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Rice University
Practical Integer-Forcing Linear Receivers:from OFDM Signal Architecture to WARP Experimental
Validation
by
Corina Ioana Ionita
A Thesis Submittedin Partial Fulfillment of theRequirements for the Degree
Doctor of Philosophy
Approved, Thesis Committee:
Dr. Behnaam Aazhang, ChairJ.S. Abercrombie Professor of Electricaland Computer Engineering, Rice Univer-sity
Dr. Joseph R. CavallaroProfessor of Electrical and Computer En-gineering, Rice University
Dr. T.S. Eugene NgProfessor of Computer Science and Elec-trical & Computer Engineering, Rice Uni-versity
Dr. Bobak NazerProfessor of Electrical and Computer En-gineering, Boston University
Houston, TexasDecember 2017
Abstract
Practical Integer-Forcing Linear Receivers:
from OFDM Signal Architecture to WARP Experimental Validation
by
Corina Ioana Ionita
This thesis presents the first practical implementation of the integer-
forcing (IF) linear receiver. Instead of treating interference as noise, the IF
linear receiver decodes linear combinations of all transmitted signals. In
theory, this promising approach has been shown to improve the overall per-
formance of wireless networks by using the interference in a constructive
way. However, in practice, little is known about the actual performance
of IF linear receivers, because they pose significant practical challenges.
In this work, I introduce solutions to the practical challenges of IF lin-
ear receivers and I implement and test these solutions in an experimental
environment. First, I identify the transmitter’s coded and uncoded sig-
nal architectures which enable the use of IF linear receivers in practical
orthogonal frequency-division multiplexing (OFDM) systems. Also, I de-
velop the receiver’s signal architecture and derive the corresponding prob-
ability density functions of the received linear combinations of messages.
Secondly, I use the Wireless Open Access Research Platform (WARP) and
the WARPlab 802.11 OFDM framework to build the IF linear receiver for
a 2 × 2 multiple input multiple output (MIMO) wireless network. This
work validates in a practical wireless network the theoretical symbol er-
ror rate and code error rate improvements of the IF linear receiver. For
the uncoded framework, I show that the IF linear receiver performs ar-
bitrarily close to the optimal maximum likelihood (ML) receiver and al-
ways better than conventional linear receivers, (the zero-forcing (ZF) and
the minimum-mean square error (MMSE) linear receivers). Furthermore,
when combined with typical LDPC codes, the performance gap between
the IF and ZF linear receiver increases drastically. Third, I also pro-
pose new ways of reducing the complexity of the IF linear receiver. This
complexity reduction introduces some performance loss. However, I show
through experimental results that, even with this decrease in performance
the IF linear receiver still outperforms conventional ones. Together, these
contributions demonstrate that IF linear receivers can be indeed applied
in practical Wi-Fi networks and can lead to significant performance im-
provements.
Acknowledgements
My deepest gratitude goes to my advisor, Behnaam. Thank you for
being there for me in all the good and the bad moments I went through
on my way to graduation. Your strong belief in my abilities has given
me courage to keep pushing and reaching the finish line. Thank you for
teaching me how to stand on my own as a researcher. A great amount of
appreciation goes to my host and collaborator Bobak Nazer from Boston
University. Thank you for teaching me so many things about how to
do good research and be the most exigent critic of my own work. Your
attention to detail has thought me to always double and triple check my
work. A huge thank you, goes to my collaborator Chen Feng from the
University of British Columbia, I will dare to call you a friend. Your
positive attitude has always lifted my spirits and gave me enough energy
to always find a solution when I was stuck and the light at the end of the
tunnel was difficult to see. Thank you for all the technical talks we had
over Skype, even from a distance you helped me understand the inns and
outs of the mapping of integer linear combinations of symbols.
My years of being a graduate student have led me to meet some ex-
traordinary people, some of which became really good friends. Thank you
all for the part you played in making me a stronger, happier and better
person. Thank you Sam, my office mate at Rice, you have been like a
big sister to me in my earlier graduate years and someone I can reach
v
out anytime and feel like yesterday was the last time we talked. Thank
you Pedro, Melissa, David K, Dash, Gareth, Matt for being such strong
students that I had to try my best to raise to your level. My group col-
leagues, Nancy, David, Joe, and all, for being such a friendly and united
group. Thank you Ryan and Drew, my office mates at Rice for being the
nicest office mates I could have possibly have. I will always miss my time
with Rajoshi and G, and Mihaela and Aida. Going for a coffee with you
was always the best break from doing work. Finally, Islam (and Eman),
my office mate at BU. Our chats about research and personal things have
always lifted my morale. A big thank you for hearing me and helping me
see my problems from a different perspective.
Last but not least, I would like to thank my family, my mum and dad
and my brother for always being so understanding in this long distance
relationship. My mother and father in law for being there for me and
helping us out when we needed the most. And finally, I would like to
thank my daughter Diana for bringing laughter to our lives every day
since you have been with us. The most special thank you goes to my
husband Cosmin. You have always believed in me no matter what and
you were my strength when I was ready to give up. Thank you for all the
sacrifices you made and for always believing in me no matter what. Te
iubesc si te voi iubi la nesfarsit!
To: Cosmin, Diana and our baby boy . . .
Contents
Abstract ii
Acknowledgements iv
1 Introduction 1
2 Related work 4
3 System model 7
3.1 OFDM signalling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.2 Frame structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.3 OFDM architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.3.1 Uncoded . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.3.2 Coded . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
4 Conventional receiver architectures 13
4.1 Joint maximum-likelihood(ML) receiver . . . . . . . . . . . . . . . . . 13
4.2 Linear receivers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
4.2.1 Zero-forcing linear receiver . . . . . . . . . . . . . . . . . . . . 14
4.2.2 Minimum-mean square error (MMSE) linear receiver . . . . . 15
5 Integer-Forcing Receiver Overview 16
6 Integer-forcing linear receivers: codes and constellations 20
6.1 Coded framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
6.1.1 Constellation and mapping . . . . . . . . . . . . . . . . . . . . 22
viii
6.1.2 Transmitter signaling: . . . . . . . . . . . . . . . . . . . . . . 24
6.1.3 Coding details . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
6.2 Uncoded framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
6.3 eIF linear receiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
6.4 Expected behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
7 Experimental aspects 35
7.1 Experimental characteristics . . . . . . . . . . . . . . . . . . . . . . . 36
7.2 Channel estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
7.3 Estimation of the signal-to-noise ratio (SNR) . . . . . . . . . . . . . . 40
7.3.1 N0 per subcarrier . . . . . . . . . . . . . . . . . . . . . . . . . 42
7.3.2 N0 per frame from training OFDM symbols . . . . . . . . . . 43
7.3.3 N0 per frame from subcarrier training pilots . . . . . . . . . . 43
7.3.4 N0 per subcarrier from data symbols . . . . . . . . . . . . . . 44
7.4 LDPC codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
7.5 WARP settings and framework code . . . . . . . . . . . . . . . . . . 47
7.5.1 Sampling frequency offset(SFO) correction . . . . . . . . . . . 47
7.5.2 Automatic gain control(AGC) . . . . . . . . . . . . . . . . . . 48
7.5.3 Code release . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
8 Results 50
8.1 SER and CER results . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
8.2 SNR estimation results: . . . . . . . . . . . . . . . . . . . . . . . . . . 56
8.3 Receiver complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
8.4 Degenerative cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
8.5 Comparison of 4QAM and 9QAM . . . . . . . . . . . . . . . . . . . . 65
9 Conclusions 68
Appendices 70
A WARP and AGC 71
B Frame synchronization 75
References 76
List of Figures
3.1 Frame format for a transmitter with two antennas. Both antennas send
short-time symbols (STS) for signal power adjustments and long-time
symbols (LTS) for frame synchonization and channel estimation. . . . 8
3.2 Frame format after FFT is applied and the result is shaped into a ma-
trix of NSC ×N) tones. A frame contains Nt OFDM training symbols
and N ODFM data symbols. . . . . . . . . . . . . . . . . . . . . . . . 9
6.1 Evolution of the constellations with respect to the mapping function.
In blue squares, A denotes the finite field based alphabet, and in green
circles the shifted constellation Co denotes the complex constellation
before the power adjustment, while the final constellation set C is given
in red. Subfigure a) shows the evolution for 4QAM modulation while
subfigure b) shows the one for 9QAM constellation. . . . . . . . . . . 23
6.2 Transmitter signal architecture for a 2×2 MIMO wireless network that
enables the use of the IF linear receiver. Each real dimension of the
signal transports a codeword. . . . . . . . . . . . . . . . . . . . . . . 25
6.3 The IF linear receiver’s signal architecture for a 2× 2 MIMO wireless
network. Single user decoders are used for each real dimension of the
signal, which contain a linear combination of the real and imaginary
parts of the sent signals. . . . . . . . . . . . . . . . . . . . . . . . . . 27
7.1 The experimental setup of the 2× 2 MIMO wireless network in a wide
indoor open space at the Boston University. . . . . . . . . . . . . . . 36
x
7.2 The experimental setup of the 2×2 MIMO wireless network in a normal
indoor open space on the hallways of the Boston University. . . . . . 37
7.3 Channel profile for the 2 × 2 MIMO wireless channel of frame num-
ber 2900 from one of the experiments. In this figure, each bar plot
shows the magnitude of all NSC = 48 data subcarriers correspond-
ing to each of the 4 individual channels of the experimental wireless
network: h11, h12, h21 and h22. . . . . . . . . . . . . . . . . . . . . . . 37
7.4 Variance of the channel magnitude over an experiment with a total of
5800 frames. Each plot shows the magnitude(straight blue line) and
variance(the red bars) of all NSC = 48 data subcarriers for each of the
4 channels of the experimental wireless network. . . . . . . . . . . . . 38
7.5 Evolution of channel magnitude and it’s phase over time for subcarriers
3, 6, and 9. The second graph shows the difference between the phase
of the channel and the phase of the channel of the first frame. . . . . 39
7.6 Evolution of channel magnitude and it’s phase over time for subcarriers
3, 4, 5 and 6. The second graph shows the difference between the angle
of the channel and the angle of the channel of the first frame. . . . . 40
7.7 Frame format after FFT is applied and the result is shaped into a
matrix of (NSC +N tSC)× (N +Nt) tones. A frame contains Nt OFDM
training symbols and N ODFM data symbols. . . . . . . . . . . . . . 41
7.8 This picture will change to show only three estimates: Evolution of
noise estimates for frame number 4500 from experiment number 48. 42
8.1 Received SNR evolution of the experiment 48 for which the results in
the following chapter were obtained. SNR at antenna 1 and at antenna
2 are shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
8.2 SER for 4QAM modulation, N = 500 uncoded symbols with Nt = 10
OFDM training symbols for noise estimation, OFDM channel is 9. . . 53
8.3 CER for 4QAM modulation, N = 500, rate 1/2 binary LDPC(3, 6)
code, OFDM channel is 9. . . . . . . . . . . . . . . . . . . . . . . . . 54
8.4 Normalized histogram of the frames shown for the SER in both Figure
8.2 and Figure 8.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
8.5 Histogram of the frames shown for the SER in both Figure 8.2 and
Figure 8.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
xi
8.6 SER for 4QAM modulation with various SNR estimates (per frame
from OFDM training symbols Npf0 and from subcarrier pilots Npf
0 and
per subcarrier Npsc0 ), N = 500 and OFDM channel is 9. . . . . . . . . 56
8.7 Histogram of the frames shown for the CER in Figure 8.6. . . . . . . 56
8.8 CER for 4QAM modulation with various SNR estimates (per frame
from OFDM training symbols Npf0 and from subcarrier pilots Npf
0 and
per subcarrier Npsc0 ), N = 500, binary LDPC(3, 6) code, OFDM chan-
nel is 9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
8.9 Histogram of the frames shown for the CER in Figure 8.8 and 8.10. . 57
8.10 CER for 4QAM modulation with various SNR estimates for both IF
and eIF linear receivers (per frame from OFDM training symbols Npf0
and from subcarrier pilots Npf0 and per subcarrier Npsc
0 ), N = 500,
binary LDPC(3, 6) code, OFDM channel is 9. . . . . . . . . . . . . . 58
8.11 SER for 4QAM modulation comparing the case when A is calculated
for each subcarrier f , with the case when the same A is used on two
adjent subcarriers, with N = 500 and the OFDM channel is 9. . . . . 59
8.12 Histogram of the 5800 frames used for the SER results shown in Figure
8.11. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
8.13 CER for 4QAM modulation comparing the case when A is calculated
for each subcarrier f , with the case when the same A is used on two
adjent subcarriers. N = 500, 1/2 rate binary LDPC(3, 6) code, the
OFDM channel is 9. . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
8.14 Histogram of the 5800 frames used for the CER results shown in Figure
8.13. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
8.15 SER for 4QAM modulation comparing the case both signals are used
to identify the sent signals with the case when only one of the two
antennas is able to receive the signal. (N = 500, OFDM channel is 9) 62
8.16 Histogram of the 5800 frames used for the SER results shown in Figure
8.15. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
8.17 CER for 4QAM modulation comparing the case both signals are used
to identify the sent signals with the case when only one of the two
antennas is able to receive the signal. (N = 500, OFDM channel is 9) 63
8.18 Histogram of the 5800 frames used for the CER results shown in Figure
8.17. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
xii
8.19 Channel profile for the experiment in which 4QAM and 9QAM were
compared. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
8.20 SER results of ML and IF receivers for an experiment during which
both 4QAM and 9QAM symbols have been sent alternatively. The
channel profile is shown in Figure 8.19. . . . . . . . . . . . . . . . . 66
8.21 Histogram of the frames used for the SER results shown in Figure 8.20,
for which 9QAM is shown in blue(or the first set of bars from left to
right) and for 4QAM is shown in red (or the second set of bars from
left to right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
8.22 SNR evolution for the 4QAM experiment for which the SER results
are shown in Figure 8.20. . . . . . . . . . . . . . . . . . . . . . . . . 67
A.1 Evolution of the transmitter’s and receiver’s RF gain for an experiment
with 7500 frames when the AGC is turned on. . . . . . . . . . . . . 72
A.2 Evolution of the transmiter’s and receiver’s BB gain for an experiment
with 7500 frames when the AGC is turned on. . . . . . . . . . . . . 72
A.3 Evolution of RSSI for an experiment with 7500 frames when the AGC
is turned on. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
A.4 Evolution of received SNR for an experiment with 7500 frames when
the AGC is turned on. . . . . . . . . . . . . . . . . . . . . . . . . . . 73
A.5 SER behavior for an experiment that uses 9 QAM modulation with
7500 frames when the AGC is turned on. . . . . . . . . . . . . . . . 74
A.6 Histogram for the experimental results shown in Figure A.5. . . . . . 74
B.1 Frame preamble showcasing the LTS peaks after FFT has been ap-
plied. This specific figure, corresponds to that of frame that has been
droppeed by the system. . . . . . . . . . . . . . . . . . . . . . . . . . 75
Chapter 1
Introduction
The wireless medium enables the receiving of not only the signal of interest but also
of the interfering signals. Conventional receivers treat interference as a destructive
signal. The design of low-complexity MIMO linear receivers (such as zero-forcing
and linear MMSE) implies first mitigation of interference through equalization and
only afterwards single-user decoders are used to recover each data stream. However,
the integer-forcing (IF) linear receiver [1] takes the opposite approach and takes ad-
vantage of this “free” quality of the wireless medium. First it removes the noise by
recovering linear combinations of messages by using single-user decoders. Next, it
identifies the sent data streams by solving the set of independent linear combina-
tions of messages. Using the same structured codebooks (like linear codebooks) and
constellations (like quadrature amplitude modulation (QAM) ) at the transmitters
guarantees that integer linear combinations of codewords are themselves codewords.
