Transient Nonlinear Figures of Merit forWireless Communications

Jose Pedro Borrego1,2, Nuno Borges Carvalho1

1Instituto de TelecomunicacoesUniversidade de Aveiro

3810-193 Aveiro, PortugalTel: +351 234377900 Fax: +351 234377901 e-mail: nbcarvalho@ua.pt

2ANACOM - Autoridade Nacional de ComunicacoesCentro de Monitorizacao e Controlo do EspectroAlto do Paimao - 2730-216 Barcarena, Portugal

Tel: +351 214348500 Fax: +351 214348590 e-mail: jose.borrego@anacom.pt

Abstract—This paper presents a new view on typicalfigures of merit for nonlinear distortion evaluation, mainlyACPR, for new wireless communication systems, that arebased on time switched signals. Those systems are timeswitched, either due to the standard specifications as TDDsystems, or due to new forms of radio communications ascognitive radio technologies.The most significant differences between conventional tran-sient nonlinear figures of merit are explained, as well as,possible ways on how to measure them. Finally, this workis concluded with some comments upon future standards.

Index Terms—Nonlinear Figure of Merit, Cognitive Ra-dio, ACPR.

I. INTRODUCTION

Nowadays, wireless communications systems com-monly transmit/receive RF signals which are not timeconstant. Most of them are, indeed, pulsed or switched,not only due to power consumption restrictions, but alsodue to new emerging opportunistic paradigms such ascognitive radio. For instance, the GSM technology isinherently time varying, either for being TDD (TimeDivision Duplex) based, or for using circumstantiallyFH (Frequency Hopping) schemes. But many othertransient signals or modulations could be identified:OFDM (Orthogonal Frequency Division Multiplexing),QAM (Quadrature Amplitude Modulation) associated tochanges in amplitude of signal, a CDMA (Code DivisionMultiple Access) signal that is not constantly transmitted,etc. In the past, spectrum assignment was technologybased, i.e. a predetermined band was allocated to aspecific radio service, and all technical parameters, suchas modulation, bandwidth, transmitted power, etc. werevery well known in advance. This had the advantage offoreseeing, with a great degree of confidence, the behav-ior of a certain radio system, making easier the spectrumplanning processes. Nevertheless, the most recent trendson the spectrum policies tend to achieve a more flexiblespectrum usage, adopting, from the very beginning, theeminent principle of ”Technological Neutrality”. Accord-ing to this, some radio bands, in particular those oneswhich were or will be released as consequence of the”Digital Dividend”, can be used in a non-discriminatorybasis, being allowed the introduction of any kind of

technology inside. It is up to the telecommunicationsoperator to choose, at any time, the most suitable tech-nology or modulation scheme to assure the provisioningof its service in accordance with the required degree ofquality. As can be seen, in the limit, a wide range oftransient RF signals easily share the same or adjacentchannels, with a strong potential of interference, arising,for instance, from nonlinear distortion phenomena. Thisfact impose that today wireless communication systemsdemand for time switching signals, being those systemsbased on Time Division Duplex [1], TDD, on powerconsumption needs, or more futuristic paradigms likecognitive radio, including the opportunistic radio ap-proaches. On the other side nonlinear distortion figures ofmerit as Adjacent Channel Power Ratio, ACPR, has beenheavily used for nonlinear distortion characterization [3]of nonlinear devices, when excited by modulated RFsignals. Nevertheless, those figures are based on typicalmeasurement approaches, usually returned by traditionalspectrum analyzers, assuming that the signals, underevaluation, are time invariant. However, new wirelesssystems, based on TDD approaches, and opportunisticradios, demand for an intrinsic transient behavior, sincethe signals are continuously changing, not only becausethe system is always changing the frequency of operation,but also due to power variations. Thus, the value ofACPR that would be measured in these situations can besignificantly affected by the transient effect of frequencychanges or amplitude variations. Thus a carefully studyof this new paradigms should be definitely studied care-fully for new and emerging RF technologies.

Another very important effect arising from this tran-sient behavior will happen in new opportunist radios, thatare actually changing frequency of operation, in orderto be able to cover frequency holes during the decisionon where to transmit. In this case, the system actuallywill work in a similar way as a frequency hoppingsystem, where the center frequency changes with time,and thus the ACPR will be significantly different fromthe traditional approaches.

This paper will try to discuss these problems, andwill describe some typical problems in multi-sine and

978-1-4244-7412-7/10/$26.00 ©2010 IEEE

modulated signals nonlinear distortion.

II. ANALYSIS OF TRANSIENT SIGNALS

The most correct way to capture the nonlinear distor-tion, that has been generated, is to evaluate the outputsignal in a short time burst, corresponding to the periodwhen the signals are being transmitted. That can be doneby using a simple short time Fourier Transform, usingafterwards a spectrogram to capture all the informationalong the time scale, equation (1).

STFT ≡ X(τ, ω) =

∫ ∞−∞

x(t)w(t− τ)e−jωt dt (1)

where w(t− τ) is a sliding time window that definesthe time slot for the Fourier calculation.

This formula actually focus on an important theme,where the engineer should start to specify the spectranot only with a frequency domain grid, but also with atime domain grid.

III. MULTI-SINE ANALYSIS

Typically, the spectrum of a multi-sine signal, whencaptured by a regular spectrum analyzer, is representedin the Fig. 1. If the signal is continuous, this could bea faithful representation. However, if the signal is timevarying, this kind of analysis could not be very suitable,and possible misleading interpretations based on thosespectra should be taken in consideration.

