016 - 115

Signal Processing - January 2016 - 116

Signal Processing - January 2016 - 117

Signal Processing - January 2016 - 118

Signal Processing - January 2016 - 119

Signal Processing - January 2016 - 120

Signal Processing - January 2016 - 121

Signal Processing - January 2016 - 122

Signal Processing - January 2016 - 123

Signal Processing - January 2016 - 124

Signal Processing - January 2016 - 125

Signal Processing - January 2016 - 126

Signal Processing - January 2016 - 127

Signal Processing - January 2016 - 128

Signal Processing - January 2016 - 129

Signal Processing - January 2016 - 130

Signal Processing - January 2016 - 131

Signal Processing - January 2016 - 132

Signal Processing - January 2016 - 133

Signal Processing - January 2016 - 134

Signal Processing - January 2016 - 135

Signal Processing - January 2016 - 136

Signal Processing - January 2016 - 137

Signal Processing - January 2016 - 138

Signal Processing - January 2016 - 139

Signal Processing - January 2016 - 140

Signal Processing - January 2016 - 141

Signal Processing - January 2016 - 142

Signal Processing - January 2016 - 143

Signal Processing - January 2016 - 144

Signal Processing - January 2016 - 145

Signal Processing - January 2016 - 146

Signal Processing - January 2016 - 147

Signal Processing - January 2016 - 148

Signal Processing - January 2016 - 149

Signal Processing - January 2016 - 150

Signal Processing - January 2016 - 151

Signal Processing - January 2016 - 152

Signal Processing - January 2016 - 153

Signal Processing - January 2016 - 154

Signal Processing - January 2016 - 155

Signal Processing - January 2016 - 156

Signal Processing - January 2016 - 157

Signal Processing - January 2016 - 158

Signal Processing - January 2016 - 159

Signal Processing - January 2016 - 160

Signal Processing - January 2016 - 161

Signal Processing - January 2016 - 162

Signal Processing - January 2016 - 163

Signal Processing - January 2016 - 164

Signal Processing - January 2016 - 165

Signal Processing - January 2016 - 166

Signal Processing - January 2016 - 167

Signal Processing - January 2016 - 168

Signal Processing - January 2016 - Cover3

Signal Processing - January 2016 - Cover4


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