Significant noise reduction is attained by choosing the coefficients of the linear combi-
nations to approximate the channel matrix while minimizing the effect of equalization
on the noise.
The IF linear receiver generates high computation rates [1] especially in the very
high SNR regime by using the interference in a constructive way. However, in prac-
2
tice, little is known about the actual performance of the IF linear receivers since they
pose significant practical challenges.. This thesis seeks to provide the first look at the
feasibility and performance of the IF linear receiver in a practical wireless network.
In this work, I build an experimental framework, address some of the most impor-
tant challenges and show that the IF linear receiver maintains in practice similar
performance gains as predicted by theory.
One of the first challenges emerges at the signal level. The receiver identifies
linear combinations of codewords [2]. Since in practice the signal lies in the complex
domain, that implies that integer linear combinations of both the real and imaginary
parts of the codewords should result in codewords. Therefore, the architecture of the
sent signal is being formed: each real dimension of the sent symbols has to carry a
codeword. Hence, similar to the conventional linear receivers, single user encoders are
used but for each real dimension of the complex signal. This affects correspondingly
the design of received signal.
In practice, an important challenge is to obtain the channel state information
required by the receiver. In this work, I assume only channel state information at the
receiver. The IF linear receiver requires both estimates of the channel and the signal
to noise ratio (SNR) in order to identify the optimal set of linear combinations to be
decoded. However, when only the estimate of the channel is available a suboptimal set
of linear combinations can still be identified that depends only the channel estimates.
In this case, the IF linear receiver can still be applied in the form of exact-IF (eIF)
linear receiver. In practice, the channel estimate is obtained from the preamble of
the frame. For noise power estimation, I introduce OFDM training symbols in the
structure of the frame. Furthermore, I identify the corresponding mapping function
that enables the correlation between linear combinations of constellation points and
linear combinations of finite field symbols.
3
Last, when compared with other conventional receivers, the IF linear receiver
requires the extra step of identifying the coefficients of the linear combinations to be
decoded. The current algorithm used for this task, identifies only an approximate of
the optimal set of linear combinations. As expected, the complexity of this algorithm
increases as the number of transmit antennas increases. However, I identified at
least one way to reduce the number of times this algorithm is used. By taking
advantage of the coherence of the channel over time and frequency. For example the
same set of linear combinations can be used over two consecutive subcarriers of the
OFDM frame, which introduces some performance loss. However, I show through
experimental symbol error rate (SER) and code error rate (CER) results that even
with this decrease in performance the IF linear receiver still outperforms conventional
ones. I further identify and test several other cases that showcase the advantages of
using the IF linear receiver in practice.
Chapter 2
Related work
The optimal receiver for a MIMO network has been shown to be the joint optimal
maximum likelihood (ML) receiver [3], which is quite impractical due to the high com-
plexity especially when the transmitted signal is encoded. The complexity of the ML
receiver can be significantly reduces by using the sphere decoder [4–10] or by exploit-
ing the structure of the lattice-aided reduction algorithm [11–15]. For high SNR the
ML receiver has been shown to be optimal [16] in terms of the diversity-multiplexing
tradeoff (DMT). Conventional receivers with a smaller complexity than that of the
ML receiver are the zero-forcing(ZF) linear receiver and the minimum-mean square
error (MMSE) linear receiver [3, 17]. The performance of these linear receivers has
thouroughly analyzed and compared like in [18]. Specific practical algorithmic details
for these receivers can be found in [19] and [20].
All these conventional linear receivers use the classical approach of treating in-
terference as noise. However, interfering signals might not always be as harmful
as it was previously thought. Recently, the idea of using interfering signals in a
constructive way was introduced under the form of the compute-and-forward (CF)
protocol [21, 22]. It was shown that significantly higher rates are achieved with the
compute-and-forward protocol. The key idea of CF is that intermediate nodes in the
5
network should decode linear functions of transmitted messages according to their
observed channel coefficients rather than ignoring the interference as noise. The
CF framework was later extended and analyzed for systems with unequal transmit
power [23].
When applied to the MIMO network, the CF protocol produces the IF linear
receiver [1, 2]. Its performance and diversity analysis for various wireless network
setups was presented in [24]. While, several aspects of the IF linear receiver for the
MIMO wireless network have been highlighted in the literature, such as the coding
architecture [25], the dedicated codes [26, 27] and the optimal lattice codes [28].
Compared to conventional linear receiver, the IF linear receiver uses one extra
step at the destination, that is the identification of the coefficients of the linear com-
binations. A lattice reduction algorithm is used to identify the optimal coefficients of
the linear combinations. One such algorithm known to produce fairly good approxi-
mates is the Lenstra-Lenstra Lovasc (LLL) algorithm [29]. Several lattice reduction
algorithms have been developed for various wireless networks that adapt the LLL
algorithm to the specifics of the IF linear receiver such as the ones in [30] and [31].
Furthermore, the IF linear receiver requires knowledge of both the channel coef-
ficients and the SNR of the wireless network. When only information of the channel
is available, the IF linear receiver is applied and it is known as the exact IF linear
receiver [1]. However, when the signal sent is uncoded, the eIF linear receiver is
known in a slightly different format as a lattice-reduction-aided detector [11,32]. The
similarities to the lattice-reduction-aided detector being further investigated in [33].
The results presented in the works discussed above are based on theoretical anal-
yses and computer-based simulations. Whereas theoretical models typically assume
simplified assumptions for mathematical tractability. Furthermore, they assume all
the information needed at the receiver is perfectly known. In contrast, real-world
6
experimental frameworks can reveal unexpected challenges and sensitivities to the
accuracy of the channel state information. Also, most of the theoretical results em-
phasize the significant gains for high to very high SNR regimes which are not very
common scenarios in practice. This is the main motivation for this work. I describe
the experimental framework I build with which I evaluate the performance of the IF
linear receiver. For the experimental framework, out of the many existing software
defined radio (SDR) platforms, I choose to work with the WARP [34] platform, be-
cause it offers an IEEE 802.11 [35] OFDM reference design. The WARP platform
enables the estimation of the SNR needed for the IF linear receiver, in the form of a
post-processing SNR [36, 37] or based on the error-vector magnitude (EVM) [38].
The framework I constructed can be used for both the coded and uncoded transmit
signals. When combined with appropriate space-time codes, the IF linear receiver can
attain capacity of the MIMO network within a constant gap [25]. Lattices constructed
on a combination of p2 quadrature amplitude modulation (QAM) and p-ary linear
codes [39] in combination with modulo operations would enable the use of IF linear
receivers. This scheme was shown to operate close to capacity in high SNR regimes
[40]. Combined with modern codes (such as LDPC codes, [41]) the performance
of both the CF portocol and the IF linear receiver has been analyzed for various
codes [26, 33,42–48].
Chapter 3
System model
Let R denote the reals, C denote the complex numbers, Z denote the integers, Z[j]
the set of Gaussian integers and Fp denote the finite field of size p, with p prime. I use
the notation xR to denote the vector containing only the real part of the elements of
the vector x, and correspondingly xI for its imaginary part. I use boldface lowercase
x ∈ C1×N to denote a row vector and boldface uppercase X ∈ C to denote a matrix.
Let X−1 denote the inverse of the matrix, X† its transpose and XH its Hermitian
transpose.
This work considers a simple two node MIMO wireless network: a transmitter with
MT transmit antennas and a receiver with MR antennas. The focus is on both the
uncoded and coded OFDM wireless frameworks. Next, I introduce only the OFDM
concepts relevant to my theoretical description of the proposed framework, all the
other details are left to be given in Chapter 7.
3.1 OFDM signalling
Following the IEEE 802.11 standard [35], a frame is formed of two parts (see Figure
3.1), a preamble which contains training symbols (short-time symbols (STS) and
8
Figure 3.1: Frame format for a transmitter with two antennas. Both antennas sendshort-time symbols (STS) for signal power adjustments and long-time symbols (LTS)for frame synchonization and channel estimation.
long-time symbols (LTS)) and the payload which contains the data. Following the
MIMO standard setup, the LTS symbols are used for both channel estimation and
synchronization. For channel estimation, each antenna emits an LTS symbol, while
the other remains silent. For synchronization, these symbols are used as well as an
LTS symbol that is emitted concurrently by all antennas (see Appendix B for more
details on how synchronization is achieved and when are frames dropped).
3.2 Frame structure
The standard OFDM signalling scheme is used and I refer the reader to [3] for a
detailed discussion of the preamble and FFT blocks. In the frequency domain, after
the fast Fourier transform (FFT) is applied to the received signal, the payload can
be seen as a resource block, a matrix of dimension NSC × N , as shown in Figure
3.2. I use NSC to denote the number of subcarriers that carry data, and N denotes
the total number of OFDM data symbols. Each entry xi[f, k] ∈ C of this matrix of
OFDM symbols represents a complex symbol, also known as a tone (see the black
entry marked in Figure 3.2); where f is the subcarrier index (or the frequency index),
k is the OFDM symbol index (or the time index), i denotes the transmit antenna and
C denotes the complex constellation of symbols. The row vector xi[f ] ∈ C1×N is used
to denote all the tone symbols containing data sent on subcarrier f , from antenna
i for the duration of a frame. Equal average transmit power is assumed from each
9
Figure 3.2: Frame format after FFT is applied and the result is shaped into a matrixof NSC×N) tones. A frame contains Nt OFDM training symbols and N ODFM datasymbols.
antenna i,
Pi =1
NSCN
N∑k=1
NSC∑f=1
||xi[f, k]||2 = Ptx. (3.1)
Without loss of generality I assume unit transmit power, which means that the trans-
mit signal is normalized by 1/PC, with PC being the average power of the complex
constellation C, PC = 1/|C|∑
x∈C ||x||2.
3.3 OFDM architecture
3.3.1 Uncoded
For the uncoded wireless network, the messages mi are row vectors with elements from
the alphabet A, mi ∈ A1×N (for example for a binary general setup, the alphabet can
be A = F2). A message is mapped to constellation points forming the sent signal, xi
10
by using ϕ(·), a mapping function,
xi = ϕ(mi). (3.2)
3.3.2 Coded
In modern communications systems forward error correction codes have become an
integral part of current standards. For a general coded framework with typical block
codes, the message mi ∈ A1×K of the i-th transmit antenna is first encoded
ci = Encoder(mi), (3.3)
and then mapped to constellation points to form the transmit signal
xi = ϕ(ci), (3.4)
where the rate of the code used is determined by the encoding and mapping techniques
[3, 39].
After the FFT has been applied on the received time samples, and the data block
has been formed, the transmitter’s architecture for the OFDM wireless MIMO net-
work is built based on the following model of the received signal for the k-th tone at
subcarrier f :
yr[f, k] =
MR∑i=1
hr,i[f ]xi[f, k] + zr[f, k], (3.5)
where yr[f, k] is the k-th received OFDM tone signal, on subcarrier f at antenna r,
with r ∈ {1, . . . ,MR}, similarly xi[f, k] is the sent OFDM tone signal from antenna
i, with i ∈ {1, . . . ,MT}, and z`[f, k] denotes the noise at receiver’s antenna r. The
11
channel from antenna i to antenna r is given by hr,i[f ], which is coherent over all
OFDM symbols for subcarrier f . The received signal at one antenna, the `-th one, is
given by
y`[f ] = h`[f ]X[f ] + z`[f ] =
MR∑i=1
h`,i[f ]xi[f ] + z`[f ], (3.6)
where h`,i is the channel coefficients from antenna i to antenna `, and h` is the `-th
row vector of the channel matrix H.
That is, for subcarrier f the received signal is given by
y1[f ]
...
yr[f ]
...
yMR[f ]
= H[f ]
x1[f ]
...
xi[f ]
...
xMT[f ]
+
z1[f ]
...
zr[f ]
...
zMR[f ]
(3.7)
where, yr[f ] ∈ C1×N is the row vector of all data bearing received OFDM tones
on subcarrier f at antenna r; H[f ] ∈ CMR×MT is the complex matrix modeling the
MIMO channel between the transmitter and receiver corresponding to subcarrier f ;
xi ∈ C1×N is the row vector of OFDM data tones sent from antenna i and zr ∈ C1×N
is the complex valued noise at antenna r. To simplify the notation, the frequency
index f is dropped, which simplifies the channel model notation
Y = HX + Z, (3.8)
until the proposed experimental setup is discussed in Chapter 7.
