Efficiency Optimization for Burst-Mode Multilevel ...

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Efficiency Optimization for Burst-Mode Multilevel Radio Frequency Transmitters Shuli Chi, Student Member, IEEE, Peter Singerl, and Christian Vogel, Senior Member, IEEE

Abstract—The utilization of a burst-mode power amplifier (PA) together with pulse-width modulation (PWM) is a promising concept for achieving high efficiency in radio frequency (RF) transmitters. Nevertheless, such a transmitter architecture requires bandpass filtering to suppress side-band spectral components to retrieve the wanted signal, which reduces the transmit power and the transmitter efficiency. High efficiency can only be expected with the maximum transmit power and signals with low peak-toaverage-power ratios (PAPRs). To boost efficiency for signals with high PAPRs and signals at variable transmit power levels, the burst-mode multilevel transmitter architecture has been widely discussed as a potential solution. This paper presents an efficiency optimization procedure of burst-mode multilevel transmitters for signals with high PAPRs and signals at variable transmit power levels. The impact of the threshold value on the transmitter efficiency is studied, where the optimum threshold value and the maximum transmitter efficiency can be obtained according to input magnitude statistics. In addition, the relation between the threshold value and the efficiency expression of burst-mode multilevel transmitters and those of Doherty PAs is investigated. It is shown that the obtained optimum threshold value, although originally designed for burstmode transmitters, can also be applied to Doherty and multistage Doherty PAs to achieve maximum transmitter efficiency. Simulations are used to validate the efficiency improvement of the optimized burst-mode multilevel transmitters compared to twolevel and non-optimized multilevel transmitters. Index Terms—Burst-mode operation, efficiency optimization, multilevel radio frequency (RF) transmitter, probability density function (pdf), pulse-width modulation (PWM)

I. I NTRODUCTION Modern wireless communication systems have been developed to provide higher data rate services to an increasing number of subscribers. To this end, new standards such as Worldwide Interoperability for Microwave Access (WiMAX) [1] and Digital Video Broadcasting - Second Generation Terrestrial (DVB-T2) [2] are utilized, where complex modulated

signals such as quadrature amplitude modulation (QAM) modulated orthogonal frequency-division multiplexing (OFDM) signals are employed for increased data rates and spectral efficiency. Nevertheless, complex modulation techniques lead to envelope-varying signals, resulting in high peak-to-averagepower ratios (PAPRs). A signal with high PAPR requires the power amplifier (PA) to operate mostly at the low-power region, i.e., a large backoff from the maximum transmit power, where the efficiency of conventional linear PAs, such as Class A or Class AB [3], is very low. Besides high PAPRs, the average transmitter efficiency can also be reduced by operating the PA with different average transmit power levels due to the power control strategy [4]. The power control strategy is an important factor in terms of the interference management and energy management. It is used to adjust the transmit power levels of mobile devices and base stations to improve system capacity, providing each user with an acceptable connection while at the same time minimizing the interference to other users and reducing the device’s power consumption. The transmit power level is set according to the distance of the mobile device to the base station, and to the neighboring cells. For example, for 3rd generation (3G) systems which adopt the code division multiple access (CDMA) technique, power control is one effective way to avoid the so-called near-far problem, thus increasing system capacity and reducing intra-cell and inter-cell interferences [5]. In Long Term Evolution (LTE) systems, the uplink power control mitigates inter-cell interference, which is the dominant interference source [6]. Due to the power control strategy, it is common for the PA to operate at 10 dB-40 dB backoff from the maximum average transmit power [7]. Consequently, the PA operates mostly in the low-efficiency region, resulting in a low average efficiency. To avoid possible efficiency degradation issues mentioned above, it is desirable in PAs to provide higher efficiency in the low-power region. Doherty amplifier [8] and Envelope Tracking (ET) [9] techniques have been proposed to boost efficiency for the conventional linear PA operation. Outphasing [10], [11] and Envelope Elimination and Restoration (EER) [12] techniques have been proposed to enhance efficiency by employing saturated or switched-mode PAs [3]. The concept of PAs operated in burst mode [13], [14] is another promising approach towards highly efficient radio frequency (RF) transmitters. The burst-mode PAs can be switched-mode PAs of Class D and Class E [3], or conventional PAs of Class AB and Class B driven into the saturated region. In order to drive such burst-mode PAs in an efficient way, appropriate

Manuscript received XXXX, 20xx. The research leading to these results has received funding from the Zukunftsfonds Steiermark/Land Steiermark under grant agreement n°5061 and the European Union’s Seventh Framework Programme (FP7/2007-2013) under grant agreement n°248277. The Competence Center FTW Forschungszentrum Telekommunikation Wien GmbH is funded within the program COMET Competence Centers for Excellent Technologies by BMVIT, BMWA, and the City of Vienna. The COMET program is managed by the FFG. Shuli Chi is with the Signal Processing and Speech Communication Laboratory, Graz University of Technology, Austria (e-mail:[email protected]). Peter Singerl is with Infineon Technologies AG, Villach, Austria (email: [email protected]). Christian Vogel is with the Telecommunications Research Center Vienna (FTW), Austria (email: [email protected]). Digital Object Identifier XXXXXXXX Copyright ©(c) 2012 IEEE. Personal use of this material is permitted. However, permission to use this material for any other purposes must be obtained from the IEEE by sending an email to [email protected].

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modulation techniques such as Σ∆ modulation [15], [16] and pulse-width modulation (PWM) [17]–[19] are employed to encode the envelope-varying input signal into bursts with constant envelope, where the duty-cycle varies depending on the magnitude information of the input signal. One drawback of this modulation technique is the generated out-of-band spectral components, which need to be filtered out in order to fulfill the transmission spectrum requirements leading to the overall efficiency degradation of the burst-mode transmitter. The authors in [20], [21] used the term coding efficiency to describe the performance of the Σ∆ modulator and to estimate the efficiency for Σ∆ based burst-mode transmitters. In this paper, we adopt the concept of describing the efficiency of PWM based burst-mode transmitters by the coding efficiency and the PA efficiency in burst-mode operation [13], [22]. In contrast to conventional burst-mode two-level transmitters, the utilization of burst-mode multilevel transmitters is able to significantly reduce the undesired out-of-band spectral components. Moreover, the multilevel technique also has the potential to ease the requirements for the bandpass filter (BPF) [18], [19], [23]. A preliminary work on the coding efficiency optimization for three-level burst-mode transmitters has been published in [24]. A. Contributions In non-optimized burst-mode multilevel transmitters, each path is designed to process a signal whose values are bounded by two equally spaced threshold values. In a three-level transmitter, for instance, the normalized input magnitude is separated according to the threshold values [0, 0.5, 1]. This paper presents an extension to the non-optimized transmitter by finding the optimum threshold values to enhance the efficiency for signals with high PAPR and signals at variable average transmit power levels. • Mathematical expressions for the coding efficiency of burst-mode multilevel transmitters with constant input signals are derived. Based on the derived equations, the average coding efficiency is calculated according to the probability density function (pdf) of the input magnitude. The average transmitter efficiency is then described by the average coding efficiency and the PA efficiency in burst-mode operation. • The efficiency optimization procedure for signals with high PAPR and signals at variable average transmit power levels is presented. A set of optimum threshold values is obtained leading to the maximum average transmitter efficiency according to input magnitude statistics. It is shown that the optimized transmitters also relax the requirements on the bandpass filtering. • The optimum threshold values and efficiency expressions of burst-mode multilevel transmitters are related to those of the Doherty and multistage Doherty PA, which have been widely adopted for efficiency enhancement over the last few years [7], [8], [25]–[28]. It is shown that the presented efficiency optimization approach is also suitable for efficiency optimization of Doherty and multistage Doherty PAs.

