Distortion mitigation in cognitive radio receivers

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Distortion Mitigation in

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DISTORTION MITIGATION IN COGNITIVE

RADIO RECEIVERS

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Chairman:

Prof. Dr. P.M.G. Apers University of Twente Secretary:

Prof. Dr. P.M.G. Apers University of Twente Promotor:

Prof. Dr. ir. B. Nauta University of Twente Assistant Promotor:

Dr. ing. E.A.M. Klumperink University of Twente Members:

Prof. Dr. ir. F.E. van Vliet TNO / University of Twente Prof. Dr. Henrik SjÖland Lund University

Prof. Dr. ir. C.H. Slump University of Twente

Prof. Dr. ir. P.G.M. Baltus Eindhoven University of Technology

This research is supported by the Dutch Technology Foundation STW, which is part of the Netherlands Organisation for Scientific Research (NWO) and partly funded by the Ministry of Economic Affairs (08081).

CTIT Ph.D. Thesis Series No. 15-348

Centre for Telematics and Information Technology P.O. Box 217, 7500 AE

Enschede, The Netherlands.

Title: Distortion Mitigation in Cognitive Radio Receivers ISBN: 978-90-365-3846-6

ISSN: 1381-3617 (CTIT Ph.D. Thesis Series No. 15-348 ) DOI: 10.3990/1.9789036538466

http://dx.doi.org/10.3990/1.9789036538466

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DISTORTION MITIGATION IN COGNITIVE RADIO

RECEIVERS

DISSERTATION

to obtain

the degree of doctor at the University of Twente, on the authority of the rector magnificus,

prof.dr. H. Brinksma,

on account of the decision of the graduation committee, to be publicly defended

on Friday the 20th March 2015 at 12:45

by

DLOVAN HOSHIAR MAHROF

born on the 26th of December 1973 in Al-Sulaimaniya, Iraq

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Promotor: Prof. Dr. ir. Bram Nauta

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To Almas,

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Abstract

The exponential increase of wireless communication increasingly leads to spectrum congestion. Attempts are being made to increase RF spectrum utilization efficiency by introducing Cognitive Radio (CR) concept. A CR tries to intelligently solve the congestion problem via Dynamic Spectrum Access (DSA), i.e. determine which frequencies are temporarily and locally free, and exploit this free spectrum. Especially in the TV broadcasting bands below 1 GHz, such CR possibilities are being explored. Ideally, a CR receiver should be able to operate directly adjacent to the primary service users, e.g. Digital TV channels, which use high power levels and often leave the adjacent channels unused. Under such conditions no or very low up-front filtering of the interferer is possible. Consequently, a CR receiver must tolerate the existence of strong interferers, i.e. have a very high linearity front-end.

This thesis examines CR receiver linearity requirements and explores techniques that mitigate distortion. The thesis starts by analyzing the linearity requirements for DSA. As the level and the spectral location of the interferers can change instantaneously per location, it is relevant to monitor the spectrum and find suitable opportunities for communication. The analysis is applied to a channelized spectrum in which a number of interferers exist, and a CR tries to exploit any free channel. It is derived how the CR linearity requirement depends not just on the power levels of the interferers but also on their spectral locations around the desired CR frequency/channel. It is shown that the linearity requirement can be relaxed by tens of dBs levels of 3rd order InterModulation product (IM3). The analysis also exploits the prediction of

the distortions in different channels for DSA. This prediction algorithm is denoted here as DPrA (i.e. Distortion Prediction Algorithm). It processes the spectrum sensing information about the power level and the spectral locations of the interferers to derive the linearity requirements for each potential CR channel. Based on this information, a CR can choose the most suitable channel compatible with its linearity capability.

A receiver with better linearity can work under worse interference conditions, and hence maximizing linearity of a CR receiver is important. To increase the linearity of a CR receiver, CMOS receiver front-ends with high linearity are explored. Receivers that exploit linear V-I conversion at RF, followed by passive down-mixing and an OpAmp-based Transimpedance Amplifier at baseband, show high linearity potential. However, it is shown that due to nonlinearity and finite gain in the OpAmp, the virtual ground

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is imperfect, resulting in distortion currents. The concept of a negative conductance is proposed to cancel such distortion currents. Through a simple intuitive analysis, the basic operation of the technique is explained. By mathematical analysis the optimum negative conductance value is derived and related to feedback theory. It is shown that a finite value of the negative conductance is needed to make the feedback loopgain theoretically approach infinity, which is practically not easily possible by increasing the gain of the OpAmp block. The technique is applied to linearize an RF receiver and a prototype is implemented in 65nm technology. Measurement results show an increase of the in-band IIP3 from 9dBm

to >20dBm, and IIP2 from 51 to 61dBm, at the cost of an increase of the noise figure from 6 to 7.5dB and

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Samenvatting

De exponentiële groei van draadloze communicatie leidt steeds meer tot congestie van het spectrum. Pogingen zijn ondernomen om het gebruik van Radio Frequentie (RF) spectrum efficiënter te maken door de invoering van Cognitieve Radio (CR) concept. Een CR probeert de congestie-probleem via Dynamic Spectrum Access (DSA) op te lossen. DSA bepaalt welke frequenties tijdelijk en lokaal vrij zijn om die toe te wijzen aan CR. De CR operatie mogelijkheden wordt vooral onderzocht in de TV uitzending banden onder 1 GHz. In ideale geval moet een CR ontvanger kunnen ontvangen direct in de kanalen naast de primaire servicegebruikers, zoals digitale TV zenders, die hoog vermogen sturen en vaak verlaten de aangrenzende kanalen ongebruikt. Onder dergelijke omstandigheden zal er geen of zeer lage pre-filteren van digitale TV signalen plaatsvinden. Bijgevolg moet een CR ontvanger het bestaan van sterke interfereren tolereren. Dat wil zeggen dat de CR ontvanger een zeer hoge lineariteit moet hebben.

Deze thesis onderzoekt de benodigde eisen aan de CR ontvanger lineariteit en verkent de technieken, die de gevolge vervorming/distorsie verzachten.

Het proefschrift begint met het analyseren van de lineariteit voor DSA. Het vermogen niveau en de spectrale locatie van de interferentie signalen verandert voortdurend per elke locatie, daarom is het relevant om het spectrum voortdurend te meten en de geschikte vrij frequenties te bepalen voor CR communicatie. De analyse wordt toegepast op een gekanaliseerd spectrum waarin een aantal kanalen zijn bezet met interferentie signalen. Een CR probeert om van die vrij kanalen te benutten. Het is afgeleid hoe afhankelijk de CR lineariteit is van niet alleen de vermogen van de interferentie signalen maar ook van hun spectrale locaties rond de gewenste CR frequentie/kanaal. Het is aangetoond dat de lineariteit met tientallen dBs niveaus van 3e orde intermodulatie product (IM3) kan worden versoepeld.

De analyse exploiteert het gebruik van de voorspelling van de verstoringen in verschillende kanalen voor DSA. Deze voorspelling algoritme wordt ontwikkelt en aangeduid als DPrA (d.w.z. Distortion Prediction Algorithm) hier. Het verwerkt de spectrum informatie over het vermogen en de spectrale locatie van de interferentie signalen voor het afleiden van de lineariteit eisen voor elke potentiële CR kanaal. Op basis van deze informatie, kan een CR het meest geschikt kanaal voor zijn lineariteit kiezen.

