RF and Microwave Power Amplifier and Transmitter Part 3 Technologies —

RF and Microwave Power Amplifier and Transmitter Part 3 Technologies —
High Frequency Design
From September 2003 High Frequency Electronics
Copyright © 2003 Summit Technical Media, LLC
RF POWER AMPLIFIERS
RF and Microwave Power
Amplifier and Transmitter
Technologies — Part 3
By Frederick H. Raab, Peter Asbeck, Steve Cripps, Peter B. Kenington,
Zoya B. Popovich, Nick Pothecary, John F. Sevic and Nathan O. Sokal
T
he building blocks
used in transmitters are not only
power amplifiers, but a
variety of other circuit
elements including oscillators, mixers, low-level
amplifiers, filters, matching networks, combiners,
and circulators. The
arrangement of building blocks is known as
the architecture of a transmitter. The classic
transmitter architecture is based upon linear
PAs and power combiners. More recently,
transmitters are being based upon a variety of
different architectures including stage
bypassing, Kahn, envelope tracking, outphasing, and Doherty. Many of these are actually
fairly old techniques that have been recently
made practical by the capabilities of DSP.
Transmitter architectures is
the subject of this installment of our continuing
series on power amplifiers,
with an emphasis on
designs that can meet
today’s linearity and high
efficiency requirements
7a. LINEAR ARCHITECTURE
The conventional architecture for a linear
microwave transmitter consists of a baseband
or IF modulator, an up-converter, and a poweramplifier chain (Figure 20). The amplifier
chain consists of cascaded gain stages with
power gains in the range of 6 to 20 dB. If the
transmitter must produce an amplitude-modulated or multi-carrier signal, each stage must
have adequate linearity. This generally
requires class-A amplifiers with substantial
power back-off for all of the driver stages. The
final amplifier (output stage) is always the
most costly in terms of device size and current
consumption, hence it is desirable to operate
the output stage in class B. In applications
requiring very high linearity, it is necessary to
use class A in spite of the lower efficiency.
34
High Frequency Electronics
Mixer
RF/
Baseband
Exciter
3-stage PA
LO
RX
Figure 20 · A conventional transmitter.
The outputs of a driver stage must be
matched to the input of the following stage
much as the final amplifier is matched to the
load. The matching tolerance for maintaining
power level can be significantly lower than
that for gain [60], hence the 1-dB load-pull
contours are more tightly packed for power
than for gain.
To obtain even modest bandwidths (e.g.,
above 5 percent), the use of quadrature balanced stages is advisable (Figure 21). The
main benefit of the quadrature balanced configuration is that reflections from the transistors are cancelled by the action of the input
and output couplers. An individual device can
therefore be deliberately mismatched (e.g., to
achieve a power match on the output), yet the
quadrature-combined system appears to be
well-matched. This configuration also acts as
an effective power combiner, so that a given
power rating can be achieved using a pair of
devices having half of the required power performance. For moderate-bandwidth designs,
the lower-power stages are typically designed
using a simple single-ended cascade, which in
some cases is available as an RFIC. Designs
with bandwidths approaching an octave or
High Frequency Design
RF POWER AMPLIFIERS
0º
90º
Figure 21 · Amplifier stages with quadrature combiners.
more require the use of quadraturebalanced stages throughout the
entire chain.
Simple linear-amplifier chains of
this kind have high linearity but only
modest efficiency. Single-carrier
applications usually operate the final
amplifier to about the 1-dB compression point on amplitude modulation
peaks. A thus-designed chain in
which only the output stage exhibits
compression can still deliver an
ACPR in the range of about –25 dBc
with 50-percent efficiency at PEP.
Two practical problems are frequently encountered in the design of
linear PA chains: stability and low
gain. Linear, class-A chains are actually more susceptible to oscillation
due to their high gain, and singlepath chains are especially prone to
unstable behavior. Instability can be
subdivided into the two distinct categories: Low-frequency oscillation and
in-band instability. In-band instability is avoided by designing the individual gain stages to meet the criteria for unconditional stability; i.e.,
the Rollet k factor [61] must be
greater than unity for both in-band
and out-of-band frequencies. Meeting
this criterion usually requires sacrificing some gain through the use of
absorptive elements. Alternatively,
the use of quadrature balanced
stages provides much greater isolation between individual stages, and
the broadband response of the
quadrature couplers can eliminate
the need to design the transistor
36
High Frequency Electronics
stage itself with k>1. This is another
reason for using quadrature coupled
stages in the output of the chain.
