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Impact of Nonlinear Power Amplifiers on the

Performance of Precoded MIMO Satellite Systems

Vito Dantona1, Thomas Delamotte2, Gerhard Bauch2 and Berthold Lankl1

1 Institute of Communications Engineering 2 Institute of Information Processing

Munich University of the Bundeswehr, Neubiberg, Germany

Email:{vito.dantona, thomas.delamotte}@unibw.de

AbstractWe investigate the applicability of linear precod-ing schemes based on Singular Value Decomposition (SVD) toMultiple Input Multiple Output (MIMO) single carrier satellitecommunication systems, where the need for high power efficiencyrequires to operate nonlinear high power amplifiers (HPA) as

close as possible to saturation. Precoded transmit signals have ahigh peak-to-average power ratio (PAPR), which might be poten-tially detrimental for the system performance on the nonlinearchannel. We compare in this work the uncoded performance ofa 2 2 MIMO and an equivalent SISO system operating withequal data rate and total transmit power. Our simulations showthat SVD precoding does not require the MIMO system to beoperated with a higher power back-off than its SISO counterpart.On the contrary, gains of almost 3 dB can be observed for somemodulation schemes.

I. INT ROD UC TION

As the demand for higher bandwidth efficiency in satel-

lite communication systems is steadily increasing, MIMO

technologies are recently emerging as a promising approach

towards this goal. However, due to the different characteristics

of the satellite channel, the translation of MIMO techniques

developed for terrestrial channels is not straightforward, as

they usually rely on the assumption of a rich scattering

environment, which is not the case for the fixed satellite

scenario. The latter is in fact usually characterized by a strong

Line of Sight (LOS) component, which would compromise the

feasibility of spatial multiplexing due to rank-deficiency of the

MIMO channel matrix.

As a possible workaround several authors [1], [2] have

proposed to employ, especially for land mobile satellite (LMS)

systems, co-located antennas working with different wave

polarizations, instead of the spatially displaced antennas used

in traditional MIMO systems. This approach provides analternative degree of freedom which can be exploited to build

a full-rank channel matrix.

Nevertheless, our previous works [3], [4] pointed out that

spatial MIMO can indeed guarantee high channel capacity

even in the LOS satellite scenario, as long as the antenna

spacing both at the satellite and at the ground terminal is

carefully chosen in order to generate an orthogonal channel

matrix. On the other hand, several impairments exist, which

would degrade the perfect orthogonality of the channel matrix.

This work was supported by the Deutsche Forschungsgemeinschaft (DFG).

They include for instance atmospheric conditions [5] and

movements of the satellites within their station-keeping box

[6]. As soon as the channel matrix deviates from orthogo-

nality, not only the channel capacity drops, but also more

complex receiver algorithms become necessary to perform

MIMO detection [7]. If a feedback link is provided, MIMO

precoding techniques can be beneficial for these situations. In

this paper we focus on a simple precoding approach based

on the Singular Value Decomposition (SVD) of the MIMO

channel matrix, which is well known to limit the receiver

complexity by performing a virtual diagonalization of the

MIMO channel. We restrict ourselves to the downlink segment

of a 2 2 MIMO system consisting of 2 transmit antennasplaced on a single satellite and 2 receive ground antennas,

which may be placed several kilometers apart and connected

by a wired backbone.

We perform an analysis of the system performance in the

presence of the nonlinear High Power Amplifers (HPA) com-

monly used on satellites. Actually, state-of-the-art Single InputSingle Output (SISO) satellite systems employ high order

Amplitude-Phase Shift Keying (APSK) signal constellations,

which can effectively cope with nonlinear power amplifiers

driven close to saturation, due to their low Peak-to-Average

Power Ratio (PAPR). Furthermore, even if predistortion mech-

anisms are provided, an input back-off (IBO) is usually

required, i.e. the transmission power is reduced by a certain

amount below the amplifiers saturation point in order to set

a trade-off between power efficiency and nonlinear distortion.

