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TamKang University, New Taipei City, Taiwan [email protected] Yun-Kai Bai2, Richard H.-H. Yang3,. 2 National Sun Yat-Sen University, Kaohsiung City, ...

2015 International Symposium on Intelligent Signal Processing and Communication Systems (ISPACS) November 9-12, 2015

Semi-blind Channel Estimation Scheme with Bayesian DFE for PRP-OFDM System Yun-Kai Bai2, Richard H.-H. Yang3,

Shiunn-Jang Chern1, Kelvin Kuang -Chi Lee1 1


Department Electrical and Computer Engineering, TamKang University, New Taipei City, Taiwan [email protected]

side [3][4]. That could be used to achieve the orthogonality among subcarriers (or tones), and eliminating the inter-blockinterference (IBI) at the same time. In practice, the selection of redundancy-length (CP) is significant to affect the system performance, and should be trade-off with the spectral efficiency.

Abstract— This paper presents a novel block-based Bayesian decision feedback equalization (DFE) receiver for the PseudoRandom Postfix (PRP) Orthogonal Frequency Division Multiplexing (OFDM) (PRP-OFDM) systems, with semi-blind channel estimation. In conventional PRP-OFDM receiver for semi-blind channel estimation, an order-one statistics of the received signal is developed, for sufficient redundant symbols (PRP). In this paper, we consider the PRP-OFDM system, where the length of PRP (redundant symbols) is less than the channel order, named as the insufficient PRP-OFDM system. To deal with this problem and maintain reasonable complexity of receiver, first the maximum shortening signal-to-noise-ratio (MSSNR) time-domain equalizer (TEQ) is applied to reduce the effect of inter-block-interference (IBI). After that the block-based Bayesian DFE equalizer is developed for alleviating the intersymbol interference (ISI), hence to achieve better system performance, in terms of bit-error rate (BER). Computer simulation results verify that the proposed scheme could achieve BER improvement over the conventional block-based equalizers, viz., the linear zero-forcing (ZF) equalizer, non-linear ZF equalizer (with oblique projector) and the minimum mean square error DFE. Keywords—PRP-OFDM; redundant symbol; ISI; IBI; bolckbased DFE; zero-forcing equalizer; Oblique projector; BER.

I. INTRODUCTION Orthogonal Frequency Division Multiplexing (OFDM) system is one of the multi-carrier modulation (MC) techniques. In conventional OFDM systems, we divide the original data stream into parallel sub-streams, and transmitted simultaneously on different corresponding sub-carriers. The spectra of individual sub-carriers are overlapped with specific orthogonality constraints imposed to facilitate separation of sub-channels at receiver. Because the original data stream of OFDM system is partition into constructive equal-size data blocks, it is named as the block transmission systems. The OFDM system approach has the ability of converting the channel of frequency-selective fading into flat fading, that has been widely adopted for high-speed wireless communications [1]-[4] [14]. The inverse Fourier transform (IFFT) is considered to be the most efficient digital implementation of OFDM systems. In cyclic prefix (CP) OFDM system, it copies the last part signals as CP, which is not shorter than channel order, and put it in front of transmission block at the transmitter

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National Sun Yat-Sen University, Kaohsiung City, Taiwan 3 National Kaohsiung First University of Sci. and Tech. Kaohsiung City, Taiwan