Large RF power devices typically
have very high transconductance, and
this can produce low-frequency instability unless great care is taken to
terminate both the input and output
at low frequencies with impedances
for unconditional stability. Because of
large separation from the RF band,
this is usually a simple matter requiring a few resistors and capacitors.
At X band and higher, the power
gain of devices in the 10 W and above
category can drop well below 10 dB.
To maintain linearity, it may be necessary to use a similarly size device
as a driver. Such an architecture
clearly has a major negative impact
upon the cost and efficiency of the
whole chain. In the more extreme
cases, it may be advantageous to consider a multi-way power combiner,
where 4, 8, or an even greater number of smaller devices are combined.
Such an approach also has other
advantages, such as soft failure, better thermal management, and phase
linearity. However, it typically consumes more board space.
7b. POWER COMBINERS
The need frequently arises to
combine the outputs of several individual PAs to achieve the desired
transmitter output. Whether to use a
number of smaller PAs vs. a single
larger PA is one of the most basic
decisions in selection of an architec-
ture [60]. Even when larger devices
are available, smaller devices often
offer higher gain, a lower matching Q
factor (wider bandwidth), better
phase linearity, and lower cost. Heat
dissipation is more readily accomplished with a number of small
devices, and a soft-failure mode
becomes possible. On the other hand,
the increase in parts count, assembly
time, and physical size are significant
disadvantages to the use of multiple,
smaller devices.
Direct connection of multiple PAs
is generally impractical as the PAs
interact, allowing changes in output
from one to cause the load impedance
seen by the other to vary. A constant
load impedance, hence isolation of
one PA from the other, is provided by
a hybrid combiner. A hybrid combiner
causes the difference between the
two PA outputs to be routed to and
dissipated in a balancing or “dump”
resistor. In the event that one PA
fails, the other continues to operate
normally, with the transmitter output reduced to one fourth of nominal.
The most common power combiner is the quadrature-hybrid combiner.
A 90° phase shift is introduced at
input of one PA and also at the output of the other. The benefits of
quadrature combining include constant input impedance in spite of
variations of input impedances of the
individual PAs, cancellation of odd
harmonics, and cancellation of backward-IMD (IMD resulting from a signal entering the output port). In
addition, the effect of load impedance
upon the system output is greatly
reduced (e.g., to 1.2 dB for a 3:1
SWR). Maintenance of a nearly constant output occurs because the load
impedance presented to one PA
decreases when that presented to the
other PA increases. As a result, however, device ratings increase and efficiency decreases roughly in proportion to the SWR [65]. Because
quadrature combiners are inherently
two-terminal devices, they are used
in a corporate combining architecture
High Frequency Design
RF POWER AMPLIFIERS
Figure 22 · Multi-section Wilkinson combining architecture.
(Figure 21). Unfortunately, the physical construction of
such couplers poses some problems in a PC-board environment. The very tight coupling between the two quarter-wave transmission lines requires either very fine gaps
or a three-dimensional structure. This problem is circumvented by the use of a miniature co-axial cable having a
pair of precisely twisted wires to from the coupling section or ready-made, low-cost surface mount 3-dB couplers.
The Wilkinson or in-phase power combiner [62] is
often more easily fabricated than a quadrature combiner.
In the two-input form (as in each section in Figure 22),
the outputs from two quarter-wavelength lines summed
into load R0 produce an apparent load impedance of 2R0,
which is transformed through the lines into at the load
impedances RPA seen by the individual PAs. The difference between the two PA outputs is dissipated in a resistor connected across the two inputs. Proper choice of the
balancing resistor (2RPA) produces a hybrid combiner
with good isolation between the two PAs. The Wilkinson
concept can be extended to include more than two inputs
[63].
Greater bandwidth can be obtained by increasing the
number of transforming sections in each signal path. A
single-section combiner can have a useful bandwidth of
about 20 percent, whereas a two-section version can have
a bandwidth close to an octave. In practice, escalating circuit losses generally preclude the use of more than two
sections.
