ML Estimation and Cancellation of Nonlinear Distortion in OFDM Systems Ali Behravan and Thomas...

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ML Estimation and Cancellation of Nonlinear

Distortion in OFDM Systems

Ali Behravan and Thomas Eriksson

Communication Systems Group, Department of Signals and Systems,

Chalmers University of Technology, SE-412 96 Gteborg, Sweden

Abstract

In this paper we study distortion cancellation and optimum detection of a nonlinearly-distorted orthogonal frequency division

multiplexed (OFDM) signal. We derive the ML optimum estimation of the signal by incorporating the reliability information of

the received subcarriers. The nonlinearity is assumed to be either a deliberate clipping of the signal to limit the signal envelope

fluctuations, or a nonlinear high power amplifier at the transmitter front-end. It will be shown that using iterative estimation of

the distortion and the proposed ML estimation, considerable performance improvement compared to the hard decision case can be

achieved, over an AWGN and a Rayleigh fading channel.

Keywords

Orthogonal frequency division multiplexing (OFDM), nonlinear distortion, maximum likelihood estimation.

I. INTRODUCTION

Orthogonal frequency division multiplexing (OFDM) has been shown to be an effective modulation in

high data rate transmission over time dispersive wireless channels [1]. This is due to the use of long symbol

intervals compared to the channel delay spread. However, it is well known that a major disadvantage of

OFDM is the strong envelope fluctuations of the signal [2], leading to out-of-band radiation and also to

intercarrier interference at the receiver.

Different solutions have been proposed to limit the range of signal fluctuations or to compensate for the

nonlinearity distortion. These methods either work in the digital or analog part of the system. Because of the


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simplicity of digital signal processing implementation, the digital methods are favorable.

Methods such as partial transmit sequence [3], selective mapping [4], and block coding [5] are among the

most widely used transmitter side solutions. They are all based on limiting the set of possible signals that

can be transmitted. Therefore they require side information to be transmitted to the receiver. This reduces the

information throughput. Furthermore, some of these methods are far too complex to implement in a system

with reasonable number of subcarriers.

Predistortion is another distortion reduction technique, where the attempt is to compensate the nonlinearity

by modifying the input signal. If predistortion is done on a Nyquist-sampled signal, the resulting signal after

the nonlinearity will have less in-band distortion. However, in order to mitigate both in-band and out-of-band

distortions, predistortion has to be done on the oversampled signal. Due to practical limitations such as A/D

speed, high oversampling is not feasible in a high data rate transmission system. Moderate oversampling of

order 8 is typical in OFDM systems [6]. The simplest predistortion method is deliberate clipping, where the

envelope of the input signal is clipped to a predetermined value. In this case post-processing at the receiver is

required to mitigate the distortion. We will use a deliberate clipping as part of the solution later in this paper.

Several receiver-side solutions have also been studied in the literature. Some of the most well-known ones

are nonlinear equalization [7], Bayesian inference for signal recovery [8], and estimation and cancellation

of the nonlinear distortion [9], [10]. In [9], an algorithm to cancel the clipping noise in a deliberately-

clipped OFDM signal is introduced. This method is only used for non-filtered signals, and only for a static

intersymbol interference channel. In [10], it is shown that this method is an iterative implementation of the

ML detector for nonlinearly distorted OFDM systems. The problem with this method is that for low clipping

level, where the number of clipped samples in one OFDM block is large, the improvement in the BER is very

small [9]. This is also shown by simulations in Section IV of this paper.

In this paper we apply nonlinear distortion cancellation based on ML estimation of the subcarriers. The

method is robust against severe nonlinearities as well as weak ones. The distortion in the frequency domain

is estimated, then it is regenerated at the receiver and subtracted from the received signal. We compare


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Baseband

modulator

Input

bits

demodulator

Baseband

bits

Output

Clippingmodulator

OFDM

Fading

channel

OFDM

demodulatorequalizer

Channel

Channelestimation

Sk sn sn s(t)

n(t)

gT(t)

gR(t)

Rk

Fig. 1. Baseband equivalent of an OFDM system with nonlinearity

hard and soft decision of symbols and show that more improvement in the performance can be achieved by

incorporating the reliability information of the soft symbols. By performing several iterations, the estimation

of the distortion becomes more accurate, and ideally in a low noise condition the entire nonlinear distortion

can be canceled. Derivation of the ML estimator and simulations are performed for both AWGN and Rayleigh

fading channels. Later in the simulations part, we consider the case of a system with no amplitude clipping,

but with a nonlinear high power amplifier.

