An Adaptive Transmultiplexer by Fast RLS Algorithms and its Application to Multicarrier Communications

Dah-Chung Chang and Da-Long Lee Department of Communication Engineering, National Central University

Jhongli City, Taoyuan 320, Taiwan E-mail: dcchang@ce.ncu.edu.tw

Abstract — We explore an adaptive transmultiplexer (TMUX) technique used for multicarrier communication systems and find the Wiener solution and adaptive reconstruction algorithm. In order to obtain good performance and low computational complexity, a fast recursive least squares (RLS) algorithm is developed. By using the polyphase decomposition method, the adaptive receiver in a filter bank system can be formulated as a multichannel filtering problem. Simulations show that the proposed algorithm, called the block step-up step-down (B-SUSD) algorithm, has the convergence rate close to the standard RLS algorithm. Finally, the new adaptive TMUX receiver is applied to the WLAN system compared to the conventional IEEE 802.11a OFDM system to show the superior performance for a filterbank-based muticarrier communication system.

I. INTRODUCTION

The transmultiplexer (TMUX) is a bandwidth-efficient communication technique that can simultaneously transmit narrowband signals through a single wideband channel. The conventional implementation of TMUXs used the discrete Fourier transform (DFT) [1-2]. Since the filter bank theory has been well developed in signal processing, the TMUX performed by modulated filter banks has received great attention [3-7]. The fundamental operation of TMUXs is similar to that of frequency-division multiplexing. The difference is that the TMUX system uses filter banks to modulate/demodulate transmitted signals, all digitally. The filter bank of a TMUX can be obtained from that of a subband system. Koilpillai, Nguyen, and Vaidyanathan [6] have shown elaborate theories to obtain a crosstalk-free TMUX from an aliasing-free QMF bank. However, the noise and channel distortion, which always exist in a communication system, are not concerned.

Chen and Lin [7] applied Kalman filters to reconstruct the transmitted signals. The drawback of this approach is that parameters required in Kalman filters have to be known in advance, which may be difficult in practical applications. Ramachandran and Kabal [3] added a compensation filter, which acts like an equalizer, in front of each separation filter. Although their method can exactly cancel crosstalks between different channels, residual intersymbol interference remains. In this paper, we explore the adaptive algorithm for the separation filter bank and find the Wiener solution. The recursive least squares (RLS) is a well-known adaptive algorithm with fast converges rate, but it requires extensive computations. Although fast RLS algorithms,

which can substantially reduce computations, have been developed, they cannot be applied directly for this system. This is due to the sampling rate conversion involved in the TMUX system. By exploring polyphase structures, we found that signal reconstruction in a TMUX system can be viewed as a multichannel filtering problem. Using this formulation, we not only simplify the derivation of the Wiener solution, but also can apply multichannel fast RLS algorithms. We develop our algorithm in light of the stabilized block step-up step-down fast RLS (B-SUSD FRLS) algorithm [8]. It is shown that the convergence rate of the stabilized B-SUSD FRLS algorithm is close to that of the standard RLS algorithm, and its performance approaches to the Wiener filter.

The Orthogonal Frequency Division Multiplexing (OFDM) technology has been widely adopted in modern wireless communication systems, like WLAN, DVB-T, and ultra-wideband (UWB) transmission. Although OFDM can implement multicarreir communication systems by fast IDFT/DFT algorithms, the spectrum sidelobe is very high so that the subcarrier orthogonality can be seriously destroyed due to noise, interference, imperfect synchronization and channel estimation. The TMUX can use filter coefficients with low sidelobe such that diminishes the influence existing in the OFDM system. The adaptive TMUX receiver is applied to the IEEE 802.11a system in the simulation. In comparison with the conventional OFDM system, the adaptive receiver effectively reduces the effect of losing orthogonality and better performance is achieved.

II. THE MMSE RECEIVER FOR THE TRANSMULTIPLEXER

A. Polyphase Formulation

f1

+

M f M −1

:

:

Combining Filters

x n0 ( )

x n1( )

x nM−1( )

+

Mh0

Mh1

MhM −1

:

:

Separation FiltersTransmission Channel

y m( )

z m( )

v m( )

x n0 ( )M

M

f0

x n1( )

x nM−1( )

^

^

^

c

Figure 1. An M-band TMUX system

This work was supported in part by the National Science Council

of Taiwan under the contract 94-2220-E-008-003.

