IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS, VOL. 60, NO. 2, FEBRUARY

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1 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS, VOL. 60, NO. 2, FEBRUARY Aliasing-Free Digital Pulse-Width Modulation for Burst-Mode RF Transmitters Katharina Hausmair, Student Member, IEEE, ShuliChi, Student Member, IEEE, Peter Singerl, and Christian Vogel, Senior Member, IEEE Abstract Burst-mode operation of power amplifiers (PAs) is a promising concept towards higher power efficiency in radio frequency (RF) transmitters. Such transmitters use pulse-width modulation (PWM) to create the driving signal for the PA, and a reconstruction filter after amplification to obtain the transmission signal. However, conventional digital pulse-width modulated signals contain a large amount of distortionthatcannotberemovedbythe reconstruction filter in a satisfactory manner. This paper introduces a method for digital PWM that is free of destructive aliasing distortion. First, a set of mathematical closedform equations fully describing all baseband processing steps required in conventional PWM-based RF transmitters is developed. Analysis of the equations leads to the conclusion that destructive distortion in digital PWM systems originates from aliasing induced by the infinite bandwidth of pulsed signals that is entailed by the nonlinear operation of the pulse-width modulator. Based on this knowledge, a PWM method is developed that ensures that the generated signals are bandlimited and hence completely avoids destructive aliasing distortion. Simulations as well as measurements demonstrate the improvement that can be achieved with the proposed method compared to conventional methods. The results indicate that by using the proposed method it becomes feasible to implement filters that allow for obtaining satisfying transmission signal quality. Index Terms Aliasing distortion, burst-mode operation, Fourier series expansion, pulse-width modulation (PWM), radio frequency (RF) transmitters. I. INTRODUCTION MODERN wireless communication systems have evolved to provide new services that perpetually demand higher data rates. Therefore, future telecommunication standards like Long Term Evolution (LTE) [1] or Digital Video Manuscript received March 20, 2012; accepted March 28, Date of publication December 13, 2012; date of current version January 24, This work was supported in part by the European Union s Seventh Framework Programme (FP7/ ) under Grant agreement no , and in part by the Land Steiermark s Zukunftsfond Steiermark under grant agreement no The Competence Center FTW Forschungszentrum Telekommunikation Wien GmbH is funded within the program COMET Competence Centers for Excellent Technologies by BMVIT, BMWFJ, and the City of Vienna. The COMET program is managed by the FFG. This paper was recommended by Associate Editor M. Laddomada. K. Hausmair and S. Chi are with the Signal Processing and Speech Communication Laboratory, Graz University of Technology, Graz 8010, Austria ( hausmair@tugraz.at; shuli.chi@tugraz.at). P. Singerl is with Infineon Technologies Austria AG, Villach 9500, Austria ( Peter.Singerl@infineon.com). C. Vogel is with the Telecommunications Research Center Vienna (FTW), Vienna A-1220, Austria ( c.vogel@ieee.org). Color versions of one or more of the figures in this paper are available online at Digital Object Identifier /TCSI Broadcasting-Second Generation Terrestrial (DVB-T2) [2] utilize spectrally efficient modulation schemes like quadrature amplitude modulation (QAM) in combination with orthogonal frequency-division multiplexing (OFDM) that result in transmission signals with rapidly changing envelopes. Maintaining high signal quality and avoiding adjacent-channel interference for these complicated types of signals poses tight spectrum emission and linearity requirements on radio frequency (RF) power amplifiers (PAs) employed in wireless transmitters. To fulfill these requirements for signals with high peak-to-average power ratio (PAPR), conventional linear RF PAs, such as Class-A or Class-AB [3], are operated in power backoff. This kind of operation results in unnecessary power consumption and a low average efficiency [4]. Hence, many solutions have been proposed to increase the poor efficiency of conventional PAs, like for example the Doherty amplifier [5], or envelope tracking techniques [3], [6]. A promising approach towards higher power efficiency is the concept of burst-mode RF transmitters [7]. It can be applied to switched-mode power amplifiers (SMPAs) like Class-D or Class-E [3], [8], as well as to linear PAs like Class-A or Class-AB to enhance the average efficiency. Similar as in related concepts like envelope elimination and restoration (EER) and outphasing [3], [9], the idea is to operate a PA in only two states: on, where the PA is driven into saturation reaching peak efficiency, and off, where no power is consumed or wasted at all. Hence, in burst-mode RF transmitters, the PA is operated with a constant-envelope signal consisting of bursts of a phase-modulated carrier, meaning that, while no restrictions are imposed on the phase information, the amplifier driving signal should not contain any amplitude information [10]. The information that is usually encoded in varying signal envelopes has to be encoded in the duration of the carrier bursts, which are obtained by modulating a pulsed signal with the phase-modulated RF carrier. Consequently, the concept of burst-mode RF PAs requires techniques to transform amplitude information into a pulsed signal. The varying-envelope bandpass signal can be retrieved from the amplified bursted-carrier signal by a bandpass filter. Common methods for encoding amplitude information into a square-wave signal are -modulation [11] and pulse-width modulation (PWM) [12], where the investigation of the latter is the focus of this paper. The research presented here includes mathematical equations characterizing all baseband processing steps in PWM-based RF transmitters, as well as a detailed analysis of these equations. A novel method to generate high-quality, high-dynamic range digital pulse-width modulated signals suitable for large-bandwidth RF amplifiers is introduced and verified via simulations and measurements /$ IEEE

