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Nonlinear Amplify and Forward Distributed
Estimation over Non-Identical Channels
Robert Santucci, Student Member, IEEE, Mahesh K. Banavar, Member, IEEE,
Cihan Tepedelenlioğlu, Senior Member, IEEE, and Andreas Spanias, Fellow, IEEE

Abstract— This paper presents the use of nonlinear distributed estimation in a wireless system
transmitting over channels with random gains.
Specifically, we discuss the development of
estimators and analytically determine their attainable variance for two conditions: (i) when full
channel-state information is available at the transmitter and receiver; and (ii) when only channel
gain statistics and phase information are available. For the case where full channel-state information
is available, we formulate an optimization problem to allocate power amongst each of the
transmitting sensors while minimizing the estimate variance. We show that minimizing the estimate
variance when the transmitter is operating in its most nonlinear region can be formulated in a
manner very similar to optimizing sensor gains with full channel-state information and linear
transmitters. Furthermore, we show that the solution to this optimization problem in most scenarios
is approximately equivalent to one of two low-complexity power allocation systems.
Index Terms—distributed estimation, predistortion, amplifiers, nonlinearity, energy efficiency
I. INTRODUCTION
Distributed estimation uses multiple inexpensive sensors to estimate a single quantity.
In previous
literature, many methods have been developed for transmitting measured data back to a fusion center
[1,2,3,4]. Analysis has been done to determine how to allocate power amongst the sensors for a variety of
channel conditions for similar problems in [5]. One common algorithm used in the literature is amplifyCopyright (c) 2013 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 a request to pubs-permissions@ieee.org.
R. Santucci, C. Tepedelenlioglu, and A. Spanias are with the Sensor and Signal Information Processing (SenSIP) Center within the School of Electrical,
Computer, and Energy Engineering at Arizona State University, Tempe, AZ 85287 USA (e-mails: {bob.s, cihan, spanias}@asu.edu respectively). M.K. Banavar
is with the Department of Electrical and Computer Engineering at Clarkson University, Potsdam, NY 13699 USA (email: mbanavar@clarkson.edu). This work
was supported in part by the National Science Foundation FRP Award 1231034.
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and-forward (AF) distributed estimation, where sensors transmit a scaled version of a measured noisy
parameter. When these amplifiers transmit simultaneously over a shared channel, that method is called AF
over a coherent multiple access channel (MAC) [6,7,8] and shown in Figure 1. For most analyzed AF
systems, linear transmitter gains have been assumed. While linear amplifiers make analytical optimization
of transmitted power tractable, they have poor power-added efficiency (PAE). In [9], we demonstrated a
technique utilizing efficient nonlinear amplifiers in an AF system with coherent MAC where the channels
had been equalized to appear as equivalent-gain additive white Gaussian noise (AWGN) channels.
1
Σ
1
Noisy Sensor
(Tx)
Channel
Modulator 1 ( 1)
Measurement
(noisy)
2
Parameter
Σ
2
Pre-amp and PA
Noisy Sensor
(Tx)
Channel
Modulator 2 ( 2)
Measurement
(noisy)
Σ
h1
h2
Fusion Center
(Rx)
Channel and
Receiver Noise
Σ
Estimator
Estimate
Pre-amp and PA
Noisy Sensor
(Tx)
Channel
Modulator
( )
Measurement
(noisy)
h
Pre-amp and PA
Figure 1: Distributed Estimation Topology
In this paper, estimators are derived and analyzed for utilizing nonlinear transmitters in an AF system
operating over AWGN channels that have not been equalized. Real world systems utilizing AWGN
channels of varying gains may exist where the fusion center has a line-of-sight to each sensor as the
dominant propagation path. This approach merges the transmitter power allocation techniques from
existing literature [2,3,4,8,10] with the techniques for utilizing efficient nonlinear amplifiers introduced in
[9]. These environments have low multipath levels. This paper derives formulae to quantify estimator
performance when nonlinear AF transmissions are used over unequal AWGN channels in two scenarios: (i)
where channel gains are known exactly; and (ii) where only gain statistics are available. For the case of
channel gains being known exactly, it is shown that optimizing the sensor power allocation to minimize
estimate variance for the most nonlinear measured parameter is an equivalent problem to allocating power
in linear sensors. It is shown that in practical scenarios, performance close to the optimal solution may be
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attained by using one of two low-complexity power allocation strategies with a changeover point
determined by the channel gain distribution and measurement to channel noise ratio.
