Time-and-frequency synchronization for
one-bit MU-MIMO
Feasibility and performance study

Master’s thesis in Communications Engineering

CARL LINDQUIST

Department of Electrical Engineering
CHALMERS UNIVERSITY OF TECHNOLOGY
Gothenburg, Sweden 2019





Master’s thesis

Time-and-frequency synchronization
for one-bit MU-MIMO

Feasibility and performance study

CARL LINDQUIST

Department of Electrical Engineering
Division of Communication and Antenna Systems

Chalmers University of Technology
Gothenburg, Sweden 2019



Time-and-frequency synchronization for one-bit MU-MIMO
Feasibility and performance study
CARL LINDQUIST

© CARL LINDQUIST, 2019.

Supervisor: Sven Jacobsson, Ericsson Research and Chalmers University of Technology
Examiner: Giuseppe Durisi, Chalmers University of Technology

Master’s Thesis
Department of Electrical Engineering
Division of Communication and Antenna Systems
Chalmers University of Technology
SE-412 96 Gothenburg
Telephone +46 31 772 1000

Typeset in LATEX
Gothenburg, Sweden 2019

iv



Time and Frequency Synchronization for One-bit MU-MIMO
Feasibility and performance study
CARL LINDQUIST
Department of Electrical Engineering
Chalmers University of Technology

Abstract
In this thesis, an uplink multiuser multiple-input multiple-output (MU-MIMO) system is consid-
ered. In the system, a number of user equipments (UEs) are communicating with a single base
station (BS). The BS is fitted with several antennas and in each receiver chain, one-bit analog-to-
digital converters (ADC) are used to convert the incoming signal from the analog to the digital
domain. At each BS antenna, there will be an uncertainty regarding the timing as well as the
carrier frequency of the received signal, typically referred to as symbol timing offset and carrier fre-
quency offset. This thesis investigates the effect of these offsets on the communication system and
provides an analytic expression for the signal-to-interference-noise-and-distortion ratio (SINDR).
Moreover, an overview into the topic of synchronization itself is provided and some standard syn-
chronization methods are described and evaluated in the context of one-bit MU-MIMO. The thesis
demonstrates that despite the nonlinear distortion introduced by the one-bit ADCs, the system
can still be synchronized. Lastly, the overall system performance in the presence of synchronization
errors is discussed, as well as some ideas for future research.

Keywords: quantization, one-bit analog-to-digital converter (ADC), multiuser multiple-input multiple-
output (MU-MIMO), symbol timing offset (STO), carrier frequency offset (CFO), synchronization

v





Acknowledgements
This thesis concludes my studies at Chalmers University of Technology, where I have been enrolled
in the Communication Engineering Master’s programme as the final part for the degree of Master
of Science in Electrical Engineering. I am most grateful to everyone that has provided me with
the necessary tools to complete this degree and this thesis.

The thesis has been carried out in collaboration with Ericsson Research at Lindholmen and I
would like thank everyone that has contributed to my thesis in any way. In particular, I would like
to extend a major thank you to my main supervisor, Sven Jacobsson at Chalmers University of
Technology and Ericsson Research, for countless hours of assistance with this topic. I would also
like to thank Dr. Keerthi Nagalapur for your patience with my questions and as well as valuable
input.

Carl Lindquist, Gothenburg, April 2019

vii





Contents

List of Figures xi

1 Introduction 1
1.1 Spectral congestion and mmWave . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Massive MIMO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4 Outline of thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 System model 9
2.1 Orthogonal Frequency Division Multiplexing . . . . . . . . . . . . . . . . . . . . . 9
2.2 Channel input-output model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 Impact of imperfect synchronization . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3.1 Symbol timing offset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3.2 Carrier frequency offset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3.3 One-bit quantizers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3 Analysis 21
3.1 Bussgang’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2 Received signal due to STO and CFO . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.2.1 Infinite-precision case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2.2 One-bit quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2.3 ZF equalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4 Synchronization in OFDM systems 37
4.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.2 Cyclic prefix-based synchronization . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.3 Schmidl and Cox synchronization . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.3.1 Morelli and Mengali extension . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.4 LTE-like synchronization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5 Performance 69

6 Discussion 73
6.1 Ideas for future research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

Bibliography 77

ix



Contents

x



List of Figures

1.1 Some quantizers with different parameters. . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Walden’s Figure of Merit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1 The system model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2 A depiction of the correct placement of the DFT window. . . . . . . . . . . . . . 14
2.3 Effect of residual STO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.4 The effect of the CFO on an OFDM symbol with 256 subcarriers, single-user sce-

nario with AWGN and high SNR. . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.5 Around 500 realizations of â+ n̂, with SNR = 5 dB. . . . . . . . . . . . . . . . . . 18
2.6 32 OFDM symbols with 256 subcarriers transmitted over AWGN and quantized

and averaged at the receiver, SNR = 0 dB. . . . . . . . . . . . . . . . . . . . . . . 19

3.1 A visualization of Bussgang’s theorem . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2 Illustration of δu > 0 and δu < 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.1 The optimum rule for detecting a known sequence under the influence of AWGN. 38
4.2 Probability of correctly locating a sequence, optimal rule . . . . . . . . . . . . . . 40
4.3 M-sequence generator in LTE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.4 A length 13 Barker sequence (top) and a length 31 m-sequence (bottom) and their

respective correlation spectra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.5 The window of received samples in the CP-based method . . . . . . . . . . . . . . 44
4.6 Log-likelihood function in cyclic prefix-based method . . . . . . . . . . . . . . . . 46
4.7 MSE, cyclic prefix-based estimation . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.8 Histograms and cumulative histograms, cyclic prefix . . . . . . . . . . . . . . . . . 48
4.9 The relationship between the missed synchronization probability and the false de-

