Turbo Equalization for FMT Systems Satyam Srivastava(1), Santosh Jadhav(2), Chirag Jain(3), and V. M. Gadre(4) (1) Department of Electrical Engineering Indian Institute of Technology, Bombay Powai, Mumbai, India 400076 E-mail: [email protected] (2) As (1) above, but Email: [email protected] (3) As (1) above, but Email: [email protected] (4) As (1) above, but Email: [email protected] ABSTRACT Filtered Multitone (FMT) modulation is a filter bank based scheme where modulation filter has a nearly rectangular frequency domain amplitude characteristics. In this paper, we propose a new turbo equalization based scheme for FMT system. We provide comparison between various equalization schemes in terms of bit error rate (BER). Although, turbo equalization based scheme requires higher computational complexity, it gives very superior performance. Moreover, it gives excellent tradeoff between performance and complexity. The system is suitable to counter the effects of realworld impairments, such as frequency and timing offsets, due to high spectral containment and effective equalization with reasonable complexity. Hence, it may be a valid alternative to OFDM in wireless applications. 1. INTRODUCTION Filtered Multitone (FMT) is based on non overlapping spectral partitioning by a uniform filter bank. Here each of the transmitter pass band filters consists of a frequency shifted version of a low-pass prototype {h(n)}, n = 0,1,..., M γ − 1 , whose frequency response is zero outside the interval |f| ≤1/2T where T is the FMT symbol period. Parameter γ is called the overlap. Such a system can be efficiently implemented by means of a fast Fourier transform (FFT) and a network of polyphase filters [1]. The prototype filter is designed to have nearly rectangular frequency-domain amplitude characteristics. This causes inevitable intersymbol interference (ISI) at the receiver, while interchannel interference (ICI) between subchannels is almost negligible. The absence of the cyclic prefix and reduced crosstalk interference makes it an attractive modulation technique. FMT has been proposed for both wired and wireless communications [1], [2] as an alternative to most widely adopted Orthogonal Frequency Division Multiplexing (OFDM).

However, equalization is required at the receiver to combat ISI due to the transmit filter and transmission channel. In [2], some simple equalization schemes based on a fixed decision feedback equalizer (DFE) were proposed. The transmission channel was equalized (adaptively) by a one tap per subchannel filter, assuming frequency response of channel over each subchannel to be flat, i.e., constant amplitude and phase. Receiver architectures based on DFE are limited by the well known error propagation phenomenon, which lowers the system's performance. Also, coded modulation cannot be applied in a straightforward manner. To overcome these problems, precoding at the transmitter using a Tomlinson Harashima Precoder (THP) [3], is a practical solution. In this technique, the feedback filter of DFE is moved to the transmitter together with the insertion of modulo operator. Although, THP does not propagate errors but it is more sensitive to errors because of bounded partition. Also, an increase in Peak to Average Power Ratio (PAPR) is observed, which itself is a major problem in multicarrier systems. The combination of trellis coding with THP has been a topic of recent research. In this paper, we propose turbo equalization for FMT systems as a viable solution. Turbo Equalization is an iterative equalization and decoding technique [4], which can provide impressive gains for communication systems that requires data transmission over ISI channels. The maximum a posteriori probability (MAP) based equalizer suffers from high computational load for channel with long memory or large constellation sizes. Recently, reduced complexity structure of soft-in soft-out (SISO) equalizer were introduced using linear equalizer (LE) and DFE based on minimum mean squared error (MMSE) criteria [4], [5]. The paper is organized as follows. A brief definition of the system model is given in Section 2. In Section 3, we describe the proposed turbo equalization scheme for FMT systems. In Section 4, we present the simulation results that illustrate the performance of different receiver structures. Section 5 concludes the paper.

2. SYSTEM MODEL

Fig. (1) depicts the basic system investigated in the paper. The (binary) data dj is encoded with a (binary) convolutional encoder yielding code symbols xj, which are mapped to the alphabet B of the signal constellation. In this paper, for simplicity we assume binary phase shift keying (BPSK), i.e., B є {-1,+1}. The interleaver permutes the code symbols xj and outputs symbols xn. This operation is denoted by xn = П(xj), where П(.) is a fixed random interleaver. The permutation П-1(.), the de-interleaver, reverses the П(.) operation. The noise is modelled as additive white Gaussian noise (AWGN).

