Inverted Wrap-Around Limiting with Bussgang Noise Cancellation Receiver for OFDM Signals

Abstract

Orthogonal frequency division multiplexing (OFDM) is a widely used multicarrier modulation technique. High peak-to-average power ratio (PAPR) is a major drawback of OFDM systems which leads to power inefficiency and signal distortions. The simplest approach is to clip the high amplitudes at some predefined threshold before the signal is applied to the amplifier. The clipping causes in-band distortion as well as out-of-band radiation. Though the out-of-band radiation is reduced using filtering, it causes peak regrowth. The peak regrowth is taken care of using the repeated clipping and filtering. This paper presents the use of a different form of limiting, inverted wrap-around (IWrap). IWrap is presented and evaluated in terms of PAPR reduction and peak regrowth. The results show that IWrap limiting is much better for PAPR reduction as compared to conventional clipping technique. The performance of the proposed limiting technique is investigated with Bussgang noise cancellation iterative receiver for OFDM signals. The results of this iterative receiver with IWrap limiting are presented in terms of bit error probabilities.

1 Introduction

Orthogonal frequency division multiplexing (OFDM) has significant advantages over conventional single-carrier modulation techniques that make it the desired modulation technique for the latest wireless communication systems. OFDM splits a high-rate single carrier into multiple lower-rate narrowband subcarriers. All these subcarriers are orthogonal to each other, so they can transmit and receive data simultaneously without interference. The idea has been taken from frequency division multiplexing in which there are guard intervals in between the subcarriers to avoid interference. In OFDM, because of orthogonality there is no need to add guard intervals between carrier frequencies. The orthogonality eliminates inter subcarrier interference (ICI) and is achieved by the power of digital signal processing (DSP) techniques, in particular, by the use of the fast Fourier transform (FFT) and its inverse.

All the subcarriers are modulated by conventional single-carrier modulation techniques such as binary phase-shift keying (BPSK), quadrature phase-shift keying (QPSK) and quadrature amplitude modulation (QAM). The Inverse FFT achieves the multicarrier modulation at the transmitter in a highly elegant and efficient way. These modulated subcarriers are summed up by using IFFT and then transmitted on the channel. At the receiver side, an FFT is applied to get back the modulated subcarriers.

OFDM has been deployed successfully in digital audio/video broadcasting (DAB/DVB), IEEE 802.11 Wi-fi, IEEE 802.16 WiMAX and 3GPP long-term evolution (LTE). The main advantage of OFDM over conventional single-carrier systems is its spectral efficiency and invulnerability to multipath effects. OFDM has a major drawback of having a non-constant signal envelope with high peak-to-average power ratio (PAPR). The high peaks produce nonlinear distortions in the signal when applied to a high-power amplifier (HPA). A highly linear amplifier with a wide dynamic range is required to minimize the distortion. However, for user mobile equipment this is likely to make inefficient use of the limited battery power of mobile devices [23].

There are many approaches in the literature to either reduce high PAPR or compensate its effects at the receiver. Some prominent techniques are soft clipping [3, 18, 22], coding [12, 27], partial transmit sequences [10, 20], selected mapping [4, 13] and active constellation extension (ACE) [15], equation method [7, 8] tone reservation [26] and random variable transformation [25]. A simple way of reducing the PAPR is to limit the high-amplitude peaks of a signal before it is applied to the amplifier of the transmitter. This soft limiting introduces out-of-band and in-band distortions in the signal resulting in an increase in the bit error rate (BER). Some hybrid techniques are also available in the literature which use clipping technique along with other technique in order to reduce the BER caused by the clipping [1, 17, 24].

Soft clipping is a simple and low-complexity PAPR reduction technique [3, 18, 22]. Limiting the peaks of OFDM symbols causes memoryless nonlinear distortion that becomes in-band distortion with Nyquist sampling. If clipping is applied to an oversampled signal, then it causes the in-band and out-of-band distortion as well [22].

This paper proposes a new way of soft limiting, inverted wrap-around (IWrap), and studies the effect of soft limiting (e.g. soft clipping) as applied to base-band and oversampled time-domain signals to eliminate the possibility of high-peak amplifier clipping. There are a few methods in the literature to compensate for the effect of the in-band clipping at the receiver to reduce BER. Iterative receivers such as Bussgang noise cancellation (BNC) and decision-aided reconstruction (DAR) can reduce clipping noise effects at the receiver and ultimately reduce bit errors [6, 9, 14]. The performance of the BNC receiver with IWrap limiting at the transmitter is investigated in this paper.

