issn 1991-1696 africa research journal -...

June 2010Volume 101 No. 2www.saiee.org.za

Africa Research JournalResearch Journal of the South African Institute of Electrical Engineers

Incorporating the SAIEE Transactions

ISSN 1991-1696

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(SAIEE FOUNDED JUNE 1909 INCORPORATED DECEMBER 1909)AN OFFICIAL JOURNAL OF THE INSTITUTE

ISSN 1991-1696

Secretary and Head OfficeMs Gerda GeyerSouth African Institute for Electrical Engineers (SAIEE)PO Box 751253, Gardenview, 2047, South AfricaTel: (27-11) 487-3003Fax: (27-11) 487-3002E-mail: [email protected]

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INTERNATIONAL PANEL OF REVIEWERSW. Boeck, Technical University of Munich, Munich, GermanyW.A. Brading, AustraliaProf. G. De Jager, Dept. of Electrical Engineering, University of Cape Town, Cape Town, South AfricaProf. B. Downing, Dept. of Electrical Engineering, University of Cape Town, Cape Town, South AfricaDr W. Drury, Control Techniques Ltd, United KingdomP.D. Evans, Dept. of Electrical, Electronic & Computer Engineering, The University of Birmingham, Birmingham, UK Prof. J.A. Ferreira, Electrical Power Processing Unit, Delft University of Technology, Delft, The NetherlandsO. Flower, University of Warwick, UK Prof. H.L. Hartnagel, Dept. of Electrical Engineering and Information Technology, Technical University of Darmstadt, Darmstadt, GermanyC.F. Landy, Engineering Systems Inc., USAD.A. Marshall, ALSTOM T&D, FranceDr M.D. McCulloch, Dept. of Engineering Science, Oxford, United KingdomProf. D.A. McNamara, University of Ottawa, Ottawa, CanadaM. Milner, Hugh MacMillan Rehabilitation Centre, CanadaProf. A. Petroianu, Dept. of Electrical Engineering, University of Cape Town, Cape Town, South AfricaProf. K.F. Poole, Holcombe Dept. of Electrical and Computer Engineering, Clemson University, United States of AmericaProf. J.P. Reynders, Dept. of Electrical & Information Engineering, University of the Witwatersrand, Johannesburg, South AfricaI.S. Shaw, University of Johannesburg, SAH.W. van der Broeck, Phillips Forschungslabor Aachen, GermanyProf. P.W. van der Walt, University of Stellenbosch, Stellenbosch, South AfricaProf. J.D. van Wyk, Dept. of Electrical and Computer Engineering, Virginia Tech, United States of AmericaR.T. Waters, UKT.J. Williams, Purdue University, USA

Additional reviewers are approached as necessary

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Vol.101(2) June 2010 SOUTH AFRICAN INSTITUTE OF ELECTRICAL ENGINEERS 45

VOL 101 No 2June 2010

SAIEE Africa Research Journal

SAIEE AFRICA RESEARCH JOURNAL EDITORIAL STAFF ...................... IFC

An adaptive receiver for STBC in frequency selective channel with improved robustness and pilot requirements by J. Mathew, H. Xu and F. Takawira ........................................................46

Optimal power control of a three-shaft brayton cycle based power conversion unit by K.R. Uren, G. van Schoor and C.R. van Niekerk...................................60

Universal decremental redundancy compression with fountain codes by F.P.S. Luus, A. McDonald and B.T. Maharaj ........................................68

NOTES FOR AUTHORS ...................................................................................IBC

Vol.101(2) June 2010SOUTH AFRICAN INSTITUTE OF ELECTRICAL ENGINEERS46

AN ADAPTIVE RECEIVER FOR STBC IN FREQUENCY SELECTIVE CHANNEL WITH IMPROVED ROBUSTNESS AND PILOT REQUIREMENTS J. Mathew, H. Xu and F. Takawira School of Electrical, Electronic and Computer Engineering, University of Kwa-Zulu Natal, King George V Avenue, Durban 4041, South Africa E-mail: xuh, [email protected] Abstract: The semi-blind recursive least squares (RLS) based adaptive receiver has been designed to perform joint interference suppression and equalization for space time block codes (STBC) in a frequency and time selective channel. In order to decrease pilot requirements in the training block, this paper introduces a linear predictor (LP) algorithm to do a forward prediction of the channel coefficients based on a smaller pilot block. We then introduce a QR-decomposition (QRD) based algorithm to improve the performance of the receiver at higher Doppler frequencies. The simulation results show that the addition of the LP does not affect the frame error rate (FER) of the overall system. Linear prediction requires a smaller number of pilot symbols in order to provide the channel estimates for a given burst. Hence the LP increases the overall throughput of the system by decreasing the pilot symbols required. The simulation results show that this is a more effective method for improving the system performance as the overall FER of the combined QRD-LP receiver is decreased significantly. Finally, by comparing the QRD based receiver and the RLS based receiver at higher Doppler frequencies, we verify that the QRD receiver has superior FER performance under these conditions. Keywords: Space time block code, frequency selective channel, adaptive receiver, LP, QR decomposition.

1. INTRODUCTION

Multiple-input multiple-output (MIMO) systems have been proven to significantly increase the capacity in rich scattering environments. The space time block code (STBC), which was first introduced in [1] and later generalized in [2], is an attractive MIMO technique due to its simple linear processing at receiver and because it requires no additional bandwidth. The Alamouti scheme is a special case full rate STBC with two transmit antennas and one receive antenna. The time reversal (TR) STBC [3], orthogonal frequency division multiplexed (OFDM) STBC [4] and single carrier frequency domain equalized (SC-FDE) STBC [5-7] have been proposed in the literature for frequency-selective channels. An overview and comparison of these schemes can be found in [8]. Since OFDM-STBC requires additional outer-coding and interleaving to fully exploit multipath diversity, which results in additional rate loss [9], OFDM-STBC scheme will not be considered in this paper. The other two schemes, which are extensions of the Alamouti’s STBC scheme, are getting particular attention because of their ability to exploit the orthogonal structure of STBC over frequency selective channels, keeping the receiver complexity at a manageable level. In [10], TR-STBC was applied to the WCDMA downlink, incorporating chip equalization to suppress multiple access interference (MAI). In [11], TR-STBC was applied to the broadband fixed wireless access systems. More recently, TR-STBC was extended to quasi-orthogonal time-reversal space-time block coding [12]. In the SC-FDE STBC scheme, frequency domain linear equalization has been incorporated into TR-STBC [6-8].

This motivates the use of SC-FDE STBC as a more attractive option. This paper mainly focuses on the SC-FDE STBC scheme. In the SC-FDE scheme, receivers require channel state information (CSI) at the receiver. One approach is to use training sequences embedded in each block to estimate the CSI. This, however, results in increased system overhead. In addition to this, the mobility of the users may cause the channel impulse response to vary rapidly and hence the quasi-static assumption of the channel becomes void. The use of longer blocks, which is required to reduce the system overhead in such training based schemes, can not be practical in such cases. W. M. Younis in [15-16] developed an efficient low complexity adaptive receiver with fast tracking abilities. Although the original scheme proposed in [16] is bandwidth efficient in that it does not require the inclusion of pilot symbols in every block, the two key problems identified are as follows:

There is still the requirement of an entire block of training data to initialize the recursive least squares (RLS) algorithm. An entire block of training data decreases the bandwidth efficiency. Actually, there is a trade off between the performance gain required and the bandwidth efficiency in the scheme proposed in [16].

The RLS algorithm is not robust, particularly at high Doppler frequencies where there is a significant performance penalty. This is evident from the simulation results in section IV of [15].


The first problem motivates us to develop an algorithm or method to decrease the number of pilot symbols and thus improve the bandwidth efficiency during the re-training interval. The second problem motivates the search for a more robust algorithm at high Doppler frequencies. The rest of this paper is organized as follows: we begin with the description of the adaptive RLS based receiver [15-16] in section 2. We then describe the linear prediction (LP) algorithm in Section 3. Section 4 details the QRD based adaptive receiver that is used to provide better performance to the RLS algorithm at high Doppler frequencies. We provide simulation results in section 5 and conclude in section 6. Notations: The following notations are used for the rest of this paper. Upper case letters denote matrices, lower case letters stand for column vectors;

* H, and T represent conjugate, transpose and

Hermitian, respectively; E stands for expectation and

NI denotes an identity matrix of size N N . NF is for an N N discrete Fourier transform (DFT) matrix.

denotes the Euclidean norm. Re stands for the real part of the term.

2. RLS BASED ADAPTIVE RECEIVER

2.1 Single User Transmission In the description to follow, it is assumed that each user is equipped with two transmit antennas and one receive antenna. Let the N -symbol block including cyclic prefix (CP) transmitted from the first and second antennas at block time 2i be given by 1x and 2x , respectively. At block time 2 1i , permuted conjugate versions *

2- xP and *1xP are sent from the first and second antennas,

respectively. Assuming the original data sequence length and channel length is R and L , respectively, the permutation matrix P is a circular reversal matrix given by

1 0 0

0 0 1=0 1 0

0 1 0

R

L

PP

P (1)

The (th)k received blocks, 2 , 2 1k i i , in the presence of noise is given by

1 1 2 2y( ) H ( ) ( ) H ( ) ( ) n( )k k x k k x k k (2) where n( )k is the noise term which is assumed zero mean Gaussian with white power spectrum, and 1H ( )k and

2H ( )k are the circulant channel matrices from antenna one and two, respectively, to the receive antenna. This

index implies that the channels are not assumed to be quasi-static. It should be noted that the original Toeplitz channel structure is converted to a circulant structure by the addition of the cyclic prefix of length L defined in (1). The circulant channel results from the addition of the cyclic prefix of length L . Next, applying the N N DFT matrix NF , to the received sequence we get

1 1 2 2Y( ) y( ) ( )X ( ) ( )X ( ) N( )Nk k k k k k kF (3) where X( ) ( )Nk x kF , N( ) n( )Nk kF and 1( )k and

2 ( )k are the diagonal matrices of 1H ( )k and 2H ( )k , respectively. The ( , )thi k element of the DFT matrix NF is defined as

2

,

1F , 1, , 1,2, , 1j ik

NN i k

e j i k NN

(4)

Using the properties of DFT, the terms for the 2 1i block can be written as

*1 2

*2 1

X (2 1) X (2 )

X (2 1) X (2 )

i ii i

(5)

Combining (3) and (5), we get

1 2 1* ** *2 1 2

quatrionic structure

X (2 )Y(2 ) N(2 )=

X (2 )Y (2 1) N (2 1)ii iii i

Y (6)

It should be noted that the choice of the permutation matrix (1) alleviates the need of post multiplying the received sequence by the P as in [5]. It is clear from the Alamouti-like diagonal channel matrix in (6) that by forming the unitary matrix ,

1 2* *2 1

HH (7)

The data sequence can be decoupled by the following unitary operation,

0 1

0 2

0 X (2 )=

0 X (2 1)H i

iY Y W (8)

where

* *0 1 1 2 2( ) and *= N(2 ) N (2 1)

TH i iW . Next we define the minimum mean square estimator (MMSE) equalizer [7] as

11SNR

HN2I (9)

The SNR at the receiver is given by 2 2SNR= /x w , where

w is the covariance of noise, x is the signal covariance. The MMSE estimate of X is then given as

11

2

X̂ (2 ) 1X̂= =ˆ SNRX (2 )

H H HN

i

i2Y I Y (10)


2 (2 )X i

1ˆ (2 )X i

Stack to form data matrix

(training)

(direct-decision)

*�

*�

*�

*�

Adaptive Equalizer

(2 2)iE

FormU (2 2) (2 )i iE

Form

( )D i

Form

( )D i

(2 )y i DFT

IFFT

IFFT

FFT

1 (2 )X i

2ˆ (2 )X i

(2 1)y i

1(2 )X i

2(2 )X i

(input)

filter output

Slicer

Slicer

DFT

FFT

Fig. 1 Block diagram for the single user adaptive receiver with two-transmit one-receive antennas

2.2 Adaptive Implementation The interference cancellation and equalization technique described in the previous section requires the knowledge of the channel state information (CSI) at the receiver. This is usually accomplished by the addition of a training (pilot) sequence to each transmitted block in order to estimate the channel at the expense of additional bandwidth requirement. In order to decrease the overhead requirements, such pilot aided schemes will require longer blocks which become impractical in channels with fast variations. Furthermore blind channel estimation based on second [20] and higher order statistics results in higher complexity at the receiver and is not suitable for online implementation. Hence the need of an online adaptive receiver under such conditions becomes imperative. The RLS algorithm provides fast tracking , and due to the special quadtronic structure of space-time block codes, the complexity can be reduce to that of an LMS algorithm. The overall adaptive receiver is shown in Fig. 1. Defining the combined matrix of the MMSE in (10) as

HB , it can be shown that this matrix has an Alamouti (quadtronic) structure i.e.