The channel is assumed to be coherent for the entire length of a frame, that is
in the frequency domain, H is coherent for all N OFDM symbols on a subcarrier,
12
while as I observed in my experiments and as it is further discussed in Chapter 7, the
channel is not coherent over frequency, for each f there is a unique channel matrix
H[f ].
Chapter 4
Conventional receiver architectures
4.1 Joint maximum-likelihood(ML) receiver
For the uncoded framework the best performance in terms of symbol error rate (SER)
is attended by the joint maximum-likelihood (ML) receiver [3]. For this channel
model, an estimate of the sent signals from all antennas on a subcarrier at a time
instance k is the result of the minimum decision rule
x†[k] = arg minx†||y†[k]−Hx†|| (4.1)
where || · || is the Euclidean norm of a vector, y[k] = [y1[k] . . . yMT[k]] is the vector
of the k-th element of each received signal, x ∈ C1×MR and x[k] = [x1[k], . . . , xMR[k]]
is a row vector of the estimates of the sent tones. For the uncoded framework, the
ML rule is a joint decision rule, which for a p2- QAM modulation is equivalent to
having to compare p2MR pairs for each complex symbol or tone. However, even if the
joint ML receiver is the optimal receiver, it has an exponentially high computational
complexity for the case when the transmit signal is encoded, making this approach
impractical.
14
4.2 Linear receivers
Conventional linear receivers for MIMO wireless networks use an equalization matrix
B ∈ CMT×MR to form the effective received signal Y,
Y = BY. (4.2)
Each row y` of the effective channel Y corresponds to the `-th received effective
channel at antenna `. For example, for the case when the signal is coded, the effective
received signal is then used by single user decoders (demodulators for the uncoded
case) to estimate the transmitted message from each antenna,
m` = Decoder(y`). (4.3)
4.2.1 Zero-forcing linear receiver
This linear receiver attempts to cancel all the interfering signals from the other an-
tennas when the rank of the channel matrix is rank(H) = MT . The corresponding
equalization matrix is given by
BZF = H−1, (4.4)
with which the effective channel becomes
Y = H−1Y = X + H−1Z. (4.5)
That is at each receiving antenna `, the signal is given by
y` = x + z` (4.6)
15
where z` denotes the efective noise observed at the `-th antenna, and it is the `-th
row of the matrix H−1Z.
4.2.2 Minimum-mean square error (MMSE) linear receiver
The MMSE linear receiver balances canceling the interfering signals with minimizing
the effect of equalization on the noise. The corresponding equalization matrix depends
on both the channel H and the Signal to Noise Ratio (SNR),
BMMSE = SNRHH(I + SNRHHH)−1 (4.7)
where I is the identity matrix, and SNR is the transmit average SNR per antenna
defined as SNR = Ptx/σ2z , with σ2
z the variance of the noise.
Both the ZF and the MMSE linear receivers have been used for both uncoded
and coded transmit signals in practice [19, 20]. When the signal x is uncoded, the
effective channel, y, is used to evaluate the decision rule at the receiver. While for
a coded signal, both the ZF and the MMSE receiver architectures, yield practical
linear receiver architecture [18] that uses y to extract the corresponding probabilistic
information that is used as the input to the decoder.
Chapter 5
Integer-Forcing Receiver Overview
Before diving into the challenges of building the experimental framework, we need to
first understand the concept of integer-forcing and it’s advantages over conventional
MIMO receivers.
The integer-forcing linear receiver architectures use interfering signals from other
antennas in a constructive way. First, using the equalized effective channel Y, the
receiver identifies a set of linear combinations of all transmitted signals. Next, the set
of independent linear combinations is solved for the estimates of the sent messages
mi.
After equalization, the effective channel resulted from an integer-forcing linear
receiver is
Y = BHX + BZ, (5.1)
which can also be written in a form that showcases the set of linear combinations to
be identified
Y = AX + (BH−A)X + BZ︸ ︷︷ ︸Z, effective noise
, (5.2)
17
where the last two terms form the effective noise, Z ∈ CMT×N , that is given by
Z = (BH−A)X + BZ. (5.3)
The equalization matrix for the IF linear receiver depends on all available infor-
mation [1, 23] at the receiver, that is both channel state information and transmit
SNR
B = SNRAHH(I + SNRHHH)−1, (5.4)
where A ∈ Z[j]MT×MT , is the matrix of Gaussian integer coefficients of the linear
combinations of the transmitted symbols that are to be evaluated at the receiver. An
approximate matrix A, that balances the influence of the channel and the noise in the
MMSE sense [1], can be obtained by applying the LLL basis reduction algorithm [29]
to the matrix
(SNR−1I + HHH)−12 , (5.5)
where I is the identity matrix of dimenson MT . As noted in [1], when the matrix of
Gaussian integer coefficients of the linear combinations is identity A = I, then the
equalization matrix of the MMSE linear receiver shown in (4.7) is obtained. That
makes, the MMSE linear receiver a special case of the IF linear receiver.
Next, the newly formed effective channel (5.2) is used as input to single user
decoders [1] to obtain estimates of the linear combinations of messages
v` = Decoder(y`),∀` ∈ {1, . . . ,MT} (5.6)
where y` is the `-th row of the effective channel Y and v` is the estimate of the
18
`-th linear combination with coefficients given in the row vector a` ∈ Z[j]1×MT of the
messages
v` = (a`[m1, . . . ,mMT]†) mod p. (5.7)
Lastly, the set of linear combinations,
[v1, · · · , vMT]† = A[m1, · · · , mMT
]† (5.8)
is solved to obtain the estimates mi of the sent messages.
A special case of the IF linear receiver is obtained when the rank of H is equal to
the number of transmit antennas MT , rank(H) = MT , and the receiver does not have
access to the SNR information. For this receiver, also known as the exact integer-
forcing (eIF) linear receiver [1], the matrix of coefficients of the linear combinations
AeIF is obtained by applying the LLL algorithm to H−1 and yeilds the following
equalization matrix
BeIF = AeIFH−1. (5.9)
The effective channel becomes
Y = AeIFX + AeIFH−1Z (5.10)
while all the other steps taking by the IF linear receiver remain the same. Similarly,
to the IF linear receiver, when the matrix of coefficients of the linear combinations is
identity, AeIF = I, the equalization matrix of the ZF linear receiver (4.4) is obtained,
making ZF a special case of the eIF linear receiver.
However, this information theory view of the IF linear receiver does not provide all
19
the details needed for an OFDM wireless practical system. In the following section, I
will introduce a transmitter and receiver architecture that enables the use of IF linear
receivers in practical OFDM based wireless networks.
Chapter 6
Integer-forcing linear receivers: codes and
constellations
First, the coded framework for the IF linear receivers is introduced, since the uncoded
case can be treated as a special case of the coded one.
6.1 Coded framework
In order to understand the design of the transmitter and receiver architecture lets
take a more detailed look at the linear combinations of messages and codewords. In a
practical framework, the sent symbols are in the complex domain. Therefore, I extend
the result of the linear combinations of messages given in (5.8). I use the notation
of vR` and vI
` to denote the real and imaginary parts of the result of the `-th linear
combination of messages
vR` =
MT∑i=1
(aR`,im
Ri − aI`,imI
i
)mod p (6.1)
vI` =
MT∑i=1
(aR`,im
Ii + aI`,im
Ri
)mod p. (6.2)
21
The basis of the IF linear receiver stands in the construction of messages and their
algebraic properties. It is shown in [1] that if the transmitted messages are part of
a lattice denoted by Λ, an integer linear combination of lattice points is a lattice
point, b1λ1 + b2λ2 = λ ∈ Λ, with integer numbers b1, b2 ∈ Z. Following this setup,
implies that when mRi ,m
Ii ∈ Λ ⊂ F1×K
p , the results of the linear combinations are
also vR` ,v
I` in the same lattice Λ. For the specific details and characteristics of the
lattices and lattice codes that enable the use of integer-forcing please see [1] and [28].
In other words, integer linear combinations of the real and imaginary parts of the sent
messages (which themselves belong to a lattice), result in a lattice point. Furthermore,
the real and imaginary parts of the corresponding encoded messages, are themselves
codewords
cRi = Encoder(mR
i ) (6.3)
cIi = Encoder(mI
i ) (6.4)
with cRi , c
Ii ∈ F1×N
p , giving the rate of the code to be R = K/N log2 p. Therefore, the
result of the `-th integer linear combinations of the real and imaginary parts of the
coded signal are given by
wR` =
MT∑i=1
(aR`,ic
Ri − aI`,icI
i
)mod p (6.5)
wI` =
MT∑i=1
(aR`,ic
Ii + aI`,ic
Ri
)mod p. (6.6)
are themselves codewords, wR` ,w
I` ∈ F1×N
p . Hence, the alphabet of the transmitted
messages that enables the use of the IF linear receiver in a practical framework is
given by A = {Fp + jFp }, with the time symbol mi[k] ∈ A.
The algebraic linearity of the messages and codewords is preserved further on in
22
the transmitted signals. This is achieved through the mapping function ϕ(·) (as shown
in [1, 23, 49]) and the design of the constellation, C. That means, that even at the
symbol level, the result of the integer linear combinations of the real and imaginary
parts of the transmitted symbols corresponds to its equivalent result of the integer
linear combinations of the real and imaginary parts of the sent codewords respectively.
Therefore, the constellation set C has certain specific properties [28]. For the IF linear
receiver, p2-Quadrature Amplitude Modulations(QAM) can be used, when p is prime.
That facilitates the use in a practical framework of standard type of modulations like
4QAM, which is based on the F2 finite field, and atypical ones like 9QAM, which is
a F3 based modulation. Higher binary based modulations require multilevel coding
techniques as identified in [2] and [48].
Both the mapping function ϕ(·) : A → C and it’s inverse ϕ(·)−1 : C → A are
bijective functions. That is, in general for a one dimensional signal (for example
mi ∈ AK , with A = Fp and C ⊂ R (not a complex based alphabet and constellation
set)), integer linear combinations of finite field symbols are mapped to integer linear
combinations of real symbols
(a1m1 + a2m2 + . . .) mod p = ϕ−1((a1ϕ(m1) + a2ϕ(m2) + . . .) mod p), (6.7)
where ai ∈ Z[j], and mod taken in the second part is defined with respect to the
constellation set C.
6.1.1 Constellation and mapping
In order to accurately define the mapping function, first define the setRp ∈ R1×p, and
then the corresponding original complex constellation set Co = {Rp + jRp}. With
these notations the mapping function is defined as ϕ : Fp → Rp, and its inverse
23
-1 -0.5 0.5 1 1.5 2
-1
-0.5
0.5
1
1.5
2
-1 -0.5 0.5 1 1.5 2
-1
-0.5
0.5
1
1.5
2
Figure 6.1: Evolution of the constellations with respect to the mapping function. Inblue squares, A denotes the finite field based alphabet, and in green circles the shiftedconstellation Co denotes the complex constellation before the power adjustment, whilethe final constellation set C is given in red. Subfigure a) shows the evolution for 4QAMmodulation while subfigure b) shows the one for 9QAM constellation.
ϕ−1 : Rp → Fp. The exact values of the set Rp is dependent on the type of the
constellation and the modulation order used p. In the following, I present the exact
mapping functions for the two specific types of modulations that have been used in
the experiments: 4QAM and 9QAM, while the inverse of the mapping function is
presented in Section 6.2.
Example 1: For 4QAM,R2 ={−1
2, 1
2
}, and the the mapping function ϕ : F2 → R2
(as shown in Figure 6.1), is defined by using a shift s = −1/2
ϕ(b) = b− s (6.8)
where the purpose of the shift is to preserve the linearity or messages in the real
domain of the constellation as given in equation (6.7). The resulted alphabet A =
{F2 + jF2} and the constellation Co = {R2 + jR2} are shown in Figure 6.1 a).
24
Example 2: For any p 6= 2 prime order QAM, the set of real symbols is given by
Rp ={−⌊p
2
⌋,⌊p
2
⌋+ 1,
⌊p2
⌋+ 2, · · · ,−
⌊p2
⌋+ (p− 1)
}.
For 9QAM, for which p = 3, the set of real symbols is given by R3 = {−1, 0, 1} (and
the resulted constellation Co is shown in Figure 6.1, b) ). The mapping function for
9QAM, ϕ : F3 → R3 is a bijective function given by
ϕ(b) =
0 , if b = 0
1 , if b = 1
−1 , if b = 2
. (6.9)
The resulted alphabet A = {F3 + jF3} and the constellation Co = {R3 + jR3} are
shown in Figure 6.1 b). For more details on the specific construction of the mapping
functions for constellations compatible with the IF approach see [49].
6.1.2 Transmitter signaling:
The last step in my proposed transmitter architecture is power scaling. At the trans-
mitter, each tone is obtained from the corresponding finite field symbol by using the
mapping function introduced above and then it is scaled to its final constellation
symbol. That is,
xi =Ptx
PCo(ϕ(cR
i ) + jϕ(cIi )) (6.10)
where ϕ is applied element wise to the row vectors, PCo is the average power of the
complex constellation Co, w.l.o.g. the average transmit power Ptx = 1 and the resulted
sent signal has elements from the constellation C, xi ∈ C1×N . The evolution from the
alphabet A to the final constellation C for both 4QAM and 9QAM modulations is
25
Figure 6.2: Transmitter signal architecture for a 2 × 2 MIMO wireless network thatenables the use of the IF linear receiver. Each real dimension of the signal transportsa codeword.
shown in Figure 6.1 a) and b).
This form of the signal, induces a specific transmitter signaling architecture as
shown in Figure 6.2. That is each real and imaginary part of the signal carries a
codeword, and single user encoders are used for encoding. This is an important
constraint for the integer-forcing linear receiver. Multiple streams could be encoded
together or at separate rates using the multilevel coding approach as shown in [50].
