A Fullband Error Adaptation Scheme for Subband Adaptive Filters 1

Katholieke Universiteit Leuven

Departement Elektrotechniek ESAT-SISTA/TR 1999-103

A Fullband Error Adaptation Scheme for Subband Adaptive Filters ¹

Koen Eneman, Marc Moonen ² December 1999

1

This report is available by anonymous ftp from ftp.esat.kuleuven.ac.be in the directory pub/SISTA/eneman/reports/99.103.ps.gz

2

ESAT (SISTA) - Katholieke Universiteit Leuven, Kardinaal Mercier- laan 94, 3001 Leuven (Heverlee), Belgium, Tel. 32/16/321809, Fax 32/16/321970, WWW: http://www.esat.kuleuven.ac.be/sista. E-mail:

koen.eneman@esat.kuleuven.ac.be. Marc Moonen is a Research Associate with the F.W.O. (Fund for Scientic Research { Flanders). This research work was carried out at the ESAT laboratory of the Katholieke Universiteit Leuven, in the frame of the Belgian State, Prime Minister's Oce { Federal Oce for Sci- entic, Technical and Cultural Aairs { Interuniversity Poles of Attraction Pro- gramme { IUAP P4{02 (1997{2001) : Modeling, Identication, Simulation and Control of Complex Systems, the Concerted Research Action MIPS (`Model{

based Information Processing Systems') and GOA{MEFISTO{666 (Mathemat-

ical Engineering for Information and Communication Systems Technology) of

the Flemish Government and Research Project F.W.O. nr. G.0295.97 (`Design

and implementation of adaptive digital signal processing algorithms for broad-

band applications') The scientic responsibility is assumed by its authors.

A Fullband Error Adaptation Scheme for Subband Adaptive Filters

Koen Eneman Marc Moonen ESAT { Katholieke Universiteit Leuven

Kardinaal Mercierlaan 94, B{3001 Heverlee { Belgium phone : +32/16321809 +32/16321060

fax : +32/16321970

email : koen.eneman@esat.kuleuven.ac.be marc.moonen@esat.kuleuven.ac.be

Abstract

For many years now, subband and frequency{domain adaptive ltering techniques have been proposed for the identication of high{order FIR systems. Standard fullband algorithms are less attractive as the imple- mentation cost is higher and their convergence behaviour is worse.

Subband processing has many desirable properties. However, when used to implement adaptive lters, various side eects occur which reduce per- formance. On the other hand, frequency{domain adaptive lters, such as the PBFDAF, do not suer from these problems despite being (nearly) equivalent to subband adaptive lters be it with \poor" lter banks.

In this paper an alternative fullband error based adaptation scheme for subband adaptive systems will be proposed. It will be shown that the weight updating mechanism of the so{called unconstrained PBFDAF is closely related to the proposed fullband error adaptation algorithm.

1 Introduction

Subband and frequency{domain adaptive schemes have been a topic of interest for many years now. They are employed to identify high{order FIR systems and are a promising alternative for standard fullband adaptation algorithms such as LMS. Still, with the available multirate techniques, it is dicult to meet all the requirements.

Frequency{domain techniques are well understood [1] and hence their perfor- mance is very tractable. On the other hand, subband adaptive lters |at rst sight| may have a lower complexity and a better performance. Unfortunately, this picture of the subband approach is certainly too optimistic.

In this paper we will focus on an alternative fullband error adaptation algorithm

for subband adaptive systems and show that the unconstrained PBFDAF uses

this fullband error adaptation scheme to update its lter weights. This is an attempt to generalise and extend the frequency{domain aliasing{compensation techniques to subband adaptive systems.

2 Subband Adaptive Filtering

2.1 General setup

The general setup for a subband adaptive system is shown in gure 1. The loud-

...

...

...

... ...

+ -

-

+ +

-

near-end signal

far-end signal

error signal

adaptive filters

analysis filter bank synthesis filter bank

+ +

+ +

e

⁰

¹

M^;1 d⁰ d¹

dM^;1 y⁰ y¹

yM^;1

L

L

L

L

L L

L L L

d=^s+^w?x

s

x

f f

f i = 0 i = 1

i = M^;1

H0

H0

H1

H1

HM^;1

HM^;1

G0

G1

GM^;1

F0

F1

FM^;1

w[^k]

Figure 1: Subband adaptive lter with ideal lter banks : echo cancellation setup

speaker and microphone are added for convenience, indicating how the adaptive lter structure may be employed in an acoustic echo cancellation setup. While acoustic echo cancellation has been a driving application for many researchers in this eld, all results obviously apply to other applications too.