In CP-OFDM system, the re-movement of the corrupted CP at receiver is capable to avoid the effect of IBI, at the same time it also makes the channel convolution from linear to circular. In consequences, each truncated block is processed using the fast Fourier transform (FFT) [1][4]. The help of diagonalizing the channel matrix by taking the FFT offers a powerful tool for devising simple effective schemes (equalizers) that compensates the degradation due to the inter-symbolinterference (ISI). Different criterion, such as the zero-forcing (ZF) and minimum mean square error (MMSE) are proposed for designing the equalizers, viz., linear equalizers and decision-feedback equalizers (DFE) to reduce the effect of ISI. Because the CP-OFDM system is very sensitive to channel null locations, close to the subcarriers, it results in performance degradation. The above mentioned problem of CP-OFDM can be solved by introducing the zero-padding (ZP) or null samples to replace the CP, hence named as the ZP-OFDM system; the price paid is to increase the receiver complexity [16]. In [7][8] an alternative approach is considered, it replaces the time domain redundancy by a pseudo-random postfix (PRP) weighted deterministic sequence, and is called the PRP-OFDM systems. It retains the advantages of ZP-OFDM system, but does not require the pilot tones, preambles, etc., hence do not have any loss in throughput or spectral efficiency compared to the CP-OFDM systems. Usually, in conventional PRP-OFDM system, sufficient PRP redundant symbols are adopted and can be applied at receiver for performing semi-blind channel estimation, with an order-one statistics of the received signal, and to track the channel variations [7]. In this paper, the PRP-OFDM system with the length of PRP (redundant symbols) to be less than the channel order is considered, and is named as an insufficient PRP-OFDM system. It can achieve better spectral efficiency compared with the one with sufficient PRP. To maintain the desired performance under the case of insufficient PRP-OFDM system, also to remain reasonable receiver complexity, we first apply the maximum shortening signal-to-noise-ratio (MSSNR) time-

2015 International Symposium on Intelligent Signal Processing and Communication Systems (ISPACS) November 9-12, 2015

domain equalizer (TEQ) for shortening channel order. It shortens the channel impulse response into the redundancy range to reduce effect of IBI [6]. After that the block-based Bayesian DFE equalizer [12]-[14] is developed to deal with ISI for improving the bit-error rate (BER) performance, compared with conventional approaches. It achieves the full potential of symbol-by-symbol equalizer. II.

to be greater than the channel order, i.e. L  D . With above assumptions, the IBI-induced channel can be modelled as [3]:


1, …, M-1. Consequently, the vector form of the n signal is denoted as sM (n)


defined as  h ( lP  P  1) · § h (lP ) ¨ ¸    ¨ ¸ h ( lP ) © h ( lP  P  1)  ¹ PuP



uq ( n )

¦ s (n)e


, 0 d q d M 1

H1,P contains nonzero elements only in its L u L top right



In a matrix form, we have T

[u ( nM ), u ( nM  1),  , u ( nM  M  1)] H



F s M (n) where F is the fast Fourier transform (FFT) matrix. Also, in (2) T H (x) and (x) are denoted as the transposition and Hermitian operations, respectively. In the PRP-OFDM modulator, with P = M + D (where D > 0) the corresponding P u 1 block signal to be transmitted is expressed as ucP ( n )



§0 ¨ ¨0 ¨ ¨ ¨0 ©

 h( L)  0      0

 h (1) ·   ¸ ¸  h( L) ¸   ¸  0 ¸¹ PuP

Also, parameters P, M, and L is chosen to satisfy: P ! M and D ! L . First, under noise free environment, the zeroforcing (ZF) solution of received block signals of (6)

FZP s M ( n )  D ( n )c P


(neglecting the term of D ( n )c D in (3)) is given by [3]

where D redundant symbols of PRP are introduced into each original symbol block to form a new P u 1 block signal, using the

H 0,P ucP ( n )  H1,P ucP ( n  1) and v P ( n ) is the


k 0

u M (n)


column being denoted as [ h (0), h (1) } , h ( L ),..., 0] , while

2 S qk


x P (n)  v P (n)

background noise vector, which is assumed to be the additive white Gaussian (AWGN) of block time n. Also, in (6) channel matrix H 0,P is denoted as a low-triangular matrix, with its first

In OFDM system each block signal consists of a sum of subcarriers that are modulated by PSK or QAM. The qth sample of an OFDM symbol is represented as j


In consequence, the received output vector, which is the convolution of channel matrix and (3) can be written as

where x P ( n )

[ s0 ( n ), s1 ( n ),  , s M 1 ( n )]

M 1

H l ,P

y P (n)


[ s ( nM ), s ( nM  1),  , s ( nM  M  1)]


Where the corresponding P u P channel matrix H l ,P is


Let us consider the discrete-time input data stream s ( n ) that is converted into M parallel sub-streams. The mth branch signal before the up-sampler is denoted as sm ( n ) s ( nM  m ) , m = 0,

H 0, PG ( l )  H1, PG ( l  1)

H l ,P


Q 1

¦G x

sˆ M ( n )