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een CR ontvanger te vergroten worden CMOS ontvanger met hoge lineariteit onderzocht. Ontvangers die gebruikmaken van lineaire spanning naar stroom conversie bij RF, gevolgd door het passieve frequentie translatie en een OpAmp gebaseerde transimpedantie versterker op lage frequentie, toont hoge potentiële lineariteit. Het is aangetoond dat als gevolg van niet lineariteit en beperkte versterking factor in de OpAmp, de virtueel grond onvolmaakt is waarmee distorsie stromingen resulteert. Het concept van een negatieve weerstand wordt voorgesteld om die distorsie stromingen te annuleren. Door middel van een eenvoudige intuïtieve analyse wordt de basiswerking van het techniek uitgelegd. Door wiskundige analyse is de optimale waarde van de negatieve weerstand afgeleid en gerelateerd aan de terugkoppeling theorie (Regel Techniek). Het is hier bewezen dat een eindige waarde van het negatieve weerstand is nodig om de versterking loop van de terugkoppeling oneindig te maken (theoretisch gezien), die praktisch door het verhogen van de OpAmp versterking niet gemakkelijk is. De techniek wordt toegepast om de lineariteit van een RF IC chip ontvanger te verhogen. Een prototype is geïmplementeerd in 65nm technologie. Meetresultaten tonen een toename van de IIP3 in-band van

9dBm naar > 20dBm, en IIP2 van 51 naar 61dBm, ten koste van een ruis toename van 6 naar 7.5dB en

een extra 10% vermogen gebruik. De chip bereikt een dynamic range (d.w.z. Spurious-Free Dynamic Range) van 85dB in 1MHz.

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List of Main Abbreviations

RF: Radio Frequency

FCC: Federal Communications Commission

DTV and DVB-T: Digital TV broadcasting and Digital Video Broadcasting Terrestrial DSA: Dynamic Spectrum Access

SR/SDR/CR: Software Radio/Software Defined Radio/Cognitive Radio ADC/DAC: Analog-to-Digital Convertor/Digital-to-Analog Convertor OFDM: Orthogonal Frequency Division Multiplexing

QPSK: Quadrature Phase Shift Keyed SAW: Surface Acoustic Wave

NF: Noise Figure of the receiver IIP3: 3rd order Input Intercept Point

IIP2: 2nd order Input Intercept Point

IM3: 3rd order InterModulation product

IM2: 2nd order InterModulation product

XM3: cross-modulation product

DPrA: Distortion Prediction Algorithm LPF/ BPF: Low Pass Filter/ Band Pass Filter

X/⊗: mathematical operation of Multiplication operation/ Convolution operation OpAmp: Operation Amplifier

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Chapter 1. Introduction

This chapter introduces the motivations and the main research questions regarding the importance of distortion mitigation in Cognitive Radio. Section 1.1 begins with Cognitive Radio motivation to solve the Frequency spectral utilization efficiency via Dynamic Spectrum Access. Afterwards in section 1.2, the Cognitive Radio receiver hardware imperfections are discussed. Then the research scope summary is presented in section 1.3 with the main research question. The chapter ends with the outline of the thesis in section 1.4.

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1.1 Cognitive Radio Motivation

The wireless communication revolution has led to a tremendous amount of applications and services in our everyday life. Examples are the early success of GSM cellular telephony, DECT telephones at home, Bluetooth technology for direct data exchange between devices, Wi-Fi internet access for local area high data rate radio operation, GPS navigation, and currently the rapid expansion of wireless video and cloud service access. To each application, government bodies allocate a specific radio frequency range in the spectrum, called frequency band. Each band is derived by specific standards. Figure 1.1 shows the frequency allocation made by the National Telecommunications and Information Administration (NTIA [1]) in the USA. The Radio Frequency (RF) spectrum between 3kHz – 300GHz is divided in band(s) per application, assigned through administrative licensing.

Given the crowdedness of the spectrum plot of Figure 1.1 alongside with the observation that the demand for wireless communication is growing exponentially, shortage of spectrum is foreseen, and in this regard the chairman of the Federal Communications Commission (FCC) remarked in 2010 [2]: “Our data shows there is a looming crisis. We may not run out of spectrum tomorrow or next month, but it’s coming and we need to do something now”. In IEEE Spectrum [3], this problem is named “The great

spectrum famine”.

Different independent measurement campaigns (e.g. [4], [5]) found that most frequency bands were not occupied continuously in time, i.e. spectral utilization efficiency is often poor. Although cellular network bands are used intensively in most parts of the world, this is not true for many other frequency bands. Measurement shows that less than 20% of the RF spectrum is actively used at any given time and place [6]. Figure 1.2 presents an example of a spectrum measurement [4] for a part of the Digital TV (DTV)

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broadcasting (Appendix A): channels 52 – 69, where each channel has 6MHz bandwidth, at the Republican National Convention, New York City, USA.

Figure 1.2: Spectrum occupancy measurements [4], Location: Republican National Convention, New York City, USA In an effort to improve the spectral utilization efficiency, FCC allows since 2008 [7] unlicensed (i.e. secondary) users to opportunistically use TV broadcast channels that are licensed for TV broadcasters (i.e. primary users), but are temporally locally unused.

Such free channels or bands are referred to as “white spaces” [7] or “spectrum holes” [8]. Free channel can for instance be utilized as shown in Figure 1.3, referred to as “Dynamic Spectrum Access (DSA)” [9], given the dynamic nature of this spectrum use. More efficient use of such DTV frequency bands will be a key target of this thesis.

The DSA concept of operation poses several requirements on the radio hardware:

1. Flexibility and reconfigurability of the radio hardware to allow for dynamic switching between different white spots across the frequency spectrum.

2. Supporting different standards, specified for different unoccupied primary users spots. 3. Sensing spectrum to detect the white spot instantaneously at any location.

Software (Defined) Radio towards Cognitive Radio:

Software Radio (SR) may achieve the ultimate level of flexibility, reconfigurability and can support all different standards. A general schematic of a SR transceiver is shown in Figure 1.4. The SR involves implementing all of the radio functionality in the software of the digital back-end (i.e. Digital Signal

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Processing DSP) as described by Mitola [10]. The propagating Radio Frequency (RF) signal, with frequency fC, is analog. This means that both the transmitted and received RF signals, via the antenna,

are also analog. Therefore a Digital-to-Analog Convertor (DAC) in the transmitter and an Analog-to-Digital Convertor (ADC) in the receiver are required. The focus of this thesis is on the hardware requirements of the receiver side. The transmitter part was the scope of another partner Ph D project [11] inside the same research program at the University of Twente. A true SR receiver is hard to implement, because of the tough requirements on the ADC resolution vs power consumption for mobile wireless applications [12].

A solution that is more feasible is a radio receiver where part of the flexibility and configurability is achieved by flexible analogue hardware instead of by software as shown in Figure 1.5, which is called the Software Defined Radio (SDR) receiver.

