par reduction in mimo broadcast ofdm systems using tone...
TRANSCRIPT
PAR Reduction in MIMO Broadcast OFDM
Systems using Tone Reservation
Maja Taseska
Jacobs University Bremen
Electrical and Computer Engineering
A thesis submitted for the degree of
Bachelor of Science
May, 2010
Supervisor: Prof. Dr.-Ing Werner Henkel
Abstract
The combination of multicarrier modulation (OFDM, DMT) and Mul-
tiple-Input Multiple-Output (MIMO) transmission significantly improves
the quality and efficiency of modern communication systems. The peak-
to-average ratio problem in multicarrier modulation requires additional
transmitter-sided processing which may become considerably complex in
the case of MIMO transmission. This problem is particularly emphasized
in MIMO broadcast scenarios (MIMO BC), where the necessary precoding
block at the transmitter side severely affects the complexity of the PAR
reduction algorithms. Many of the most important PAR reduction schemes
that have been developed for SISO (Single-Input Single-Output) and MIMO
systems, become too complex and inadequate for practical implementations
when precoding is performed at the transmitter. Herein, on the one hand,
we will refer to Selected Mapping (SLM) and Selected Sorting (SLS) PAR
reduction schemes, which operate by generating multiple signal represen-
tations in the frequency domain. On the other hand we will analyze the
performance of the Tone Reservation Method (TR), originally proposed by
Jose Tellado. We show that the Tone Reservation method, which operates
entirely in time domain, significantly improves the PAR in MIMO scenarios
as well, while retaining low complexity.
Contents
1 Introduction 1
2 The Principles of OFDM 4
2.1 Introduction: Single-Carrier Modulation
vs OFDM? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 OFDM Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2.1 Discrete Fourier Transform (DFT) . . . . . . . . . . . . . . . . 5
2.2.2 The Cyclic Extension . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 Peak-to-Average Power Ratio in OFDM . . . . . . . . . . . . . . . . . 10
3 Multilple-Input Multiple-Output Transmission (MIMO) 14
3.1 Overview of MIMO systems . . . . . . . . . . . . . . . . . . . . . . . . 14
3.1.1 MIMO Channel Model . . . . . . . . . . . . . . . . . . . . . . . 15
3.2 SVD-based Two-Sided Processing in a Single-User (Point-to-Point) Sce-
nario . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.3 MIMO Multi-User Scenario . . . . . . . . . . . . . . . . . . . . . . . . 17
3.3.1 QR Decomposition for MAC (upstream) Interference Cancellation 18
3.3.2 Precoding for the Broadcast Channel . . . . . . . . . . . . . . . 19
3.4 MIMO-OFDM System . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4 PAR Reduction in SISO and MIMO Systems 23
4.1 Overview of the Existing Methods . . . . . . . . . . . . . . . . . . . . . 23
4.1.1 Selected Mapping (SLM) . . . . . . . . . . . . . . . . . . . . . . 24
4.1.2 Partial Transmit Sequences (PTS) . . . . . . . . . . . . . . . . . 26
4.1.3 Tone Reservation . . . . . . . . . . . . . . . . . . . . . . . . . . 27
ii
CONTENTS
5 PAR Reduction by the Tone Reservation Method in a MIMO Broad-
cast Scenario 31
5.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
5.2 The System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
5.3 Simulation and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
5.3.1 The PAR Reduction . . . . . . . . . . . . . . . . . . . . . . . . 34
5.3.2 Comments on Complexity . . . . . . . . . . . . . . . . . . . . . 36
6 Conclusions 38
iii
Chapter 1
Introduction
Designing wireless digital communication systems that efficiently exploit the spatial
domain of the transmission medium provides a significant improvement of the spectral
efficiency. Multiple-Input Multiple-Output (MIMO) systems employ spatial multiplex-
ing by using multiple transmit and receive antennas and provide significant capacity
gains over single path systems. The combination of the recent advances in MIMO
systems with the numerous well analyzed advantages of multicarrier modulation, par-
ticularly Orthogonal Frequency Division Multiplexing (OFDM), gave rise to the MIMO
OFDM systems which are the most prominent systems in modern wireless communi-
cations technology. OFDM as the most popular technique for digital transmission over
frequency selective channels, is a crucial part of the MIMO systems, because it trans-
forms the wideband frequency selective MIMO channel into a number of parallel flat
fading subchannels. This allows for less complex signal processing, needed for reliable
support of multiple users.
While OFDM transmission over mobile communication channels alleviates the dis-
tortions induced by the channel, and constitutes a crucial block of the MIMO systems,
current research efforts are focused on dealing with the inherent drawbacks of the
multi-carrier modulated systems, one of them being the high Peak-to-Average Power
ratio (PAR). This problem is expressed more than in single carrier systems due to the
fact that the independent subcarriers of the OFDM are added together increasing the
probability of forming large peaks which significantly exceed the average power value.
This in turn, leads to non-linear distortion at the power amplifier and out-of-band
1
radiation. There are two approaches suggested to mitigate the problem that the PAR
of an OFDM system can cause: either developing techniques to reduce the PAR, or
improving the amplification stage of the transmitter. Regarding the first solution, sev-
eral distortionless peak-power reduction techniques have been developed which offer
satisfactory improvement at the cost of additional signal processing at the transmitter.
There are well developed schemes for PAR reduction in single antenna systems (SISO),
and the same are being modified and adjusted for MIMO systems. The Selected Map-
ping scheme (SLM), proposed by Fischer, Huber and Bauml, and its modification for
MIMO broadcast channels, the Selected Sorting (SLS), proposed by Fischer and Siegl
in [4], operate in frequency domain by allowing for a selection of an optimal signal
representation. The Tone Reservation method (TR) is based on subtraction of time
domain Dirac-like signals from the original multicarrier symbol and operates fully in
time domain.
In this work, we concentrate on the PAR reduction problem in MIMO OFDM sys-
tems, particularly in the MIMO broadcast channel case (downlink). Highest complexity
of the PAR algorithms can be observed in the case of a broadcast channel, where due
to the physical separation of the users at the receiving end, joint processing is only
possible at the transmitter. Non-linear precoding is necessary to reduce spatial inter-
ference among the multiple users. SLM and SLS have been applied on this transmitter
structure of the BC scenario in [4], and PAR improvement is observed at very large
computational cost. In our work, we simulate the MIMO BC transmitter structure
and we apply Tellado’s Tone Reservation Method (TR) after the precoding block. The
benchmarks presented in the results section are the complexity of TR versus the com-
plexity of SLM, and the increase in complexity of TR when used in the MIMO BC
scenario as compared to the TR employed in SISO. Regarding the performance of the
TR algorithm in MIMO systems, we compare two different approaches of the algorithm
application, based on the selection of the reduced peak. We show that the Tone Reser-
vation Method is applicable in MIMO OFDM systems as well, and offers very good
performance at lower complexity than the other PAR reduction methods.