Fig. 1: traditional view of multi-sine spectra

The joint time-frequency representation is one possibleway to evaluate such transient or non-stationary RFsignals, by using a downfall 2D plot, showing the signalenergy distribution, coded in a color scale, in the time-frequency plane. This is a very intuitive solution to visu-alize the time-spectral characteristics of the signal. Thereare several possible techniques to generate this joint time-frequency representation, being the spectrogram the mostcommon one. The spectrogram depicted in Fig. 2 givespart of the multi-sine signal evolution along the time axis,showing the sequential jumps in frequency, and now itis also possible to discriminate which carrier or carriersare part of the signal at any time instant.

It was considered a multi-sine signal with 20 tones,starting with a single tone assuming, in an ascendingorder, each frequency, and ending with all 20 tonessimultaneously.

Fig. 2: Spectrogram of a multi-sine with single tone switchingfollowed by a 20 tones multi-sine

The 2D spectrogram introduced before is, in fact, a 3Drepresentation of the signal, as the amplitude is codedusing a color scale, equivalent to the spectrogram of theFig. 3.

Fig. 3: 3D Spectrogram of a multi-sine with single tone switch-ing followed by a 20 tones multi-sine

Considering that the multi-tone signal described abovepasses through a nonlinearity y(t) = a1x(t)+a2x(t)

2+a3x(t)

3 it is possible to observe in the spectrogram ofthe Fig. 4 the distortion generated by y(t) when all 20tones are present, because new adjacent spectral contentappear now surrounding the 20 tones input signal.

A 3D view (Fig.5) gives a better illustration of the re-sponse provided by the nonlinear system, distinguishingalso the second harmonics components of distortion.

From the figure it is possible to see that the multi-sinecan be actually built as a real summation of sines coinci-dent in time or as a sine that is hopping from frequencyto frequency with time, either way the nonlinear outputdistortion will be completely different, and in the case ofthe continuous over time summation of sines, the in-banddistortion is observed, while in the case of the frequencyhopping no in-band distortion is visible.

Fig. 4: Spectrogram of the output of a multi-sine with singletone switching followed by a 20 tones multi-sine whentraversing a nonlinear system

Fig. 5: 3DSpectrogram of the output of a multi-sine with singletone switching followed by a 20 tones multi-sine whentraversing a nonlinear system

IV. MODULATED SIGNALS

Consider now, for example, the case of a 256-QAMsignal, that is continuously transmitted, and another casewhere the same signal is not continuously on air, beingswitched according to the need for transmitting or not.Each case will have a different response over time.Actually, the 256-QAM switched signal, considered inthe latter case, results from the multiplication of the input256-QAM signal with a square wave. In the frequencydomain, this corresponds to convolve the original signalwith a sinc shaped waveform. Fig. 6 presents the com-parison between the two spectrum waveforms, for thecontinuous and the switched 256-QAM. As can be seen,in terms of spectrum, they are quite coincident, and theintegrated power is, actually, almost the same.

If both signals traverse a nonlinearity, for instance y =G[x(t) + 0.1x(t)3], where G is the gain of the device.

The nonlinear distortion generated by the device willbe as the one presented in Fig. 7.

The Fig. 7 shows that the switched version has astronger value of ACPR. This suggests that when atypical ACPR measurement is preformed, then a worstdistortion will be captured, which is not necessarily right,but can be a miss interpretation of the measurement

Fig. 6: Timed switched and continuous 256-QAM input signal

Fig. 7: Timed switched and continuous 256-QAM output sig-nal, after traverse a nonlinear Power Amplifier

system. Actually, the statistical behavior of the differentsignals are presented on Fig. 8 and Fig. 9, and can clearlyshow the main differences between those signals.

Fig. 8: Statistical behavior of the 256-QAM continuous signal

Nevertheless if the spectra is measured using thespectrogram as presented before, the results are muchmore interesting, and an improvement in the figures ofmerit understanding is much more clear now, Fig. 10 andFig. 11.

The spectrogram of the switched signal is depicted inFig. 12, where the switching behavior is now seen.

Fig. 9: Statistical behavior of the 256-QAM time switchedsignal

Fig. 10: Spectrogram of a QAM input signal

Fig. 11: Spectrogram of a QAM ouput signal

Fig. 12: Spectrogram of a switched QAM signal

V. CONCLUSIONS

In this paper a new light over switched signal figuresof merit measurements was given, and the concept ofspectrogram use for nonlinear evaluation was presented,much more work is now needed for opening the pathsfor future emerging technologies FOM’s.

REFERENCES

[1] G. J. Mazzaro, M. B. Steer, K. G. Gard and A. L. Walker,”Response of RF Networks to Transient Waveforms: Interferencein Frequency-Hopped Communications”, IEEE Transactions onMicrowave Theory and Techniques, Vol. 56, No. 12, Dec. 2008.

[2] M. R. Portnoff, ”Time-Frequency Representation of Digital Signalsand Systems Based on Short-Time Fourier Analysys”, IEEE Trans-actions on Acoustics, Speech, and Signal Processing, Vol. ASP-28,No. 1, Feb. 1980.

[3] J. C. Pedro, and N. B. Carvalho, Intermodulation Distortion inMicrowave and Wireless Circuits, Artech House, Inc., 1st Edition,2003.

[4] V. C. Chohan, and J. K. Fidler, ”Generalised transient responseof bandpass transfer functions to FSK and PSK-type signals”,Electronics Letters, Vol. 9, No. 14, Jul. 1973.

[5] H. J. Blinchikoff, and A. I. Zverev, Filtering in the Time andFrequency Domains, Raleigh, NC: SciTech, 2001.

[6] H. Hartley, ”Transient response of narrow-band networks to nar-rowband signals with applications to frequency-shift keying”,IEEE Transactions on Communication Technology, Vol. 14, No.4, Aug. 1966.

[7] H. Salinger, ”Transients in frequency modulation”, ProceedingsIRE, Vol. 30, No. 8, Aug. 1942.