The RF band filter, the channel filter and the clock frequency to the mixer, which provide the frequency down shift (or conversion) functionality, are all flexible and reconfigurable. The bandwidth of the RF Amplifier can be wide to cover all the required bands, on condition that the RF filter

has enough out of band interferer rejection, so that the interferer will not distort the desired signal. Extending the SDR and combining it with the Spectrum Sensing provides a radio that is aware of its frequency environment and autonomously adjusts its communication parameters to fully realize DSA (see Figure 1.3). This concept is called a Cognitive Radio (CR) [13]. The concept of CR was first proposed by Joseph Mitola in a seminar at the Royal Institute of Technology in Stockholm (KTH) in 1998 and published in an article by Mitola and Gerald Q. Maguire, Jr. in 1999 [14] .

In summary, the CR concept promises to increase the spectrum utilization efficiency via DSA. The hardware physical implementation of the CR is based on SDR receiver plus Spectrum Sensing device. In

Figure 1.4: Software Radio receiver front-end

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requirements will be explained. This thesis aims to mitigate the linearity requirements challenges by benefiting from the presence of spectrum sensing device, which is assumed to be on board.

1.2 CR receiver hardware imperfections

Imperfections of practical receivers decrease the quality of the received desired signal, via distortion and noise [15].

The DTV interferers will often lead to distortion in practical receivers with a limited degree of up-front filtering and linearity, which will be explained in Chapter 2. In traditional narrowband RF receivers, out of band strong interferers can greatly suppressed by a Surface Acoustic Wave (SAW) RF band filter (a fixed center frequency version of the RF band filter shown in Figure 1.5) after the antenna. This will greatly reduce the distortion level of the received signal at the ADC input, i.e. the required out of band linearity of the receiver will be also reduced. The SAW RF filter with a fixed center frequency does not help to immune the CR receiver from the DTV interferers that actually exists in-band, hence the in-band linearity becomes a critical specification for CR receiver. The ultimate solution is a flexible and reconfigurable RF channel filter with high rejection (i.e. very high order filtering) of the interferers from the adjacent channel(s). Note that guard channels are not desired to maximize the spectral utilization efficiency. Such RF channel filters are very difficult to be implemented, especially on-chip. Different implementations of flexible and reconfigurable RF filters are presented as example [16] and [17]. However their application as flexible and reconfigurable RF channel filter in the DTV band is bounded by their limited adjacent channel(s) interference rejection ratio. As a consequence of the poor rejection performance of RF channel filter, the in-band linearity of the receiver must be increased to tolerate the received DTV interferers that actually exists in-band.

Any receiver contributes noise. The noise contribution, which is quantified by the Noise Figure (NF), increases the noise floor. Hence the sensitivity level of the receiver that must be above the noise floor by the minimum SNR requirement will decrease. Reducing the radio range (i.e. cell size) and/or increasing the transmitted signal power helps always in reducing the NF requirement of the receiver, note that this is not the case with linearity requirement, hence distortion becomes more important and noise less, especially for the near future map of communication, which develops towards small cells.

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The following section summarizes the research scope and presents the main research questions of the thesis.

1.3 Research Scope summary and questions

In summary, a CR requires flexibility in receive frequency and such flexibility comes at the cost of less up-front filtering, hence distortion products will be increased, especially because the in-band linearity is normally limited. The high distortion products generated across the unoccupied DTV channels will finally limit the spectral utilization efficiency. Hence distortion mitigation for CR is important regarding the main idea of increasing the spectrum usage via DSA in the DTV spectrum band. It will be assumed that a spectrum sensing device is available on-board, and options will be explored to benefit from the presence of such a device. The main research questions are as follows:

1. How to analyze the distortion level (i.e. linearity requirement) across the unoccupied DTV band for Dynamic Spectrum Access CR radio receiver?

2. How to predict the distortion level in Dynamic Spectrum Access, given Spectrum Sensing data? 3. How to increase the linearity of CR receiver hardware, given less up-front filtering?

Finally, the CR concept of operation demands both, (RF) analog and digital signal processing that continuously interacts with each other. This makes CMOS technology to be the chosen candidate in this thesis for possible monolithic integration between those two domains. In the following section, the outline of the thesis will be discussed.

1.4 Outline of the Thesis

In this section, the outline of the thesis is presented in addressing the main research questions:

Chapter 2 investigates the first question:

Due to the existence of in-band strong DTV interferers and the nonlinearity of a radio receiver, different types of distortion products will be generated inside a CR-receiver, namely intermodulation products, cross-modulation products and self-interference products as analyzed in chapter 2. Using analysis based on a memory-less nonlinearity model, it will be shown that CR receiver is always limited by cross-modulation and self-interference products in any white spots. While the level and the spectral location

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of the intermodulation products across the white spots depends on the level and the spectral location of the DTV signals. The cross-modulation and self-interference products are typically much weaker than the intermodulation products. Thus it makes sense to monitor the level and spectral location of interferes via spectrum sensing device, and classify the white spots into two types, namely “intermodulation spots” and “intermodulation-free spots”. For the latter, the linearity requirement of a CR receiver is more relaxed compared to the intermodulation spots. The analysis will be verified by measurements.

Chapter 3 investigates the second question:

Based on the analysis in chapter 2, chapter 3 describes the development of a Distortion Prediction Algorithm (DPrA) to quantify the level and the spectral location of the distortion products across the white spots for Dynamic Spectrum Access. Then, according to the DPrA output, the CR receiver linearity requirement can be quantified for different white spots as shown in chapter 3. Hence, the CR receiver autonomously can select the most suitable white spots (i.e. unoccupied channel), taking into account knowledge of linearity performance of its own CR receiver. Note, CR receiver with better linearity performance can exploit more of the available white spots, hence higher spectral utilization efficiency (i.e. the original motivation behind DSA). Therefore, increasing the CR receiver linearity is an important target in this research.

Chapter 4 and 5 investigate the third question:

Chapter 4 focuses on increasing the linearity of CR receiver hardware. High linearity CMOS radio receivers often exploit linear V-I conversion at RF, followed by passive down-mixing and an OpAmp-based Transimpedance Amplifier at baseband. Due to nonlinearity and finite gain in the OpAmp, virtual ground is imperfect, inducing distortion currents. A negative conductance concept will be introduced that allows for cancelling such distortion currents. Through a simple intuitive analysis, the basic operation of the technique will be explained. By mathematical analysis the optimum negative conductance value is derived and related to feedback theory. In- and out-of-band linearity, stability are also analyzed.

The technique is applied to linearize the RF receiver in chapter 5, and a prototype is implemented in 65nm technology. Measurements show an increase of in-band linearity, quantified by IIP3 and IIP2, with

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to 7.5dB and <10% power penalty. In 1MHz bandwidth, a Spurious-Free Dynamic Range of 85dB is achieved at <27mA up to 2GHz for 1.2V supply voltage.

Chapter 6 summarizes the main research work:

Finally, Chapter 6 summarizes the main conclusion, makes a list of the main contributions of this work, and provides recommendations for future work.

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Chapter 2. Cognitive Radio Receiver Linearity Requirements [18]

This chapter investigates the first research questions “How to analyze the distortion level (i.e. linearity requirement) across the unoccupied DTV band for Dynamic Spectrum Access CR radio receiver?” The DSA operation inside the DTV band means that the power level and spectral location of the DTV signals around the CR signal may instantaneously change; hence the traditional linearity analysis [15] cannot be directly applied. The analysis of this chapter decomposes the complex view of the distortion products into basic roots/types. Based on that, the linearity requirements for CR receiver are derived. The content of this chapter is structured as follows. Section 2.1 introduces the problem. Then section 2.2 describes the receiver model and nonlinearity model. The spectral location analysis of the 3rd order distortion products is done in Section 2.3. The effect of this spectral location of the distortion on the linearity requirement is analyzed in Section 2.4. Section 2.5 verifies our theoretical analysis and presents results from practical measurements. Finally, conclusions will be drawn in Section 2.6.

