# A New-Trend Model-Based to Solve the Peak Power Problems in OFDM Systems

Authors: Ashraf A. Eltholth; Adel R. Mekhail; A. Elshirbini; M. I. Dessouki; A. I. Abdelfattah
Journal: Modelling and Simulation in Engineering
 (2008)

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## Abstract

The high peak to average power ration (PAR) levels of orthogonal frequency division multiplexing (OFDM) signals attract the attention of many researchers during the past decade. Existing approaches that attack this PAR issue are abundant, but no systematic framework or comparison between them exists to date. They sometimes even differ in the problem definition itself and consequently in the basic approach to follow. In this paper, we propose a new trend in mitigating the peak power problem in OFDM system based on modeling the effects of clipping and amplifier nonlinearities in an OFDM system. We showed that the distortion due to these effects is highly related to the dynamic range itself rather than the clipping level or the saturation level of the nonlinear amplifier, and thus we propose two criteria to reduce the dynamic range of the OFDM, namely, the use of MSK modulation and the use of Hadamard transform. Computer simulations of the OFDM system using Matlab are completely matched with the deduced model in terms of OFDM signal quality metrics such as BER, ACPR, and EVM. Also simulation results show that even the reduction of PAR using the two proposed criteria is not significat, and the reduction in the amount of distortion due to HPA is truley delightful.


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## Body

## 1. Introduction

In OFDM systems, the  combination  of different signals
with different phase and frequency gives a large dynamic range that is used to
be characterized by a high PAR, which results in severe clipping effects and
nonlinear distortion if the composite time signal is amplified by a power
amplifier, which have nonlinear transfer function. This degrades the
performance of an OFDM system. A measure of the degradation can be very helpful
in evaluating the performance of a given system, and in designing a signaling
set that avoids degradation. The high PAR sets strict requirements for the
linearity of the PA. In order to limit the adjacent channel leakage, it is
desirable for the PA to operate in its linear region. High linearity
requirement for the PA leads to low-power efficiency and therefore to high-power
consumption. PAs are divided into classes according to the biasing used. A
class (A) amplifier is defined as an amplifier that is biased so that the
current drawn from the battery is equal to the maximum output current. The
class (A) amplifier is the most linear of all amplifier types, but the maximum
efficiency of the amplifier is limited to 50%. In reality, due to the fact that
the amplitude of the input signal is most of the time much less than its
maximum value, the efficiency is much less than the theoretical maximum, that
is, only a few percent. This poor efficiency causes high-power consumption,
which leads to warming in physical devices. This is a problem especially in a
base station where the transmitted power is usually high. To achieve a better
efficiency, the amplifier can be biased so that current flows only half the
time on either the positive or negative half cycle of the input signal. An
amplifier biased like this is called a class (B) amplifier. The cost of the
increased efficiency is worse linearity than in a class (A) amplifier. High
demands on linearity make class (B) unsuitable for a system with high PAR. On
the other hand, the large scale of the input signal makes it difficult to bias
an amplifier operating in class (A). In practice, the amplifier is a compromise
between classes (A) and (B), and is called a class (AB) amplifier [1].Several options appear in the literature related with
OFDM systems and nonlinearities. PAR reduction using clipping or coding or
phase optimization techniques or a combination of any two of them [2], are the
tools to combat nonlinearities used in the transmitter. Also a good work have
been done in modeling the performance of
OFDM systems with power amplifiers in [1], but it has assumed the OFDM
signal to have Gaussian distribution which is not very accurate description of
the composite time OFDM signal. In this paper, the effect of power amplifier
nonlinearities is modeled in OFDM systems. We showed that the distortion due to
these effects is highly related to the dynamic range itself rather than the
clipping level or the saturation level of the nonlinear amplifier, and thus we
propose two criteria to reduce the dynamic range of the OFDM, namely, the use of
N-MSK modulation and the use of Hadamard transform. Computer simulations of the
OFDM system using Matlab are completely matched with the deduced model in terms
of OFDM signal quality metrics such as BER, ACPR, and EVM. Also simulation
results show that even the reduction of PAR using the two proposed criteria is
not significant, the reduction in the distortion due to HPA is significant.
Section 2 depicts OFDM signal statistical properties, while
Section 3 introduces
power amplifier models. The derivation of clipping noise and distortion is
presented in Section 4, OFDM signal quality metrics are discussed in
Section 5.
Simulation results are included in Section 5. Finally, the conclusions are
drawn.