All power-combining techniques all suffer from circuit
losses as well as mismatch losses. The losses in a simple
two-way combiner are typically about 0.5 dB or 10 percent. For a four-way corporate structure, the interconnects typically result in higher losses. Simple open
microstrip lines are too lossy for use in combining structures. One technique that offers a good compromise
among cost, produceability, and performance, uses suspended stripline. The conductors are etched onto doublesided PC board, interconnected by vias, and then sus38
High Frequency Electronics
Figure 23 · Power consumption by
PAs of different sizes.
pended in a machined cavity. Structures of this kind allow
high-power 8-way combiners with octave bandwidths and
of 0.5 dB.
A wide variety of other approaches to power-combining circuits are possible [62, 64]. Microwave power can
also be combined during radiation from multiple antennas through “quasi-optical” techniques [66].
7c. STAGE SWITCHING AND BYPASSING
The power amplifier in a portable transmitter generally operates well below PEP output, as discussed in
Section 4 (Part 1). The size of the transistor, quiescent
current, and supply voltage are, however, determined by
the peak output of the PA. Consequently, a PA with a
lower peak output produces low-amplitude signals more
efficiently than does a PA with a larger peak output, as
illustrated in Figure 23 for class-B PAs with PEP efficiencies of 60 percent. Stage-bypassing and gate-switching techniques [67, 68] reduce power consumption and
increase efficiency by switching between large and small
amplifiers according to signal level. This process is analogous to selection of supply voltage in a class-G PA, and
the average efficiency can be similarly computed [69].
A typical stage-bypassing architecture is shown in
Figure 24. For low-power operation, switches SA and SB
route the drive signal around the final amplifier.
Figure 24 · Stage-bypassing architecture.
High Frequency Design
RF POWER AMPLIFIERS
Figure 25 · Adaptive gate switching.
Simultaneously, switch SDC turns-off
DC power to the final amplifier. The
reduction in power consumption can
improve the average efficiency significantly (e.g., from 2.1 to 9.5 percent in
[70]). The control signal is based upon
the signal envelope and power level
(back-off). Avoiding hysteresis effects
and distortion due to switching transients are critical issues in implementation.
A PA with adaptive gate switching
is shown in Figure 25. The gate width
(hence current and power capability)
of the upper FET is typically ten to
twenty times that of the lower FET.
The gate bias for the high-power FET
keeps it turned off unless it is needed
to support a high-power output.
Consequently, the quiescent drain
current is reduced to low levels unless
actually needed. The advantages of
this technique are the absence of loss
in the switches required by stage
bypassing, and operation of the lowpower FET in a more linear region
(vs. varying the gate bias of a single
large FET). The disadvantage is that
the source and load impedances
change as the upper FET is switched
on and off.
7d. KAHN TECHNIQUE
The Kahn Envelope Elimination
and Restoration (EER) technique
(Figure 26) combines a highly effi40
High Frequency Electronics
Figure 26 · Kahn-technique transmitter.
cient but nonlinear RF power amplifier (PA) with a highly efficient envelope amplifier to implement a highefficiency linear RF power amplifier.
In its classic form [73], a limiter eliminates the envelope, allowing the constant-amplitude phase modulated
carrier to be amplified efficiently by
class-C, -D, -E, or -F RF PAs.
Amplitude modulation of the final RF
PA restores the envelope to the phasemodulated carrier creating an amplified replica of the input signal.
EER is based upon the equivalence of any narrowband signal to
simultaneous amplitude (envelope)
and phase modulations. In a modern
implementation, both the envelope
and the phase-modulated carrier are
generated by a DSP. In contrast to
linear amplifiers, a Kahn-technique
transmitter operates with high efficiency over a wide dynamic range
and therefore produces a high average efficiency for a wide range of signals and power (back-off) levels.
Average efficiencies three to five
times those of linear amplifiers have
been demonstrated (Figure 27) from
HF [74] to L band [75].
Transmitters based upon the
Kahn technique generally have excellent linearity because linearity
depends upon the modulator rather
than RF power transistors. The two
most important factors affecting the
linearity are the envelope bandwidth
and alignment of the envelope and
phase modulations. As a rule of
thumb, the envelope bandwidth must
be at least twice the RF bandwidth
and the misalignment must not
exceed one tenth of the inverse of the
RF bandwidth [76]. In practice, the
drive is not hard-limited as in the
classical implementation. Drive
power is conserved by allowing the
drive to follow the envelope except at
low levels. The use of a minimum
drive level ensures proper operation
of the RF PA at low signal levels
where the gain is low [77]. At higher
microwave frequencies, the RF power
devices exhibit softer saturation
characteristics and larger amounts of
amplitude-to-phase
conversion,
necessitating the use of predistortion
for good linearity [78].