6.1.3 Coding details
Now, that the transmitter’s architecture has been established let us move on to de-
scribing the receiver’s architecture, with an emphasis on what enables the use of codes
with integer-forcing linear receivers. The `-th effective received signal (as derived from
equation (5.2)) is given by
y` = a`X + z`, (6.11)
26
where z` is the `-th row of the effective noise matrix Z. The effective received signal
can be further split into the real and imaginary parts
yR` =
MT∑i=1
(aR`,ix
Ri − aI`,ixI
i
)+ zR
` (6.12)
yI` =
MT∑i=1
(aR`,ix
Ii + aI`,ix
Ri
)+ zI
` . (6.13)
The next step is to decode the estimates of the integer linear combinations of
messages vi, obtained as shown in (5.6). First we identify the probability density
function (pdf) needed as the input to the decoder (or the corresponding log-likelihood
ratio (LLR) for the F2 finite field). Since, w.l.o.g. I assume all symbols are equally
likely, the pdf is evaluated at the symbol level. That is, y`[k], the k-th symbol of the
`-th effective channel contains a linear combination of codewords and it is given by
y`[k] =
MT∑i=1
a`,ixi[k] + z`[k], k ∈ {1, ..., N}, (6.14)
where xi[k] denotes the k-th symbol sent from antenna i, and z`[k] is the k-th element
of the effective noise row vector z`. Next, split (6.14) into its real and imaginary parts
yR` [k] =
MT∑i=1
(aR`,ix
Ri [k]− aI`,ixIi [k]
)+ zR` [k] (6.15)
yI` [k] =
MT∑i=1
(aR`,ix
Ii [k] + aI`,ix
Ri [k]
)+ zI` [k]. (6.16)
Since a codeword is formed on each real dimension of the received signal, a single
user decoder is applied to each dimension of the effective channel,
vR` = Decoder(yR
` ), (6.17)
vI` = Decoder(yI
` ). (6.18)
27
Figure 6.3: The IF linear receiver’s signal architecture for a 2 × 2 MIMO wirelessnetwork. Single user decoders are used for each real dimension of the signal, whichcontain a linear combination of the real and imaginary parts of the sent signals.
The IF linear receiver architecture of the signal is shown in Figure 6.3 for a 2× 2
MIMO wireless network. We show all the blocks needed to gather the information
that is required to form the effective channel y`, that is to form the equalization
matrix given in (5.4). This means that besides SNR and channel estimation, the
LLL algorithm would need to be implemented together with the individual blocks
that form the conditional probability density function for each real dimension of the
effective channel.
For example, for the real part of the received effective channel yR` [k], the decoder
requires for each symbol to have knowledge of the probability that the result of the
corresponding linear combination of codewords, wR` [k] = q, equals the finite field
symbol q, for each q ∈ Fp. Similarly, the equivalent probability is required by the de-
coder applied to the imaginary received signal. These conditional probability density
functions (pdf) are defined as follows for each finite field symbol, q,
p(wR
` [k] = q | yR` [k])
=p(yR` [k] |wR
` [k] = q)p(wR
` [k] = q)
p (yR` [k])(6.19)
p(wI
` [k] = q | yI` [k])
=p(yI` [k] |wI
` [k] = q)p(wI
` [k] = q)
p (yI` [k]). (6.20)
28
In the above pdf functions, the Bayes’ rule is used to expand the conditional probabil-
ity density function.Equally likely finite symbols are assumed, hence, p(wR
` [k] = q)
=
1p,∀q ∈ Fp. As a consequence, the pdfs p
(yR` [k]
)and p
(yI` [k]
)do not depend on the
exact finite field variables wR` [k], respectively wI
` [k],
p(yR` [k]
)=∑q∈Fp
p(yR` [k] |wR
` [k] = q)p(wR
` [k] = q)
(6.21)
p(yI` [k]
)=∑q∈Fp
p(yI` [k] |wI
` [k] = q)p(wI
` [k] = q). (6.22)
That means, the conditional probability density functions given in (6.19) and (6.20)
are proportional to
p(wR
` [k] = q | yR` [k])∼ p
(yR` [k] |wR
` [k] = q)
(6.23)
p(wI
` [k] = q | yI` [k])∼ p
(yI` [k] |wI
` [k] = q). (6.24)
Therefore, next I focus on these conditional probability density functions p(yR` [k] | wR
` [k] = q)
and p(yI` [k] | wI
` [k] = q). In order to obtain a closed form of these pdfs, first define
WRq (a`) to be the set of vectors w ∈ A1×N that yield the same result q ∈ Fp of the
real part of the linear combination with coefficients a`
WRq (a`) =
w =
wR1 + jwI
1
...
wRi + jwI
i
...
wRMT
+ jwIMT
†
: (aR` (wR)† − aI
`(wI)†) mod p = q,
with wRi , w
Ii ∈ Fp,∀i ∈ {1, . . . ,MT}
}. (6.25)
In a similar manner define WIq(a`) to be the set of vectors w that yield the same
29
result q ∈ Fp of the imaginary part of the linear combination with coefficients a`
WIq(a`) =
w =
wR1 + jwI
1
...
wRi + jwI
i
...
wRMT
+ jwIMT
†
: (aR` (wI)† + (aI
`(wR)†) mod p = q,
with wRi , w
Ii ∈ Fp,∀i ∈ {1, . . . ,MT}
}. (6.26)
With these notations defined, the conditional probability density function of the
`-th received effective signal, given the result q for each real and imaginary part of
the linear combination can be identified as follows for the real part,
p(yR` [k] | wR` [k] = q, a`) =
1∣∣WRq (a`)
∣∣ ·∑w∈WR
q (a`)
1√2πNzR`
exp
(1
2NzR`
(yR` [k]−
(µzR`
+ aR` ϕ((wR)†)− aI
`ϕ((wI)†)))2
),(6.27)
and for the imaginary part
p(yI` [k] | wI` [k] = q, a`) =
1∣∣WIq(a`)
∣∣ ·∑w∈WI
q(a`)
1√2πNzI`
exp
(1
2NzI`
(yI` [k]−
(µzI`
+ aR` ϕ((wI)†) + aI
`ϕ((wR)†)))2
). (6.28)
where I use µzR`, µzI`
, NzR`and NzI`
to denote the mean and variances of the corre-
sponding effective noise elements shown in (6.15) and (6.16). Using the formulation
of effective noise given in (5.3), µzR`is the mean of real part of the random variable
30
of the effective noise random variable
z`[k] = (b`H− a`)w + b`[z1[k], . . . , zMR[k]]† (6.29)
where I assume zr[k] is Gaussian noise with zero mean and variance N0,r, with r ∈
{1, . . . ,MR} and b` is the `-th row of the equalization matrix B shown in (5.4).
Furthermore, using (6.29), the variance of the effective noise is given by
NzR`=
MR∑r=1
||b`[r]||2N0,r (6.30)
since all the other terms shown in (6.29) are constants and do not affect the variance
of the random variable z`[k]. In a similar manner, the mean and variance of the
imaginary part of the effective noise are obtained.
For each element k of the `-th linear combination the vectors of probability density
functions for all elements in Fp are given by
pR` [k] =
[p(yR` [k] | wR
` [k] = 0), . . . , p(yR` [k] | wR` [k] = p− 1)
]†(6.31)
pI` [k] =
[p(yI` [k] | wI
` [k] = 0), . . . , p(yI` [k] | wI` [k] = p− 1)
]†. (6.32)
With this information the decoder returns the estimates vR` , v
I` ∈ F1×K
p of the `-th
linear combination of messages
vR` = Decoder
([pR` [1], . . . ,pR
` [k], . . . ,pR` [N ]
])(6.33)
vI` = Decoder
([pI` [1], . . . ,pI
` [k], . . . ,pI` [N ]
]). (6.34)
31
With these estimates, the set of linear equations
vR1 + jvI
1
...
vR` + jvI
`
...
vRMR
+ jvIMR
= A
mR1 + jmI
1
...
mRi + jmI
i
...
mRMR
+ jmIMR
(6.35)
is solved to obtain the estimates of the messages sent, mRi , m
Ii ∈ F1×K
p .
With these estimates, a codeword error is declared when an error occurs in either
of the elements in the set
(mR1 , m
I1, . . . , m
RM , m
IM) 6= (mR
1 ,mI1, . . . ,m
RM ,m
IM) (6.36)
6.2 Uncoded framework
For the uncoded case, the sent signal is a direct mapping of the finite field message
xi =Ptx
PCo(ϕ(mR
i ) + jϕ(mIi )) (6.37)
with mRi ,m
Ii ∈ F1×N
p , and similarly to (6.10), the average signals’ transmit power is
adjusted to unity, with PCo the constellation resulted from the mapping function as
defined in the previous subsection. The effective channel is formed, as shown in (5.2).
Next the receiver identifies the real and imaginary parts of the `-th estimate of the
linear combination of messages vR` , v
I` ∈ F1×N
p ,
vR` = ϕ−1(
⌈yR`
⌋mod p) (6.38)
vI` = ϕ−1(
⌈yI`
⌋mod p) (6.39)
32
for all ` ∈ {1, . . . ,MR}, where the round operation quantizes the received signal to
integers. For the uncoded receiver architecture, the inverse of the mapping function
plays a significant role, that is it maps symbols from the integer domain Rp to the
finite field Fp, ϕ−1 : Rp → Fp. Similar to the mapping function, its inverse differes
depending on the order of the modulation.
Example 1 cont.: For 4QAM modulation which is based on F2, similar to (6.8),
the shift s = −12
is being added back
ϕ−1(b) = b+ s. (6.40)
More specifically, the shift is added back to the signal, scaled by the coefficients of
the corresponding linear combination
ϕ−1(dyR` [k]c) = dyR` [k]c+
MT∑i=1
(aR`,is− aI`,is
)(6.41)
ϕ−1(dyI` [k]c) = dyI` [k]c+
MT∑i=1
(aR`,is+ aI`,is
)(6.42)
Example 2 cont.:For modulations with p 6= 2, and prime, the inverse mapping is
simply the inverse of the mapping function ϕ(), a bijective function, ϕ−1 : Rp → Fp.
For example, for 9QAM, the inverse mapping is given by
ϕ−1(b) =
0 , if b = 0
1 , if b = 1
2 , if b = −1
. (6.43)
Next, similar to (6.35), the set of linear equations of uncoded messages is solved
to obtain the message estimates mRi , m
Ii ∈ F1×N
p for all i ∈ 1, . . . ,MR.
For the uncoded framework, a symbol error is observed if at tone k of the i-th
33
transmitted message the following is true
(mR1 [k], mI
1[k], . . . , mRM [k], mI
M [k]) 6= (mR1 [k],mI
1[k], . . . ,mRM [k],mI
M [k]). (6.44)
6.3 eIF linear receiver
The architecture for the eIF linear receiver uses the same steps as the ones described
for the IF linear receiver with the equalization matrix given in (5.9) that uses the
matrix AeIF. More specifically, for the coded framework, the major change is in the
actual probability density function of the received linear combinations. That is, as
shown in equation (5.10) the effective noise for the `-th linear combination depends
on the inverse of the channel matrix and the corresponding set of Gaussian integer
coefficients of the linear combinations.
6.4 Expected behavior
The behavior of the eIF linear receiver is expected to not be influenced by the accuracy
of the SNR, as it does not use this information, as shown in (5.9). Therefore, expect as
theory deduced for the behaviour of the eIF linear receiver to be on average worse than
that of the IF linear receiver since it does not take advantage of all the information
of the system. Also, the MMSE linear receiver uses all the information available, see
(4.7), but identifies the same set of linear combinations, given by A = I. That is,
on average IF should perform better than MMSE, or in the worst case as good as
MMSE. Furthermore, since the ZF linear receiver is a special case of the eIF linear
receiver (4.4), and eIF choose the optimal A matrix to minimize the effect of H−1
on the noise. Therefore, expect that the eIF linear receiver will outperform the ZF
linear receiver on average. All these trends have been observed in a practical system
34
and are confirmed by the SER and CER experimental results shown in Chapter 8.
However, before showing the results, lets first enumerate some of the technical
challenges. Then lets provide the solutions I used in building this experimental frame-
work with which evaluates the performance of these receivers.
Chapter 7
Experimental aspects
In this chapter I describe the general experimental setup, the specifics of the exper-
iments as in the frame structure and the approach to estimate the noise variance. I
also introduce the types of error correcting codes that were used in the actual frame-
work and discuss a few more WARP settings. Lastly, I provide the link to the website
were I shared the experimental data and the actual code used to generate the results
presented in the following section.
The experimental setup I build is for a 2× 2 MIMO wireless network for which I
used two WARP v3 Kit SDR platforms [34], correspondingly in my notation MT = 2
and MR = 2. Each node was equipped with two single-band (2.4 GHz) omnidirec-
tional antennas (model RE07U-SM). I used the WARPlab 7.7.0 framework which
facilitates the interaction between the WARP hardware and a PC that runs MAT-
LAB. The purpose of the computer is to trigger transmissions and enable the receivers,
while it also generates, sends and saves data to and from the buffers of the WARP
boards. Starting with the WARPlab IEEE 802.11 standard compliant frame structure
and using MATLAB I implemented the proposed receiver and transmitter signaling
architectures shown in Figures 6.2 and 6.3. This transmit signal architecture enables
the use of both the IF and eIF linear receivers. The same signaling architecture is used
36
Figure 7.1: The experimental setup of the 2 × 2 MIMO wireless network in a wideindoor open space at the Boston University.
to implement the receivers used as the baseline of comparison: the optimal joint-ML
receiver, the zero-forcing (ZF) and the corresponding minimum-mean square error
(MMSE) linear receivers.
The WARPlab framework facilitates the off-line processing of the received signal.
Therefore, for each of these experiments, the receivers were applied to the same data
set that was saved for off-line processing at the end of each over-the-air experiment.