## 2. OFDM Signal Statistical Properties

It is well known that according to
the central limit theory that the real and imaginary parts of the OFDM signal
completely agree with the normal distribution and consequently its absolute agrees with the Rayleigh
distribution with probability density function expressed by(1)P(x)=xse−x2/2s2,x∈[0,∞].where s is a parameter, and mean μ=sπ/2, and variance σ2=((4−π)/2)s2.Figure1 explicitly shows that the measured amplitude
histogram of the inphase component/quadrature component for a 256 subcarrier OFDM
signal.Figure 1
Inphase/quadrature
component histogram of a 256 subcarrier OFDM signal.While Figure2 shows the histogram of the OFDM signal absolute.Figure 2
Amplitude histogram of a
256 subcarrier OFDM signal.It is clear that the distribution in Figure1 obeys a
Gaussian distribution, while that in Figure 2 obeys a Rayleigh distribution.


## 3. Power Amplifier Model

A
short description of power amplifier models will be given in this section.
Consider an input signal in polar coordinates as [1] (2)x=ρejφ.
The output of the power amplifier can be written as (3)g(x)=M(ρ)ej(ϕ+P(ρ)), where
M(ρ) represents the AM/AM conversion and P(ρ) the AM/PM conversion characteristics of the
power amplifier.Several
models have been developed for nonlinear power amplifiers, the most commonly
used ones are as follows.


## 3.1. Limiter Transfer Characteristics

A
Limiter (clipping) amplifier is expressed as [3] (4)M(ρ)={ρ,|ρ|<A,A,|ρ|≥A, where A is the clipping level. This model does not consider AM/PM conversion.


## 3.2. Solid-State Power Amplifier (SSPA)

The conversion characteristics of solid-state power
amplifier are modeled by Rapp’s
SSPA with characteristic [3]: (5)vout=vin(1+(|vin|/vsat)2p)1/2p,where vout and vin are complex i/p & o/p, vsat is the output at the saturation point (vsat=A/2), and P is “knee factor” that controls the
smoothness of the transition from the linear region to the
saturation region of
characteristic curve (a typical value of P is 1).Figure3 shows the AM/AM conversion of the two described models with  A=1. It is clear from the figure; as the value of knee factor increases the SSPA model approaches the limiter model.Figure 3
AM/AM
conversion of HPA.One
problem with these methods is that the special model of nonlinear device
requires expensive and time consuming experimental measurements to identify
model parameters. On the contrary, simple measures of nonlinearity directly
related to a low-order polynomial model, such as third- and fifth-order
intersect points, are usually available to a system designer at the early stage
of specification definition or link budget analysis. The
third-order nonlinearity model can be described by the Taylor  series as [4] (6)y(t)=ao+a1x(t)+a2x2(t)+a3x3(t), where x(t) is the input, y(t) is the output signal, and {an} are
Taylor  series coefficients.The nonlinearity of radio frequency circuits is often expressed in
terms of the third-order intercept point (AIP3). It can be
shown that AIP3 and parameters of third-order nonlinearity
model are related as [4] (7)AIP3=4a1/3|a3|.In this study, we only consider the third-order model of nonlinearity,
since the third-order nonlinearity is usually dominated in real systems.In this paper, we have simulated Rapp’s SSPA model, and then
deduced the third-order nonlinearity model to formulate the relation betweenvsat in Rapp’s SSPA model and AIP3 in dB widely used to
express nonlinearity and we have got (8)vsat=p1(AIP3)3+p2(AIP3)2+p3(AIP3)+p4,where (9)p1=0.001936,p2=−0.09044,p3=1.438,p4=−5.197.