Figure 27 · Efficiency of Kahn-tecnique transmitters.
High Frequency Design
RF POWER AMPLIFIERS
Figure 28 · Class-S modulator.
Figure 29 · Class-G modulator.
Class-S Modulator
A class-S modulator (Figure 28) uses a transistor and
diode or a pair of transistors act as a two-pole switch to
generate a rectangular waveform with a switching frequency several times that of the output signal. The width
of pulses is varied in proportion to the instantaneous
amplitude of the desired output signal, which is recovered
by a low-pass filter. Class S is ideally 100 percent efficient
and in practice can have high efficiency over a wide
dynamic range. Class-S modulators are typically used as
parts of a Kahn-technique transmitter, while class-S
amplifiers are becoming popular for the efficient production of audio power in portable equipment. A class-S modulator can be driven by a digital (on/off) signal supplied
directly from a DSP, eliminating the need for intermediate conversion to an analog signal.
Selection of the output filter is a compromise between
passing the infinite-bandwidth envelope and rejecting
FM-like spurious components that are inherent in the
PWM process. Typically, the switching frequency must be
six to seven times the RF bandwidth. Modulators with
switching frequencies of 500 kHz are readily implemented using discrete MOSFETs and off-the-shelf ICs [74],
while several MHz can be achieved using MOS ASICs or
discrete GaAs devices [75].
(typically 90 percent) is achieved by a diplexing combiner.
Obtaining a flat frequency response and resistive loads
for the two PAs is achieved by splitting the input signals
in a DSP that acts as a pair of negative-component filters
(Figure 30) [79]. The split-band modulator should make
possible Kahn-technique transmitters with RF bandwidths of tens or even hundreds of MHz.
7e. ENVELOPE TRACKING
The envelope-tracking architecture (Figure 31) is similar to that of the Kahn technique. However, the final
amplifier operates in a linear mode and the supply voltage is varied dynamically to conserve power [81, 82]. The
RF drive contains both amplitude and phase information,
and the burden of providing linear amplification lies
entirely on the final RF PA. The role of the variable power
supply is only to optimize efficiency.
Typically, the envelope is detected and used to control
a DC-DC converter. While both buck (step-down) or boost
(step-up) converters are used, the latter is more common
as it allows operation of the RF PA from a supply voltage
higher than the DC-supply voltage. This configuration is
Class-G Modulator
A class-G modulator (Figure 29) is a combination of linear series-pass (class-B) amplifiers that operate from different supply voltages. Power is conserved by selecting the
one with the lowest useable supply voltage [69] so that the
voltage drop across the active device is minimized.
Split-Band Modulator
Most of the power in the envelope resides at lower frequencies; typically 80 percent is in the DC component.
The bandwidth of a class-S modulator can therefore be
extended by combining it with a linear amplifier. While
there are a number of approaches, the highest efficiency
42
High Frequency Electronics
Figure 30 · Split-band modulator.
Figure 31 · Envelope-tracking architecture.
also more amenable to the use of npn or n-channel transistors for fast switching. The result is a minimum VDDRF
corresponding to the DC-supply voltage and tracking of
larger envelopes with a fixed “headroom” to ensure linear
operation of the RF PA. If the RF PA is operated in class
A, its quiescent current can also be varied.
In general, excess power-supply voltage translates to
reduced efficiency, rather than output distortion. In principle, perfect tracking of the envelope by the supply voltage preserves the peak efficiency of the RF PA for all output amplitudes, as in the Kahn technique. In practice,
efficiency improvement is obtained over a limited range of
output power.
A high switching frequency in the DC-DC converter
allows both a high modulation bandwidth and the use of
smaller inductors and capacitors. The switching devices
in the converter can in fact be implemented using the
same same transistor technology used in the RF PA.
Converters with switching frequencies of 10 to 20 MHz
have recently been implemented using MOS ASICs [80],
GaAs HBTs [83, 84] and RF power MOSFETs [85].