In Chapter 2, the principles of multicarrier modulation (OFDM) are revised and the
PAR problem is formalized, Chapter 3 is an overview of the variants of MIMO systems
2
(Point-to-Point, Multiple Access - MAC, Broadcast - BC) and a detailed description
of the precoding procedure in the downlink case is given. In Chapter 4, we look more
closely at the theory and the ongoing research into the PAR reduction methods men-
tioned above, and their performance and complexity. In Chapter 5, our transmitter
model is described, where we apply the Tone Reservation Method for PAR Reduc-
tion. The results of the simulation are presented along with complexity observations.
Chapter 6 presents the conclusions of the work.
3
Chapter 2
The Principles of OFDM
2.1 Introduction: Single-Carrier Modulation
vs OFDM?
Modern communication services are in continuous demand for higher data rates.
The reliability of the data transmitted at high rates highly depends on the digital
transmission infrastructure. In single-carrier systems, as higher rates are used, the
symbol duration becomes shorter and thus the system becomes more susceptible to
information loss from impulse noise, and Intersymbol Interference (ISI) becomes more
pronounced.
OFDM extends the concept of single-carrier modulation by using multiple subcarriers
within the same channel. The total data rate is then divided among the subcarriers.
The serial data stream is passed through a S/P converter and is split into data blocks of
size N , where N is the number of parallel channels (subcarriers) of the system. Each
data channel is used to modulate a separate carrier at different frequencies. Since
every N th symbol is applied to a carrier, this procedure has an interleaving effect on
the symbols. At the receiver, the OFDM signal is demultiplexed and the N separate
carriers are demodulated. Then, the baseband signals are deinterleaved using a P/S
converter to obtain the original data. In practice, OFDM is implemented using the
Discrete Fourier Transform (DFT). The sinusoids of DFT form an orthogonal basis set
and a signal in the vector space of the DFT can be represented as a linear combination
4
2.2 OFDM Structure
of the orthogonal sinusoids. The Inverse Discrete Fourier Transform (IDFT) is used at
the OFDM transmitter to map the input signal onto a set of orthogonal subcarriers,
and similarly, the DFT is used again at the OFDM receiver to recover the received
subcarriers. The orthogonality of the subcarriers offers higher spectral efficiency than
the conventional Frequency Division Multiplexed (FDM) systems (more about FDM
in [10, 11]).
In single-carrier modulation, the data stream is applied to a single carrier, and
the modulated signal occupies the whole bandwidth available for the system. This
means that impulse noise, or deep fades of the channel are likely to cause bursts of
transmission errors. In contrast, in multi-carrier system, during N symbol periods
each of the subcarriers transmits only one symbol, each with N -times longer duration.
Therefore, an identical channel fade would only affect a fraction of the duration of
each symbol. In this case, the system is more likely to recover the partially distorted
N symbols, and is thus less susceptible to impulse noise and channel fades. Moreover,
as the symbol duration is increased, the ISI decreases.
The usage of a cyclic prefix in the multi-carrier system implementation offers further
enhancement of the performance of OFDM, namely, a very effective ISI mitigation and
a possibility of simple equalization techniques. The length of the cyclic prefix, Lcp
should be at least LC− 1 samples long, in order to achieve complete removal of the ISI
(LC is the length of the impulse response of the channel). The cyclic prefix of length
Lcp, is realized by appending the last Lcp time domain samples of the previous OFDM
symbol to the new symbol. The next section explains the usage of the cyclic prefix, as
well the overall structure of the OFDM in more detail .
2.2 OFDM Structure
2.2.1 Discrete Fourier Transform (DFT)
The key components of a modern OFDM system implementation are the IDFT
and the DFT at the transmitter and the receiver respectively. For a system with N
5
2.2 OFDM Structure
subcarriers, these components have to implement the following operations:
dn =1√N
N−1∑k=0
DkWknN , n = 0, 1, ..., N − 1 , (2.1)
Dn =1√N
N−1∑k=0
dkW−knN , n = 0, 1, ..., N − 1 , (2.2)
where WN is the N th root of unity:
WN = ej2π/N . (2.3)
The DFT essentially correlates the input with each of the sinusoidal basis functions.
If the input signal has energy at a certain frequency, there will be a peak in the
correlation of the input with the sinusoid at that frequency. Since the basis functions of
the DFT are orthogonal (uncorrelated), the correlation performed in the DFT domain
for a given subcarrier only sees the energy for that subcarrier. The energies from the
other subcarriers do not contribute.
The formation of a single OFDM symbol is as follows: the IDFT block takes a
vector of length N , which contains frequency domain complex data symbols drawn
from a particular constellation (usually the standard PSK or QAM):
D = [D0, D1, . . . , DN−1] (2.4)
In the IDFT block, this vector is converted to a time domain vector by using Eq. (2.1):
d = [d0, d1, . . . , dN−1] (2.5)
The IDFT correlates the frequency domain input with its orthogonal basis functions
and maps it onto the sinusoidal subcarriers. The IDFT output signal is actually a
summation of all sinusoidal basis functions. The output vector of N time-domain
samples makes up a single OFDM symbol. At the receiver, a DFT is used to perform the
reverse operation according to Eq. (2.2), i.e, to transform the time domain vector back
into frequency domain. The IDFT and the DFT blocks are practically implemented
by using efficient FFT algorithms.
6
2.2 OFDM Structure
2.2.2 The Cyclic Extension
The method of introducing a cyclic prefix prior to transmission in ODFM allows for
easy ISI removal and enables efficient channel equalization. In order to understand the
principle, consider a linear time-invariant channel (LTI) with a finite number of taps
represented by a vector of length L:
h = [h0, h1, . . . , hL−1] , (2.6)
and recall the important property of continuous time and discrete time Fourier trans-
forms (FT and DFT):
FT{d(x) ? h(x)} = FT{d(x)} × FT{h(x)} , (2.7)
DFT{d[n]⊗ h[n]} = DFT{d[n]} ×DFT{h[n]} , (2.8)
where ?, ⊗, and × denote linear convolution, circular convolution, and multiplication,
respectively.
In DFT domain, the duality between convolution in time domain and multiplication
in frequency domain does not hold for the linear convolution, but for the circular one.