That is, the analog processing is done on the WARP board, and after upconversion,
the raw time samples of the received signal are saved, and all the mentioned receivers
and linear receivers are applied.
7.1 Experimental characteristics
Two WARP boards are used in the experimental setup: one is the transmitter, while
the second one is the receiver. In these experiments, the two antennas at each board
are situated no more than 40cm apart, approximately the typical width of a laptop.
All of the experiments were performed in a line-of-sight typical indoor office space or
open indoor space environment as shown in Figures 7.1 and 7.2, while the distance
37
Figure 7.2: The experimental setup of the 2× 2 MIMO wireless network in a normalindoor open space on the hallways of the Boston University.
1->1 Channel Estimates (Magnitude)
-10 -5 0 5
Baseband Frequency (MHz)
0
1
2
31->2 Channel Estimates (Magnitude)
-10 -5 0 5
Baseband Frequency (MHz)
0
1
2
3
2->1 Channel Estimates (Magnitude)
-10 -5 0 5
Baseband Frequency (MHz)
0
1
2
32->2 Channel Estimates (Magnitude)
-10 -5 0 5
Baseband Frequency (MHz)
0
1
2
3
Figure 7.3: Channel profile for the 2 × 2 MIMO wireless channel of frame number2900 from one of the experiments. In this figure, each bar plot shows the magnitudeof all NSC = 48 data subcarriers corresponding to each of the 4 individual channelsof the experimental wireless network: h11, h12, h21 and h22.
between the nodes varied between 6m and 10m.
Following the IEEE 802.11(a,g,n) standards frames specifications [35, 51], my
frame design uses a total of 64 subcarriers, out of which NSC = 48 are data sub-
carriers and 4 are used for carrier frequency phase offset estimation, while the rest
are used for minimizing inter-symbol interference. For transmission the center carrier
frequency is 2.4GHz, and as per IEEE 802.11 standard specifications I use one of the
available 12 channels, of 20 MHz bandwidth each. Each frame contains a total of
N = 500 OFDM data symbols and an additional Nt = 10 OFDM training symbols,
38
-10 -5 0 5
Baseband Frequency (MHz)
0
1
2
3
1->1 Channel Estimates (Magnitude)
-10 -5 0 5
Baseband Frequency (MHz)
0
1
2
3
1->2 Channel Estimates (Magnitude)
-10 -5 0 5
Baseband Frequency (MHz)
0
1
2
3
2->1 Channel Estimates (Magnitude)
-10 -5 0 5
Baseband Frequency (MHz)
0
1
2
3
2->2 Channel Estimates (Magnitude)
Figure 7.4: Variance of the channel magnitude over an experiment with a total of5800 frames. Each plot shows the magnitude(straight blue line) and variance(the redbars) of all NSC = 48 data subcarriers for each of the 4 channels of the experimentalwireless network.
which is equivalent to a 2% overhead.
7.2 Channel estimation
The channel coefficients are estimated using typical standard channel estimation tech-
niques [51], that is using the orthogonal long-time symbols (LTS) sent from each
antenna as described in Section 3.1 and shown in Figure 3.1. One concurrent LTS
symbol and two orthogonal LTS symbols are received, both transmitters send the con-
curent LTS symbols, while each antenna will send one LTS symbol while the other
antenna will send the null symbols. The channel for the MIMO wireless network
is estimated for each subcarrier f , and I denote its estimate by H. In Figure 7.3,
the channel profile for one frame from a typical experiment is shown. The Channel
profile refers to the channel magnitude for each subcarrier for all four channels of the
2 × 2 MIMO wireless network. Furthermore, in the majority of my experiments I
observed that the channel remains fairly coherent in time if there are no changes in
the environment. For example, in Figure 7.4, I show the variance of the magnitude
of all channels for all the frames of one entire experiment, that lasted approximately
39
10 hours. It is clear that the majority of frames have the same channel profile, which
implies that the channel is fairly coherent over time.
However, over time a similar trend is observed in the magnitude but not in phase
as shown in Figure 7.5. Furthermore, consecutive subcarriers were observed to show
a more similar evolution of both magnitude and phase over time as shown in Figure
7.6 for subcarriers 3 to 6. Please note, that a time delay of approximately 3 seconds
was introduced between frames to accommodate the transfer of the raw time samples
between the buffers of the WARP boards and the computer. Therefore, in a real
system the coherence of the wireless channel might be even more higher over time in
both magnitude and phase.
Future directions : The coherence of the channel might be an important charac-
teristic of the wireless channel for developing integer-forcing linear receiver. If the
channel is coherent over time, there might be huge advantages to reducing the number
of times the LLL algorithm is required to be applied, especially for MIMO wireless
systems with a high number of antennas.
10 20 30 40 50 60 70 80 90 100
frame number (time)
0
2
4
|HA
A|
subcarrier 3
subcarrier 6
subcarrier 9
10 20 30 40 50 60 70 80 90 100
frame number (time)
-5
0
5
an
gle
(HA
A)
- a
ng
le(H
AA[1
])
subcarrier 3
subcarrier 6
subcarrier 9
Figure 7.5: Evolution of channel magnitude and it’s phase over time for subcarriers3, 6, and 9. The second graph shows the difference between the phase of the channeland the phase of the channel of the first frame.
40
10 20 30 40 50 60 70 80 90 100
frame number (time)
0
2
4
|HA
A|
subcarrier 3
subcarrier 4
subcarrier 5
subcarrier 6
10 20 30 40 50 60 70 80 90 100
frame number (time)
-5
0
5angle
(HA
A)
subcarrier 3
subcarrier 4
subcarrier 5
subcarrier 6
Figure 7.6: Evolution of channel magnitude and it’s phase over time for subcarriers3, 4, 5 and 6. The second graph shows the difference between the angle of the channeland the angle of the channel of the first frame.
7.3 Estimation of the signal-to-noise ratio (SNR)
In order to form the equalization matrix (5.4) for the IF linear receiver, two types of
information about the wireless channel are needed: the channel state information (an
estimate of the channel, H) and statistical information about the noise, like mean and
variance N0. The uncoded IF linear receiver requires knowledge of SNR, but not the
uncoded eIF linear receiver. Similarly, the uncoded MMSE linear receiver requires
knowledge of SNR while the ZF one does not. However, for the coded framework,
all receivers require knowledge of SNR in order to compute the probability density
function needed for decoding. Therefore, in this framework I use dedicated OFDM
symbols [52] to estimate the power of the noise. These OFDM symbols are known
as OFDM training symbols. The OFDM training symbols are known at both the
transmitter and receiver. I denote by Id the set of indexes of the OFDM symbols
that carry data and by It the set of indexes of OFDM training symbols used for
estimating the SNR (as shown in Figure 7.7), the length of these sets is given by
Nt = |It| and N = |Id|. The set of indexes of the OFDM training symbols It is
the same for each transmit antenna, i ∈ {1, . . . ,MR}. The OFDM training symbols
from all antennas are emitted concurrently, facilitating the estimation of the total
post processing received SNR. Therefore, after obtaining the channel estimates H,
41
Figure 7.7: Frame format after FFT is applied and the result is shaped into a matrixof (NSC +N t
SC)× (N +Nt) tones. A frame contains Nt OFDM training symbols andN ODFM data symbols.
for each subcarrier f , the receiver first forms an estimate of the received observed
noise on each tone
z`[f, k] = y`[f, k]−MT∑i=1
H`,i[f ]s`,i[f, k], ∀k ∈ It,∀f ∈ {1, . . . , NSC} (7.1)
where s`,i[f, k] denotes the k-th OFDM training symbol sent from transmitter’s an-
tenna i to the receiver’s antenna `, on subcarrier f . In a similar manner, y`[f, k]
denotes the received effective channel, while H`,i[f ] is the estimate of the channel
between antenna ` and receiver’s antenna i, which is assumed to be coherent for the
duration of the frame, that is H`,i[f ] = H`,i[f, k],∀k ∈ {Id ∪ It}.
Using these training symbols two types of noise estimates can be formed: one
noise estimate per subcarrier, Npsc0 [f ], that uses only the Nt tones sent on subcarrier
f to estimate the power of the noise; and a noise estimate per frame, Npf0 that uses all
training tones from all data subcarriers, a total of NSC×Nt to form a single estimate
of the power of the noise.
As shown in Figure 7.7, a frame contains NSC data subcarriers and N tSC pilot
42
0 5 10 15 20 25 30 35 40 45
SC number
0
0.02
0.04
0.06
0.08
0.1
0.12
N0,1
(not dB
)
N0,1psc
N0,1acc
N0,1pf
N0,1
from SC pilots
0 5 10 15 20 25 30 35 40 45 50
SC number
0
0.01
0.02
0.03
0.04
0.05
0.06
N0,2
(not dB
)
N0,2psc
N0,2acc
N0,2pf
N0,2
from SC pilots
Figure 7.8: This picture will change to show only three estimates: Evolution of noiseestimates for frame number 4500 from experiment number 48.
subcarriers used for carrier frequency offset estimation (the row of tones shown in grey
color). The number of subbcarriers used for data and CFO is given by the length
of the set of the indexes of the subcarriers used for data NSC = IdSC , respectively
used for training N tSC = ItSC . In my experiments, I use the typical standard format
of the frame, that is the set indexes of the training subcarriers is given by ItSC =
{8, 22, 44, 58}, where the total number of subcarriers including the ones that are zero
and are used for padding is 64.
7.3.1 N0 per subcarrier
Using the noise estimate formed in (7.1), the receiver creates a noise variance estimate
for each subcarrier f of the frame
Npsc0,` [f ] =
1
Nt
∑k∈It
(z`[f, k]− µpsc` [f ])2 , (7.2)
43
where µ`[f ] denotes the estimate of the mean of the noise received on subcarrier f at
antenna `. This estimate is formed using all the Nt tones from the OFDM training
symbols on subcarrier f ,
µpsc` [f ] =
1
Nt
∑k∈It
z`[f, k]. (7.3)
7.3.2 N0 per frame from training OFDM symbols
When all the tones of the OFDM training symbols from a frame are used, a single
estimate of the noise variance per frame can be formed
Npf0,` =
1
NtNSC
∑f∈IdSC
∑k∈It
(z`[f, k]− µpf
`
)2
, (7.4)
where µpf` is the estimate of mean of the noise calculated using all tones of the OFDM
training pilots on all subcarriers received at antenna `
µpf` =
1
NtNSC
∑f∈IdSC
∑k∈It
z`[f, k]. (7.5)
7.3.3 N0 per frame from subcarrier training pilots
Each frame uses NSC subcarriers to carry data and N tSC subcarriers that carry pilots
used for carrier frequency offset (CFO) estimation. The number of subcarriers N tSC is
given by the size of the set of indexes of the subcarriers used for CFO, N tSC = |ItSC |.
All the tones (Nt +Nd) from all these subcarriers can be used to form an estimate of
the noise power,
Npfs0,` =
1
(Nt +Nd)N tSC
∑f∈ItSC
∑k∈It∪Id
(z`[f, k]− µpfs
`
)2
, (7.6)
44
where µpfs` is the estimate of the mean of the noise calculated based on the tones used
for CFO estimation
µpfs` =
∑f∈ItSC
∑k∈It∪Id
(z`[f, k]− µpfs` ). (7.7)
7.3.4 N0 per subcarrier from data symbols
Since WARPlab facilitates the decoding of the received signals offline, lets take advan-
tage of knowing all the randomly generated sent symbols at the receiver and compute
the most accurate noise estimate for each subcarrier using all the data. That is,
Nacc0,` [f ] is calculated as in (7.2), with the summation being done over all OFDM
symbols (data and training) k ∈ {It ∪ Id}:
Nacc0,` [f ] =
1
Nd
∑k∈Id
(z`[f, k]− µacc` [f ])2 , (7.8)
with µacc` [f ] the estimate of the mean of the noise for each subcarrier f given by
µacc` [f ] =
∑k∈Id
z`[f, k]. (7.9)
As expected and shown in Figure 7.8, Npsc0,` (f) reflects better the variation of SNR
over subcarriers, while Npf0,` uses a larger number of tones, but it does not capture the
variation of the noise over subcarriers.
For the rest of the experimental results presented, the transmit SNR is evaluated
for each subcarrier using the per subcarrier estimate of the noise power Npsc0,` ,
SNR[f ] = 10 log10
PTx
Npsc0,` [f ]
, (7.10)
where PTX is the unit transmit power, I use a number of Nt = 10 OFDM training
45
symbols at each transmit antenna. Compared to the length of the sent signal, which
in each experiment is usually N = 500, this represents a 2% overhead. I have tried
different values for Nt in my experiments and noticed that more than 10 OFDM
training pilots do not improve significantly the accuracy of the estimate of the noise
power.
For each experiment, the grouping of the frames is done based on the mean received
SNRm at all antennas,
SNRm =
∑MR
r=1 SNRr
MR
(7.11)
where SNRr is the received SNR at antenna r, given by
SNRr = 10 log10
PRx
Npf0,r
, (7.12)
where PRx is the total estimated received power at antenna r given by
PRx =1
NtN tSC
∑f∈ItSC
∑k∈It
(M∑i=1
Hr,i[f ]sr,i[f, k]
)2
. (7.13)
As I show in Chapter 8, the per subcarrier estimate of the noise power, provides
the best performance of the IF linear receiver.
7.4 LDPC codes
In my experiments I use low-density parity check (LDPC) codes [39] and in particu-
lar (3, 6) LDPC codes. These codes are suitable for IF linear receivers since they are
linearly build codes, which preserves the property that integer linear combinations
of codewords yeild another codeword. Even if this code is one of the most common
46
LDPC codes in the literature [39,53], it is powerful enough for general purposes. The
generator matrix for each code used is constructed using progressive edge growth
algorithm [54], which maximizes the length of the shortest cycle in the generator
matrix and it generates codes that can be used with the message-passing decoding
algorithm. Since, IF linear receivers were initially build to work with prime order
based constellations, I present experimental results for both binary and ternary (3, 6)
LDPC codes. That is most of the results are presented for either a 9-QAM constella-
tion based transmission with F3 ternary LDPC codes, or QPSK (4-QAM) modulation
with the binary (F2) LDPC codes. However, as mentioned in [42,50,55], the IF linear
receiver can be used with non-prime order based constellations, when combined with
a multilevel coding scheme.
