FREQUENCY DOMAIN PROBABILISTIC DATA ASSOCIATION EQUALIZER FOR OFDM SYSTEMS WITHOUT CYCLIC-PREFIX Saeedeh Parsaeefard , Hamidreza Amindavar Amirkabir University of Technology, Tehran, Iran
[email protected],
[email protected] z
Orthogonal frequency division multiplexing (OFDM) is a subdivision of multi carrier modulation (MCM) that uses Nc orthogonal subcarriers to transmitted data, it has recently been widely used in wired and wireless high data rate communications due to robustness to multipath fading channel and bandwidth efficiency compared to single carrier modulation. In the conventional OFDM systems; cyclic prefix(CP) with minimal length at least equal the length of channel impulse response,used to avoid ICI and ISI. But an OFDM system with CP, using ν samples, reduces the spectral efficiency by Nc /(Nc +ν) to cause loss of transmitter power. Consequently, there exists increasing interest in OFDM systems with short or no CP. But if CP is removed because of ICI and ISI the orthogonality of subcarriers is lost reducing the system performance significantly. A well known technique to cancel out these effects is the use of time domain equalizer (TEQ) in the receiver. TEQ is a FIR filter that the length of the cascaded channel-equalizer impulse response is shorter than the desired CP length. TEQ design problem has been extensively studied [1, 2]. In TEQ the noise is enhanced because the total BER in each subcarrier is dependent upon SNR, therefore, BER is reduced significantly. In order to overcome this shortcoming, we utilize the PDA equalizer in the frequency domain assuming that ISI and ICI are modelled as Gaussian random variates, hence, the we estimate the symbol in each subcarrier. PDA is a highly successful algorithm to target tracking in RADAR systems, and it has been recently used as a near optimal multiuser detector [3, 4] assuming binary symbols only, and for channel equalization and estimation [5, 6] assuming zero padding to compensate for ISI. In a PDA algorithm, it is assumed that the transmitted signal is a random variable with an associated a posteriori probability to enforce the channel noise and any undesired effect, e.g. ISI, to be a Gaussian random variable. An iterative procedure is used to provide the convergence to a Gaussian
1-4244-0549-1/06/$20.00 ©2006 IEEE.
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+
Channel
ym
FFT
1. INTRODUCTION
Xi
IFFT
In this paper we use probabilistic data association (PDA) equalizer in OFDM signal without cyclic prefix (CP) to remove intersymbol interference (ISI) and intercarrier interference (ICI) in the frequency domain. In order to cancel out the ISI and ICI in OFDM systems, CP with a length longer than the channel length is used. However, this causes performance degradation due to increased required bandwidth, hence, a lower bit rate, and uncalled for power utilization in the OFDM transmitter for CP. Research on reducing or even cancelling out the CP in the transmitter has focused on using bandlimiting filters in time domain. We use probabilistic data association (PDA) strategy to equalize for ISI and ICI effects in the frequency domain by considering them both as a Gaussian process in each subcarrier to cancel their effect without any noise enhancement.
Yi
PDA Equalizer
ABSTRACT
Fig. 1. OFDM system using the frequency domain PDA equalizer without CP.
probability density function based on the received signal. In this paper, the joint effect of ISI and ICI in no CP OFDM signal cancelled by PDA algorithm. This paper is organized as follows, in section 2, a short description of OFDM system is provided, in section 3, we develop the new OFDM equalizer based on PDA algorithm, and in section 4, the performance evaluation of the proposed algorithm in terms of BER is compared against the training TEQ, at the end, some concluding remarks are provided. 2. SYSTEM DESCRIPTION The schematic diagram for the frequency domain PDA equalizer is shown in Figure 1 with no CP. Consider an OFDM system with Nc subcarriers. The complex valued data symbols X that are chosen from PSK, or QAM modulation, are converted from serial to parallel format to produce a Nc × 1 vector Xm where m represents the OFDM symbol number. The OFDM transmitter employs an inverse discrete Fourier transform (IDFT) of size Nc for modulation producing the Nc × 1 vector xm represented by ∗
xm = W Xm /Nc
(1)
∗
where W is an Nc × Nc IDFT matrix that is the complex conjugate of W whose entries are Wnk = exp(−j2πnk/Nc ) for n, k = 0, 1, · · · , Nc − 1. Then the resulting time domain symbol xm is parallel to serial transformed and transmitted over the channel. We assume that the channel during a block of OFDM can be modelled as a LTI system with FIR impulse response hm of length r and additive white Gaussian noise zm with variance Rz . In OFDM system, we can assume that the length of channel is shorter than the symbol length Nc therefore ISI is caused only by the pervious symbol. At the receiver, initially, the incoming data is transformed from serial to parallel format [7] xm−1 + zm , ym = [ Λ0 , Λ1 ] (2) xm where zm is Nc × 1 vector of channel noise and Λ0 , Λ1 are matrix channel that shown the ISI and ICI in each subcarrier and have Nc ×
Nc dimensions: ⎡ ⎢ ⎢ ⎢ ⎢ Λ0 = ⎢ ⎢ ⎢ ⎣
0 .. . .. . .. . 0
···
···
⎡
h0 .. ⎢ ⎢ . ⎢ ⎢ Λ1 = ⎢ h ⎢ r−1 ⎢ .. ⎣ . 0
hr−1 .. . .. . .. . 0
h0 .. . hr−1 ···
··· .. . .. . ···
··· .. . .. . .. . hr−1
h1
⎤
⎥ ⎥ ⎥ ⎥ h1 ⎥ ⎥ .. ⎥ ⎦ . 0
··· .. .