However, in the transmission process, the OFDM signal is linearly convolved with the
channel impulse response. The cyclic prefix is realized in such a way that for every
block of length N denoted by
d = [d0, d1, . . . , dN−1] , (2.9)
we form an input block of length N + L− 1:
d = [dN−L+1, dN−L+2, . . . , dN−1, d0, d1, . . . , dN−1] . (2.10)
This creates pseudo-periodicity of the input which enables the linear convolution to be
equivalent to a circular one. The beneficial result is that the effect of the channel in
frequency domain becomes multiplicative, according to Eq. (2.8). This means that the
spectrum of the OFDM signal (i.e. the frequency domain constellation points) will be
multiplied with the frequency response of the channel. Each subcarrier symbol will be
multiplied by a complex factor corresponding to the channel response at the subcarrier’s
frequency. The effects of the channel can be undone by designing a frequency-domain
7
2.2 OFDM Structure
equalizer which is much simpler than time domain equalizers. It takes place after the
FFT block of the receiver, and compensates for the phase and amplitude distortion of
the channel by performing a single complex multiplication per subcarrier. In addition,
the ISI is easily removed, because it extends only into the first L− 1 received symbols,
which are anyhow discarded at the receiver as they correspond to the cyclic prefix. The
relevant data is ISI - free. The simplification of dealing with ISI comes at the cost of
reduced power and data rate efficiency. Namely, part of these two resources has to be
allocated for the cyclic prefix and therefore cannot be used for data transmission. The
fraction of data rate and energy that is lost is LL+N
. The overhead can be reduced by
increasing the number of carriers N as much as possible. However, due to the time-
varying nature of the channels in practical implementations, the maximum number of
carriers that can be used is limited.
INPUT BITSTREAM QAM/PSK
MAPPING
IDFT
P/S
+
CYCLICPREFIX
CHANNEL
S/P
REMOVEPREFIX
DFT+
D1
D2
DN dN
d2
d1 d1
d2
dN
dN-L+1 yN-L+1
y2
yN
y1 y
1
y2
yN
Y1
YN
Y2
Figure 2.1: OFDM Block Diagram
Looking into the OFDM system in terms of matrices gives useful insights regarding
the relation between frequency selective channel and MIMO channel discussed in the
next chapter. In order to further clarify the concept behind OFDM, we will rehearse
the standard approach of a matrix description of the circular convolution and the DFT,
used in the standard Signals and Systems textbooks. The starting point for the analysis
is the duality between circular convolution and DFT, expressed in Eq. (2.8). Let d
be a time-domain vector of length N . If the channel impulse response has a length L
(L ≤ N), for convenience, we apply zero-padding of length N − L in order to be able
to use the channel impulse response in the matrix operations. Thus we obtain vector
8
2.2 OFDM Structure
h of length N
h = [h0, h1, . . . , hL−1, 0, . . . , 0]
If we take T to be the Toeplitz matrix of the channel (circular convolution matrix),
and y the signal after the channel (at the receiver), then the convolution of the input
with the noiseless channel can be represented by the linear operation:
y = Td ; (2.11)
Now, Eq. (2.8) can be equivalently written as:
WTd = ΛWd , (2.12)
where W is the DFT matrix,
1 1 1 · · · 11 W W 2 · · · WN−1
1 W 2 · · · · · · W 2(N−1)
.... . . . . .
......
. . . . . ....
1 WN−1 · · · · · · W (N−1)(N−1)
(2.13)
and Λ is a diagonal matrix with diagonal entries equal to the DFT coefficients of the
channel. Now, by simply reordering Eq. (2.12), we obtain:
T = W−1ΛW , (2.14)
which means that the Toeplitz matrix is diagonalized in the coordinate system defined
by the columns of the DFT matrix and the eigenvalues of T are the DFT coefficients
of the channel impulse response h. Finally, for noiseless channel, we have the following
input-output relation:
y = Td = WΛW−1d . (2.15)
When the channel is Gaussian, with AWGN (Additive Gaussian White Noise), the
input-output relation is:
y = Td + n = WΛW−1d + n . (2.16)
where both n and n are AWGN vectors.
9
2.3 Peak-to-Average Power Ratio in OFDM
Utilizing this matrix analysis in the context of the OFDM structure, where D is
a frequency-domain input vector, T is the Toeplitz matrix of the channel and Y is
the frequency domain vector on the receiver side after the DFT block, we obtain the
following equation:
Y = W−1TWD
⇒ Y = ΛD (2.17)
Under the assumption that a cyclic prefix is used in the process, the last equations show
that the OFDM decomposes the ISI channel into N orthogonal flat fading subchannels.
This is accomplished by using the IDFT and DFT rotations as pre and post-processing
matrices. Knowledge of the channel frequency response matrix H is not necessary
neither at the transmitter nor at the receiver.
2.3 Peak-to-Average Power Ratio in OFDM
The major drawback of the multicarrier modulation is the possibility of extreme
amplitude excursions of the signal in time domain. The basic problem is that depending
on the transmitted symbol sequence, the subcarriers can constructively interfere to yield
amplitudes that scale with the number of subcarriers N . Thus, the peak power scales
as N2, whereas the average scales only as N . This creates the possibility that the
PAR reaches values as high as N . For example, if N = 256, this means that the PAR
can reach up to 24 dB, imposing the requirement that the power amplifier stays in a
linear region over a range of 24 dB higher than operating power required on average.
However, the PAR value of N occurs with extremely low probabilities of order 10−5.
Thus, the best way to describe the PAR is by assigning probability distribution to
it. For that purpose, the statistical model of the transmitted time-domain samples
d0, d1, . . . , dN should be analyzed.
As described in the previous sections, the time domain OFDM signal is obtained by
taking the IDFT of the vector D of frequency-domain symbols. Thus, it represents
a sum of complex exponential functions, whose amplitudes and phases depend on the
data contained in D. The symbols from the most commonly used PSK and QAM con-
stellations satisfy the following statistical properties: E{Dk} = 0 and E{|Dk|2} = 1,
10
2.3 Peak-to-Average Power Ratio in OFDM
where E{♦} denotes the expected value. Assuming that the different symbols Dk are
independent, the time domain samples dk represent linear combination of N indepen-
dent identically distributed random variables (IDFT is a linear map). By the central
limit theorem, dk can be approximated by Normal Gaussian complex, i.e, Rayleigh dis-
tributed random variables. If the variance is normalized to σ = 1√2, then the cumulative
distribution function (cdf) of a Rayleigh random variable is:
P{x ≤ x0} = 1− e−x0 (2.18)
Moreover, due to the row orhtogonality of the IDFT matrix, the random variables dk
are uncorrelated to each other, which in the case of Gaussian random variables implies
that they are independent. We have now presented all the statistical analysis necessary
to derive the PAR in an OFDM system. The expression that defines the PAR is:
PAR =maxk{|dk|2}E{|dk|2}
. (2.19)
From the analysis of the distributions of dk, it follows that the random variables Pk ≡|dk|2 can be modeled as Rayleigh random variables, where by convenience, they can be
normalized so that E{Pk} = 1. Taking into account this normalization in Eq. (2.19),
we obtain PAR = maxk{Pk}. Thus, the cumulative distribution function (cdf) of the
PAR can be computed as follows:
P [PAR > γ] = 1− P [maxk{Pk} ≤ γ]
= 1− P [P0 ≤ γ, P1 ≤ γ, . . . , PN−1 ≤ γ]
= 1− (1− e−γ)N (2.20)
Here, we used the independence of dk. It is clear from the expression that the PAR
problem becomes more severe as the number of subcarriers, N , increases. However,
the above mentioned PAR value N for a system with N subcarriers is very pessimistic
for properly randomized data. The PAR in practical systems is usually always below
15 dB, which is still considered undesirably high for the system even though it occurs
with very low probabilities.