1 2* *2 1

B B=

B -BB (11)

where the entries of B , given the block size N , are given by

1

*1 1

0 01

2 20 0

1B diag ( , )+1 SNR

1B diag( , )+1 SNR

N

nN

n

n n

n n

(12)

Hence (10) can be written as

1 1 2* *2 12

X̂ (2 ) B Bˆ B -BX (2 )

i

iY (13)

This can be rearranged into the following form.

*1 1

**22

quadtronic structure

X̂ (2 ) Ediag(Y(2 )) diag(Y (2 1))=U( )

ˆ E-diag(Y(2 1)) diag(Y (2 ))X (2 )

i i ii

i iiE (14)

The vectors 1E and 2E contain the diagonal entries of

1B and 2B , respectively. The essence of this derivation lies in the manner in which the quadtronic structure is maintained in (14), which is in the form of the RLS problem [12-13]. The adaptive RLS solution is now applied. The equalizer coefficients E are adaptively updated for every two blocks using the following recursion [15]:

(2 2)(2 ) (2 2)U (2 2) D(2 2) U(2 2) (2 )H

ii Q i i i i i

EE E

(15)

where 1 1

112

(2 2) [ (2 ) (2 )U(2 2)

I U(2 2) (2 )U (2 2) U (2 2) (2 )]H HN

Q i Q i Q i i

i Q i i i Q i (16)


It is easy to see that the inverse term in (16) is given by a diagonal matrix

112

(2 2) 0I U(2 2) (2 )U (2 2)

0 (2 2)H

N

ii Q i i

i (17)

D(2 2)i in (15) is referred to as the desired response and is given by

1*2

1*2

X (2 2), for training

X (2 2)D(2 2)

X (2 2), for decision-direct tracking.

X (2 2)

iii

ii

(18)

The parameters of the algorithm are initialized as follows: (0) 0E and 2(0) I NQ , with being a large number (usually 610 [18]). is called the forgetting factor and is in the range 0,1 . The adaptive receiver operates in two modes.

*1 2X (2 2) X (2 2)

Ti i denotes the pilot block

during training mode and *

1 2X (2 2) X (2 2)T

i i denotes the slicer output of

the received data during the decision direct mode as shown in Fig. 1.

(2 2)Q i has the following diagonal structure. (2 2) 0

(2 2)0 (2 2)

T iQ i

T i (19)

The recursive updates of (2 2)T i is given by

1 1(2 2) (2 ) (2 ) (2 2) (2 )T i T i T i i T i (20) where

2 2(2 2) (2 2)diag Y(2 ) Y(2 1)i i i i (21)

The diagonal matrix (2 2)i in (17) is given by

12 21(2 2) I (2 )diag (2 ) (2 1)Ni T i Y i Y i (22)

Hence we get

112 2 1(2 2) diag Y(2 ) Y(2 1) (2 )i i i T i (23)

Finally the RLS equalizer is then given by

(2 2) 0(2 2) (2 ) U (2 2)

0 (2 2)

D(2 2) U(2 2) (2 ) .

HT ii i i

T i

i i i

E E

E

(24)

At first glance the RLS algorithm may appear computationally complex, due to the number of matrix

inversions required in the algorithm. However, due to quadtronic structure of Y( )i and the diagonal structure of ( )T i , the matrix inversions in (22) and (23) are in fact scalar inversions. This results in the algorithm giving an LMS complexity. We will now describe a new method of decreasing the need of an entire pilot block during training mode as shown in (18). The solution is based on the forward LP and the knowledge of the autocorrelation function for the broadband channel.

3. LINEAR PREDICTION ALGORITHM

3.1 Continuous Time Channel Model The received signal for the Rayleigh frequency-flat fading channel is modelled as [21]:

( ) Re ( ) ( ) 2 exp 2 ( )c c cr t v t h t j f t n t (25)

where ( )v t is the transmitted baseband signal, ( )cn t is the AWGN noise, ( )h t is the continuous-time fading process and cf is the frequency of the fading process. The fading process is modelled as a zero-mean, wide-sense stationary, complex Guassian random process. The amplitude ( )h t has a Rayleigh distribution. The phase of the fading process, which represents the carrier phase error, is uniformly distributed over 0, 2 . The fading process is a time variant process

which makes the fading estimation problem a difficult one. Added to this is the fact that the amplitude, ( )h t , can have a small value for extended periods of time (this represents a deep fade). Assuming the fading process is correlated in time, the autocorrelation function is given as:

*1( ) ( ) ( )2F t h t t h tE (26)

A commonly used model to represent the land mobile fading process is the Jakes model [17]. The autocorrelation function for this model is given as:

01( ) (2 )2F dt J B t (27)

where 0 ( )J is the order zero Bessel function of the first kind, and dB is the Doppler spread of the channel. We assume that the channel follows the Rayleigh fading model with the autocorrelation function given by (27). Next, we use the LP to decrease the pilot requirements for the RLS equalizer.

3.2 Applying Linear Prediction to the RLS Algorithm The thM -order linear forward predictor for the channel coefficients is given by [21]:


1*

10

1ˆM

m m mms

h c h (28)

where ˆ

mh are the channel coefficients and mc are the predictor coefficients. The predictor coefficients are the solution to the Wiener-Hopf equation [19] given by:

0

1

1

(0) (1) ( 1) ( )(1) (0) ( 2) ( 1)

( 1) ( 2) (0) (1)

s F s F s F s F

s F s F s F s F

s F s F s F s FM

M McM Mc

M M c

(29)

where s is the normalized energy for the channel and

( )F m is as defined in (27). The size of the window must be at-least greater than or equal to the length of the channel. For each given tap, we need to get 1L previous values, where L is the length of the channel. Hence we send a pilot sequence of at least length 1L . We will once again use the example of two antennas and one receive antenna. Let the channel circulant matrix be given by:

0, 1, ,

1, ,

0, ,

, 1, 0, 1,

1, , 1, 0,

00

H = 1, 2,0

L L

L L

L

L L L

L L

h h hh h

h hh h h h

h h h h

(30)

in terms of the channel impulse response sequence

0, 1, ,, , , Lh h h h� , where H is an R R circulant matrix if R samples of the received sequence are taken into account. The index, , represents the two different paths. It should be noted that the original Toeplitz channel structure is converted to a circulant structure by the addition of the cyclic prefix of length L defined in (1). The circulant channel results from the addition of the cyclic prefix of length L . Let 1 1

Tx x , 2 2Tx x , 1 1

TH H , 2 2TH H ,

1 1TY Y and 2 2

TY Y . The received sequence given in (2) can be re-written as

1 2 11 1* *2 1 22 2

HX

Y HY= =

Y Hx x n

x x nP P (31)

where the size of 1Y , 2Y , 1x , 2x , 1n and 2n is 1 R . In training mode, the algorithm proceeds as follows. Ignoring the noise components we get Y=X H (32)

where X and H are as defined in (31). Since the pilot sequence is known, the matrix X is reconstructed at the receiver. The length of the pilot sequence corresponds to the window size of the LP. If the channel length is given by L , then M L . The zero

forcing (ZF) solution for the two channels is given by [6]:

11

2

HH= X X X Y

HH H (33)

It must be noted, that there is no DFT and IDFT operations in the training period just described. Hence, the tap components required for the initial window of the linear estimator are retrieved. The algorithm then switches to direct-decision mode which has the DFT and IDFT operations. We define the diagonal matrices as

*1 1

*2 2

* *0 1 1 2 2

F H F ,

F H F ,

( )

N N

N N (34)

where 1H and 2H represent circulant channel matrices. The tap values, and hence the circulant matrices 1H and 2H , are then estimated for the next block using (28), with the predictor coefficients for the previous window of size M already defined from the training period. The diagonal matrices are then derived using (34). Using (11), we can solve for the channel matrix entries

1B and 2B , and hence solve for the channel vector E in (15). To summarize, the algorithm is implemented as follows:

1. The shortened pilot stream is sent and using 1

1 2H= H H X X X YT H H the initial

estimates of the channel taps are updated.

2. The LP then estimates the channel coefficients for the next block using (28). Hence we update

1H and 2H .

3. By solving 0 in (34) and 1B and 2B in (11), the vector channel vector, E in (15), is updated. The values of ( )i and ( )T i in (23) and (20), respectively, are then updated.

4. The RLS algorithm then switches to direct-

decision mode, until it is re-trained again. The pilot requirement for the system is therefore decreased to the predefined window size M . This key result improves the system performance in the following two ways:

Given that the addition of the linear prediction algorithm does not depreciate the system performance in terms of frame error rate (FER), the resultant system has a greater throughput than the system without linear prediction. This is because, although the FER for the new system is


the same as the old system, the decrease in number of pilot symbols required increases the overall data sent. Hence there is an improvement in performance in terms of bandwidth efficiency.

In [15] it is shown that by decreasing the re-training interval for an adaptive algorithm, the overall performance in terms of FER improves. The reduction in the number of pilot symbol requirements in the linear prediction system allows the re-training interval to be decreased while still using the same bandwidth as the old system. Hence the overall system performance in terms of FER can be improved by sacrificing some of the bandwidth efficiency gained using linear prediction.

It is also mentioned in [15] that the RLS algorithm suffers in terms robustness particularly at higher Doppler frequencies. To achieve better robustness at higher Doppler frequencies, an adaptive QRD based receiver is derived in the next section.

4. MODIFIED QRD BASED RECEIVER 4.1 QRD receiver The QR solution to the least squares problem is documented in [19]. In order to apply the algorithm to our receiver, we recall from (14) that the least squares problem is given by

*1 1

**22

Quadtronic structure

X̂ (2 ) Ediag(Y(2 )) diag(Y (2 1))=U( )

ˆ E-diag(Y(2 1)) diag(Y (2 ))X (2 )

i i ii

i iiE (35)

where 1 2E E TE is the effective channel coefficient vector after the FFT operation and U( )i is the received observation sequence. Once again we define the desired response as

1*2

1*2

X (2 2), for training

X (2 2)D(2 2)

X (2 2), for direct-decision tracking

X (2 2)

ii

iii

(36)

The algorithm proceeds with the following four steps:

1. The channel response vector, E , is initialized to zero.

2. Perform QR decomposition of the U received sequence matrix in (35). Hence we get

(2 2) (2 2)=U(2 2)Q i R i i

which is computed using the Givens rotation described in [19].

3. The error signal is computed as (2 2) D(2 2) U(2 2) (2 )i i i iP E

4. Finally, the recursive update is given as

1

(2 2)(2 ) (2 2) (2 2) (2 2)i

i R i Q i iE

E P

The overall block diagram for the QRD based receiver is shown in Fig. 2. The only modification to the receiver block diagram of Fig 1 is the addition of the QRD block. In the next subsection, the LP is added to the QR-based receiver in order to shorten the pilot requirement for the system. 4.2 QRD receiver with LP To incorporate the LP, we pursue the exact same steps we described in section 3. Using (26)-(34), we estimate the initial values for the channel vector E . The algorithm then switches to direct-decision mode and the values of E are then updated recursively using (37), (38) and (39).

1. The shortened pilot stream is sent, and using 1

1 2H= H H X X X YT H H the

initial estimates of the channel taps are updated.

2. The LP then estimates the channel coefficients for the next block using (28). Hence we update 1H and 2H .

3. By solving 0 in (8) and 1B and 2B in (11), the vector channel vector, E in (15), is updated. The values of ( )i and ( )T i in (23) and (20), respectively, are then updated.