Decoder complexity
One might argue that since we now use single user decoders at the receiver the
complexity of the receiver would be affected by the decoder and the use of the LLL
algorithm. However, single user encoders and decoders come at a complexity that
increases in the length of the message encoded. Therefore, the use of single user
encoders increases the complexity of the decoder by a multiplication factor given by
the number of data subcarriers NSC . For example, when LDPC codes are used the
complexity at the decoder is given by
NSC2MT O(N)︸ ︷︷ ︸Single user
decoder complexity
+NSCO(2MT )︸ ︷︷ ︸LLL complexity
(7.14)
where MT is the number of transmit antennas and hence individual messages, N is the
length of the codeword, and the number 2 is due to the fact that we have a codeword
on each real dimension of the complex signal (the real and imaginary parts of the
47
complex signal). In the above complexity formulation, we used the most common
complexity of the LLL algorithm. However, even with the highest complexity of the
LLL algorithm [56,57] obtained when the complex version is used
NSCO(4M2T log(2MT )). (7.15)
the total complexity of the IF linear receiver is more affected by the complexity of
the single user decoders since usually the length of the codeword is much higher than
the number of transmit antennas, N >>> MT .
7.5 WARP settings and framework code
7.5.1 Sampling frequency offset(SFO) correction
In OFDM systems signals are sent on narrow-band subchannels, which overlap and
are orthogonal. However, the mismatch in sampling frequencies between transmitter
and receiver can lead to serious degradation due to loss of orthogonality between
subcarriers. Therefore, there are different adjustments implemented at the receiver
to compensate for this mismatch. The first is done in time domain, that adjusts
the carrier frequency offset of the time samples. However, sometimes CFO is not
enough and further correction needs to be done after the FFT has been applied, that
is to correct the mismatch in sampling frequency between transmitter and receiver
of the OFDM symbols in the frequency domain. There have been several techniques
that use the pilot subcarriers (these are the pilot tones on each OFDM symbol that
are used for CFO estimation). However, the current method used in the WARPlab
reference design for sampling frequency offset correction after the FFT has been
applied seems to introduce more error after the correction is realized. Therefore, for
48
these experiments, I turned the SFO off. What I observed is that with the SFO
mechanism on, more errors were observed in all receivers used. I believe this is due
to a higher loss in accurary of the channel estimates and it’s quality is important at
the receiver.
7.5.2 Automatic gain control(AGC)
The IEEE 802.11 reference design bitstream used on each WARP board has built in
mechanisms to adjust the receivers gains through the automatic gain control (AGC)
algorithm which is implemented in the FPGA fabric. However, through the WARPlab
framework access is given to disable and enable the AGC and manually set the re-
ceived LNA gain (denoted as the RF gain) and the VGA gain (denoted as the base-
band gain). This algorithm adjusts the two received gains based on the power of the
signal evaluated through the RSSI value given by the MAX2829 transceiver in the
WARP’s radio such that it maximizes the measured error vector magnitude (EVM)
using the datasheet data of the transceiver’s chip. What I observed in experiments is
that when AGC is enabled, it sets the received gains for both antennas independently
with the scope to obtain a unit magnitude of the channel on both received signals.
This behaviour hides the actual received power of the signal on the four channels,
which damages the channel estimate and hence the identification of the optimal inte-
ger coefficients of the linear combinations. Therefore, for my experiments I disabled
the AGC and manually set the same received gains in order to maintain the rela-
tive difference in power on all four channels (see Appendix A for a more in depth
explanation of the influence of the AGC).
This motivates the following remark: next generation wireless receivers that use
Interger-Forcing linear receivers have to use a new AGC that either sets the received
gains of all antennas such that it maintains the relative power of all the channels or
49
provides extra information about the actual received power of the received signals at
each antenna relative to each other.
7.5.3 Code release
In the next chapter, Chapter 8, I present results of the experiments I performed with
the IF linear receiver in an 802.11 IEEE standard compatible framework. The frame-
work for the experiments was build in Matlab using the WARPlab framework [34]. I
am releasing all the code used for the transmitter and receiver that I have used to run
the experiments and analyze the data. With this thesis I am releasing the code used
to put the data together and group the received frames by the relevant SNR experi-
mental value. Everything can be found on github at the link: https://github.com/
corinai0/WARPlab_IF, for which I highly recommend you start with the wiki page
and the single file that is commented and explained in much greater details then all the
other files released: w_lin_rec_mimo_ofdm_txrx_dithers_08_05_repEXP_noAGC.m.
Since the amount of data for each experiment performed uses more than 25Gb
of space I am also releasing the most relevant experiment. The entire data set for
which most of the results in the following chapter were obtained are shared pub-
licly using the Amazon S3 data service and can be followed at the link: http://
warpintegerforcing.s3-website.us-east-2.amazonaws.com/ (if the site is down
please check s3.amazonaws.com/warpintegerforcing/ and the github link given
above). Also, a shorter data set of the same experiment (just two Matlab workspaces
of the dataset) is shared on the data.world website:
https://data.world/corinai/warp-integer-forcing-linear-receiver. The pur-
pose of this data set is for testing such that to make it easily accessible for parties that
want to try out the receiver’s code directly on the experimental data very quickly.
Chapter 8
Results
In this chapter we present the results of our experiments run with WARPlab frame-
work build to study and analyze the performance of the IF linear receiver. The
evaluation metrics used are the symbol-error-rate (SER) for the uncoded framework
(the signal is not coded) and the codeword-error-rate (CER) for coded framework (the
signal si encoded with a (3,6) LDPC code for both the binary case and the ternary
case). For this setup the SER is calculated as the sum of all symbol errors that occur
on a per tone basis, as defined in (6.44) from all subcarriers of a received frame. In
a similar manner, the CER for a frame is the sum of all codeword errors (as defined
(6.36)) that occur on any of the 48 data subcarriers of the frame. That is, after each
frame is received and decoded / demodulated a SER / CER number is associated
with it and its corresponding mean /SNR value.
In order to introduce the results lets first present the method for putting together
the data. Unlike a simulation environment, in an experiment there is no control over
the channel or its specifics. Therefore, I choose the metric of the post-processing
SNR to identify each frame. More specific, the total received SNR is evaluated for
each frame [58,59]. Furthermore, since in this experimental framework, there are two
receiving antennas, I use the mean of the two received SNR values at each antenna,
51
SNRm = (SNR1 + SNR2)/2, where
SNR` = 10 log10
(P`,rx
µN0,`
), ` ∈ 1, . . . ,M, (8.1)
where Prx is the total received power at antenna ` and µN0,`is the mean of the noise
power estimates over all subcarriers.
Once a SER / CER value is calculated for each frame accompanied by an /SNR
value, the frames with the /SNR in a specific range (s1 <= SNR < s2) are grouped
together. For each such group, one SER/CER value is generated by taking the mean
of all the SER/CER values of all the frames that fall within this set.
Therefore, all the following plots are accompanied by a histogram that shows the
number of frames or the normalized percentage of frames over which the SER or
CER value was computed for that specific SNR point shown on the x-axis. For all the
results presented in this chapter and the following one, the entire range of the mean
SNR of all the frames received within an experiment has been binned into intervals
of equal length. Therefore, the histogram has the purpose of showing the amount of
data that was used to generate each point on the SER / CER plot and in the same
time provides a value for the confidence level in the trends observed in the behavior
of the plotted curves. For fairness of comparison, the same set of intervals (hence the
set group of frames) have been used for all the types of receivers presented on the
same SER or CER plot.
Note: All the results shown in Sections 8.1, 8.2 , 8.3 and 8.4 have been obtained by
applying different receivers to the same data set obtained from the same experiment
that is also shared on Amazon S3. The evolution of the SNR for these frames is
shown in Firgure 8.1. Even if the transmit BB gains and RF gains were kept fixed
for the duration of the entire experiment, it is clear that the estimated SNR does not
remain constant due to slight changes in the wireless medium. The results presented
52
in Section 8.5 were obtained from a different experiment for which the details are
given within the section.
0 1000 2000 3000 4000 5000 6000
frame number (time)
12
14
16
18
20
22
24
26
28
30
tota
l re
ceiv
ed S
NR
(dB
)
at 1st antenna
at 2nd antenna
Figure 8.1: Received SNR evolution of the experiment 48 for which the results in thefollowing chapter were obtained. SNR at antenna 1 and at antenna 2 are shown.
53
8.1 SER and CER results
The symbol error rate (SER) of both eIF and IF linear receivers is shown in Figure
8.2 (histogram shown in Figure 8.5) for an over-the-air experimental setup which uses
uncoded 4QAM modulation. These results were obtained for a line of sight experiment
with the distance between transmitter and receiver of 10m and the distance between
antennas of 40cm. The length of the uncoded message is of 500 symbols, with the
channel profile of this experiment shown in Figure 7.4.
22 22.5 23 23.5 24 24.5 25 25.5 26 26.5 27
SNRm
[dB]
10-5
10-4
10-3
10-2
10-1
SE
R
ML
IF
MMSE
eIF
ZF
Figure 8.2: SER for 4QAM modulation, N = 500 uncoded symbols with Nt = 10OFDM training symbols for noise estimation, OFDM channel is 9.
Note, that for this specific channel profile, there is significant improvement in
SER for the IF linear receiver when compared to the MMSE one. Similarly, significant
improvement is also shown by the eIF linear receiver when compared to the ZF one.
Furthermore, with a SNR estimate obtained with only 2% overhead, the performance
of the IF linear receiver is significantly close to the performance of the optimal joint
ML receiver. That is, in a practical OFDM wireless network, the IF linear receiver
can significantly outperform the MMSE linear receiver utilizing the same amount of
54
22 22.5 23 23.5 24 24.5 25 25.5 26 26.5 27
SNRm
[dB]
10-5
10-4
10-3
10-2
10-1
CE
R
IF
MMSE
eIF
ZF
Figure 8.3: CER for 4QAM modulation, N = 500, rate 1/2 binary LDPC(3, 6) code,OFDM channel is 9.
Figure 8.4: Normalized histogram of the frames shown for the SER in both Figure8.2 and Figure 8.3.
Figure 8.5: Histogram of the frames shown for the SER in both Figure 8.2 and Figure8.3.
information at the receiver: channel state and SNR information.
Similar results and similar behaviour of the linear receivers is observed when a
coded system is used, as shown in Figure 8.3. The setup for the experiment providing
these results is the same as the one used for the results in Figure 8.2. For this
55
experiment, a 1/2 rate LDPC(3, , 6) binary code with a 500 long codeword has been
used for each real and imaginary dimension of the complex symbol. Based on the gap
between the SER, CER respectively of the IF linear receiver and the other ones, we
can draw the conclusion that in a practical environment, the IF linear receiver and
the eIF linear receiver can significantly outperform conventional ones.
The accuracy of these experimental results is measured by the amount of data
(frames) used to compute each metric. That is for the results shown in Figures 8.2
and 8.3, I show in Figure 8.4 the corresponding normalized histogram of the 5800
frames resulted from the experiment and in Figure 8.5 I show the histogram of
the corresponding number of frames. Since I do not have control of the wireless
channel, the height of each bar denotes the amount of frames used to generate the
corresponding SER or CER values, with the width giving the specific SNR range.
This also denotes the confidence and accuracy of the trends and behaviour of the
presented linear receivers.
56
8.2 SNR estimation results:
In Figure 8.6 the evaluation of IF and MMSE linear receivers is done for various
types of SNR estimates. The main difference is in the estimate of the noise power N0
(estimate of the variance of the noise).
21 21.5 22 22.5 23 23.5 24 24.5 25 25.5 26
SNR
10-5
10-4
10-3
10-2
Figure 8.6: SER for 4QAM modulation with various SNR estimates (per frame fromOFDM training symbols Npf
0 and from subcarrier pilots Npf0 and per subcarrier Npsc
0 ),N = 500 and OFDM channel is 9.
Figure 8.7: Histogram of the frames shown for the CER in Figure 8.6.
With only Nt = 10 OFDM training symbol and when the signal is uncoded, both
IF and the MMSE linear receivers are sensitive to accuracy of the SNR, hence noise
variance estimate N0, as shown in Figure 8.6 (with the histogram shown in Figure 8.7).
However, when the signal is coded, IF is significantly more sensitive to the accuracy
57
22.5 23 23.5 24 24.5 25 25.5 26
SNR
10-5
10-4
10-3
10-2
10-1
Figure 8.8: CER for 4QAM modulation with various SNR estimates (per frame fromOFDM training symbols Npf
0 and from subcarrier pilots Npf0 and per subcarrier Npsc
0 ),N = 500, binary LDPC(3, 6) code, OFDM channel is 9.
Figure 8.9: Histogram of the frames shown for the CER in Figure 8.8 and 8.10.
of the SNR estimate compared to both the MMSE and ZF linear receivers as shown
in Figure 8.8 (and histogram of the frames shown in Figure 8.9). Furthermore, when
the sent signal is coded, the performance of eIF is also affected by the accuracy of
the SNR estimate since it influences the form of the pdf of the effective noise used to
compute the input to the decoder.
58
22.5 23 23.5 24 24.5 25 25.5 26
SNR
10-5
10-4
10-3
10-2
Figure 8.10: CER for 4QAM modulation with various SNR estimates for both IFand eIF linear receivers (per frame from OFDM training symbols Npf
0 and fromsubcarrier pilots Npf
0 and per subcarrier Npsc0 ), N = 500, binary LDPC(3, 6) code,
OFDM channel is 9.
As shown in Figure 8.10 , when the signal is coded, the eIF linear receiver will be
affected by the quality of the SNR estimate since it is used in the computation of the
pdf at each decoder.