B. Outline This paper is organized as follows: Section II describes the principle of operation for burst-mode multilevel transmitters. In Section III, the critical power efficiency aspects are described in detail. First, efficiency expressions of burst-mode multilevel transmitters for constant input signals are derived. Then, an analysis of the average efficiency is presented, and the efficiency optimization procedure is derived according to the pdf of the input magnitude. Finally, the efficiency relation between the burst-mode multilevel transmitter and the (multistage) Doherty PA is discussed. Simulation results are given in Section IV to verify the performance of the presented optimization approach. A discussion on the practical implementation considerations is given in Section V before we draw our conclusions in Section VI. II. B URST-M ODE M ULTILEVEL T RANSMITTERS Fig. 1 depicts a block diagram of a burst-mode multilevel transmitter with M signal paths, i.e., (M +1) levels. A complex baseband input signal x(t) is separated into a magnitude signal a(t) and a phase signal φ(t), which can be expressed in polar form as x(t) = a(t)ejφ(t) (1) where ejφ(t) contains the phase information and the magnitude information a(t) is assumed to be normalized to a(t) ∈ [0, 1]. Then a(t) is encoded by the multilevel PWM by comparing a(t) with the reference signal rm (t) in the form of triangular waveform, where the adjacent reference signals are required to be phase shifted by 180◦ to maintain the phase continuity and avoid pulse skipping [23]. The index m denotes the path index of the transmitter and is given with m ∈ [1, M ]. In addition, the reference signal rm (t) is bounded by [Vm−1 , Vm ] where Vl is a set of threshold values defined as  for l = 0   = 0, ∈ (0, 1), for l ∈ [1, M − 1] Vl (2)   = 1, for l = M where l ∈ [0, M ] denotes the index of the threshold values. Applying the multilevel PWM, a series of baseband PWM signals apm (t) based on the magnitude level of a(t) is generated. Accordingly, when a(t) is higher than 0 but lower than the threshold value V1 , only the main PWM operates and generates the main baseband PWM signal ap1 (t) with the magnitude of V1 but varying pulse widths. When a(t) exceeds the threshold value V1 , the first auxiliary baseband PWM signal ap2 (t) consisting of pulses with the magnitude of (V2 − V1 ) is generated. In the meantime, the main PWM outputs the constant value of V1 , indicating that the duty cycle of the main baseband PWM signal is 100%. The operation of other auxiliary PWMs follows the same procedure. Because the PWM leads to a modulated signal consisting of the desired signal and out-of-band spectral components [29], [30], the desired signal can be retrieved later on by a BPF [3]. Fig. 2 illustrates PM=3 an example of a four-level baseband PWM signal m=1 apm (t) with the threshold values of V0 = 0, V1 = 0.3, V2 = 0.5 and V3 = 1. The peak-to-peak magnitude values

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a (t )

r1 ( t )

x ( t ) = a ( t ) e jφ ( t ) Component Separator

Comparator

a p1 ( t )

x p1 , PB ( t )

a pm (t )

x pm , P B (t )

a pM (t )

x p M , PB ( t )

Main

PA

Main

rm ( t ) Comparator

Auxiliary (m-1)

x p , PB ( t )

PA

Combiner

BPF

x PB (t )

Auxiliary (m-1)

rM ( t ) Comparator

Auxiliary (M-1)

PA

Auxiliary (M-1)

RF PAs Pulse-width modulators

φ (t ) Fig. 1.

fc

Phase Modulator

cos(2π f c t + φ ( t ))

Block diagram of a burst-mode multilevel transmitter incorporated with M pulse-width modulators and M RF PAs.

levels compared to the conventional two-level transmitter, because the multilevel transmitter generates less out-of-band power [24]. The optimized transmitter generates the minimum out-of-band power resulting in the maximum transmitter efficiency. The analytical interpretation will be given in detail in the following section.

1 0.9 0.8

Magnitude

0.7 0.6

III. P OWER E FFICIENCY

0.5 0.4 0.3 0.2

Magnitude Input 4-level PWM Reference Signals

0.1 0 1.7

1.8

1.9

2 2.1 Time (µs)

2.2

2.3

Fig. 2. Time domain PWM signals. An (M +1)-level baseband PWM signal P M m=1 apm (t) (solid black) is generated by comparing the input magnitude a(t) (dashed black) with M reference signals rm (t) (light grey) with certain threshold values. These threshold values are V0 = 0, V1 = 0.3, V2 = 0.5 and V3 = 1 for the given four-level (M = 3) example.

of the reference waveforms are 0.3, 0.2 and 0.5, respectively. Note that, in general, the output voltage levels in the multilevel PWM are not necessarily equally spaced. A set of upconverted PWM signals xpm ,P B (t) is obtained by mixing the baseband PWM signals with a phase modulated carrier cos(2πfc t + φ(t)), which is provided by the phase modulator as shown in Fig. 1. This set of upconverted PWM signals is used to drive burst-mode PAs. The main PA is driven by the main upconverted PWM signal xp1 ,P B (t). When the input magnitude exceeds the threshold value V1 , the first auxiliary upconverted PWM signal xp2 ,P B (t) is generated and drives the first auxiliary PA while the main PA remains on. When the input magnitude exceeds the threshold value V2 , the second auxiliary PWM and PA will be activated and so on. A lossless power combiner is assumed to combine the PA outputs, and to pass the generated signal to the BPF. Using the above principle, the multilevel transmitter is able to provide higher efficiency for a large range of output power

OF

B URST-M ODE T RANSMITTERS

Efficiency is a critical factor evaluating the transmitter performance. For burst-mode transmitters, coding efficiency, denoted as ηc , together with the PA efficiency, denoted as ηPA , are considered for evaluating the transmitter efficiency, denoted as η. In this section, the efficiency analysis for constant input signals will be given first. Then the analysis of the average efficiency is presented. The average efficiency is represented by the pdf of the input magnitude, where the input signal can be a signal at the maximum average transmit power level as well as a combination of signals at different average transmit power levels. Then, the average efficiency optimization is performed for burst-mode multilevel transmitters. Finally, the efficiency relation between burstmode multilevel transmitters and (multistage) Doherty PAs is investigated. A. Transmitter Efficiency The transmitter efficiency η describes the ratio of the desired RF inband signal power to the consumed dc power [31]. For burst-mode transmitters, the efficiency can also be described by the PA efficiency and the coding efficiency as [22] η(a)=

Ps (a) = ηPA · ηc (a) Pdc (a)

(3)

where η(a) represents the efficiency for constant input signals, i.e. x(t) = a, a ∈ [0, 1]. The powers Ps (a) and Pdc (a) describe the desired inband power of xP B (t) and the consumed dc power for constant input signals, respectively. The term ηPA represents the efficiency of the RF PA in burst-mode operation. An ideal burst-mode PA operates in two states: “on” and “off”. During the “on” state, the PA is driven into saturation and reaches the peak efficiency ηPA . During the “off” state, the PA is completely off and no power is consumed. For example, for Class B or multilevel Class B PAs, the PA efficiency is

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ηPA = π/4 = 78.5% [32]. For PAs like Class E and Class F−1 , the PA efficiency ηPA is theoretically 100% and measured in the range of 70% − 80% [17], [21]. The coding efficiency for constant input signals is denoted as ηc (a) and will be introduced in the following subsection.

r(t)

