Iterative Correction And Decoding

ITERATIVE CORRECTION AND DECODING OF OFDM SIGNALS AFFECTED BY CLIPPING Wolfgang Rave, Peter Zillmann, and Gerhard Fettweis Dresden University of Technology, Vodafone Chair Mobile Communications Systems Mommsenstrasse 18, 01062 Dresden, Germany [email protected]

Abstract

1.

We investigate iterative receive algorithms for orthogonal frequency division multiplexing (OFDM) transmission affected by clipping at the transmit power amplifier (PA). One algorithm aims at minimizing the Euclidean distance to the received sequence, while the other reconstructs the clipping noise. Both methods significantly reduce the BER compared to simple zero forcing, even for severe clipping. The iterative clipping noise correction is also investigated with a ’soft’ sequence and a Soft-Input Soft-Output (SISO) decoder with coded OFDM. The gains with a soft sequence are moderate due to burst errors, but the number of iterations can be reduced compared to a correction based on hard decisions.

Introduction

The transmit signal in multicarrier data transmission systems is the superposition of many narrowband signals. This results in an approximately Gaussian distribution of the I- and Q-components of the complex baseband signal (central limit theorem). Consequently, multicarrier systems like OFDM are often impaired by clipping at the transmit PA. This paper presents several approaches to receiver-based iterative correction of clipped OFDM signals.

2.

System Model

The system model used throughout the paper represents a simplified, discrete time, baseband equivalent OFDM system. A vector X = [X0 , X1 , . . . , XN −1 ]T of length N is formed at the transmitter, and Xk ∈ S ∀ k ∈ N, 0 ≤ k ≤ N − 1, where the symbol set S = (S0 , S1 , . . . , SM −1 ) contains all M = 2p , p ∈ N possible complex transmit symbols.

W. Rave, P. Zillmann, and G. Fettweis

X is then converted to the time domain vector x = [x0 , x1 , . . . , xN −1 ]T by means of an Inverse Discrete Fourier Transform (IDFT) of length N . The elements of x are approximately complex Gaussian distributed, with variance Px . The time domain vector is then distorted by a nonlinear function g(·) to deliver z = [z0 , z1 , . . . , zN −1 ]T . In this paper, the baseband equivalent memoryless nonlinearity g(·) is assumed to be a soft limiter (SL), which distorts the magnitude but not the phase of the elements of x. This models the baseband equivalent of an ideally predistorted nonlinear PA. The magnitude rz,k = |zk | is then given by ½ rx,k rx,k ≤ A rz,k (rx,k ) = , (1) A rx,k > A where A is the clipping level of the nonlinear device. After the nonlinearity, a vector n = [n0 , n1 , . . . , nN −1 ]T of complex AWGN further corrupts z to yield y = z + n. (2) This system model is shown in Fig. 1. The complex Gaussian random x (t )

g (⋅)

z (t )

y (t )

+

Figure 1.

System and channel model

n (t )

variables nk are assumed zero-mean with variance 2σn2 (σn2 per real dimension) and uncorrelated. The Input Power Backoff (IBO) for the soft limiter device is given by IBO = A2 /Px .

(3)

It is well known that the output of a memoryless nonlinearity with Gaussian input can be described by means of the Bussgang decomposition [1]. The signal zk can be written as zk = αxk + dk ,

(4)

where α is a scaling factor depending on the nonlinearity and Px , and dk is a clipping noise term uncorrelated with xk . For complex Gaussian input and real-valued g(·), α can be determined from 1 α= Px

Z∞ rx prx (rx )g(rx ) drx ,

(5)

0

with rx being the magnitude of x. The Rayleigh distribution for prx (rx ) leads, for the SL, to

Iterative Correction and Decoding of OFDM Signals affected by Clipping

√ ¡ ¢ 1√ α = 1 − e−IBO + π IBO erfc( IBO) , (6) 2 where erfc(·) is the complementary error function. Furthermore, it¢is eas¡ ily shown that, in our special case considered here Pz = 1−e−IBO Px = βPx , from where the power of the clipping noise Pd can be computed as Pd = Pz − α2 Px = (β − α2 )Px .