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2.1 Problem Definition

Cognitive Radio is a new emerging radio communication paradigm [14] aiming at improving the utilization efficiency of the scarce spectral resources. It senses unused “white spots” in the radio spectrum that is licensed to a primary user and adapts its communication strategy to use these parts while minimizing interference to the primary service.

This chapter studies requirements on the 3rd order linearity of a CR receiver for wireless applications operating in the DTV bands [7]. Cognitive Radio requires high programmability for radio transmission and reception because the position of the white spots changes dynamically with time (i.e. dynamic spectrum). Traditional radio receivers are narrowband and are typically highly dedicated to a specific RF-band, especially due to the fixed RF-filter (see Figure 2.1).

Figure 2.1: Traditional narrow-band RF receiver with one RF “Front-End”

The high out-of-band rejection of the RF-filter, usually a high-linearity highly selective Surface Acoustic Wave (SAW) filter, suppresses the interference of out-of-band interferers. Traditional radio standards define in detail how the radio-band is used, e.g. by specifying a multiple access method (TDMA, FDMA, CDMA) and blocker profiles. Consequently in-band signals are “under control”, while RF-filtering reduces the out-of-band interference. In such narrowband systems, typically third order intermodulation products (IM3) produced by nonlinearities in the receiver dominate as is shown in Figure 2.2. Both the

signal and the IM3 product will be converted to the baseband and their ratio must be higher than the

minimum Signal to Distortion Ratio that is required to demodulate the transmitted information with acceptable bit error rate.

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Figure 2.2: Corruption of the desired signal due to intermodulation products caused by two interference signals (Conventional narrow-band Radio)

For CR, the distinction between “out-of-band” and “in-band” signals more or less vanishes, as bands are no longer reserved for one or a few radio standards. Also, the power of radio signals is not restricted to a low level, as in ISM bands, in which more freedom is allowed provided low power levels are used (typically 10-100mW). If we want to maximize spectral efficiency in the TV-bands, we like to use white spots between DTV-signals and preferably even in the proximity of DTV signals. In such cases, very steep high-order RF-filters would be needed, which are difficult to implement even with high-Q SAW filters. If the frequency-distance between DTV signals is small, narrowband RF-filters would be needed. To cover a significant part of the TV bands, many filters would be needed. From these observations we conclude that a broadband RF receiver with relaxed RF-filter requirements is highly desired. In this chapter we look for ways to benefit from spectral sensing which is available in a CR anyway. We will look at a broadband receiver, but still limit the bandwidth to below an octave so that second order distortion products are largely suppressed by the RF-filter and the 3rd order distortion is the main remaining problem.

A broadband RF receiver implies the reception of undesired interference as shown in Figure 2.3. The example of a DTV spectrum in Figure 2.3 consists of 20 numbered frequency spots: 5 spots contain interference (i.e. DTV signals at spots 1, 4, 11, 16 and 19) and one spot contains the desired signal (i.e. CR signal at spot 8).

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Figure 2.3: An DTV instantaneous spectrum scenario at a wideband RF bandpass filter output with 20 frequency spots In [19], the use of spectral sensing to relax linearity requirements of a receiver is addressed. The paper assumes a dense spectrum with a large number of interferers. According to the law of large numbers, the distribution of the intermodulation products due to the many interferers becomes even across the spectrum and can be seen as a distortion induced (noise) floor. However, a different picture arises when the spectrum is dominated by a few high power interference signals in the band of the RF bandpass filter. According to our analysis and measurements, different spectral distributions of the interference produce quite different distributions of the distortion products across the spots of the DTV spectrum. Consequently, the linearity requirements of the CR receiver will be different for each white spot. A prediction of the spectral location of the distortions provides the CR with the ability to choose an appropriate white spot. The required information about the level and the spectral location of the interference signals can be obtained from spectrum sensing, which is normally available anyhow in a CR. Our analysis is based on a CR receiver with direct conversion architecture. The theory can be extended to other (non-zero) IF architectures, where the image problem must be included as an extra contribution of distortion to the desired signal, but this is outside of the scope of this chapter.

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2.2 Behavior Model for Nonlinearity

It is common to model the nonlinearity of an RF receiver assuming a memory-less weakly nonlinear model (e.g. [15] and [20]) with a truncated 3rd order Taylor series around the DC operating point:

3 in 3 2 in 2 in 1 out

k

s

k

s

k

s

=

+

Equation 2.1

where k1 specifies the gain, while k2 and k3 characterize the second and third order nonlinearity of the circuit with input signal sin and output sout (s can be both a voltage or current). The sin contains all the input signals at the output of the RF bandpass filter.

Equation 2.1 contains three terms. The first term with k1 is the amplified version of sin, the linear term. The second and the third term will be called the 2nd order and 3rd order nonlinear terms, respectively. Those nonlinear terms generate different distortion products. Some of those distortion products fall into the band of interest. By using an octave RF bandpass filter [19] and RF components with differential structure, the effect of 2nd order nonlinear term can often be handled. However, the effect of the 3rd order nonlinear terms cannot be ignored because their distortion products always fall inside the desired DTV spectrum.

2.3 Analysis of the Spectral Locations of Distortion Products

A CR scans the spectrum for white spots, which have a bandwidth in the order of 6-8 MHz in the DTV spectrum. Although the DTV signals and the CR signal are wideband (i.e. the CR occupies the whole spot) signals, The analysis starts representing them as tones with a power equal to the integral of the power of the wideband signal. Afterwards, the analysis will be extended to deal with the spectral shape of the wideband signals.

Referring to Figure 2.4, sin in Equation 2.1 contains three DTV signals around ω1, ω6 and ω19. To not lose the general view, CR signal will be introduced later after specifying the spectral location of the distortion products of the DTV signals.

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Figure 2.4: An instantaneous scenario of the DTV spectrum dominated by three interference signals in the RF passband The sin can be described by the following equation:

      =

∑

=1,6,19 n t jω TV in n ne A 2 Re s Equation 2.2

where ATV is the corresponding RMS value of the DTV (single tone) signals. Substituting sin in the 3

rd

order nonlinear term of Equation 2.1 gives the following distortion products:

(

)

{

}

            + + + +             +             + + +             + + = t jω 2 TV TV 3 2 TV TV 3 3 TV 3 t jω TV TV TV 3 t jω 2 TV TV 3 t jω 2 TV TV 3 2 TV TV 3 3 TV 3 t jω 2 TV TV 3 2 TV TV 3 3 TV 3 3 in 3 19 6 19 1 19 19 14 19 6 1 11 6 1 6 19 6 1 6 6 1 19 1 6 1 1 e A A 3k A A 3k A k 2 3 2 Re e A A A 3k 2 Re e A A k 2 3 2 Re e A A 3k A A 3k A k 2 3 2 Re e A A 3k A A 3k A k 2 3 2 Re s k Equation 2.3

Figure 2.5 provides a visual way to understand the spectral location of the distortion products that are double underlined in Equation 2.3. These products are important because they fill the white spots, where a CR may operate. They represent two types of intermodulation products:

1. The intermodulation products of two interferers (observe spot 11), which is the traditional IM3

that is already mentioned in literature [15].