tection probability when using the cyclic prefix-based synchronization method. . . 49
4.10 Likelihood functions averaged over multiple antennas . . . . . . . . . . . . . . . . 50
4.11 The first synchronization symbol in the method by Schmidl and Cox. . . . . . . . 51
4.12 Timing metric is Schmidl and Cox . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.13 Statistics of the Schmidl and Cox timing metric . . . . . . . . . . . . . . . . . . . 54
4.14 MSE of the Schmidl and Cox method . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.15 Histograms and cumulative histograms, Schmidl and Cox . . . . . . . . . . . . . . 56
4.16 Missed synchronization and false detection probability for Schmidl and Cox . . . . 57
4.17 Illustration of the frequency dependency of the estimator in the one-bit quantization

case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.18 MSE of STO estimation with more antennas, Schmidl and Cox . . . . . . . . . . . 58

xi



List of Figures

4.19 The real part of the first synchronization symbol in the method by Morelli and
Mengali. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.20 The MSE of the frequency estimator, ε0 = 2.2. Results obtained via 5 · 103 simula-
tions over an AWGN channel with 512 subcarriers and q = 8. . . . . . . . . . . . . 60

4.21 Zadoff-Chu sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.22 Cross-correlation of time-domain Zadoff-Chu sequences . . . . . . . . . . . . . . . 63
4.23 MSE of STO estimation, Zadoff-Chu . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.24 Histograms and cumulative histograms, Zadoff-Chu . . . . . . . . . . . . . . . . . 64
4.25 Missed synchronization and false detection probability for Zadoff-Chu . . . . . . . 65
4.26 MSE for the different STO and CFO estimators . . . . . . . . . . . . . . . . . . . 67

5.1 STO effect on the SNR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.2 CFO effect on the SNR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.3 Effect of multiple receive antennas, SNR and STO . . . . . . . . . . . . . . . . . . 71
5.4 RMSE for the S and C algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

xii



1
Introduction

Before the advent of digital communication, the way in which information was communicated
was strongly influenced by the information type. Written words were typically transmitted as
letters and the spoken word could for example be transmitted via telephone wires. These two
systems both transported information, either via mailboxes, sorting systems and mail carriers or
rotary dials, switchboards and copper wires but clearly, the requirements of the two are vastly
different. This alludes to perhaps the main disadvantage inherent analog information transfer,
namely the lack of flexibility. The information type, such as a voltage level or letter, dictates the
requirements of the transmission channel. A phone call can not be transmitted via the postal
system and vice versa. Since the second generation of mobile networks in the 1990’s [1], the
underpinnings of communication systems have been digital technologies. These systems need not
concern themselves with the type of data to transmit — information in binary form is transmitted
in the same fashion regardless of what the bits represent. This allows us to focus on developing
strategies and techniques for data transmission in general, instead of having parallel development
tracks for different applications, With that said, however, certain performance requirements, such
as bit error rate (BER), data rate, spectral efficiency and latency of the system, will vary with the
type of information the systems transmits. These are some of the standard performance metrics
with which we benchmark systems and in terms of mobile communications, these are commonly
used to set the specification of new generations.

Mobile communication networks has transitioned to a new generation roughly every tenth year
since the first 1G network appeared in Japan [2] in the late 1970s. Largely adhering to this rule,
the commercial release date for the fifth generation (5G) mobile network is imminent [3]. In
5G, a thousandfold increase in data rates, reduced end-to-end latency and supporting a ten- to a
hundredfold increase in connected devices, are some of its targets [4, 5]. The first 5G networks
have not yet been rolled out commercially, but numerous demonstrations have taken place. For
example, KT Corp. showcased a number of applications of 5G during the 2018 Winter Olympic
Games in South Korea, including live-streaming 360° video of competing athletes [6]. In May
2017, Ericsson together with Verizon also demonstrated live-streaming of 360° video. In the trial,
a car was driving around a race track while the driver’s only visuals where streamed to a set of
virtual reality goggles from a camera mounted on the hood of the car [7].

To achieve the targets set for 5G networks, a number of technologies, both improvements of
existing technologies as well as more radical ones, will need be deployed [8]. Two of the new
technologies that will be incorporated in 5G are massive multiple-input multiple-output (MIMO)

1



1. Introduction

and millimeter-wave (mmWave). These technologies are considered key [5] and the work in this
thesis is primarily related to these two. The next section provides a brief introduction to these
topics.

1.1 Spectral congestion and mmWave

In the pursuit of higher data rates, we are intrinsically bound by the inverse relationship of symbol
time and the bandwidth. This can be intuitively understood from the fact that if the symbol time
is shortened, the signal will change more frequently, yielding a larger bandwidth. Whilst for
example higher order modulation formats can be used to increase the rates, increasing the system
bandwidth have been a trend throughout the development of mobile communication standards.

Increasing the bandwidth can lead to several challenges. First and foremost, the radio frequency
spectrum, in which all mobile communication takes place, is a limited resource. As such, increasing
the bandwidth indiscriminately is usually not an option. Below 6 GHz, where most wireless
communication takes place in current mobile communication systems, the spectrum is extremely
crowded, serving as a motivation to develop communication technologies outside of this band. One
example of this is Visible Light Communication (VLC) [9], which, as implied by the name, uses
visible light (that is, in the frequency band from 400–800 THz) as a carrier and overlays the light
with a data-carrying signal. The actual communication is not carried out in the THz-domain; the
visible light is merely used as a carrier wave. This technology is as of yet not particularly fast,
but serves as an example of a means of wireless communication outside of the conventional band.

While auxiliary technologies such as VLC might slightly unload the mobile communication network
for specific tasks, it will most likely never be the main workhorse of human communication. The
majority of research attention is given to the mobile communication network itself and going
forward, the move to higher frequencies is set to play a key part of the next mobile communications
standard. Communications will take place both in bands below 6 GHz and in a number of different
bands above 6 GHz [10], with the high frequency range in the 5G standard set to 24.25–52.6 GHz
and the highest band allocated as of yet set to 37–40 GHz [11]. These higher frequency band are
commonly referred to as mmWave, as the wavelength for frequencies ranging from 30–300 GHz
have wavelengths ranging from 10–1 mm. Shorter wavelength has a number of effects, perhaps
the most prominent being the increased susceptibility to blockages, due to shorter penetration
depth. Further, as stated in the well-known Friis transmission equation [12], the received power of
a signal transmitted in free space is inversely proportional to the square of its frequency, meaning
that even without blockages, higher frequencies lead to less received power for a given antenna
aperture.

