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Normalized Least Mean Squares (LMS) Algorithms and Block Adaptive Filters - Prof. Yingbo H, Study notes of Electrical and Electronics Engineering

University of California-RiversideElectrical and Electronics Engineering
Prof. Yingbo Hua

An in-depth analysis of various lms algorithms, including normalized lms algorithms and block adaptive filters. The standard lms algorithm, normalized lms algorithms, and block lms algorithm. It also discusses the concept of 'averaged estimate' and the choice of step size for these algorithms. Additionally, the document introduces the fast lms algorithm and frequency-domain normalization.

Typology: Study notes

Pre 2010

Uploaded on 03/28/2010

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Page 1 - Note 3 by Y. Hua
Normalized LMS Algorithm
• Recall the standard LMS algorithm:
)()()(
ˆ
)1(
ˆ*nennn uww
µ
+=+
• Normalized LMS algorithms:
)()(
)(
1
)(
ˆ
)1(
ˆ*
2nen
n
nn u
u
ww +=+
)()(
)(
~
)(
ˆ
)1(
ˆ*
2nen
n
nn u
u
ww
µ
+=+
)()(
)(
~
)(
ˆ
)1(
ˆ*
2nen
na
nn u
u
ww +
+=+
µ
• The first normalized LMS algorithm follows from "minimal
disturbance":
()
2
1
ˆ)(
ˆ
)1(
ˆ
min nn
nww
w−+
+
subject to )()()1(
ˆndnn
H=+ uw
pf3
pf4
pf5

Partial preview of the text

Download Normalized Least Mean Squares (LMS) Algorithms and Block Adaptive Filters - Prof. Yingbo H and more Study notes Electrical and Electronics Engineering in PDF only on Docsity!

Normalized LMS Algorithm

  • Recall the standard LMS algorithm:

w n + = w n + μ u n e n

  • Normalized LMS algorithms:

2

n e n n

n n u u

w + = w +

2

n e n n

n n u u

w w

2

n e n a n

n n u u

w w

  • The first normalized LMS algorithm follows from " minimal

disturbance" :

2 min (^) w ˆ n + 1 w ˆ( n + 1 )− w ˆ( n )

subject to ˆ ( n 1 ) ( n ) d ( n )

H w + u =

Block Adaptive Filters

  • Recall the definitions:

u ( n )= [ u ( n ), u ( n − 1 ),...., u ( n − M − 1 )]

w ( n ) = [ w 0 ( n ), w 1 ( n ),..., wM − 1 ( n )]

  • Block data processing:

y ( kL i ) ( ) ( k kL i )

H − = w ˆ u − ; i = 0 , 1 ,..., L − 1

  • Block LMS Algorithm:

( ) ( ) (^) ∑ ( ) ( )

−

=

1

0

ˆ 1 ˆ

L

i

w k w k μ u kL i e kL i

  • Note: (^) ∑ (^ )^ (^ )

−

=

1

0

L

i

kL i e kL i L

u is the "averaged estimate" of

the gradient E ( ( ) n e ( ) n )

− 2 u.

  • The choice of step size (from an approximate analysis):

L × tr ( R )

0 μ (Normally, L = M )

  • For each k , use " overlap-and-save FFT" to compute the

vector:

∑^ (^ ) (^ )

−

=

1

0

M

i

φ k u kM i e kM i ;

Define

( k ) circularly _ reversed _ vector _ of _ ' a ( k )

R u (^) a = u

k

a k e

e

where [ ]

T e ' ( k )= e ( kM − M + 1 ) L e ( kM )

Then

( k ) first _ M _ elements _ of _ ( ( k ) 2 M ' a ( k ))

R

φ = u a ⊗ e

  • Note that:

2

IFFT FFT k FFT k

IFFT FFT k FFT k

k k

a a

a

R a

M a

R a

u e

u e

u e

  • How much is the computational saving?

Frequency-Domain Normalization

  • Define

v (^) a ( k )=ˆ FFT ( u (^) a ( k ))

, 2 1

, 0

v k

v k

k

a M

a

v a M

where each element corresponds to a frequency bin.

  • To track the power in the i-th frequency bin:

2

Pi ( k )= γ P i ( k − 1 )+( 1 − γ) va , i ( k ) ; i = 0 , 1 ,..., 2 M − 1

  • To ensure (approximately) equal rate of convergence for all

frequency components (i.e., modes ):

, , P k

v k v k i

ai a i ←

where "←^ " here means "replaced by".