4. The error signal is computed as (2 2) D(2 2) U(2 2) (2 )i i i iP E

5. Finally, the recursive update is given as 1(2 2) (2 ) (2 2) (2 2) (2 2)i i R i Q i iE E P

The overall block diagram for the QRD-based receiver with linear prediction (QRD-LP) is shown in Fig 2.


2 (2 )X i

1ˆ (2 )X i

Stack to form data matrix

(training)

(direct-decision)

*�

*�

*�

*�

Adaptive Equalizer

(2 2)iE

Form

U (2 2) (2 )i iEForm

( )D i

Form

( )D i

(2 )y i DFT

IFFT

IFFT

FFT

1 (2 )X i

2ˆ (2 )X i

(2 1)y i

1(2 )X i

2 (2 )X i

(input)

filter output

Slicer

Slicer

DFT

FFT

Perform QRD

Linear Predictor

Fig. 2 Equivalent receiver model with the QRD and LP

Table 1 A typical urban (TU) channel model

Delay ( sec) 0.0 0.2 0.5 1.6 2.3 5.0

Strength (dB) -3.0 0.0 -2.0 -6.0 -8.0 -10

5. SIMULATION RESULTS

We now present the simulation results for the modified RLS algorithm that includes the LP. The main objective of the simulations is to verify that there is no significant performance loss incurred by using the LP. Provided that there is no performance loss, we are able to increase the re-training interval while still maintaining the same effective number of pilot symbols. This is once again shown in the simulation results. Computer simulations are also carried out to compare the performance of the QRD and RLS algorithms with respect to the tracking (non-stationary) and steady state (stationary) performance. We then verify through simulations that there is no significant performance loss when combining QRD based receiver with the LP. Finally, we show results of the new QRD-LP receiver and compare this to the RLS receiver in [16]. These results summarize our achievements by showing how the new QRD-LP receiver outperforms the RLS receiver in terms of bandwidth efficiency and robustness at higher Doppler frequencies.

5.1 Simulation Environment The enhanced data rate for global evolution (EDGE) simulation environment is used to test the performance of this adaptive receiver. The EDGE typical urban (TU) channel with 8-PSK modulation is considered. Equalization for EDGE poses a challenging problem due to the use of 8-PSK modulation unlike its predecessor that uses binary modulation in GSM. The TU channel impulse response generally has a non-minimum phase characteristic and the guassian minimum shift keying (GMSK) transmit filter used to combat this adds additional ISI. The delay profile of the TU channel is presented in Table 1 [15, 16]. The symbol duration of 3.69 s sT and 0.3 is proposed in 3rd generation TDMA cellular standard EDGE (2.5G). The overall TU channel impulse response (CIR) after GMSK pulse shaping is given is Fig. 3.


-1 0 1 2-25

-20

-15

-10

-5

0

Tap delay ( in symbols)

Tap

Gai

n (d

B)

Fig. 3 Equivalent CIR for a TU channel with GMSK pulse shaping.

From Fig. 3, the overall channel has a length of four symbols and hence the channel memory is 3L . All channels are assumed independent. The performance of the RLS algorithm is shown for different frequencies. The Jakes model is used to generate the Rayleigh fading coefficients. All taps are assumed independent and identically distributed (i.i.d.) Gaussian. Next, we present the simulation results for the modified RLS algorithm that includes the LP. The main objective of the simulations is to verify that there is no significant performance loss incurred by using the LP. Provided that there is no performance loss, we are able to increase the re-training interval while still maintaining the same effective number of pilot symbols. This is once again verified in the simulation results.

5.2 New Linear Prediction Based RLS Receiver 5.2.1 Performance Penalty The initial set of simulation results are done to investigate if there is any performance loss incurred by using the new linear prediction based algorithm. The

original RLS receiver is set with frequencies of 10 Hz and 40 Hz. The block size of 32 symbols is used. The re-training interval is set to 50 blocks in all cases and the window size of the LP, M , is set to 10 blocks. The new receiver with the LP is also set with the same frequencies and re-training interval. Fig. 4 shows that the performance penalty incurred is negligible, hence making this an attractive method of improving the overall performance of the adaptive receiver. The auto-correlation function in (27) is an accurate model for the Rayleigh fading process used. It is shown in [19] that the forward LP performance is dependant on the accuracy of the auto-correlation function. This is the reason for the performance penalty being minimal. 5.2.2 Performance Using Equivalent Pilot Symbols The LP decreases the overhead required by the system during the training period of the algorithm. Fig. 5 shows the RLS algorithm with the training period that requires an entire block of pilot symbols as compared to the new receiver with linear prediction that shortens this requirement. The block size of 40 symbols and Doppler frequency of 40Hz are used. The re-training


5 10 15 20 2510

-5

10-4

10-3

10-2

10-1

100

Eb/N0 (dB)

BE

R

RLS-LP 10Hz

RLS 10HzRLS-LP 40Hz

RLS 40Hz

Fig. 4 Comparison of performance penalty with the linear predictor

5 10 15 20 2510

-3

10-2

10-1

100

Eb/No

BE

R

Eqivalent RLS with LP 40 Hz

40 Hz RLS

Fig 5 Performance of the linear predictor with equivalent number of pilot symbols.


interval is set to 40 blocks. The window size for the LP is set to 10. In order to use the equivalent number of pilot symbols, the re-training interval is decreased to 10 blocks for the new system. From [16], it was concluded that increasing the retraining interval increases the overall system performance. Therefore, using the LP improves the performance of the adaptive receiver without the additional bandwidth requirement. From these results, an adjustment to the window size for the LP can result in better performance as the re-training interval can be decreased further. Since M L , the knowledge of the channel length is required for this as it limits the window size that can be used. 5.3 QR Based Adaptive Receiver with Linear Prediction The main purpose in this section is to investigate the following:

Performance at high Doppler frequencies. Tracking and estimation error performance. Performance of the QR receiver with linear

prediction. 5.3.1 Performance at High Doppler frequency The main problem associated with the RLS algorithm is its lack of robustness, particularly at high Doppler frequencies. To achieve better performance at higher Doppler frequencies, an adaptive QRD based receiver was derived in section 4. The result of computer simulation for the new QRD receiver is shown in Fig.6. The robustness of the new receiver as compared to the RLS based receiver is evident from the results in Fig. 6. The Doppler frequency of 100 Hz is used. The block size and re-training interval is set to 96 bits and 50 blocks, respectively. Hence, by using the orthogonal transformation, the extent to which the ill-conditioned problem affects the estimates is reduced and hence the overall performance at high user mobility is improved. 5.3.2 Tracking and Estimation Error Performance The purpose of this sub-section is to compare the tracking error and estimation error performance as the Doppler frequency is varied. Fig. 7 shows the results at Doppler frequencies of 60 Hz, 40Hz and 0 Hz. We choose 60 Hz and 40 Hz to compare the tracking performance of the two algorithms at lower frequencies. The result at 0 Hz can be used to compare the estimation error performance of the two algorithms. The block size and re-training interval is set to 96 bits and 50 blocks, respectively.

The QRD based receiver provides a more robust solution as shown in the results for the Doppler frequencies of 60 Hz and 40 Hz. The results imply that the QR based solution has a better lag error or tracking performance than the RLS based receiver. The results at 0 Hz for both algorithms also show that the estimation error or steady state performance of the RLS algorithm is better than that of the QR based algorithm. The RLS algorithm provides a better steady state performance. Since there are no transformations involved in this algorithm, the estimate error for the RLS algorithm in a stationary environment is more accurate than the QRD based approach. This is shown in Fig 7 for the Doppler frequency of 0 Hz where the RLS algorithm outperforms the QRD based receiver. In order to further improve the performance of the receiver, the LP is combined with the QR based algorithm. The performance improvement can be achieved in terms of bandwidth efficiency and overall FER performance. The next sub-section presents the simulation results that show these two performance improvements. 5.3.3 Performance of the QRD-LP For the first set of simulations, the block size is set to 96 bits. The Doppler frequency of 60 Hz is used. The re-training interval is set to 50 blocks. The channel model and signal constellation is the same as previous section. The results in Fig. 8 shows that the addition of LP doesn’t affect or hamper the performance to a significant extend. This emphasizes that the LP is an attractive method to shorten the preamble system requirements. In order to verify this, we compare the QR receiver with fixed pilot symbols, and the QR-LP receiver with equivalent number of fixed pilot symbols.

For the second set of computer simulations in Fig. 9, the block size and Doppler frequency is set to 120 bits and 60 Hz, respectively. The re-training interval is set to 40 blocks for the RLS and QR algorithms. The window size for the LP is set to 40 bits and hence the equivalent re-training interval for the QR-LP is set to 10 blocks. As opposed to its RLS predecessor, the new QR based receiver provides a more robust solution at higher frequencies. Combining the LP with the QRD based receiver results in decreasing the preamble requirements. This in-turn allows the system re-training interval to be decreased, hence increasing the overall receiver performance as shown in Fig.9. Hence


10 15 20 25 30 35 4010

-2

10-1

100

Eb/N0

FE

R

QRD 100Hz

RLS 100Hz

Fig 6 Simulation results to verify the robust property of the QRD algorithm

10 15 20 25 30 35 4010

-5

10-4

10-3

10-2

10-1

100

Eb/N0 (dB)

FE

R

QR 60Hz

RLS 60 Hz

QR 0 Hz

RLS 0 Hz

QR 40 Hz

RLS 40 Hz

Fig 7 Simulation results of the QRD and RLS algorithms for different Doppler frequencies


10 15 20 25 30 35 4010

-3

10-2

10-1

100

Eb/N0 (dB)

FE

R

QR-LP 60 Hz, Pilot size =96 Bits

QR-LP 60 Hz, Pilot size =30 Bits

Fig 8 Comparison of QRD and QRD with LP at 60 Hz

10 15 20 25 30 35 4010

-3

10-2

10-1

100

Eb/E0 (dB)

FE

R

QR-LP 60 Hz, pilot size = 40 bits, retrain interval = 10 Blocks

QR 60 Hz, pilot size = 120 bits, retrain interval = 40 Blocks

RLS 60 Hz, pilot size = 120 bits, retrain interval = 40 Blocks

Fig 9 Overall modified QRD-LP receiver versus RLS receiver.


the results show a performance improvement for the new QRD-LP receiver when compared to its RLS counterpart.

6. CONCLUSION We have described a bandwidth efficient method of improving the overall system performance of the adaptive RLS receiver. The LP decreases the number of pilot symbols that are required during the re-training period. Hence, the re-training interval can be decreased which results in the improved performance. The simulation results also show that there is no additional penalty when using the LP making it an attractive scheme. The only requirement at the receiver is that the knowledge of the autocorrelation function for the channel is assumed. For the RLS based receiver in [15], the performance penalty at higher Doppler frequencies is larger when compared to the performance at lower frequencies. Hence, in this paper, we provide a more robust alternative algorithm for the adaptive receiver in [15] to perform joint interference and equalization at high Doppler frequencies. This is achieved by using a new algorithm based on QR orthogonal transformation. Finally, the QRD based receiver is combined with the LP to increase the overall system performance by decreasing the re-training interval of the adaptive algorithm. This results in a receiver that offers stable performance at high frequencies and better overall performance as opposed to its RLS counterpart. While the RLS based receiver provides an approach that does not require CSI estimation and hence reduces system overhead, the new QRD-LP based receiver provides a more robust solution while further decreasing the overhead requirements. The simulation results provide verification of the better performance for this new receiver.

7. REFERENCES [1] S. M. Alamouti, “A simple transmit diversity

technique for wireless communications,” IEEE Journal on Selected Areas in Communications, vol. 16, no. 8, pp.744-765, Aug. 1998.

[2] V. Tarokh, N. Seshadri, and A. R. Calderbank, “Space-time codes for high data rate wireless communication: performance criterion and code construction,” IEEE Trans. on Information Theory, vol. 44, no. 2, pp. 744-765, Feb. 1998.

[3] E. Lindskog and A. Paulraj, “A transmit diversity scheme for channels with inter-symbol interference,” Proceedings of IEEE Conference on International Communications, vol. 1, pp. 307-311, New Orleans, La, USA, June 2000.

[4] Z. Lui and G. B. Gannakis, “Space time block coded multiple access through frequency selective fading channels,” IEEE Trans. Communications, vol. 49, no. 6, pp.1033-1044, June 2001.