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2006 proceedings.

1-4244-0355-3/06/$20.00 (c) 2006 IEEE

3247

In Fig. 1, we depict an M-band TMUX system with noise and channel distortion. The input signals are first upsampled by M-fold expanders and interpolated by the combining filter bank. The resultant signals are then combined to a single channel signal, which is transmitted over a communication channel. In the receiver, the separation filter bank and decimators are used to reconstruct original signals.

Let the length of the separation and combing filters be L , the time index before upsampling be n , and that after upsampling be m . The relation of m and n can be written as m=Mn+l or m=Mn-l, where l=0,1,…, M-1. The reconstructed signal )(ˆ nxi in Fig. 1 can be expressed as

∑−

=

−=1

0

)()()(ˆL

ii Mnzhnxτ

ττ

where )(nhi denotes the nth coefficient of the ith separation filter. As we can see from (1), the input vector for )(⋅h does not have the time-shift property. In other words, the present input vector and the previous input vector do not overlap L-1 elements. As a consequence, fast RLS algorithms cannot be applied. Although we can still apply fast RLS filtering before downsampling, it will require M times computations. Here, we propose a structure that can overcome this problem. Using this structure, derivation of the Wiener solution also becomes simpler.

Let lMk +=τ and MLK /= be an integer. Then, we have

∑ ∑

∑∑−

=

−

=

−

=

−

=

−=

−−+=

1

0

1

0

1

0

1

0

)()(

)()()(ˆ

M

l

K

klli

M

l

K

kii

knzkh

lMkMnzlMkhnx

where )()( lMnhnh ili += are the polyphase

components of )(mhi , and )()( lMnznzl −= are the

polyphase components of )(mz . Equation (2) can be interpreted as a result of multichannel filtering. There are M-channel input signals )(nzl and M filters ),(nhli

.1,...,1,0 −= Mi This leads to the reconstruction structure depicted in Fig. 2. The device marked by “MMSE” is some adaptive algorithm used for adjusting filter coefficients as that we will develop in the next section. To facilitate the derivation of the Wiener solution, we rearrange

)(nhli and ),(mzl 1,...,1,0 −= Ml , and express (2) as a product of two vectors. Define

TL

TKi

Ti

Tii 11,1,0, ] [ ×−= hhhh L

TL

TS

TS

TS Knnnn 1)]1()1( )([)( ×+−−= zzzz L

where TMiMiiki khkhkh 1,110, )]()( )([ ×−= Lh TMMS nznznzn 1110 )]()( )([)( ×−= Lz

Thus, (2) can be written as )()(ˆ nnx T

ii zh= B. The Wiener Solution

Let the channel impulse response be{ }110 −Nccc L , the

polyphase components of )(my be )()( lMnynyl −= ,

and the polyphase components of )(mv be )( lMnvl − .

As we can see from Fig. 1, )(my is the transmitted signal and )(mv is channel noise. For simplicity of representation, we assume that MLNP /)1( −+= is an integer. Define

TLN

TS

TS

TS Pnnnn 1)1()]1()1( )([)( ×−++−−= yyyy L

TL

TS

TS

TS Knnnn 1)]1()1( )([)( ×+−−= vvvv L

where TMMS nynynyn 1110 )]()( )([)( ×−= Ly

TMMS nvnvnvn 1110 )]()( )([)( ×−= Lv

Then, we have the following equation )()()( nnCn vyz +=

where

=

−

−

−

110

110

110

00

0000

N

N

N

ccc

cccccc

C

LK

MOM

LL

LL

We next relate )(nyl to the input signal )(nx . The

signal )(my can be expressed as follows:

∑ ∑−

=

∞

−∞=

−=1

0

)()()(M

i kii kxMkmfmy

where )(mfi denotes the mth coefficient of the ith combining filter. Using the polyphase representation, we can rewrite (8) as