ª FH º « » with ¬ 0 DuM ¼

q P (n


[GQ 1

G 0 ] x Q ( n ) (7)

q 0


dimension P u M . In (3) c P [01uM c ] weighted by a pseudo random value D ( n )  C is appended to the H H D

vector FZP s M ( n ) , and c D : [c0 ,}cD -1 ] . Without loss of

where x Q ( n )



[ x P ( n  Q  1),  , x P ( n )] is a block vector. In

order to reconstruct the desired block signals s M ( n ) from (7),


generality, the elements of vector u M ( n ) are assumed to be 2

i.i.d. and zero mean random variables with variance V u

we need to design a filter-bank matrix [GQ 1

G0 ] ,

where Q is the order of filter-bank. The relationship of P, M,


which are independent of D ( n )c D . The elements of vector

and Q is as follows: P t M  L / Q

ucP ( n ) are then sent through the channel, that is modeled as an FIR filter of order L, where h (0) and h ( L ) are not nulls. Usually, in PRP-OFDM system the postfix duration is chosen

integer. L/Q











redundancy in each original data block [3].

is ceiling at


2015 International Symposium on Intelligent Signal Processing and Communication Systems (ISPACS) November 9-12, 2015

Now, for the PRP-OFDM system, from Fig. 1 and with (3), the received data block of (6) can be rewritten as

u M ( n ) are assumed to be zero-mean. In (11), the first L  1

( H 0, P  J n H1, P ) ucP ( n )  v P ( n )

y cP ( n )

samples of y P ( n ) , and its last L  1 samples, are denoted as

§ F s (n) · ¨¨ M ¸¸  v P ( n ) © D ( n )c D ¹



HJ n

y 0,L 1 ( n ) and y1,L 1 ( n ) , respectively, and are exploited for

easily retrieving the channel matrices, that are relying on the

In (8) scalar parameter is defined as J n channel matrix is denoted as HJ n

As it is described earlier the elements of the transmitted signal

D ( n  1) / D ( n ) , and

deterministic nature of the postfix:

( H 0, P  J n H1, P ) , in which

the second term is the interference due to past block caused by H1,P , while the first term H 0,P models the effect of ISI in

yˆ c ,1, L 1

the current block. Substituting (3) into (8), and rewrite it as y P (n)

y cP ( n ) - D ( n ) HJ c P



With (9) in the sequel we shall show that HJ n c P can be retrieved by simply averaging (i.e. mean value calculation) of

ª y1,L1 ( n ) º » ¬ D (n) ¼



H 0,L 1c L1

Add the above two terms together, we obtain yˆ c , L 1


the received samples if the OFDM data symbols F s M ( n ) are assumed to be zero-mean. That is, channel can be extracted by simply de-convolution of (9).

ª y 0,L1 ( n ) º » H1,L1c L1 ¬ D ( n - 1) ¼

yˆ c ,0,L 1

yˆ c ,0, L 1  yˆ c ,1, L1

H CIR,L1c L 1


C L 1h L 1

Where matrix C L 1 is a circular matrix with its first column

A. Semi-blind Channel Estimation with Oder One Statistics First, we introduce the semi-blind channel estimation with the PRP sequence, and assume that channel impulse response (CIR) is time-invariant, where the length of PRP is chosen to be D=P-M =L+1, and L is the order of CIR. It can be easily showed that channel matrix H CIR , L 1 H 0, L 1  H1, L 1 is


being defined as [ c0 , c1 , ,..., cL 1 , cL ] , and in (13) channel vector is defined as h L 1


[ h (0), h (1) } , h ( L )] . Finally, we

obtain an estimate of channel vector, that is, hˆ L 1


C L 1yˆ c ,L 1 .