The remainder of the paper is organized as follows. In Section II, we present the analysis and derivation

of nonlinear distortion in an OFDM system. The analysis is then used to derive the optimum detection of

symbols in Section III. In Section IV we present simulation results on an AWGN and a Rayleigh fading

channel. Concluding remarks is given in Section V.

I I . SYSTEM MODEL

Figure 1 shows the basic block diagram of an OFDM system. The N baseband modulated symbols are

first transformed by means of an IFFT to an OFDM symbol which will be a set of channel symbols. The

OFDM modulator includes a serial to parallel block, an IFFT and a parallel to serial block. The signal is then

clipped to limit the high envelope samples. We consider Nyquist rate signal clipping, but the analysis holds

for upsampled signals provided that the signal processing at the receiver side is done on the upsampled signal

as well. The pulse shaping filter gT(t) is a root-raised cosine filter.


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The Nyquist-sampled signal at the output of the OFDM modulator can be described as

sn =1N

N1

k=0

Skexpj2kn

Nn = 0, 1, . . . ,N1, (1)

where Sk is a QPSK symbol and sn is a sample of the time domain signal.

The baseband nonlinearity is a deliberate clipping with input-output profile as

y =

x |x| < Amax

Amaxx|x| |x| Amax

(2)

where Amax is the clip level.

The OFDM modulated signal is the sum of N independent and identically distributed random processes.

According to the central limit theorem, if the number of subcarriers is large enough the signal can be approxi-

mated as a complex Gaussian distributed random process. From the Bussgang theorem and by extending that

to complex Gaussian processes, the output of a memoryless nonlinearity with a Gaussian input can be written

as the sum of a scaled input replica and an uncorrelated distortion term as [11]

sn = sn + dn, (3)

where dn is the distortion term and is a constant described as

=E[snsn

]

E[|sn|2] . (4)

In Figure 1, at the receiver and after matched filtering and sampling, the time domain signal rn is passed

through a FFT block which makes a set of received decision variables. At this stage we assume that the

channel is transparent and add no noise to the signal. In this case the decision variable Rk

consists of a useful

signal term plus a nonlinear distortion term approximated as an uncorrelated Gaussian noise term [12, 13].

From (3), the expression for subcarrier Rk can be written as

Rk = Sk+Dk(S0, S1, , SN1) k= 0, 1, ,N1, (5)

where the capital letters indicate the corresponding signals after the FFT block. Dk(S0, S1, , SN1) is the

distortion in the kth subcarrier, which is a function of all subcarriers.


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III. NONLINEAR DISTORTION CANCELLATION

In this section we attempt to estimate and remove the nonlinear distortion term in (5). The structure of

receiver to implement the distortion cancellation is shown in Figure 2.

OFDM

demodulator

Hard / Soft

decision modulator

OFDM

demodulator

BasebandOutput

bits

_

_

Clipping

gR(t)rnRk

Rk

+

+

Fig. 2. Block diagram of the distortion cancellation method

Channel observations {Rk} are used as the first estimate of the transmitted subcarriers. The OFDM mod-

ulator and the clipping block in the feedback branch generate an estimate of the transmitted signal, which is

used to find an estimate of the distortion.

We use the terms hard decision and soft decision for the two cancellation methods we study here, due to

the resemblance to hard / soft decision in a standard modulation and coding context. With hard decision,

the variables

Rk are chosen by standard ML detection of the observation Rk. With soft decision, we instead

derive an ML-optimal estimate of the observation Rk without rounding off to the closest symbol. For both

soft and hard decision, the detected / estimated subcarriers Rk are used to compute a distortion variable dn

that is subtracted from the received signal.