One way to mitigate these issues is to use a technique known as beamforming. By using more than
one antenna, we can adjust the shape of the beam according to some design criterion. Commonly,
the beam is designed to interfere either constructively or destructively at one or more locations.
Next, we will take a look at Massive MIMO, the technology underpinning beamforming strategies.

2



1. Introduction

1.2 Massive MIMO

It is well-known that using more than one antenna can yield considerable performance improve-
ments. Intuitively, this can be understood by considering a simple system where a message is
transmitted to a single antenna through a noisy environment. The noise might have a very strong
influence on some parts of the message, making it impossible for the receiver to completely re-
cover what was transmitted. If, however, the receiver uses more than one antenna to pick up the
message, the receiver would then have multiple versions of the same message. Now, if parts of
the message strongly affected by noise in one version are unaffected in others, the receiver could
then combine all versions of the message to hopefully recover the full message. A system where
the receiver uses more than one antenna to pick up a signal from a single source, is known as
single-input multiple-output (SIMO). MIMO refers to the case where both the transmitter as well
as the receiver are equipped with more than one antenna. If designed properly, a MIMO system
enable increased throughput, as it can establish parallel communication streams by adding the
spatial dimension as a scheduling resource. A scenario where a single user is equipped with several
antennas is sometimes referred to as single user (SU) MIMO and further, we can also consider
many single-antenna users, known as multi user (MU) MIMO. Both SU-MIMO and MU-MIMO
have been a part of both the Long Term Evolution (LTE) and Worldwide Interoperability for
Microwave Access (WiMAX) standard for roughly ten years and has played a significant part in
reaching target data rates [13].

Massive MIMO, as implied by the name, means that the number of antennas is high. Typically, the
number of antennas at the base station (BS) is significantly higher than the number of antennas
at the user equipments (UE). A key technique in Massive MIMO is beamforming, where the many
antennas facilitates accurate direction of beams to discrete points in space. Massive MIMO has
been an active research topic for several years and is expected to become an integral part of
the next generation of mobile systems. Some theoretical benefits of Massive MIMO are capacity
gains, increased robustness, and highly improved energy efficiency [14, 15]. Whether or not all of
these theoretical promises can be realized at a reasonable cost remains to be seen, but regardless,
Massive MIMO is a highly promising technology. The major interest in bringing this to market
has spawned a number of new research areas related to the challenges associated with having large
antenna arrays. In the following section, we will outline some of these challenges and provide the
necessary motivation for this work.

1.3 Challenges

Having a large antenna array intrinsically entails an increased number of hardware components.
Each antenna will require a hardware chain and for the elements to cooperate in some fashion,
additional controlling units are needed. To keep the power consumption from skyrocketing, con-
siderable research interest has been devoted to simplifying some hardware components. Examples
of such, include the analog-to-digital converters (ADC) and digital-to-analog converters (DAC).
As implied by their name, these devices are responsible for converting the output of a digital

3



1. Introduction

signal processor (DSP) to signal that can be fed to an antenna, or vice-versa. In the case of a
ADC, these work by taking a snapshot of the analog signal at discrete timing instants and storing
the amplitude of signal as closely as allowed by the resolution of the ADC. More precisely, an
ADC performs both sampling (time-discretizing) and quantization (amplitude-discretizing). The
discretizing operations are necessary as a digital device has finite memory, meaning the analog
values can not be stored with infinite-precision. This thesis will focus primarily on the quantizing
part of the ADC and the sampled and nonquantized signal will be commonly be referred to as the
infinite-precision case.

Mathematically, we express the sampling part of the ADC as ys[n] = yc(nTs), where ys is the
sampled version of the continuous signal yc at times nTs, n ∈ Z≥0. The sampling time Ts specifies
the time between two successive samples. The inverse of the sampling time Fs = 1/Ts is called
the sampling frequency. Next, the quantization operation can be described as r[n] = Q(ys[n]),
where r[n] is the quantized version of the discrete signal ys[n], according the- rules specified by
the quantizer Q. The number of possible outputs from the quantizer is known as the number of
quantization levels. The number of quantization levels L is linked to the resolution of an ADC
as L = 2b, where b is the resolution in bits. For example, a three-bit quantizer has L = 23 = 8
possible output values, represented digitally as every permutation of a three-bit binary number.

In Figure 1.1, a number of examples of the sampling speed Fs and resolution b are shown. Beginning
in the top left corner, the time-continuous signal yc has been sampled with some speed Fs and
then quantized with a three-bit quantizer. The red dots depicts the digital representation r[n]
of the signal and the quantization error is defined as ys[n] − r[n], i.e. the difference between the
nonquantized and the quantized signal in the sampling instants. As the number of quantization
levels is limited to 23 = 8 levels, the quantization error is clearly discernible. In Figure 1.1, we
have also added a zero-order hold reconstruction of the digital values, represented by the solid
black line. This is a model of how the analog signal can be reconstructed from the digital values
using a DAC that simply holds the digital value until the next sampling instant, producing a
square-like output. Clearly, a using a first-order hold filter that linearly interpolates between
successive sampling points would have produced an output that looks more similar to the original,
but the purpose here is to demonstrate the effect of increasing the sampling speed. This is done
in Figure 1.1b, where the sampling speed is 3Fs. Comparing the quantization error in Figure 1.1a
and 1.1b, we see that the magnitude of error is comparable, but the reconstructed line is much
closer to the original. Though not evident from Figure 1.1b, there are additional benefits to
increasing the sampling rate besides in the reconstruction phase, which will be mentioned later.