## 4. Nonlinearity Distortion Analysis

Here, we will analyze the effect of
nonlinear amplifier on the OFDM signal, first: we will consider the NLA as a
limiter that is expressed by(10)g(x)={x,x≤a,a,x>a, and thus the distortion due to the
NLA as a limiter can be represented by an  extra Gaussian noise with variance σlimiter2,
where  (11)σlimiter=∫a∞(x−a)2P(x)dx=∫a∞(x−a)2xse−x2/2s2dx,∴σlimiter=1s(e−x2/2s2(−x2s2−(a2+2s2)s2+2axs2)−a2πs3erf(x2s))a∞=1s(0−a2πs3)−1s(e−a2/2s2(−a2s2−(a2+2s2)s2+2a2s2)−a2πs3erf(a2s)).This leads to (12)σlimiter=a2πs2(erf(a2s)−1)+2s3e−a2/2s2.And finally (13)σlimiter=2s3e−a2/2s2+a2πs2erfc(a2s).If we consider the
solid-state power
amplifier model given by Rapp (14)g(x)=x(1+(x/vsat)2p)1/2p,and let p=1 and vsat=a, the NLA distortion can be represented
by (15)σSSPA=∫0∞(x−g(x))2P(x)dx=∫0∞(x−x1+(x/vsat)2)2xse−x2/2s2dx,σSSPA=12s(−2a2s2+4s4+a4ea2/2s2Ei[−a22s2]−2a2e−a2/2s22πs(a2−s2)erf[a2s]+2a2e−a2/2s22πs(a2−s2)),  where Ei(x) is the exponential
integral, is defined as Ei(x)=∫−∞x(et/t)dt.
It is plotted as in Figure 4. (16)∴σSSPA=12s(2ae−a2/2s22πs(a2−s2)(1−erf[a2s])+a4ea2/2s2Ei[−a22s2]−2a2s2+4s4), and finally (17)σSSPA=(ae−a2/2s22π(a2−s2)erfc[a2s]+a42sea2/2s2Ei[−a22s2]−a2s+2s3).When
plotting the  deduced distortion models in (13), (17) versus the distribution
parameter (s) with saturation level a=2 (AIP3=10 dB), we  notice
as shown in Figure 5  that the distortion due to the SSPA nonlinearity
is much more larger than that of its limiting effect, also it is obvious that
the distortion is highley senstive to any variation of the parameter (s) as the
slopes of the curves show.Figure 4
Exponential integral.Figure 5
Nonlinear amplifier
distortion versuss, with saturation level a=2
(AIP3=10 dB).On the other hand, when plotting
distortion model versus the saturation level with    parameter (s=0.08)  as
depicted in Figure 6, it is shown that the distortion decays as the value of
saturation level increases.Figure 6
NLA distortion versus saturation levela, with s=0.08.And when plotting distortion model versus
the saturation level with parameter (s=0.8), Figure 7 shows a great increase
in the distortion despite of the constant value of (A/s).Figure 7
NLA distortion versusa,
with s=0.8.And finally when plotting distortion model versus (A/s) as shown in Figure 8
that shows that the distortion is reduced as the value of (A/s) increases.Figure 8
NLA distortion versusA/s.From Figures5, 6, 7, and 8,
it is clear that the distortion due to these effects is highly related to (s)
the distribution parameter, which controls the dynamic range itself, rather
than the clipping level or the saturation level of the nonlinear amplifier.


## 5. OFDM the Signal Quality Metrics

## 5.1. Error Vector Magnitude

The
modulation accuracy of the OFDM signal is measured by error vector magnitude.
EVM is a measure for the difference between the theoretical wave and modified
version of the measured waveform. The measured waveform is modified by first
passing it through a specified receiver measuring filter.The
waveform is further modified by selecting the frequency, absolute phase,
absolute amplitude, and clock timing so as to minimize the error vector. The
EVM result is defined as the square root of the ratio of the mean error vector
power to the mean reference signal power expressed as a percentage.
Mathematically, the error vectore can be written as(18)e=y−x; where
y is the modified measured signal and x the ideal transmitted signal. EVM can
be defined as (19)EVMrms=E[|e|2]E[|x|2].


## 5.2. Adjacent Channel Leakage Power Ratio (ACPR)

Another figure of merit, specific to evaluate
the out of band behavior of the HPA, is the ACPR; it should stay below the
value specified. The ACPR is the ratio of the transmitted power to the power
after a receiver filter in the adjacent channel.In
order to evaluate the ability of HPA models to reproduce the ACPR, we will use(20)ΔACPR=ACPR(S(f))−ACPR(S(f)), where
S(f) is the PSD of the output of HPA while
S(f) is the true output.The
ACPR can be defined as(21)ACPR=∫f2f3S(f)df∫f1f2S(f)df+∫f1f4S(f)df∫f1f2S(f)df,where
f1 and f2 are the frequency limits of the main channel,
and f2 and f3 are the limits of the upper adjacent
channel, and f1 and f4 are the frequency limits of the
lower adjacent channel as shown in Figure 9.Figure 9
OFDM signal spectrum.