For the cases where only channel gain statistics and phase information are known, this paper summarizes
the estimator performance impact when using nonlinear transmitters with both analytic formulae and
simulated results. In this case, because individual sensor properties are not available, it is not possible to
assign sensor gains individually.
The rest of this paper is organized as follows. In section II, we derive analytical expressions for estimate
variance with full channel state information for AWGN channels. Then, we minimize the estimate variance
under a total sensor power bound in section III. The results from sections II and III are verified, and the
low-complexity power allocation system performance is shown via simulations in section IV. In section V,
we derive formulas for estimate variance where only gain statistics are known and validate them via
simulation. Finally, concluding remarks are presented in section VI.
II. SYSTEM MODEL
A typical distributed estimation topology for utilizing AF distributed estimation over a coherent MAC is
shown in Figure 1. In this example, the network contains 𝐿 sensors measuring a single parameter, 𝜃, in the
presence of noise. The channel gains for the 𝑙 th sensor, ℎ𝑙 , are i.i.d. random variables with distribution
𝑝(ℎ). The channels and the receiver have additive Gaussian noise, cumulatively given as 𝜈. Each sensor’s
limiting output level, 𝑔𝑙 , is assumed dependent upon ℎ𝑙 and independent from each other. The limiting
output power is adjusted using variable-voltage bias as done in [11,12] . Measurement noise, 𝜂𝑙 , are i.i.d.
and independent from both sensor and channel gains. The amplifier inputs are predistorted to compensate
for manufacturing process, voltage, temperature, and aging variation and gain-scaled in proportion with
maximum output power, to fit a gain-scaled version of a common maximum output-power normalized
model, 𝑔𝑙 𝑠(𝑥), as in [9]. Limiting amplifier model function requirements are outlined in [9] and [13].
When full channel and phase gain information is available at the sensors, and sensor phases are set such
that the signal phases align at the FC, then the value received at the FC is:
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𝐿
𝑦(𝜃) = ∑ ℎ𝑙 𝑔𝑙 𝑠(𝜃 + 𝜂𝑙 ) + 𝜈.
(1)
𝑙=1
The expected value at the FC for the parameter 𝜃 is:
𝐿
𝐸[𝑦(𝜃)] = 𝐸[𝑠(𝜃)] (∑ ℎ𝑙 𝑔𝑙 ) + 𝜇𝜈 ,
(2)
𝑙=1
where 𝜇𝜈 is the mean channel noise. Because channel and sensor gains are fully known, a scaling shown in
the denominator of (3) is applied to the value received at the FC:
𝑦
𝑧= 𝐿
.
∑𝑙=1 ℎ𝑙 𝑔𝑙
(3)
Using this scaling parameter and assuming zero-mean channel and receiver noise we get:
𝜁 (̅ 𝜃) = 𝐸[𝑧(𝜃)] = 𝐸[𝑠(𝜃)].
(4)
Figure 2: For a fixed receiver noise, using a more nonlinear amplifier results in more estimate variance. The scaled
Tanh refers to the hyperbolic tangent model described in (11), and the Cann model is described in [14].
The estimate can be computed by substituting the actual value received at the FC in place of the expected
value, and solving for the parameter 𝜃, as was done in [9]. The variance of 𝑧 can be evaluated to be:
Var(𝑧) =
(𝐸[𝑠 2 (𝜃)] − 𝐸 2 [𝑠(𝜃)]) ∑𝐿𝑙=1 ℎ𝑙2 𝑔𝑙2 + 𝜎𝜈2
,
(∑𝐿𝑙=1 ℎ𝑙 𝑔𝑙 )2
(5)
where 𝜎𝜈2 is the variance of the channel noise, 𝜈.