The distortion cancellation in Figure 2 can be done for a few iterations to get a better estimate of the

distortion and as a consequence more improvement in the detection performance.

When the clipping level is low, the algorithm may falsely replace some of the unclipped samples [9], which

degrades the performance of the method. However, as will be shown later, if the ML estimation is done on

the received subcarriers, the algorithm can work down to vary low clipping levels.

In the following subsections we will derive the optimum ML soft decision by incorporating a measure of

reliability of the symbol, for an AWGN and a Rayleigh fading channel.


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A. Nonlinear Distortion Cancellation in AWGN Channel

In the case of an AWGN channel, the expression in (5) will contain a noise term. The algorithm in this case

is the same as the one for the noise-free channel. It is shown in Appendix A that for QPSK-OFDM signal in

AWGN channel the maximum likelihood soft decision on subcarriers can be found from

R(m)k = tanh(

(m)k R

(m)k ), (6)

where (m)k is derived in Appendix A as

(m)k =

Es/2

N04

+E[|D

(m)k

|2]2

(7)

In (7), m is the number of iterations, Es is the energy per symbol and N0/2 is the variance of the white

Gaussian channel noise. D(m)k is the Fourier transform of the distortion that is computed in the feedback loop

in Figure 2. As can be seen from (7), the parameter (m)k is a function of both SNR and the nonlinearity profile

.

Therefore the new estimate of the soft symbols can be found from

R(m+1)k =

1

R

(m)k

D(m)k ( R(m)0 ,

R(m)1 , . . .,

R(m)N1)

(8)

B. Nonlinear Distortion Cancellation in Multipath Fading Channel

Let us consider the following general representation of the channel

h(t,) =L1

l=0

l(t)(l(t)), (9)

where l(t) is a complex zero mean Gaussian process. We assume that the channel is frequency-selective

and slowly-varying with a coherence time longer than the OFDM block duration. Therefore the channel is

approximately time-invariant over each OFDM block.

It can be shown that by proper choice of the guard interval and assuming perfect channel state information

at the receiver, the equivalent channel between the baseband modulator and demodulator can be modeled as


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a linear fading channel for each subcarrier [14]. The kth subcarrier signal after a zero forcing (ZF) channel

equalizer has the form of [14]

Rk = Sk+Dk

+

Nk

Hkk= 0, 1, ,N1, (10)

where Hk is Fourier transform of the channel coefficients l ,

Hk =L1

l=0

l exp(j2kl

T), (11)

and T is the duration of one OFDM block.

If an MMSE equalizer is used, the output signal is just a scaled version of (10) [14], therefore the results

here can be extended to the system with an MMSE equalizer as well.

Since we assume that the channel is perfectly known at the receiver, the coefficient Hk for each OFDM

block is known. The ML optimum estimate of the decision variable Rk in (10) can then be found similar to

the case of AWGN channel. In this case R(m)k is the same as (6), but the expression for

(m)k , according to the

Appendix A, is

(m)k =

Es/2

N0

4|Hk|2 +E[

|D

(m)k

|2]

2

. (12)

IV. SIMULATION RESULTS

To evaluate the performance of the proposed detections, let us consider the system of Figure 1, when a 256

subcarrier OFDM system with baseband QPSK modulation is used. The clip level is 0 dB (or Amax = 1)

Figure 3 shows the average received SDNR (signal to distortion-plus-noise ratio) for different iterations as

a function of the channel SNR when the optimum soft decision of symbols is employed. The SDNR of the

subcarrier k at the mth iteration is defined as

SDNR(m)k =

E[|Sk|2]2E[|D(m)k |2] +N0

(13)

As the figure shows, for high SNR values after 2 or 3 iterations the SDNR is almost equal to SNR which

means that the nonlinear distortion has almost been totally removed. However, for low SNR values, where

the additive noise is dominant, the SDNR may even be lower compared the system with no cancellation.


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0 5 10 15 20 25 305

0

5

10

15

20

25

30

No cancellation

1st iteration

2nd iteration

3rd iteration

No clipping

SDNR

[dB]

SNR [dB]

Fig. 3. Signal to Distortion plus Noise Ratio as a function of the channel SNR.