Precoding, on the contrary, generates on each branch of the

MIMO system an arbitrary signal point displacement which is

a linear combination of the constellations chosen for the input

signal and is thus not optimized from the point of view of

power efficiency. As a consequence, one could expect that a

higher power back-off might be needed in a MIMO system

in order to cope with the higher PAPR of such constellations.

This would be a crucial objection to the feasibility of precoded

MIMO satellite systems, because a very high power back-

off could even cancel the Signal-to-Noise Ratio (SNR) gain

achievable through spatial multiplexing. In order to investigate

this, we compare in the following the impact of power back-off

in a SISO and a MIMO system through numerical simulations.

After explaining in detail the system and channel models

978-1-4673-4688-7/12/$31.00 2012 IEEE


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in Section II, we compare in Section III the performance of

a MIMO system achieving a fixed date rate with a reference

SISO system providing the same rate. Here we present and

discuss numerical results for the required power back-off;

afterwards, we evaluate the out-of-band emissions generated

by nonlinear amplifiers. We give comments and conclusions

in Section IV.

II . SYSTEM ANDC HANNEL M ODEL

A. Parametrization of the Line-of-Sight MIMO Channel

Before describing the particular system model, we briefly

introduce the general LOS MIMO channel model. Let x =[x1x2. . . xNt ]

T and y = [y1y2. . . yNr ]T be the input and

output vectors respectively of a generic Nr Nt frequency-flat MIMO channel. The input/output relationship is described

through:

y= Hx + n (1)

where n is a complex zero-mean Gaussian noise vector such

that EnnH

= 2INr and the matrix H summarizes the

Nr Nt channel coefficients hij between the j -th transmitterelement and thei-th receiver element. In the most general case,

the channel transfer matrix H is made up of a LOS compo-

nent and a non-LOS component which includes all multipath

contributions. As we are focusing on LOS MIMO channels,

we will neglect the second component in the following. The

entries of the channel matrix are thus deterministic and may

be expressed as the result of free-space propagation:

hij =aij expj 2fc

c0rij

, (2)

where c0 is the speed of light, fc is the carrier frequency

and rij is the path length for a specific transmitter/receiver

pair with j = 1, . . . , N t and i = 1, . . . , N r. The complexenvelope aij may be considered approximately constant and

equal to a= ej for all i, j. The phase is common to allmatrix entries and can be thus arbitrarily set to zero, so that

aij R+. Actually, only the phase differences betweenthe channel matrix entries are relevant.

For this reason, we will employ in the following a scaled

version H of the channel matrix with hij = hij|hij | =

hij

,

such that each entry of the new matrix has magnitude 1.Moreover, we will restrict ourselves to the particular case

where Nt = Nr = 2. This restriction allows to use thevery compact parametrization of the MIMO LOS channel

introduced in [8]. The scaled channel matrix may be writtenas:

H=

ej11 ej12

ej21 ej22

, (3)

whereij = 2fcc0 rij . We now define the phase difference ofthe two paths arriving at the i-th receiver as

i= i1 i2 (4)and finally the difference of the phase differences of the two

receivers as

= 1 2, (5)

with . The parameter is a compact measureof the orthogonality of the 2 2 MIMO channel. The case= 0will denote the keyholechannel, whereas the channelwill have full rank for = .

B. System Model

The downlink segment of a 2 2 single-satellite MIMOsystem is considered in the following (cp. Figure 1). The

transmit antennas are located on a geostationary satellite,

whereas the receive antennas constitute a distributed ground

terminal, as they may be located several kilometers apart

in order to guarantee nominal orthogonality of the channel

matrix. It is assumed that the satellite payload acts as a fully

regenerative repeater and that a wired infrastructure, such as an

optical backbone, connects the two ground antenna terminals

to a signal processing unit. The receiver perfectly knows the

channel matrix, performs a Singular Value Decomposition of

it and feeds it back to the transmitter.

Fig. 1. Schematic view of the considered scenario. Only the downlinksegment is analyzed in this work.

The block diagram of the considered MIMO system in

equivalent baseband is shown in Figure 2. A reference SISO

system is given as well for the sake of completeness.