The rest of this paper is organized as follows: Sect. 2 describes soft clipping at Nyquist sampled signals. Section 3 explains IWrap limiting. Section 4 discusses the evaluating parameters of IWrap limiting in comparison with the clipping. Section 5 of the paper draws some conclusions.

2 Clipping

The clipping is a nonlinear process and produces distortion in an OFDM signal which generally degrades bit error rate (BER) at the receiver. Clipping at the Nyquist sampling rate causes all the clipping distortion to fall in-band. Unfortunately, significant peak regrowth can happen after digital-to-analogue conversion (D/A) because of the upsampling process and the nature of the interpolation between Nyquist rate samples [21]. Clipping an oversampled signal reduces peak regrowth because the interpolation has already been done before the limiting. But it produces out-of-band (OOB) distortion which is called spectral regrowth [18]. The OOB may be removed by frequency-domain filtering as proposed by [2], though, unfortunately again, this filtering can also reintroduce some time-domain peaks. Ultimately the remaining in-band distortion can be compensated at the receiver as is the theme of this thesis.

The clipping could be applied to signals at different stages of processing at the transmitter. In this paper, we have described and used base-band clipping which is applied after the IFFT (Fig. 1). A base-band OFDM signal is expressed as:

$$\begin{aligned} x(n)=\frac{1}{\sqrt{N}}\sum _{k=0}^{N-1}X_{k}e^{\frac{j2\pi nk}{N}}, 0\le \text {n}<N-1 \end{aligned}$$

(1)

where N is the number of subcarriers and $X_k$ for $k=0,1,\ldots ,N-1$ are the complex modulated data symbols.

2.1 Clipping Nyquist Sampled Signals

Fig. 1

Clipping at Nyquist sampling

The clipping process is described by the following expression:

$$\begin{aligned} {x_c}(n)={\left\{ \begin{array}{ll} x(n),&{} \quad \text {if }\left| x(n)\right| \le A\\ e^{j\theta \left( n\right) }.A &{} \quad \text {if} \left| x(n)\right| >A \end{array}\right. } \end{aligned}$$

(2)

The amplitudes of the time-domain samples are reduced to A, keeping their phases unchanged. The clipping ratio (CR) is defined as:

$$\begin{aligned} \gamma = \frac{A}{ \sqrt{P_\mathrm{in}}} \end{aligned}$$

(3)

where $P_\mathrm{in}$ is the average energy of the OFDM signal before clipping. When $\gamma$ is high, there is no distortion in the signal. When $\gamma$ reduces, clipping distortion increases. The initial experiments show that majority of the constellations received are correct at the receiver; data in Table 1 support the statement. The estimates in Table 1 have been calculated by transmitting 10,000 OFDM symbols with 64 subcarriers each, modulated with 16-QAM modulation. The effect of clipping in time domain is the reduced amplitudes of the samples and in frequency-domain distortion of the constellation symbols on the complex plane. It is evident from the results that at moderate levels of clipping the clipping function has to limit a few samples which reduces the complexity of the clipping function. Practically, COordinate Rotation DIgital Computer (CORDIC) algorithm maybe used to obtain the amplitude which is more suitable for hardware implementation [11].

Table 1 Estimate of clipped samples

3 Inverted Wrap-Around (IWrap) Limiting

In this section, the use of the inverted wrap-around (IWrap) limiting in place of clipping is proposed. The effect of the IWrap limiting function is illustrated in Fig. 2. Soft clipping limits the peaks to a threshold and discards the overshoot, whereas IWrap wraps around the peaks inside invertedly. In other words, the components of the complex waveform whose magnitude is above the threshold are removed and added back into the signal. The limited samples of the signal are therefore made smaller than the threshold. Equation 4 shows the limiting function for IWrap.