ˆ c is obtained to get In consequence, an estimate of H Jn P

circular matrix [7], with its first column to be denoted T as [ h (0), h (1) } , h ( L )] .

y P (n)

ˆ c y cP ( n ) - D ( n ) H J P




Now, we denote u P ( n ) [u ( nP ), } , u ( nP  M - 1)] and decomposing it into two sub-vectors from, that is H (10a) u 0, L1 ( n ) [u ( nP ), } , u ( nP  L )] ,



In this section, we consider the case that length of redundant symbol (or PRP) is less than the channel order, that is, P - M D  L  1 . Under such circumstance, we are not able to obtain desired results with the approach using (13) and (14). Under the assumption that the elements of u M ( n ) are i.i.d. and

and u1, L 1

[u ( nP  M - L - 1), } , u ( nP  M - 1)]



Similarly, we may perform the same operation for the noise H vector v P ( n ) [ v0 ( n ), } , v P -1 ( n )] , as well as corresponding sub-vectors


and v1, L 1 ( n )

as v 0, L1 ( n )

[ v P - L-1 ( n ), } , v P-1 ( n )]


[ v0 ( n ), } , v L ( n )]

, respectively.


with zero-mean, we obtain the mean vector y P ( n ) y P (n )


As a

E{y P ( n )} :

E{H 0,P ucP ( n )  H1, P ucP ( n - 1)}  E{v P ( n )} H 0,P E{ucP ( n )}  H1, P E{ucP ( n - 1)}  E{v P ( n )}

consequence, the received vector can be written as y cP ( n )

H 0,P ucP ( n )  H1,P ucP ( n - 1)  v P ( n )

ª H 0,L1u0,L1 (n)  D ( n - 1) H1,L1c L1  v 0,L1 ( n ) º « »  = « » «¬ H1,L1u1,L1 ( n )  D ( n ) H 0,L1c L1  v1,L1 ( n ) »¼

( H 0,P  H1, P ) E{c P }


H Pc P

Or it can be expressed in an alternative form as

ª y 0,L1 ( n ) º « » «  » «¬ y1,L1 ( n ) »¼



2015 International Symposium on Intelligent Signal Processing and Communication Systems (ISPACS) November 9-12, 2015

ª c0 «  « «  « « cD 1 « 0 « «  «¬ 0


c D 1







c D 1



º ª«  »  « » « h( L) c D 1 » 0 » «« 0 »«  0  » « » h (0) 0 »« «  c0 »¼ « ¬ h( D ) c1

h ( D  1) º

» » » » » (15b) » » » » »¼


CPh P Consequently, the estimated channel site information hˆ P is obtained as

hˆ P


CP y P


In this section, to evaluate the system performance in terms of the bit-error-rate (BER), computer simulations are carried out, to demonstrate the merits of the PRP-OFDM system with the block-based Bayesian DFE, when it is compared with the conventional equalizers discussed earlier.


This approach is different the one discussed in (14), but is more suitable for the PRP-OFDM system, with insufficient PRP. The block diagram for semi-blind channel estimation is illustrated in Fig.1. After knowing the CSI with (16), the effect of IBI can be reduced using the MSSNR-TEQ approach [6]. Basically, the channel shortening technique proposed in [6] is an optimal shortening and a variation of least-square (LS) approach. It shortens the length of channel impulse response (L) into the redundancy range (PRP length) to reduce effect of IBI. Consequently, the block-based Bayesian DFE is applied for eliminating the ISI [5][9]12][13], it is known to perform much better than conventional equalizers, viz., non-orthogonal projector (or named as the oblique projector) with non-linear ZF-equalizer [18], linear ZF equalizer, and the MMSE DFE. As illustrated in Fig. 1, the block diagram of overall PRPOFDM system with insufficient length of PRP. If we assume that the zero-order FIR filter-banks precoder is with full column rank (M), and the corresponding FIR filter-banks equalizer is with order Q-1 [18], the transceiver in a matrix form is illustrated as in Fig. 2.