D(a)Tp

1 0.8 0.6

B. Coding Efficiency The coding efficiency, which evaluates the performance of the PWM encoder, is defined as the ratio of the desired RF inband signal power relative to the total power of the RF bursts [20], [22]. In general, the coding efficiency for constant input signals is given by   Ps (a) , for a ∈ (0, 1] ηc (a) = Ptot (a) (4)  0, for a = 0

where Ptot (a) is the total power of RF bursts xp,P B (t) for constant input signals. The output of the ideal BPF leads to the desired RF inband signal with the constant input signal as xP B (t) = a cos(2πfc t),

a ∈ [0, 1].

with a normalized load value of 1 Ω. The assumed load value makes no difference for the efficiency calculation because the ratio of the signal power to the total power is the same. Note that the signal power Ps (a) depends on the input magnitude a. The total power Ptot (a), on the other hand, changes according to the employed transmitter architecture. This will be analyzed in the following subsections. 1) Burst-Mode Two-level Transmitters (M = 1): A burstmode two-level transmitter contains only one signal path. The coding efficiency for a two-level transmitter, which has been investigated in [22], is briefly introduced here for comparison. Fig. 3 shows the time domain signals for a burst-mode twolevel transmitter. The total power of the RF bursts xp,P B (t) is calculated as Z D(a)Tp 1 Ptot (a)= (cos 2πfc t)2 dt Tp 0 1 = a, a ∈ [0, 1] (7) 2 where D(a) describes the duty-cycle of the main PWM output and is described by a ∈ [0, 1].

(8)

Therefore, according to (4) with (6) and (7), the coding efficiency for the two-level transmitter is given by ηc (a) =

1 2 2a 1 2a

= a,

a ∈ [0, 1].

x(t)

0.2 0 −0.2

Tp

ap (t)

−0.4 −0.6

xp,P B (t)

−0.8 −1 0

0.01

0.02

0.03 Time (µs)

0.04

0.05

Fig. 3. Time-domain signals for the burst-mode two-level transmitter. The period of the reference signal r(t) is represented by Tp . The duty-cycle is denoted as D(a) which is the ratio of the pulse width to the period Tp .

(5)

Assuming that the input signal is sufficiently long and the carrier frequency fc is much higher than the PWM frequency fp , where fp = 1/Tp , the signal power Ps (a) is closely approximated by Z Tp 1 1 (6) x2P B (t)dt = a2 , a ∈ [0, 1] Ps (a) = Tp 0 2

D(a) = a,

Amplitude

0.4

(9)

It can be seen that the coding efficiency linearly depends on the input magnitude and the coding efficiency reaches the maximum of 100% at the maximum a = 1. 2) Burst-Mode Multilevel Transmitters (M ≥ 2): A block diagram of a burst-mode multilevel transmitter is illustrated in Fig. 1. The coding efficiency for constant signals is categorized into M cases depending on the input magnitude as x(t) = a

for a ∈ (Vm−1 , Vm ]

(10)

where the threshold value Vl is defined in (2) in Section II. For the first case (m = 1) when x(t) = a ∈ (0, V1 ], which is illustrated in Fig. 4(a), the total power of RF bursts can be obtained by Z D1 (a)Tp 1 (V1 cos 2πfc t)2 dt Tp 0 1 = aV1 , a ∈ (0, V1 ] 2

Ptot1 (a)=

(11)

where D1 (a) describes the duty-cycle of the main PWM output and is given by D1 (a) =

a , V1

a ∈ (0, V1 ].

(12)

Therefore, for the first case, the coding efficiency is calculated according to (4) with (6) and (11) as ηc1 (a) =

1 2 2a 1 2 aV1

=

a , V1

a ∈ (0, V1 ].

(13)

It is clear that the coding efficiency for this case also linearly depends on the input magnitude with the efficiency peak obtained at a = V1 . In the second case (m = 2) when x(t) = a ∈ (V1 , V2 ], which is illustrated in Fig. 4(b), the total power of RF bursts

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can be calculated as Z 1  D2 (a)Tp Ptot2 (a)= (V2 cos 2πfc t)2 dt+ Tp 0 Z (1−D2 (a))Tp  (V1 cos 2πfc t)2 dt 0  1 a(V1 + V2 ) − V1 V2 , a ∈ (V1 , V2 ] (14) = 2 where D2 (a) represents the duty-cycle of the first auxiliary PWM output and is given by

1

V2 D1 (a)Tp

x(t)

Amplitude

V1

0

ap1 (t) −V1

Tp

D2 (a) =

xp,P B (t)

−V2

a − V1 , V2 − V1

a ∈ (V1 , V2 ].

(15)

Therefore, for the second case, the coding efficiency is given according to (4) with (6) and (14) by −1

0

0.01

0.02

0.03

0.04

0.05

Time (µs) (a) Time-domain signals with input x(t) = a ∈ (0, V1 ]. 1

ap1 (t) + ap2 (t)

D2 (a)Tp

x(t)

ηc2 (a) =

a2 , a(V1 + V2 ) − V1 V2

ηcm (a)=

Amplitude

0

−V1 −V2 xp,P B (t) −1

0

0.01

0.02

0.03

0.04

0.05

Time (µs) (b) Time-domain signals with input x(t) = a ∈ (V1 , V2 ]. x(t)

Vm

Dm (a)Tp

ap1 (t) + · · · + apm (t)

Vm−1 Amplitude

V1

(16)

which produces the maximum coding efficiency with a = V2 . The coding efficiency for other cases can be generalized from the above analysis. For the mth case as shown in Fig. 4(c), the coding efficiency is expressed as

V2 V1

a ∈ (V1 , V2 ]

a2 , a(Vm−1 + Vm ) − Vm−1 Vm a ∈ (Vm−1 , Vm ]

(17)

which gives the maximum coding efficiency with a = Vm . As a result, the coding efficiency for burst-mode multilevel transmitters for constant input signals a ∈ [0, 1] can be represented in general as  0 for a = 0     a2 (18) ηc (a) =  a(Vm−1 + Vm ) − Vm−1 Vm    for a ∈ (Vm−1 , Vm ], m ∈ [1, M ].

where M represents the total path number of the burst-mode transmitter, and Vl is the threshold value defined by (2). Fig. 5 illustrates coding efficiency curves of different burstmode transmitters for constant input signals. For a (M + 1)level transmitter, there will be M efficiency peaks. The dash-dotted line shows the efficiency curve for the two-level transmitter. As shown, the coding efficiency of multilevel transmitters is better than the two-level case for all input values. Consequently, multilevel transmitters are able to provide higher efficiency compared to the two-level case.

0

C. Average Efficiency The average efficiency is defined as the ratio of the average desired RF inband power P s to the average consumed dc power P dc and is given by [32]

−V1 −Vm−1 xp,P B (t) −Vm

0

0.01

0.02

0.03

0.04

0.05

Time (µs) (c) Time-domain signals with input x(t) = a ∈ (Vm−1 , Vm ]. Fig. 4.

Time-domain signals for the burst-mode multilevel transmitter.