The problem of high peaks in the amplitude is most severe at the transmitter output.
In order to transmit these peaks without clipping, the range of the D/A converter must
accommodate the peaks, and more importantly, the power amplifier must remain linear
11
2.3 Peak-to-Average Power Ratio in OFDM
over an amplitude range that includes the peak amplitudes. This implies a significant
back-off of the average operating power with respect to the maximum operating power
available, leading to poor power efficiency of the amplifier. In order to avoid the
large back-offs, saturation or clipping of the amplifier must be allowed. However, this
additional non-linear distortion leads to increased bit-error rates (BER) and causes
spectral widening of the signal and adjacent channel interference.
Linear
Region
NonlinearRegion
Vin
Vout
Figure 2.2: Power Amplifier Response
Therefore, a different approach and additional transmitter signal processing are
required to reduce the PAR in an OFDM system. The most popular and the most
efficient techniques for PAR reduction are analyzed in detail in Chapter 4.
12
2.3 Peak-to-Average Power Ratio in OFDM
Figure 2.3: Probability of the peak amplitude per OFDM symbol being above the
threshold value
Figure 2.4: Probability of the instanteneous signal amplitude of the time-domain
OFDM signal being above the threshold value (N = 128)
13
Chapter 3
Multilple-Input Multiple-Output
Transmission (MIMO)
3.1 Overview of MIMO systems
Multiple-Input Multiple-Output (MIMO) systems operate with multiple antennas
at the transmitter and the receiver. The multiple antennas provide additional spatial
dimensions for communication and can be used to increase data rates through mul-
tiplexing as well as improve performance through diversity. Multiplexing is realized
by exploiting the structure of the channel gain matrix to obtain independent signaling
paths that can be used to send independent data, thus increasing channel capacity.
The multiple antennas can also be used to obtain diversity, by sending the same sym-
bol via many antennas. The performance gain of the MIMO systems comes with an
increased cost of the multiple antennas used and their space and power requirements,
as well as increased signal processing complexity.
MIMO communication systems have gained a lot of popularity mostly because of
their usage in single-user point-to-point scenarios. However, recent research has proven
that multi-user systems can exploit the multiple spatial dimensions offered by the mul-
tiple antennas even more efficiently than the single-user systems. In addition to the
multiplexing and diversity gains as in the single-user case, multiple antennas allow the
base station to simultaneously transmit and receive data from the different users. In
a multi-path environment, the channel affects the different input signals differently,
14
3.1 Overview of MIMO systems
and therefore, the output signal needs to be decomposed in order to obtain the orig-
inal signals corresponding to the different transmitter antennas. The most common
mathematical tools used for MIMO channel decomposition are the Singular Value De-
composition for the point-to-point (single-user) scenarios where two-sided processing is
applicable, and the QR decomposition for the multi-user scenarios, where due to spatial
separation of either the transmitters or the receivers, only one-sided signal processing
is possible. These techniques will be explained in the subsequent sections.
3.1.1 MIMO Channel Model
In the most general model, the MIMO channel is considered to cause temporal (ISI)
and spatial (multiuser) interference. Suppose that the impulse response has length L.
Then the channel can be represented by the matrix polynomial:
H(z) =L−1∑k=0
Hkz−k , (3.1)
where each matrix Hk is an Nr×Nt matrix, containing the complex channel coefficients
of delay k between all pairs of antennas. For reliable data transmission over this kind
of channel, the following is required:
• Temporal interference (ISI) is eliminated by employing multi-carrier modulation
(OFDM) in the system. The available bandwith is thus partitioned into many
orthogonal bands which can be modeled as narrowband channels by Nr×Nt chan-
nel coefficient matrices. Denote the channel matrix of subcarrier n as H(n). Then
the narrowband transmission for that subcarrier can be modelled by the equation
y(n) = H(n)x(n) + n , (3.2)
where H(n) is a Nr×Nt matrix of the channel gains H(n)ij from the ith transmit
to the jth receive antenna at subcarrier n, x(n) is Nt×1 input vector, y is Nr×1
output vector, and n is the noise vector. The noise samples are assumed to be
white with variance N2
.
• Spatial interferences which occur in each of the narrow-band subchannels are
eliminated by additional transmitter and/or receiver side processing, reviewed in
the next sections.
15
3.2 SVD-based Two-Sided Processing in a Single-User (Point-to-Point)Scenario
3.2 SVD-based Two-Sided Processing in a Single-
User (Point-to-Point) Scenario
The multiplexing gain of a MIMO channel results from the fact that it can be decom-
posed into R parallel independent subchannels where R = rank{H}. By multiplexing
independent data onto these independent channels, an R-fold increase in data rate can
be obtained, in comparison to a system with just one antenna at the transmitter and
receiver.
The parallel decomposition of the channel is obtained by defining a transformation on
the channel input and output x and y through transmitter pre-processing and receiver
post-processing. The underlying concept is the singular value decomposition of the
channel matrix,
H = UΣVH, (3.3)
where U and V are Nr×Nr and Nt×Nt unitary matrices, respectively, and Σ is a
Nr×Nt diagonal matrix, with entries the singular values σi of the channel matrix H,
i = 1, 2, . . . , R , i.e, the square roots of the eigenvalues of HHT = HTH.
In this analysis we assume the channel to be narrowband, so that it can be modeled
by the matrix of complex channel coefficients H. When the SVD of the channel matrix
is computed, the pre- and post-processing are realized by the linear transformations:
x = Vx (3.4)
y = UHy (3.5)
The diagonalization of the channel matrix is clearly explained through the following
equations:
y = UH (Hx + n)
= UH(UΣVHx + n
)= UH
(UΣVHVx + n
)= UHUΣVHVx + UHn
= Σx + n (3.6)
Since unitary linear transformations do not change the average power of white Gaussian
noise, the input noise n and the output noise vector n follow the same distribution.
16
3.3 MIMO Multi-User Scenario
Η = UΣVHV UHx x~ y~ y
Figure 3.1: SVD Channel Decomposition
In this way, the presented method of transmitter pre- and receiver post-processing,
transforms the channel into R parallel subchannels with input vector x, output vector
y and channel gains σi, i = 1, 2, . . . , R. The parallel subchannels are then used for
data multiplexing. It can be noticed that the SVD decomposition of a MIMO channel
is very similar to the OFDM transmission explained in Section 2.2.2. Both transform
the channel into parallel subchannels by unitary matrix operations at the transmitter
and the receiver. However, in the SVD decompositon, the channel matrix has to be
known on both sides.