3.

(7)

Clipping Correction Algorithms

3.1

Simplified ML Detection Algorithm

In the following, we assume that the nonlinear function g(·) is known at the receiver. The ideal ML detector for the communications system ˆ i , where 0 ≤ under consideration forms all possible transmit vectors X N i ≤ M − 1. Then, these vectors are converted to time domain and ˆi . Finally, the distance of distorted by g(·), resulting in the vectors z ˆ each zi to the received vector y is computed, and the one with minimum distance is selected. Thus the ML detector solves the following problem: ¯¯ £ © ª¤¯¯2 ˆ = arg min ¯¯y − g IDFT X ˆ i ¯¯ . X ˆi X

(8)

It is apparent that such a detector is prohibitively complex for practical values of N and M . The following approach is proposed: Instead of generating all possible hypotheses for the vector X in parallel, we look for the symbol which minimizes the above distance metric in the time domain for each element of X sequentially. The complete detection algorithm then consists of the following steps: Sequential Mean Square Error Reduction 1 Convert the received time domain vector y/α into the frequency domain vector Y by means of a Discrete Fourier Transform (DFT), with the scaling factor α depending on IBO. Then, quantize Y according to the decision boundaries corresponding to the symbol alphabet S (hard detection). This operation ˆ 1 as a first estimate of X. delivers X ˆ (0) to X ˆ (0) based on X ˆ 1 by in2 Generate M new frequency domain vectors X 1,1 1,M serting all possible symbols from S as the first vector element. The superscript denotes the index of the element which is varied. 3 For 1 ≤ m ≤ M compute the distances ¯¯ £ © (0) ª¤¯¯2 (0) ˆ ¯¯ . d1,m = ¯¯y − g IDFT X 1,m

(9)

and choose the symbol which results in the smallest distance. ˆ 1 . Thus, the symbol decisions are 4 Repeat steps 2 and 3 for all N elements of X refined sequentially.

W. Rave, P. Zillmann, and G. Fettweis

When no nonlinear distortion is present, this algorithm is equal to the complete ML solution, because the DFT and IDFT are orthogonal transforms, and reducing a possible error term in the frequency domain inevitably results in a reduced time domain error. The nonlinearity destroys orthogonality, which makes the above algorithm suboptimum. However, it reduces exponential to linear complexity. Only M N hypotheses are tested per OFDM symbol. The loss of orthogonality implies that the order, in which the subcarriers are processed, can affect the convergence behavior. Furthermore, several iterations of the whole algorithm can potentially improve performance. Note that, for each hypothesis, only a discrete increment in frequency domain has to be transformed into the time domain. A complete IFFT is not necessary. Performance Results for AWGN Channel: The algorithm was tested for uncoded QAM transmission with N = 64. No guard or pilot subcarriers were simulated, and the transmission channel model was AWGN. The SL backoff was varied from 0-2 dB, modelling severe clipping (the clipping probability pclip = e−IBO amounts to 0.368, 0.284 and 0.205 for IBO values of 0, 1 and 2 dB.). Fig. 2 (left) shows the performance gains after one iteration obtained for 16-QAM for these IBO values when the sequential MSE reduction algorithm is applied. 10

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Figure 2. Performance of sequential MSE reduction for 16-QAM and IBO values of 0, 1 and 2 dB (left) and for QAM modulation with different constellation size (right).

On the right, the performance of a simplified version of the proposed algorithm for different constellation size after two iterations is compared. The hypotheses were now generated in a hierarchical manner by first selecting the quadrant which minimizes the MSE. Then, the actual QAM symbol in that quadrant was selected. Thus, only 8 instead of 16 (16QAM) or 12 instead of 64 (64-QAM) hypotheses were tested per subcarrier. The figure shows how a 2nd iteration of the algorithm further improves performance. The hierarchical performance for 16-QAM and this IBO is identical to the full search.