η=

Ps = ηPA η c P dc

(19)

where η c describes the average coding efficiency by [24] R1 2 a fa (a)da Ps ηc = = R 10 2 , a ∈ [0, 1] (20) a P tot f (a)da 0 ηc (a) a

6

D. Efficiency Optimization

100 90

Coding Efficiency (%)

80 70 60 50 40 30

4-level, V1 = 0.25, V2 = 0.5 3-level, V1 = 0.25 3-level, V1 = 0.5 2-level

20 10 0

0

0.2

0.4 0.6 Constant Input a

0.8

1

Fig. 5. Coding efficiency with constant input signals for burst-mode (M +1)level (M = 1, 2, 3) transmitters.

where fa (a) is the pdf of the input magnitude and ηc (a) represents the coding efficiency for constant input signals given in (18). Note that the pdf fa (a) is not limited to representing the distribution of the input magnitude at the maximum average transmit power level, i.e., with input magnitude bounded by [0, 1]. It can also describe the magnitude distribution of signals at different average transmit power levels, or a combination of the above-mentioned signals due to the power control. For example, two sets of signals with the same PAPR are considered, where the average transmit power levels are the maximum and 20 dB backoff from the maximum. Therefore, the signals are described within intervals of [0, 1] and [0, 0.1], respectively. The resulting pdf of the combined signal will have more occurrences at the lower magnitude values, which leads to a higher overall PAPR than the given PAPR. A detailed example of signals at variable average transmit power levels will be given by the end of the following subsection. The average transmitter efficiency is given by combining (19) and (20) R1 2 a fa (a)da η = ηPA R 10 a2 , a ∈ [0, 1]. (21) f (a)da 0 ηc (a) a We can thus calculate the average efficiency using the pdf fa (a), the PA efficiency ηPA , and the coding efficiency ηc (a). For example, the average transmitter efficiency for a twolevel transmitter is R1 2 a fa (a)da η = ηPA R01 , a ∈ [0, 1] (22) afa (a)da 0

obtained from (21) with (9) . Note that, for the generic (M +1)level (M ≥ 2) case, the coding efficiency ηc (a) in (21) depends on the threshold value Vl . Consequently, the choice of Vl has an impact on the average efficiency η. This will be investigated in the following subsection, leading to the optimization concept for burst-mode multilevel transmitters.

In order to study the impact of the threshold values on the average efficiency, first we expand the expression for the average efficiency (21) with the coding efficiency (18). We will then show that it is possible to find a set of threshold values maximizing the average efficiency. We will begin with the efficiency optimization of a burst-mode three-level transmitter and then extend the concept to general burst-mode transmitters. 1) Burst-Mode Three-level Transmitters (M = 2): For a burst-mode three-level transmitter, according to (18), the coding efficiency is given as  a  for a ∈ [0, V1 ]  V 1 (23) ηc (a) = a2   for a ∈ (V1 , 1].  a(V1 + 1) − V1 Substituting ηc (a) in (21), the average efficiency for the burstmode three-level transmitter can be expressed by R1 2 a fa (a)da , a ∈ [0, 1] (24) η = ηPA 0 G(V1 )

where G(V1 ) describes the total power of RF bursts as Z 1 Z 1 afa (a)da− G(V1 ) = V1 afa (a)da+ 0

V1

V1

Z

1

fa (a)da.

(25)

V1

R1 Note that in (24), the average signal power ( 0 a2 fa (a)da) is independent of the threshold values. We can thus obtain the maximum average efficiency by finding the minimum of G(V1 ). To this end, the first-order and second-order derivatives of G(V1 ) with respect to V1 are derived Z 1 Z 1 ∂G fa (a)da (26) = afa (a)da − ∂V1 0 V1 and ∂2G = fa (V1 ). ∂V12 where we used the relation Z 1 ∂ y(ξ)dξ= −y(V1 ). ∂V1 V1

(27)

(28)

It is assumed that the pdf fa (·) is smooth and is non-zero over the definition domain. Hence it follows from (27) that fa (V1 ) > 0, and (25) can provide the minimum of G(V1 ) if there exists a Vˆ1 in (26) such that Z 1 Z 1 fa (a)da= afa (a)da. (29) Vˆ1

0

This value is given by [24] Vˆ1 = F −1 (1 − E(a))

(30)

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Thus the optimum threshold value V1 of three-level transmitters can be analytically determined from the input magnitude pdf according to (30). Using this threshold, the maximum average efficiency is reached as in (31). 2) Burst-Mode (M +1)-level Transmitters (M ≥ 3): In this case, the average efficiency can be expressed by (21) with (18) as R1 2 a fa (a)da η = ηPA 0 , a ∈ [0, 1] (32) G(V1 , ..., VM−1 )

0.2

0.1

0.05

0 −40

Fig. 6. 12 dB.

PAPR (dB) Vˆ1

4-level

The obtained optimum threshold values according to the uniform distribution are the same as the typically chosen non-optimized threshold values. Therefore the non-optimized transmitter only gives the maximum efficiency for uniformly distributed input magnitudes. Typical communication signals have different distributions, which require to solve the optimization problem numerically, where optimization methods such as line searching [33] can be used. The pdf of the input power of discrete multitone (DMT) signals with 7 dB and 12 dB PAPR is shown in Fig 6. A high PAPR leads to more occurrences at low input power. Table I gives the optimum threshold values of three-level and four-level transmitters for DMT signals with different PAPRs when a(t) ∈ [0, 1]. The optimum threshold value Vˆ1 of the three-level transmitter is obtained according to (30), and the optimum threshold values (Vˆ1 , Vˆ2 ) of the four-level

−25 −20 −15 −10 Normalized Input Power (dB)

−5

0

Vˆ1 Vˆ2

7

8

9

10

11

12

0.42

0.40

0.38

0.35

0.33

0.30

0.30

0.27

0.25

0.23

0.21

0.20

0.57

0.53

0.50

0.46

0.44

0.41

TABLE II AVERAGE TRANSMITTER EFFICIENCY FOR DMT SIGNALS WITH DIFFERENT PAPR S WHEN a(t) ∈ [0, 1].

Vm−1

For uniformly distributed signals, the optimum threshold values and the maximum efficiency can be analytically determined as l Vˆl = , l = 0, ..., M (35) M and 1 η max = ηPA (36) 1 . 1 + 2M 2

−30

Pdf of the input power of DMT signals with PAPRs of 7 dB and

3-level

G(V1 , ..., VM−1 ) M Z Vm X (a(Vm−1 + Vm ) − Vm−1 Vm ) fa (a)da. (33) = The maximum average efficiency can be obtained by finding a set of optimum threshold values (Vˆ1 , ..., VˆM−1 ) leading to the minimum of (33). The maximum efficiency is given by R1 2 a fa (a)da , a ∈ [0, 1]. (34) η max = ηPA 0 G(Vˆ1 , ..., VˆM−1 )

−35

TABLE I O PTIMUM THRESHOLD VALUES FOR DMT SIGNALS WITH DIFFERENT PAPR S WHEN a(t) ∈ [0, 1].

where G(V1 , ..., VM−1 ) describes the total power of the RF bursts as

m=1

PAPR 7 dB PAPR 12 dB

0.15 Pdf

where E(·) is the expectation operator and F −1 (·) is the inverse cumulative distribution function (cdf). With (24), (25), (29) and (30), the maximum average efficiency is given by R1 2 a fa (a)da η max = ηPA R01 ˆ1 afa (a)da V R1 2 a fa (a)da = ηPA R 1 0 . (31) afa (a)da F −1 (1−E(a))

PAPR (dB)

7

8

9

10

11

12

2-level

39.7

35.4

31.6

28.4

25.5

22.5

η

3-level (Non-opt.)

64.4

61.3

57.6

53.0

48.9

44.1

(%)

3-level (Opt.)

65.5

63.5

61.7

60.2

58.4

57.2

4-level (Non-opt.)

71.6

70.0

68.1

65.9

62.9

59.3

4-level (Opt.)