2. The intermodulation products of three interferences (spot 14), which is 6 dB higher than the traditional 2-tone IM3 for interferes with equal power. This type of distortion will also be called

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Figure 2.5: Spectral location of the intermodulation products for the case of Figure 2.4

For simplicity, we will keep the power of the DTV signals equal in this paper. Of course, using Equation 2.3 case with different power can be analyzed easily.

Repeating the same analysis for another DTV spectrum scenario with three large interference signals at different locations, gives the distortion distribution that is depicted in Figure 2.6.

Figure 2.6: Spectral location of the intermodulation products for another case

Comparing Figure 2.5 with Figure 2.6, we recognize that the first figure presents 15 white spots without intermodulation distortion (i.e. IM3-free spots) and 2 white spots with IM3 (i.e. IM3-spots), while the

second figure presents 13 IM3-free spots and 4 IM3-spots. For all the possible spectral locations of the

three interference signals, the number of the IM3-free spots can be counted and a histogram of it can be

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Figure 2.7: Probability density of IM3-free spots for all possible scenarios with 3 interferers (Figure 2.6 depict one scenario)

Note that the fact that a spot is IM3-free does not mean that there is no 3rd order nonlinearity

requirement anymore for the receiver. Actually, the interferers will cause cross-modulation on top of the desired signal as shown in Figure 2.8.

Figure 2.8: Cross-modulation products (XM3) due to interferers which cross-modulate the CR signal

To analyze the effect of cross-modulation, the mathematical expression for sin including a CR at spot 17 is written as follows:       + =

∑

=1,6,19 n t jω TV t jω CR in n n 17 ₂ _A _e e A 2 Re s Equation 2.4

where ACR is the corresponding RMS value of the CR signal. Substituting sin in Equation 2.1 and retaining just the terms that exist in the desired band gives:

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(

)

                                  + + + + = jω t products distortion order 3 modulation -Cross 2 TV 2 TV 2 TV CR ce interferen -Self 3 CR 3 signal CR amplified Desired CR 1 Band Desired out 17 rd 19 6 1 A A e A 3A A 2 3 k 2 A k 2 Re S 4 4 4 4 4 4 4 4 4 3 4 4 4 4 4 4 4 4 4 2 1 4 4 4 4 4 3 4 4 4 4 4 2 1 3 2 1 43 42 1 Equation 2.5

where the first term on the right hand is the amplified version of CR signal, the second term is the self-interference product [21] and the last three terms are the mentioned 3rd order cross-modulation [22] products (XM3) of the DTV signals. The self-interference product can usually be neglected in comparison

to the XM3 products because the DTV signals are usually much stronger than the desired signal. The

same analysis is done to show the spectral location of the distortion products of 4 interference signals with high power and the results can be seen in Figure 2.9 and Figure 2.10 (Expected value = 7).

Figure 2.9: Spectral location of the intermodulation products for a scenario with 4 interferers

Figure 2.10: Probability density of the existence of given number of free spots without intermodulation distortions for scenarios with 4 interferers

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As mention previously, both the DTV signals and the CR signal are wideband signals with about the same bandwidth. Therefore their shape can be approximated by square blocks in the frequency domain. The mathematical representation in the time domain of those signals and their distortion products is written in the same way as mentioned previously with the exception that ATV and ACR now represent the complex envelope of the DTV signal(s) and CR signal, respectively. In the time domain, the intermodulation and the cross-modulation products are nothing else than a multiplication of three signals, where the involved signals are indicated with arrows. E.g. three arrows constitute the inputs to the distortion circles in Figure 2.5, Figure 2.6 and Figure 2.8, in some cases from two frequencies, in other cases from three frequencies. In the frequency domain, this corresponds to two times convolution of the signals. The result of this convolution is shown in Figure 2.11 for the interference scenario of Figure 2.5, but now with more realistic block-shaped spectra instead of single tones.

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A Bell shape results, occupying three spots as shown in Figure 2.11. Analysis shows that more than 66% of the distortion power is concentrated in the frequency spot in the center of the Bell, while the power spilled over to the adjacent frequency spots is less than 17% of the distortion power each. Referring to Equation 2.3, the bell shapes around spots 1, 6 and 19, each contain a self-interference product and two cross-modulation products (e.g. the Bell shape around spot 6 contains its self-interference product and the cross-modulation products of the other two interferers of spots 1 and 19 on top of the interferer of spot 6). As a consequence of spreading, the distortion to the adjacent spots of 1, 6 and 19, the spots 2, 5, 7, 18 and 20 will be called adjacent-distortion spots (i.e. the distortion that is adjacent to the interferers). The linearity requirements in those adjacent-distortion spots of Figure 2.11 will be higher than that of Figure 2.5. Additionally, the bell shape of the IM3 products (i.e. around spot 11 and 14) will

reduce the number of IM3-free spots by 4 spots in comparison to Figure 2.5, as spots 10, 12, 13 and 15

will contain 17% of the total IM3 power of the Bell shape, hence those spots will be also called IM3 spots.

Paper [23] presents an analysis about the interference effects in Multi-Band - Orthogonal Frequency Division Multiplexing (MB-OFDM) systems. It proves that the power of the IM3 resulting from two OFDM

signals (Quadrature Phase Shift Keyed (QPSK) modulation) each with a large number of sub-carriers is higher than that of two tones that have the same power as the OFDM signals:

[dBm]

∆

P

_IM _IM _Wideband 3 3

=

+

+ Equation 2.6

where PIM3+ and PIM3 represent the intermodulation distortion of two wideband signals and two tones, respectively. PIM3

+

is distributed over three spots (e.g. see spots 10, 11, 12 of Figure 2.11). ∆Wideband represents a correction term. Instead of dealing with wideband complex signals, Equation 2.6 provides the possibility to analyze the nonlinearity with the traditional two-tone test method and simply add a correction term.

As an example that can be applied to our case, a MatLab simulation setup has been built to simulate the nonlinearity with two real OFDM signals at RF. QPSK modulation was used and the frequency difference between the sub-channels is 4.46 kHz satisfying the Digital Video Broadcasting Terrestrial (DVB-T) standard. The resulting IM3 was simulated. The simulation result is shown in Figure 2.12, where the

number of the sub-carriers per OFDM signal has gradually been increased from 1 to 900. Thus we can compare the IM3 generated for two (narrowband) single tones with that for a truly wideband OFDM

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Figure 2.12: Comparison of the IM3 power caused by two (single) tones and two multi-tone OFDM signals with a number

of sub-carriers from 1 to 900

Figure 2.12 shows a ∆Wideband of about 3 dB. Changing the modulation of the two OFDM signals to 16QAM, we find 3.5 dB. By further increasing the number of the QAM modulation to higher orders, the signal characteristic of the two OFDM signals resembles the case of having two band limited noise signals. In this case the factor is increased to 4.5 dB. Table 2.1 summarizes the simulation results:

Table 2.1: CORRECTION FACTOR ∆WIDEBAND FOR TWO OFDM SIGNALS WITH DIFFERENT MODULATION CHARACTERISTICS

Two Signals Max. ∆Wideband [dB]

Two OFDM: QPSK 3

Two OFDM: QAM16 3.5

Two Band limited Noise 4.5

As the purpose of this paper is to compute the linearity requirements for CR, the highest value of the maximum ∆Wideband will be taken, which is 4.5 dB.