[5] S. Zhou and G. B. Giannakis, “Space-time coding with maximum diversity gains over frequency-selective fading channels,” IEEE Signal Processing Letters, vol. 8, no. 10, pp. 269-272, Oct. 2001.

[6] S. Zhou and G. B. Giannakis, “Single-carrier space-time block coded transmissions over frequency-selective fading channels,” IEEE Trans. on Information Theory, vol. 49, no.1, pp. 164-179, Jan. 2003.

[7] N. Al-Dhadhir, “Single-carrier frequency domain equalization for space time block coded transmission over frequency selective fading channels,” IEEE Communication Letters, vol. 5, pp. 304-306, July 2001.

[8] N. Al-Dhadhir, “Overview and comparison of equalization schemes for space-time-coded signals with application to EDGE,” IEEE Trans. Signal Processing, vol. 50, no.10. pp. 2477-2488, Oct. 2002.

[9] H. Mheidat, M. Uysal and N. Al-Dhahir, “Time-and frequency-domain equalization for Quasi-orthogonal STBC over frequency-selective channel,” IEEE International conference on communications, Vol.2, pp. 697-701, June 2004.

[10] G. White, J. Correia, Y. Zakharov and A. Burr, “Time-reversal space-time block coded WCDMA receiver in urban and suburban environments,” IEE Proc-Commun., Vol. 152, No.6, pp.1047, 1054, Dec. 2005

[11] P. Xiao, R. Carrasco and I. Wassell, “Time reversal space-time block coding for FWA systems”, International conference on wireless and mobile communication, pp.51-55, July 2006.

[12] H. Mheidat, M. Uysal and N. Al-Dhahir, “Quasi-orthogonal time-reversal space-time block coding for frequency-selective fading channels,” IEEE Trans. On Signal Processing, vol. 55, no. 2, pp. 772-778, Feb. 2007.

[13] Y. Zhu and K. B. Letaief, “Single-carrier frequency-domain equalization with decision-feedback processing for time-reversal Space-time block coded systems”, IEEE Trans. on Communications, vol. 53, no. 7, pp.1127-1131, July 2005.

[14] T. W. Yune, C.H. Choi, and G. H. Im, “Single carrier frequency-domain equalization with transmit diversity over mobile multipath channels,” IEICE Trans. Commun., vol. E89-B, pp. 2050-2060, July 2006.

[15] W. M. Younis et al, “Efficient adaptive receivers for joint equalization and interference cancellation in multi-user space-time block-coded systems,” IEEE Trans. on Signal Processing, vol. 51, no.11. pp. 2849-2861, Nov. 2003.


[16] W. M. Younis, “Efficient receivers for space

time block coded transmission over broadband channels," PhD Thesis, UCLA, USA, 2004.

[17] W. C. Jakes ,“Microwave mobile communica-tions,” NY, Wiley, 1974.

[18] S. Haykin, “Adaptive filter theory,” Prentice Hall, 1986.

[19] T. K. Moon, and W. C. Stirling, “Mathematical methods and algorithms” Prentice Hall, 2000.

[20] H. H. Zheng and L. Tong “Blind channel estimation using second order statistics: algorithms,” IEEE Trans. on Signal Processing, vol.45, no.8. pp.1919-1930, 1997.

[21] I. D. Marsland , “Iterative noncoherent detection of differentially encoded M-PSK,” PhD Thesis, University of British Colombia, Canada, 1999.


OPTIMAL POWER CONTROL OF A THREE-SHAFT BRAYTON CYCLE BASED POWER CONVERSION UNIT K.R. Uren*, G. Van Schoor* and C.R. Van Niekerk* * School of Electrical, Electronic and Computer Engineering, North-West University, Potchefstroom Campus, P/Bag X6001, Potchefstroom, 2520, South Africa. Email: [email protected] / [email protected] / [email protected]

Abstract: This paper discusses the development of a control system that optimally controls the power output of a Brayton-cycle based power conversion unit. The original three shaft design of the Pebble Bed Modular Reactor (PBMR) power plant is considered. The power output of the system can be manipulated by changing the helium inventory to the gas cycle. The helium inventory can be manipulated in four ways: Injecting helium at the high-pressure side of the system by means of a booster tank; extracting helium at the high-pressure side of the system; injecting helium at the low-pressure side of the system and lastly opening and closing the bypass control valve. The control system has to intelligently generate set point values for each of the four helium manipulation mechanisms to eventually control the power output. In this paper two control strategies are investigated namely PID control and Fuzzy PID (FPID) control. The FPID control strategy is a linear type Fuzzy controller, but can progressively be made nonlinear if nonlinearities exist in the system. An optimal control system is derived by applying an optimisation technique to the gain constants of the controllers. A Genetic Algorithm (GA) is used to optimise the gain constants of both the PID and FPID controllers. The GA uses the ITAE performance index as an objective function. Key words: Brayton-cycle, PID control, Fuzzy PID control, Genetic Algorithms, Pebble Bed Modular Reactor.

1. INTRODUCTION

In this paper a power generation system will be considered that can produce up to 110 MW of electrical power. This system is called a module and can operate in a stand-alone mode, or as part of a power plant that can have more of these units [ HYPERLINK \l "MCN02" 1 ]. Figure 1 gives a schematic layout of this power generation module. This module contains a graphite-moderated, helium-cooled reactor and uses the Brayton direct gas cycle to convert the heat, which is generated in the core by nuclear fission. The heat is then transferred to the coolant gas (helium), and converted into electrical energy by means of a gas turbo-generator. The ideal Brayton cycle consists of two isentropic and two isobaric processes. In Figure 2 a temperature vs. entropy graph of the ideal Brayton cycle is given. Starting at (1), gas at a low pressure and temperature is compressed in an isentropic process to a higher pressure (2). From (2) to (3), the gas is heated in an isobaric (constant pressure) process to the maximum cycle temperature. From (3) to (4), the hot high-pressure gas is expanded isentropically in a turbine to a lower pressure and temperature. The cycle is completed from (4) to (1) by cooling the gas at constant pressure. By adding to the gas inventory of the cycle, the electrical power generated will be increased and by removing inventory the power generated will be decreased. This is the primary method of controlling power.

Core

High-pressurecompressor

Low-pressurecompressor

Intercooler Pre-coolerRecuperator

Gas cycle bypasscontrol valve

Power turbine

Low-pressureturbine

High-pressureturbine

Generator

12

3 4

Low-pressure injection

High-pressure injection and extraction

Figure 1: Schematic layout of the Brayton cycle based power conversion unit [1]

1

2

3

4

Entropy

Tem

pera

ture

Figure 2: Temperature vs. entropy graph of the ideal

Brayton cycle [2]


The power control system constitutes four helium manipulation mechanisms:

Gas bypass Low-pressure injection High-pressure extraction High-pressure injection (by means of a booster

tank) An existing linear Simulink® model [3,4] of the system shown in Figure 1 is used to illustrate to the reader the effect the four helium manipulation mechanisms have on the power output of the system. The linear model is used as a test platform for the control system. Opening the gas cycle bypass control valve will reduce the power and closing it will increase the power as shown in Figure 3 and Figure 4 respectively. Extraction of gas at the high-pressure side results in an instant decrease in the power of the system. The power response due to extraction is given in Figure 5. A limited amount of helium can be injected at the high-pressure side of the system depending on the pressure in the booster tank. Figure 6 shows the instant increase in power during boosting (high-pressure injection). Injection of gas at the low-pressure side of the system does not result in an instant increase in the power output of the system. The power first decreases and then starts to increase as shown in Figure 7. This phenomenon is called the non-minimum phase effect [1].

Figure 3: Bypass valve opening

Figure 4: Bypass valve closing

Figure 5: High-pressure extraction

Figure 6: Booster tank high-pressure injection

Figure 7: Low-pressure injection

A control system needs to be designed that will intelligently generate set point values for each of the four helium manipulation mechanisms as shown in Figure 8.

Control system

Bypass valve

Low-pressure injection

High-pressure extraction

Booster tank

Power plant

ErrorDesired power Grid power+

-

Figure 8: Power control system configuration


2. CONTROL SYSTEM DESIGN 2.1 Control methodology The control system comprises four individual controllers, each generating a set point value for a specific helium manipulation mechanism. The control system is simulated for both PID and Fuzzy PID control strategies. A schematic layout of the control system is given in Figure 9. The difference between the power reference value, refP , and the actual electrical power generated, P , called the power error pe , determines whether helium should be added to, or removed from the cycle. Although helium injection and extraction are normally used as the main control mechanisms, the bypass valve is mainly used to control the power output of the system in this particular case. pe is the input to a controller that generates a bypass valve set point value, BPV-sp. The efficiency of the system depends greatly on the setting of the bypass valve. If the valve opening it too large, a great amount of helium will be re-circulated through the compressors, rendering the system very inefficient. The non-minimum phase effect can be avoided by closing the bypass valve while injecting helium at the low-pressure side of the system. If the bypass valve opening is too small it would not be possible to avoid the non-minimum phase effect. The bypass valve therefore has to be kept at a predefined reference to allow for a certain amount of reserve capacity without degrading efficiency too much. This predefined reference is called the bypass valve reference,

refBPV . The bypass valve set point value is subtracted from the bypass valve reference value to obtain the bypass valve error, BPVe . BPVe is the input to three other controllers that generate set point values for boosting, low-pressure injection and high-pressure extraction. When the bypass valve operates away from its reference point, these three controllers will generate set point values that will restore the bypass valve to its reference value.

The activation system determines which set point value may be ported to the system. When the power error is positive the low-pressure injection set point is connected to the system and when the power error is negative the high-pressure extraction set point is connected. Boosting is only activated when the power error is positive and above a specified value. The bypass valve set point is always connected. This activation system eliminates conflicting set points among the helium manipulation mechanisms. For example it is not desirable to inject helium and extract helium at the same time. 2.2 Fuzzy PID control A Fuzzy controller can be regarded as a superset of linear controllers [5-8]. Under certain assumptions it is possible for the Fuzzy controller to emulate a PID controller. In conventional PID controllers, the control variable, )(tu is defined in terms of the error, )(te between a reference value, refy and the process output, )(ty :

)()()()(0

tedtdGdtteGteGtu d

t

ip (1)

where pG , iG and dG are proportional, integral and derivative gains respectively. In order to emulate a PID controller by means of a linear Fuzzy controller, the summation in the PID control equation has to be replaced by a Fuzzy rule base acting like a summation [9]. A Fuzzy PID (FPID) controller uses the variables error, e, change of error, ce, and integral of error, ie, in the antecedent of IF-THEN rules and the control variable, u, as consequent [10-12]. A Fuzzy controller based on the Mamdani-type Fuzzy inferences would consist of rules having the form:

perefP

refBPV BPVe

P

Figure 9: Schematic layout of the power control system


(2)

where n is the rule number and jiA , , nB are Fuzzy sets. A Fuzzy controller can be represented as an input-output mapping. In the general case it may result in a non-linear shaped control hyper surface. When three inputs

iecee , , and one output u are considered, this mapping takes the form given in (3). (3)

However assumptions need to be made to allow the Fuzzy rule base to act like a summation resulting in a linear mapping given by (4). (4)

A Fuzzy controller becomes linear by making the following assumptions with respect to the input universes, rules, membership functions and Fuzzy connectives [9]:

The input universes of the Fuzzy controller must be large enough for the input to stay within the limits (saturation is not allowed). The input sets must be triangular and cross their neighbouring sets at the membership value 5.0 ; their peaks thus being equidistant. Any input value can thus be a member of at most two sets; and its membership of each is a linear function of the input value.

The terms of the rules has to be combined by the AND operator (outer product) to ensure completeness. The output sets should preferably be singletons equal to the sum of the peak positions of the input sets. The output sets may also be triangular and symmetric about their peaks, but singletons simplify defuzzification.

Linearity is also ensured by choosing the algebraic product for the AND connective.