72.3

71.5

70.8

69.9

69.1

68.9

transmitter are obtained by finding the minimum of (33). For the DMT signal with the same PAPR but a different a(t)max , the optimum threshold values can be calculated by scaling the values in the Table I with a(t)max . Table II presents the average efficiency for two-level, nonoptimized and optimized three-level and four-level transmitters. The PAs are assumed in ideal Class B operation and the efficiency is therefore ηPA = 78.5%. For two-level transmitters the average efficiency is calculated from (22). For non-optimized transmitters the average efficiency is obtained by (32) with equally spaced threshold values. For optimized three-level and four-level transmitters the average efficiency is calculated according to (31) and (34) with the optimum threshold values given in Table I, respectively. It is shown that the efficiency of the optimized transmitters outperforms two-level and non-optimized transmitters, especially for signals with high PAPRs. For the signal with 12 dB PAPR, the optimized three-level transmitter achieves an efficiency improvement of 13.1% points compared to the non-optimized case, and 34.7% points compared to the two-level case. For the four-level case, the improvements are 9.6% points and 46.4% points, respectively. Fig. 7(a) shows a pdf (power profile) of a representative

8

0.1

according to Fig. 7(a). The PAPR of the combined signals is 22 dB. The corresponding optimum threshold values and average efficiency are shown in Table III and Table IV. The ideal Class B operation efficiency is assumed for the calculation. The efficiency improvement obtained by the optimized three-level transmitter is 17.4% points compared to the nonoptimized case and 28.8% points compared to the two-level case. For the four-level case, the improvements are 24.3% points and 45.3% points, respectively.

0.08

Pdf

0.06 0.04 0.02 0 −50

−40 −30 −20 −10 Normalized Average Transmit Power (dB)

0

(a) Pdf (power profile) of the normalized average transmit power of a representative CDMA mobile transmitter. 0.12 0.1

Pdf

0.08 0.06 0.04 0.02 0 −70

−60

−50 −40 −30 −20 Normalized Input Power (dB)

−10

0

(b) Pdf of the input power of the combined DMT signals with different average transmit power levels according to (a). Fig. 7. Pdf of (a) a representative CDMA mobile transmitter average transmit power and (b) the input power of the combined DMT signals with different average transmit power levels according to (a). TABLE III O PTIMUM THRESHOLD VALUES FOR THE GIVEN POWER PROFILE . 3-level Vˆ1 4-level (Vˆ1 , Vˆ2 )

0.16 (0.08, 0.31)

TABLE IV AVERAGE TRANSMITTER EFFICIENCY FOR THE GIVEN POWER PROFILE .

2-level η (%)

12.2

3-level

3-level

4-level

4-level

(Non-opt.)

(Opt.)

(Non-opt.)

(Opt.)

23.6

41.0

33.2

57.5

CDMA mobile transmitter average transmit power according to [7], [34]. In order to perform the efficiency optimization for signals at variable average transmit power levels, it is assumed that the transmit power changes linearly with the input power [27], and the power profile describes a combination of DMT signals with the same PAPR of 7 dB but different average transmit power levels, i.e., different input magnitude intervals. The occurrence of the DMT signal at a certain average transmit power level can be found in Fig. 7(a). Therefore a pdf describing the distribution of overall input magnitudes of the combined DMT signals can be obtained, denoted as fa (a), and is used for the efficiency optimization. Fig. 7(b) gives the pdf of the input power of the combined DMT signals with different average transmit power levels

E. Efficiency Relation to the Doherty PA The Doherty PA is well known for its potential to deliver high efficiency [8], [27]. Many publications have investigated extended Doherty PAs [7], [26] and multistage Doherty PAs [25], [28] to further enhance the efficiency over classic Doherty PAs. For an ideal Doherty PA based on ideal Class B PAs, according to [32], the Doherty PA efficiency for a constant output voltage vo is  1 vo   , for vo ∈ [0, αvmax ] ηPA · ·   α vmax   o )2 ( vvmax (37) ηDPA (vo ) = η · , PA  vo  (α + 1) − α  v  max   for vo ∈ (αvmax , vmax ]

where vmax is the maximum output voltage level, and the PA efficiency is ηPA = 78.5%. The design parameter α represents the voltage division ratio. This ratio determines the contribution of the main PA and the auxiliary PA to the output power of the Doherty PA. When a Doherty PA is backed off from full power to lower power output levels, it is assumed that the output power changes linearly with the input power [27]. Therefore, according to (37), the output voltage vo can be first normalized to the maximum output voltage vmax , and then related to the normalized input voltage a. This leads to the Doherty PA efficiency for constant input signals x(t) = a, a ∈ [0, 1] of  a  for a ∈ [0, α]  ηPA · α (38) ηDPA (a) = a2   ηPA · for a ∈ (α, 1] a(α + 1) − α where ηDPA (a) represents the efficiency for the Doherty PA. For a classic Doherty PA, the ratio α is set to 0.5. If α is set to values lower than 0.5, the main amplifier reaches saturation at a lower power level. This extended Doherty PA has been used for efficiency enhancement for transmit power over a wide range of power levels [7]. In order to show the efficiency relation between the burstmode multilevel transmitters and the Doherty PAs, the efficiency for a three-level burst-mode transmitter is given according to (3) and (18) as  a  for a ∈ [0, V1 ]   ηPA · V 1 (39) η(a) = a2   for a ∈ (V1 , 1].  ηPA · a(V1 + 1) − V1

9

+Vdc Combiner

Combiner Preprocessing

a (t ) r1 (t )

Comparator

a p1 (t )

RTL =

Gain

Main

λ

V1

x (t ) = a(t )e jφ ( t ) Component

4

+Vdc

R0 V1

transmission line

Separator

+ Load

r2 (t )

Comparator

a p2 (t )

R0

φ (t ) Fig. 8.

-

Delay

Auxiliary

fc

x p , PB (t )

Phase Modulator

1 4 fc

Simulation setup of a burst-mode three-level transmitter.

Based on the analysis before, we know that the threshold value V1 also determines the input power to the main and auxiliary PAs. Therefore, the design parameter α in Doherty PAs and the threshold value V1 in burst-mode multilevel transmitters is considered to have the same functionality. Comparing (38) with (39), it is clear that the efficiency expressions of the Doherty PA and the burst-mode three-level transmitter are identical for constant input signals. For a three-stage Doherty PA, the efficiency curve can be described as [28]  a  ηPA · for a ∈ [0, α]   α    2  a ηDPA (a) = ηPA · a(α + β) − αβ for a ∈ (α, β] (40)     a2   for a ∈ (β, 1]  ηPA · a(β + 1) − β

where α and β, which are the inverse of γ2 and γ1 in [28], determine the contribution of the PAs to the overall output power, and the transition points where the auxiliary PAs are turned on. Note that in (40) the output voltage is already related to the normalized input. The efficiency expressions for the three-stage Doherty PA and the four-level burst-mode transmitter are the same when we compare (40) to (18) with M = 3, and with the knowledge of α, β having the same functionality with V1 , V2 . This relation can be generalized for other multistage Doherty PAs and burst-mode multilevel transmitters. IV. S IMULATION R ESULTS A. Simulation Setup Simulations were carried out in Matlab and Simulink environment using the SimPowerSystems toolbox. The SimPowerSystems toolbox is designed for simulating and modeling electrical power systems [35]. Fig. 8 illustrates the setup of the simulated burst-mode threelevel transmitter. The main and the auxiliary PA are in Class B operation. The transistors of the PAs are modeled as voltage driven current sources including the nonlinear knee effect. The capacitors are ideally modeled. The inductors connected with the supply voltage are modeled with a small ohmic loss mainly to guarantee the simulator convergence. The output signals from the main and auxiliary PAs are combined through a