59
8.3 Receiver complexity
One way to reduce the complexity of the IF receiver is to reduce the number of times
the LLL algorithm is used. The IF linear receiver is based on decoding a set of
independent linear combinations. The MMSE concept provides a way to identify the
best linear combination that maximize the computation rate [1]. However, the IF
linear receiver is not constrained to only this optimal set, any set can be decoded.
Therefore, in order to reduce the number of times the LLL algorithm is used, that
is use the matrix A on d adjacent subcarriers. For example, for d = 2 adjacent
subcarriers, once a set of Gaussian integer coefficients A[f ] has been identified for
the subcarrier f , on the following subcarrier the same coefficients are used, that is
A[2f ] = A[2f − 1],∀f ∈{
1, . . . ,NSC
2
}. (8.2)
21 21.5 22 22.5 23 23.5 24 24.5 25 25.5 26
SNR
10-5
10-4
10-3
10-2
10-1
Figure 8.11: SER for 4QAM modulation comparing the case when A is calculated foreach subcarrier f , with the case when the same A is used on two adjent subcarriers,with N = 500 and the OFDM channel is 9.
60
Figure 8.12: Histogram of the 5800 frames used for the SER results shown in Figure8.11.
22.5 23 23.5 24 24.5 25 25.5 26
SNR
10-5
10-4
10-3
10-2
10-1
Figure 8.13: CER for 4QAM modulation comparing the case when A is calculated foreach subcarrier f , with the case when the same A is used on two adjent subcarriers.N = 500, 1/2 rate binary LDPC(3, 6) code, the OFDM channel is 9.
Figure 8.14: Histogram of the 5800 frames used for the CER results shown in Figure8.13.
61
As shown in both Figure 8.11 and Figure 8.13, when not the optimal Gaussian
coefficients are used there is a loss in performance as expected. The histograms
for both plots are given in Figure 8.12 and Figure 8.14. However, even with this
loss, the performance of both IF and eIF linear receivers outperforms the one of the
conventional ones. As for the other results, the corresponding histograms are given
in Figure 8.12 and Figure 8.12.
Therefore, we can infer, that even in a practical system, both IF and eIF linear
receiver can provide significant gain, even when the set of linear combinations decoded
is not the optimal one.
62
8.4 Degenerative cases
There are cases when in a practical system one or more antennas can mail function,
reducing the number of independent streams received. Even in such cases, multiple
linear combinations can be decoded from the same received signal. For example, for
this experimental setup of a 2×2 MIMO wireless network, if only one of the antennas
is active, both ZF and eIF linear receiver ca no longer decode any of the received
signals. However, the two best linear combinations can be chosen from one received
signal, such that the signals of interest can still be decoded.
21 21.5 22 22.5 23 23.5 24 24.5 25 25.5 26
SNR
10-4
10-3
10-2
10-1
100
Figure 8.15: SER for 4QAM modulation comparing the case both signals are used toidentify the sent signals with the case when only one of the two antennas is able toreceive the signal. (N = 500, OFDM channel is 9)
With this strategy of using for decoding the coefficients of the best and the second
best linear combinations identified for the case only one signal is received at the
destination. Using these Gaussian integer coefficients a new matrix A ∈ Z[j]2×2 is
formed. With this new matrix A, as expected the IF linear receiver yields a much
lower SER when compared to the case when both received signals were successfully
63
Figure 8.16: Histogram of the 5800 frames used for the SER results shown in Figure8.15.
22.5 23 23.5 24 24.5 25 25.5 26
SNR
10-3
10-2
10-1
Figure 8.17: CER for 4QAM modulation comparing the case both signals are used toidentify the sent signals with the case when only one of the two antennas is able toreceive the signal. (N = 500, OFDM channel is 9)
Figure 8.18: Histogram of the 5800 frames used for the CER results shown in Figure8.17.
64
captured. However, as shown in Figure 8.15, even with one signal available, the
IF linear receiver can still provide estimates of the sent signal with a significant
performance improvement over the MMSE linear receiver. The histogram for the
uncoded setup is given in Figure 8.16. Furthermore, for this specific channel profile,
when the same strategy is applied to the coded framework the improvement is not as
significant as for the coded case as shown in Figure 8.17, for which the histogram is
show in Figure 8.18. This decrease in performance improvement may be due to the
fact that the error correcting code used in both the IF and the MMSE linear receiver
is strong enough to correct enough errors on it’s own. However, as it was mentioned
before, this result is shown for a specific channel profile. Thus, it does not affect the
statement that on average the IF linear receiver improves the overall performance of
wireless communication system.
65
8.5 Comparison of 4QAM and 9QAM
In order to compare the performance of 4QAM and 9QAM, the two modulations for
which the framework was developed, the same experiment has to be used. That is,
during the same experimental setup, frames with 4QAM and 9QAM were sent in an
alternative way, such that the same channel profile is observed by both modulations.
For this experiment the channel profile, that is the magnitude of the two direct
channels and the magnitude of the two cross channels is shown in Figure 8.19.
1->1 Channel Estimates (Magnitude)
-10 -5 0 5
Baseband Frequency (MHz)
0
2
4
1->2 Channel Estimates (Magnitude)
-10 -5 0 5
Baseband Frequency (MHz)
0
2
4
2->1 Channel Estimates (Magnitude)
-10 -5 0 5
Baseband Frequency (MHz)
0
2
4
2->2 Channel Estimates (Magnitude)
-10 -5 0 5
Baseband Frequency (MHz)
0
2
4
Figure 8.19: Channel profile for the experiment in which 4QAM and 9QAM werecompared.
The same average transmit power has been used for both modulations, which
means the maximum transmit power for 4QAM is higher than for 9QAM. Therefore,
the SER performance of the setup with 4QAM modulation should be better than the
one of the setup with 9QAM. This behavior was observed as expected and it is shown
in Figure 8.20, with the histogram given in Figure 8.21. The performance of the ML
receiver is shown next to the performance of the IF linear receiver. The performance
of the IF linear receiver closely follows the one of the ML receiver and a similar
difference is forming in with respect to the MMSE linear receiver, this behavior is
consistent for both types of modulations 4QAM and 9QAM.
Remark: Unfortunately higher SNR has not been observed during the approxi-
66
18 19 20 21 22 23 24
SNRm
[dB]
10-5
10-4
10-3
10-2
10-1
100
SE
R
ML - 9QAM
IF - 9QAM
MMSE - 9QAM
ML - 4QAM
IF - 4QAM
MMSE - 4QAM
Figure 8.20: SER results of ML and IF receivers for an experiment during which both4QAM and 9QAM symbols have been sent alternatively. The channel profile is shownin Figure 8.19.
Figure 8.21: Histogram of the frames used for the SER results shown in Figure 8.20,for which 9QAM is shown in blue(or the first set of bars from left to right) and for4QAM is shown in red (or the second set of bars from left to right).
mate 5000 frames received for each setup which limits the range of SER behavior for
the MMSE linear receiver. The SNR observed for the setup with 4QAM modulation
is shown in Figure 8.22 for the first 4000 frames received at the destination WARP
board.
67
0 500 1000 1500 2000 2500 3000 3500 4000
frame number (time)
-5
0
5
10
15
20
25
30
tota
l re
ceiv
ed S
NR
(dB
)
at 1st antenna
at 2nd antenna
Figure 8.22: SNR evolution for the 4QAM experiment for which the SER results areshown in Figure 8.20.
Chapter 9
Conclusions
This thesis studies the practicality of integer-forcing linear receivers: from the design
of the OFDM signal to the validation of the performance gains through over-the-air
experiments. I presented the transmitter’s signal architecture that enables the use of
IF linear receivers in practice. Furthermore, I identify the steps at the receiver that
conserve the linearity of the linear combinations of messages. The exact form of the
probability density function is derived for both the real and imaginary parts of each
linear combination.
The main challenges of building the experimental framework are highlighted. One
of the first ones is the conservation of linearity, both at the symbol level and codeword
level. I provide the exact mapping functions that were used to map finite field symbols
to constellation points for both F2 and F3 finite fields. Furthermore, experiments
were done for two different types of modulations 9QAM and 4QAM modulations in
combination with both binary and ternary LDPC codes. The results were provided for
both the coded and uncoded frameworks with respect to both SER and CER. These
experimental results indicate the IF linear receiver can provide significant gains when
used in a real wireless network. In particular, for the uncoded case, the IF linear
receiver performs arbitrarily closed to the ML linear receiver as theory predicted.
69
Furthermore, I showed that the complexity of the decoder can be reduced without
significant loss in performance by decoding the same set of linear combinations over
adjacent subcarriers. Also, by using different estimates of the noise power, I establish
that the noise power varies in frequency over a frame. In cases when the number
of received signals is smaller than the number of sent signals, both the ZF and eIF
linear receiver can not be used. However, I show, that enough independent linear
combinations of messages can be decoded from the received signals that enable the
full recovery of the sent signals. Experimental results, show that the performance
gains of the IF linear receiver still outperform the ones of the MMSE linear receiver.
This work, takes the first step towards confirming that significant gains can be
attained by IF and eIF linear receivers in practical systems. I believe that the chal-
lenges and the solutions I proposed will encourage the new generation WiFi systems
in using interference in a constructive way since it is a free consequence of the wireless
medium.
Appendices
Appendix A
WARP and AGC
The main reason that lead me to further investigate the influence of the AGC on
the accuracy of the noise and SNR estimates came from the behavior of the received
signal strength indicator (RSSI) value. The RSSI value is obtained from the current
WARPlab 802.11 reference design that is on the FPGA of the WARP board. As
mentioned in WARPlab online documentation [34] , for the WARP SDR platform,
RSSI is mainly affected by the AGC. That is with the AGC turned on, for an exper-
iment with aproximately 7500 frames, the evolution of SNR is shown in Figure , the
evolution of the transmitter’s and receiver’s baseband gain and RF gain are shown in
Figure A.2 and A.1.
In order to increase the range of SNR, for this experiment I went through a wide
range of values for the two gains (BB and RF), that set the transmit power. It is
clear, that as the transmit power increases, the degradation of the received signal
with respect to the noise increases as shown in Figure A.4. Furthermore, the current
AGC implementation in the WARP 802.11 OFDM reference design, adjusts the RSSI
of each antenna individually. For an accurate signal evaluation for the IF linear
receiver, the RSSI of the signals received at all antennas should be set together, not
independently. Since, IF benefits from using the actual strength of the cross channel
72
0 1000 2000 3000 4000 5000 6000 7000
frame number (time)
0
10
20
30
40
50
60
gain
valu
e
RX BB gain at A
RX BB gain at B
TX BB gain
Figure A.1: Evolution of the transmitter’s and receiver’s RF gain for an experimentwith 7500 frames when the AGC is turned on.
0 1000 2000 3000 4000 5000 6000
frame number (time)
0
0.5
1
1.5
2
2.5
3
3.5
4
ga
in v
alu
e
RX RF gain at A
RX RF gain at B
TX RF gain
Figure A.2: Evolution of the transmiter’s and receiver’s BB gain for an experimentwith 7500 frames when the AGC is turned on.
and direct channels in a MIMO wireless network.
Since this behavior of the AGC affects the quality of the SNR and relative received
power for each channel of the wireless network, the behavior of the IF, eIF and ZF
receivers is directly affected as shown in Figure A.5 (with the corresponding histogram
showed in Figure A.6). That is, there is a range of received SNR when ZF appears to
behave much better than other linear receivers. This indicated something was either
not estimated correctly, or the received power of the signal was not treated in a fair
73
0 1000 2000 3000 4000 5000 6000 7000
frame number (time)
300
350
400
450
500
550
600
650
700
750
800
RS
SI
at A
at B
Figure A.3: Evolution of RSSI for an experiment with 7500 frames when the AGC isturned on.
0 1000 2000 3000 4000 5000 6000 7000
frame number (ordered by time received)
-5
0
5
10
15
20
25
30
35
SN
R (
dB
)
SNR1
SNR2
Figure A.4: Evolution of received SNR for an experiment with 7500 frames when theAGC is turned on.
manner. It was clear that the power of the interfering channels (also known as cross
channels) was not treated fairly, since all other receivers use the information on the
interfering channels except the ZF one. Once, the AGC was turned off and all the
74
gains were set manually this behavior of the ZF linear receiver with respect to the eIF
and IF linear receivers was never observed in the rest of the experiments performed.
10 12 14 16 18 20 22 24 26 28 30
SNR2 [dB]
10-5
10-4
10-3
10-2
10-1
100
SE
R t
ota
l
Frames averaged based on SNR2
ML
IF
MMSE
eIF
ZF
Figure A.5: SER behavior for an experiment that uses 9 QAM modulation with 7500frames when the AGC is turned on.
Figure A.6: Histogram for the experimental results shown in Figure A.5.
Appendix B
Frame synchronization
As mentioned in Section 3.1 the preamble of the system is used for synchronization.
However, there are cases when the synchronization does not happen and a frame has
to be dropped. As mentioned in [34], when either the peaks of the LTS symbols do
not go above the imposed threshold (see Figure B.1), or the spacing in between the
peaks is not as expected and more than 3 peaks are observed.
0 100 200 300 400 500 600 700 800 900 1000
Sample Index
0
1
2
3
4
5
6
7LTS Correlation and Threshold
Figure B.1: Frame preamble showcasing the LTS peaks after FFT has been applied.This specific figure, corresponds to that of frame that has been droppeed by thesystem.