2 416 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS, VOL. 60, NO. 2, FEBRUARY 2013 Fig. 1. Simplified illustration of the spectrum of a pulsed signal containing the baseband version of the desired signal that can be recovered by a filter. The dynamic range (DR) is the distance between average power of the desired signal and the power of the surrounding distortion. A. Problem Statement and Related Research A pulse-width modulator encodes a nonconstant-envelope signal into a train of rectangular pulses with varying widths, such that the amplitude information of the input signal is represented by the widths of the pulses. The phase-modulated and RF upconverted pulsed signal can be used to drive an RF PA in burst-mode operation. After amplification, the desired signal, which is the amplified passband equivalent of the original baseband signal, has to be recovered by a bandpass filter. This is necessary, since in the process of creating the pulsed signal, a large amount of additional frequency content is added to the original baseband signal, as is illustrated in Fig. 1. The filtering operation is feasible only if the distortion around the desired signal is low enough. A measure of the amount of distortion is the dynamic range (DR), which is defined as the distance between the average power of the desired signal and the power of the surrounding distortion given in db. It is therefore of utmost importance that the amplified signal exhibits a sufficiently good DRtoenablesatisfyingsignalrecoveryquality and to fulfill the tight spectrum emission requirements posed by modern communication standards. This is only possible, if the driving signal supplied to the PA provides excellent spectral characteristics. With the growing trend of implementing transceivers as software-defined radios, it is desirable to downscale processes in the analog domain and to move as much functionality to the digital domain as possible [13], [14]. Hence, an entirely digital implementation of the PWM process is desirable [15]. While analog and digital implementation of various PWM techniques have been investigated and widely used for low-bandwidth applications as for example in power electronics [16] and digital audio [17], comparatively little research has been published on digitally implemented PWM for large-bandwidth RF applications. Especially the spectral characteristics of pulse-width modulated signals have to be thoroughly investigated to assess the actual applicability of the process in burst-mode RF transmitter architectures for modern telecommunication standards. There are several publications on determining the output signal spectrum of a pulse-width modulator and explaining problems arising from different aspects of the PWM process. In [18], [19] the authors present general analytical solutions for the harmonic contents of the spectrum of pulse-width modulated signals, while digitally generated PWM is not addressed. In [12], exact analytical expressions for the spectra of pulsed signals for different types of PWM are derived and the authors of [20] analyze PWM spectra to obtain design criteria for choosing an appropriate PWM switching frequency, but again the effects of sampling are not discussed. Determining the minimum average switching frequency for inband-error-free encoding is also the topic of [21]. When generating PWM digitally, a considerable amount of distortion can be observed in and around the band of the original input signal, which prevents perfect signal recovery and tightens the requirements posed on the bandpass recovery filter [22]. The origin of this destructive power has been investigated before, although with different methods and conclusions than we provide in this paper. In [23], the effects of sampling on the switching instants of pulsed signals due to finite temporal resolution in digital systems are analyzed from a different point of view than presented in this paper, while aliasing distortion due to the sampling process is discussed in [24], although based on simulation studies only. In [25], problems with the low DR of pulse-width modulated signals due to the image effect after RF upconversion, for large-bandwidth modern communication standard input signals, are mentioned. The authors of [26] discuss the image problem in more detail and also offer several countermeasures. The low DR is also a topic in [22], where the authors reach the conclusion that PWM is hardly suitable for the use in burst-mode RF transmitters and therefore propose -modulation as a more suitable choice. -modulation essentially suffers from similar problems, but offers the possibility to improve the DR by using noise-shaping methods. The authors of [27] [29] also focus their work on -modulators using different techniques to improve the DR. A noise cancellation circuit that necessitates additional analog hardware is proposed in [27]. In [28] a digital feedforward error correction technique requiring a filter is used. This technique is analyzed in detail in [30]. Optimal noise-shaping employing complex gradient-based search algorithms is applied in [29]. B. Contribution of This Paper This paper offers a new view on distortion present in digital pulse-width modulated signals, where digital in this paper refers to discrete-time continuous-amplitude signals: We show that distortion in digital pulse-width modulated signals is caused by aliasing, which is also a conclusion the authors of [24] have reached. In contrast to them, however, we present a set of analytical closed-form equations based