In Figure 1.1c, the sampling speed is again set to Fs and the resolution increased to 10 bits. With
b = 10, the number of quantization levels increases to 210 = 1024, yielding a quantization error
that is virtually zero.

In Figure 1.1a, 1.1b and 1.1c, we have used what is known as an automatic gain control (AGC). This
is a device which attenuates the input signal so that it fits within the range of the quantizer levels.
Defining q to be the difference between two adjacent quantization levels, this implementation of
the AGC ensures that the maximum input value is q/2 above the highest quantization level. Then,
the maximum size of the error will be q/2, regardless of the signal level. Without the AGC, the
quantizer would have had to select its highest or lowest value as soon as the input went outside its

4



1. Introduction

0 5 10

−1

−0.5

0

0.5

1

1.5

n

yc

r[n]

Quantization error

Reconstructed signal

(a) b = 3, sampling speed is Fs.

0 5 10

−1

−0.5

0

0.5

1

1.5

n

yc

r[n]

Quantization error

Reconstructed signal

(b) b = 3, sampling speed is 3Fs.

0 5 10

−1

−0.5

0

0.5

1

1.5

n

yc

r[n]

Quantization error

Reconstructed signal

(c) b = 10, sampling speed is Fs.

0 5 10

−2

−1

0

1

2

3

n

yc

r[n]

Quantization error

Reconstructed signal

(d) b = 1, sampling speed is Fs.

Figure 1.1: Some quantizers with different parameters.

range, yielding significant errors at the extremes. This is an example of saturation (or clipping)
and is a well-known effect in for example amplifiers, arising when the dynamic range of the input
exceeds the dynamic range of the output.

Lastly, in Figure 1.1d, the resolution is set to a single bit. Note that having an AGC or not
does not make a difference, as only the sign of the incoming signal is stored; the magnitude is
of no importance. Consequently, we can remove the AGC from the receiver chain, which is one
of the reasons why one-bit quantizers have generated significant research interest in the last few
years. Fascinatingly, and perhaps rather counter-intuitively, we will see that the very apparent
quantization error in Figure 1.1d is not enough to render a MIMO system unusable. In fact,
research has demonstrated quite the opposite, as studies has demonstrated not only that systems
using low-resolution converters does in fact work — only a few bits are required to get similar

5



1. Introduction

performance to that of the infinite-precision case [16].

As is clear from Figure 1.1, in order to improve the performance of a ADC, we can either sample the
signal more often or store the value with greater precision. Unfortunately, increasing the sampling
rate and resolution of an ADC also increases the power consumption. For a given sampling rate
Fs Hz and resolution b bits, the power dissipation scales as roughly [17]

P ∝ 2beffFs (1.1)

where beff is the effective number of bits. It is a standard way of measuring the true resolution of
an ADC, taking into account the distortion introduced by the ADC circuit itself.

From (1.1), we see that for each additional bit of resolution, the power consumption scales by
roughly a factor two. Moreover, in order to compare different architectures and identify trends, a
figure of merit (FOM) was proposed by Walden in [18] as

FW = 2beffFs
P

(1.2)

This FOM, now referred to as Walden’s FOM, is in commonly in use and in Figure 1.2, the
inverse of (1.2) with each data point divided with its resolution is shown. The envelope have been
constructed from the average of the five best (with relation to their respective FOM) designs [19].

102 103 104 105 106 107 108 109 1010 1011 1012
10−16

10−15

10−14

10−13

10−12

10−11

10−10

10−9

Fs [Hz]

FO
M

W
[J

/c
on

ve
rs

io
n

st
ep

]

ISSCC 2018

VLSI 2018

ISSCC 1997-2017

VLSI 1997-2017

Envelope

Figure 1.2: Walden’s Figure of Merit. Data gathered from the International Solid-State Circuits Con-
ference (ISSCC) and the Very Large Scale Integration Symposium (VLSI). Data fetched from [19].

We see that increasing the speed of an ADC can indeed be costly in terms of power consumption.
High speed converters can bring a number of benefits, such as radio frequency (RF) sampling, where
the received signal is sampled and converted into the digital domain at passband directly, removing

6



1. Introduction

a number of analog components required in the mixing stage [20]. This reduces complexity and
cost, as well as decreases power consumption. Further, it has been shown for example in [21] that
oversampling can yield performance benefits if low-resolution converters are used.

With future mobile communication network likely to operate at wider bandwidths than today, it
seems likely that the demand for fast ADCs is not going to diminish. Consequently, reducing the
resolution of the converters could be a solution. For example, a typical ADC deployed in a BS
today has a resolution of 10–14 bits [22]. Replacing those converters with ADCs and DACs with
lower resolution has the potential to yield significant energy savings, at least when examining the
converters in isolation. As shown in Figure 1.1, the quantization error grows as the resolution
is decreased, introducing more distortion to the system. Adding more antennas, as is the core
concept in Massive MIMO, will provide array gain that to some degree mitigates the effect of
increased quantization distortion, while at the same time increasing the total power consumption.
This thesis exclusively compare the infinite-resolution case and the extreme one-bit quantized case
and for a more in-depth discussion into this topic of low-resolution converters and their role in
future Massive MIMO systems, see for example [23].

For one-bit ADCs to be a viable technology in future communication systems, it needs to be
shown that acceptable performance can still be achieved despite the major impairments that they
bring. Research so far has demonstrated the viability of low-resolution quantizers in a number
of aspects, indicating that this indeed a promising path forward. A topic currently receiving
some attention with promising results [24, 25, 26, 27] concerns the synchronization of systems
employing low-resolution converters. Demonstrating that systems employing one-bit ADCs can
still be synchronized is an important step in the research of this technology and the main motivation
for this thesis.

