A Comprehensive Analysis of the Clipping Effects on Signals with Different Statistical Patterns

Pedro Miguel Cruz and Nuno Borges Carvalho

Institute of Telecommunications – University of Aveiro, DETI - UA, Aveiro, Portugal Abstract — This paper presents a detailed study on the

impact of two different types of clipping, soft- and hard-clipping, effects on several signals that present different probability density functions (PDFs), and thus, different peak-to-average power ratios. A mathematical approximation to calculate the integrated output power based on the input signal PDFs will be shown. Afterwards, the clipping effects will be evaluated in terms of fundamental signal compression, adjacent channel power leakage, and in-band signal-to-noise ratio, which are very important figures of merit regarding the actual communications systems.

Index Terms — Clipping function, nonlinear distortion, probability density function, signal-to-noise ratio.

I. INTRODUCTION

New wireless technologies are demanding for improved signal modulation formats such as, quadrature amplitude modulation (QAM) on top of orthogonal frequency division multiplexing (OFDM), which allow an increase in bit rate speeds simultaneously with a reduction or at least without an increase in bandwidth. Unfortunately, these new modulation formats originate signals with high values of peak-to-average power ratio (PAPR) that degrade completely the overall system, since they become to work closer of nonlinear regions.

On the other side, recent advances are permitting the implementation of very low power devices, having no battery back-up, to operate by means of power being transmitted through the wireless environment, which is then converted into DC voltage at the battery-less device terminals. This fact imposes that, in the transmitter side, a careful attention should be taken in order to design an output stage that send as much RF signal power as possible in a way to power up the passive device as far as possible. In conjunction with the RF signal generation a signal with data information should be added to configure or detect the battery-less device. As an example, in [1] and [2], the authors are using multisine excitation signals and chaotic signals on top of data information, respectively, to extend the reading range of commercial RF identification readers.

In the abovementioned scenarios, it could happen that the values of PAPR in the created RF signals become very high, close to 20dB in certain cases, forcing the nonlinearity of the wireless system to have a strong clipping effect in this type of signals, when one want to maximize the transmitter operating range to provide efficiency and output power as feasible.

Those clipping functions can be effectively originated in the power amplifier stage when approaching saturation or in

analog-to-digital converters (ADCs) when they are drove above the configured reference voltage. The amplifier version of the clipping can be considered a sort of soft-clipping, since it clips the signal in a smooth and continuous way, while in the case of the ADC it is considered hard-clipping, since the incoming signal is clipped in an almost instantaneous way.

This paper will firstly analyze the two usual types of clipping and its impact on the behavior of the output signals. Then, in section II, it is explained why differently distributed signals do not compress to the maximum output integrated power at same rate/slope, based in some probability density function (PDF) concepts and in some corroborating simulations. Section III will show some simulated results of different statistically distributed multisine signals and a 16-QAM modulated signal when they are applied in both soft-clipping (hyperbolic tangent) and hard-clipping situations. Then, several figures of merit are calculated, including adjacent channel distortion and co-channel distortion representative of adjacent channel power ratio and in-band signal-to-noise ratio, respectively. Finally, some conclusions will be drawn.

II. MATHEMATICAL EXPLANATION OF CLIPPING EFFECTS

In a broad view, current electronic nonlinear components can be separated in two groups when considering the saturation or clipping constraint. One that is termed hard-clipping and is imposed by a transfer function that limits the output signal right after the input signal traverses a certain threshold and another that is termed soft-clipping, in which the input signal starts to clip smoothly with the increase of the input signal power. Representative curves of each kind of clipping format are shown in Fig. 1.

It is known that in the case of soft-clipping does always exists some small-signal distortion, while in the case of hard-clipping no small-signal distortion is detectable, being that limited by the noise floor level.

Fig. 1 – Soft-clipping function (left) and hard-clipping (right).

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In order to follow with this analysis several multisine having different amplitude and phase arrangements to produce different statistically-shaped signals have been designed, which includes a uniform and normal shaped multisines and a phase aligned one (constant phase). Additionally, a more realistic 16-QAM modulated waveform has also been considered. Table I and Fig. 2 illustrates the details of each realized signal through the individual PDF and associated PAPR measure.

Then, the integrated output power has been determined by means of the integration (sum) of the created time-domain signal. Fig. 3 shows the obtained results for both multisines and for a pure square wave function. The constant-phase multisine starts to clip at a lower input power due to the fact that its higher voltage peaks are cut sooner than the low-value peaks of the uniform signal. However, as the input power increases, the uniform multisine output power tends rapidly to a saturated value. This is due to the fact that its time-domain waveform characteristic moves rapidly to a square waveform once clipping occurs, which represents the upper bound of the integrated output power as expected.

This can be mathematically validated by considering that the integrated input power can be calculated by the second-

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Fig. 2 – Probability density functions for each of the created signals.

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Fig. 3 – Output integrated power as a function of the input integrated power applied to a hyperbolic tangent (clipping function) of a square wave and two of the designed multisines (uniform and constant phase). In addition, the output integrated power calculated by means of equation (6) is presented for a set of input power levels.