h0 ···
0 .. . .. . h0
(3)
gi := E(hi Xi |Y ), ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
(4)
YNc ×1 = H Nc ×Nc X + INc ×1 + ZNc ×1
(5)
I represents ISI caused by the pervious OFDM symbol, Z is DFT of channel noise in each subcarrier and H represented as channel matrix after DFT block is given by ∗
H = W Nc Λ1 W Nc /Nc
(6)
3. REMOVING ISI-ICI USING PDA In this section we develop the algorithm based on PDA to equalize ISI and ICI in OFDM without CP. In this paper to cancel ISI and ICI effects with PDA, we model them as a Gaussian process in each received symbol. Typically, the transmitted symbol in each subcarrier (Xi ) takes its values from a finite alphabet set, [a1 , · · · , aM ], of M -array PSK or QAM modulation. Because the transmitted symbols are equally likely, thus, I in the following is zero mean and its variance is as follows: ∗ var(I) = E {H1 Xm−1 Xm−1 H1∗ } = var(X)H1 H1∗ ∗
H1 = WNc Λ0 WNc /Nc
(7) (8)
thus we assume the effect of ISI, and AWGN as the effective zero
and variance of Rz˜ = Rz + var(I). Subsequently, mean noise Z in order to develop the algorithm, we consider the ith element of X that is represented by Xi and rewriting (5), for i = 1, · · · , Nc as: Y = hi X i +
Nc
hi Xj + Z,
(9)
j=1,j=i
where hi and hj are column vectors, the ith and jth columns of H in (6). With respect to the received signal Y , there are M posteriori probability values associated with each digital input that are denoted by: ηi,m = Pr(Xi = am |Y ),
m ∈ [1, M ], i ∈ [1, Nc ].
(10)
To estimate these probabilities in the PDA algorithm, we assume that the transmitted signal in each subcarrier is considered as a Gaussian random variable. After passing the signal through the channel;
Ri := var(hi Xi |Y )
(11)
as the conditional mean and variance of hi Xi . The association of Xi to the vector hi is important because it eliminates the need to invert the channel matrix to estimate Xi thus it dose not enhance noise in the first subchannel and the performance of system is improved. By assuming the transmitted signals are IID, we derive the required parameters in (11) as follows: Nc
gi = Y −
then, ym is demodulated by the DFT block, the output of this block in a vector form is represented by (for simplicity we eliminate the index of OFDM symbol number m:
where
which is an FIR filter, the posteriori PDF of Xi is almost Gaussian due the central limit theorem, thus if the mean and variance of signal in each subcarrier is calculated conditioned on the received signal Y , then the statistics of the transmitted signal is fully characterized. In order to evaluate the conditional mean and variance, we define the following variables:
E(Xj |Y )hj
(12)
var(Xj |Y )hj hH ˜ j + Rz
(13)
j=1,j=i
Ri =
Nc
j=1,j=i
E(Xj |Y ) =
Xj P r(Xj |Y ) =
Xj ∈[a1 ,··· ,aM ] M
M
ηj,m |am |2 |E{Xj |Y }|2
var(Xj |Y ) =
ηj,m am
(14)
m=1
(15)
m=1
since the mean and variance are determined in (13-15), now, we can express PDF for the conditional vector hi Xi |Y :
−(hi Xi −gi )Ri−1 (hi Xi −gi )H exp 2 √ P (hi Xi |Y ) = (16) (2π)Nc /2 Ri and if we insert the values of am in (16) the values of ηi,m in (10) are determined and the soft decision can be derived with these values using MAP detection. In order to simplify the calculations of ηi,m , defined: λi,m := ηi,m /ηi,1 (17) due to the probability normalization, m ηi,m =1,we obtain 1 , m=1 1+ M m=2 λi,m (18) ηi,m = m = 2, · · · , M λi,m , finally, the MAP estimate of Xi is determined to be the am value that renders η the largest. Because gi and Ri are dependent on mean and variance of other subcarriers, inter-relationship between probabilities of the unknown symbols in each subcarrier create an iterative multistage procedure to estimate ηi,m . In each stage gi and Ri are calculated using the pervious values of the probabilities of other subcarriers and the new probability is obtained using (15), and (16). This procedure continues until all the subcarrier probabilities, for all subcarriers i ∈ [1, Nc ], converge, this means one of the probability is close to one. In this iterative procedure only the calculation of Ri is computationally cost Nc − 1 times per subcarrier in each iteration, to speed up the calculations, in time domain PDA algorithm [3] two auxiliary variables g and R similar to (12,13) without the index i