The input signals x and d are fed into identical M {band analysis lter banks.

After subsampling with a factor L , (mostly LMS{based) adaptive ltering is

done in each subband. The outputs of the subband adaptive lters are recom-

bined in the synthesis lter bank and fed to the output. The ideal characteristics

of the analysis bank lters H i and synthesis bank lters G i are shown (ideal

bandpass lters). Due to aliasing eects, this setup will only work for M

^>

L .

+

... ... ...

+

... ...

... ...

+

F0

H (z)

H (z)

JG T

(z) L

L

L

L

L

L

L L L

z

;1

z

;1

z

;1

z

;1

z

;1

z

;1 j= 0

j= 0 x d

e j=L^;1

j=L^;1

F1

F

M;1

Figure 2: Subband adaptive lter : polyphase implementation

2.2 Polyphase implementation

The output of the analysis bank lters is immediately downsampled. Hence it is cheaper to do all lter operations at the downsampled rate. By re{arranging gure 1 we obtain gure 2.

^H

( z ) and

^G

( z ) are respectively called the analysis and synthesis polyphase matrix. J is the anti{diagonal matrix. Element ( i;j ) of

^H

( z ) is

[

^H

( z )] _ij = H _i

^j:L

( z )



i = 0

^!

M

^;

1

j = 0

^!

L

^;

1 (1) H i

^j:L

( z ) is the j {th out of L polyphase component of the i {th subband lter h i [ k ], in other words the z {transform of h i [ j + Lk ]. Similarly,

[

^G

( z )] _ij = G i

^j:L

( z )



i = 0

^!

M

^;

1

j = 0

^!

L

^;

1 (2)

2.3 DFT modulated subband adaptive lters

Subband adaptive systems are often based on DFT modulated lter banks. M subband lters are derived from a single prototype lter h

⁰

[ k ] :

h _i [ k ] = h

⁰

[ k ] e

^;

^j

^{2 k iM}

i = 0

^!

M

^;

1 (3)

,

H i ( z ) = H

⁰

( e ^j

^{2 iM}

z ) (4) It appears that the analysis lters are frequency shifted versions of each other and each lter covers a part of the frequency spectrum.

DFT modulated lter banks may be implemented eciently using polyphase

decomposition and fast signal transforms. In [2] a general framework for DFT

+

... ...

... ... ...

... ...

+ +

F

F

F

;1

F0

B(z)

B(z)

C(z) L

L

L

L

L

L

L L L

z

;1

z

;1

z

;1

z

;1

z

;1

z

;1 j= 0

j= 0 x d

e j=L^;1

j=L^;1

F1

F

M;1

Figure 3: DFT modulated subband adaptive lter

modulated subband systems was proposed. A DFT modulated lter bank with L {fold downsampling can be implemented as a tapped delay line of size L fol- lowed by a structured M

^

L {matrix

^B

( z ), containing polyphase components of the prototype h

⁰

, and an M

^

M {DFT matrix

^F

. For DFT modulated lter banks gure 2 can be redrawn resulting in gure 3. The synthesis bank is con- structed in a similar fashion with an L

^

M {matrix

^C

( z ).

By splitting signals into subbands and subsequent subsampling faster (initial) convergence and better tracking properties are hoped for. As the adaptive computations as well as the lter bank convolutions can be done at a reduced sampling rate, the subband approach is supposed to give a better performance at a lower cost. Unfortunately, this picture of the subband approach is certainly too optimistic [4].

3 Frequency{domain Adaptive Filtering

As a cheaper alternative to LMS, the frequency{domain adaptive lter (FDAF) was introduced, which is a direct translation of Block LMS in frequency domain [1]. The implementation cost for the FDAF is low, but the input{output delay introduced by the algorithm is typically too high.

The FDAF can be extended by splitting the acoustic impulse response in equal parts. In this way a kind of mixed time{ and frequency{domain adaptive lter is obtained, called the Partitioned Block Frequency{Domain Adaptive Filter (PBFDAF) [3][6]. Here block lengths can be adjusted, resulting in a cheap adaptive lter with acceptable processing delay.