Iterative Correction and Decoding of OFDM Signals affected by Clipping

It can be seen that the performance with clipping and sequential Mean-Square Error (MSE) reduction for QPSK actually surpasses linear AWGN performance in certain Eb /N0 regions. Two effects are responsible for this behavior: First, clipping reduces the transmit power by a factor of β = 1−e−IBO , which results in a rescaling of the Eb /N0 axis for a fair comparison. For an IBO of 0 dB, the resulting SNR gain is 2 dB. Secondly, the nonlinear distortion introduces dependencies between the frequency domain symbols, which help in recovering the correct symbols.

3.2

Iterative Correction with Hard Detection

Instead of approximating the complete search in some suboptimum way, an alternative is to estimate the clipping noise signal. α

nonlinearity -

xˆ

zˆ

FFT n z

y

Y

FFT

D

iFFT

Ycorr . 1/α

demapper & decoder

hard dec.

Xˆ

Tellado et al. [2], as well as Chen and Haimovich [3], proposed to compute the clipping noise term according to the Bussgang decomposition. This noise term is

then subtracted in the frequency domain according to the following steps (see the sketch of the algorithm above for the uncoded case): Iterative Clipping Correction ˆ ˆ based on hard decisions X 1 compute an estimate of the transmit sequence x derived from the demodulated equalized symbol vector Y. ˆ = g(ˆ 2 calculate the difference d = αˆ x − g(ˆ x) between the clipped sequence z x) and the attenuated unclipped sequence αˆ x. 3 add the Fourier transform D of this difference as a spectral correction to the demodulated symbols.

These steps can be applied iteratively to improve the reconstruction of the time domain sequence and the spectral correction respectively. Performance Results for AWGN Channel: Again, severe clipping and a medium constellation (16-QAM with an IBO of 0 dB) is the point of reference. The correction capability of the algorithm for different IBOs is shown in Fig. 3 (left). Comparing the zero forcing solution to the first and third iteration of the algorithm, we note that for all IBO levels the performance is improved by two orders of magnitude. The dependence on constellation size is demonstrated in Fig. 3 (right). The initial error rate for 64-QAM is high (' 12%), so that the algorithm does not improve the BER. On the other hand, for a robust constellation like QPSK, the first iteration step suffices to improve the zero forcing

W. Rave, P. Zillmann, and G. Fettweis

10

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Figure 3. Performance of iterative clipping correction for varying IBO (16-QAM transmission, AWGN channel) (left) and different constellation size (right).

result greatly. A better than unclipped AWGN performance was not observed, though.

3.3

Modified Algorithm using a soft Sequence

As is well known from decision feedback algorithms and decoding, the performance of an algorithm can be improved if soft instead of hard decisions are used. In our case, other sequence hypotheses can be incorporated into the correction term used for feedback. In the uncoded case, we compute a weighted sum of sequences instead of a single hard decided sequence NX cand 1 Pdist,n (ˆ xn ) x ˆn . n Pdist,n

x ˆsof t = P

(10)

n=0

The weights Pdist,n (ˆ xn ) are calculated according to the euclidean distance between the sequence hypotheses and the observed sequence. The conditional probability for observing the estimated sequence of N symbols in complex gaussian noise including the clipping noise dk with vari2 ) is calculated according to ance N0 = 2(σn2 + σnd Pdist

N n ||g(ˆ Y 1 xl ) − yl ||2 o = exp − . (πN0 )N N0

(11)

l=1

g(ˆ xl ) and yl denote samples of the clipped reconstructed sequence and the corresponding samples of the observed sequence. To select the candidate sequences we adopted the obvious idea to use metric values of the detected symbols to select those bits as candidates to be flipped, that have the smallest absolute likelihood values. Starting from the sequence