A. Sufficient Case ( P  M t L  1 ) First, we consider the case of sufficient redundancy PRPOFDM system. The block-based Bayesian DFE is investigated in [5][9][11], and compared with the conventional zero-forcing equalizers [18]. In sufficient case, a specific parameter set (P, M, L) is chosen to be (37, 32, 4), P  M t L  1 , with known FIR channel h1 that has four zeros located at -0.8, 1, 0.9 exp( j 9S / 20) and 1.1exp(  j 9S / 20) . In our case, the postfix sequence is chosen to be cD = [1.5649-0.0356i 1.14040.2923i -1.1347+0.3148i 1.5316+0.1681i 1.6562+0.2440i]T, that is derived following the approach indicated in [8]. The magnitude of original complex channel is given in Fig. 3 (a), while Fig. 3(b) is the channel estimation result with the proposed semi-blind approach using the PRP information of PRP-OFDM modulator. For fair comparison, first we would like to investigate the accuracy of channel estimation of PRPOFDM system for sufficient redundancy case, P  M t L  1 . The results of channel estimation are an averaging window over 200 OFDM symbols. Next, Fig. 4 shows an example of an OFDM system with sufficient redundancy, P  M ! L , where the parameter set is chosen to be (P, M, L) = (10, 5, 4) with known FIR channel, h3 = [0.2504 -0.52905+j0.049463 0.58922-j0.089033 -0.50888+j0.03957 0.19831]T. It shows that the block-based Bayesian DFE performs the best among all equalizer schemes. B. Insufficient Case ( P  M  L  1 ) Recalled that the pseudo randomly weighted sequence D ( n ) is time varying, the multiplication of it with c P , i.e. D ( n )c P , used in this paper avoids this issue. Let us consider the insufficient case, for instances, M 8 , P 12 and L 7 (channel order), and for channel vector denoted as h2 = [1.0438 -2.2054-0.42008i 2.9661+0.95208i -3.1599-1.4517i 1.8777+j.5986 -0.8628-j0.95076 0.34051+j0.44546 j0.1736]T. Due to the fact that in this case the semi-blind channel estimation with length L  1 could not be used as that in sufficient PRP case. Instead, we have to use the estimated channel impulse response with length, P, which is the total length of transmitted signal per block, if the information of channel order is available. As suggested in [8], the postfix sequence is chosen to be cD =[0.0789+0.4082i -0.043-0.2456i 0.0926-0.1566i -0.0587-0.2248i]T. Again, the following result of channel estimation is an averaging window over 200 OFDM symbols. Similarly, Fig. 5(a) shows the magnitude of the original channel impulse response and Fig. 5(b) is the estimation result using equation (16).

Fig. 2 The block diagram of the transceiver in a matrix form

Next, for parameter set of P, M, and L are chosen to be 12, 8, and 7, respectively, and channel vector is h2, from Fig.6, we


2015 International Symposium on Intelligent Signal Processing and Communication Systems (ISPACS) November 9-12, 2015

observed that all the equalizer with MSSNR-TEQ outperform those equalizers just described above for Q 2 (see (7)). Among them the MSSNR-TEQ with Bayesian DFE has the best performance, and MSSNR-TEQ with the MMSE-DFE outperforms the MSSNR-TEQ with linear ZF equalizer. Here, we would like to point out that the MSSNR-TEQ with the linear ZF equalizer performs slightly better than the one with MMSE-DFE with Q 2 . V.


In this paper, for PRP-OFDM system, with sufficient redundancy, in the receiver first the PRP was used to alleviate the effect of IBI. Consequently, after channel impulse response was estimated with the pseudo randomly weighted postfix sequences, the block-based Bayesian DFE was applied to the PRP-OFDM receiver to improve the system performance, in terms of capability for suppressing the ISI, over the conventional ZF and MMSE approaches. For the case of insufficient PRP-OFDM system, we proposed a novel channel estimation scheme with MSSNR-TEQ scheme for shortening the order of channel impulse response. Again, to further improve the system performance, the Bayesian DFE was applied to reduce the effects of ISI. It means that after performing the channel estimation and removing the effect of IBI with MSSNR TEQ, the Bayesian DFE could be applied for eliminating the ISI. Simulation results shown in computer simulation result confirmed the above-mentioned observation. We then concluded that the new proposed PRP-OFDM with the block Bayesian DFE associated with MSSNR-TEQ channel shortening technique had the advantages over the conventional equalizer schemes. It requires a minimum pilot overhead, lowcomplexity channel tracking, and achieved better performance achievement ACKNOWLEDGMENT The partial financial support of this work by the Ministry of Science and Technology, Taiwan (R.O.C.), under contract MOST 103-2221-E-032 -021 is of great acknowledged.