1.4 Outline of thesis

In this thesis, some aspects of one-bit converters will be investigated. Specifically, it will be
examined whether systems employing these types of converters can be synchronized using standard
methods. In Chapter 2, the system model and channel input-output will be discussed. along with
an overview of the effects of imperfect synchronization. In Chapter 3, the necessary mathematical
tools will be mentioned and a more in-depth analysis of the effects mentioned in Chapter 2 will
be performed. Expressions for the power in the received signal and signal-to-interference-noise-
and-distortion-ratio (SINDR) will also be derived. In Chapter 4, some well-known synchronization
strategies will be discussed and their performance in the non-quantized and the one-bit quantized
case will be compared. In Chapter 5, the overall performance of the system with synchronization
errors present will be examined via simulations and lastly, in Chapter 6, the results from this
thesis will be discussed. Some ideas for future research into this field will also be suggested.

7



1. Introduction

8



2
System model

This chapter explains the system model, the assumptions made regarding the communication
system and the effects of imperfect synchronization. Here, the aim is to give a complete charac-
terization of the problem at hand, before delving into the analysis.

2.1 Orthogonal Frequency Division Multiplexing

Before going into the channel input-output model, it is fitting to say something with regards to the
modulation format used in modern communication systems. The medium through which the signal
is transmitted is called the channel and can be characterized by a number of parameters. Noise
introduced by for example hardware, is usually modeled as an additive effect. As this in intended
to capture a wide array of noise sources into a single additive effect, it can by the central limit
theorem be approximated as a Gaussian random variable. In addition to this, wireless channels
also have a multiplicative effect known as fading. A natural distinction between different fading
environments is whether the channel has the same effect on all frequencies in the signal bandwidth
or if its frequency response varies. The former is usually referred to as a frequency-flat channel,
arising in narrow-band scenarios where the communication bandwidth is smaller than the channel
coherence bandwidth. The latter case is referred to as a frequency-selective channel and usually
arises in environments where the reflection and refraction of a transmitted signal must be taken
into account.

With growing signal rates, the system will occupy a larger bandwidth, introducing additional
design concerns. In a single-user scenario, no other user is competing for the same frequency
resource, but if, however, multiple users are present, some sort of scheduling in order to prevent
collisions between messages must be implemented. One solution would be to designate a slot in
time for each user, in which the user is granted access to the channel. This idea is the basis of
time-division multiple access (TDMA) and was the main multiple access technique in early mobile
systems. It is still used today, for example in LTE where it is used in conjunction with other
techniques [28].

While TDMA indeed solves the issue of multiple users simultaneously accessing the channel, but
does nothing to alleviate another issue, namely the frequency-selectiveness of the channel. If the

9



2. System model

signal bandwidth is larger than the channel coherence bandwidth, it means that the spectral gain is
not constant over the entire signal bandwidth, making equalizing more complex. A response to this
is to use orthogonal frequency division multiplexing (OFDM), which converts a single wideband
channel into several narrowband channels with lower rates. Dividing the available frequency
spectrum into several channels, known as a multi-carrier modulation, had been around for a while,
but the idea of modulating each symbol with signals from an orthogonal set is normally credited
to [29]. Using an orthogonal set of signals, we are able overlap the frequency bands without
incurring any inter-carrier interference (ICI), thus maximizing the transmission rate. Further,
frequency resources can be dynamically allocated based on the current conditions of the sub-
channels, thereby avoiding energy being wasted on unusable parts of the spectrum. The first
major application of the format was the in asymmetric digital subscriber line (ADSL) in the
1990’s [30]. By then, efficient implementation had been made available via fast DSP units and
fast Fourier transforms (FFT). For a more complete description of the development of OFDM, see
e.g. [31].

In the modulation process, the inverse discrete Fourier transform (IDFT) is used to modulate
each data symbol x̂[k] ∈ C with a complex exponential function drawn from an orthogonal set.
Specifically, for N sub bands

x[n] = 1√
N

N−1∑
k=0

x̂[k] ej2πnk/N , n = 0, 1, . . . , N − 1 (2.1)

where x̂[k] is the data symbols drawn from some constellation and x[n] the time-domain signal to be
transmitted. The scaling factor 1/

√
N in front of the summation is there to ensure that the average

power of the signal is not affected by the IDFT operation. The exponentials e(j2πnk/N), ∀n ∈
{0, 1, . . . , N − 1} in (2.1) are commonly referred to as subcarriers and are what the data to be
transmitted is modulated with. The phase difference of the complex exponential functions being
multiplied with x̂ is 1/N and to verify that the set spanned by these complex is orthogonal, note
that

N−1∑
n=0

ej2πnk/N · e−j2πnp/N

=
N−1∑
n=0

ej2π
k−p
N

n

= 0 , ∀ k 6= p.

Not all N values of x̂[k], k = 0, 1, . . . , N − 1 needs to contain a data symbol. Some zeros are
commonly inserted at specific points in order to increase the robustness of the system. This will
however not be considered in this thesis and all subcarriers will be assumed to carry data symbols,
which will be referred to as symbol-sampling rate.

In the demodulation process, the frequency-domain symbols are retrieved by computing the dis-

10



2. System model

crete Fourier transform (DFT) of the time-domain signal, defined as

x̂[k] = 1√
N

N−1∑
n=0

x[n] e−j2πnk/N , k = 0, 1, . . . , N − 1 (2.2)

where the scaling 1/
√
N again ensures that the average power after the DFT is unchanged.

In a fading channel, some concessions regarding the transmission must be made in order for the
orthogonality between subcarriers to be preserved. Specifically, each OFDM symbol is commonly
cyclically extended via the insertion of a cyclic prefix (CP). The CP consists of the last P ≤ N
samples of the original OFDM symbol, bringing the total length of the OFDM symbol to N + P .
Naturally, as some parts of the symbol are repeated, this redundancy lowers the total rate. Another
issue with OFDM is the peak-to-average ratio (PAPR), which captures how much the amplitude
of the time-domain signal varies. In OFDM, compared to a single-carrier system, this ratio tends
to be fairly high, spelling difficulty for the amplifiers. Typically, an amplifier is only linear in a
limited region, meaning that we are more likely to incur nonlinear distortion for signals with large
amplitude variations. This thesis will assume that this effect, as well as a number of additional
hardware-related concerns, can be disregarded. For more details on these matters, see e.g. [32].