The next step in the design process is to derive the Fuzzy gain constants (see Figure 10) from the PID gain constants. The Fuzzy PID controller emulates the PID controller if the following equation holds [5]: (5)

By comparing the gain constants of the FPID controller with the gains of the conventional PID controller in (5), the following relations can be derived [5]:

(6)

(7)

(8)

If it is assumed that the error is within the range [-E,E] and the input universe of the Fuzzy controller is for example [-100,100], the Fuzzy gain FGp can be derived as follows [5]: (9)

e FGp

dtd / FGd

FGi

FGu u

Figure 10: Fuzzy PID controller

A rule base with three inputs, however, easily becomes very large and rules concerning the integral action becomes troublesome. It is therefore common practice to separate the integral action to form a Fuzzy PD+I (FPD+I) controller as shown in Figure 11.

e FGp

dtd / FGd

FGi

FGu u

Figure 11: Fuzzy PD+I controller

The controller function is separated into two additive parts as given by (10). (10)

2.3 Membership function definitions A Mamdani inference system is used. Consider a universe of discourse, [-E,E]. The membership functions for the inputs and output are defined as shown in Figures 12 and 13. The input membership functions for both the

) is (then ) is ( and ) is ( and ) is ( if : ,3,2,1

n

nnnn

BuAieAceAeR

),,( ieceefu

ceGieGeGu dip

ieFGuFGiceFGuFGdeFGuFGpFGuieFGiceFGdeFGp

ieGceGeGu idp

][

pp GFGp

FGuGFGuFGp 1

p

dd G

GFGpFGdGFGuFGd

p

ii G

GFGpFGiGFGuFGi

EFGp

EFGpEFGpeFGpEEe

100]100 ,100[],[

],[

)(),( iefceefuuu IFPDIFPD


error, e, and change of error, ce, are triangular. The input space is partitioned into three Fuzzy sets called negative (N), about zero (AZ) and positive (P). Singleton membership functions are chosen to define the output control variable u. The output space is partitioned into five Fuzzy sets called Negative big (Neg_Big), Negative small (Neg_Small), Zero (Zero), Positive small (Pos_Small) and Positive big (Pos_Big).

1N AZ P

-E E0 c,ce

Figure 12: Input membership functions

1

-2E E0-E 2E

Neg_Big Neg_Small Zero Pos_Small Pos_Big

u Figure 13: Output membership functions

2.4 Rule base definition The rule base of the Fuzzy controllers consists of nine rules. These rules link two inputs namely the error and the change of the error to a control output. The rules are defined in the following table.

Table 1: Rule base of Fuzzy controllers

Change in error Inputs Negative About Zero Positive

Negative Neg_Big Neg_Small Zero

About Zero Neg_Small Zero Pos_Small

Err

or

Positive Zero Pos_Small Pos_Big

3. CONTROLLER OPTIMISATION 3.1 Genetic algorithm optimisation Genetic Algorithms (GAs) are general search algorithms that imitate natural biological evolution. The idea is to evolve populations of individuals that are better adapted to their environment than the individuals from which they are created. GAs operate on a population of potential solutions applying the principle of survival of the fittest to produce successively better approximations to a solution. At each generation of a GA a new set of approximations is created by the process of selecting individuals according to their level of fitness and reproducing them using operators borrowed from natural genetics [13]. The performance of both the PID and FPD+I controller can be improved by adapting the gain values of each controller according to some objective function [14-17]. A strong characteristic of GAs is that they are able to optimise a large amount of parameters simultaneously. In the case of PID control, 4 PID controllers, each having 3 gains, will be optimised. This will result in 12 gain constants that will be optimised simultaneously. In the case of FPD+I control each controller has 4 gains resulting in a total of 16 gains to be optimised simultaneously. The GA used to optimise the controllers make use of real-valued genes instead of binary encoded genes. Consider for example the four Fuzzy controllers presented in Table 2. As previously stated, all four gain values of each Fuzzy controller will simultaneously be optimised by the GA. Let indN be the number of individuals in the population and indL the number parameters that needs to be optimised. In Table 2 the number of individuals used can vary but the number of parameters is fixed at 16. The initial population is therefore an indind LN matrix shown in Table 2. 3.2 Objective function The power output response of the system controller by both PID and FPD+I controllers is evaluated by means of the ITAE performance index as given by:

(11)

The upper limit T is a finite time chosen somewhat arbitrarily so that the integral approaches a steady-state value and )(tep

is the power error. It is usually convenient to choose T as the settling time, sT . The lower the value of the performance index the better the performance.

.)(0

dttetITAET

p


Table 2: Initial GA population

This specific performance index was chosen because it reduces the contribution of the large initial error to the value of the performance integral, and it emphasizes errors occurring later in time.

4. RESULTS The optimal gain values for the PID and FPD+I control strategies after 100 generations are summarised in Tables 3 and 4.

Table 3: Optimal gain values of the PID control strategy PID controller GE GCE GIE

1 (BPV-sp) 47.67 37.75 0

2 (LPINJ-sp) 0 0 0

3 (HPEXT-sp) 134.61 199.61 0

4 (BOOST-sp) 194.69 62.16 157.91

Table 4: Optimal gain values of the FPD+I control strategy

FPD+I

controller GE GCE GIE GU

1 (BPV-sp) 14.51 83.97 0 0.31

2 (LPINJ-sp) 0 0 0 0.17

3 (HPEXT-sp) 0 61.01 0 0.79

4 (BOOST-sp) 19.35 62.34 0 0.69

As can be seen the GA chose the integral gains close to zero. This shows that proportional derivative control is sufficient. The GA penalises low-pressure injection by giving the proportional and derivative gains values of zero. This means that according to the objective function low-pressure injection leads to undesirable responses. Figures 14 and 15 show the plots of the objective function values of the fittest individual in each generation. It can be seen that after approximately 20 generations the objective function value converges.

Figure 14: Objective value of the fittest individual in each

generation (PID strategy)

Figure 15: Objective value of the fittest individual in each

generation (FPD+I strategy)

The performance improvement of the optimised controllers are now illustrated by testing the system with a specific power reference sequence. The response of a non-optimal system is given in Figure 16 and that of the systems optimised for the PID and FPD+I control strategies are given in Figures 17 and 18 respectively.

Individuals 1

Fuzzy controller 1 Fuzzy controller 2 GE GCE GIE GU GE GCE GIE GU

40.23 17.7 28.95 15.38 20.45 33.61 10.2 15.56 82.06 13.26 13.35 0.09 70.21 8.01 11.61 0.02 52.43 25.64 15.20 2.54 30.12 11.3 14.71 0.62 47.5 49.10 9.09 10.65 80.33 22.27 1.56 4.32

90.50 13.46 25.63 0.89 9.13 2.98 20.88 17.77 47.21 25.29 7.89 10.48 43.2 15.21 8.63 18.23

Fuzzy controller 4 GE GCE GIE GU

20.45 33.61 10.2 15.56 70.21 8.01 11.61 0.02 30.12 11.3 14.71 0.62 80.33 22.27 1.56 4.32 9.13 2.98 20.88 17.77 43.2 15.21 8.63 18.23

90.50 13.46 25.63 0.89 9.13 2.98 20.88 17.77 52.43 25.64 15.20 2.54 30.12 11.3 14.71 0.62

20.45 33.61 10.2 15.56 43.2 15.21 8.63 18.23

1indN

indN

Parameters 1 indL


Figure 16: Non-optimal system output (ITAE value of

77.7)

0 20 40 60 80 100 120 140 160100

101

102

103

104

105

106

107

Time (s)

Pow

er (

MW

)

P

Pref

Figure 17 Optimal PID control (ITAE value of 23.03)

Figure 18 Optimal FPD+I control (ITAE value of 24.62)

The GA was able to derive optimal gain values after 100 generations. The objective function value of 77.7 for a non-optimal system was reduced to values of 23.03 and 24.62 for the optimal PID and FPD+I control strategies respectively. This shows that the GA is an effective parameter optimisation technique.

5. CONCLUSION In this study both PID and FPD+I control strategies consisting of four controllers each were developed to optimally control the power output of a Brayton cycle based power conversion unit. The performance of these control strategies was optimised by using a GA.

The optimised control systems showed superior performance compared to the non-optimal control system. The ITAE objective function proved to be very effective. However, further work on the objective function is needed to take other constraints into account such as the reserve capacity and system stresses. Fuzzy controllers that simulate PID control were used. These linear Fuzzy controllers can be converted to non-linear Fuzzy controllers by using Gaussian membership functions. Some research can still be done on GAs and their use in relation to Fuzzy systems. Different parts of the Fuzzy system can be optimised by means of a GA. The effect of optimisation of the rule base and the membership function parameter values warrants valuable future work [18,19].

6. REFERENCES [1] M.C. Nieuwoudt and L.P. Cilliers, "Demonstration

Plant Operations Summary," PBMR, Pretoria, Tech. Rep. 2002.

[2] A.P. George, "Introduction to the Pebble Bed Modular Reactor (PBMR)," PBMR, Pretoria, Tech. Rep. 2002.

[3] J.F. Pritchard, "Development of a linear model of the Brayton cycle," PBMR, Pretoria, Tech. Rep. 2002.

[4] O. Rubin and J. Pritchard, "Dynamic modelling for control of a new generation nuclear power station," in IFAC, 2002.

[5] V.Georgesco. Fuzzy Control Applied to Economic Stabilization Policies. [Online]. http://www.ici.ro/ici/revista/sic2001_1/art5.htm

[6] K.C Sio and C.K. Lee, "Stability of Fuzzy PID controllers," IEEE Transactions on systems, man and cybernetics - Part A: Systems and Humans, vol. 28, no. 4, pp. 490-495, July 1998.

[7] H. Chen, "Optimal Fuzzy PID controller design of an active magnetic bearing system based on adaptive genetic algorithms," in Proceedings of the Seventh International Conference on Machine Learning and Cybernetics, Kunming, 2008, pp. 2054-2060.

[8] X. Cheng, Z. Lei, and Y. Junqiu, "Fuzzy PID controller for wind turbines," in Second International Conference on Intelligent Networks and Intelligent Systems, 2009, pp. 74 - 77.

[9] J. Jantzen. Tuning of Fuzzy PID Controllers. [Online]. www.iau.dtu.dk/~jj/pubs/fpid.pdf

[10] H. Cho and K. Cho and B. Wang, "Fuzzy-PID hybrid control: Automatic rule generation using genetic algorithms," Fuzzy sets and Systems, vol. 92, pp. 305-316, 1997.

[11] K. Jong-Hwan and K. Kwang-Choon and E.K.P Chong, "Fuzzy Precompensated PID Controllers," IEEE Transactions on control systems technology, vol. 2, no. 4, pp. 406-411, December 1994.


[12] H. Baogang and G.K.I. Mann and R.G. Gosine,

"New Methodology for Analytical and optimal design of Fuzzy PID controllers," IEEE Transactions on Fuzzy systems, vol. 7, no. 5, pp. 521-539 , October 1999.

[13] P. Flemming, H. Pohlmein and C. Fonsesca A. Chipperfield. Genetic Algorithm Toolbox for Matlab® Ver. 1.2. [Online]. http://www.shef.ac.uk/content/1/c6/03/35/06/manual.pdf

[14] T.S. Li and M. Shieh, "Design of a GA-based Fuzzy PID controller for non-minimum phase systems," Fuzzy sets and systems, vol. 111, pp. 183-197, 2000.

[15] R. et al. Bandyopadhyay, "Autotuning a PID controller: A Fuzzy-genetic approach," Journal of Systems architecture, vol. 47, pp. 663-673, 2001.

[16] A. Varsek, T. Urbancic, and B. Filipic, "Genetic

Algorithms in Controller Design and Tuning," IEEE Transactions on systems, man, and cybernetics, vol. 23, no. 5, pp. 1330-1339 , Sept 1993.

[17] D. Leitch and P.J. Probert, "New Techniques for Genetic Development of a Class of Fuzzy Controllers," IEEE Transactions on systems, man and cybernetics - Part C: Applications and reviews, vol. 28, no. 1, pp. 112-123, February 1998.

[18] F. Herrera and L. Magdalena. Genetic Fuzzy Systems: A tutorial. [Online]. http://www.mat.upm.es/~llayos/papers/GFS.ps

[19] L. Magdalena and F. Monasterio-Huelin, "A Fuzzy Logic Controller with learning through the evolution of its knowledge base," International Journal of Approximate Reasoning, vol. 5, pp. 335-345, 1997.