3 HAUSMAIR et al.: ALIASING-FREE DIGITAL PULSE-WIDTH MODULATION 417 Fig. 2. Continuous-time system model of a burst-mode RF transmitter. on Fourier series decomposition characterizing all complex baseband processing steps required in a burst-mode transmitter. The equations, comprising analog and digital PWM processes including phase modulation, allow us to explain the origins of the aliasing in pulse-width modulated signals. We show that the bandwidth of pulse-width modulated signals is infinite, which is an inevitable consequence of the inherently nonlinear operation of the pulse-width modulator. This implies that ordinary PWM cannot be implemented digitally without suffering from low DR due to aliasing. Based on the equations, we have developed a method that eliminates all distortion originating from the sampling process. The proposed method improves the DR and eliminates distortion in the input signal band of digital pulse-width modulated signals and therefore, unlike the authors of [22] state, makes PWM suitable for the generation of pulsed signals in burst-mode RF transmitters. We verify our analytical results through numerical simulation and measurements. The results are comparable to those obtained by the authors of [28], [30] since both methods suffer from similar side effects. However, our method directly changes the mode of operation of the pulse-width modulator, while in [28], [30] digital post-processing techniques are used. This paper is organized as follows: In Section II we explain the burst-mode RF transmitter system model. Next, we derive and analyze a mathematical description of the baseband processing steps required in such atransmitterinsectioniii. By introducing the concept of sampling in Section IV, we present and analyze the equations for an equivalent digital implementation of the baseband processing. Based on these equations, in Section V we propose the concept of bandlimited PWM that allows to perform PWM digitally without causing destructive aliasing distortion. Finally, the discussed theory is verified through numerical simulations and measurements in Section VI before we draw our conclusions in Section VII. II. BURST-MODE RF TRANSMITTER SYSTEM MODEL A system model of a burst-mode transmitter is depicted in Fig. 2. The complex baseband input signal, which is bandlimited to,isgivenby (1) and can be any type of communications standard signal. The signal is split into a magnitude signal and a phase signal. The magnitude signal is encoded into a pulsed signal by a pulse-width modulator. The pulsed signal is then modulated with the complex exponential function of the phase signal to obtain. This signal is upconverted to the passband by modulation with an RF carrier with frequency in rad/s. The resulting signal consists of varying-duration bursts of the phase-modulated carrier. After the passband burst signal is amplifiedbythepa,theamplified passband version of the original input signal isrecoveredbya bandpass filterandpassedontotheantenna. It should be mentioned here, that there are several ways to implement a burst-mode RF transmitter. Since the focus of this paper is on the generation of pulsed signals using digital PWM, and because the basic mathematical principles are true for any RF transmitter employing PWM, the most convenient system model for that purpose has been chosen. The discussion of general advantages and drawbacks of different architectures, as for example investigated in [10], [31], is not considered here. III. ANALOG BASEBAND PROCESSING From the system model in Fig. 2, it can be seen that the baseband processing steps comprise the pulse-width modulator operating on the magnitude signal, and complex phase modulation performed by multiplying the output of the pulse-width modulator with the complex exponential function of the phase signal. A. Principle of Pulse-Width Modulation The pulse-width modulator encodes a nonconstant-envelope signal into a train of two-level pulses with varying widths, such that the widths of the pulses represent the magnitude of the aforementioned signal. The input signal to the pulse-width modulator, i.e., the modulating signal, can be demodulated by integration over the pulsed signal, which can be achieved by using an adequate filter. A well-established PWM method is asymmetric double-edge PWM [12]. This method is depicted in Fig. 3. It can be seen that the pulsed signal consists of asymmetrical pulses that are centered around the midpoint of a fixed interval, the pulse period. The transitions between the two levels 0 and 1, and vice versa, are called leading edge (LE) and trailing edge (TE), respectively. Both edges are determined

4 418 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS, VOL. 60, NO. 2, FEBRUARY 2013 be used to derive an analytical closed-form description of the asymmetric double-edge PWM process [32], [33]. By introducing two different time variables and,the PWM operator can be written as a two-dimensional operator, where the signal in (5) is obtained for (6) Fig. 3. Illustration of analog asymmetric double-edge PWM, where the LEs and TEs of the pulsed signal are located at the intersection points of the modulating signal with the reference wave. Since the reference wave is periodic in with period,it follows that. Thus, for any is a periodic function of and can be expanded into a Fourier series [34, p. 284] by finding the intersection points of the modulating signal with a triangular reference waveform,where and are bounded by and has to satisfy some constraints concerning its slope [12]. The higher the amplitudes, the