2.2 Channel input-output model

In this work, we consider an uplink baseband model where U single-antenna UEs are communi-
cating with a BS equipped with B antennas. Operations such as filtering, up and down conversion
and mixing will be assumed to be ideal and therefore not explicitly considered.

An overview of the considered MU-MIMO OFDM system is depicted in Figure 2.1. The raw
bits coming into our system are assumed to be uniformly distributed and fed to a modulator
unit, generating U streams of Gray-coded complex symbols drawn from a quadrature phase-shift
keying (QPSK) alphabet, i.e. x̂u[k] ∈ {±1/

√
2± 1/

√
2j} for u ∈ {1, 2, . . . , U}. Next, the symbols

x̂[k] = [x̂1[k], x̂2[k], . . . , x̂U [k]]T ∈ CU are transformed into the time-domain via the IDFT defined
in (2.1), a cyclic prefix is added and the time-domain signal x[n] = [x1[n], x2[n], . . . , xU [n]]T ∈ CU

is transmitted over a channel H ∈ CB×U . In this thesis, the channel will be assumed to be
frequency-flat, i.e. the number of channel taps, modelling the delay spread, is assumed to be one.
Upon arrival at the receiver, there will be some uncertainty regarding the absolute timing as well
as the carrier frequency of the signal. These effects are called the symbol timing offset (STO) and
carrier frequency offset (CFO), respectively, and are a major focus of this thesis.

At the receiver, the received signal y[n] = [y1[n], y2[n], . . . , yB[n]]T ∈ CB is fed to 2B ADCs, one
pair for each receiver chain, where the real and imaginary part of the signal at each antenna
element will be quantized separately. The resolution is limited to one bit, effectively only storing
the sign of the incoming signal. Next, the STO and CFO will be estimated and compensated via
some method, examined in more detail in Chapter 4. If the estimation of the STO and CFO is

11



2. System model

not perfect, then there will be residual STO and CFO after the compensation stage, potentially
affecting the performance of the system. The precise effect of the residual STO and CFO will be
examined in Chapter 3.

After the STO and CFO estimation and compensation stage, the CP is removed and a DFT is
performed. Lastly, to obtain an estimate x̂est[k] ∈ CU of the transmitted symbols x̂[k], the signal
is passed through a equalization stage. After equalization, the estimated symbols are then fed to
the decision and decoding units of the system, finally yielding an estimate of the transmitted bits.

IDFT CP
UE 1

x̂1 x1

IDFT CP
UE 2

x̂2 x2

IDFT CP
UE U

x̂U xU

... ... ...

Freq.-flat.
channel

Base station

STO/CFO
Im ADC
Re ADC

r2 STO/CFO
est.

CP
DFT

y2

STO/CFO
Im ADC
Re ADC

r1 STO/CFO
est.

CP
DFT

y1

STO/CFO
Im ADC
Re ADC

rB STO/CFO
est.

CP
DFT

yB

... ... ... ...

EQ
Comb

...

x̂est
1

x̂est
2

x̂est
U

Figure 2.1: The system model for U single-antenna transmitting UEs and B receive antennas.

Assuming that the users are transmitting a continuous stream of OFDM symbols, the received
signal at the bth antenna yb[n] can be expressed as

yb[n] =
U∑
u=1

ej2πε0,un/N hb,u xu[n+ δ0,u] + wb[n]. (2.3)

In (2.3), ε0,u and δ0,u represents the CFO and STO between the BS and the uth UE, respectively.
The scalar value hb,u represents the channel from user u to antenna b and xu[n] is the transmitted
signal from user u. The term wb[n] stands for the thermal noise at antenna b, wb[n] ∼ CN (0, N0),
where N0 is the noise power spectral density.

The quantized signal rb[n] is defined as rb[n] = Q (yb[n]), where Q(·) describes the nonlinear
quantizer operation. It is defined as

Q(yb[n]) = 1√
2

(
sgn (Re{yb[n]}) + j sgn (Im{yb[n]})

)
(2.4)

where sgn(·) is the signum function. The scaling
√

1/2 is chosen so that E[|rb[n]|2] = 1.

Lastly, an estimate of the transmitted symbols from user u, x̂est
u [k], is obtained as

12



2. System model

x̂est
u [k] =

B∑
b=1

au,br̂b[k] (2.5)

where au,b is the (u, b)th entry in the equalization matrix A ∈ CU×B and r̂b[k] is the DFT of rb[n].

In this thesis, a zero-forcing (ZF) equalizer is considered, which attempts to invert the effect of
the channel via the psuedo-inverse of the channel, i.e.

A = (HHH)−1 HH . (2.6)

The analysis of the received signal is performed in Chapter 3.

We will use this model as the basis for our investigation and next, we will give an overview of the
effects of STO, CFO, and one-bit quantization on the signal.

2.3 Impact of imperfect synchronization

This section will provide an overview of the adverse effects STO and CFO will have on the received
signal. In a real-world system, there are additional synchronization-related matters to consider,
such as the sampling clock offset, but we will limit ourselves to the STO and CFO. For a discussion
on the effects of other synchronization offsets, see e.g. [33].

This section intends to provide the intuition for the effects of STO and CFO. For a more complete
mathematical analysis, please refer to Section 3.

2.3.1 Symbol timing offset

Generally speaking, the receiver in a communication system does not know in advance when the
transmitter will send something. This means that the designer of the system will need to devise a
way for the receiver to automatically detect when a signal is present in the channel. This process
is known as frame detection and in Section 4, some strategies for accomplishing frame detection
are mentioned. Determining exactly when a transmitted message has arrived at the receiver is not
trivial and uncertainty in that process can lead to STO, the topic of this section.