The precoded vector x is the linear combination of data

symbols stacked in a vectors = [s1s2]T, obtained by splitting

an input bit stream and mapping each resulting substream

to a finite signal constellation. Data symbols of different

substreams may belong to different complex constellations,

such that sl Al, where Al denotes the symbol alphabet forthe l-th substream,l = 1, 2. It must be verified EssH= I2.Thus, the vectorx writes as

x= Bs (6)

where B represents the 2 2 precoding matrix such thattrBBH

Pt, Pt being the maximum mean transmittedpower.

By considering the SVD of the MIMO channel H =UVH, with U C22, V C22 unitary matrices and C22 diagonal matrix containing the singular values l,the precoding matrix is then given by

B= VP (7)


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Fig. 2. Block diagram of a reference SISO system (top) and the considered MIMO system (bottom)

The matrix V contains as columns the right singular vectors

of the MIMO channel matrix, while P= diag

pl

, l= 1, 2is determined according to some power allocation strategy.1

We neglect for the moment the power back-off and data

predistortion stages, which will be introduced later on. A root-raised-cosine (RRC) filter with impulse response g(t)and roll-off factor is used as pulse shaping filter on each transmitter

branch. The equivalent lowpass signal is fed to the nonlinear

HPA, which is modeled as in Section II-C and connected to

the antenna element of the respective branch. The propagation

model is a pure LOS model as described above. We recall

that the free-space loss is approximately equal for each pair

of Tx-Rx antennas and is thus neglected in the system model.

Additive White Gaussian Noise (AWGN) with variance

2(M) is added on each receiver branch. It must be noted here

that the system will not be free from intersymbol interference

(ISI), due to the presence of the nonlinear HPA between thetwo RRC filters.

The left singular vectors ofH will be used to process thecomplex values y sampled at the output of the matched filters

such that

s= UHy= Ps +UHn (8)

with = diag(l) , l= 1, 2. The considered SVD-basedprecoding strategy hence allows to decouple the system into

orthogonal eigenmode subchannels. We assume that the re-

ceiver performs independent Maximum Likelihood detection

on each subchannel.

For the simulations described in this paper, the power allo-

cation has been performed through the Mercury-Waterfillingstrategy [9], which gives the optimal solution for parallel

AWGN channels and fixed arbitrary signal constellations2.

C. Power Amplifier Model

The most widespread baseband model for the nonlinear

response of high-power amplifiers, which dates back to Saleh

1We remark that power allocation is intended for the virtual subchannelsbefore the multiplication with V, while the two resulting precoded streamsalways have equal power.

2Due to the nonlinearity of the HPAs, the optimality in our scenario is ofcourse no more guaranteed.

[10], will be used in the following. Letrin(t) =in(t)ejin(t)

be the complex envelope signal at the input of the HPA.

Magnitude and phase of the output complex envelope rout(t)write as:

out(t) = A [in(t)] (9)

out(t) = in(t) + [in(t)] (10)

respectively, where

A() = a

1 +a2 (11)

() =

2

1 +2. (12)

The two latter equations are known as AM-to-AM and

AM-to-PM conversion respectively, and have two parameters

each. Several values of the model parameters can be found in

the literature, deriving from response measurements of real

amplifiers. Some examples are given in Table I with the

respective sources. Without loss of generality, we will use

throughout the paper the parameter set reported in the first

column, adopted from [11] and many other works.

TABLE ISOME PARAMETER SETS USED FOR THE S ALEH MODEL

Parameter [11], [12] [10], [13] [14]

a 2 1.9638 2.1587a 1 0.9945 1.1517 /3 2.5293 4.0033 1 2.8168 9.1040

The characteristics described by equations (11-12) are plot-

ted in Figure 3. The solid and dashed line represent AM-AM

and AM-PM conversion respectively. As shown in the figure,

Salehs equations assume that the output power is normalized

to the saturation power of the HPA. Similarly, input power

is normalized to the input power which causes saturation, so

that the maximum of the AM-AM conversion curve is always

located at the point(0dB, 0dB), if logarithmic units are used.Generally speaking, the presence of nonlinear amplifiers has

two main effects on a digital communication system:


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Fig. 3. Salehs model [10] for AM-to-AM and AM-to-PM characteristic ofHPAs (left and right vertical axis respectively).