The L FB {taps fullband adaptive lter w n [ k ] at block index n is partitioned in

L P

^FB

equal parts

^w

p

ⁿ

of length P each :

¹

w

p

^{n 8}

= p

2

6

4

w n [ pP ] w n [( p + 1) ... P

^;

1]

3

7

5

; p = 0 : L FB

P

^;

^{1 (5)}

W

p

^{n 8}

= p

^F



w

p

ⁿ

0



l

P

l

L

^;1

(6)

The equations dening the PBFDAF are

²

:

X

p

^{n 8}

= diag p

8

<

: F

2

6

4

x

[(

ⁿ

+ 1)

^L;pP;M

+ 1]

...

x

[(

ⁿ

+ 1)

^L;pP

]

3

7

5 9

=

; x

?

?

?

?

y

M (7)

y

=



0

P

^{;1 0}

0 I

L



F

;1 L

FB

P

;1

X

p

^{=0 X}

p

^nW

p

ⁿ

(8)

d

=



0

d

n



l

P

^;1

l

L ;

^d

n =

2

6

4

d

[

^nL

+ 1]

...

d

[(

ⁿ

+ 1)

^L

]

3

7

5 x

?

?

?

?

?

y

L (9)

e

=

^d;y

(10)

W

p

^{n+1 8}

= p

^W

p

ⁿ

+

^F



I

P

⁰

0 0

L

^;1



F

;1

^X

_Hp

^nFe

(11) In each iteration, L new x {samples are taken in, and L new lter output sam- ples are produced. L is called the block length, the corresponding input/output delay is 2 L

^;

1.

^F

is an M

^

M DFT matrix,
= 2diag(  m ) contains the sub- band dependent step sizes and M = P + L

^;

1. If P is divisible by L (which is typically the case),

^X

p

ⁿ

=

^X0n;pP=L

and hence equation 7 requires only 1 DFT operation, which corresponds to p = 0. The other

^X

p

ⁿ

can be recovered from previous iterations. Very often M is chosen equal to M = P + L

^;

1 + s = 2 ^r with s > 0 and r an integer. In most practical applications P = L .

There exists two variants of this algorithm, called the constrained and the un- constrained PBFDAF. For the unconstrained version the weight update com- pensation is left out resulting in the following weight update equation :

W

p

^{n+1 8}

= p

^W

p

ⁿ

+

^X

_Hp

^nFe

(12) The unconstrained updating requires 3 DFTs whereas the constrained PBFDAF is more expensive, having an extra

²

^L _P

^FB

DFTs to compute. The latter on the other hand has better convergence properties.

The PBFDAF turns out to be a special subband adaptive lter having inter- esting convergence properties at a low implementation cost [5]. Despite the low

1

We assume that

^LFPB

is integer.

2

For signal conventions : see gure 3

frequency{selective lter banks upon which the PBFDAF is based the algorithm doesn't seem to suer from aliasing eects and hence the convergence properties are comparable to those of LMS.

4 Alternative adaptation scheme

In an attempt to generalise and extend the frequency{domain error correction to subband adaptive systems, we now focus on an alternative adaptation scheme for subband adaptive systems, which adjusts the subband lters F _m (see gure 4) based on the fullband error e instead of using the subband error signals

 m = d m

^;

y m as is normally done in a standard subband scheme (see gure 1).

- - -

... ...

... ...

... ...

+ +

+

+

e z

;1

z

;1

L L L

e0

eL;2

eL;1

d0

d1

d

M;1

x0

x1

xM;1

y0

y1

yM;1

F0

F1

FM^;1

JG T

(z)

JG T

(z)

Figure 4: Oversampled subband adaptive lter : synthesis part

4.1 Fullband error adaptation scheme

Dene :

²

6

4

Y

⁰

^k Y _kM ...

^;1

3

7

5

| {z }

Y

k

=

2

6

4

X

⁰

^k ::: 0 ... ... ...

0 ::: X _kM

^;1

3

7

5

| {z }

X

k

2

6

4

F

⁰

^k F _kM ...

^;1

3

7

5

| {z }

wk

; (13)

in which (see gure 4) X _km =

2

6

4

x m [ k

^;

L f + 1] ::: x m [ k

^;

L f

^;

L SB + 2]

... ... ...

x _m [ k ] ::: x _m [ k

^;

L _SB + 1]

3

7

5 x

?