We will assume that frame detection and STO estimation has already been performed and that
the timing uncertainty is in the order of ±N , the length of the OFDM symbol. As mentioned in
Section 2.2, this remaining STO will be referred to as residual STO and we define the residual STO
from the uth user to be δu = δ0,u−δest

u . From Figure 2.1, the next step after synchronization offset
estimation is the removal of the CP and a DFT operation. The continuous stream of received

13



2. System model

samples is divided into blocks of size N , which are fed to an N -point DFT. We refer to the ith
block as the ith DFT window. This is illustrated in Figure 2.2. If δu = 0, then the ith DFT
window contains precisely the N samples from the ith symbol. Clearly, if there is no residual
STO, no additional interference will be introduced.

OFDM block:
· · · CP Data

i − 1
CP Data

i

CP Data
i + 1

· · ·
DFT window

Figure 2.2: A depiction of the correct placement of the DFT window.

If, however, δu 6= 0, the residual STO might cause interference in the system. There are several
cases to consider, as the DFT window will either have its starting point inside the CP of the ith
symbol or contain some samples from either the previous symbol i − 1 or from following symbol
i + 1. These cases are denoted Case (i), Case (ii) and Case (iii), respectively, and in the last
two cases, as samples belonging to a different OFDM symbol are included in the DFT window,
inter-symbol interference (ISI) is introduced. Moreover, the misaligned DFT window also causes
self-interference in the form of ICI.

In the following discussions, a single-user scenario transmitting over an additive white Gaussian
noise (AWGN) channel with no quantization (i.e. infinite-resolution) is considered where the
signal-to-noise ratio (SNR) is set very high, such that (2.3) simplifies to

y[n] = ej2πε0n/N x[n+ δ0]. (2.7)

The corresponding frequency-domain symbols is then found via the DFT, i.e. ŷ[k] = DFT{y[n]}.
The purpose of disregarding thermal noise and MU interference is to demonstrate only the effect
of the STO. Since there is only a single user, the subscript u in δu and εu will be temporarily
dropped and lastly, throughout this thesis, δu > 0 will be taken to mean that the DFT window
contains samples from the next symbol.

Case (i): −P < δ < 0

OFDM block:
CP Data

i − 1
CP Data

i

CP Data
i + 1

DFT window

δ

In this case, the starting point of DFT window is taken too early, but still within the part of
the cyclic prefix. We will miss a few samples of the symbol in the end, but as the cyclic prefix
contains the information that we missed, we will able to perfectly reconstruct the transmitted
data. A cyclic shift in the time domain will appear as a linear phase shift for all subcarriers in the
frequency domain, which can be rectified with an equalizer. The effect is depicted in Figure 2.3a.

14



2. System model

We clearly see that there is no added ISI or ICI. In Figure 2.3, the axes are the in-phase (I) and
the quadrature (Q) part of the signal.

Case (ii): −N ≤ δ ≤ −P

OFDM block:
CP Data

i − 1
CP Data

i

CP Data
i + 1

DFT window

δ

In this case, the DFT window will miss δ samples of symbol i and instead get δ samples from the
(i− 1)th symbol. The presence of erroneous samples in the DFT window will cause both ISI and
ICI and since some samples of the desired part are missing, there will be an attenuation of the
desired part of the signal. The misaligned DFT window will also cause a rotation of the received
constellation, depicted in Figure 2.3b.

Case (iii): 0 < δ ≤ N

OFDM block:
CP Data

i − 1
CP Data

i

CP Data
i + 1

DFT window

δ

The effect in this case is in the frequency-flat channel model identical to Case (ii). The situation
is depicted in Figure 2.3c, where δ > 0. The received frequency-domain symbols are rotated and
scaled and from the spreading of the symbols, it is evident that interference has been introduced.

−1 0 1
−1

0

1

I

Q

(a) Case (i), δ = −13

−1 0 1
−1

0

1

I

Q

(b) Case (ii), δ = −18

−1 0 1
−1

0

1

I

Q

(c) Case (iii), δ = 2

Figure 2.3: The effect to a QPSK constellation in some cases of residual STO in a single-user AWGN
channel with high SNR, N = 64, P = 16.

2.3.2 Carrier frequency offset

In order to properly decode a received message, the receiver will need to have accurate knowledge
of the carrier frequency. The data is modulated onto a carrier wave with some frequency fc and

15



2. System model

without knowledge of this parameter, the receiver will not be able to shift the signal down the
baseband. Due in part to imperfect oscillators at the transmitter and receiver, there will be a
difference between the reference clocks in either end. Additionally, if the receiver or transmitter
is moving with respect to the other, the Doppler effect will cause a change to the frequency of
the transmitted signal. The difference in the carrier frequency at the transmitter and receiver is
called CFO. This can have a detrimental effect on our received signal if not mitigated, especially
in systems employing OFDM, where exact knowledge of the center frequency in each sub band
is key to preserve the orthogonality between subcarriers. In the context of OFDM, the CFO is
commonly measured in relation to the subcarrier spacing, ∆f , and we define the CFO associated
with the uth UE as ε0,u = (fc,u − f ′c,u)/∆fu, where f ′c,u denotes the carrier frequency estimate in
the receiver. As in the case of the STO, the CFO will be estimated and compensated before the
DFT operation and we denote the residual CFO after estimation as εu = ε0,u − εest

u .

To explain the effect of different values for the CFO, we begin by defining the set of exponentials
At = {exp(j2πnk/N) : k = 0, 1, . . . , N − 1}. These are each of the complex exponentials
from (2.1) and represent the center frequencies of the sub bands in the transmitted signal.