## 6. Simulation Results

An
OFDM system is implemented using 512 carriers with cyclic prefix length equal
to 4. Each carrier is modulated using 16-QAM constellation. AWGN noise is
included. BER simulations compared with theoretical results considering the power
amplifier distortion models deduced above in (13), (17) are shown in Figure 10.Figure 10
BER of OFDM system with HPA.From
this figure, it is possible to see that it was predicted in the previous
analysis. The effect of nonlinear power amplifier is illustrated, where a
limiter amplifier is included in the simulations with clipping levels of 12 dB.
The harmful effect of the nonlinearity can also be clearly seemed in this figure.
Finally the figure shows that the computer simulations of BER are completely
matched with the deduced models both the limiting and the nonlinearity effect.Table1 shows The ACPR value for both limiting and nonlinear effects with different
limiting values relative to the maximum absolute value of the OFDM composite
time signal Ymax.Table 1
ACPR and EVM for different limiting values.
Sat. level/YmaxACPR (dB)EVMLimitingSSPALimitingSSPA1−44.7094−28.72681.3e−0160.09292−44.7094−31.04971.3e−0160.027310−44.7094−32.87911.3e−0160.00120.75−30.4841−27.25940.01260.14480.5−23.4736−25.06240.09690.24740.25−14.5232−22.35490.39720.4741It is
clear that as the limiting value decreases, the ACPR increases. It can also be
noted that the effect  of nonlinearity on ACPR value is negligible as compared
to that of limiting as the clip level varies; this is due to the fact that the
spectral leakage that causes the ACPR to increase is mainly due to the clipping
that can be viewed as windowing the spectrum by rectangular window. Table1
shows also The EVM value for both limiting and nonlinear effects with different
limiting values, it is clear that as the limiting value decreases, the EVM
increases. And as the EVM is a measure of the total distortion, it is highly
affected by the nonlinearity rather than the limiting effect.


## 6.1. OFDM Using Different Modulators

We have
simulated an OFDM system shown in Figure11 with 256 carriers and
oversampling factor of 2, with different modulation techniques, namely, M-QAM
(with M=4, 8, and 16) and MSK.Figure 11
Block diagram of an OFDM system.Table2 Shows PAR, standard
deviation δ, and the dynamic range of the OFDM signal with
the above mentioned modulation techniques. It is again in agreement with the
above results. It is clear that although the PAR reduction due to the use
of MSK instead of QPSK is
slightly small, the true gain is the reduction in the dynamic range by 3 dB, which enables us to use a
low linearity and high-efficiency power amplifiers.Table 2
PAR, STD, and dynamic range of OFDM signal.
ModulationPAR(dB)STD (δ)Absolute signal rangeMSK8.39020.04270.0642QPSK8.70010.05620.0912≡MSK+3dB8-QAM9.00630.09650.1470≡QPSK+4dB16-QAM9.29890.12470.2213≡8QAM+3.5dBIn addition, a new indicator arises in the
table which is the standard deviationδ, it is obvious from the table that MSK has the
lowest δ while
for M-QAM, as M increase δ increases also, since we can deal with OFDM
signal as a narrowband Gaussian noise with a mean of zero and variance of δ2, then 68% of amplitude values
ranges in [−δ–δ] and 99.994% of amplitude values ranges in [−4δ–4δ], this can be a good indicator for clipping
efficiency.When
applying SSPA amplifier usingvsat=0.1 on the OFDM signal, we have noticed the
advantage of MSK over QPSK, as shown in Figure 12(a). Also, we have
applied the amplifier on the OFDM signal using M-QAM with M=4, 8, and 16. Using vsat=0.2, we noticed that as M increases, the
distortion due to NLA increases and so the BER as shown in Figure 12(b).(a)
Effect of NLA on MSK and QPSK. (b)
Effect of NLA onM-QAM.
(a)(b)As regarding to our previous
results, it can be noticed that the MSK modulation gives us the lowest PAR when
used in OFDM, besides its main advantage that it ignores any fading-introduced
amplitude fluctuation present in the received signal, and hence facilitates the
utilization of power efficient class (C) amplifier [5].


## 6.2. Walsh Hadamard Transform

The
Walsh-Hadamard transform (WHT) is perhaps the best known of the nonsinusoidal
orthogonal transforms. The Walsh-Hadamard transform of a signalx, of size N=2n, is the matrix-vector product   WHTN⋅x, [6] where (22)WHTN=⊗i=1nDFT2=DFT2⊗⋯⊗DFT2︷n,DFT2=(1/2)[111−1] is the 2-point DFT matrix, and ⊗ denotes the tensor or Kronecker product. The
tensor product of two matrices is obtained by replacing each entry of the first
matrix by that element multiplied by the second matrix. Thus, for example, (23)WHT4=12[111−1]⊗12[111−1]=14[11111−11−111−1−11−1−11].The WHT has gained prominence in digital signal processing
applications, since it can be computed using additions and subtractions only. Consequently,
its hardware implementation is simpler.