?

?

?

?

y

L

^f

; L

^SB !

(14)

F _km =

2

6

4

f _km [0]

f _km [ L SB ...

^;

1]

3

7

5 x

?

?

?

?

?

y

L

^SB

; Y _km =

2

6

4

y m [ k

^;

L f + 1]

y m ... [ k ]

3

7

5 x

?

?

?

?

?

y

L

^f

(15)

w

k is the adaptive lter vector. L SB is the length

³

of the subband adaptive lters, L f is the length of the synthesis bank polyphase lters. Further dene

2

6

4

e

⁰

[ k ] e L

^;1

... [ k ]

3

7

5

| {z }

E FB

k

=

2

6

4

S

^00:L

::: S _M

^;10:L

... ... ...

S

^0L;1:L

::: S M

^;1L;1:L

3

7

5

| {z }

S T

2

6

4

 ^k

⁰

 _kM ...

^;1

3

7

5

| {z }

E SB

k

(16)

in which

²

6

4

 ^k

⁰

 _kM ...

^;1

3

7

5

| {z }

E SB

k

=

2

6

4

D ^k

⁰

D _kM ...

^;1

3

7

5

| {z }

D

k

; 2

6

4

Y

⁰

^k Y _kM ...

^;1

3

7

5

| {z }

Y

k

; (17)

D _km =

2

6

4

d m [ k

^;

L f + 1]

d _m ... [ k ]

3

7

5 x

?

?

?

?

?

y

L

^f

;  _km =

2

6

4

 m [ k

^;

L f + 1]

 _m ... [ k ]

3

7

5 x

?

?

?

?

?

y

L

^f

; (18) S _m

^l:L

=

^

g m

^l:L

[ L f

^;

1] ::: g m

^l:L

[0]

^

:

L

^f !

(19)

E

FB k is the fullband error vector,

^E

^SB _k contains the subband errors,

^S

represents the synthesis lter bank and g m

^l:L

[ k ] = g m [ kL + l ] is the l {th out of L polyphase components of the m {th synthesis lter g _m [ k ].

Optimal error suppression is obtained when Eq. 16 is as small as possible : min

w

k E

n

jjE

FB k

^jj22

o

(20)

The optimal

^w

is

w

opt = argmin

wk jjS

T

D

k

^;S

T

X

k

^w

k

^jj22

: (21)

3

We assume that all lters are FIR.

In practice the optimal subband lters are estimated adaptively using a steepest descent algorithm. The gradient with respect to

^w

k is

rw

k = 2

^X

_Hk

^SS

^T (

^X

k

^w

k

^;D

k ) : (22) Therefore,

^w

opt can be found in an iterative way using a fullband error block adaptation algorithm :

w

k

⁺¹

=

^w

k + 2

^X

_Hk

^SS

^T (

^D

k

^;X

k

^w

k ) (23)

=

^w

k + 2

^X

_Hk

^SS

^T

^E

^SB _k (24) with
= diag(  m ) a diagonal matrix with subband dependent stepsizes. The subband errors are passed through the synthesis bank twice before they are combined with the far{end signal x .

In a standard subband adaptation scheme the subband errors are minimised, leading to

min

w

k E

n

jjE

SB k

^jj22

o

: (25)

The gradient with respect to

^w

k is

rw

k = 2

^X

_Hk (

^X

k

^w

k

^;D

k ) : (26) Therefore, the optimal

^w

can be found in an iterative way using subband errors:

w

k

⁺¹

=

^w

k + 2

^X

_Hk

^E

^SB _k : (27) Equation 27 corresponds to a standard subband weights updating algorithm using a block{LMS algorithm in each band. The compensation matrix

^SS

^T is absent here. It is hoped that employing the fullband error adaptation scheme will lead to improved performance.

4.2 PBFDAF weight updating

The alternative adaptation scheme studied in paragraph 4.1, leading to a weight updating equation based on fullband errors (Eq. 24) is now applied to the PBFDAF.