UNIVERSAL DECREMENTAL REDUNDANCY COMPRESSION WITH FOUNTAIN CODES F.P.S. Luus*, A. McDonald** and B.T. Maharaj*** * Sentech Chair in Broadband Wireless Multimedia Communications, University of Pretoria, Lynnwood Road, 0002, Tshwane, South Africa. E-mail: [email protected] ** Dept. of Electrical, Electronic & Computer Engineering, University of Pretoria, Lynnwood Road, 0002, Tshwane, South Africa. E-mail: [email protected] *** Sentech Chair in Broadband Wireless Multimedia Communications, University of Pretoria, Lynnwood Road, 0002, Tshwane, South Africa. E-mail: [email protected] Abstract: A new universal noise-robust lossless compression algorithm based on a decremental redundancy approach with Fountain codes is proposed. The binary entropy code is harnessed to compress complex sources with the addition of a preprocessing system in this paper. Both the whole binary entropy range compression performance and the noise-robustness of an existing incremental redundancy Fountain code compression technique are exceeded. A new autocorrelation-based symbol length estimator, the Burrows-Wheeler block sorting transform (BWT) and Move-to-Front transformation (MTF) with a new entropy ordered MTF indices transformation reduces the binary entropy of a universal data source. The preprocessed input source is coded with a new modified incremental degree LT-code (Luby Transform) and a low-complexity decremental redundancy algorithm is used to compress the Fountain-coded source. The improved compression and robustness against transmission errors with our novel incremental degree puncturing decremental redundancy algorithm is shown. The universal (complex memory source) compression performance of the proposed system is shown to achieve appreciable compression. Key words: symbol length estimator, EOI, entropy ordered indices, decremental redundancy, incremental degree LT-code, LT-IDP, LT-CLID, adaptive successive rate refinement.

1. INTRODUCTION

An integral part in the determination of the overall performance and usage efficiency in communication is source compression [1]. However, in wireless networks with residual channel noise and errors, the traditional data compression techniques, including Huffman [2], Lempel-Ziv [3], Run-length, Tunstall [4], arithmetic coding and entropy encoders that describe compressed data with variable-length symbols suffer from catastrophic error propagation for even single errors [5-6]. Reducing the input block length of these compressors, to minimise the effect of error propagation, results in poorer compression. This paper extends the noise-robust binary entropy coder of our previous work in [7], which uses error correction codes to diminish the effects of catastrophic error propagation. A preprocessing system is added to produce binary compressible data from a source with complex memory so that the compressor in [7] can be harnessed as a universal compressor. Linear error correction codes can achieve a coding rate equal to the source entropy for asymptotically large discrete memoryless sources [8]. Non-universal attempts for compression of memoryless sources with linear codes This research was funded by the Sentech Chair in BWMC, UP, Lynnwood Road, 0002, Tshwane, South Africa. An earlier version of this paper was first published in IEEE WiMOB'08, Avignon, France, and with approval, this is an extended article.

[9-11] performed poorly compared to traditional compression algorithms. Sparse-graph low-density parity check (LDPC) block error correcting codes have been used for lossless compression of processed sources [12-15] and lossy compression of binary symmetric sources [16]. The variable length coding rates supported by Fountain codes [17] lend itself toward a more natural compressor implementation than the fixed-rate LDPC codes, so the focus in this paper is on rateless Fountain codes. Belief-propagation decoding (BP) and a priori log-likelihood ratio information can be used for decompression, and the incremental redundancy iterative CLID (Closed-loop iterative doping) algorithm can utilise the BP variable node reliability information to improve compression speed and ratio. A novel two-stage approach in conjunction with CLID was presented [17] to reach comparable performance to the LDPC compressor. The possibility of using only LT-codes (Luby Transform), which are patented [18], and CLID is mentioned in [17] as an inferior technique (LT-CLID), as well as the use of a decremental-redundancy approach with an LT-code and systematic precode (Raptor code). Rectangular puncturing matrix and adaptive successive rate refinement strategies are proposed [19] for decremental redundancy compression with forward error correcting codes. Turbo codes have been used for


decremental redundancy source coding [1], where encoded data were punctured sufficiently to produce compressed data that were losslessly decodable. Universality is ensured with a new autocorrelation-based symbol length estimator (SLE), the Burrows-Wheeler block sorting transform (BWT) and Move-to-Front transformation (MTF) with a new entropy ordered MTF indices (EOI) transformation. The binary entropy of a universal source is reduced by exploiting higher order source entropy with a combination of these preprocessing techniques. The SLE improves the exploitation of source memory with the BWT and the proposed EOI after-MTF transform further reduces the binary entropy. The decremental redundancy approach to compression with binary Fountain codes, originally proposed by us in [7], is included to make this paper self-contained. The purpose of the decremental redundancy approach is to increase the robustness to transmission errors over that of the existing incremental redundancy closed-loop iterative doping (LT-CLID) algorithm [17]. The noise-robust decompression of corrupted compressed data should minimize the symbol error probability in the decompressed data. The noise-robust lossless compression algorithm in [7], which is based on a decremental redundancy approach with Fountain codes, is used for compressing binary memoryless sources. The whole binary entropy range compression performance and the noise-robustness of LT-CLID are again shown to be exceeded by the decremental redundancy approach. It is shown that appreciable compression of the Calgary and Canterbury compression benchmark corpora [20] can be achieved with our universal compressor when it uses the proposed entropy coder. Section 2 explains the different algorithms and transforms involved and how they are combined to operate as a universal data compressor. Section 3 introduces the SLE for determining a suitable symbol length for a BWT on a binary represented source with unknown content. A new after-MTF transform, namely the EOI transform, is proposed in Section 4 for improving the overall compression. The use of Fountain codes in a decremental redundancy setting and the performance variations with a systematic precode, a constant input degree distribution and a low-complexity puncturing distribution are revisited in Section 5. Finally, in Section 6, the combined universal compression performance of the proposed scheme is evaluated against existing compressors and algorithms.

2. COMPRESSION SYSTEM OVERVIEW A memoryless entropy coder can be used in a universal data compressor as shown in Fig. 1, where the necessary preprocessing stages reduce the binary entropy of the

input. As a universal data compressor this system processes a binary string input, with no a priori knowledge of the nature of the source, to give an approximately memoryless output, which is then segmentally compressed. The entropy of the input source is reduced for appreciable compression with the Fountain code and redundancy-varying algorithm. This study was limited to binary Fountain codes, and hence the binary entropy is reduced to increase the achievable compression.

Block Sorting Transform

(SLE+BWT)

Entropy reduction

(MTF+EOI)Segmentation

Fountain encoder

Redundancy-varying algorithm

Precoding

Belief-propagation Fountain decoding

Inverse Redundancy-

varying algorithm

Inverse Precoding

Inverse BWT

Inverse MTF+EOI

Com

pres

sion

Dec

ompr

essi

on

Figure 1: Universal compression system The BWT [21] is used to shift redundancy in the source memory to redundancy in the marginal distributions. An SLE determines a source symbol bit length, which will increase the exploitation of memory with the BWT. The MTF [22] reduces the binary entropy of the BWT output, whilst a one-dimensional EOI transformation further lowers the entropy of the MTF transform output. Compression is achieved through entropy encoding of the preprocessed source. A Fountain code in the form of a binary LT-code [23] is used to produce output symbols, in addition to a decremental redundancy algorithm to remove redundancy in the output, to produce the compressed data. The decompression involves the mapping of the compressed data to a reconstructed bipartite graph which is decoded with belief-propagation (BP). Lossless compression can be verified since decompression at the compressor is possible.

3. SYMBOL LENGTH ESTIMATION The digital input source, for the universal compression application, can be primitively represented as a bit string. The BWT should receive a string of symbols that best reveal the memory present in the source. Most uncompressed real-world data sources are described with q -ary (non-binary) symbols, e.g. ASCII characters are 8 bit symbols. The type of input data is not known a priori, so a mechanism is required to choose a suitable source symbol bit length, which produces the most memory in the source. The SLE produces an estimate of the symbol length which shows the highest correlation in the input source, and it is a heuristic approach. Estimation of the symbol length is based on an autocorrelation over fixed sized blocks into which the input bit string is divided. The usefulness of a symbol


length to reveal source memory is determined, for a low-order SLE, as the correlation magnitude calculated for adjacent symbols. A higher-order SLE correlates non-adjacent symbols in a structured progressive manner. The algorithm details for a low-order estimator are explained as follows. a) The input bit string is divided into M blocks of

length 2N bits, for detecting symbol lengths of up to N bits. The last block can be padded to 2N bits, since M is generally a large number, thus the adverse effect on the accuracy of the symbol length estimation algorithm is negligible.

b) Replace all binary 0 symbols with 1 , to ensure an antipodal autocorrelation.

c) Starting with the first block :1m m M , calculate the autocorrelation sum 2

1( ) ,N m m

m n nnb b

where mnb is the value of the thn bit of the thm block,

for all symbol lengths :1 N . d) Keep a mean symbol length autocorrelation sum

11

( ) ( ),mm kk

m for each symbol length after processing each block m . A mean symbol length autocorrelation is kept rather than a cumulative autocorrelation to facilitate an implementation that requires less memory.

e) Find : 1arg max ( )m m and update a symbol length frequency counter ( ) ( ) 1m m after processing each block m .

f) After all M blocks have been processed, find the symbol length estimate : 1ˆ arg max ( ) . This is the symbol length that has the highest autocorrelation sum most frequently.

By calculating a mean autocorrelation function ( )m , the algorithm is able to distinguish a smaller constantly occurring autocorrelation peak among larger peaks with an average value of zero (when averaged over all blocks). By using the most frequently occurring peak over M functions, instead of solely using the peak of the final mean autocorrelation function ( )M , we avoid a situation in shorter files where a few blocks with very large autocorrelation peaks at the incorrect lag, lead to incorrect estimates. This situation might occur in shorter 24 bits-per-pixel image files with a header that contains strongly correlated data with 8 bits per symbol. The performance of the preprocessing stages in the exploitation of memory, for different source symbol bit lengths for the Calgary and Canterbury corpora [20], were determined. This performance dependency on symbol length was used as a benchmark to evaluate the accuracy of the symbol length estimator. The statistics of

1( ) ( ) ( )m t

Ns m t

are shown in Fig. 2 where multiples of 8 bit symbols are most prominent. Accurate source symbol length estimation ensures that the BWT will remove as much memory-based redundancy as

possible, which maximises the compression with a binary fountain code and redundancy-varying algorithm.

0 5 10 15 20 25 300

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Symbol length [bits]

Mea

n

0 10 20 300

0.1

0.2

0.3

0.4

0.5


Sta

ndar

d de

viat

ion

Figure 2: Statistical analysis on the symbol estimation results The best symbol length is the one that allows for the maximal removal of redundancy. The SLE results in Fig. 2 show a good correlation with the removed redundancy in Fig. 3, showing the effectivity of the symbol length estimation used in the preprocessing.

1 8 16 24 320.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7


Mea

n ra

tio o

f red

unda

ncy

rem

oved

0 5 10 15 20 25 300

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Symbol length [bits]S

tand

ard

devi

atio

n of

red

unda

ncy

rem

oved

Figure 3: Binary entropy reduction performance for the Calgary and Canterbury corpora for varying symbol lengths

4. ENTROPY ORDERED INDICES TRANSFORM The MTF transform replaces the normally high binary entropy representations of the symbols in the BWT output, with low value number representations. An optimal preprocessor for the binary Fountain code compressor should minimize the binary entropy of the input. The MTF transform with entropy ordered indices replaces the normally high binary entropy representations with low binary entropy representations.

4.1 Entropy ordered recurrence list creation As a recurrence time coding scheme, the MTF transform replaces a symbol with the number of distinct symbols that have appeared since the previous occurrence of the current symbol. For a source with memory, the BWT output contains segments of a repeated symbol so that the MTF transform produces a low value number output, with the most common index value being 0 . The EOI transform replaces the recurrence index values in the MTF transform output with entropy ordered index values. A one-dimensional transformation vector must be generated to map the normal MTF transform output values to low entropy values.