## 6.3. Fast Walsh-Hadamard Transform (FWHT)

As the FFT is an algorithm to compute
the DFT efficiently, similarly the FWHT is an algorithm to compute the WHT
efficiently. The FWHT can be expressed as [6] (24)W(u)=1N∑x=0N−1f(x)g(x,u),where g(x,u)=[∏i=0n−1(−1)bi(x)bn−1−i(u)].The FWHT can be derived using matrix factoring or matrix partitioning
techniques. The signal flow graph for a 4 point FWHT is shown in Figure13.Figure 13
Four-point FWHT.In  a digital system, the 1/4 multiplier can be simply implemented in two
arithmetic shifts. The number of additions and subtractions needed to compute
the four WHT coefficients is4×log24=8.


## 6.4. OFDM System Using FWHT

We have examined the use of FWHT with OFDM with different lengths.The idea: since the peak value of the OFDM symbol results at the instant
of coherent addition of subcarriers, it is possible to reduce this peak by
modifying the subcarriers initial phases so as to avoid this condition. This
can be done via successive phase shifts, we have tried when summing up a number
of orthogonal sinusoidal waveforms with successive phase shifts with different
phase shift step in each time we try and we have got the optimum case atphase
shift step = π which is equivalent to modifying the OFDM symbol
representation to be as follows: (25)yn=∑k=0N−1(−1)kXkej2π(nk/N).The above equation can be approximately implemented as shown in Figure
14 in the proposed block diagram of OFDM system.Figure 14
OFDM system block diagram.In the simulations, 16 QAM modulation scheme is selected in OFDM with 512
subcarriers and an oversampling factor of 2, that is, IFFT length = 1024.Figure15 depicts the constellation
diagram of an OFDM mapped signal with and without FWHT, it is clear that the
FWHT can be viewed as an active constellation extension (ACE) method, one of
the latest PAR reduction, which alter or introduce new signal constellations to
combat large signal peaks [7], but our concern here is its effect on the
dynamic range of the OFDM signal. Simulations show that when using the OFDM
shown in Figure 14 with 16-QAM without FWHT, the PAR is found to be
10.8 dB,
while when using the FWHT (2 and 4 points), the PAR equals 10.5 and 10.15 dB, respectively,
provided that the dynamic range is reduced by 3 and 6 dB in the same cases,
respectively.Figure 15
Constellation diagram of OFDM signal.The  BER performance as a function of the
signal-to-noise ratio (SNR)Eb/No in OFDM systems with and without FWHT in the
presence of NLA with vsat=0.1 is depicted in Figure 16.Figure 16
Effect of NLA on BER
with/without FWHT.It can be noticed that the use of FWHT (4 points) greatly avoids the
distortion due to NLA as compared to the 16 QAM case, despite the negligible
reduction in the PAR. It is clear that the real reason in this good performance
is the reduction in the dynamic range of about 6 dB, again in a complete
agreement with the above results.


## 7. Conclusions

In this paper, the effects of
nonlinearities in the power amplifier over OFDM systems were analyzed and
simulated. We can conclude that(i)
the distortion due to high-power
amplifier, either limiting or nonlinearity effects, is highly related to the
distribution parameter (s) that controls the dynamic range itself rather than
the clipping level or the saturation level of the amplifier;(ii)
also it is noticed that the
effect of nonlinearity on ACPR value is negligible as compared to that of
limiting as the clip level varies; this is due to the fact that the spectral
leakage that causes the ACPR to increase is mainly due to the clipping that can
be viewed as windowing the spectrum by rectangular window;(iii)
the EVM is a measure of the
total distortion, it is highly affected by the nonlinearity rather than the
limiting effect. And generally as the limiting value decreases, the EVM
increases;(iv)
although the PAR reduction due
to the use of MSK instead of QPSK is slightly small, the true gain is the
reduction in the dynamic range by 3 dB, which enables us to use a low linearity
and high-efficiency power amplifiers like class (B) or (C);(v)
Walsh-Hadamard transform is used
with OFDM systems as an intelligent scaling factor to reduce the dynamic range
of the OFDM signal without the risk of amplifying the noise when restoring the
signal to its original level. This technique offers an excellent solution to
all of peak power problems in OFDM systems and without any loss in terms of
spectral efficiency and without any side information being transmitted, and can
be applied with low computational complexity;(vi)
the use of WHT with OFDM system enables the
use of the high-efficiency class (C) amplifier without affecting the BER
performance.



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