∑ ∑

∑ ∑∑ ∑

−

=

−

=

−

=

∞

−∞=

−

=

∞

−∞=

−=

−=

−−=

1

0

1

0

1

0

1

0

)()(

)()(

)()()(

M

i

K

kili

M

i kili

M

i kiil

knxkf

kxknf

kxlMkMnfny

S/P +

MMSE

−

( )z m

1( )Mz n−

1( )z n

0 ( )z n

ˆ ( )ix n

( )ix n1,M ih −

1,ih

0,ih

Figure 2. The adaptive TMUX receiver structure

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

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3248

where )()( lMnfnf ili −= are the polyphase

components of )(mfi . To have a vector representation of (9), we first define

TL

TMl

Tl

Tll 11,1,0, ] [ ×−= ffff L

TL

TM

TTS nnnn 1110 )]()( )([)( ×−= zxxx L

where TKlililiil Kfff 1, )]1()1( )0([ ×−= Lf

TKiiii Knxnxnxn 1)]1()1( )([)( ×+−−= Lx

Then, (9) can be written as )()( nny S

Tll xf=

Define an extended input vector as TPL

TS

TS

TS Pnnnn 1)]1()1( )([)( ×+−−= xxxx L

Then, )(ny can be obtained as:

)()( nn xy F= where

PLLNF

FF

×−+

=

)1(

OF

and [ ]TLMMF ×−= 110 fff L .

Finally, using (5), (6), and (14), we obtain )()()(ˆ nnCnx T

iTii vhxh += F

The Wiener solution for the separation filters is to find the optimal ih that minimizes the mean square error between

the desired signal )(ndi and the reconstruction signal

)(ˆ nxi . The Wiener solution can be obtained in the following matrix equation:

izdzzi RR 1−=h

where zzR is the correlation matrix of )(nz and izdR is the

cross-correlation matrix of )(nz and )(ndi . From (14) and

(6), we have that )()()( nnCn vxz += F . Then zzR can be calculated by

vTT

xx

Tzz

RCRCnnCnnCER

+=++=

FFF

})]()()][()([ { vxvx F

where }{⋅E denotes the expectation operation, vR is the

covariance matrix of the channel noise, and I2vvR σ=

where vσ is the standard deviation of the noise and I is an

LL × identity matrix. The desired signal )(ndi is the delayed version of the corresponding input signal, i.e.,

)( Dnx − , where D accounts for the total effect of filter banks delay and channel delay. Hence,

})()({ )}()]()({[

DnxnECDnxnnCER

izd

−⋅=−+=

xvy

F

We can find the Wiener solution of ih by (16)-(18).

III. ADAPTIVE RECONSTRUCTION BY FRLS ALGORITHMS

To find the Wiener solution in (16) may require

extensive computations when the input or the environment is time-varying. An adaptive filtering scheme will be more useful in this case. For fast convergence, we consider the RLS algorithm here. Using our formulation, we can apply the multichannel fast RLS algorithm called the block step-up step-down (B-SUSD) algorithm [8]. This algorithm is a multichannel extension of the FAEST algorithm [9]. Since the stability problem may arise when computations are performed in finite precision, a stabilized procedure is then adopted. We summarize the stabilized TMUX FRLS algorithm in Table I. Some notations are explained as follows. The vector )(nLz is the input vector in (2) and defined as

TL

TM

TTL nnnn 1110 )]( )( )([)( ×−= zzzz L

where TKiiii Knznznzn 1)]1()1( )([)( ×+−−= Lz .

By this representation, the corresponding separation filters are

TTiM

Ti

Tii ] [ ,1,1,0 −= hhhh L

where Tlililiil Khhh )]1()1( )0([, −= Lh as shown in

Fig. 2. The two permutation matrices T and S are defined such that

−

=

−

=+ )(~)(

)1()(~

)(Ln

nn

nn

M

LT

L

MTML z

zz

zz ST

where T

MM nznznzn )]()( )([)(~110 −= Lz

Note that 1−= TT T and 1−= SST . The main idea of the stabilized B-SUSD FRLS algorithm is to update the Kalman gain )(nLw by a blockwise step-up step-down procedure, i.e.,

)1()1()( +→+→ + nnn LMLL www The computational complexity of the stabilized TMUX FRLS algorithm is on the order of )6( MLO [8] while that

of the standard RLS algorithm is )2( 2LO . Thus, when the separation filter length is long, the stabilized TMUX FRLS algorithm can save significant computations.