lossless quarter wavelength transmission line (TL) to achieve load modulation. The characteristic impedance of the TL is R0 RTL = (41) V1 where R0 = 25 Ω is the load impedance used in the simulations, and V1 is the threshold value. In the low-power operation when only the main PA is turned on, the impedance seen by the main PA is R0 l (42) Rmain = 2. V1 In the high-power operation when both PAs are on, the impedances seen by the main PA and the auxiliary PA are R0 R0 h h Rmain = , Raux = . (43) V1 (1 − V1 ) The varying impedances result in the gain fluctuation and the efficiency degradation as described in [25], [36], [37]. A practical method to improve the gain flatness is to divide the input power in an unequal way [7], [36]. For burstmode transmitters, the reduction of the input magnitude to the PWM will not decrease the burst magnitude to the PA as required but the burst duration. Therefore the method introduced in [38], indicated by dashed boxes, is used to achieve the unequal input voltage division in each operation region, leading to proper power combining. The gain inside the combiner preprocessing block is set to V1 according to the currents from PAs in different power operation regions, which are inversely proportional to the impedances. When only the main PA is turned on, the magnitude of the signal after the combiner preprocessing block in the main path is V12 . When both PAs are turned on, the magnitudes of the signals in the main and auxiliary paths are V1 and (1 − V1 ), respectively. By doing so, the gain flatness and efficiency for burst-mode transmitters are improved. A delay block in the auxiliary path is used to compensate the delay (phase shifting) between the PA outputs, where the delay value is calculated as λ 4

1 (44) c 4fc where λ is the wavelength of the radio wave at the carrier frequency fc = 1 GHz, and c is the speed of light, set to ∆=

=

10

RTL1 =

R0 , V1 V2

RTL2 =

R0 V2

(45)

where (V1 , V2 ) are the threshold values of four-level burstmode transmitters and R0 = 16.7 Ω. In the low-power operation, the impedance seen by the main PA is l Rmain =

80 70

Transmitter Efficiency (%)

3 × 108 m/s. The desired signal is retrieved later on by ideal bandpass filtering. The simulation setup for other burst-mode multilevel transmitters can be generalized from this burst-mode three-level transmitter architecture. For a 4-level burst-mode transmitter, the impedance of the TL that connects the main PA and the first auxiliary PA, and the impedance of the TL that connects two auxiliary PAs and the load R0 are [28]

60 50 40

20 10

R0 . V12

(46) 0

In the medium-power operation, the impedances seen by the main PA and the first auxiliary PA are m Rmain =

R0 , V1 V2

m Raux1 =

R0 . V2 (V2 − V1 )

(47)

Ps (a) Pdc (a)

(49)

where Ps (a) is the signal power of xp,P B (t) in load lying in the signal band of interest. The average efficiency for the DMT signal at the maximum average transmit power level is Ps η= . (50) P dc A complex bandwidth of Bs = 5 MHz is used for the DMT signal, therefore P s is the signal power of xp,P B (t) in load lying in 5 MHz bandwidth. For signals at variable average transmit power levels, the average efficiency is evaluated by R1 2 a fa (a)da η = R 10 2 (51) a 0 η(a) fa (a)da where η(a) represents the simulated transmitter efficiency for constant input signals with different threshold values. B. Efficiency for Constant Input Signals Fig. 9 and Fig. 10 show the simulated efficiency curves of two-level, three-level, and four-level burst-mode transmitters with different threshold values for constant input signals. The dashed lines depict the efficiency curves for the non-optimized transmitters. The solid lines depict the efficiency curves of the

0.4 0.6 Constant Input a

0.8

1

70

Transmitter Efficiency (%)

η(a) =

0.2

80

R0 R0 R0 h h , Raux1 = .(48) , Raux2 = V1 V2 V2 (1 − V1 ) 1 − V2

The combiner preprocessing is performed according to the different impedances seen by the PAs, where the gain in the main path and the first auxiliary path is set to V1 and V2 , respectively. The delays in the first and the second auxiliary paths are ∆ and 2∆, respectively. The transmitter efficiency for constant input signals is

0

Fig. 9. Simulated efficiency curves of two-level and three-level transmitters with different threshold values for constant input signals.

In the high-power operation, the impedances seen by the main PA and two auxiliary PAs are h Rmain =

Non-optimized, V1 = 0.5 Vˆ1 = 0.42 for PAPR 7dB Vˆ1 = 0.40 for PAPR 8dB Vˆ1 = 0.38 for PAPR 9dB Vˆ1 = 0.35 for PAPR 10dB Vˆ1 = 0.33 for PAPR 11dB Vˆ1 = 0.30 for PAPR 12dB 2-level

30

60 50 40

Non-optimized, (V1 , V2 ) = (0.33, 0.67) (Vˆ1 , Vˆ2 ) = (0.30, 0.57) for PAPR 7dB (Vˆ1 , Vˆ2 ) = (0.27, 0.53) for PAPR 8dB (Vˆ1 , Vˆ2 ) = (0.25, 0.50) for PAPR 9dB (Vˆ1 , Vˆ2 ) = (0.23, 0.46) for PAPR 10dB (Vˆ1 , Vˆ2 ) = (0.21, 0.44) for PAPR 11dB (Vˆ1 , Vˆ2 ) = (0.20, 0.41) for PAPR 12dB 2-level

30 20 10 0

0

0.2

0.4 0.6 Constant Input a

0.8

1

Fig. 10. Simulated efficiency curves of two-level and four-level transmitters with different threshold values for constant input signals.

optimized transmitters for DMT signals with different PAPRs. The optimum threshold values are given in Table I. In the figures, it is shown that the efficiencies of the optimized and non-optimized multilevel transmitters are considerably higher than the two-level case. It is also shown that the efficiency curves of the optimized transmitters reach the initial peaks at lower input compared to the non-optimized cases. Therefore, the optimized transmitters can provide higher efficiency for signals with high PAPRs, which will be demonstrated in the following subsection. C. Average Efficiency for DMT Signals with Different PAPRs Fig. 11 and Fig. 12 show the simulated average efficiency for three-level and four-level burst-mode transmitters with

11

5 0

55

−5

50 Normalized Power (dBc)

Average Transmitter Efficiency (%)

60

45 40 35

PAPR PAPR PAPR PAPR PAPR PAPR

30 25 20 0.2

0.3

7dB, Vˆ1 = 0.42 8dB, Vˆ1 = 0.40 9dB, Vˆ1 = 0.38 10dB, Vˆ1 = 0.35 11dB, Vˆ1 = 0.33 12dB, Vˆ1 = 0.30

0.4

−10

DR

−15 −20 −25 −30 −35 −40

3-level non-optimized 3-level optimized

−45

0.5 0.6 0.7 Threshold Value V1

0.8

0.9

−50 0.95

1

Fig. 11. Average transmitter efficiency of three-level burst-mode transmitters for DMT signals with different PAPRs. Markers represent the average efficiency obtained with the calculated optimum threshold value Vˆ1 . The nonoptimized cases are shown with V1 = 0.5, and the two-level cases are included with V1 = 1 for the comparison purpose.

1 Frequency (GHz)

1.05

Fig. 13. Output spectra of the optimized and the non-optimized three-level transmitters with the DMT signal of 12 dB PAPR. The DRs are in the same range of 30 dB.