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[3] A. Scaglione, G. B. Giannakis, and S. Barbarossa, “Redundant filterbank precoders and equalizers—Part I: Unification and optimal designs,” IEEE Trans. Signal Processing, vol. 47, pp. 1988–2006, July 1999. [4] K. Abend and B. D. Fritchman, “Statistical detection for communication channels with intersymbol interference,” Proc. IEEE, vol. 58, pp. 779– 785, May 1970. [5] A. Stamoulis, G. B. Giannakis, A. Scaglione, “Block FIR DecisionFeedback Equalizers for Filterbank Precoded Transmissions with Blind Channel Estimation Capabilities,” IEEE Trans. Comm., Vol. 49, no. 1, pp.69-83, Jan. 2001. [6] P. J. melsa, R. C. Younce, and C.E. Rohrs, “Impulse response shortening for discrete multitone transceiver,” IEEE Trans. Comm. Vol. 44, No. 12, pp. 1662-1672, Dec. 1996 [7] M. Muck, M. de Courville, and P. Duhamel, ȾA Pseudorandom Postfix OFDM Modulator-Semi-Blind Channel Estimation and Equalization,ȿ IEEE Trans. Signal Process., vol. 54, no. 3, pp.1005-1017, Mar. 2006 [8] M. Muck, M. De Courville, and P. Duhamel, “Postfix design for pseudo random postfix OFDM modulators,” presented at the 9th Int. OFDM Workshop, Dresden, Germany, Sep. 2004. [9] D.Williamson, R. A.Kennedy, and G.W. Pulford, “Block decision feedback equalization,” IEEE Trans. Commun., vol. 40, pp. 255–264, Feb. 1992. [10] S. Chen, B. Mulgrew, and S. McLaughlin, “Adaptive Bayesian equalizer with decision feedback,” IEEE Trans. Signal Processing, vol. 41, pp. 2918–2927, Sept. 1993. [11] S. Chen, S. McLaughlin, B. Mulgrew, and P. M. Grant, “Bayesian decision feedback equalizer for overcoming co-channel interference,” Proc. Inst. Elect. Eng.—Commun., vol. 143, no. 4, pp. 219–225, 1996. [12] S. Chen, S. McLaughlin, B. Mulgrew, “Complex-valued radial basis function network, Part II: application to digital communication channel equalization,” EURASIP Signal Processing., Vol. 36, pp. 175-188, 1994. [13] S. Chen, B. Mulgrew, S. McLaughlin, and P. M. Grant, ”Adaptive Bayesian decision feedback equalizer for dispersive mobile radio channel, ” IEEE Trans. Commu, Vol. 43, no. 5, pp. 1937-1946, May 1995. [14] S. J. Chern, Y.-D. Lee, and Richard H.-H Yang, “Performance of the MIMO CS-PRP-OFDM System with Complementary Codes,” The 2011 International Symposium on Intelligent Signal Processing and Communications Systems, Thailand, Dec. 07- 09, 2011. [15] S. M. Kay, Fundamentals of Statistical Signal Processing: Decition Theory. NJ: Prentice Hall, 1998. [16] B. Muqut, Z.Wang, G. B. Giannakis, M. de Courville, and P. Duhamel. “Cyclic Prefixing or Zero padding for Wireless Multicarrier Transmissions?, ” IEEE Trans. Comm., Vol. 50, no. 12, pp. 2316-2148, Dec. 2002. [17] R.T. Behrens and L.L. Scharf, “Signal processing applications of oblique projection operators,” IEEE Trans. Signal Processing, vol.42, no. 6, pp. 1413-1424, June 1994. [18] C. H. Wu, S. J. Chern, “A Novel Zero-Order FIR Zero-Forcing Filterbanks Equalizer Using Oblique Projector Approach for OFDM Systems,” IEICE TRANS. COMMUN., VOL E88-B NO. 12, Dec. 2005.

2015 International Symposium on Intelligent Signal Processing and Communication Systems (ISPACS) November 9-12, 2015

(a) The PRP-OFDM transceiver

(b) Bayesian DFE equalizer Figure 1: (a) Discrete-time model of the PRP-OFDM transceiver, (b) Bayesian DFE equalizer.

Figure 3: Channel information (a) Original channel information (b) Estimation result.

Figure 5: Channel information (a) Original channel information (b) Estimation result with (16).

Fig. 4 BER performance of different equalization for the PRP-OFDM system with sufficient redundancy.

Figure 6: BER performance of different equalization for the PRP-OFDM system with insufficient redundancy.


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