The estimate of 𝜃, denoted by 𝜃̂, can be computed as [15]:
𝜃̂ = argmin({𝑧 − 𝜁 (̅ 𝜃)}(Var(𝑧))−1 {𝑧 − 𝜁 (̅ 𝜃)}),
𝜃
(6)
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which may not always have a closed form solution. From Var(𝑧) and the slope of the 𝐸[𝑠(𝜃)], the variance
of 𝜃̂ can be found without explicitly computing the value of 𝜃̂ [13,15,16]:
Var(𝜃̂) =
Var(𝑧)
𝐸 2 [𝜕𝑠(𝜃)/𝜕𝜃
]
,
(7)
which can be written as:
Var(𝜃̂) =
(𝐸[𝑠 2 (𝜃)] − 𝐸 2 [𝑠(𝜃)])( ∑𝐿𝑙=1 ℎ𝑙2 𝑔𝑙2 ) + 𝜎𝜈2
.
𝐸 2 [𝜕𝑠(𝜃)/𝜕𝜃 ] (∑𝐿𝑙=1 ℎ𝑙 𝑔𝑙 )2
(8)
It is shown in Figure 2 for several receiver models that increased amplifier nonlinearity results in increased
estimate variance for a fixed amount of receiver noise [14].
To verify the analytical expressions presented, an example system was utilized. Measurement noise was
chosen to be bounded by ±𝑏, with 𝑏 ∈ ℝ+ such that 𝑛𝑙 ~𝑈[−𝑏, 𝑏], to simulate the effects of quantization
noise when measurements are taken by the sensor. The hyperbolic tangent limiting amplifier model from
[17] was used. It has parameters for maximum output power, 𝐿𝑜 , and nominal gain, 𝑔𝑜 , and is given by:
𝑔𝑜
(9)
𝑣𝑖𝑛 ) .
𝐿𝑜
When both parameters are scaled by 𝑔𝑙′ and the sensor scales physical measurement, 𝑥, to produce input
𝑣𝑜𝑢𝑡 = 𝐿𝑜 tanh (
voltage, 𝑣𝑖𝑛 , by positive real factor, 𝑘𝑜 ∈ ℝ+ , according to 𝑣𝑖𝑛 = 𝑥/𝑘𝑜 , the output is:
𝑔𝑙′ 𝑔𝑜 𝑥
(10)
𝑣𝑜𝑢𝑡𝑠𝑐𝑎𝑙𝑒𝑑 = 𝐿𝑜 tanh ( ′
).
𝑔𝑙 𝐿𝑜 𝑘𝑜
′
By letting 𝑔𝑙 = 𝑔𝑙 𝐿𝑜 and 𝑘 = 𝐿𝑜 𝑘𝑜 /𝑔𝑜 , output is made into the form of a scaled common limiting
𝑔𝑙′
amplifier model, 𝑣𝑜𝑢𝑡 𝑠𝑐𝑎𝑙𝑒𝑑 = 𝑔𝑙 𝑠(𝑥) with:
(11)
𝑠(𝑥) = tanh(𝑥/𝑘).
The estimator is found similarly to [9], using 𝑚 = exp(2𝑏𝑧/𝑘) when 𝑚 ∈ [0,1] and z is given by (3) to be:
𝑏
sinh (𝑘) (1 + 𝑚)
𝜃̂ = 𝑘 cosh
−1
.
2𝑏
(
√2𝑚 cosh ( ) − 1 − 𝑚2
𝑘
(12)
)
The variance for this estimator has been derived in [9]. The analytical expressions derived in this section
are validated via simulations in section IV.
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III. SENSOR POWER ALLOCATION STRATEGY
With an analytical solution in hand for estimate variance from (8), a minimization problem can be formed
to intelligently allocate power between the individual transmitting sensors in order to achieve the best
performance while maintaining a specific performance level. The problem can be posed as an optimization
problem of the sensor gains. Given an a priori probability of the underlying physical values to be
measured, 𝑝𝜃 (𝜃), a function 𝑃𝑎𝑚𝑝 (𝑔𝑙 , 𝑥𝑙 ) which is the power consumed by a transmitting a noisy
measurement 𝑥𝑙 with sensing gain of 𝑔𝑙 , and a transmitter power budget of 𝑃𝑏𝑢𝑑𝑔𝑒𝑡 , the best average
estimate variance that is possible is:
minimize ∫
(𝐸[𝑠 2 (𝜃)] − 𝐸 2 [𝑠(𝜃)]) ∑𝐿𝑙=1 ℎ𝑙2 𝑔𝑙2 + 𝜎𝜈2
{𝑔1 ,…,𝑔𝐿 }
𝐸2 [
𝜕𝑠(𝜃)
]
𝜕𝜃
(∑𝐿𝑙=1 ℎ𝑙 𝑔𝑙 )2
𝑝𝜃 (𝜃)𝑑𝜃,
(13)
𝐿
subj.