longer the arising pulse widths. The triangular reference waveform is given by where the coefficients are given by (7) (8) where denotes the th interval, such that LE and TE of the th pulse occur at the time instants [12] (2) In Fig. 4 an example of the pulse-train obtained with the operator for a specific value of the modulating signal is depicted. It can be seen that the width of the pulse is directly proportional to the modulating signal. With the aid of Fig. 4, the coefficients in (8) can be derived as (3) Asymmetric double-edge PWM can be implemented by using a comparator [12]. An operator describing the PWM can then be defined as (4) Inserting the result of (9) into (7), the Fourier series expansion of can finally be computed as (9) (10) such that the pulsed signal is obtained by (5) From (6) it follows that the PWM operator,and the pulsed signal, can be obtained from (10) by setting as Note that there are similar PWM methods, e.g., trailing-edge PWM or leading-edge PWM [12], that are based on the same principles as asymmetric double-edge PWM, but use different reference waves. Asymmetric double-edge PWM is the method of choice here because of its favorable spectral properties [12]. Nevertheless, the derived principles are true for any other of these similar PWM methods. B. Time Domain Analysis A detailed analysis of the signal generated by a pulse-width modulator requires an analytical description of the PWM process. Hence, it is demonstrated that a Fourier series can (11) where is the angular PWM frequency and the Taylor series has been used to obtain a description that allows for a convenient derivation of the frequency domain equations later on. After pulse-width modulation, phase modulation is performed by multiplying the pulsed signal with the complex

5 HAUSMAIR et al.: ALIASING-FREE DIGITAL PULSE-WIDTH MODULATION 419 The continuous-time Fourier transform of the phase-modulatedpulsedsignal in (12) is given by with (15) Fig. 4. The two-level pulse train obtained by the operator for a specific, i.e., a specific modulating signal, such that the pulse widths are. exponential function of the phase signal phase-modulated signal is given by.using(1),the (12) From (12) it can be seen that the phase-modulated pulsed signal consists of the complex input signal,andthe sum of multiplied by weighted even powers of the modulating signal, modulated by the harmonics of the PWM frequency. In contrast to -modulators, there is no amplitude quantizer function involved in asymmetric double-edge PWM. Hence, there is no amplitude quantization error present like in -modulators. Another difference compared to -modulators is that there is no feedback loop required in the PWM method considered here. The output of the pulse-width modulator and the phase modulated signal only depend on the current value of the modulating signal and the phase signal, respectively. No knowledge of future or past values is required. These observations are true for any PWM method based on the same principles as asymmetric double-edge PWM. C. Frequency Domain Analysis Using the continuous-time Fourier transform [35] and (11), the frequency domain description of the pulsed signal can be derived as where and denotes the Fourier transform. (13) (14) (16) where denotes the convolution. From (15) it can be seen that the phase-modulated signal spectrum contains the input signal spectrum, which can later be recovered by a filter, and additional components located around the input signal. Even though the factor in (16) assures that decreases with increasing and, the spectrum of the pulsewidth modulator output is not bandlimited, since tends to infinity. Furthermore, examining the content of more closely shows that this spectral component consists of the sum of scaled Fourier transformed even powers of the modulating signal convoluted with the input signal spectrum, located at harmonics of the PWM frequency. Each of these harmonic components is also of unlimited bandwidth, since tends to infinity. This leads to distortion in the frequency band of the input signal that can prevent the perfect recovery of the input signal. However, by choosing large enough, this type of distortion can be kept negligibly low. Finding a proper PWM frequency for an input signal with bandwidth is investigated in [20], [21] and in the simulations in Section VI-A. IV. DIGITAL BASEBAND PROCESSING The system model of a burst-mode transmitter employing digital baseband processing, so that the PWM is implemented in the digital domain, is illustrated in Fig. 5. To determine the characteristics of the signals occurring in such a transmitter, an analytical description of all digital baseband processing steps has to be found that is equivalent to the previously obtained analog baseband description. Note that the figure merely depicts a system model that allows for finding an analytical description of the digital processing steps. For an actual implementation of a burst-mode transmitter, also a polar transmitter architecture as shown in [28] [31] can be used. In such an architecture, the transmitter can be realized with only a single bit digital-to-analog converter (DAC) for the pulsed signal. Nevertheless, the basic observations presented here are true for any system employing digitally generated pulse-width modulated signals. A. Time Domain Analysis Though in general the input signal of a pulsed transmitter is generated digitally, to obtain a digital system that is equivalent to the system described in Section III, the concept of sampling is introduced here. The digital signal is obtained by sampling the analog signal, such that,and,where is the