For DFT modulated lter banks the synthesis bank polyphase matrix

^G

( z ) can be written as

F

;

T

C

T ( z ) J: (28)

For the PBFDAF it can be proven [4] that when P is a multiple of L

C

( z ) =

^{ I}

L

⁰

L :::

⁰

L

^{0 }

| {z }

M

=

^C

(29)

is independent of z and hence also

^G

( z ) is independent of z . This means the syn-

thesis polyphase lters have length L f = 1. The PBFDAF is indeed a subband

as dened in Eq. 16 then equals

^G

( z ) =

^S

=

^F;

^T

^C

^T J . For the PBFDAF the update equation 24 becomes

w

k

⁺¹

=

^w

k + 2
M

^X

^Hk

^FC

^T

^CF;1

⁽

^D

^k

^;X

^k

^w

^{k ) (30)}

=

^w

k + 2
M

^X

^Hk

^F



I

L

⁰

0 0

M

^;

L



F

;1

E

SB k : (31)

E

SB k are the subband errors,

^FC

^T

^CF;1

does the error correction. It can be proven (see appendix A) that the weight update equation of a so{called uncon- strained PBFDAF [5][6] for which P is a multiple of L

W

p

^{n+1 8}

= p

^W

p

ⁿ

+

^X

_Hp

^nFe

(32) is closely related to equation 31, derived here based on fullband error feedback.

One can observe that the subband lters are adapted based on corrected sub- band errors. The correction implies passing the subband errors through the synthesis lter bank in order to obtain a fullband, i.e. an aliasing{free, error signal. This in fact refers to the overlap{save or overlap{add procedure upon which frequency{domain techniques are based. It is believed that a more gen- eral class of subband adaptive systems which are based on a lter bank set fullling 3 realisation conditions of frequency selectivity, perfect reconstruction and time{invariance modelling [4], and provided with the adaptation mecha- nism presented in paragraph 4.1, could approximate the performance of the frequency{domain approach.

5 Conclusions

For the identication of high{order FIR systems ecient adaptive algorithms, such as subband and frequency{domain adaptive lters, are required. Frequency{

domain adaptive algorithms turn out to have many desirable properties. We presented an adaptation scheme that feeds back the fullband error, which can be considered as a generalisation and extension of frequency{domain error cor- rection techniques to subband adaptive systems.

6 Acknowledgements

Marc Moonen is a Research Associate with the F.W.O. (Fund for Scientic Re- search { Flanders). This research work was carried out at the ESAT laboratory of the Katholieke Universiteit Leuven, in the frame of



the Belgian State, Prime Minister's Oce { Federal Oce for Scientic,

Technical and Cultural Aairs { Interuniversity Poles of Attraction Pro-

gramme { IUAP P4{02 (1997{2001) : Modeling, Identication, Simulation

and Control of Complex Systems



the Concerted Research Action MIPS (`Model{based Information Pro- cessing Systems') and GOA{MEFISTO{666 (Mathematical Engineering for Information and Communication Systems Technology) of the Flemish Government



Research Project F.W.O. nr. G.0295.97 (`Design and implementation of adaptive digital signal processing algorithms for broadband applications') The scientic responsibility is assumed by its authors.

References

[1] J. Shynk, \Frequency{Domain and Multirate Adaptive Filtering," IEEE Sig- nal Processing Magazine, pp. 15{37, January 1992.

[2] Z. Cvetkovic, \Oversampled Modulated Filter Banks and Tight Gabor Frames in l

²

(

^Z

)," in Proceedings of the 1995 IEEE Int. Conf. on Acoust., Speech and Signal Processing, (Detroit, Michigan, USA), pp. 1456{1459, May 1995.

[3] J.{S. Soo and K. Pang, \Multidelay Block Frequency Domain Adaptive Fil- ter," IEEE Trans. Acoust., Speech and Signal Processing, vol. 38, pp. 373{

376, February 1990.

[4] K. Eneman and M. Moonen, \Hybrid Subband/Frequency{Domain Adap- tive Systems," Tech. Rep. 99{80, ESAT{SISTA, Katholieke Universiteit Leu- ven, Belgium, December 1999.

[5] K. Eneman and M. Moonen, \Filter Bank Constraints for Subband and Frequency{Domain Adaptive Filters," in Proceedings of the 1997 IEEE Workshop on Applications of Signal Processing to Audio and Acoustics, (New Paltz, New York, USA), October 19{22 1997.