2.4 Estimation of Linearity Requirements

The linearity of a receiver must be adequate to preserve a minimum Signal to Distortion Ratio needed at the demodulator. As previously depicted in Figure 2.5, we distinguish two types of white spots: IM3-free

spots and IM3-spots. We will show that the CR requires higher linearity if it operates at IM3-spots than

for IM3-free spots with only self-interference and the XM3. This section begins with introducing an

estimation of the linearity requirement for the case that the CR is operating at an IM3-free spot and the

interferers are modeled by tones. Afterwards the linearity equation will be extended to cover the case of wideband interference signals. Finally, this linearity requirement will be compared to that of a CR

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Referring to Figure 2.8 and Equation 2.5, the Signal to Distortion Ratio can be written as follows:

(

)

2 1,6,19 n 2 TV CR 3 2 CR 1 n A A 3k A k D S               =

∑

= Equation 2.7

where the self-interference has been neglected. The linearity requirement can be presented as the ratio between k1 and k3:













×

=

∑

=1,6,19 n 2 TV 17 spot 3 1 n

A

D

S

3 k

k

Equation 2.8

where S/D is the required Signal to Distortion Ratio for the CR receiver output. It is proportional to the sum of the power of the DTV signals, which will cross-modulate the receiver CR-signal.

Equation 2.6 presents the correction factor ∆Wideband for the intermodulation power. The same ∆Wideband is

valid for the XM3 power (the ∆Wideband is just the relative difference between the modulation products of

the two tones in relation to that of the two wideband signals). Therefore Equation 2.6 provides a useful tool to extend the derivation of Equation 2.8 to deal with wideband interference signals as explained in Appendix B:

∑

=

×

=

1,6,19 n 2 TV 10 ∆ 17 spot 3 1 n Wideband

A

0.66

10 D

S

3 k

k

Equation 2.9

Using the mentioned maximum ∆Wideband of 4.5dB gives:

∑

=

×

=

1,6,19 n 2 TV 17 spot 3 1 n

A

1.4 D

S

3 k

k

Equation 2.10

It is instructive to relate Equation 2.10 to the traditional IIP3 specification. According to the definition of

IIP3, the RMS value of IIP3 is related to k1/k3 as follows [15]:

[V]

k

6

4

2 k

k

3

4 A

3 1 3 1 IIP₃

=

Equation 2.11

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Substituting Equation 2.10 in Equation 2.11 gives the following equation:

[W]

P

1.4 D

S

2 P

1,6,19 n TV 17 spot IIP₃

∑

_n =

×

=

Equation 2.12

We recognize that dealing with wideband signals instead of tones, the IIP3 requirement will be increased

by around 1.5 dB (i.e. 10*log(1.4)). To get a sense of what Equation 2.12 presents, let’s assume that the interference signals of Figure 2.11 have a power of -10 dBm and the required Signal to Distortion Ratio is 10 dB. Then the required linearity to keep XM3 low enough expressed in terms of IIP3 would be 4 dBm.

Increasing the power of the interferences by 10 dB increases the requirement also by 10 dB. Equation 2.12 can be generalized as follows:

[W]

P

1.4 D

S

2 P

n TV IIP₃

=

×

∑

_n Equation 2.13

where n refers to all the interference signals that exist in the RF bandpass filter of the CR receiver. As a final step, let us compare IIP3 of Equation 2.12 (i.e. IIP3XM3) to that IIP3 requirement when the CR

operates in IM3-spot (IIP3IM3) like for example spot 11 in Figure 2.11. In this case the distortion products

will contain the IM3 plus the XM3 products:

                                                + + =

∑

= t jω products distortion order 3 ation Intermodul 2 TV TV modulation -Cross 1,6,19 n 2 TV CR 3 signal CR amplified Desired CR 1 Band Desired out 11 rd 6 1 n A A e 2 3 A 3A k 2 A k 2 Re s 4 4 4 4 4 4 4 3 4 4 4 4 4 4 4 2 1 43 42 1 4 4 3 4 4 2 1 43 42 1 Equation 2.14

Writing down the Signal to Distortion Ratio gives:

(

)

2 ation Intermodul 2 TV TV 3 modulation -Cross 1,6,19 n 2 TV CR 3 2 CR 1 6 1 n ₂k A A 3 A A 3k A k D S                        +       =

∑

= ₁_{4 2}₄ _{4 3}₄ 4 4 4 3 4 4 4 2 1 Equation 2.15

The resulting linearity requirement in terms of k1/k3 becomes:

        + × =

_∑

= CR 2 TV TV 1,6,19 n 2 TV 11 spot 3 1 A A A 2 1 A D S 3 k k 1 6 n Equation 2.16

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Expressing the linearity requirements in IIP3 and extending the derivation to wideband signals gives the

following two equations for spot 11 and 10, 12 (see Figure 2.11):

[W] P P P 0.66 10 2 1 P 0.66 10 D S 2 P 6 1 Wdeband n Wdeband 3 TV CR TV 1.4 10 ∆ 1,6,19 n TV 1.4 10 ∆ 11 spot IIP           × + ∑ × × = = 144424443 4 4 4 3 4 4 4 2 1 Equation 2.17 [W] P P P 0.17 10 2 1 P 0.66 10 D S 2 P 6 1 Wideband n Wideband 3 TV CR TV 0.7 10 ∆ 1,6,19 n TV 1.4 10 ∆ 10,12 spot IIP         × + × × =

∑

= 14 24 4 34 4 4 3 4 4 2 1 Equation 2.18

By comparing between Equation 2.17 and Equation 2.18, the IIP3 requirement of spots 10 or 12 (i.e. the

spots adjacent to the Bell center spot that contains 17% of the total IM3 power) is 3 dB lower than the

IIP3 requirement of spot 11 (i.e. the center spot of the Bell shape that contains 66% of the total IM3

power), hence

0 .

17 /

0.66 =

1 /

2

.

As mentioned previously, the CR requires also linearity requirement in the adjacent-distortion spots (see Figure 2.11). Repeating the same analysis as previously done and using Equation 2.3, Equation 2.5 and Equation 2.11 results: [W] P P ) P P P 2 1 ( 0.17 10 P 0.66 10 D S 2 P CR TV TV TV TV 0.7 10 ∆ 1,6,19 n TV 1.4 10 ∆ 2 spot IIP 1 19 6 1 Wideband n Wideband 3     + + × +     × × =

∑

= 14 24 4 34 4 4 3 4 4 2 1 Equation 2.19

To get a sense about the linearity requirement of Equation 2.12, Equation 2.17, Equation 2.18 and Equation 2.19 consider the same numbers as for the previous example, while assuming that the CR signal is equal to -60 dBm. Then the different IIP3 requirements are summarized in Table 2.2.