6 420 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS, VOL. 60, NO. 2, FEBRUARY 2013 Fig. 5. System model of a burst-mode transmitter with digital complex baseband processing. sampling period and and are the sampling frequency in Hz and the angular sampling frequency in rad/s, respectively. Using (11), the pulse-width modulated signal that is produced by the digital PWM operator can be expressedinthesamewayas output and the spectrum of the phase-modulated digital pulsed signal, which are given by [35] (20) and (17) In this equation, with, the scaled frequency,which is commonly used to indicate a discrete-time system [35], is introduced. The important difference between analog and digital PWM becomes clear when looking at the time instants of the rising and falling pulse edges in the digital system, which are (18) As a result of the sampling, pulse edges can only occur at discrete sampling instances. Therefore, the switching instants and of the digital pulsed signal do not exactly match the switching instants and of the analog pulsed signal. Similar to (17), the digital phase-modulated signal can be obtained using (12), resulting in (19) B. Frequency Domain Analysis The impact of implementing PWM digitally becomes also evident when examining the spectrum of the digital PWM (21) Sampling causes aliases of the analog spectrum to appear around every th multiple of the sampling frequency [35]. Since the analog pulse-width modulator produces an output spectrum that is of infinite bandwidth, sampling will inevitably lead to aliasing [35]. This causes the DR of the digital pulse-width modulator output to deteriorate compared to ideal analog PWM. The distortion remains even when converting the digital pulsed signal to an analog signal, since the overlap of the aliased spectrum components with the spectrum in the frequency range of interest can never be removed again. The explained time and frequency domain effects of sampling on an analog pulsed signal are illustrated in Figs. 6 and 7. For the purpose of better illustration, the phase-modulated signals are depicted in the spectra plot, while the signals without phase modulation are shown in the time domain. It is imperative to understand that the aliasing is not actually causedbysamplingthesignal, but by sampling the nonlinear PWM operator. As a consequence, the aliasing effect is unavoidable, since the PWM operator itself is discretized in time and therefore aliasing is generated by this operator, directly within the digital system. Such an aliasing effect cannot be prevented or eliminated by an anti-aliasing lowpass filter. Even the use of very large sampling frequencies can never completely eliminate the distortion in digital PWM systems. It is, however, possible to limit the spectral extent of the PWM output by bandlimiting the PWM operator. Using the presented analysis, this can be achieved elegantly by slightly modifying the derived equations describing the PWM operator. PWM is then generated directly from the adapted equations instead of using a comparator.

7 HAUSMAIR et al.: ALIASING-FREE DIGITAL PULSE-WIDTH MODULATION 421 Fig. 6. Simplified illustration of the time domain signals in analog and digital PWM systems: (a) analog modulating signal, digital modulating signal and triangular reference wave, (b) analog pulsed signal,(c)digital pulsed signal. Sampling causes the pulse edges of to appear at slightly different time instants as the edges in. V. BANDLIMITED PWM It is clear from the sampling theorem that larger sampling frequencies will lead to a lower amount of aliasing distortion and therefore yield better results. At the same time it is, however, desirable to keep the sampling frequency as low as possible to ensure feasibility for hardware implementation and to minimize digital power consumption [36]. We therefore propose asimple,flexible and efficient method to limit the bandwidth of the signal generated in the PWM process in order to completely eliminate the aliasing distortion. It can be seen from (13), (15) and (16) that the unlimited number of harmonic components in is responsible for the infinite bandwidth of the pulse-width modulated signal. By limiting the number of harmonic components to a proper amount,suchthat Fig. 7. Simplified illustration of frequency domain signals in analog and digital PWM systems: (a) analog input signal spectrum and the aliases that occur in the digital input signal spectrum at multiples of the sampling frequency, (b) analog pulsed signal spectrum, (c) digital pulsed signal spectrum where the representation according to [35] is used. The digital signal spectrum consists of the analog signal spectrum and aliases around each multiple of the sampling frequency. the bandwidth of the pulsed signal becomes limited and the aliasing effect is no longer present, as is illustrated in Fig. 8. This can be achieved by using a finite Fourier series approximation [37] instead of the complete decomposition for the PWM operator, such that (23) where is given in (8). Comparing (23) with (7), it can be seen that both PWM operators are obtained by using the exact same Fourier coefficients, but the bandlimited PWM operator only comprises coefficients from indices to instead of to. Hence, performing the same steps as in Section III IV, the bandlimited digital pulse-width modulated signal is derived as (24) (22) Using the finite series approximation for the derivation of the PWM operator yields a pulsed signal that is bandlimited and therefore the aliasing effect can be completely eliminated in the digital implementation of the PWM. Note that