0 2 4 6 8 10 12 14 16 18 2010

7

106

105

104

103

102

101

100

Eb/N

0[dB]

BER

No cancellation

Hard, 1 iter

Hard, 2 iter

Soft, 1 iter

Soft, 2 iterNo clipping

Fig. 4. BER of the proposed method on an AWGN channel with hard and optimum soft decision on symbols

Figure 4 shows the BER as a function of channel SNR for different iterations, with hard decision and

optimum soft decision on symbols. For the hard decision case, no further improvements can be achieved


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0 1 2 3 4 5 68

9

10

11

12

13

14

15

16

17

18

# iterations

SNR[

dB]

Harddecision

Softdecision

Lower bound (linear)

No cancellation

Fig. 5. Required SNR to achieve BER= 104 in AWGN channel as a function of the number of iterations

with higher number of iterations since the estimation of the distortion in the feedback loop is almost the same

after the second iteration. In contrast, for soft decision case, using more iterations is advantageous since the

estimate of the distortion becomes more and more accurate.

As long as the number of symbol errors in the detection of one OFDM block is reasonably small, most of

them can be remapped to the original quadrant. Our extensive simulations show that if the error rate is less

than 1%, which is the case in most practical systems, the algorithm always improves the system performance

by using more iterations.

In order to see the capability of the method in improving the performance, we study the required SNR to

achieve a specific BER for different iterations. Figure 5 shows the required SNR to achieve BER= 104,

as a function of the number of iterations. As we mentioned earlier, with hard decision the performance

improvement after 2 iterations is not significant, while with optimum soft decision the required SNR always

reduces with more iterations.

In the case of a fading channel we consider Rayleigh fading with an exponentially decaying power delay


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profile with normalized delay spread equal to 2. The convolutional code used here is a rate 1/2 with generators

(5, 7)8.

Figure 6 shows the performance of the method for different iteration with hard and soft decision on the

received symbol. As can be seen from the figure, in the case of hard decision, the improvement in the

performance is limited and the floor in the bit error rate curve still exists after 2 iterations. Almost no

improvement can be attained after 2 iterations. However, when ML soft decision is used, the floor in the

BER curve disappear and after 2 iterations at BER= 104 the SNR gap from the linear case is about 3 dB.

0 5 10 15 20 25 30 3510

7

106

105

104

103

102

101

100

Eb/N

0[dB]

BER

No cancellation

Hard, 1 iter

Hard, 2 iterSoft, 1 iter

Soft, 2 iter

No clipping

Fig. 6. BER of the proposed method on a fading channel with hard and optimum soft decision on symbols

Figure 7 shows the required SNR to achieve BER= 104, as a function of the number of iterations. Ac-

cording to the figure, an average gain of 3 dB in SNR can be achieved, if soft decision is used compared to

the hard decision case.

We now consider the problem of an OFDM system with no baseband clipping, and with a nonlinear high

power amplifier at the RF front-end. The nonlinear amplifier at the transmitter front-end will distort the

bandpass signal, but here we use the baseband equivalent of the amplifier. In this case in order to be able


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0 1 2 3 4 5 619

20

21

22

23

24

25

26

27

28

29

# iterations

SNR[

dB]

Harddecision

Softdecision

Lower bound (linear)

No cancellation

Fig. 7. Required SNR to achieve BER= 104 in Rayleigh fading channel as a function of the number of iterations

to use the distortion cancellation method we need to know the nonlinearity profile. Let us assume that the

nonlinear amplifier is a Saleh model with AM/AM and AM/PM characteristics as [15]

F(|x|) = A2max|x|

|x|2 +A2max(14)

(|x|) = 3

|x|2|x|2 +A2max

. (15)

Figures 8 and 9 show the performance of the cancellation methods on this system for the AWGN and the

fading channel described earlier. As expected with optimum soft decision significant performance improve-

ment can be achieved in both cases. Note that the BER in this case is higher than the case where the baseband

signal is clipped. The IBO used in this case is 3 dB for both setups.