Iterative Correction and Decoding of OFDM Signals affected by Clipping

based on hard decisions, again only a scaled DFT vector (according to the modified subcarrier symbol) has to be added to the current sequence to obtain the new hypothesis for which the euclidean distance is computed. In contrast to the recently proposed idea to check all possible combinations of a set of unreliable bits (’reduced state sequence estimation’, see [4]), we found it more efficient to test the bits in the candidate set sequentially and to make immediately a step to the modified sequence, if the distance decreased. The list of bits is passed twice which led to improved performance. Steps in the wrong direction could be made less probable by testing the bits sorted according to their absolute L-values in ascending order. Soft Iterative Clipping Correction (uncoded case) Initialization: calculate metric values from the received symbols Y to obtain a list of unreliable bits together with their symbol and bit positions. Set the current and soft sequences to the reconstructed sequence based on hard decisions. 1 select the next bit in the list and compute a modified sequence with respect to the current best sequence and its euclidean distance. 2 add the modified sequence according to its weight to the soft sequence and update the total probability of all sequences accordingly. 3 replace the current sequence by the modified sequence, if a distance decrease was observed. 4 go to step 1, if end of list is not reached or the list is passed for the first time. 5 final step: normalize the soft sequence by the total probability (sum weight of all sequences).

Performance Results for AWGN Channel: Performance was tested for AWGN transmission of 16-QAM with IBO 0 dB as shown in Fig. 4. Sequence alternatives for the 64 smallest absolute metric values were tested (this number had been found to give a reasonable compromise between performance improvement and complexity). Two

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Figure 4. Comparison of hard and soft iterative clipping correction for different IBO (16-QAM, left) and different numbers of subcarriers (16-QAM, IBO 0 dB, right).

W. Rave, P. Zillmann, and G. Fettweis

iterations with sequentially optimized sequences achieve a similar performance as the original algorithm with about three iterations. In addition a gain in the transition region above the error floor is visible. Similar results were obtained when the number of subcarriers was varied (see Fig. 4, right). Again two iterations with soft sequences perform slighly better than three iterations with the original algorithm.

4.

Iterative Correction with Soft Decoding

Correction Algorithm: We extended the approach from the previous section to a coded system by using reliability information about the coded bits gained from a MAP decoder operating according to the BCJR algorithm [5]. A bit-interleaved coded OFDM system suffering from clipping distortion can be considered as a serial concatenation of a channel and an outer code. Therefore, a SISO [6] decoder can be employed to get reliability information not only about information bits but 1/α

z

n

y

Y

FFT

Ycorr .

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La Lsig SISO

decoder

D

FFT -

g (xˆ )

soft limiter

α

xˆ soft

iFFT

ˆ L% X soft sequence code

search

Π

Linfo Lcode

û

also about the coded bits that can be fed back to compute a soft estimate of the transmitted sequence. The soft sequence was again optimized sequen-

tially to avoid exponential complexity, generalizing slightly the approach of the uncoded case by incorporating a-priori information obtained from the decoder. The probability that the observed sequence y ˆ is due to the estimated sequence x ˆ after decoding is written as the product of the probability for the euclidean distance between hypothesis and observation and the likelihood derived from the decoder: Pseq (ˆ x) = Pdist (y|ˆ x) · Pf lip (ˆ x) .

(12)

The conditional probability Pdist for observing the received sequence y after transmitting x ˆ is computed as in eq. (11). The second factor in eq. (12) represents a-priori knowledge gained from decoding. It is calculated relative to the probability P0 (set to P0 = 1) of the most likely (hard decided) sequence ˆ c = sgn(Lcode (c)) after SISO decoding. Taking other candidate sequences into account reduces the sequence probability according to the probabilities of the coded bits cl . The probability for a sequence in which one or more bits are flipped is Y Y 1 − P (cl = cˆl ) = P0 · e−|L(cl )| . (13) Pf lip (ˆ x) = P0 · P (cl = cˆl ) l

l

Iterative Correction and Decoding of OFDM Signals affected by Clipping

To limit the sequence search to tractable complexity, a set size Ncand of unreliable bits has to be fixed. Finally a ’soft sequence’ is calculated as the weighted average of all candidate sequences: NX cand 1 Pseq,l (ˆ xl ) x ˆl . l Pseq,l