8 422 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS, VOL. 60, NO. 2, FEBRUARY 2013 Fig. 8. Limited-bandwidth PWM spectrum without aliasing. VI. RESULTS In this section, the proposed method is evaluated by presenting results obtained in simulations and measurements. For assessing the applicability and performance of the bandlimited PWM in burst-mode RF transmitters, it is more convenient to use the ordinary frequency instead of the angular frequency and the scaled angular frequency, respectively. The wellknown relation between these frequency parameters is,and [35], where is the system sampling frequency in Hz. Fig. 9. Limited-bandwidth PWM spectrum with non-destructive aliasing. this approach also solves the image problem of RF upconverted pulse-width modulated signals discussed in [26]. A side effect of the proposed method is that it induces ripples in the amplitude of the ideal time domain switching signal, which then no longer consists of two levels only. This effect is known as the Gibbs phenomenon [35]. Regardless of the chosen transmitter architecture, due to this effect the proposed method requires the use of a multibit DAC in order to adequately convert the rippled pulses into an analog signal. In addition to that, the signal does not have ideal switching characteristics anymore. When using such a signal for burst-mode operation to drive a linear PA or a single-level SMPA, the non-ideal switching characteristics can lead to nonlinear distortion. This isshowninsectionvi-bforaclass-abpa,andin[30]foran SMPA. However, also the switching characteristics of a digital comparator-based or an analog pulsed signal deteriorate before reaching the PA. This is due to the bandlimitation of the analog circuits of the upconverter and, depending on the PA design, the amplifier input matching networks [3], [8]. A possible way to minimize the amplitude ripples, is to intentionally choose a number of components that induces a certain amount of aliasing. As long as the components do not leak into or near the input signal band, signal recovery will not be affected, as is illustrated in Fig. 9. Another way to mitigate the Gibbs phenomenon is to apply an appropriate filter to the pulsed signal. Note that the effect of amplitude ripples in the time domain pulsed signal is also reported by the authors of [28], [30] after the subtraction of a correction signal from the ideal switching signal. Just as for the ideal analog PWM, the only distortion remaining in the input signal band of the pulsed signal generated with the digital bandlimited PWM, is distortion originating from the overlap with the first harmonic component, which can be kept insignificantly low by an adequate choice of the PWM frequency,asisshowninsectionvi-aandin[20],[21].the proposed PWM method is therefore practically distortion-free in the frequency band of interest. A. Simulation Results In order to verify the performance of the proposed method, the bandlimited PWM was modeled and simulated in Matlab and Simulink environment. The general test settings for all presented simulation results were chosen as follows: The input signal was generated as a discrete-multitone (DMT) signal, since this is a general form of a bandlimited complex baseband signal with nonconstant envelope. The input signal bandwidth was set to MHz, where denotes that the input signal spectrum for.thepapr,i.e.,thecrestfactor,of was db. The sampling frequency was set to MHz. To achieve a frequency domain bin spacing of khz, the number of simulated time domain samples was chosen according to. For the performance evaluation, the normalized dynamic range is used, which is given by (25) where denotes that lies within the input signal band, i.e.,,and denotes that is bounded by,and corresponds to the phase-modulated pulsed signals given in (15) and (21) respectively. 1) Comparison of Bandlimited PWM to Comparator-Based PWM: Fig. 10 shows an example of a spectrum of the phase-modulated pulsed signal generated with the proposed PWM method using harmonics, compared to the spectrum of a phase-modulated signal generated with conventional PWM, which was obtained by comparing the magnitudes of the digital input signal with a digitized triangular reference signal. The PWM frequency was set to MHz. By doing so, the first harmonic component does not produce detectable distortion in the band of interest. The figure illustrates that with a proper, there is no aliasing distortion in the band of interest when using the proposed PWM method. The DR of the conventional pulsed signal is db, while with the proposed method the DR improves to db. On the other hand, the limited number of harmonic components leads to the Gibbs phenomenon in the time domain signal, which can be seen in Fig. 11, where the output signals of both pulse-width modulators without phase modulation are shown. By using a smaller PWM frequency, the number of harmonics that can be used without causing aliasing can be in-

9 HAUSMAIR et al.: ALIASING-FREE DIGITAL PULSE-WIDTH MODULATION 423 TABLE I AVERAGE DRS FOR MHZ FORCOMPARATOR-BASED PWM AND BANDLIMITED PWM TABLE II AVERAGE DRS OFBANDLIMITED PULSED SIGNALS FOR DIFFERENT FIXED-POINT RESOLUTIONS MHZ FOR Fig. 10. Normalized power spectra of the signals, generated with the proposed bandlimited PWM and the conventional comparator-based PWM, respectively. The frequency resolution is khz. Fig. 11. A comparison of the time domain signals of the proposed bandlimited PWM method and the conventional PWM implemented by a comparator. creased, while at the same time the ripples in the time domain signal will decrease. Also, for the comparator PWM a smaller PWM frequency can lead to a better DR, because the aliasing distortion reaching into the band of interest will contain less power. However, the DR of the bandlimited pulsed signal will decrease due to distortion originating from the first harmonic component. Therefore, the ratio between the PWM frequency and the signal bandwidth has to be chosen carefully. This is verified by the results given in Table I, where the performance evaluation of both bandlimited and conventional PWM for different ratios is shown, where the average DRs over different