3.3 MIMO Multi-User Scenario
In order to support multiple users, the signal space dimensions of a multi-user system
must be allocated to the different users. There are multiple ways to distribute the
resources among the different users, resulting in a capacity region for the system, which
determines at which rates can the users communicate with arbitrarily small probabiltiy
of error. In this work, the relevant aspect of the multi-user scenarios is the transmitter
or receiver-side signal processing, necessary for achieving cancellation of the spatial
interference. The spatial separation of the users at the receiver side for MAC or at the
transmitter side for BC scenarios, do not allow for a double-sided processing as the
one based on SVD. In the MAC case, only post-processing and in the BC case only
pre-processing can be performed. The QR channel decomposition leads to a spatial
decision feedback equalizer at the MAC receiver, or a spatial Tomlinson-Harashima
precoder (THP) at the BC transmitter.
17
3.3 MIMO Multi-User Scenario
3.3.1 QR Decomposition for MAC (upstream) Interference
Cancellation
In this section, a very short outline of this scenario is given. The underlying concept
is to consider the QR decomposition of the channel matrix:
H = QR , (3.7)
where Q is a unitary and R an upper triangular matrix. The post-processing of the
received signal y is done by first multiplying with the matrix QH, where the superscriptH denotes Hermitian transpose. We assume the channel to be narrowband
and the channel matrix to be a U×U matrix, i.e, there are U users each equipped
with a single antenna, and U antennas at the base station.
y = QH QRx︸ ︷︷ ︸y
+n = Rx + n (3.8)
I - [diag(R)]-1R
decisiondiag(R)-1QH
yy~
x
Figure 3.2: Spatial DFE based on QR decomposition
Due to the fact that R is a triangular matrix, the estimates xk of the transmitted
symbols xk can be easily obtained:
xk =1
Rk,k
· yk −U∑
j=k+1
Rk,j
Rk,k
· xj (3.9)
18
3.3 MIMO Multi-User Scenario
3.3.2 Precoding for the Broadcast Channel
The relevant scenario for this work is the one involving communication from a base
station to a number to geographicaly separated users. Due to the geographical separa-
tion, joint processing at the receiver is not feasible. Therefore, signal processing based
on the QR decomposition is performed at the transmitter. The concept is similar to
the one presented for the MAC scenario. For the model we consider again the coeffi-
cient matrix H as described in the previous section. The first step is to obtain the QR
decompostion of the Hermitian transpose of matrix H.
HH = QR (3.10)
Denote as x the input to the channel, obtained from the information vector x by
appropriate preprocessing, which is now going to be analyzed. Then the following
equation holds:
y = RHQHx + n . (3.11)
Q is a unitary matrix, so multiplication by Q simply rotates the signal space without
causing an energy increase. Therefore, we can define x′ such that x = Qx′. Now, Eq.
(3.11) becomes:
y = RHx′ + n (3.12)
For the information vector x, reception without interference can be achieved if the
following equation holds:
diag(RH)x = RHx′ , i.e., (3.13)
r11x1r22x2
...rUUxU
=
r11 0 · · · 0r21 r22 · · · 0...
.... . .
...rU1 rU2 · · · rUU
·x′1x′2...x′U
Due to the triangular structure of the matrix RH, the equation leads to the following
precoding operation:
19
3.4 MIMO-OFDM System
x′1 = x1
x′2 = x2 −r21r22
x′1
...
x′U = xU −rU,U−1rUU
x′U−1 − · · ·rU,1rUU
x′1
This computation may lead to very large energy increase of x′ and thus modulo opera-
tion has to be included in order to bound the energy of the transmitted signal. Finally,
the precoding operation can be described as follows:
x′1 = x1
x′2 = ΓM2
[x2 −
r21r22
x′1
]...
x′U = ΓMU
[xU −
rU,U−1rUU
x′U−1 − · · ·rU,1rUU
x′1
] (3.14)
ΓMi[x] is defined as:
ΓMi[x] = x−Mid
⌊x+ Mid
2
Mid
⌋, (3.15)
where Mi is the PAM constellation size for user i, d is the constellation point spacing,
and x is a real number. If M -QAM is employed, as it will be done in the relevant
simulations for this work, x is a complex number and then ΓMi[x] is defined as:
Γ√Mi[Re(x)] + jΓ√Mi
[Re(x)] (3.16)
3.4 MIMO-OFDM System
We have seen in the previous sections that many of the explained methods for pro-
cessing the MIMO channel matrix were based on the assumption of a narrowband
channel. A way to remove this constraint is by employing multi-carrier modulation
20
3.4 MIMO-OFDM System
I - [diag(RH)]-1RH
modulo Qx x' x~
Figure 3.3: Precoder based on QR decomposition (spatial THP)
(OFDM or DMT) and then perform the described signal processing techniques to each
subcarrier separately. The key requirement of the overall system is that the cyclic
prefix is at least as long as the channel impulse response, as described in Section 2.2.2
and that the IDFT blocks of all users are synchronized. Then the model described
with Eq. (3.2) holds for each subcarrier and each can be pre- and/or post-processed.
In the context of this work, the relevant scenario is combining the precoding proce-
dure described in the previous section, with OFDM modulation. The way to do this
is to regroup the U information vectors
Xu = [X(1)u , X(2)
u , . . . , X(N)u ] , (3.17)
where X(n)u is the data of user u at subcarrier n, and to form N new vectors
D(n) = [D(n)1 , D
(n)2 , . . . , D
(n)U ] , (3.18)
where D(n)u denotes the data of user u at subcarrier n, and N is the number of sub-
carriers. After having formed the vectors D(n) for each subcarrier, then the precoding
can be applied on each subcarrier, separately.
21
3.4 MIMO-OFDM System
PRECODER 1
PRECODER 2
PRECODER N
ANTENNA 1
ANTENNA 2
ANTENNA U
CHANNEL
RECEIVER 1
RECEIVER 2
RECEIVER U
.
.
.
.
.
.
.
.
.
D(1)
D(2)
D(N)
Figure 3.4: MIMO BC OFDM System
22
Chapter 4
PAR Reduction in SISO and
MIMO Systems
4.1 Overview of the Existing Methods
The simplest technique to reduce the PAR is to clip the signal such that the peak
amplitude is reduced to the desired maximum level. Since large peaks occur with low
probability (see Eq. (2.20)), in some cases clipping could be an effective technique
for PAR reduction. However, clipping is a nonlinear process which may cause inband
distortion and out-of-band noise. These severely degrade the system performance, and
cause interference to other systems. The clipping technique can be improved by using
a different pulse shaping function instead of a rectangular one. In any case, in order
to reduce out-of-band radiation, the window should be as narrowband as possible.
This implies a longer window in time domain, which adversely affects the bit-error rate
(BER). By making an acceptable trade-off, in some cases clipping and signal windowing
can be effective techniques of PAR reduction.