Fig. 1: Sytem Model

In FMT, equalization due to the transmission channel, transmit and receive filter is required at the receiver, after demodulation. Let us assume the frequency response of the transmission channel over each subchannel is flat. Hence, the transmission channel can be adaptively equalized by one tap per subchannel equalizer (as for OFDM systems with cyclic prefix) [2]. For the equalization of transmit filter we propose turbo equalization. 3. PROPOSED EQUALIZATION SCHEME

Turbo equalizer for FMT consists of M SISO Equalizers for each subchannel and a SISO decoder, operating in an iterative manner. The L-value operator L(x), called log likelihood ratio (LLR), is applied to quantities x є {-1,+1},and is given by P( x = +1) L( x) = log (1) P( x = −1) The equalizer inputs a priori L1(xn) LLR and outputs a posteriori LLR minus a priori LLR called the extrinsic LLR LE(xn). For the initial equalization step, no a priori information is available and hence we have L1(xn)=0. The equalizer output after de-interleaving is considered to be a priori LLR L2(xj) for the decoder. The decoder also computes extrinsic LLRs LD(xj) which after interleaving is given to the equalizer as input. It should be noted that L2(xj)= П-1(LE(xn)) and L1(xn)= П(LD(xj)). The SISO decoder also computes the data bit estimates. A suitably chosen termination criterion stops the iterative process. The resulting turbo equalization scheme for FMT system is shown in Fig. (2).

Fig. 2: Turbo FMT Equalizer

3.1 SISO Equalizer

We have not shown the subchannel dependence in many equations so as not to make them very confusing. The signal received at the mth subchannel can be represented in a compact form as (index m has been omitted): zn = Hxn + wn ,

where zn is the channel observation, xn is the transmitted signal, H is the channel convolution matrix and wn is AWGN with variance σ 2 = N 0 / 2 per dimension. The overall channel response, {heq( m ) (k )}, k = 0,..., 2γ − 2 of the mth subchannel due to transmit and receive filters is independent of subchannel index. Thus, the overall channel memory is µ = 2γ − 2 . 1 M γ −1 (2) ∑ h(n)h(kM − n − 1) M m=0 We have considered both MMSE-LE and MMSE-DFE, as developed in [4], [5], for the SISO Equalization task. Since overall channel impulse response is designed to be real, therefore we will have only real tap coefficients for the equalizer. In exact implementation, we will have time varying coefficients. We can also use approximate implementation in which we simply use time invariant coefficients, thus reducing computational complexity at slight performance loss. In this case, the equalizer coefficients can be computed offline, so the cost of initialization of equalizer will be zero. heq( m ) =

3.1 SISO Decoder For the decoding stage, we consider a binary rate-1/n convolutional code with constraint length ĸ. The BCJR decoding algorithm [6] is known to yield the optimal symbol estimate with minimum symbol-error rate. The essential part is the BCJR algorithm's ability to yield soft information in the form of a posteriori LLRs for coded and information bits. Using the same notation as in [6], for stage t of the code trellis transiting from state St −1 = s ' to St = s associated with input dt and output x j = ( xt1 xt2 ...xtn ) ({x j } ↔ {xtk } with j = (t − 1) n + k for a rate-1/n convolutional code), we have Λ ( x j ) = log

∑α

t −1

( s ')γ t ( s ', s ) β t ( s )

∑α

t −1

( s ')γ t ( s ', s ) β t ( s )

Sk+

Sk−

+ k

(3)

where S is the set of pairs ( s ', s ) such that k coded bit at state t is 1 and Sk− is corresponding set for -1, and th

Λ (dt ) = log

∑α

t −1

( s ')γ t ( s ', s ) β t ( s )

∑α

t −1

( s ')γ t ( s ', s ) β t ( s )

U k+

U k−

(4)

where U k+ is the set of pairs ( s ', s ) such that kth information bit at state t is 1 and U k− is corresponding set for 0. The terms α t and β t are defined with forward and backward recursions as ( τ denotes the frame length of information bits),

α t ( s ) = ∑ α t −1 ( s ') γ t ( s ', s ),1 ≤ t ≤ τ and β t ( s ) = ∑ β t +1 ( s ') γ t +1 ( s, s '),τ − 1 ≥ t ≥ 0 s'

(5)

s'

γ t ( s ', s ) denotes the transition probability for the branch s ' → s for the stage t of the code trellis n

n

1 l =1 l =1 2 So, the extrinsic information produced by the decoder can be computed as

γ t ( s ', s ) = P( St = s | St −1 = s ') = ∏ P( xtl ) = ∏ 1 + xtl tanh( L2 ( xtl )) 

∑α LD ( x j ) = log

Sk+

t −1

( s ') β t ( s )

∑α t −1 (s ') β t (s) Sk−

n

∏

l =1, l ≠ k n

∏

l =1, l ≠ k

(6)

P(xtl ) = Λ D ( x j ) − L2 ( x j )

(7)

P( xtl )

where again {x j } ↔ {xtk } with j = (t − 1) n + k . This after interleaving is passed to the SISO equalizer.