: ∫ (∑ ∫ 𝑃𝑎𝑚𝑝 (𝑔𝑙 , 𝜃 + 𝜂𝑙 )𝑑𝜂𝑙 ) 𝑝𝜃 (𝜃)𝑑𝜃 ≤ 𝑃𝑏𝑢𝑑𝑔𝑒𝑡 .
to
𝜂𝑙
𝑙=1
This optimization is not easily evaluated using convex optimization techniques, so as part of making the
optimization more tractable, a very similar dual problem was posed similar to the techniques in [8]. The
dual problem minimizes the power consumed, subject to a specified average variance level, Υ𝑡 :
𝐿
minimize ∫ (∑ ∫ 𝑃𝑎𝑚𝑝 (𝑔𝑙 , 𝜃 + 𝜂𝑙 )𝑑𝜂𝑙 ) 𝑝𝜃 (𝜃)𝑑𝜃,
{𝑔1 ,…,𝑔𝐿 }
subj. to: ∫
𝑙=1 𝜂𝑙
(𝐸[𝑠
2(
𝜃)] − 𝐸 [𝑠(𝜃)]) ∑𝐿𝑙=1 ℎ2𝑙 𝑔2𝑙
2
𝜕𝑠(𝜃)
∑𝐿𝑙=1 ℎ𝑙 𝑔𝑙 )
𝐸2 [
]
(
𝜕𝜃
2
(14)
+
𝜎2𝜈
≤ Υ𝑡 .
This optimization problem is itself still computationally intensive to solve using convex optimization
techniques, so the optimization problem was reposed with a slightly different target. In this revision of the
dual problem, it is decided to optimize against the worst case variance produced by the desired estimator.
It is observed in [9], that the worst-case estimation performance occurs when the limiting amplifier model
is most compressed. This compression occurs at the largest value of 𝜃. Given all noise measurements
𝑥𝑙 = 𝜃 + 𝜂𝑙 are relatively similar and that dynamic biasing in utilized to set the amplifier output level
according to 𝑔𝑙 , it is assumed that the amplifier’s efficiency will be similar for all sensors, and thus, the
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function 𝑃𝑎𝑚𝑝 (𝑔𝑙 , 𝑥𝑙 ) can be reduced to a constant factor, 𝜙. Similarly, if only the worst case of 𝜃, 𝜃𝑤𝑐 , is
considered, the probability density function 𝑝𝜃 (𝜃) = 𝛿(𝜃𝑤𝑐 ), and Υ𝑤𝑐 is the worst case estimate variance.
This reduced problem can be considered as:
𝐿
minimize ∑ 𝜙𝑔𝑙2 ,
{𝑔1 ,…,𝑔𝐿 }
st:
𝑙=1
− 𝐸 2 [𝑠(𝜃𝑤𝑐 )])( ∑𝐿𝑙=1 ℎ𝑙2 𝑔𝑙2 )
𝜕𝑠(𝜃 )
𝐸 2 [ 𝜕𝜃𝑤𝑐 ] (∑𝐿𝑙=1 ℎ𝑙 𝑔𝑙 )2
(𝐸[𝑠 2 (𝜃𝑤𝑐 )]
+
𝜎𝜈2
(15)
= Υ𝑤𝑐 .
By introducing a slack variable, 𝜌, and ignoring the constant factor 𝜙 which has no bearing on the
minimization, the problem can be rearranged to a similar form as solved in [8] provided the amplifier
modeling function 𝑠 satisifies the conditions for consistent estimators established in [9,13]:
𝐿
minimize ∑ 𝑔𝑙2 .