Representative results for an envelope-tracking transmitter based on a GaAs FET power amplifier are shown in
Figure 32. The efficiency is lower at high power than that
of the conventional amplifier with constant supply voltage
due to the inefficiency of the DC-DC converter. However,
the efficiency is much higher over a wide range of output
power, with the average efficiency approximately 40 percent higher than that of the conventional linear amplifier.
Spurious outputs can be produced by supply-voltage
ripple at the switching frequency. The effects of the ripple
can be minimized by making the switching frequency sufficiently high or by using an appropriate filter. Variation
of the RF PA gain with supply voltage can introduce distortion. Such distortion can, however, be countered by predistortion techniques [to be covered in Section 8 (Part 4)].
Figure 32 · Efficiency of a GaAs FET envelopetracking transmitter.
7f. OUTPHASING
Outphasing was invented during the 1930s as a
means of obtaining high-quality AM from vacuum tubes
with poor linearity [86] and was used through about 1970
in RCA “Ampliphase” AM-broadcast transmitters. In the
1970s, it came into use at microwave frequencies under
the name “LINC” (Linear Amplification using Nonlinear
Components) [87].
An outphasing transmitter (Figure 33) produces an
amplitude-modulated signal by combining the outputs of
two PAs driven with signals of different time-varying
phases. Basically, the phase modulation causes the
instantaneous vector sum of the two PA outputs to follow
the desired signal amplitude (Figure 34). The inverse
sine of envelope E phase-modulates the driving signals
for the two PAs to produce a transmitter output that is
proportional to E. In a modern implementation, a DSP
and synthesizer produce the inverse-sine modulations of
the driving signals.
Hybrid combining (Figure 33) isolates the PAs from
the reactive loads inherent in outphasing, allowing them
to see resistive loads at all signal levels. However, both
PAs deliver full power all of the time. Consequently, the
efficiency of a hybrid-coupled outphasing transmitter
varies with the output power (Figure 35), resulting in an
average efficiency that is inversely proportional to peakto-average ratio (as for class A). Recovery of the power
from the dump port of the hybrid combiner offers some
improvement in the efficiency [88].
The phase of the output current is that of the vector
sum of the two PA-output voltages. Direct summation of
the out-of-phase signals in a nonhybrid combiner inherently results in reactive load impedances for the power
amplifiers [89]. If the reactances are not partially cancelled as in the Chireix technique, the current drawn from
the PAs is proportional to the transmitter-output voltage.
September 2003
43
High Frequency Design
RF POWER AMPLIFIERS
Figure 33 · Hybrid-combined outphasing transmitter.
Figure 36 · Chireix-outphasing transmitter.
This results in an efficiency characteristic similar to that of a class-B PA.
The Chireix technique [86] uses
shunt reactances on the inputs to the
combiner (Figure 36) to tune-out the
drain reactances at a particular
amplitude, which in turn maximizes
the efficiency in the vicinity of that
amplitude. The efficiency at high and
low amplitudes may be degraded. In
the classic Chireix implementation,
the shunt reactances maximize the
efficiency at the level of the unmodulated carrier in an AM signal and produce good efficiency over the upper 6
dB of the output range. With judicious choice of the shunt susceptances, the average efficiency can be
maximized for any given signal [89,
90]. For example, a normalized susceptance of 0.11 peaks the instantaneous efficiency at a somewhat lower
amplitude, resulting in an average
efficiency of 52.1 percent for an ideal
class-B PA and a 10-dB Rayleighenvelope signal (vs. 28 percent for lin-
Development of the Doherty technique in 1936 [92] was motivated by
the observation that signals with significant
amplitude
modulation
resulted in low average efficiency.
The classical Doherty architecture
(Figure 37) combines two PAs of
equal capacity through quarter-wave-
Figure 34 · Signal vectors in outphasing.
Figure 35 · Efficiency of outphasing
transmitters with ideal class-B PAs.
44
High Frequency Electronics
ear amplification).
Virtually all microwave outphasing systems in use today are of the
hybrid-coupled type. Use of the
Chireix technique at microwave frequencies
is
difficult
because
microwave PAs do not behave as ideal
voltage sources. Simulations suggest
that direct (nonhybrid) combining
increases both efficiency and distortion [91]. Since outphasing offers a
wide bandwidth and the distortion
can be mitigated by techniques such
as predistortion, directly coupled and
Chireix techniques should be fruitful
areas for future investigation.