and R is the are defined. they are the conditional mean of noise Z conditional variance matrix of Y , respectively. gi Ri
= =
g + E(Xi hi ) R − var(Xi |Y
(19) )hi hH i
(20)
Parameters DFT size Number of zero subcarrier Guard type Guard length Sampling rate Signal constellation
Specification 64 8 Cyclic Extension 16 20 MHz BPSK, QAM, 16QAM ,64QAM
addition of some CP to the PDA algorithm produces the similar results of a full CP OFDM system. In Figure 6, the performance of the PDA equalizer, TEQ, and ZF equalizers in CP OFDM for an autoregressive dispersive channel is considered, we observe that the performance of the PDA equalizer is almost the same as the CP OFDM system, and is better than TEQ. This shows the PDA equalizer has acceptable performance in long dispersive channels; AR channel, as well.
Table 1. Simulations parameters. 5. CONCLUSION and by applying the matrix inversion lemma [8] for R, we have R−1
=
Ri−1 −
−1 var(Xi |Y )R−1 hi hH i R . −1 h 1 + var(Xi |Y )R−1 hH i i R
(21)
PDA method introduced in this paper can partially or fully dispenses with CP, on the other hand, PDA in time domain [3] is traditionally used as an alternative as the maximum likelihood detector and equalizer with lower computational complexity. The computational cost for the maximum likelihood equalizer is exponential with M , M Nc , but for the PDA equalizer the cost is of order Nc3 in each iteration. Furthermore, the introduced frequency domain multistage PDA structure in an OFDM system uses a posteriori probabilities, ηi,m in (8), hence, it is a soft interference canceller for ISI and ICI. 4. SIMULATIONS AND RESULTS For the simulation we use the OFDM system parameters according to IEEE 802.11a that is summarized in Table 1. We analyze the convergence speed for PDA. For wireless applications it is not desirable if the convergence is slow and the iterative process is large leading to an increased computational complexity. The pervious work on PDA [5] reported an acceptable convergence speed, in our frequency domain PDA we evaluate the mean square in the number of iterations required so that the smallest subcarrier probability (18) attains 1 for different SNR, this is shown in Figure 2, we observe that in high SNR almost initially mean square distance is low showing that most of subcarriers had converged, and for low SNR after 3 iterations a distinguishable change in the mean square distance is not observed; a sign of convergence. Thus in our application the maximum iteration used is 3. We use the following four different scenarios in the OFDM systems: OFDM system with CP and perfect zero forcing equalizer, OFDM system without CP and the PDA equalizer, OFDM system without CP with channel shortening TEQ, and partial CP OFDM system in higher level modulation scheme with PDA equalizer. We also consider two other different types of channels: the multipath Ricean fading channel with 14 paths and 28 delays [9] and autoregressive dispersive (AR) channel H(z) = 1/(1 + 0.5z −1 ). Furthermore, we assume to have perfect synchronization in the receiver, the performance metric of interest is BER for different SNR. In the simulation, we perform the shortening channel using TEQ [2] until errors in the training stages vanish, and in PDA equalizer for the AR channel, we use truncated Taylor series expansion of H(z) to obtain {h0 , h1 , · · · , hr−1 }. In Figures 3, 4, and 5, the system performance for (4,16,64)-QAM in Rician multipath fading channel, are shown, these include different OFDM system modes. As it is obvious from these figures, their performances for 4 QAM are close to each other but the major difference starts to reveal itself in higher level QAM where TEQ does not demonstrate good performance because of enhanced noise but PDA still is close to CP OFDM. In Figure 4, the