5

0.7

0

60

0.5

40

0.4

30

0.3

20

0.2

Normalized Power (dBc)

Threshold Value V2

50

Average Transmitter Efficiency (%)

−5

0.6

−10

−20 −25 −30 −35 −40

−50 0.95

0.2

0.3 0.4 0.5 Threshold Value V1

0.6

0

0.7

Fig. 12. Average transmitter efficiency of four-level burst-mode transmitters for the DMT signal with 12 dB PAPR. Contour levels of 25% − 60% with an interval of 5%, and 63.5% are shown in the figure. The average efficiency of 63.5% is obtained with the optimum threshold value (Vˆ1 , Vˆ2 ) = (0.20, 0.41). TABLE V S IMULATED AVERAGE TRANSMITTER EFFICIENCY FOR DMT SIGNALS WITH DIFFERENT PAPR S WHEN a(t) ∈ [0, 1]. PAPR (dB) η (%)

4-level non-optimized 4-level optimized

−45

10

0.1 0.1

DR

−15

7

8

9

10

11

12

2-level

37.2

33.2

29.8

26.8

24.2

21.5

3-level (Non-opt.)

59.4

56.7

53.7

49.9

46.3

42.2

3-level (Opt.)

60.1

58.4

57.2

56.1

54.6

53.7

4-level (Non-opt.)

64.8

63.7

62.6

61.1

58.8

55.9

4-level (Opt.)

65.3

64.4

64.4

63.9

63.5

63.5

1 Frequency (GHz)

1.05

Fig. 14. Output spectra of the optimized and the non-optimized four-level transmitters with the DMT signal of 12 dB PAPR. The DRs are in the same range of 30 dB.

different threshold values for signals with different PAPRs. In Fig. 11, the markers represent the average efficiency obtained with the calculated optimum threshold values Vˆ1 . The nonoptimized cases are shown with V1 = 0.5, and the two-level cases are included with V1 = 1 for the comparison purpose. For the DMT signal with 12 dB PAPR, an improvement of 11.5% points is obtained with the optimized three-level transmitter compared to the non-optimized case. Compared to the two-level case, an improvement of 32.2% points can be achieved. In Fig. 12, contour levels of 25% − 60% with an interval of 5%, and 63.5% are shown. The average efficiency of 63.5% is obtained with the calculated optimum threshold

12

where Xp,P B (f ) is the spectrum of the combined PA output signal xp,P B (t). The frequency f ∈ fin denotes that f lies within the signal band of interest, i.e. |f − fc | ≤ Bs /2, and f ∈ fout denotes that f is bounded by Bs /2 < |f −fc | ≤ 2Bs . Fig. 13 shows the frequency spectra Xp,P B (f ) with the 12 dB PAPR DMT signal of non-optimized and optimized three-level transmitters. Note that signals to be transmitted to the antenna require bandpass filtering of the shown signals in the figures. The Hann window is used to mitigate the leakage effect. It can be seen that the DRs of these two output signals are comparable, which are around 30 dB. With the same signal applied to the four-level transmitters, nonoptimized and optimized transmitters give the DRs of around 30 dB as shown in Fig 14. In general, as observed from the simulations, DRs of the PA output from non-optimized and optimized transmitters are in the same range. Linearization techniques such as feed-forward or digital predistortion can be applied to further improve the linearity [3], [26], [39]. It can be seen from the figures that the optimized transmitters produce fewer out-of-band spectral components, which improve the coding efficiency and therefore the overall efficiency. Moreover, the use of the optimized transmitters can relax the requirements of the BPF. With more levels, the out-of-band spectral components can be further reduced, however, at the cost of an increased hardware complexity. D. Average Efficiency for Average Transmit Power over a Wide Range of Power Levels The power profile and the generated pdf of the combined signals with different average transmit power levels used in this subsection are shown in Fig. 7. The simulated efficiency curves η(a) for constant input signals are shown in Fig. 15. It can be seen that the first efficiency peak for the four-level case is slightly lower due to the lower PA efficiency for low-power input signals with small threshold values. Table VI shows the average efficiency for two-level, nonoptimized and optimized three-level and four-level transmitters according to (51). It is shown that the efficiency of the optimized three-level transmitter gives an improvement of 15.5% points compared to the non-optimized case. Compared to the two-level case, the optimized three-level transmitter achieves an improvement of 26.6% points. With the optimized fourlevel transmitter, the efficiency improvement is 19.3% points compared to the non-optimized case, and 39.9% compared to the two-level case. Note that the optimized three-level transmitter outperforms the non-optimized four-level transmitter,

80 70

Transmitter Efficiency (%)

values of (0.20, 0.41). For the DMT signal with 12 dB PAPR, an improvement of 7.6% points can be obtained compared to the non-optimized case, and 42% points compared to the two-level case. The results are summarized in Table V. The difference between the simulation results in Table V and the calculated results in Table II is due to the PA losses. In order to evaluate the linearity performance of the output spectrum, the normalized dynamic range (DR) is used i h  maxf ∈fin |Xp,P B (f )|2 i h DR = 10 log10  (52) 2 maxf ∈fout |Xp,P B (f )|

60 50 40 30

2-level 3-level 3-level 4-level 4-level

20 10 0

0

0.2

optimized non-optimized optimized non-optimized

0.4 0.6 Constant Input a

0.8

1

Fig. 15. Simulated efficiency curves of two-level, three-level and four-level transmitters with constant input signals for the given power profile. TABLE VI E VALUATED AVERAGE TRANSMITTER EFFICIENCY FOR THE GIVEN POWER PROFILE .

2-level η (%)

12.0

3-level

3-level

4-level

4-level

(Non-opt.)

(Opt.)

(Non-opt.)

(Opt.)

23.1

38.6

32.6

51.9

where the use of the three-level transmitter will circumvent the design complexity and cost compared to the four-level case. The optimum threshold values (V1 , V2 ) = (0.08, 0.31) will result in a higher impedance seen by the main PA in the low-power operation than in the high-power operation. It is known that too large impedance ratios should be avoided since they will lead to efficiency degradation in power backoff as discussed in [37]. A possible way to mitigate this problem is to obtain a new set of threshold values that gives a higher value of V1 . For example, with (V1 , V2 ) = (0.1, 0.34), the obtained average transmitter efficiency is 52.7%, which is higher than the average transmitter efficiency with the optimum threshold values due to a higher transmitter efficiency in the power backoff. If the value of V1 is further increased, the average transmitter efficiency may decrease due to a lower average coding efficiency because the threshold values are far away from the optimum threshold values. For example, with (V1 , V2 ) = (0.15, 0.4) and (V1 , V2 ) = (0.2, 0.45), the obtained average efficiencies are 49.6% and 44.8%, respectively, which still have 17% and 12.2% points efficiency improvements compared to the non-optimized cases, 37.6% and 32.8% points efficiency improvements compared to the two-level cases. V. P RACTICAL I MPLEMENTATION C ONSIDERATIONS From the analysis in Section III and Section IV, it can be seen that the threshold value has a very important impact on the transmitter design for efficiency enhancement. First, it determines when the auxiliary PA is activated. Accordingly,