Table 2.2: LINEARITY REQUIREMENT OF EQUATIONS (12), (17), (18) AND (19) FOR THE SCENARIO OF Figure 2.11

Equation Spot no. Spot Description IIP3 [dBm]

Equation 2.12 17 IM3-free spot 4

Equation 2.17 11 IM3-spot 21

Equation 2.18 10, 12 IM3-spot 18

Equation 2.19 2 Adjacent-distortion spot 25

Clearly, there is a very significant benefit in looking for IM3-free spots. Such spots like spot 17 can well

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2.5 Measurement Results

The spectral location of the distortion products is the consequence of modeling a memory-less weakly nonlinear device with a Taylor series as presented in Equation 2.1. The model has been verified by building a setup where a Low Noise Amplifier (LNA) from the Mini-Circuits (i.e. ZFL-1000LN) is tested with three interference signals. The power and the spectral location of the distortion products are measured. The LNA has gain of 18 dB, IIP3 of -6 dBm, Output 1 dB compression point of 3 dBm and its

operating region is from 0.1 to 1000 MHz (i.e. VHF/UHF band). The interference signals are three carriers with QPSK modulation, which are generated by vector signal generators from Agilent technology. In the first experiment, the power of each interferer is around -26 dBm and their frequency location is around the following values: 548, 563 and 600 MHz. This case looks like the case shown in Figure 2.5 and the measurement result is shown in Figure 2.13.

Figure 2.13: Spectral location of the distortion products in case of three interferers at 548, 563 and 600 MHz Comparing to Figure 2.5, Figure 2.13 indeed contains two distortion products:

1. The first interferer at 548 MHz and the second interferer at 563 MHz generates the traditional IM3 at 578 MHz (i.e. 2*563-548)

2. The combination of all the three interferers generates the previously mentioned new IM3 at

frequency 585 MHz (i.e. 600-(563-548) and (600-563)+548). Shown in Figure 2.13, this type of IM3 is higher than the first IM3 by 6 dB as our theory has predicted (see Equation 2.3).

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The power of those IM3 products are measured and compared to the results of our equations as shown

in Table 2.3.

Table 2.3: Comparison between the measurement and the theory results of the IM3 distortion products

IM3 frequency [MHz] Theory [dBm] Measurement

[dBm]

578 -39.4 -38.7

585 -33.4 -32.5

Another experiment has been done with a scenario that looks like the case shown in Figure 2.6. The measurement result is shown in Figure 2.14.

Figure 2.14: Spectral location of the distortion products in case of three interferers at 554, 560 and 590 MHz Consequently, the measurement results verify our theory about the effect of interference locations on the spectral location of the distortions. The IM3 spots and the IM3-free spots are clearly shown in Figure

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2.6 Conclusion

This chapter studied the 3rd order nonlinearity requirements for a CR receiver with a wideband RF bandpass filter that operates in the DTV spectrum, where several high powers DTV signals fall in the RF filter-band. Due to the nonlinearity of the RF receiver and the existence of strong interference signals, different types of 3rd order distortion products will be produced. The spectral location of these distortion products depends on the spectral locations of the interference signals. Analysis and measurements show that white spots can be classified into two types:

1. IM3-spots, for which the distortion is dominated by IM3 products. The IM3 products are much

stronger than XM3 and self-interference if the interferers are much stronger than the desired

signal.

2. IM3-free spots where the CR signal will only be distorted by XM3 and self-interference.

If an IM3-free spot is selected for CR operation, its third order linearity is relaxed. In a scenario with

three interferers at -10dBm and a CR-signal at -60dBm, the resulting IIP3XM3=4dBm, while the IIP3IM3 is

18-25 dBm. Since the linearity of any radio design is limited, typically to not much more than 0dBm in CMOS, spectral sensing information about the level and the spectral location of the strong interferers can be used to predict the location of the IM3-free spots. Such a prediction mechanism seems to be

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Chapter 3. Distortion Prediction Algorithm and Frequency Selection

This chapter investigates the second research question: “How to predict the distortion level in Dynamic Spectrum Access, given Spectrum Sensing data?”

The coexistence of a Cognitive Radio signal among the primary users of the DTV spectrum increases the linearity requirements of a wideband CR receiver considerably to very high numbers (see Table 2.2) as derived in the previous chapter. The analysis in chapter 2 has classified the DTV white spots into IM3

-spots and IM3-free spots, depending on the existence of IM3 distortion products at a particular white

spot. It was concluded that the IIP3 requirements on the CR receiver will be different across the white

spots, and since the linearity of any radio design is limited, this means that not all the white spots are equally appropriate for CR operation. This chapter aims at developing a Distortion Prediction Algorithm (DPrA) that calculates the distortion level across the white spots. Based on the DPrA results in combination with the theory in chapter 2, the linearity requirement per white spot can be quantified. Section 3.1 presents the DPrA concept. Then, section 3.2 presents the verification of the DPrA concept by applying it to the DSA1_{scenarios of the previous chapter and some other application examples test its}

ability to recognize the construction pattern of complex distortion products. Finally, the chapter ends with conclusions in section 3.3.

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3.1 Distortion Prediction Algorithm (DPrA)

The analysis starts in representing the wideband1 DTV signals and the wideband CR signal (i.e. the CR occupies the whole channel) as single frequency with a power equal to the integral of the wideband signal power. Afterwards, the analysis will be extended to deal with the spectral shape of the wideband signals.

The target of the DPrA is to derive the instantaneous level and spectral location of the 3rd order distortion products across the white spots by using spectral sensing information. An example of the instantaneous level and the spectral location of some DTV signals is shown in Figure 3.1. The calculation of the output spectrum in this figure is already explained in chapter 2 (see Figure 2.5 and Equation 2.3).

Figure 3.1: Example of the operation of the 3rd order Distortion Prediction Algorithm (DPrA) As indicated in Figure 3.1, the DPrA boils down to evaluate the following mathematical expression:

( )

t

k

[s

( )

t

s

( )

t

s

( )

t

]

s

k

₃ 3_in

=

₃ _in

×

_in

×

_in Equation 3.1

where k3 characterizes the third order nonlinearity term of the circuit. Note that sin contains all the input signals passing through the RF bandpass filter. Equation 3.1 involves two times multiplication in the time domain of sin. In this case, it consists of three signals at channel numbers: 1, 6 and 19, i.e.:

( )

        ∑ = =1,6,19 n t jω TV in t Re 2 A ne n s Equation 3.2

where n represents the corresponding channel number and ATV is the corresponding RMS value of the

DTV (single tone) signals. 1

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Repeating the procedure of the previous chapter in substituting sin in Equation 3.1 produces the following distortion products:

( )

[

]

(

)

{

}

            + + + +             +             + + +             + + = × × t jω 2 TV TV 3 2 TV TV 3 3 TV 3 t jω TV TV TV 3 t jω 2 TV TV 3 t jω 2 TV TV 3 2 TV TV 3 3 TV 3 t jω 2 TV TV 3 2 TV TV 3 3 TV 3 in in in 3 19 6 19 1 19 19 14 19 6 1 11 6 1 6 19 6 1 6 6 1 19 1 6 1 1 e A A 3k A A 3k A k 2 3 2 Re e A A A 3k 2 Re e A A k 2 3 2 Re e A A 3k A A 3k A k 2 3 2 Re e A A 3k A A 3k A k 2 3 2 Re t s t s t s k Equation 3.3

Expanding the total number of the channels, and the number and spectral locations of the interferers will increase the complexity of Equation 3.3. Thus it is not practical to implement this time domain derivation, especially in case of a dynamic spectrum scenario that changes continuously. However, it will be shown below that the analysis can also be done in the frequency domain in a systematic way using matrix operations. In the frequency domain, the multiplications in Equation 3.1 correspond to convolutions:

( )

f S

( )

f S

( )

f ] [S Domain Frequency in the Products Distortion Order Third DPrA in in in ⊗ ⊗ ⇒ ⇒ _{Equation 3.4}

where Sin(f) is the frequency domain representations of sin(t).