[6] J. Paez Borrallo and M. Garca Otero, \On the implementation of a parti- tioned block frequency domain adaptive lter (PBFDAF) for long acoustic echo cancellation," Signal Processing, vol. 27, pp. 301{315, June 1992.

A PBFDAF vs. fullband error adaptation scheme

We rst give a slightly dierent, but equivalent description of the fullband error adaptation algorithm presented in paragraph 4.1. The denition of

^X

k ,

^w

k and F _kl are dierent and were obtained by re{arranging the corresponding equations in paragraph 4.1 :

2

6

4

Y

⁰

^k Y _kM ...

3

7

5

=

^

X [ k ] ::: X [ k

^;

L SB + 1]

^

| {z }

2

6

4

F 

⁰

^k

 ...

F _kL

3

7

5

; (33)

in which

X [ n ] =

2

6

4



⁰

[ n ] ::: 0 ... ... ...

0 :::  _M

^;1

[ n ]

3

7

5 x

?

?

?

?

?

y

ML

^f

; M

!

(34)

 _m [ n ] =

2

6

4

x _m [ n

^;

L _f + 1]

x m ... [ n ]

3

7

5

; (35)

F  _kl =

2

6

4

f

⁰

^k [ l ] f _kM

^;1

... [ l ]

3

7

5

; Y _km =

2

6

4

y m [ k

^;

L f + 1]

y _m ... [ k ]

3

7

5

(36)

The other equations 16,17,18 and 19 are left unchanged. Apparently,

w

k =

^Pw

 k ;

^X

k = 

^X

k

^P

T ; (37) with

^P

a permutation matrix, such that 

^w

k being a vector with L SB blocks of M elements can be projected on

^w

k having M blocks of L SB elements. It appears that

^Y

k = 

^X

k

^w

 k =

^X

k

^w

k has not changed. Hence, this set of equations is equivalent to the denitions given in paragraph 4.1. Also the weight updating equations 24 and 31 are still valid :



w

k

⁺¹

= 

^w

k + 2 

M

^X

^{ H k}

^F



I

L

⁰

0 0

M

^;

L



F

;1

E

SB k : (38) Equation 38 holds for L f = 1 and

^C

( z ) =

^

I L 0

^

which is the case for the PBFDAF as will be shown.

Recall the equations dening the unconstrained PBFDAF :

X

p

^{k 8}

= diag p

8

<

: F

2

6

4

x

[(

^k

+ 1)

^L;pP;M

+ 1]

...

x

[(

^k

+ 1)

^L;pP

]

3

7

5 9

=

;

; p = 0 : L FB

P

^;

^{1 (39)}

y

=



0

P

^;1+

s

⁰

0 I

L



F

;1 L

FB

P

;1

X

p

⁼⁰

X

p

^kW

p

^k

(40)

d

=



0

P

^;1+

s

⁰

0 I

L

 2

6

4

d [( k + 1) L

^;

M + 1]

d [( k + 1) ... L ]

3

7

5

(41)

e

=

^d;y

(42)

W

p

^{k +1 8}

= p

^W

p

^k

+

^X

_Hp

^{kF e}

; M = P + L

^;

1 + s (43)

If P is a multiple of L it can be proven [4] that the unconstrained PBFDAF

is a subband adaptive system for which in each of the M subbands there is an

L P

^FB

{taps adaptive FIR lter.

^X

p

^k

then contains all the subband samples at time lag p _PL w.r.t. the actual time k + 1.

By time{reversing equation 40, J

^y

=



I

L

⁰

0 0

P

^;1+

s



J

^F;1

L

FB

P

;1

X

p

⁼⁰

X

p

^kW

p

^k

(44)

=



I

L

⁰

0 0

P

^;1+

s



F

;1 L

FB

P

;1

X

p

^{=0 F}

J

^F;1X

p

^kW

p

^k

: (45) Now, the following holds :

M 1

^F

J

^F

=

2

6

4

e ^j

^{2 0M}

::: 0 ... ... ...