7g. DOHERTY TECHNIQUE
length lines or networks. The “carrier” (main) PA is biased in class B
while the “peaking” (auxiliary) PA is
biased in class C. Only the carrier PA
is active when the signal amplitude is
half or less of the PEP amplitude.
Both PAs contribute output power
when the signal amplitude is larger
than half of the PEP amplitude
Operation of the Doherty system
can be understood by dividing it into
low-power, medium-power (load-modulation), and peak-power regions
[96]. The current and voltage relationships are shown in Figure 38 for
ideal transistors and lossless matching networks. In the low-power
region, the instantaneous amplitude
of the input signal is insufficient to
overcome the class-C (negative) bias
of the peaking PA, thus the peaking
PA remains cut-off and appears as an
open-circuit. With the example load
impedances shown in Figure 37, the
carrier PA sees a 100 ohm load and
operates as an ordinary class-B
amplifier. The drain voltage increases
linearly with output until reaching
supply voltage VDD. The instantaneous efficiency at this point (–6 dB
from PEP) is therefore the 78.5 percent of the ideal class-B PA.
As the signal amplitude increases
into the medium-power region, the
carrier PA saturates and the peaking
PA becomes active. The additional
current I2 sent to the load by the
peaking PA causes the apparent load
impedance at VL to increase above
High Frequency Design
RF POWER AMPLIFIERS
Figure 37 · Doherty transmitter.
the 25 ohms of the low-power region.
Transformation through the quarterwavelength line results in a decrease
in the load presented to the carrier
PA. The carrier PA remains in saturation and acts as a voltage source. It
operates at peak
efficiency,
but
delivers an increasing amount
of power. At PEP
output, both PAs
see 50-ohm loads
and each delivers
half of the system
output power. The
PEP efficiency is
that of the class-B
PAs.
The resulting
instantaneousefficiency curve is
shown in Figure
39. The classical power division (α =
0.5) approximately maximizes the
average efficiency for full-carrier AM
signals, as well as modern single-carrier digital signals. The use of other
power-division ratios allows the lower
Figure 38 · Ideal voltage and current relationships in Doherty transmitter.
efficiency peak to be shifted leftward
so that the average efficiency is
increased for signals with higher
peak-to-average ratios. For example,
α = 0.36 results in a 60 percent average efficiency for a Rayleigh-envelope
signal with a 10-dB peak-to-average
ratio, which is a factor of 2.1 improvement over class B. Doherty transmitters with unequal power division can
be implemented by using different
PEP load impedances and different
supply voltages in the two PAs [97].
Much recent effort has focused on
accommodating non-ideal effects
(e.g., nonlinearity, loss, phase shift)
into a Doherty architecture [93, 94,
95]. The power consumed by the quiescent current of the peaking amplifier is also a concern. The measured
ACPR characteristics of an S-band
Doherty transmitter are compared to
those of quadrature-combined classB PAs in Figure 40. The signal is IS95 forward link with pilot channel,
paging channel, and sync-channel.
The PAs are based upon 50-W
LDMOS transistors. Back-off is varied to trade-off linearity against output. For the specified ACPR of –45
dBc, the average PAE is nearly twice
that of the quadrature-combined PAs.
In a modern implementation, DSP
can be used to control the drive and
bias to the two PAs, for precise control and higher linearity. It is also
possible to use three or more stages
to keep the instantaneous efficiency
relatively high over a larger dynamic
range [96, 98]. For ideal class-B PAs,
the average efficiency of a three-stage
Doherty can be as high as 70 percent
for a Rayleigh-envelope signal with
10-dB peak-to-average ratio.
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Figure 39 Instantaneous efficiency
of the Doherty system with ideal
class-B PAs.
46
High Frequency Electronics
Figure 40 · Measured ACPR performance of an S-band Doherty transmitter.
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High Frequency Design
RF POWER AMPLIFIERS
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Author Information
The authors of this series of articles are: Frederick H. Raab (lead
author), Green Mountain Radio
Research, e-mail: f.raab@ieee.org;
Peter Asbeck, University of California
at San Diego; Steve Cripps, Hywave
Associates; Peter B. Kenington,
Andrew Corporation; Zoya B. Popovic,
University
of
Colorado;
Nick
Pothecary, Consultant; John F. Sevic,
California Eastern Laboratories; and
Nathan O. Sokal, Design Automation.
Readers desiring more information
should contact the lead author.
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