Let 1 2 2, , ,W w w w denote the entropy ordered

index list, where the amount of unique symbols in the BWT output is A and 2log 1A . For a binary

one biased transform, the elements iw of list W are ordered according to the amount of binary ones, 1 in w , in the binary representation of iw . The output of concatenated indices has as few binary ones as possible to minimise the binary entropy. a) Initialize the list W so that 1iw i for 1,2i . b) To sort W according to element-wise entropies,

create an intermediate vector 0 1 2 1, , ,G g g g ,

where 1 1 , for 0,2 1 .i ig n w i

i) Start with 0 0g , since 1 1 0n w . ii) Increment the list index counter, i.e. 1i i . iii) Generate the next elements ig to 2 1ig so that

1, for 0 .i j jg g j i iv) Update the index counter to the new size of G ,

that is 2i i . v) Repeat the generation process from iii) to v)

until 2 1i is satisfied. This algorithm is 296·2 2·2 lo/ g ·2 48 times less computationally complex than a brute-force list generation, for 32 bit integer index values.

c) Sort 1

: , for 1 2i iw wW g g i according to G .

4.2 Entropy ordered indices transformation The entropy ordered indices transform output is

1 2, , , ,AV v v v where A is also the length of the

MTF transform output. Set each element iv of V to the value jw in W , where the original index value is

1iu j , in the MTF transform output U , such that

1, for 1 .ii uv w i A

4.3 Inverse transformation The entropy ordering of the index values is a repeatable process, and an inverse transform exists. The original MTF transform output U can be obtained from the entropy ordered indices transform V as follows. a) Generate the same entropy ordered index list W at

the receiver. Determine 1 2 2, , ,W w w w where

1 , for 1 2 .iww i i

b) Reconstruct the original MTF transform output 1: 1, for 1 2

ii vU u w i from V and 'W .

4.4 Analysis The EOI transform will reduce the entropy of the MTF transform as shown in the following theorems. Lemma 1: There exists a set of number pairs

( , ) :J a b a b , where 1J and

2log 1 4a , so that 1 1( ) ( )n a n b . Proof: Let the binary equivalent the pair ( , )a b J be

represented in 2 , and ( 1)2 2( , , , ) : q s rh p q r s denote

the binary value formed by the bits : [ , ]kp k r s corresponding to positions ( 1)2 k in 2

qp . The set of lengths of most significant sub-symbols { [1, 2] : ( , , 1, ) ( , , 1, ) 1}h a h b ensures that a b , where 1( ( , , 1, ))n h a

1( ( , , 1, )) 1n h b . The size of the set ( )

2 1: ( )Y y n y m with m

binary ones is ( )! ( )! !Y m m . Then for

1 1( ( , ,1, )) ( ( , ,1, )) 2n h b n h a the size of

J is 2 2

1 2 0

( )! ( )!( )! ! ( )! !

i

i j

Ji i j j

.

Thus 1 1( ) ( )n b n a for a b , and 1J for 4 . Theorem 1: For MTF transform U with the normal list

[1,2 ]1:iq ii of index values, the relation

11

1 1

( )( )A Aji

i j

n un vH V H H U H

A A holds.

Proof: By design, the entropy ordered list indices iw have the relation 1 1( ) ( ), for i jn w n w i j . According to Lemma 1 the relation for the normal MTF transform indices iq are 1 1( ) ( ), for ( , )i j j in q n q q q J . For U with a ratio ir of occurrences of index value iq

we have 2

11ii

r . The MTF transform is characterised

by 1 i ir r , and the following reasonable assumptions 1

1

2 2

1 2 1 i ji j

r r and 12

11 ( ) 0.5i ii

r n q are

made. Since 1i ir r the MTF transform indices will then result in a transform with more binary ones, for a binary one biased approach, and consequently a higher entropy such


that 2 21 11 1( ) ( ) .i i i ii i

r n w r n q An

approximation of 2 2

1 11 11 ( ) ( )i i i ii i

r n w r n q is 1

1

1 1

1

1 1

2 2 211 1

1 1 2

2 2 21

11 2 1

11

2 21 1

1 1

( )( ) ( )1 1

( )1 since

( )0.5 si

0.5

nce 0.5, for 1 2

( ) ( )since

ji i ii j

i i j

ii i j

ii j

i

i ii i

i i

n wr n w n wr r

n wr r r

n wi

n q n qr r

1

1

2 2 21

1 1 2 1

( )since .i

i i ji i j

n qr r r

The relation between the normal list indices iq and

entropy ordered indices iw is then given as 2 2 2

1 1 1

1 1 1

( ) ( ) ( )1i i i

i i ii i i

n w n q n wr r r .

For the binary entropy function (·)H and the above relation, the theorem holds for the assumptions made.

4.5 Experimental results The binary entropy reductions H U and H V were measured for the Calgary and Canterbury corpora [20] with the application of the preprocessing, as shown in Table 1. The SLE, BWT and MTF transform were used in addition to the entropy ordered indices transform (EOI) for determining H V . The binary entropy values in Table 1 are multiplied with the ratio of the transform filesize to the original filesize. For a sufficiently large input source with an original BWT symbol bit length being greater than 2log 1A , the MTF transform filesize is smaller than the original filesize. Table 1: Preprocessing binary entropy reduction comparison Corpus MTF transform MTF+EOI transform H U Std.

Dev. H V Std.

Dev. Calgary 0.4468 0.1984 0.3947 0.2054 Canterbury 0.4149 0.0845 0.3639 0.0824 Calgary 0.0521 Canterbury

Additional binary entropy reduction with EOI transform 0.0510

The EOI transform further reduces the binary entropy of real-world sources by more than 5% when compared to the use of only the MTF transform as post-BWT stage. The preprocessing appreciably reduces the binary entropy

from more than 0.9 to less than 0.4, such that a binary entropy coder can achieve meaningful compression. The effectivity of the EOI transform in reducing binary entropy of the Canterbury corpus is displayed in Fig. 4.

1 2 3 4 5 6 7 8 9 10 110

0.2

0.4

0.6

0.8

1

Canterbury corpora file number

Bin

ary

entr

opy

Initial binary entropyReduced binary entropy (BWT+MTF+EOI)Reduced binary entropy (BWT+MTF)

Figure 4: Binary entropy reduction of Canterbury corpus with preprocessing stages

5. DECREMENTAL REDUNDANCY COMPRESSION WITH FOUNTAIN CODES

The Fountain code compressor that we developed in [7] is described in detail in this section, and the benefits of the code are shown. Entropy encoding of the preprocessed source is required for compression to be achieved. A Fountain code in the form of a binary LT-code [23] is used to produce output symbols in addition to a decremental redundancy algorithm to remove redundancy in the output to produce the compressed data. During decompression the compressed data is mapped to a reconstructed bipartite graph which is decoded with belief-propagation (BP).

5.1 Belief-propagation decoding Appreciable compression is possible with the soft-decoding of the LT-code when a priori log-likelihood information is included [17]. The BP algorithm is a message-passing algorithm, and the natural log-likelihood domain is used for all message calculations. The log-likelihood ratio of a random variable nx , describing the bit value associated with a node n , is

( ) ln ( 0) / ( 1)n n nL x p x p x . The compressed data includes the quantized uniform a priori log-likelihood ratio value ( )iL x for the original input source. The log-likelihood ( )oL x for each individual output node o is set at the start of the decoding based on the unpunctured output values as ( )oL x L , or ( ) 0oL x if punctured. The theoretical maximum achievable compression with a binary entropy encoder in terms of the a priori log-likelihood ratio is 1 1 iL xH e .

Messages sent in the th iteration are denoted by ( )m . The message from output node o to input node i is denoted by ( )

oim , and ( )iom denotes the message from the

input node i to the output node o . Message updates during each iteration for BP are given as follows:


( )

( ) 1

{ }

( )2 tanh tanh tanh

2 2o

o i ooi

i I i

L x mm (1)

( ) ( 1)

{ }

, if =0, if( )

)

0

(

i

i

io i o io O o

L xm L x m (2)

where iO is the set of output nodes incident on input node i and oI the set of input nodes incident on output node o .

The value 1 ( )sign ( )i

i i oio Ox L x m is the bit

value of an input node i for the isomorphism : (2) { 1, 1}GF , where (0) 1 and (1) 1 .

5.2 Compressor The procedure followed for compression of a memoryless source is explained as follows. 1) Choose a coding rate Z , where 1Z is the

proportional output overhead of the LT-code. 2) Create a robust soliton degree distribution sampling

pool and construct the bipartite graph, with K input nodes and KZ output nodes. A pseudo-random number sequence is used with the seeding value included in the compressed data.

3) Map all source input bits ix to each associated input node i .

4) Encode all output values ox , according to (2)GF addition, so that

oo ii I

x x .

5) Determine the uniform a priori input log-likelihood ratio ( )iL x for the entire input source.

6) Puncture or remove uniformly at random, a fraction 1 of the output values. In practice is adjusted with adaptive successive rate refinement [19], which is a binary search algorithm, of depth .

i) Choose two puncturing extremes with 1H

and 1 1 iL xL H e .

ii) Determine the mean 0.5 L H and randomly puncture (1 )KZ output values.

iii) With the decompressor, verify that lossless decompression is possible. In the case that it is not, set L , else set H .

iv) Repeat the search 1 times with the updated puncturing ratio extremes.

7) Form the compressed data with the unpunctured output values and input log-likelihood ratio, in addition to necessary overhead information such as the original source length and utilized compression parameters.

5.3 Decompressor The decompressor receives the compressed data as input which contains the unpunctured output values and input a priori log-likelihood ratio. Lossless decompression is ensured for perfect transmission of the compressed data. The decompression procedure is given as follows. 1) Recreate the bipartite graph by using the same

seeding value and creation parameters. 2) Map the available output values to the graph by

retracing the puncturing order. 3) Execute a limited amount of BP iterations. 4) Calculate the associated input node values, which

will be the preprocessed source input. 5) Determine the validity of the input by calculating and

comparing parity or checksum information included in the compressed data header.

5.4 Robust Soliton optimisation for compression The Robust Soliton c and variables were optimised as in Fig. 5, for all fountain codes used. A varying binary entropy source was generated for this experiment, where 10000 bit blocklengths were compressed, such that it contained 16 blocks in the useful binary entropy range of 14% to 88%.

0

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00.2

0.40.6

0.81

0.73

0.74

0.75

0.76

0.77

0.78

δ [probability]c [constant of order 1]

Com

pres

sion

rat

io o

f

vary

ing

bina

ry e

ntro

py s

ourc

e

Figure 5: Detailed optimising of Robust Soliton variables for maximum compression. For all the robust soliton output degree distribution implementations 0.05c and 0.4 were used, as it is in the good performance region of Fig. 5. 5.5 Puncturing influence on decoding The decremental redundancy approach involves the use of puncturing to increase the compression ratio, but the removal of output values in the compressed data affects the bipartite graph and belief-propagation decoding. The amount of incident opposing nodes is called the degree nd of a node n . Let the total amount of edges emanating from the input nodes be denoted by E . If iO contains an output node o with a punctured value, the input node i has modified degree 1'i id d and the


new effective set of incident output nodes is { }'i iO O o , the amount of bipartite graph edges are

reduced to E E . The reason is that the log-likelihood ratio for output nodes with punctured values is ( ) 0oL x , such that ( )

oim cannot contribute to the recovery of the value of any node present in the bipartite graph. Theorem 2: The probability of an input symbol value being unrecoverable is increased during puncturing so that ( 0) (' 0)i ip d p d for any input node. Proof: Let the resulting amount of edges, after puncturing be denoted by 'E . The input degrees are Poisson distributed for a large E , so that the probability of an

unconnected input node is 0/

p0!

' 0/E K

i

e E Kd

/ / ( 0)E K E Kie e p d .