1) As the output amplitude and phase shift depend on the

input signal amplitude, the received signal constellationis distorted. This effect is known as constellation warp-

ing;

2) As already stated above, inter-symbol interference arises

at the receiver, even if the cascade of transmit and

receive filter satisfies the first Nyquist criterion. This

results in clusteringof the received signal points, which

are scattered in the vicinity of the nominal constellation

points, even in absence of other noise sources.

Due to the limited energy resources on satellites, it is usually

desired to exploit the HPA as efficiently as possible, i.e. to

operate it as close as possible to the saturation point. On

the other hand, the distortion introduced by the HPA in thevicinity of this point can become prohibitive for the system

performance. In practice, the actual average transmit power

is reduced from the value that would cause saturation of the

amplifier by a certain amount, which is called Input Back-

Off (IBO). This results of course in a reduction of the output

average power, defined as Output Back-Off (OBO). It is

easy to understand that signal constellations which exhibit a

high PAPR need a higher power back-off than more com-

pact constellations. In this sense, Phase Shift Keying (PSK)

constellations are advantageous over Quadrature Amplitude

Modulation (QAM) schemes. On the other hand, high-order

PSK constellations are impractical due to their small minimum

Euclidean distance. For this reason, Amplitude Phase Shift

Keying (APSK) modulation is widely used in state-of-the-art

SISO satellite systems, as a good trade-off between robustness

against nonlinear distortion and against additive noise [12],

[15], [16].

For instance, the 32-points constellation shown in the right

part of Figure 4 can be used to transmit 5 bits/symbol. If a

22MIMO architecture with SVD precoding is used instead,the same data rate can be achieved by using 8-PSK on the

stronger virtual subchannel and QPSK on the weaker one. The

actual signal points which will be fed to the pulse shaping

filters will result from a linear combination of the points of

the two PSK constellations, as in (6). The precoding matrix

will be dependent on the MIMO channel state, which we

have described above through the parameter. For instance,

if = 0.7 is assumed, the precoded constellations on thetwo branches of the MIMO system will look like in the left

part of Figure 4. Assuming a RRC transmit filter with roll-

off factor0.3, the PAPR of the precoded MIMO signal3

willbe significantly higher (up to 0.6 dB) than the PAPR of theAPSK-modulated signal, as shown in Figure 5.

Fig. 4. Signal constellations resulting from SVD precoding at the twotransmit antennas of a MIMO system with = 0.7 (left) and for a state-

of-the-art SISO APSK system (right).

Fig. 5. Simulated PAPR for a 2 2 MIMO system with SVD precodingand a SISO system (RRC transmit filter with roll-off factor0.3).

D. Data Predistortion

Besides the simple introduction of a power back-off, a

number of advanced techniques exist to contrast the impact of

nonlinear distortions, both at the receiver side (equalization)

and at the transmitter side (predistortion)4. We assume in the

following that the system under analysis is provided with a

data predistorter, which inverts the known characteristic of the

HPA and maps the signal points to a modified constellation, sothat the original constellation is recovered at the HPA output

except for an amplitude scaling factor0, i.e.A [R()] = 0and [()] = 0, where R() and () describe the predis-torter response in amplitude and phase. If the HPA is operated

beyond the saturation point, its AM-AM characteristic cannot

be inverted anymore. The effective cascade of predistorter and

HPA thus shows a saturation value and it still makes sense to

3The PAPR is defined here asPAPR = max{|x(t)|2}/E|x(t)|2

, where

x(t)is the complex low-pass equivalent signal at the output of a transmit filter.4A complete survey on this topic woud be beyond the scope of the present

work; the interested reader might consult [17].


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Fig. 6. Total degradation curves for the MIMO system without data predistortion for different values of the parameter andPB = 103, = 0.3. Thecurve for the reference SISO system is shown in black. Subfigures (a),(b),(c) refer to the system configurations A, B, C defined in Table II respectively.