To build an intuitive understanding of the matrix implementation of the DPrA, let’s start with just two tones, as shown in Figure 3.2 and expressed as follows:

( )

= ∑ + ∑         ∑ = + + = + − = = m 2, 3 t jω m 2,-3 m t jω -m 2,3 n t jω TV in t Re 2 A ne n A e m A e m s Equation 3.5

where m represents the channel number for double side spectrum (i.e. positive and negative frequencies) and Am is the MAX value of the DTV (single tone) signals at channel m. For simplicity, let’s

assume that all the occupied channels contain equal signal amplitude, equal toATV/ 2 as follows:

2 / A A A A A = = ₊ = ₊ = Equation 3.6

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Figure 3.2: Frequency spectral content of the of two tones (two single frequencies)

The frequency spectrum span of Sin(f) is equal to 7 channels (N=7 channels: m=-3,-2,-1,0,+1,+2,+3;) around the zero-frequency (see Figure 3.2), each channel has a bandwidth equal to the DTV channel bandwidth, namely fCH (for the clarity of later figures, the label fCh on x-axis will be removed). Assuming

pure tones, Sin(fm) can be written in the discrete form:

( )

[

(

)

(

)

]

[

(

)

(

)

]

(

) (

)

[

m ch m ch

]

TV

[

(

m ch

) (

m ch

)

]

TV ch m 3 ch m 3 ch m 2 ch m 2 m in f 3 f δ f 3 -f δ A 2 2 f 2 f δ f 2 -f δ A 2 2 f 3 f δ A f 3 -f δ A f 2 f δ A f 2 -f δ A f S + + + + + = + + + + + = ₊ ₋ ₊ ₋ Equation 3.7

SCONV(fm) in Equation 3.8 presents the mathematical expression of the first convolution process between the square brackets in Equation 3.4:

( )

( ) (

)

(

)

( ) (

)

(

)

∑

∞ + ∞ − = ∞ + ∞ − =

−

=

−

=

⊗

=

.. ρ m ch in m in .. ρ m m in m in m in m in m CONV m m

ρ

f

m

S

ρ

S

ρ

f

S

ρ

S

f

S

f

S

f

S

Equation 3.8

Where ρm is a dummy variable required to perform the convolution operation [25]. This convolution operation involves a multiplication between the Sin(ρm) spectrum and its flipped

1

version, shifted to the right by |m|fCH, written as Sin(m fCH - ρm). Figure 3.3 visualizes this convolution operation that calculates the frequency content of SCONV(fm) at m=0..+6. For the negative frequency side of SCONV(fm) (i.e. at m=-6..-1), a similar plot can be created, by multiplying the spectrum of Sin(ρm) with its flipped version, now shifted to the left by |m|fCH. Figure 3.3 shows that seven shifting operations are required to calculate

the frequency content of SCONV(fm) at m=0..+6: one shifting operation for the frequency content at the origin (i.e. at m=0) and another six shifting operations for the positive frequency content (i.e. at m=+1..+6). To complete the spectrum content of SCONV(fm) with the negative frequency content (i.e. at m=-6..-1), still extra 6 shifting operations are needed.

1

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Figure 3.3: Visualizing the convolution operation of Equation 3.8

Hence for the total spectrum view, 13 shifting operation (i.e. in general form 2N-1, where N=7) from the left to the right are required. The DPrA implements this convolution operation via Matrix multiplication as shown in Figure 3.4, where the Qm elements of the S_CONV

(

mf_ch

)

is derived. First, we form a vector,

calledS_in . This vector corresponds to Sin(fm) (see the 7 channels in Figure 3.2) and contains 7 elements. The index and the value of those 7 elements correspond to the spectral location and the amplitude of

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Figure 3.4: Matrix implementation of the convolution operation of Figure 3.3 The Sin(m fCH - ρm) is constructed in a matrix form as follows:

• Its middle column (i.e. 7th column) is nothing else than the flipped version of S_in shifted by zero (i.e. Sin(- ρm)). Then multiplying S_in elements by the elements of this column calculates the

signal

(

)

ch CONV mf

S quantity at m=0.

• The columns at the right of the 7th column (i.e. 8th - 13th columns) represent the flipped version ofS_in , right shifted by m=+1 to +6, respectively (i.e. Sin(m fCH - ρm)). Multiplying S_in with those columns provides the signal

(

)

ch CONV mf

S quantity at m=+1 to m=+6.

• The columns at the left of the 7th column (i.e. 1st - 6th columns) represent the flipped version of

in

S , left shifted by m=-6 to -1, respectively (i.e. Sin(m fCH - ρm)). Multiplying S_in with those columns provides the signal

(

)

ch CONV mf

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The content of the Qm values of the signal S_CONV

(

mf_ch

)

, which are calculated according to Figure 3.4 are

listed in the following table:

Table 3.1: Frequency content of

(

)

ch CONV mf

S at m=-6 to m=+6

Qm elements Qm construction Qm amplitude level

Q-6 ( A-3 )2 ½ ( ATV )2 Q-5 2 ( A-3 A-2 ) ( ATV )2 Q-4 ( A-2 )2 ½ ( ATV )2 Q-3 0 0 Q-2 0 0 Q-1 2 ( A-3 A+2 ) ( ATV )2 Q0 2 ( A-3 A+3 )+ 2 ( A-2 A+2 ) 2 ( ATV )2 Q+1 2 ( A-2 A+3 ) ( ATV )2 Q+2 0 0 Q+3 0 0 Q+4 ( A+2 )2 ½ ( ATV )2 Q+5 2( A+2 A+3 ) ( ATV )2 Q+6 ( A+3 )2 ½ ( ATV )2

The second column of Table 3.1 specifies how the delta functions of the two tones will contribute to the Qm elements of the signal S_CONV

(

mf_ch

)

. Substituting the amplitude of the two tones, defined in Equation

3.6, the value of the distortion component is evaluated (see third column of Table 3.1). In general, the vector S_in has a dimension 1x7 (i.e. 1xN) and the dimension of the resulting matrix is 7x13 (i.e. N x (2 N-1).

The second convolution operation in Equation 3.4 can be explained and plotted in the same way as we did for the first convolution. The mathematical expression of this convolution is written as follows:

( )

(

( ) (

)

( ) (

)

(

)

∑

∞ + ∞ − = ∞ + ∞ − =

−

=

−

=

⊗

.. ρ m ch in m CONV .. ρ m m in m CONV m in m CONV m m

ρ

f

m

S

ρ

S

ρ

f

S

ρ

S

f

S

f

S

Equation 3.9

This convolution process is implemented by multiplying the result of the previous convolution SCONV(mfCH) with the flipped and shifted version of S_in . Figure 3.5 visualizes this convolution operation,

(50)

Figure 3.5: Visualizing the convolution operation of Equation 3.9

In this figure, one can recognize the conventional IM3 products [15] at m=±1 and m=±4 around the

Distortion mitigation in cognitive radio receivers