0 ::: e ^j

^{2 (M;1)M}

3

7

5

=  : (46) Call

v

p =

^F;1X

p

^kW

p

^k

; (47) then

F

J

^v

p = M 

^F;1v

p = 

^Fv

p : (48) Equation 45 becomes

J

^y

=



I

L

⁰

0 0

P

^;1+

s



F

;1 L

FB

P

;1

X

p

⁼⁰



^Fv

p (49)

=



I

L

⁰

0 0

P

^;1+

s



F

;1 L

FB

P

;1

X

p

⁼⁰

(

^Fv

p )

^

: (50)

v

p =

^v

_p because d [ k ] and x [ k ] and the unknown system in between are assumed to be real{valued, such that

J

^y

=



I

L

⁰

0 0

P

^;1+

s



F

;1 L

FB

P

;1

X

p

⁼⁰



^X

_p

^kW

_p

^k

: (51) Based on Eq. 39 and 46, we obtain

diag

0

B

@F

J

2

6

4

x [( k + 1) L

^;

pP

^;

M + 1]

x [( k + 1) ... L

^;

pP ]

3

7

5 1

C

A

= 

^X

_p

^k

; (52) so that nally

J =



L

^

L

FB

P

;1

 (53)

with



X

p

^k

= diag

0

B

@ F

2

6

4

x [( k + 1) L

^;

pP ] x [( k + 1) L

^;

... pP

^;

M + 1]

3

7

5 1

C

A

= 

^X

_p

^k

: (54) By time{reversing also the other equations of the PBFDAF

J

^d

=



I

L

⁰

0 0

P

^;1+

s

 2

6

4

d [( k + 1) L ] d [( k + 1) L ...

^;

M + 1]

3

7

5

; (55)

J

^e

= J

^d;

J

^y

(56)

Eq. 54, 53, 55 and 56 replace Eq. 39, 40, 41 and 42. All vectors appear to be time{reversed and the weight vector is complex conjugated. Applying the same transformation to the weight update equation 43 :

W

p

^{k +1 8}

= p

^W

p

^k

+

^X

_Hp

^kFe

(57)

W



p

^{k +1 8}

= p

^W

_p

^k

+

^X

_Tp

^kF

JJ

^e

: (58) By applying formula 46 and 54

W



p

^{k +1 8}

= p

^W

_p

^k

+

^X

_Tp

^k



^F

J

^e

(59) one can observe that the weight update formula becomes

W



p

^{k +1 8}

= p

^W

_p

^k

+


^X

^H _p

^kF

J

^e

: (60) By stacking the previous equation

⁸

p , we obtain

2

6

6

4 W



0

...

k +1 W



L

FB

P

;1

k +1 3

7

7

5

=

2

6

4 W



0

...

k W



L

FB

P

;1

k 3

7

5

+

2

6

6

4



X

H

0

...

k



X

H

L

FB

P

;1

k 3

7

7

5

F

J

^e

(61) in which

J

^e

=



I

L

⁰

0 0

P

^;1+

s



F

;1

E

(62)

and

E

=

^F

2

6

4

d [( k + 1) L ] d [( k + 1) L ...

^;

M + 1]

3

7

5

;

h ^X



⁰k

:::

^XL

^

^FB

P

;1

k i

2

6

4 W



0

...

k W



L

FB

P

;1

k 3

7

5

:

(63)

Equation 61 corresponds to equation 38 if we take P = L and dene



w

k =

2

6

4 W



...

0k W



L

FB

P

;1

k 3

7

5

(64)



X

k =

^{h X}



⁰k

:::

^XL

^

^FB

P

;1k

i

(65)

and hence

X [ k

^;

l ] = 

^X

l

^k

(66)

F  _kl =

^W

_l

^k

(67)

D

k =

^F

2

6

4

d [( k + 1) L ] d [( k + 1) L ...

^;

M + 1]

3

7

5

(68)

E

SB k =

^E

(69)

Y

k =

^{h X}



0k

:::

^X

^

^LFB

P

;1

k i

2

6

4 W



...

0k W



L

FB

P

;1

k 3

7

5

(70)

L f = 1 (71)

L SB = L FB

P : ⁽⁷²⁾

This proves that apart from a slightly dierent denition of the adaptation matrix
the weight update equation of the unconstrained PBFDAF, as it is mostly used in practice with P = L , corresponds to equation 38, which was derived based on fullband error feedback.

If P is multiple of L with _PL > 1, the equations do not match exactly anymore.

In that case it suces to dene



X

k =

^

X [ k ] X [ k

^;

_PL ] ::: X [ k

^;

_PL ( L SB

^;

1)]

^

: (73)