5.6 Coding amendments Changes to the input- and output degree distributions are presented in this section to reduce the probability

( 0)'ip d of a zero degree input node i . Several degree distribution modifications were investigated as follows. Systematic precoding: The average input degree is increased with a systematic precode that adds high degree output symbols. The systematic precode output consists of the original input and additional output symbols. The precode is simultaneous since the LT-code input involves only the systematic part of the precode output. The number of unique input nodes participating in each of the additional high degree output nodes is maximized to ensure the lowest ( 0)'ip d . Effectively, with the LT-coding done only on the systematic part, the combination of the precode and LT-code produces a modified coding rate and output degree distribution. The effective bipartite graph can be used for simultaneous decoding of the precode and the LT-code. Input degree equalization: The LT-code input degree distribution can be changed from a Poisson distribution, for a large amount of edges E , to a constant distribution. The input nodes will have an approximate equal degree of

/id E K , so that output value puncturing for this amendment will produce a minimum ( 0)'ip d for an unchanged output degree distribution and coding rate. Precoding and equalization: The modification of the input degree distribution to a constant distribution used in conjunction with a systematic precode to increase the probability of lossless decoding of the input for the same puncturing rates. Incremental degree puncturing compressor: A low-complexity puncturing distribution can be used with a

modified LT-code to significantly improve the decremental redundancy compression performance. This useful puncturing distribution is used in the LT-code incremental degree puncturing compressor (LT-IDP). An incremental degree puncturing distribution produces much better compression than a random or decremental degree puncturing distribution. With an incremental degree puncturing distribution, (1 )KZ values of the lowest degree output nodes are removed. The iterative puncturing algorithm searches for an unpunctured value associated with an output node with the lowest degree of all the output nodes with unpunctured values. Normally output degrees are sampled uniformly at random from a degree pool for LT-coding, so that: 1) For an increasing output size, the output degree

distribution will be approximately representative of the robust soliton degree distribution.

2) Transmission burst errors will not as severely reduce the lossless decoding probability.

In order to reduce the linear computational complexity of finding the lowest degree output node with an unpunctured value to a constant complexity, an incremental degree LT-code can be used. LT-code output degrees are sampled incrementally and a random interleaver may be used prior to transmission to increase lossless decoding probability in the case of burst-type noise channels. The addition of a precode and/or the modification of the input degree distribution does not increase the performance of this puncturing distribution. 5.7 Performance comparison The binary compression performance of the various discussed existing and proposed compressors are compared in this subsection.

0 0.05 0.1 0.15 0.2 0.25 0.30

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1

Bernoulli source constant (ratio of binary one symbols to total amount of bits)

Com

pres

sion

/ B

inar

y en

trop

y

Binary entropy.Systematic precode andequalised input degreeswith random puncturing.Systematic precode withrandom puncturing.Equalised input degrees withrandom puncturing.Random puncturing.LT CLID.LT IDP.

Figure 6: Comparative memoryless source compression performance The comparative memoryless source compression performance for 10000K bit input blocks, 1.1Z (except for LT-CLID), with an LT-code with a robust soliton output degree distribution with 0.05c and

0.4 (as previously optimised), belief propagation


decoding with a maximum LLR of 25.0L and 8 a priori LLR bits, is given in Fig. 6. The source is a collection of Bernoulli sources with varying binary entropies. The LT-CLID compressor uses 300 BP iterations (per epoch in a 7 binary search) with a doping delay of

10dD iterations and a doping frequency of 1fD iterations per bit. The remaining compressors use 40 BP iterations (per epoch in a 7 binary search). The systematic precodes produce 100 additional output nodes of degree 100od , with edges to input nodes such that each input node has approximately 1 edge incident on one of the extra output nodes. The graphs are 10th degree polynomial approximations of the achieved compression ratios, including overhead information, for 5 repetitions of 200 input blocks of length 10000 bits over the entire binary entropy range. The mean compression ratio and standard deviations for the entire binary entropy range of the varying entropy Bernoulli sources for the investigated amendments are shown in Table 2. The systematic precode improves the compression performance of the random puncturing algorithm although not as much with the equalized input degrees. The LT-CLID incremental redundancy algorithm performs significantly better than the random puncturing decremental redundancy algorithm although the LT-IDP compressor outperforms the LT-CLID algorithm with a reduced computational complexity. Table 2: Comparative statistics of memoryless source compression performance Algorithm Mean Std. Dev. Systematic precode and random puncturing

0.907 0.164

Equalized input degrees and random puncturing

0.893 0.163

Systematic precode and equalized input degrees

0.896 0.176

Random puncturing 0.913 0.148 LT-CLID 0.857 0.189 LT-IDP 0.828 0.244 5.8 Noise robustness The purpose of this compression approach is to achieve meaningful compression whilst being more robust against error propagation and transmission errors than conventional compressors. The ability of the proposed LT-IDP compressor to recover from data corruption during decompression is compared with that of LT-CLID. An input source with 10 corruption repetitions, in a 10 times 10000 bit block, over the 0.141 to 0.722 binary entropy range was used. The compressed header was not corrupted, as it has a small size relative to the payload.

The LT-IDP compressor has an LT-code with 1.1Z , robust soliton output degree distribution with 0.05c and 0.4 , Poisson input degree distribution with 40 BP iterations (per epoch in 7 binary search). The LT-CLID compressor is optimized for higher noise robustness with a doping delay of 50dD iterations and a doping frequency of 5fD iterations per bit, for 300 BP iterations (per epoch in 7 binary search). Both compressors use belief propagation decoding with a maximum LLR of 25.0L and 8 a priori LLR bits. The noise robustness on a flat fading AWGN (Additive White Gaussian Noise) channel for binary antipodal modulation is shown in Fig. 7.

2 3 4 5 6 7 8 9 10 11

105

104

103

102

101

Eb/N

0

Bit

erro

r ra

te

LT IDP (Decremental redundancy)LT CLID (Incremental redundancy)Bit error probabilityLT CLID AverageLT IDP Average

Figure 7: Noise robustness on a flat fading BI-AWGN channel The graphs for LT-CLID and LT-IDP are the BER of the decompressed data and the line BER graph is the corruption ratio of the compressed data. The BSC (Binary Symmetric Channel) noise robustness is compared in Fig. 8 for the same compressor specifications as for AWGN.

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.0110

5

104

103

102

101

BSC p parameter [ratio]

Bit

erro

r ra

te

LT IDP (Decremental redundancy)BSC p parameterLT CLID (Incremental redundancy)LT CLID AverageLT IDP Average

Figure 8: Noise robustness on the BSC channel For the BSC channel, only LT-IDP is able to sustain the error fraction of the compressed data and for low p values it can reduce the error fraction in the decompressed data. The graph means, for BI-AWGN and BSC respectively, are 19.08% and 36.64% higher for LT-CLID. For both channels the higher noise-robustness of LT-IDP is apparent from Figs. 7 and 8.


6. UNIVERSAL COMPRESSION PERFORMANCE The file compression of the Calgary and Canterbury corpora were measured with the Bzip2 (Huffman coding), Gzip (Lempel-Ziv coding), PAQ8L [24] (predictive arithmetic coding) and the Fountain code compressors.

5 10 15 20 250

0.1

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0.9

1

1.1

1.2

Corpora file number

Com

pres

sion

rat

io

BZIP2GZIPPAQ8LLT CLIDLT IDP

Figure 9: Compressor performance comparison for Calgary and Canterbury corpora The BWT with symbol lengths determined by the SLE, the MTF and EOI transforms were used to pre-process the corpora files for the Fountain code compressors. The LT-CLID incremental redundancy and LT-IDP decremental redundancy algorithms were both separately used to compress the preprocessed files. For both of the redundancy-varying algorithms, blocks of 10000 bits were individually compressed. The LT-CLID algorithm was optimized for best compression with a doping frequency of 1fD iterations per bit and a doping delay of 10dD iterations for 300 BP iterations (per epoch in a 7 binary search). The LT-IDP algorithm used only 40 BP iterations (per epoch in a 7 binary search). The results are compared for the Calgary and Canterbury corpora in Table 3. Table 3: Compression performance statistics BZip2 GZip PAQ8L LT-

CLID LT-IDP

Mean 0.299 0.328 0.223 0.592 0.512 Std. Dev.

0.102 0.108 0.092 0.149 0.179

The LT-IDP compressor has a compression ratio for the Calgary and Canterbury corpora which is approximately 0.08 less than the ratio achieved by the LT-CLID compressor. There was only one benchmark file, namely geo in the Calgary corpus, which could not be compressed by either the LT-IDP or LT-CLID compressors. The preprocessor appreciably translated the original complex entropy, approximately represented by the performance of PAQ8L to the binary entropy of the preprocessor output. The Bzip2, Gzip and PAQ8L compressors performed significantly better than the Fountain code compressors as shown in Fig. 9.

7. CONCLUSION

A new method of improving the memory exploitation with the BWT, by determining a suitable symbol length with an SLE, was introduced. A new preprocessing strategy for translating complex source entropy to binary entropy was used to allow for universal compression with a non-universal binary entropy coder. This strategy included a new after-MTF stage transform, the EOI transform, which further reduced the binary entropy by more than 5%. A novel low-complexity implementation for the EOI transform is given, with a complexity which is 48 times less than that of a brute-force approach. Compression with LT-codes with a decremental redundancy-varying algorithm, as proposed by us in [7], was revisited, and the binary compression performance of a basic random puncturing scheme was shown to be improved by 97.82%, with a low-complexity incremental degree puncturing distribution combined with a modified LT-code. This decremental redundancy algorithm (LT-IDP) has an increased compression performance of 20.88% over that of the existing LT-CLID method. Amendments to the application of the LT-code were discussed to improve the compression performance of a random puncturing. The critical property of resistance to catastrophic error propagation during decompression with error-correction codes was considered. It was numerically shown that the LT-IDP algorithm provides greater noise robustness than the LT-CLID method for the BI-AWGN and BSC channels over the useful binary entropy range. The combination of the preprocessing and the binary Fountain code entropy coder as a universal compressor removed 62.79% of the maximum redundancy according to the best possible compression estimate (PAQ8L), for the Calgary and Canterbury corpora.

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[7] F.P.S. Luus and B.T. Maharaj, “Decremental redundancy compression with fountain codes”, IEEE Int. Conf. on Wireless and Mobile Computing (Wimob 2008), pp. 328-332, 12-14 Oct. 2008.

[8] I. Csiszar and J. Korner, Information Theory: Coding Theorems for Discrete Memoryless Systems, Academic, New York, 1981.

[9] P. E. Allard and A. W. Bridgewater, “A source encoding technique using algebraic codes”, Proceedings: 1972 Canadian Computer Conference, pp. 201-213, June 1972.

[10] H. Ohnsorge, “Data compression system for the transmission of digitalized signals”, Conf. Rec. IEEE Int. Conf. on Communications, vol. II, pp. 485-488, June 1973.

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[12] G. Caire, S. Shamai and S. Verdu, “A new data compression algorithm for sources with memory based on error correcting codes”, 2003 IEEE Workshop on Information Theory, pp. 291-295, March 30-April 4, 2003.

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[14] G. Caire, S. Shamai and S. Verdu, “Universal data compression with LDPC codes”, Third International Symposium On Turbo Codes and Related Topics, pp. 55-58, Brest, France, September 1-5, 2003.

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[16] A. Braunstein, R. Zecchina and F. Kayhan, “Efficient LDPC codes over GF(q) for lossy data compression”, IEEE International Symposium on Information Theory (ISIT 2009), pp. 1978-1982, June 28-July 3 2009.

[17] G. Caire, S. Shamai, A. Shokrollahi and S. Verdu, “Fountain Codes for Lossless Data Compression,” DIMACS Series in Discrete Mathematics and Theoretical Computer Science, 2005. [Online]. Available: http://www.princeton.edu/~verdu/reprints/dimacs2005.pdf

[18] A. Shokrollahi, S. Lassen, and M. Luby. “Multi-stage code generator and decoder for communication systems”. U.S. Patent Application #20030058958, 2003.

[19] N. Dutsch, “Code optimisation for lossless compression of binary memoryless sources based on FEC codes”, Euro. Trans. Telecomms, 17, pp. 219-229, Wiley InterScience, 2006.

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[21] M. Burrows and D. J. Wheeler, “A block-sorting lossless data compression algorithm,” Digital Systems Res. Ctr., Palo Alto, CA, Tech. Rep. SRC 124, May 1994.

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[24] M. Mahoney, S. Osnach and B. Pettis, PAQ8L software, 2007. [Online]. Available: http://cs.fit.edu/~mmahoney/compression/paq8l.zip


Notes

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