Fig. 2

IWrap limiting

$$\begin{aligned} c_n={\left\{ \begin{array}{ll} x_n,&{} \quad \text {if }\left| x_n\right| \le A\\ e^{j\theta \left( n\right) }.\left( 2*A-\left| x_n\right| \right) ,&{} \quad \text {if} \left| x_n\right| >A \end{array}\right. } \end{aligned}$$

(4)

where $x_n$ is the original unclipped time-domain complex signal, $c_n$ is the signal after IWrap limiting, A is the predefined magnitude threshold, $\theta \left( n\right)$ is the argument of the complex sample $x_n$ and $\left| x_n\right|$ is the magnitude of $x_n$. The computational complexity of IWrap limiting is approximately same as conventional clipping with an extra subtraction for each clipped sample. However, in case of repeated clipping and filtering, IWrap limiting is better which does not require any repetitions.

Bussgang theorem for memoryless non-linearities states that if a Gaussian signal x(n) passes through a memoryless non-linearity, a coefficient $\alpha$ may be found such that the output $x_c(n)$ may be written as:

$$\begin{aligned} {x_c}(n)=\alpha x(n) + d(n) \end{aligned}$$

(5)

where d(n) is a Gaussian signal which is uncorrelated with x(n). $\alpha$ is a scalar value which minimizes the energy of d(n) over all possible values of $\alpha$. To find this optimal value of $\alpha$, multiply both sides of Eq. 5 by x(n) and take expectations to obtain [3], [5]:

$$\begin{aligned} E\left\{ x_c(n) x(n)\right\} = \alpha E\left\{ x(n)^2 \right\} + E\left\{ x(n)d(n)\right\} \end{aligned}$$

(6)

If $E\left\{ x(n)d(n)\right\} =0$, then x(n) and d(n) are uncorrelated and

$$\begin{aligned} \alpha = \frac{E \left\{ {x_c(n)x(n)}\right\} }{E \left\{ x(n)x(n)\right\} } \end{aligned}$$

(7)

Auto-correlation of a signal at zero gives average power of the signal [24]. Equation 7 may be written as:

$$\begin{aligned} \alpha = \frac{E \left\{ {x_c(n)x(n)}\right\} }{ \sigma ^{2}} \end{aligned}$$

(8)

The nonlinear distortion factor $\alpha$ for IWrap limiting is derived using the Bussgang theorem. The non-linearity is applied to the envelope characteristics of the signal; therefore, as shown in [19], the distortion factor $\alpha$ can also be calculated using the envelope characteristics as:

$$\begin{aligned} \alpha = \frac{E\left\{ {z(n)f(z(n))}\right\} }{ \sigma ^{2}} \end{aligned}$$

(9)

where z(n) is the amplitude of the complex OFDM signal x(n), f(z(n)) is the non-linearity applied to the z(n). The distortion factor $\alpha$ (derived in “Appendix”) can be calculated using the following equation:

$$\begin{aligned} \alpha =1-2\text {e}^{-{\gamma }^2} + \sqrt{\pi }.{\hbox {erfc}(\gamma )}.{\gamma } \end{aligned}$$

(10)

where $\gamma$ represents the clipping ratio and erfc is the complementary error function defined as:

$$\begin{aligned} \hbox {erfc}(x) = \frac{2}{\sqrt{(\pi )}}\int _x^{\infty }e^{-t^2}\text {d}t. \end{aligned}$$

(11)

It is also clear that the limiting process reduces the output power. According to the central limit theorem, with the higher values of N, the distribution of the real and imaginary values of the time-domain OFDM signal can be normally assumed to be Gaussian. If we assume OFDM signal as a zero-mean complex Gaussian process, then the output power of the limited signal may be calculated as [5]:

$$\begin{aligned} {{P_x}_c}_n = \left( 1-2\sqrt{\pi }.\hbox {erfc}(\gamma ).\gamma \right) {P_x}_n \end{aligned}$$

(12)

where $P_{xn}$ represents the power of the signal before limiting.

4 Evaluating Parameters

The PAPR reduction schemes may be evaluated through the following parameters: PAPR reduction, power spectral density (PSD) and BER analysis.