data sets are given. Except for the varying PWM frequency,thesettingswerethesameasintheexamplegiven above. Clearly, in all cases the proposed bandlimited PWM outperforms the conventional PWM. 2) Fixed-Point Implementation of the Bandlimited PWM: When implementing the proposed PWM method in hardware, the signals can assume only a discrete number of amplitude levels, such that amplitude quantization is introduced. In order to estimate requirements for a fixed-point hardware implementation, the bandlimited pulse-width modulator was implemented using the Matlab Simulink fixed-point environment. The PWM frequency was set to MHz and again harmonics were used. Simulations were performed for different numbers of fixed-point bit resolutions.the resulting DRs, averaged over different data sets, are given in Table II. Comparing these results to the results given in Table I, it can be seen that even for computation in low-resolution fixed-point arithmetic, the DRs of bandlimited pulsed signals outperform the comparator generated pulsed signals. B. Measurement Results In order to verify the simulation results by measurements, a lab demonstrator was set up. Measurements were performed to assess the feasibility of the bandlimited PWM with regard to the requirements posed on the DAC, and to evaluate the performance of the proposed PWM method when applied to an RF PA. The PA that was used for the measurements is a linear Class-AB PA, while similar measurements presented in [30] suggest that the PWM method proposed here is also suitable for burst-mode operation of a single-level RF SMPA. 1) Measurement Setup: A block diagram of the measurement setup is depicted in Fig. 12 and a photograph of the lab demonstrator is shown in Fig. 13. The phase-modulated pulse-width modulated signal was created in a Matlab simulation. It was then transferred to an FPGA board, which is connected to a high-bandwidth, high-resolution DAC evaluation module. The FPGA-DAC board combination (TI TSW3100, TI DAC5682Z EVM), courtesy of Texas Instruments, which was used for the setup, provides two channels with MHz input data rate each and offers bit resolution. Since the DAC output is transformer coupled, it cannot handle dc output. Therefore, the baseband signal had to be digitally upconverted to a low intermediate frequency in Matlab, before it was transferred to and processed by the evaluation module. The DAC output signal was then upconverted to MHz RF frequency using a vector signal generator as an IQ-mixer to obtain the real bandpass signal. Due to the low output power of the mixer, the bandpass signal hadtobepre-amplified by a highly linear

10 424 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS, VOL. 60, NO. 2, FEBRUARY 2013 Fig. 12. Block diagram of the measurement setup. The test signal is generated in Matlab and transferred to a pattern generator FPGA board that is connected to a 16 bit dual-channel DAC evaluation module ( MHz per channel). The DAC output is upconverted to the passband, pre-amplified and passed to the PA. Measurements are taken after the mixer and after an attenuator that is connected after the PA. Power supplies are not depicted. Fig. 13. Photo of the lab demonstrator. wideband amplifier, before it was passed to the RF PA. The PA used for the measurements is an W, GHz Class-AB PA with transmission line based input/output matching, fabricated on a Rogers 4530B substrate with an Infineon VLDMOStransistor (engineering sample). The basic properties of the PA are a power added efficiency (PAE) of more than 72% over MHz bandwidth, a gain of db over MHz bandwidth, and peak power of Wover MHz bandwidth, all at MHz center frequency. After the amplification, the signal was attenuated before it could be measured with a power spectrum analyzer (PSA). The general test settings were chosen as in the simulations, with as a DMT signal with a PAPR of db, MHz and MHz. To enable a fair comparison to the simulations, the measurement filter bandwidth of the PSA was set to khz. 2) DR Measurements: In order to allow discerning between the effects of the DAC and the mixer, and the effects of the PA, measurements were taken after the mixer and after the attenuator. Figs. 14 and 15 show signals measured after the mixer. The bandlimited PWM achieves approximately db DR, while the DR of the conventional PWM is around db. Hence, the proposed PWM method shows an improvement of db compared to the comparator-based PWM. Figs. 16 and 17 show the same signals measured after the attenuator. While the comparator-based burst signal has approximately the same DR as before amplification, i.e., db, the Fig. 14. Measurement result of the comparator-based pulse-width modulator after the mixer, db for MHz. Fig. 15. Measurement result of the bandlimited pulse-width modulator after the mixer, db for MHz and. bandlimited burst signal suffers from a large decrease in DR to around db. This is most likely caused by nonlinear effects in the analog circuits of the PA and its matching networks. Due to the amplitude variations and the slower transitions times of the