We perform some simulations to demonstrate effectiveness of the proposed TMUX reconstruction. The performances of the standard RLS algorithm and the algorithm in [6] are also compared. A five band system is used and each analysis/synthesis filter has 55 taps. The prototype filters presented in the Table IV of [10] are adopted. The reconstruction performance is measured by the reconstruction signal to noise ratio (SNR). This is defined as

−−

=})](ˆ)({[

)}({log10)SNR( 2

2

10 DnxnxEnxEn

(10)

(11)

(12)

(13)

(14)

(15)

(16)

(17)

(18)

(19)

(20)

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3249

The noise is assumed to be white Gaussian and the channel's response to be {-0.077, -0.355, 0.059, 1, 0.059, -0.273}. The input is a first-order AR signal with correlation coefficient 0.8.

In the first experiment, we show the convergence characteristic of the proposed adaptive algorithm. Let the input SNR be 30 dB and the forgetting factor for RLS algorithms be 0.995. Fig. 3 shows the learning curves for all algorithms. From the figure, we see that the convergence behavior of the standard RLS and the stabilized TMUX FRLS is very close, so is the steady-state reconstruction SNR. The algorithm in [6] does not consider the channel effect resulting in poor results. In the second experiment, we compare the reconstruction SNRs for different input SNRs. The result is shown in Fig. 4. As we find that the Wiener filter outperforms the stabilized TMUX FRLS by 0.7 dB. This is due to the fact that the forgetting factor used in RLS is 0.995 instead of one. For all input SNRs, the Wiener and the adaptive filtering perform a lot better than the conventional separation filtering. At low input SNR, noise dominates the distortion. Thus, the performance difference

between conventional separation filters and the proposed filters is not as significant as that at high input SNR. At high input SNR, the channel effect dominates. The proposed separation filters act like equalizers and can achieve better performance.

IV. APPLICATION TO THE MULTICARRIER COMMUNICATION SYSTEMS

The OFDM system uses IDFT/DFT to implement a multicarrier system. Data is modulated on the subcarriers and the subcarriers are mutually orthogonal so that the system can distinguish individual data on subcarriers after removing the channel effect in the receiver. However, the orthogonality is not satisfied with channel noise and interference. Consider the following equation describing the transmitted OFDM signal:

∑∑−

=

−⋅∆⋅∞

−∞=

⋅

−Π=1

0

)(2][)(N

k

nTtkfj

n s

sOFDM

sekXTnTttx π

TABLE I. THE STABILIZED TMUX FRLS ALGORITHM

Initials:

1:)0()0( ×== LlL 0hw , MLBA ×== 0:)0()0( , 1:)0( =Mα ,

MMbM

fM ×== I:)0()0( αα , 3,2,1,1: == iKi

Recursions:

The time updating of the Kalman gain: )()()1(~)1( nnAnn L

TM

fM zze −+=+

)1()()1( 1,1 +=+ −− nnn fM

fM

fM eαλβ

)1()()(

)1( +

−

+

=+ ×+ n

nAnn f

MMM

LML β

Iw

0wT

Partition : )1(:)1()1(

+=

++

+ nnn

MLM

L wc

Sδ

)(/)1()1( nnn MfM

fM αε +=+ e

)1()()()1( ++=+ nnnAnA fTML εw

)1()()1()1( +++=+ nnBnn MLL δcw )1()()1(~)1(ˆ +−−+=+ nnBLnn L

TM

bM zze

)1()()1(~ +=+ nnn MbM

bM δλαe

3,2,1),1(~)1()1(ˆ)1(, =+−++=+ inKnKn bMi

bMi

ibM eee

)1()1()()1( +++=++ nnnn fM

fTMMLM βαα e

)1()1()1()1( ,1, ++−+=+ + nnnn MTb

MLMM δαα e 3,2),1(/)1()1( ,, =++=+ innn M

ibM

ibM αε e

)1()1()()1( ,2, +++=+ nnnBnB TbML εw

)1()1()()1( +++=+ nnnn fTM

fM

fM

fM ελαα e

)1()1()()1( ,3,3, +++=+ nnnn TbM

bM

bM

bM ελαα e

The time updating of the separation filter bank: For 0=l to 1−M Do )()1()1()1( nnndne i