We considered PDA algorithm in the frequency domain to remove the joint effect of ISI, ICI, and noise in no CP OFDM systems. In most of the TEQ designs the equalizer cost function is based on SIR criterion but in OFDM system the BER and bit rate are directly dependent on SNR but with TEQ the noise is enhanced and the amount of SNR in each subcarrier and BER is reduced. This effect is worse in higher modulation constellations where the decision boundaries are closer to each other. In this paper, we suppress noise and other undesirable effects such as ISI and ICI, without a zero forcing equalizer, by modelling them as a Gaussian process using PDA algorithm, thus the performance is greatly improved. We also examine the improvement in the new algorithm if we assume a percentage of CP is available in the OFDM system. In TEQ based algorithms and others [10], a problem is associated with scenarios when there exists at least a null in the channel frequency response causing an increased number of iterations for training phase, but, using the introduced PDA approach such a problem is also bypassed. Acknowledgement This work is supported in part by Iran Telecommuncations research center(ITRC). 6. REFERENCES [1] P. J. W. Melsa, K. R. Pattipati, P. K. Willett, F. Hasagawa, “Impulse response shortening for discrete multi tone transceiver,” IEEE Tran. on Com., pp.1662-1672, Sept. 1996. [2] I. Djokovic, “MMSE equalizer for DMT system with and without crosstalk,”Thirty-first Asilomar Conference on Signals, Systems & Computers, pp. 545-549, 1997. [3] J. Luo, K. R. Pattipati, P. K. Willett, F. Hasagawa, “Nearoptimal multiuser detection in synchronous CDMA using probabilistic data association,” IEEE COM. Letters,, Vol. 5, NO. 9, Sept. 2001. [4] D. Pham, J. Lue, J. Lue, K. R. Pattipati, P. K. Willett, “A PDA-Kalman approach to multiuser detection in asynchronous CDMA,” IEEE Com. Letter, Vol. 6, NO. 11, pp.475-477 , Nov. 2002. [5] S. Liu, Z. Tian, “Near-optimal soft decision equalization for frequency selective MIMO channels,” IEEE Tran. on signal proc., Vol. 52, NO. 3, March 2004. [6] Z. J. Wang, Z. Han, K. J. Ray Liu, “MIMO-OFDM channel estimation via probabilistic data association based TOAs ,” Globecome 2003, IEEE Global Telecom. conf., Vol. 22, NO. 1, pp. 626-630, Dec. 2003. [7] S. Trautman, T. Karp, N. J. Fliege, “Frequency domain equalization of DMT, OFDM systems with insufficient gaurd interval ,” Globecom 2002, IEEE Int. Conf. on Com. Vol. 3, pp. 1646-1650, 2002.
convergence speed of OFDM for 4 qam 0.06
10
0.04 −1
10 0.03
BER
distance beetween eta and 1
0.05
16 QAM −OFDM
0
SNR=15db SNR=25db SNR=30db
0.02
−2
10
0.01
0
1
2
3 4 Number of iteration
5
no CP OFDM \TEQ no CP OFDM \PDA CP OFDM \ZF AWGN
6 −3
10
15
20
25
30
SNR dB
Fig. 2. Convergence speed for PDA under 4-QAM. 4 QAM −OFDM
0
10
Fig. 4. Performance comparison under 16-QAM.
−1
10
64 QAM
0
−2
10
−3
10
no CP OFDM \TEQ no CP OFDM \ PDA CP OFDM \ZF AWGN
−1
10
BER
BER
10
no CO OFDM /TEQ no CP OFDM /PDA 25% CP OFDM /PDA 50% CP OFDM /PDA CP OFDM /ZF
−2
10
−4
10
15
20
25
30
SNR dB
−3
10
Fig. 3. Performance comparison under 4-QAM.
15
20
25
30
SNR
[8] G. Golub, C. .F. Van Loan, Matrix computation,” John Hopkins Univ. Press, 2nd Edition, 1990.
Fig. 5. Performance comparison under 64-QAM for no CP and different length of CP at the PDA equalizer.
[9] S. Boumard, A. Mammela, “Channel estimation versus equalization in an OFDM WLAN system,”IEEE Vehic. Techn. Conf., May 2001.
OFDM 4 QAM in AR Channel
0
10
[10] D. Kim, G. L. Stuber, “Residual ISI cancellation for OFDM with application to HDTV broadcasting ,”IEEE J. select areas in com., Vol. 16, NO. 8, pp.1590-1599, Oct. 1998. −1
BER
10
−2
10
CP OFDM / ZF EQ NO CP OFDM/ PDA EQ. NO CP OFDM / TEQ
−3
10
14
16
18
20
22
24
26
28
SNR dB
Fig. 6. Performance comparison under 4QAM for and autoregressive channel.