4.1 Bit Error Rate Analysis

The block diagram of the OFDM transmitter and receiver with the BNC iterative loop is shown in Fig. 3. Clipping strongly affects the overall performance of the OFDM system when the clipped signals passed through the AWGN channel. Figure 4 shows the OFDM bit error probability using IWrap and soft clipping both with BNC receivers at different values of CR. Since BNC is an iterative receiver, the number of iterations is kept 2 for all the simulations. These probability estimates were produced using simulations by transmitting 10,000 OFDM symbols with 16-QAM modulation. The simulations are conducted for three values of the clipping ratio, i.e. $\text {CR}=2.0, 1.8, 1.6$. For severe clipping levels such as $\text {CR}=1.6$, the limiting distortion produced by IWrap is large enough as compared to that with soft clipping. The performance of BNC algorithm with soft clipping is much better as compared to the performance of BNC with IWrap. IWrap with BNC receiver still improved the BEP at moderate levels of clipping. It is interesting to notice that IWrap and soft clipping both are non-linearities. Clipping without any correction is even better than IWrap with BNC high severe levels of clipping.

Fig. 3

OFDM system with Bussgang noise cancellation receiver. a OFDM transimitter. b OFDM receiver

Fig. 4

Bit error probabilities of IWrap and soft clipping with BNC receiver

4.2 Power Spectral Density

Fig. 5

Power spectral density of clipped and filtered OFDM signal

The power spectral density (PSD) plotted in Fig. 5 shows the effect of the clipping distortion. When clipping is applied to an oversampled signal, energy is generated at frequencies outside of the signal bandwidth. This phenomenon is called spectral regrowth. The black plot shows the filtered signal after applying a FFT/IFFT filter.

4.3 PAPR Reduction and Peaks Regrowth

The clipping operation causes in-band distortion as well as out-of-band radiation. The out-of-band power spill reduces power spectral efficiency because of the adjacent channel interference. A band-limiting filter is applied to suppress the spectral regrowth, but this filtering process increases PAPR.

Fig. 6

Probability of $PAPR > PAPR_o$ for soft clipping and IWrap limiting for three values of CR

For this purpose, repeated clipping and filtering have been proposed by [3],[16] to address the peak regrowth issues. This approach has shown that the repeated CAF process significantly reduces peaks regrowth on an expense of the number of repeats. Since the filter is based on a frequency-domain IFFT/FFT pair, every iteration costs a lot towards the computation. Keeping in mind these constraints of computation involved in repeated CAF, we have proposed IWrap limiting. Figure 6 shows a comparison of PAPR reduction using soft clipping and IWrap limiting after limiting and filtering. These results are produced for three different values of CR, i.e. CR=1.6, 1.8, 2.0. The peak regrowth using the IWrap with filtering is smaller as compared to the peaks regrowth using clipping with filtering.

5 Conclusions

Soft clipping when applied with filtering increases the PAPR after filtering. IWrap, an alternative limiting technique, has been proposed to limit the high peaks of the time-domain OFDM signal. It effectively reduces the possibility of regrowth of the high peaks after filtering. The results of CCDF have been presented in comparison with soft clipping and filtering. A slight BER degradation has been observed as compared to soft clipping. IWrap can be effectively used with other techniques in the literature which are combined to work with the soft clipping and side information is used to transmit the information of the clipped samples. Performance of IWrap with side information in the presence of fading channels is the main focus of our future work which is in progress.

Appendix

This appendix explains the derivation of nonlinear distortion factor $\alpha$ for IWrap limiting.

The limiting function f(z(n)) for IWrap could be written as follows:

$$\begin{aligned} f(z(n))={\left\{ \begin{array}{ll} z,&{} \text {if } 0\le \text {z}<A\\ 2A-z, &{} A\le \text {z}<\infty \end{array}\right. } \end{aligned}$$

(13)

According to central limit theorem, the OFDM signal may be considered Gaussian for large number of subcarriers. Hence the amplitude values of the signal follow a Rayleigh distribution. The probability density function (Pdf) of an OFDM symbol is:

$$\begin{aligned} P(z) = \frac{2z}{ \sigma ^{2}}\text {e}^{\frac{-z^{2}}{\sigma ^{2}}} \end{aligned}$$

(14)

The expectation from Eq. 9 could be written as:

$$\begin{aligned} \alpha =\frac{E\left\{ {z(n)f(z(n))}\right\} }{ \sigma ^{2}}=\frac{1}{\sigma ^{2}}\int _{0}^{\infty } zf(z)P(z)\hbox {d}z \end{aligned}$$