{𝑔1 ,…,𝑔𝐿 ,𝜌}
𝑙=1
subject to:
𝐿
(𝐸[𝑠 2 (𝜃𝑤𝑐 )]
−
𝐸 2 [𝑠(𝜃𝑤𝑐 )])
∑ ℎ𝑙2 𝑔𝑙2 + 𝜎𝜈2 = 𝜌2 Υ𝑤𝑐 .
(16)
𝑙=1
𝐿
𝜕𝑠(𝜃𝑤𝑐 )
𝐸[
] (∑ ℎ𝑙 𝑔𝑙 ) = 𝜌
𝜕𝜃
𝑙=1
IV. SIMULATION RESULTS
In this section we perform simulations to verify and explain the analytic solutions from sections II and
III. To conduct simulations, it was necessary to choose channel gains and sensors gains. Channel and
sensor gains were both chosen bounded by 0 and 1, such that ℎ𝑙 ~𝑈[0,1] and 𝑔𝑙 ~𝑈[0,1]. This choice was
made for mathematical ease to illustrate that some channels may be in an attenuated path and others not. In
these simulations, channel noise was kept small to concentrate on the sensor power allocation and channel
gains. In subsection IV-A, channel gains were randomly assigned to robustly verify the results predicted by
(8). In subsection IV-B, the optimization from (16) is calculated and performance is compared to two low-
complexity techniques.
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8
A. Estimate Variance Equations in Full CSI
To identify the difference in performance that can be expected with non-identical gain channels,
simulations were run in MATLAB. Analytically predicted values were compared to those using the
techniques for equalized channels presented in [9]. For these simulations, 𝐿 = 105 sensors were utilized to
ensure that true values were attained across noise distributions. The expectations and variances of the
received value and the estimate were compared to identical channels from [9]. Because the limiting
amplifier model from [9] includes the channel and sensor gains, the mean value at the fusion center for the
exactly-known varying channel gains case were scaled by 𝑐 = 𝐸[ℎ𝑔] = 1/4 and the variances were scaled
by 𝑐 2 to replicate including the gains in the equation. Furthermore, the variances obtained in [9] are
asymptotic and thus to have a fair comparison, an equivalent asymptotic variance, Σ(𝜃) = 𝐿Var(𝑧(𝜃)) ,
must be used for the exactly known gains.
Figure 3: Asymptotic variance of value received at FC for fully-known varying gain channels and equalized channels.
The estimate variance is determined from (8) via two components, Var(𝑧) and 𝐸[𝜕𝑠(𝜃)/𝜕𝜃]. It can be
seen by comparing equations from (11) and that once channel and sensor gains are compensated, the
expected value at the fusion center and consequently its derivative are identical for the two techniques.
Thus, any performance differences from the identical variance case are driven by changes in the variance
into the FC. It can be seen from Figure 3 that there is a small penalty paid when the channels are varying
versus identical channels. It can also be seen that our simulated results match the theoretical results.
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B. Optimal Sensor Gain Power Allocation
The convex optimization proposed in section III has been solved for a simulation scenario with 𝐿 = 12
sensors using Mathematica [18]. The optimization was performed by solving the KKT conditions [19]. In
this problem, the total channel power sum was set equal to 1 so different channel scenarios could be
simulated. The results shown below are for when channel gains were allocated so that there were 3 strong
channels with ℎ𝑙 = 0.57735 and 9 weak channels with ℎ𝑙 = 5.7735 × 10−4 .
The optimal variance was
found, and compared to three different low-complexity non-optimal strategies for sensor power allocation.
Figure 4: Comparison of sensor power allocation strategies.
In the first low-complexity approach, sensor gains were used to equalize the ℎ𝑙 𝑔𝑙 gain products. In the
second low-complexity approach, sensor power was allocated in proportion to the channel power. In the
third approach, no strategy was taken and sensor power was distributed equally amongst all sensors. The
results comparing estimate variance attainable for a given power consumption with each approach are
shown in Figure 4. It can be seen from Figure 4 that that over most variance targets one of the two nonoptimal power allocation strategies was effectively equivalent to the performance of the optimal solution.
When estimate variance requirements are loose, it makes sense to allocate power to the strongest channels.