V. CONCLUSIONS

In this paper we presented the optimum detection and estimation of the nonlinear distortion in an OFDM

system. The method utilizes ML estimation of symbols to reduce the distortion iteratively. When the clip level

is low, performing hard decision in the feedback loop only results in very small performance improvement,


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0 2 4 6 8 10 12 14 16 18 2010

7

106

105

104

103

102

101

100

Eb/N0 [dB]

BER

No cancellationHard, 1 iter

Hard, 2 iter

Soft, 1 iterSoft, 2 iter

No clipping

Fig. 8. BER of the system with nonlinear amplifier on an AWGN channel with hard and optimum soft decision on symbols

0 5 10 15 20 25 30 3510

7

106

105

104

103

10

2

101

100

Eb/N

0[dB]

BER

No cancellation

Hard, 1 iterHard, 2 iter

Soft, 1 iter

Soft, 2 iterNo clipping

Fig. 9. BER of the system with nonlinear amplifier on a fading channel with hard and optimum soft decision on symbols

while using soft decision always improves the performance by several dB. Simulations on both AWGN and

a Rayleigh fading channel show that using ML soft decision on symbols, and with only 2 iterations the SNR


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gap to the linear case is 23 dB at BER= 104. Also the SNR gain compared to the hard decision case is at

least 3 dB. Furthermore it was shown that by performing more iterations of the algorithm, more improvement

in the performance is achievable, if reliability information of the soft symbols is used. For sufficiently high

SNR values, simulations showed that the algorithm always converges and results in lower bit error rates.

APPENDIX

I. OPTIMUM SOFT DECISION OF NONLINEARLY DISTORTED OFDM SIGNAL

We derive the ML optimum estimation of symbols in a nonlinearly distorted OFDM system. It has been

shown that an OFDM system with a nonlinearity at the transmitter front-end and AWGN or slowly varying

fading channel can be modeled as a linear channel with a gain and an equivalent additive Gaussian noise [11],

[14]. Also we know that a QPSK-modulated symbol with independent additive noise components on the real

and imaginary axes can be viewed as two independent BPSK-modulated symbols on AWGN channel.

We consider the equivalent linear system with BPSK modulation. Figure 10 shows the transmission of

BPSK symbols over an AWGN channel. The equivalent additive noise is zero-mean Gaussian with variance

2, and the BPSK symbols si {A, +A}.bi si siri

nN(0,2)

BPSKmodulator

Softdecision

Fig. 10. Block diagram of a BPSK transmission system.

The problem is to find the maximum likelihood estimate of si, which is denoted by si. The ML estimation

ofsi is

si = E[si|r] (16)

= A(2Pr(si = +A|r)1) (17)

= A tanh(L(si|r)

2), (18)


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where L(si|r) is the log-likelihood ratio (LLR) of symbols si, which is defined as

L(si|r) = log Pr(si = +A|r)Pr(si = A|r) . (19)

For an additive white Gaussian noise channel the likelihood ratio is L(

si|

r) =

2Ar/2. Therefore the ML

estimate ofsi is

si = A tanh(Ar

2). (20)

Now in the system of Figure 2 and in the case of an AWGN channel, from the independence of channel

noise and the distortion we can write

R(m)k = tanh

Es/2N0

4+

E[|D(m)k

|2]2

R(m)k

. (21)

For a Rayleigh fading channel with perfect channel state information at the receiver, and assuming a zero

forcing equalizer, the ML estimate ofR(m)k is

R(m)k = tanh

Es/2N0

4|Hk|2+

E[|D(m)k

|2]2

R(m)k

. (22)

We must show that the optimum detection holds after the first iteration as well. In the block diagram of

Figure 2, after the hard / soft decision block, the symbols Rk are not Gaussian distributed anymore. However

Rks are independent and identically distributed, and from the central limit theorem, the symbols at the output

of the OFDM modulator in the feedback loop can be approximated as complex Gaussian random variables.

It follows from Bussgang theorem that the subtractive distortion term which is fed back to the received time

domain signal is Gaussian distributed and therefore the optimum receiver for all iterations ( m = 1, 2, ) isthe same as (6).

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