However, new distortionless schemes have been developed which introduce nonlinear
predistortion of the transmit signal to prevent signal peaks prior to amplification. In
this chapter, we will give an overview of the newest and most efficient distortionless
PAR reduction methods. Some of them are based on generating a set of OFDM signals
by multiplying the data vector in frequency domain with different phase vectors and
choosing the optimal one for transmission. Another group of peak amplitude reduction
23
4.1 Overview of the Existing Methods
techniques is based on modifying the time domain signal, wherever its amplitude ex-
ceeds the desired peak limit. The straightforward clipping and windowing mentioned
previously belong to this group as well. The method that we will put emphasis on is
the Tone Reservation method (TR), which is one of the most efficient PAR reduction
techniques, has very low complexity and operates entirely in time domain. All of the
mentioned techniques can be modified for usage both in SISO and MIMO systems.
In this work, we will particularly analyze the performance of the Tone Reservation
method in MIMO broadcast scenario and compare its performance to the other popu-
lar techniques when used in similar scenarios.
4.1.1 Selected Mapping (SLM)
This method benefits from the multicarrier signal sensitivity to phase shifts. The
original vector of frequency domain symbols, D, of length N , where N is the number of
subcarriers, is multiplied by M different phase vectors P(m), m = 1, 2, . . . ,M creating
M different representations of the original block. The phase vectors are of the following
type:
P(m) = [P(m)1 , P
(m)2 , . . . , P
(m)N ] (4.1)
where P(m)µ = ejφ
(m)µ , φ
(m)µ ∈ [0, 2π), µ = 1, 2, . . . , N ,
and each of the new M signal representations is:
D(m)
= D�P(m), (4.2)
where � denotes element-wise multiplication. The M different signal representations
are then converted to time domain and the block with minimum PAR is transmitted.
Ideally, M representations are independent mappings of the original vector D. In that
case the probability in Eq. (2.20) decreases exponentially with M :
P{PAR > γ} = [1− (1− e−γ)N ]M (4.3)
The reduced PAR comes with the cost of increased computational complexity in the
OFDM IDFT block, because for every transmitted OFDM frame, M IFFTs have to
be performed instead of one. At the receiver, the signal must be post-processed in
order to obtain the correct estimate D of the original block of symbols D. Thus, the
receiver needs to know the vector P(m) that was chosen at the transmitter to minimize
24
4.1 Overview of the Existing Methods
the PAR. Therefore, the index m has to be transmitted with high reliability, which
requires additional information of dlog2(M)e and additional measures to ensure reliable
transmission of m. Eventually, the decoding process at the transmitter performs the
following operation:
D = DFT{d(m)} �P∗(m) (4.4)
where � is element-wise multiplication, and ∗ denotes conjugation.
For SISO systems, the application of the SLM method is straightforward according
to the theory given above. This technique can easily be extended to MIMO systems.
• SLM in MIMO Single-User Scenario (Point-to-Point)
The described SLM technique can be applied easily in MIMO single user sce-
narios as independent implementation of the SISO SLM for every transmitted
antenna. The complexity and the additional information for the phase vectors
used now increase linearly with the number of antennas since MNT IDFTs and
NT dlog2(M)e bits of side information are necessary (NT is the number of transmit
antennas).
• SLM in MIMO Multi-User Uplink (MAC)
In this case, every user is assumed to be equipped with a single transmit antenna,
and again SISO SLM is applied on every transmit antenna.
• SLM in MIMO Multi-User Downlink (BC)
This is scenario is of biggest interest in current research, because the precoding
block at the transmitter described in Section 3.3.2 leads to very high complexity of
the system when SLM is employed. The SLM technique, when used as described,
requires receiver side channel equalization, because it is impossible to determine
the impact of the phase vector P(m)µ to the µth transmit antenna if precoding
is performed at the transmitter, since the precoding permutes the order of the
antennas (users) differently for different subcarriers. However, in BC scenarios,
joint processing at the receiver is not applicable. Another alternative is to perform
the phase rotation after the precoding block, which is also not applicable, since
the rotation would distroy the channel equalization of the precoding block. The
only possibility to apply SLM in this case is if one optimal phase vector P (u) is
25
4.1 Overview of the Existing Methods
chosen and used for each of the transmitting antennas. This will, of course, lead
to reduced efficiency in the PAR reduction, and moreover the complexity of the
precoding procedure becomes very high for practical implementation. Therefore,
the SLM method is not optimal in the case of a broadcast channel. A modification
of SLM, the Selected Sorting (SLS), proposed by Fischer and Siegl in [?], is
designed particularly for the broadcast channel. It operates in the same way as
the SLM method, but in order to reduce complexity, only selected subcarriers are
multiplied by the phase vectors.
4.1.2 Partial Transmit Sequences (PTS)
Partial Transmit Sequences represents a PAR technique derived from SLM. The
difference is that instead of rotating all subcarries by different phases as in the case
of SLM, the subcarriers are grouped and rotated block by block. The first step of the
algorithm is a subdivision of the frequency domain OFDM block D of length N into NB
disjoint blocks of length B = NNB
. Then, NB vectors, denoted as Xi, i = 1, 2, . . . , NB,
each of length N are formed. The vector Xi corresponds to the ith symbol block, where
the entries that correspond to the subcarrier indices belonging to other block, are set
to zero. In this way, we have:
X =
NB∑i=1
Xi ; (4.5)
Next, each Xi vector is rotated by constant rotation factor:
ri = ejφ φi ∈ [0, 2π) , (4.6)
to obtain the modified vector:
X =
NB∑i=1
ri ·Xi =
NB∑i=1
ejφ ·Xi ; (4.7)
The vector X is converted to time domain by the IFFT block of the OFDM:
x = IFFT{NB∑i=1
ejφ ·Xi} =
NB∑i=1
ejφ · IFFT{Xi} =
NB∑i=1
ejφ · xi ; (4.8)
The last step of the algorithm is picking optimal set of phase rotations,
{ropt1 , ropt2 , . . . , roptNB} that minimizes PAR.
26
4.1 Overview of the Existing Methods
An important advantage of PTS over SLM is the fact that the PAR optimization
can be done in time domain after the IFFT block, so no additional IFFTs are required.
This is possible due to the linearity of the IFFT, as shown in Eq. (4.8). Research
has proven that PTS has a slightly better performance than SLM. PTS can also be
modified for the MIMO scenarios, in the same way as SLM, described in the previous
section.
Data SourceX
S / P
+
Partition
intoBlocks
X1
X2
XNB
IFFT
IFFT
IFFT
x1
x2
xNB
r1 r2 rNB
OPTIMIZATION
+ x
Figure 4.1: Partial Transmit Sequences diagram
4.1.3 Tone Reservation
The Tone Reservation method is based on iteratively adding a data-dependent, time
domain signal to the original symbol to reduce its peaks. The transmitter compute the
signal by time-domain efficient methods, which results in reduced complexity of the
system as compared to the previously mentioned methods. Tone Reservation belongs
to larger class of PAR reducion methods known as Additive Methods, or Average Power
Increasing Methods, due to the fact that by adding a signal to reduce the peaks, the
average power of the transmitted symbols is increased. Instead of using signal clipping
for PAR reduction, which leads to non-linear distortion of the signal, the idea of the
Tone Reservation method makes use of the finding that the clipped portion of the
27
4.1 Overview of the Existing Methods
signal can be represented as:
x− xclip =∑i
βi · (δ → mi) , (4.9)
where βi are the values that exceed the threshold xclip, and mi are the clip locations.