4. SIMULATION RESULTS We now look at the simulation results comparing the performance of the proposed schemes for the FMT system. The channel bandwidth was taken to be 20 MHz with 64 subchannel frequency slots. BPSK modulation was used to map the bitstream into an information symbol sequence. The sampling rate was 20 Msamples/sec. FMT symbol duration (T) was taken as 3.2 µs to achieve spectral efficiency of 1 bit/sec/Hz (for uncoded system). We assumed the channel is known only to the receiver through the use of pilot tones. We used Hamming window with cutoff frequency, fcutoff = 0.315/T and the overlap factor γ was 10. We took a perfectly terminated rate-1/2 convolutional encoder having the

generator polynomial G = [7 5]. The length of the equalizer filter was 36. A random interleaver of length 640 was taken. In Fig. (3) and (4) we show the BER performance as a function of SNR for the turbo equalization based scheme using MMSE-DFE and MMSE-LE, respectively. We performed 8 iterations for each scheme. The simulations were performed on 2000 FMT blocks of length 64 each.

Fig. 3: BER performance for MMSE-DFE

Fig. 4: BER performance for MMSE-LE

For the first iteration the performance of MMSE-DFE is much better than MMSE-LE as expected. In the absence of any a priori information, MMSE-DFE which uses non-linear equalizer outperforms its linear counterpart MMSE-LE except at low SNR (0-5 dB). Over the subsequent iterations (from 2 to 8), we see that the performance of MMSE-LE improves substantially compared to MMSE-DFE. This can be explained from the fact that DFE uses a slicer in the feedback loop. This hard limiting destroys the notion of soft information in the received signal. On the other hand, LE maintains the fidelity of soft information with iterations and thus gives impressive gains. Thus, MMSE-LE achieves a gain of 2.3 dB against MMSE-DFE at BER = 10-4 after 8 iterations. We attained convergence by fifth iteration in both the schemes.

5. CONCLUSION We obtained very impressive gains by using turbo equalization based schemes in FMT system at the cost of increased computational complexity. However, we can use approximate implementation to bring down complexity at slight performance loss. The complexity of the system is mainly due to iterations and we obtain impressive gain with iterations. Thus, we obtained a simple-to-design system providing excellent tradeoff between performance and complexity. Therefore, it may be a valid alternative to OFDM in wireless applications.

6. REFERENCES [1] Giovanni Cherubini, Evangelos Eleftheriou, Sedat Ölçer, “Filtered multitone modulation for very high-speed digital subscriber lines,” IEEE Journal on Selected Areas in Communications, vol. 20, no. 5, pp. 1016-1028, June 2002. [2] N. Benvenuto, S. Tomasin, L. Tomba, “Equalization methods in OFDM and FMT systems for broadband wireless communications,” IEEE Transactions on Communications, vol. 50, no. 9, pp. 1413-1418, Sep. 2002. [3] N. Benvenuto, S. Tomasin, “Efficient pre-coding schemes for FMT broadband wireless systems,” 13th IEEE Intl. Symposium on Personal, Indoor and Mobile Radio Communications, vol. 4, pp. 1493-1497, Sep. 2002. [4] M. Tuchler, R. Koetter, A. Singer, “Turbo equalization: principles and new results,” IEEE Trans. On Communications, vol. 50, pp. 754-767, May 2002. [5] M. Tuchler, A. Singer, R. Koetter, “Minimum Mean Squared Error Equalization using a priori Information,” IEEE Transactions on Signal Processing, vol. 50, pp. 673-683, Mar. 2002. [6] L. R. Bahl, J. Cocke, F. Jelinek, J. Raviv, “Optimal decoding of linear codes for minimizing symbol error rate,” IEEE Transactions on Information Theory, vol. IT-20, pp. 284-287, Mar. 1974.