TABLE I: CALCULATED PAPR FOR EACH EXCITATION SIGNAL Multisine Distribution PAPR Value [dB] Uniform 2.72 Normal / Gaussian 7.98 Constant Phase 19.93 16-QAM 5.89

order moment of the probability density function (PDF) of the input signal.

∫+∞

∞−= dxxpdfxPIN ).(.2 (1)

Then, the integrated output power will be given by:

∫+∞

∞−= dyypdfyPOUT ).(.2 (2)

wherein one can apply the property that the output PDF of a memoryless nonlinearity can be calculated by:

dxdy

xpdfypdf )()( = (3)

This property arises immediately from the characteristic function evaluation. For instance, considering the case of a soft-clipping function represented by a hyperbolic tangent, tanh(x), using (3) and substituting in (2), it can be written:

∫+∞

∞−= dy

dxdy

xpdfxPOUT .)(.)tanh( 2 (4)

And, based in the derivative of the inverse function theorem and in the definition of differential, it can be said that:

dxdy

dxdy

dxdydy

dx =⇒= .11 (5)

So, equation (4) can now be solved and leads to:

∫+∞

∞−= dxxpdfxPOUT ).(.)tanh( 2 (6)

Because tanh(x)2 is zero at x = 0, and tends asymptotically to one, it is expected that the signals presenting a higher-valued PDF near zero will have a different lower rate slope when the clipping starts to occur, as can be seen from Fig. 3. The small differences observed in Fig. 3 for the constant-phase multisine between the simulated case and the calculated by means of equation (6) is apparently due to some misleading calculation on the obtained PDFs at the different power levels.

Similar conclusions would be achieved for the hard-clipping case, in which by the summation rule in the integration, a piecewise integral function could be employed.

Even though it is quite relevant to extrapolate the integrated output power based on the input signal PDF for a given nonlinear function, it would have much practicality the extraction of information on the truly signal quality, say in-band signal-to-noise ratio and adjacent channel leakages. Next section will exactly deal with this issue.

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Fig. 4 – Simulated output powers for different excitation signals when passing a soft-clipping (top) and a hard-clipping (bottom) functions. From left to right it is shown the fundamental output power, co-channel distortion, and adjacent channel leakage, respectively.

III. STATISTICAL ANALYSIS OF THE NONLINEAR DISTORTION

This section will pursue an approach based on the statistical properties of signals to extract meaningful information, when systems representing soft- and hard-clipping behaviors are excited by differently distributed signals that present quite different time-domain statistics, and thus PAPR values.

This theory has been proposed by Bendat [3] and was intensely utilized in [4, 5] to study the nonlinear distortion of a PA when excited by a CDMA signal, which allows the extraction of the autocorrelation functions for the input, output and cross values. Based on this strategy, the previously created excitation signals have been averaged over 100 simulated timed windows to determine the adjacent-channel and co-channel distortions.

The obtained simulated results are presented in Fig. 4 in terms of fundamental signal compression, co-channel distortion power, and adjacent-channel leakage power for a sweep of the input power. There, it can be observed hugely diverse responses appearing in-band and out-of-band for the different applied signals.

The behavior observed at low power inputs on the 16-QAM modulated signal for the adjacent channel distortion can be explained by some inherent leakage present in the original signal.

IV. CONCLUSION

In this paper a complete explanation about the signal clipping behavior has been given by means of a mathematical

explanation and simulations for the calculation of integrated powers and from autocorrelations to provide a meaningful evaluation of the signal transmission quality. We have also observed that the statistics of the input signal have a strong impact on the co-channel and adjacent channel distortions.

ACKNOWLEDGEMENT

The authors would like to acknowledge the support of this work by the Portuguese Science and Technology (FCT) under the project EXCL/EEI-TEL/0067/2012 (CREaTION) and by the Post-Doc grant (SFRH/BPD/92452/2013) given to the first author, and to the COST action IC1301 Wireless Power Transmission for Sustainable Electronics (WiPE).

REFERENCES

[1] A.J.S. Boaventura, and N.B. Carvalho, “Extending Reading Range of Commercial RFID Readers,” IEEE Trans. Microwave Theory and Techniques, vol. 61, no. 1, pp. 633-640, Jan. 2013.

[2] A. Collado, and A. Georgiadis, “Improving Wireless Power Transmission Efficiency using Chaotic Waveforms,” IEEE MTT-S International Microwave Symposium Digest, June 2012.

[3] J. Bendat, Random Data: Analysis and Measurement Procedures, 3rd ed., New York, Wiley, 2000.

[4] V. Aparin, “Analysis of CDMA Signal Spectral Regrowth and Waveform Quality,” IEEE Trans. Microwave Theory and Techniques, vol. 49, no. 12, pp. 2306-2314, Dec. 2001.

[5] R. Santos, N.B. Carvalho, and K.G. Gard, “Characterization of SNDR Degradation in Nonlinear Wireless Transmitters,” International Journal of RF and Microwave Computer-Aided Engineering, vol. 19, no. 4, pp. 470-480, July 2009.