Fig. 7. Same as Figure 6, except that data predistortion is enabled.

C. Discussion of Results

It is a common objection that increasing the PAPR of

transmitted signals would result in worse performance ina nonlinear channel. The results presented above turn this

objection upside down, as we have stated that precoded MIMO

constellations do not need additional power back-off compared

to optimized APSK SISO constellations; on the contrary, the

same BER can be expected with even lower back-off.

In order to gain a deeper insight into this behaviour, we

illustrate an example. Figure 8 shows the BER performance

of SISO 16-APSK compared to MIMO employing two QPSK

subchannels. The dotted lines show the BER on a linear

channel and the solid lines refer to the nonlinear channel with

a power back-off. No predistortion is applied and the value

of the parameter is0

.7

. The MIMO system would have a

SNR gain of 4 dB on the linear channel, due to array gain and

the use of constellations with higher Euclidean distance. For

the nonlinear channel, the optimum OBO has been selected

both for the SISO and the MIMO case. This amounts to 3 dB

and 2 dB respectively (cp. Figure 6), and the operating points

are shown as black dots in the inset figure. The CNR loss for

the SISO system at PB = 103 with respect to the linear case

is 1.5 dB, whereas the MIMO system only loses 0.9 dB. Byconsidering this gain of 0.6 dB and the OBO which is 1 dBlower, the total gain of the MIMO system results in 1.6 dB,which is the distance between the respective TD minimum

Fig. 8. Bit Error Rate of the SISO and MIMO systems (ConfigurationA) operated at the respective optimum working point without predistortion.For the MIMO channel, = 0.7. Dotted lines show the BER on a linearchannel.

points shown in the inset figure. The feared degradation due

to the higher PAPR of the MIMO constellations only comes

to the fore in the low-SNR region (CNR < 8 dB); here, theBER on the non-linear channel deviates significantly from the

linear case and is even larger than for SISO. Nevertheless, this

effect disappears at BER of practical interest.


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D. Evaluation of Out-of-band Emissions

As we have stated that MIMO systems could be operated

with reduced back-off, attention must be paid to spectral

regrowth as a side effect. This aspect is very dependent on

particular system implementations and compatibility require-

ments. As a general baseline, we considered recommendations

in [19] for out-of-band emissions.

Figure 9 shows that the power spectral density of theprecoded MIMO signal at the output of the HPA would still be

compliant with those minimal requirements for the optimum

working-point determined above (red solid line). Furthermore,

there is no significant additional spectrum regrowth due to

the use of the precoded constellations (2 8-PSK) insteadof SISO 64-APSK (green solid line). The dotted lines refer

to the spectrum obtained if the power back-off is selected

to be optimum for the SISO system. In this example, no

predistortion is employed and = 0.7 is selected for theMIMO channel; the transmit filters roll-off factor is = 0.3.Although not shown here, these general results still hold true

if different constellations are chosen and if data predistortionis enabled.

Fig. 9. Power spectral density of the signal at the HPA output for the systemconfiguration C without predistortion and roll-off factor = 0.3. Results areplotted for two values of output power back-off: 3.22 dB and 6.11 dB, whichare optimal for the MIMO and SISO respectively.

IV. CONCLUSION

In this work, we have stated through computer simulations

that the use of SVD-based precoding schemes in LOS MIMO

satellite communication systems would not require a higher

power back-off than conventional SISO systems, although

precoded signal constellations are not optimized for PAPR

reduction. On the contrary, the MIMO approach would even

theoretically allow a lower back-off operating point (with gains

up to almost 3 dB for some modulation schemes), as long as

out-of-band power requirements are satisfied.

This result is very promising, since the application of

SVD-based precoding would greatly simplify the detection

algorithms for LOS MIMO systems when the MIMO channel

matrix is not perfectly orthogonal. Furthermore, the results of

this work can be used as a starting point for the development

of link adaptation algorithms in a variable-rate scenario.

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