x ˆsof t = P

(14)

l=0

Soft Iterative Clipping Correction (coded case) ˜ sig from vector Y; deinterleave to obtain vector Lsig . 1 compute metric values L 2 soft decoding of Lsig to obtain reliability information Lcode for coded bits and information bits Linf o . ˜ code and a 3 compute hard decided sequence x ˆhard from interleaved vector L weighted average of sequences x ˆsof t , where the most unreliable bits are flipped as a soft estimate for the transmitted signal. 4 compute the spectrum D of the difference between clipped and non-clipped soft estimate of the time-domain sequence x ˆsof t and add it to Y. 5 goto step 1, if the maximum number of iterations is not reached. 6 final step: compute hard decisions for information bits using Linf o .

Performance Results for AWGN Channel: Results illustrating the performance of 16-QAM transmission for an AWGN channel and severe clipping are presented in Fig. 5. An IBO of 0 dB with uncoded transmission and ZF clipping correction leads to a BER floor at ' 0.04. A simple non-systematic terminated half rate code with constraint length 3 and a block size of one OFDM symbol was used with soft-in soft-out decoding. 0

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Figure 5. Clipping correction with soft or hard decided sequence for coded AWGN transmission of 16-QAM (left) and 64-QAM with different IBO 0 and 1 dB (right).

The performance after non-iterative decoding is still much worse than unclipped AWGN performance. The iterative correction algorithm from

W. Rave, P. Zillmann, and G. Fettweis

[2, 3] for coded transmission with hard detection leads to a significant improvement. More than 2 iterations yield only marginal additional benefit due to error propagation. The sequential optimization of the soft sequence was carried out by testing the sequence alternatives for the 64 smallest L-values. Two loops over all subcarriers were made, reducing the gap to unclipped performance by about 0.5 dB at BER = 10−4 . An even more severe clipping situation with 64-QAM and IBO values of 0 and 1 dB was studied for 64 subcarriers. Fig. 5 (right) shows the performance for both IBO values after the second iteration. It is comparable to three iterations with hard decisions; more iterations do not deliver additional benefits. One iteration in the decoder can be avoided at the expense of more effort in the detector.

5.

Conclusions

Under the premise that deliberate clipping in baseband can faithfully be reproduced at the receiver, the bit error rate can be significantly reduced by clipping correction techniques. In particular, the clipping noise reconstruction algorithm appears to be a promising approach with reasonable complexity. Using a weighted sum of hypotheses for feedback, the error rate can be reduced, but an error floor remains, and the advantage with respect to hard decisions is small. Thus, the hard decision technique remains attractive due to its simplicity.

References [1] A. Papoulis. Probability, Random Variables and Stochastic Processes. 3rd. Ed. McGraw-Hill Inc., 1991. [2] J. Tellado, L. Hoo, and John M.Cioffi. Maximum-Likelihood Detection of Nonlinearly Distorted Multicarrier Symbols by Iterative Decoding. IEEE Trans. Comm., 51(2):218–228, February 2003. [3] Hangjun Chen and Albert M. Haimovich. Iterative Estimation and Cancellation of Clipping Noise for OFDM. IEEE Comm. Lett., 7(7):305–307, July 2003. [4] Huy D. Han and Peter Hoeher. Simultaneous Predistortion and Nonlinear Detection for Nonlinearly Distorted OFDM Signals. In Proc. IST Summit 2005. [5] L. Bahl, J. Cocke, F. Jelinek, and J. Raviv. Optimal Decoding of Linear Codes for Minimizing Symbol Error Rate. IEEE Trans. Inf. Theory, IT-20:284–287, 1974. [6] S. Benedetto, G. Montorsi, D. Divsalar, and F. Polara. Soft-Input Soft-Output Modules for the Construction and Distributive Iterative Decoding of Code Networks. Europ. Trans. Telecomm., Vol. 9, pages 155–172, March 1998.