TLll hz +−+=+

)1(/)1()1( ++=+ nnen Mll αε )1()1()()1( +++=+ nnnn lLii εwhh

End For

0 100 200 300 400 500 600 700 800 900 10000

5

10

15

20

25

30

35

iterations

reco

nstr

uctio

n S

NR

, dB

(1)

(2) (3)

(4)

Figure 3. Comparison of learning curves for reconstructing AR(1) signal with the subband input SNR=30 dB: (1) the Wiener filter, (2) the standard RLS algorithm, (3) the stabilized B-SUSD FRLS algorithm, (4) the Koilpillai et. al's algorithm.

0 5 10 15 20 25 30−5

0

5

10

15

20

25

30

channel signal to noise ratio, dB

reco

nstru

ctio

n S

NR

c, d

B

(1)

(2)

(3)

(1) The Wiener Filter

(2) B−SUSD FRLS Algorithm

(3) Koilpillai et al. Algorithm

Figure 4. Comparison of reconstruction performance for AR(1) signal with noise and channel distortions.

(22)

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where sT is the symbol period, sTf /1=∆ is the subcarrier

frequency spacing, and ][⋅Π denotes the rectangular function. The OFDM signal is generated by IDFT for every N data samples. Then the continuous-time OFDM signal can be thought as multiplying the discrete-time samples by a rectangular pulse with period of N data samples. It can be shown that the spectrum of the OFDM signal is composed of a shifting summation of N sinc functions with the frequency spacing of f∆ . The amplitude attenuation of the first sidelobe for the sinc function is about 13.1dB, which is very high compared to that for a typical filterbank system. The high sidelobe value results in a significant loss of orthogonality when noise or interference is introduced.

We use the proposed adaptive TMUX proposed to compare the conventional OFDM system for the IEEE 802.11a application. The sample rate is 20 MHz, modulation format is 64-QAM, active subcarrier number is 52, and the CP length is 16 samples. The OFDM system uses two long training symbols for channel estimation by least squares (LS) and linear minimum mean squared error (LMMSE) algorithms. The 64-band filterbank system uses 128-tap coefficients for the combining filters and is designed by the method in [11] with stopband attenuation of about -25 dB. The amplitude response of the indoor channel model used in the simulation is plotted in Fig. 5.

Fig. 6 shows the bit error rate (BER) curve comparison of LS channel estimation, LMMSE channel estimation, adaptively reconstructed separation filters with 128 taps. For the adaptive TMUX system, the guard band can be released since the sidelode of filterbank is lower than that of the sinc function. From the results, we can see that the adaptive TMUX systems either with guard bands or without guard bands outperform the OFDM systems with LS or LMMSE channel estimation. The adaptive receiver minimizes the error between the received signal and the transmitted signal, which includes minimizing the loss of orthogonality due to noise as well. However, when SNR is high, the loss of orthogonality is not significant and the performance of the adaptive TMUX system approaches to that of the OFDM system.

V. CONCLUSIONS

We have obtained the adaptive TMUX algorithm and the Wiener solution by the property of polyphase decomposition in a multirate system. By the multichannel fast RLS algorithm, the B-SUSD algorithm can be used to reduce computational complexity of the adaptive TMUX system used for a multicarrier communication system. The adaptive TMUX receiver can release the influence of loss of orthogonality due to noise and imperfect channel estimation compared to an OFDM system. The new algorithm is applied to the IEEE 802.11a and the simulation shows that the performance is better than the conventional OFDM technique.

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[9] G. Carayannis, D. G. Manolakis, and N. Kalouptsidis, “A Fast Sequential Algorithm for Least-Squares Filtering and Prediction,”

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[10] K. Nayebi, T. P. Barnwell, III, and M. J. T. Smith, “Time-Domain Filter Bank Analysis: A New Design Theory,” IEEE Trans. Signal Processing, Vol. 40, No. 6, pp.1412-1428, 1992.

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Figure 5. The simulated indoor channel

Figure 6. Comparison of the OFDM receivers and the adaptve TMUX receivers for the IEEE 802.11a system

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2006 proceedings.

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