References
[1] J. Zhan, B. Nazer, U. Erez, and M. Gastpar, “Integer-forcing linear receivers,”IEEE Transactions on Information Theory, vol. 60, no. 12, pp. 7661–7685, Dec2014. 1, 2, 5, 5, 5, 5, 6.1, 6.1, 8.3
[2] ——, “Integer-forcing linear receivers,” 2010 IEEE International Symposiumon Information Theory, pp. 1022–1026, Jun. 2010. [Online]. Available:http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=5513734 1, 2,6.1
[3] D. Tse and P. Viswanath, Fundamentals of Wireless Communication. Cam-bridge University Press, 2005. 2, 3.2, 3.3.2, 4.1
[4] E. Viterbo and J. Boutros, “A universal lattice code decoder for fading channels,”IEEE Transactions on Information Theory, vol. 45, no. 5, pp. 1639–1642, Jul1999. 2
[5] N. Sommer, M. Feder, and O. Shalvi, “Closest point search in lattices usingsequential decoding,” in Proceedings. International Symposium on InformationTheory, 2005. ISIT 2005., Sept 2005, pp. 1053–1057. 2
[6] M. O. Damen, H. E. Gamal, and G. Caire, “On maximum-likelihood detectionand the search for the closest lattice point,” IEEE Transactions on InformationTheory, vol. 49, no. 10, pp. 2389–2402, Oct 2003. 2
[7] B. Hassibi and H. Vikalo, “On the sphere-decoding algorithm I. Expected com-plexity,” IEEE Transactions on Signal Processing, vol. 53, no. 8, pp. 2806–2818,Aug 2005. 2
[8] A. Burg, M. Borgmann, M. Wenk, M. Zellweger, W. Fichtner, and H. Bolcskei,“VLSI implementation of MIMO detection using the sphere decoding algorithm,”IEEE Journal of Solid-State Circuits, vol. 40, no. 7, pp. 1566–1577, July 2005. 2
77
[9] A. K. Singh, P. Elia, and J. Jalden, “Achieving a vanishing SNR gap to exactlattice decoding at a subexponential complexity,” IEEE Transactions on Infor-mation Theory, vol. 58, no. 6, pp. 3692–3707, June 2012. 2
[10] J. Jalden and P. Elia, “Sphere decoding complexity exponent for decoding full-rate codes over the quasi-static MIMO channel,” IEEE Transactions on Infor-mation Theory, vol. 58, no. 9, pp. 5785–5803, Sept 2012. 2
[11] H. Y. H. Yao and G. Wornell, “Lattice-reduction-aided detectors forMIMO communication systems,” Global Telecommunications Conference, 2002.GLOBECOM ’02. IEEE, vol. 1, pp. 424–428, 2002. [Online]. Available:http://ieeexplore.ieee.org/document/1188114/ 2
[12] M. Taherzadeh, A. Mobasher, and A. K. Khandani, “Communication over MIMObroadcast channels using lattice-basis reduction,” IEEE Transactions on Infor-mation Theory, vol. 53, no. 12, pp. 4567–4582, Dec 2007. 2
[13] ——, “LLL reduction achieves the receive diversity in MIMO decoding,” IEEETransactions on Information Theory, vol. 53, no. 12, pp. 4801–4805, Dec 2007.2
[14] Y. H. Gan, C. Ling, and W. H. Mow, “Complex lattice reduction algorithm forlow-complexity full-diversity MIMO detection,” IEEE Transactions on SignalProcessing, vol. 57, no. 7, pp. 2701–2710, July 2009. 2
[15] J. Jalden and P. Elia, “DMT optimality of LR-aided linear decoders for a generalclass of channels, lattice designs, and system models,” IEEE Transactions onInformation Theory, vol. 56, no. 10, pp. 4765–4780, Oct 2010. 2
[16] L. Zheng and D. N. C. Tse, “Diversity and multiplexing: a fundamental trade-off in multiple-antenna channels,” IEEE Transactions on Information Theory,vol. 49, no. 5, pp. 1073–1096, May 2003. 2
[17] T. M. Cover and J. A. Thomas, Elements Of Information Theory Notes, 2nd ed.Hoboken, NJ: Wiley-Intersience, 2006. 2
[18] Y. Jiang, M. K. Varanasi, and J. Li, “Performance analysis of ZF and MMSEequalizers for MIMO systems: an in-depth study of the high SNR regime,” IEEETransactions on Information Theory, vol. 57, no. 4, pp. 2008–2026, April 2011.2, 4.2.2
[19] T.-D. Chiueh, P.-Y. Tsai, and I. Lai, Baseband Receiver Design for WirelessMIMO-OFDM Communications. John Wiley & Son, 2010. 2, 4.2.2
[20] C. G. Kang, J. Kim, W.-Y. Yang, and Y. Cho, MIMO-OFDM Wireless Commu-nications with MATLAB. John Wiley & Son, 2010. 2, 4.2.2
78
[21] B. Nazer and M. Gastpar, “Compute-and-forward: Harnessing interferencewith structured codes,” 2008 IEEE International Symposium on InformationTheory, no. 1, pp. 772–776, Jul. 2008. [Online]. Available: http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=4595091 2
[22] ——, “Compute-and-forward: harnessing interference through structuredcodes,” IEEE Trans. Inf. Theory, vol. 57, no. 10, pp. 6463–6486, Oct. 2011.2
[23] B. Nazer, V. Cadambe, V. Ntranos, and G. Caire, “Expanding the compute-and-forward framework: unequal powers, signal levels, and multiple linear combina-tions,” IEEE Transactions on Information Theory, vol. 62, no. 9, pp. 4879–4909,Sept 2016. 2, 5, 6.1
[24] K. R. Kumar, G. Caire, and A. L. Moustakas, “Asymptotic performance oflinear receivers in MIMO fading channels,” IEEE Transactions on InformationTheory, vol. 55, no. 10, pp. 4398–4418, Oct. 2009. [Online]. Available:http://ieeexplore.ieee.org/document/5238756/ 2
[25] O. Ordentlich and U. Erez, “Precoded integer-forcing universally achieves theMIMO capacity to within a constant gap,” IEEE Transactions on InformationTheory, vol. 61, no. 1, pp. 323–340, Jan 2015. 2
[26] ——, “Cyclic-coded integer-forcing equalization,” IEEE Transactions on Infor-mation Theory, vol. 58, no. 9, pp. 5804–5815, Sept 2012. 2
[27] N. E. Tunali, K. R. Narayanan, J. J. Boutros, and Y.-C. Huang, “Latticesover Eisenstein integers for compute-and-forward,” 2012 50th Annual AllertonConference on Communication, Control, and Computing (Allerton), pp. 33–40,oct 2012. [Online]. Available: http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=6483196 2
[28] R. Zamir, Lattice Coding for Signals and Networks, 1st ed. Cambridge UniversityPress, 2014. 2, 6.1, 6.1
[29] A. K. Lenstra, H. W. Lenstra, and L. Lovasz, “Factoring polynomials with ra-tional coefficients,” Mathematische Annalen, vol. 261, no. 4, pp. 515–534, 1982.2, 5
[30] L. Ding, K. Kansanen, Y. Wang, and J. Zhang, “Exact SMP algorithms forinteger-forcing linear MIMO receivers,” IEEE Transactions on Wireless Com-munications, vol. 14, no. 12, pp. 6955–6966, Dec 2015. 2
[31] A. Mejri and G. R. B. Othman, “Practical implementation of integer forcinglinear receivers in MIMO channels,” IEEE Vehicular Technology Conference, pp.1–5, 2013. 2
79
[32] R. F. H. Fischer, “Lattice-reduction-aided equalization and generalized partial-response signaling for point-to-point transmission over flat-fading MIMO chan-nels,” in 4th International Symposium on Turbo Codes Related Topics; 6th In-ternational ITG-Conference on Source and Channel Coding, April 2006, pp. 1–6.2
[33] A. Sakzad, J. Harshan, and E. Viterbo, “Integer-forcing MIMO linear receiversbased on lattice reduction,” IEEE Transactions on Wireless Communications,vol. 12, no. 10, pp. 4905–4915, 2013. 2
[34] “WARP Project: http://warpproject.org.” 2, 7, 7.5.3, A, B
[35] “IEEE 802.11 Standard: https://standards.ieee.org/about/get/802/802.11.html.”2, 3.1, 7.1
[36] E. Eraslan, B. Daneshrad, and C. Y. Lou, “Performance indicator for MIMOMMSE receivers in the presence of channel estimation error,” IEEE WirelessCommunications Letters, vol. 2, no. 2, pp. 211–214, 2013. [Online]. Available:http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=6425374 2
[37] R. Measel, C. S. Lester, D. J. Bucci, K. Wanuga, G. Tait, R. Primerano, K. R.Dandekar, and M. Kam, “An empirical study on the performance of wirelessOFDM communications in highly reverberant environments,” IEEE Transac-tions on Wireless Communications, vol. 15, no. 7, pp. 4802–4812, July 2016.2
[38] M. Duarte, A. Sabharwal, C. Dick, and R. Rao, “Beamforming in MISO sys-tems: Empirical results and EVM-based analysis,” IEEE Transactions on Wire-less Communications, vol. 9, no. 10, pp. 3214–3225, 2010. 2
[39] D. J. C. MacKay, Information Theory, Inference, and Learning Algorithms,1st ed. Cambridge University Press, 2003. 2, 3.3.2, 7.4
[40] U. Erez and S. ten Brink, “A close-to-capacity dirty paper coding scheme,” IEEETransactions on Information Theory, vol. 51, no. 10, pp. 3417–3432, Oct 2005.2
[41] T. J. Richardson, M. A. Shokrollahi, and R. L. Urbanke, “Design of capacity-approaching irregular low-density parity-check codes,” IEEE Transactions onInformation Theory, vol. 47, no. 2, pp. 619–637, Feb 2001. 2
[42] O. Ordentlich, J. Zhan, U. Erez, M. Gastpar, and B. Nazer, “Practical codedesign for compute-and-forward,” in 2011 IEEE International Symposium onInformation Theory Proceedings, July 2011, pp. 1876–1880. 2, 7.4
[43] S. N. Hong and G. Caire, “Quantized compute and forward: A low-complexityarchitecture for distributed antenna systems,” in 2011 IEEE Information TheoryWorkshop, Oct 2011, pp. 420–424. 2
80
[44] ——, “Compute-and-forward strategies for cooperative distributed antenna sys-tems,” IEEE Transactions on Information Theory, vol. 59, no. 9, pp. 5227–5243,Sept 2013. 2
[45] T. Yang, I. Land, T. Huang, J. Yuan, and Z. Chen, “Distance spectrum and per-formance of channel-coded physical-layer network coding for binary-input gaus-sian two-way relay channels,” IEEE Transactions on Communications, vol. 60,no. 6, pp. 1499–1510, June 2012. 2
[46] N. E. Tunali, K. R. Narayanan, and H. D. Pfister, “Spatially-coupled low densitylattices based on construction a with applications to compute-and-forward,” in2013 IEEE Information Theory Workshop (ITW), Sept 2013, pp. 1–5. 2
[47] N. E. Tunali, Y. C. Huang, J. J. Boutros, and K. R. Narayanan, “Lattices overeisenstein integers for compute-and-forward,” IEEE Transactions on InformationTheory, vol. 61, no. 10, pp. 5306–5321, Oct 2015. 2
[48] B. Hern and K. R. Narayanan, “Multilevel coding schemes for compute-and-forward with flexible decoding,” IEEE Transactions on Information Theory,vol. 59, no. 11, pp. 7613–7631, Nov 2013. 2, 6.1
[49] C. Feng, D. Silva, and F. Kschischang, “An algebraic approach to physical-layernetwork coding,” IEEE Transactions on Information Theory, no. c, pp. 1–1,2013. [Online]. Available: http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=6582523 6.1, 6.1.1
[50] S. H. Chae, S. K. Ahn, and J. Park, “Integer-forcing receiver with practicalbinary codes and its performance analysis,” in GLOBECOM 2017 - 2017 IEEEGlobal Communications Conference, Dec 2017, pp. 1–6. 6.1.2, 7.4
[51] “IEEE standard for information technology–telecommunications and informa-tion exchange between systems. Local and metropolitan area networks–specificrequirements part 11: Wireless LAN medium access control (MAC) and phys-ical layer (PHY) specifications,” IEEE Std 802.11-2012 (Revision of IEEE Std802.11-2007), pp. 1–2793, March 2012. 7.1, 7.2
[52] G. R. G. Ren, H. Z. H. Zhang, and Y. C. Y. Chang, “SNR estimation algorithmbased on the preamble for OFDM systems in frequency selective channels,” IEEETrans. Commun., vol. 57, no. 8, pp. 2230–2234, 2009. 7.3
[53] M. C. Davey and D. J. C. MacKay, “Low density parity check codes over GF(q),”in 1998 Information Theory Workshop, Jun 1998, pp. 70–71. 7.4
[54] X.-Y. Hu, E. Eleftheriou, and D. M. Arnold, “Regular and irregular progressiveedge-growth tanner graphs,” IEEE Transactions on Information Theory, vol. 51,no. 1, pp. 386–398, Jan 2005. 7.4
81
[55] S. H. Chae, M. Jang, S. K. Ahn, J. Park, and C. Jeong, “Multilevel coding schemefor integer-forcing MIMO receivers with binary codes,” IEEE Transactions onWireless Communications, vol. 16, no. 8, pp. 5428–5441, Aug 2017. 7.4
[56] A. Akhavi, “Worst-case complexity of the optimal LLL algorithm,” in LATIN2000: Theoretical Informatics, G. H. Gonnet and A. Viola, Eds. Springer BerlinHeidelberg, 2000, pp. 355–366. 7.4
[57] M. R. Bremner, Lattice Basis Reduction: An Introduction to the LLL Algorithmand Its Applications, 1st ed. Boca Raton, FL, USA: CRC Press, Inc., 2011. 7.4
[58] S. Boumard, “Novel noise variance and SNR estimation algorithm for wire-less MIMO OFDM systems,” in Global Telecommunications Conference, 2003.GLOBECOM ’03. IEEE, vol. 3, Dec 2003, pp. 1330–1334 vol.3. 8
[59] S. He and M. Torkelson, “Effective SNR estimation in OFDM system simula-tion,” in IEEE GLOBECOM 1998 (Cat. NO. 98CH36250), vol. 2, 1998, pp.945–950 vol.2. 8