(15)

$$\begin{aligned} = \frac{1}{\sigma ^{2}}\left( \int _{0}^{A}zf(z)P(z)\hbox {d}z + \int _{A}^{\infty }zf(z)P(z)\hbox {d}z\right) \end{aligned}$$

(16)

solving Eq 16, taking first part

$$\begin{aligned} =\int _{0}^{A}zf(z)P(z)\hbox {d}z \end{aligned}$$

(17)

Putting in f(z) and P(z) into the equation from 13 and 14

$$\begin{aligned} =\int _{0}^{A}z z \frac{2z}{\sigma ^2}\text {e}^{\frac{-z^{2}}{\sigma ^{2}}}\hbox {d}z=\int _{0}^{A}\frac{z^2}{\sigma ^2}\text {e}^{\frac{-z^{2}}{\sigma ^{2}}}2z\hbox {d}z \end{aligned}$$

(18)

Evaluating the integral gives

$$\begin{aligned} \int _{0}^{A}zf(z)P(z)\hbox {d}z= & {} \left[ {\sigma ^{2}}\text {e}^t(t-1)\right] _{0}^{A} \end{aligned}$$

(19)

$$\begin{aligned}= & {} \left[ {\sigma ^{2}}\text {e}^{-\frac{{z}^2}{\sigma ^2}}\left( -\frac{z^2}{\sigma ^2}-1\right) \right] _{0}^{A} \end{aligned}$$

(20)

$$\begin{aligned}= & {} \left[ -\text {e}^{-\frac{z^2}{\sigma ^2}}\left( {z}^2+{\sigma ^2}\right) \right] _{0}^{A}\end{aligned}$$

(21)

$$\begin{aligned} \int _{0}^{A}zf(z)P(z)\hbox {d}z= & {} -\text {e}^{-\frac{A^2}{\sigma ^2}}\left( {A}^2+{\sigma ^2}\right) +\sigma ^{2} \end{aligned}$$

(22)

Solving the second part from 16

$$\begin{aligned} \int _{A}^{\infty }zf(z)P(z)\hbox {d}z= & {} \int _{A}^{\infty }z(2A-z)\frac{2z}{\sigma ^2}{e}^\frac{-z^2}{\sigma ^2}\hbox {d}z \end{aligned}$$

(23)

$$\begin{aligned}= & {} \int _{A}^{\infty }2Az\frac{2z}{\sigma ^2}{e}^\frac{-z^2}{\sigma ^2}\hbox {d}z - \int _{A}^{\infty }z^2\frac{2z}{\sigma ^2}{e}^\frac{-z^2}{\sigma ^2}\hbox {d}z \end{aligned}$$

(24)

Evaluating Eq. 24 gives

$$\begin{aligned} =2{A}^2 {e}^{\frac{-A^2}{\sigma ^2}} + {A\sigma }\sqrt{\pi } \hbox {erfc}\left( \frac{A}{\sigma }\right) - {e}^{\frac{-A^2}{\sigma ^2}}\left( A^2 + \sigma ^2\right) \end{aligned}$$

(25)

Using Eqs. 22 and 25 in Eq. 16

$$\begin{aligned} \alpha= & {} \frac{1}{\sigma ^2}\left( -\text {e}^{-\frac{A^2}{\sigma ^2}}\left( {A}^2+{\sigma ^2}\right) +\sigma ^{2} + 2{A}^2 {e}^{\frac{-A^2}{\sigma ^2}} \right. \nonumber \\&\left. + {A\sigma }\sqrt{\pi } \hbox {erfc}\left( \frac{A}{\sigma }\right) - {e}^{\frac{-A^2}{\sigma ^2}}\left( A^2 + \sigma ^2\right) \right) \end{aligned}$$

(26)

$$\begin{aligned} \alpha= & {} 1 -{2}\text {e}^{-\frac{A^2}{\sigma ^2}} + \frac{A}{\sigma }\sqrt{\pi } \hbox {erfc}\left( \frac{A}{\sigma }\right) \end{aligned}$$

(27)

Using clipping ratio $\gamma$ = A/$\sigma$,

$$\begin{aligned} \alpha =1 -{2}\text {e}^{-\gamma ^2} + \gamma \sqrt{\pi } \hbox {erfc}(\gamma ) \end{aligned}$$

(28)

Equation 28 is the distortion factor for IWrap.