13

the coding efficiency and the transmitter efficiency can be evaluated. Second, it determines the impedances seen by PAs and therefore the gain property of the transmitter. The gain fluctuation in burst-mode transmitters can be mitigated by delivering unequal input voltage to each PA at different power operation regions as described in Section IV-A. Third, the threshold value determines the transistor periphery ratio of the main PA and the auxiliary PAs [28]. For example, the transistor periphery ratio is V1 : (1 − V1 ) for a three-level case and V1 V2 : (1−V1 )V2 : (1−V2 ) for a four-level case. Note that the use of too large transistor ratios should be avoided since the efficiency in the low-power region would be decreased due to the device parasitics, matching and combining losses [37]. A possible way to mitigate this problem is to penalize small magnitudes and therefore increase the threshold values. The PA design is also important for burst-mode transmitters. Switched-mode PAs and harmonics-tuned PAs [3], e.g., in Class F mode, can be used to further improve the peak efficiency for constant input signals as well as the average efficiency for statistical signals without degrading the linearity performance. To operate the PA in a highly efficient way, burst-mode operation requires the input matching network to provide a relatively large bandwidth to ensure an approximately rectangular envelope at the transistor gate. The design of such wideband matching network is challenging especially if a high gain is required. Packaged transistors with partly built-in matching networks would be preferred to achieve a large bandwidth as well as a high efficiency [40]. Another important aspect in the optimization is the number of signal paths M used in burst-mode transmitters. On the one hand, as shown in Section III, a large M leads to a higher possible average efficiency for statistical signals. On the other hand, a large M results in an increased hardware complexity including the required signal processing, additional PAs and a more complex power combiner. As shown in Section III and Section IV, optimized transmitters with a small M can achieve a significant improvement over non-optimized transmitters for signals with high PAPRs and signals over a wide range of power levels. They can even achieve a higher efficiency compared to non-optimized transmitters with a larger M as shown in Section IV-D and therefore circumvent the hardware complexity. VI. C ONCLUSIONS In this paper, efficiency optimization of burst-mode multilevel transmitters has been investigated. For this purpose, detailed mathematical descriptions of the transmitter efficiency utilizing the PA efficiency and the coding efficiency have been presented. The maximum average efficiency can be achieved by changing the threshold values according to input magnitude statistics. The efficiency expressions of the burst-mode multilevel transmitters have been related to those of the Doherty PA and the multistage Doherty PA. It has shown the same functionality of the threshold values in burst-mode multilevel transmitters and the design parameters in (multistage) Doherty PAs. This result makes the presented optimization procedure also suitable for the efficiency optimization of (multistage) Doherty PAs.

Simulations of burst-mode multilevel transmitters have been performed to verify the analytical results. For signals with high PAPRs and signals at variable average transmit power levels, the optimized burst-mode multilevel transmitters show a major efficiency improvement compared to the non-optimized and the two-level cases. For the DMT signal with 12 dB PAPR, efficiency improvements of 11.5% and 32.2% points are obtained with the optimized three-level burst-mode transmitter compared to the non-optimized and the two-level cases. With the optimized four-level case, the improvements are 7.6% and 42% points, respectively. For the given power profile, the optimized three-level burst-mode transmitter achieves improvements of 15.5% and 26.6% points compared to the nonoptimized and the two-level cases. Efficiency improvements of 19.3% and 39.9% points are obtained with the optimized four-level case. This paper provides an approach to obtain the optimum thresholds and the optimized efficiency, which will allow the designers to estimate and predict the efficiency of the optimized transmitter in an early design stage before great effort is put into the complete design prototype. The increasing need of higher efficiency for signals with high PAPRs and signals at variable average transmit power levels makes the optimized burst-mode multilevel transmitter a promising candidate for future wireless communication networks. R EFERENCES [1] D. Pareit, B. Lannoo, I. Moerman, and P. Demeester, “The History of WiMAX: A Complete Survey of the Evolution in Certification and Standardization for IEEE 802.16 and WiMAX,” IEEE Commun. Surveys Tuts., vol. PP, no. 99, pp. 1 –29, 2011. [2] “Digital Video Broadcasting (DVB) – Frame structure channel coding and modulation for a second generation digital terrestrial television broadcasting system (DVB-T2),” European Standard (Telecommunications series) ETSI EN 302 755 V1.1.1, Sep. 2009. [3] S. C. Cripps, RF Power Amplifiers for Wireless Communications, 2nd ed. Norwood, MA, USA: Artech House, Inc., 2006. [4] M. Chiang, P. Hande, T. Lan, and C. W. Tee, Power Control in Wireless Cellular Networks. Foundations Trends Networking, 2008. [5] A. Yener, R. Yates, and S. Ulukus, “Interference Management for CDMA Systems Through Power Control, Multiuser Detection, and Beamforming,” IEEE Trans. Commun., vol. 49, no. 7, pp. 1227–1239, Jul. 2001. [6] A. Simonsson and A. Furuskar, “Uplink Power Control in LTE Overview and Performance, Subtitle: Principles and Benefits of Utilizing rather than Compensating for SINR Variations,” in IEEE 68th Veh. Tech. Conf., pp. 1–5, Sep. 2008. [7] M. Iwamoto, A. Williams, P.-F. Chen, A. Metzger, L. Larson, and P. Asbeck, “An Extended Doherty Amplifier With High Efficiency Over a Wide Power Range,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 12, pp. 2472–2479, Dec. 2001. [8] W. Doherty, “A New High Efficiency Power Amplifier for Modulated Waves,” Proc. IRE, vol. 24, no. 9, pp. 1163–1182, Sep. 1936. [9] Y. Li, J. Lopez, D. Lie, K. Chen, S. Wu, T.-Y. Yang, and G.-K. Ma, “Circuits and System Design of RF Polar Transmitters Using EnvelopeTracking and SiGe Power Amplifiers for Mobile WiMAX,” IEEE Trans. Circuits Syst. I, vol. 58, no. 5, pp. 893–901, May 2011. [10] H. Chireix, “High Power Outphasing Modulation,” Proc. IRE, vol. 23, no. 11, pp. 1370–1392, Nov. 1935. [11] D. Perreault, “A New Power Combining and Outphasing Modulation System for High-Efficiency Power Amplification,” IEEE Trans. Circuits Syst. I, vol. 58, no. 8, pp. 1713–1726, Aug. 2011. [12] L. Kahn, “Single-Sideband Transmission by Envelope Elimination and Restoration,” Proc. IRE, vol. 40, no. 7, pp. 803–806, Jul. 1952. [13] B. Franc¸ois, E. Kaymaksut, and P. Reynaert, “Burst Mode Operation as an Efficiency Enhancement Technique for RF Power Amplifiers,” in GA and Sci. Symp. URSI, pp. 1–4, Aug. 2011.

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Shuli Chi received her M.Sc. degree in electromagnetic field and microwave technology from Beijing University of Posts and Telecommunications, Beijing, China, in 2009, and is currently working toward the Ph.D. degree in electrical and information engineering at Graz University of Technology, Graz, Austria. Her interests include signal processing in wireless communication systems, and digitallyassisted highly-efficient transmitter architectures.

Peter Singerl received his M.Sc. and Ph.D. in electrical engineering from Graz University of Technology, Austria in 2000 and 2006, respectively. From 8/2003 to 3/2007 he was with the Christian Doppler Laboratory for Nonlinear Signal Processing at the Graz University of Technology, Austria. He joined Infineon Technologies Austria in 8/2000, where he is currently working as a concept engineer and engineering manager for wireless communication applications. Peter Singerl’s research interests include the identification and linearization of power amplifiers as well as system concepts for high power radio frequency transmitters. Peter Singerl received two international student best paper awards and has authored and co-authored more than 30 international publications including journal and conference papers and patents.

Christian Vogel (S’02-M’06-SM’10) was born in Graz, Austria, in 1975. He received the Dipl.-Ing. degree in telematik (summa cum laude) and the Dr. techn. degree in electrical and information engineering (summa cum laude) from Graz University of Technology, Austria, in 2001 and 2005, respectively. In 2004, he was visiting researcher at the division of Electronics Systems at Link¨oping University, Sweden, and from 2008 to 2009, he was a postdoctoral research fellow at the Signal and Information Processing Laboratory at ETH Zurich, Switzerland. He is now senior researcher at the Telecommunications Research Center Vienna (FTW) and lecturer at the Signal Processing and Speech Communication Laboratory, Graz University of Technology. His research interests include the design and theory of digital, analog, and mixed-signal processing systems with special emphasis on communication systems and digital enhancement techniques for analog signal processing systems.