→ m denotes a cyclic shift by m and δ is the Dirac vector. However, the generation
of a Dirac function requires the whole DFT frame which would occupy the whole
transmission capacity of the system. Instead, the Tone Reservation method generates
Dirac-like signals p using only some reserved carriers (frequency bins) and subtracts
them iteratively from the signal formed by the remaining carriers. This leads to the
following approximation of Eq. (4.9):∑i
βi · (δ → mi) ≈∑i
γi · (p→ mi) , (4.10)
where γi are weighting coefficients.
The frequency bins used for the Dirac-like function generation can either be chosen
at random or can be fixed. If they are chosen at random, after certain number of trials,
a set of bins is found that shows a sufficient peak compared to the sidelobes in the
time domain vector. The other option is to have the reserved bins fixed beforehand,
either by using the carriers that are not used for data transmission or where the SNR
is not satisfactory. After selecting the bins and storing the corresponding time domain
vector, the tone reservation algorithm is applied according to the following steps:
1. Initialize X to be the DFT-domain information vector when the reserved bins
are set to zero.
2. Initialize the time domain solution x(0) to x, obtained as the IFFT of X
3. Find the value x(i)m and location m for which |x(i)m | = maxk|x(i)k |.
4. If |x(i)m | < xtarget or i > imax then stop and transmit x(i), otherwise
5. Update the time-domain vector:
x(i+1) = x(i) − α ·(x(i)m − sign
(x(i)m)· xtarget
)· (p→ m) (4.11)
i := i+ 1 and go to Step 3.
28
4.1 Overview of the Existing Methods
The term α ·(x(i)m − sign(x
(i)m ) · xtarget
)is a possible realization of the weighting coef-
ficients γi in Eq. (4.10). α represents the step size and is chosen dependent on xtarget
which denotes the desired maximum value after the algorithm. The iterative applica-
tion of the algorithm tries to force the exceeding values to below the threshold xtarget.
The diagram of the Tone Reservation method is shown in the next figure. c(t) is the
time domain Delta-like function that is generated using the reserved carriers and is
added to the original signal.
CHANNEL
D0
D1
DN
C1
CN
C2
IDFT
IDFT
d(t)
c(t)
d(t)+c(t)
Figure 4.2: Tone Reservation diagram
Despite its efficiency in terms of computational complexity, the Tone Reservation
method also has several disadvantages that should be pointed out. The first and most
obvious drawback is the inevitable decrease in data rate due to the reserved carriers.
In some systems, it is possible to overcome this drawback by using carriers with very
low SNR which are anyway not used for data transmission. However, it may happen
that the generated signal by those predefined carriers does not show a significant peak
29
4.1 Overview of the Existing Methods
with respect to its sidelobes, so the problem of choosing optimal carriers for reservation
is still a challenge. On the one hand, the choice of carriers certainly affects the amount
of PAR reduction, but on the other hand, searching over all possible sets significantly
increases the complexity for practical implementation. There is a trade-off between
the number of reserved carriers and the performance of the method. The more carriers
are reserved, the larger is the PAR reduction and less power is needed for the power
reduction signal, at the expense of the other system resource, namely the data rate.
30
Chapter 5
PAR Reduction by the Tone
Reservation Method in a MIMO
Broadcast Scenario
5.1 Motivation
In the previous chapter some of the recent and most efficient PAR reduction tech-
niques have been presented. For SISO systems, these techniques have been well ana-
lyzed and optimized, and current research has been dealing with applying these tech-
niques to MIMO systems. Increasing the number of antennas at the transmitter, in-
evitably leads to an increase in the complexity of any PAR reduction technique, because
the PAR reduction algorithms need to be applied to all antennas, separately. In the
scenarios where joint signal processing can be performed at the receiver side, such as
in the single-user point-to-point and in the multi-user MAC (multipoint-to-point) sys-
tems, the complexity increase is related only to the increase in number of transmit
antennas. However, in the multi-user BC (point-to-multipoint) systems, the precoding
block at the transmitter additionaly affects the complexity, usually to an extent that
is undesirable for practical implementations.
The SLM technique, presented in Section 4.1.1 has many shortcomings when used
in broadcast scenarios. Moreover, even in the other MIMO systems, single-user, or
multi-user MAC, the multiple IFFT operations at each antenna, already impose very
31
5.2 The System Model
large computational complexity. In addition, we saw that for the BC case, there is the
restriction that the same vector has to be used for all the antennas, due to the presence
of a precoding block. Eventually, it is clear that this leads to a less efficient PAR
reduction procedure, where the complexity is severely increased, and moreover, the
amount of PAR reduction is reduced. In this chapter, we will analyze the performance
of the Tone Reservation method in MIMO systems, particularly in the MIMO broadcast
system, and we will see that this method offers a high efficiency even for this critical
scenario. The main reason is that the complexity of the algorithm is independent of the
precoding block, and the only PAR reduction complexity increase of any MIMO system
over a SISO system arises due to the larger number of antennas. In addition, the fact
that TR operates entirely in time domain, saves lot of computational complexity that
is lost with the IFFT operations when SLM algorithm is used.
5.2 The System Model
In practical situations, the most general MIMO channel model can be described by
Eq. (3.1). Therefore, as explained in the previous chapters, an OFDM modulation
is necessary to decompose the wideband MIMO channel into multiple narrowband
channels. In the MIMO broadcast scenario, a precoding block is employed at each
subcarrier as shown in Fig. 3.4. The Tone Reservation algorithm described in Section
4.1.3, is applied to the time domain signals at each transmitter, just after the IFFT
block of the OFDM. In the MIMO system, there are multiple OFDM transmitters
and TR is separately applied at each of them, before transmission. We will consider
the transmitter model discussed in Section 3.4, whereby we add a processing block
that performs the PAR reduction using TR algorithm. The full transmitter model is
presented in Fig. 5.1.
32
5.3 Simulation and Results
PRECODER 1
PRECODER 2
PRECODER N
ANTENNA 1
IFFT
ANTENNA 2
IFFT
ANTENNA U
IFFT
CHANNEL...
.
.
.
D(1)
D(2)
D(N)
d1
d2
dU
TR
Algorithm
TR
Algorithm
TR
Algorithm
d1^
d2
dU
Figure 5.1: Transmitter Model
5.3 Simulation and Results
For the simulation of the transmitter model in the previous figure, we used the
following parameters:
• NT = NR = U = 4, where U is the number of users at the receiver side;
• The length of the channel impulse response is L = 5, and the channel coefficients
are complex i.i.d. Gaussian distributed with variance 1L
;
• The randomly generated information symbols are drawn from a 16-QAM con-
stellation;
• Number of carriers of the system: N = 128;
The channel is assumed to be time-varying, and therefore the Q and R matrices used
in the precoding procedure described in (3.3.2) have to constantly be recomputed after
several OFDM blocks. After precoding each subcarrier and applying the IFFT on the
frequency domain vectors of all antennas, as shown in the previous figure, we are ready
to apply the Tone Reservation method following the algorithm in Section 4.1.3.