11 HAUSMAIR et al.: ALIASING-FREE DIGITAL PULSE-WIDTH MODULATION 425 TABLE III DRS BEFORE AND AFTER PA AND AVERAGE PA EFFICIENCIES INPUT SIGNAL WITH MHZ FORDIFFERENT FOR DMT Fig. 16. Measurement result of the comparator-based pulse-width modulator after PA and attenuator, db for MHz. DR is achieved for, which is the maximum number of harmonics that does not induce aliasing in the digital pulsed signal for the chosen settings. 3) Efficiency Measurements: The burst-mode transmitter concept is utilized for its potentially high efficiencies. Therefore, the proposed PWM method is only viable, if the PA efficiency does not suffer from its use. The efficiency that is interesting for RF transmitter applications is the efficiency for the signal that is to be transmitted, i.e.,. Hence, the average PA efficiency in % is computed as the ratio of the average PA output power lying in the signal band of interest, in W, and the average consumed dc power in W [38], [39] (26) Fig. 17. Measurement result of the bandlimited pulse-width modulator after PA and attenuator, db for MHz and. bandlimited pulsed signal, the nonlinearities result in spectral regrowth, which lowers the DR. Nonlinear distortion might also be present in the comparator-based signal, but is not observable in the DR. In any case, appropriate linearization techniques like digital predistortion [8] could ease the problem, but are beyond the scope of this work. Nevertheless, the proposed pulse-width modulator still outperforms the comparator-based modulator by db of DR. Table III gives the DRs for pulsed signals with different number of harmonics before and after the PA, where stands for comparator-based PWM. Results show that when using the aliasing-free PWM method, the DR before the PA is approximately the same for all numbers of harmonics. However, when using fewer harmonics,thedrafterthepa decreases with. This agrees with our interpretation that the amplitude variations and the slower transition times are a main cause for the spectral regrowth caused by PA nonlinearities. However, despite the spectral regrowth we can conclude from the measurements that the bandlimited PWM method has a better DR than the comparator-based PWM method in all cases. Hence, the best result for burst-mode operation in terms of where is measured using the PSA, meaning that there is no actual bandpass filter used. In Table III the average PA efficiencies depending on the number of harmonics are shown. Again, the DMT input signal from the previous sections with MHz and a PAPR of db was used. The measurement results show that there is no efficiency degradation entailed by the use of the proposed PWM method. This might seem surprising, since due to the ripples in the envelope of time domain signal, the PA is not operated in ideal on - and off -states, but also in less efficient regions around those states. There is, however, a reasonable explanation for the maintained PA efficiency. First of all, due to the bandlimitation of the analog circuits of the mixer and the PA matching networks, amplitude variations are also introduced in the envelope of the comparator-generated signal before it is actually amplified. Therefore, also the comparator-based signal is not an ideal burst signal when it reaches the amplifier. Second, of all, the proposed PWM method has a better coding efficiency than the comparator-based PWM, since due to the bandlimitation less harmonic content is generated. It has been shown in [31], [38], [39] that the coding efficiency is one of the main contributing factors for the actually achievable efficiency of burst-mode PAs. For the sake of completeness, measurement results for the PA in linear operation are also given in Table III. In conventional linear operation, db power backoff is necessary to achieve the same DR as for the burst-mode operation. As a consequence, the average efficiency is significantly reduced by more than points. In conclusion, the presented measurements show that if the system parameters are chosen appropriately, the bandlimited

12 426 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS, VOL. 60, NO. 2, FEBRUARY 2013 PWM method outperforms the comparator-based PWM in terms of DR without entailing any decrease in PA efficiency. VII. CONCLUSIONS A method for eliminating aliasing distortion in digital pulse-width modulated signals has been proposed. The method is based on a detailed analysis of a mathematical description of all baseband processing steps in a burst-mode transmitter and has been obtained via a Fourier series decomposition. As demonstrated by measurement results, the proposed bandlimited PWM shows a substantially improved DR compared to conventional comparator-based PWM. The measurements show that a DR improvement is reached even after amplification by a PA, without causing degradation of the PA efficiency. The improved DR potentially makes the reconstruction filtering operation after amplification feasible, thus making the bandlimited PWM suitable for application in burst-mode RF transmitters. However, there are certain trade-offs that have to be considered. While in the spectral domain an improvement in DR can be achieved, in the time domain an amplitude component is superimposed on the ideal pulsed signal. Irrespective of the chosen transmitter architecture, this requires the use of a multibit DAC, whereas for a comparator-based pulse-width modulator a bit DAC can be sufficient. Then again, the comparator-based PWM requires excessively high sampling rates to produce remotely acceptable results, which can be avoided by using the bandlimited PWM. This trade-off between sampling frequency and DAC resolution should be studied further in the context of feasibility of the burst-mode transmitter concept for specific applications. Future work shall include the further investigation of the bandlimited PWM when applied to an RF PA. While the proposed PWM method shows an improved DR also after amplification, effects of nonlinear distortion introduced by the analog circuits can be observed in the measurement results. Therefore, adequate linearization techniques for burst-mode RF transmitters have to be developed, in order to ensure sufficient DR and transmission signal quality after power amplification and reconstruction filtering. ACKNOWLEDGMENT The authors would like to thank Texas Instruments for generously providing the DAC evaluation module (TI DAC 5682Z EVM) and the pattern generator module (TI TSW3100) used in the measurement setup. 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