However, when the estimate requirements are tighter than the best attainable combination of strong
channels, more power must be added to the faded channels, resulting in a system closer to equalizing the
channels. The changeover point between optimal power allocation strategies depends on the ratio of
measurement to channel noise, and the channel gain distribution.
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V. NONLINEAR ESTIMATION WITH PHASE-ONLY CSI
The simulations above show that full CSI is available for each channel, good estimator performance can
be attained with nonlinear transmitters using simple strategies for sensor power allocation. We now
investigate what happens when only partial CSI is available. We consider the case of phase-only CSI,
where gain statistics are known for the channels collectively but not individually. Phase-information is
required to synchronize the transmitter phases. In this case, we revisit (2) which assumed only channel
gains were known. Revising the expectation with random i.i.d. ℎ𝑙 𝑔𝑙 and zero-mean sensing noise:
𝐸[𝑦(𝜃)] = 𝐿𝐸[ℎ𝑔]𝐸[𝑠(𝜃)].
(17)
By making the substitution 𝑧 = 𝐿𝐸[ℎ𝑔] into (17), asymptotic expressions of the expected value, 𝜁 (̅ 𝜃), and
variance at the FC can be determined where Σ(𝜃) = 𝐿Var(𝑧(𝜃)):
𝜁 (̅ 𝜃) = 𝐸[𝑧(𝜃)] = 𝐸[𝑠(𝜃)], and
𝐿𝐸[ℎ2 𝑔2 ]𝐸[𝑠 2 (𝜃)] − 𝐿𝐸 2 [ℎ𝑔]𝐸 2 [𝑠(𝜃)] + 𝜎𝜈2
Σ(𝜃) =
.
𝐿𝐸 2 [ℎ𝑔]
(18)
(19)
A parameter estimate, 𝜃̂, can be derived from (18) given the actual value received, 𝑧. Because 𝜁 (̅ 𝜃) is
the same for the case of full CSI, the estimated parameter 𝜃̂ can still be found using the estimator from (12).
Similarly estimator performance can be found from:
Σ(𝜃)
(20)
.
[𝜕𝜁 (̅ 𝜃)⁄𝜕𝜃]2
Having only gain statistics for a nonlinear distributed estimation operating over a nonlinear channel
AsVar(𝜃̂ ) =
results in a large variance at the FC. The variation in 𝑧 becomes much greater than for identical constant
channels as the transmitter becomes more nonlinear. The resulting orders-of-magnitude increase in
variance at the FC predicted by (19) and verified by simulation with gain and noise distributions identical
to section IV is shown in Figure 5. To overcome this increase in variance takes a significant increase in the
number of sensors, and thus considerably more total power and expense, as shown in Figure 6.
A second problem with nonlinear distributed estimation using only channel statistics is that is even with a
practically-large 𝐿, the value at the estimator 𝑧 ∉ [0,1]. When 𝑧 ∉ [0,1], the estimator in (12) provides a
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complex-valued estimate and fails. It is important to make sure the number of channels is large enough to
make the probability Pr[𝑧 > 1] → 0 when implementing a nonlinear distributed estimation system [20].
Figure 5: Asymptotic variance of value received at FC for statistically-known random channels and equalized channels.
Figure 6: Having gain-statistics (partial CSI) instead of having identical AWGN results in a large increase in the number of
sensors to achieve the same variance target at the fusion center.
VI. CONCLUSIONS
This paper has expanded upon our previous work in distributed estimation using efficient nonlinear
transmitters [9] by incorporating the effects of varying channel gains. When full CSI is available, we have
demonstrated that nonlinear estimation can be performed with only a limited impact on performance, where
that limitation is dependent upon the distribution of the channel gains. Nearly optimal performance can be
obtained for a given transmitter power budget by allocating transmitted power either in proportion or
inversely to channel power. When the transmitted power budget is large, inversely-proportional power
allocation can be used to equalize weak channels. This paper has further demonstrated when phase-only
CSI is available, nonlinear transmitters have a detrimental impact on estimation. When operating in an
environment where channel gains may only be statistically known, it is best to use only linear or nearly
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IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGIES
linear sensors. Future work includes study of the effects of multipath channels and frequency selective
channels, and design of optimal amplifiers for these cases.
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