33
5.3 Simulation and Results
5.3.1 The PAR Reduction
In order to assess the performance of the algorithm in the presented scenario, we
analyzed the sample Complementary Cumulative Distribution Function (CCDF) of
the PAR of the time domain signal before, and after the peak reduction. There are
two factors which generally affect the amount of peak reduction:
1. The number of iterations, i.e., how many times the Delta-like function is sub-
tracted from the original signal in order to reduce the critical peaks.
2. The percentage of reserved carriers, used for generation of the Delta-like function.
The more carriers are reserved, the more emphasized is the peak in the Delta-
function with respect to the rest of the samples. This contributes to faster and
more substantial PAR reduction of the data signal, at the expense of the data
rate. In order to present the effect of the number of subcarriers, we performed
a simulation both with 5 % and 10 % reserved carriers, and compared the PAR
reduction after 4 iterations of the algorithm.
These two factors are the common factors for evaluating the performance of the Tone
Reservation algorithm. Here, we present another, third aspect of its performance,
which is specific only for the case when the TR algorithm is used in a MIMO scenario.
3. The selection of a peak for reduction. Namely, at the output of the IFFT block
of all the NT transmitters of the system, we have a signal with high peak-to-
average ratio. As described in the Tone Reservation algorithm in Section 4.1.3, a
search should be performed to find the maximum peak, and then cyclically shift
the Delta-like function to the appropriate position. Once we set the maximum
number of iterations for the system to I, there are two ways to apply the searching
for the worst peak:
• An equal number of iterations is performed at each antenna, i.e., b INTc
• The total number of iterations is used to search all the antennas jointly, thus
selecting the worst peak among all the antennas.
The latter searching method is more efficient, because it performs the peak reduc-
tion exactly on those samples, where the PAR is most critical, and in general, the
34
5.3 Simulation and Results
critical samples are not equally distributed among all the antennas. Therefore,
giving equal treatment to all the antennas, is not the most efficient approach. The
increased efficiency when performing a joint search, comes at the cost of a more
time-consuming search, as it has to be performed on a longer array, containing
the samples from all transmit antennas.
The results of the simulation of the overal model, and the influence of the three factors
mentioned above, are shown in the next two plots.
Figure 5.2: Comparison between the two search methods, for a different number of
iterations. The number of iterations shown on the plot is the average number of
iterations performed per antenna. For example, if the number is 4, this means total of
16 iterations are applied in the system, with either of the search methods.
35
5.3 Simulation and Results
Figure 5.3: Effect of the number of reserved carriers
5.3.2 Comments on Complexity
The application of the PAR algorithms in MIMO systems is a very popular field in
current research, and the main objective is to reduce the complexity to a level that
is not problematic for practical applications. In the previous chapter, we explained
the drawback of the SLM method when it is applied in a MIMO scenario, namely
the severe increase in computational complexity due to the multiple frequency-time
domain transformations at each antenna. Moreover, in the scenario of interest, i.e., the
broadcast scenario, the complexity and the performance efficiency of the SLM method
is additionally affected by the precoder.
Having multiple antennas at the transmitter, would inevitably lead to increased
complexity of any PAR reduction method, as the algorithms have to be applied on
all transmit antennas. However, performing the TR iterations on the time domain
signal requires much lower computational cost than an IFFT operation. This already
36
5.3 Simulation and Results
offers and advantage of the TR method over SLM, where all the modified data vectors
have to be transformed in time domain by IFFT, and then the best one is selected for
transmission. However, one more important observation of this work is the fact that
the Tone Reservation method, does not suffer from additional complexity due to the
precoder block in a broadcast system. The PAR reduction is done on the time-domain
signal at each transmitter, and the effects of any additional transmitter pre-processing
are irrelevant at this stage.
37
Chapter 6
Conclusions
In this work we analyzed the performance of the Tone Reservation method for PAR
reduction in MIMO broadcast systems. In the first chapters, the importance of the
OFDM and MIMO systems was presented, along with the PAR problem of the multicar-
rier modulation. Chapter 4 gave an overview of the newest PAR reduction techniques
and their application in SISO and MIMO scenarios. The fact that the Tone Reser-
vation method has already proven to be extremely efficient in single-antenna systems,
was a motivation to try to apply this method to the case of multiple antennas, and
moreover, in broadcast systems, where the precoding block usually severely affects the
PAR reduction algorithms. We simulated the system model, where all of the system
components (precoding block, OFDM, and PAR reduction block) were included. The
simulations analyzed the statistical behavior of the time-domain signal taking all an-
tennas into account, before and after the Tone Reservation algorithm was applied. In
order to characterize the performance, we compared the PAR according to:
1. Number of iterations applied (4,8,12);
2. Searching method for the peaks (Search performed at each antenna individually,
versus, overall search on all antennas);
3. Number of reserved carriers for generation of the Delta-like signal. (5 % versus
10 %).
The amount of PAR reduction depends on all of these factors. For optimum results,
a trade-off between computational efficiency, data rate, and PAR reduction has to be
38
found. Obviously, the more iterations are applied, the more peaks are reduced and
there is a larger reduction in the PAR. However, it should be noted that regardless
of the number of iterations, the reduction power of the algorithm is limited by the
generated Delta-like function. Namely, due to the fact that this signal is generated
only by a small percentage of subcarriers, its addition to the original data signal in-
troduces undesired amplitude variations not only at the critical peak, but at the other
time-domain samples, as well. Reserving more subcarriers for the Delta-like signal,
increases its PAR reduction potential, but at the same time leads to higher losses of
data rate. Eventually, analyzing the specific factor for MIMO scenarios, namely, the
peak-search method, we conclude that performing a joint search over all antennas leads
to significantly larger PAR reduction than performing the search on each antenna, sep-
arately. The reason for this is that the first search ensures that the overall most critical
peaks will be treated first by the algorithm iterations, whereas the latter one performs
equal number of iterations on all antennas. As it may happen that the large peaks are
concentrated only at a small number of antennas, performing all the iterations only on
these antennas, would lead to much better PAR results than trying to reduce peaks at
the antennas which do not have a critical PAR ratio at all.
Finally, it is obvious that there are many aspects to be optimized and improved in
each of the PAR reduction algorithms, including the Tone Reservation, when they are
applied in MIMO scenarios. This problem represents an open and challenging field in
current research which deserves further work and effort for improvement. Designing
power- and data rate-efficient MIMO OFDM transmitters greatly contributes to the
development of modern wireless communication systems.
39
APPENDIX
The appendix for this work is not public.
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