Next, define another set of exponentials Ar = {exp(−j2πn(k + εu)/N) : k = 0, 1, . . . , N −
1}. These are the center frequencies in transmitted signal, each affected by a phase rotation
proportional to εu. These are what is used to demodulate the received signal and note that while
the demodulated symbols will be rotated if εu 6= 0, we see that if εu ∈ Z, then Ar = Ar. This
means that as long as εu is integer-valued, the received signal will be demodulated with the same
orthogonal set it was modulated with in the transmitter, however in a different order. Since
orthogonality is preserved, no ICI is introduced and given that this integer-value for εu can be
found, it will be possible to perfectly retrieve the transmitted symbols from the received signal.
However, if εu /∈ Z, then At 6= Ar. This introduces ICI and can potentially cause significant
performance degradation.

Along the lines of the previous paragraph, it is common practice to divide the CFO into an integer
part and an fractional part, as their respective impact on the received signal is very different.
In the integer case, the phase difference between two consecutive entries in either At or Ar is
precisely ∆f and we can conclude that the effect of an integer CFO is that the output of the DFT
is cyclically shifted with respect to the transmitted sequence. No ICI has been incurred and given
that we can somehow acquire the integer part of the CFO, we will be able to perfectly reconstruct
the transmitted sequence.

The fractional part of the CFO can, however, be potentially devastating. In the context of OFDM,
it will cause every entry akr ∈ Ar to differ from akt ∈ At, resulting in ICI. In Figure 2.4, we see the
effect of CFO on our received OFDM signal. The SNR is again set very high, so the only visible
effect is that of the CFO. We see that already at εu = 0.1, in Figure 2.4b, a significant amount of
noise has been added. At εu = 0.3, in Figure 2.4c, the effect is so severe that even if εu was known
and the induced rotation compensated for, the CFO will still make it unable to correctly decode
every point. These figures clearly illustrate the importance of accurate CFO estimates and the
effect will be more closely examined in Section 3.

16



2. System model

−1 0 1
−1

0

1

I

Q

(a) ε = 0.02

−1 0 1

−1

0

1

I
Q

(b) ε = 0.1

−1 0 1

−1

0

1

I

Q

(c) ε = 0.3

Figure 2.4: The effect of the CFO on an OFDM symbol with 256 subcarriers, single-user scenario with
AWGN and high SNR.

2.3.3 One-bit quantizers

A one-bit quantizer is a device that will only store the sign of the incoming signal. We will
quantize the real and imaginary part of the received baseband signal separately, as indicated
in Figure 2.1. Obviously, a lot of information is lost in that process. Assuming a single user
transmitting over an AWGN channel to a receiver consisting of only a one-bit quantizer, then, in
every sampling instant, the output of the quantizers would, regardless of the input, be a member
of the set X = γ{±1 ± j}, where γ is some scaling factor. In a single-antenna system, we could
not support any modulation format of higher order than QPSK, as there simply would not be any
additional information to utilize. However, if each transmitted symbol were received on more than
one antenna with independent noise realizations, then each received symbol could potentially be
combined to gain additional insights about the transmitted symbol. The intuition for this can be
obtained by examining Figure 2.5.

In Figure 2.5a and 2.5b, the clouds of blue dots represents a number of realizations of â+ n̂, where
â is the top-left constellation point in a 16-point quadrature amplitude modulation (16-QAM)
constellation (marked with a black square) and n̂ ∼ CN (0, 1/SNR). The plots are zoomed in
so that only the second quadrant is visible. The real and imaginary part of each point is then
quantized separately with a one-bit quantizer described by (2.4). The quantization point is marked
with the yellow diamond and lastly, all quantized points are averaged to form the orange dots.

In Figure 2.5a, we see that almost all of received symbols are in the second quadrant, and as hardly
any received points have crossed into to any quadrant other than the second, almost all points are
quantized to the same quantization point. Consequently, the average of the quantization points is
very close the quantization point itself.

In Figure 2.5b, however, a significant number of the received points show up in the other three
quadrants. The average of the quantized data is therefore shifted towards the origin, making it
possible to distinguish between the transmitted points in Figure 2.5a and 2.5b.

17



2. System model

−1 0

0

1

Re

Im
Received data

Transmitted data

Quantization point

Avg. of quant. data

(a)

−1 0

0

1

Re

Im

Received data

Transmitted data

Quantization point

Avg. of quant. data

(b)

Figure 2.5: Around 500 realizations of â+ n̂, with SNR = 5 dB.

Many realizations of a received symbol with independent noise realizations approximately describes
a multi-antenna system. Note that while the thermal noise on each antenna could plausibly be
modelled as independent, interference sources are usually not uncorrelated over all antennas, so
this description is not completely accurate. However, at least part of the noise associated with
the signal on each receive antenna can be assumed to be independent, so even if correlated noise
sources can not be completely averaged out, multi-antenna systems are able to support high order
constellations, as shown in e.g. [34]. Note that the number of data points in Figure 2.5 is highly
excessive when viewed as individual antenna elements - the illustration is only meant to clearly
demonstrate why one-bit quantization supports higher-order modulation formats.

An observation from Figure 2.5, is that it would have been impossible to resolve the 16-QAM
points if the SNR� 1, as the received symbols in Figure 2.5a and 2.5b would have been quantized
to the same point. This is a perhaps surprising result that implies that for a given set-up, there
will be an optimal level of thermal noise in terms of symbol error rate. This would also means
that the system performance could benefit from the introduction of additional noise. Framed in
the context of dithering, the intuition for this effect may become clear. Improving performance
using dithering is a well-known technique in a number of fields, such as image or audio processing.
It is not uncommon for dithering to be used as a mean to artificially increase the variability of a
data set, which, as demonstrated in Figure 2.5, is critical for the average of the quantizer output
to fall in set with higher cardinality than four.

Lastly, we note that the ability of one-bit quantizers to support higher order modulation for-
mats in a multi-antenna setup, hints that OFDM can be used as a transmission scheme. In the
time-domain, OFDM can be viewed as a higher-order modulation. As demonstrated, these can
be supported by one-bit quantizers if the number of antennas is sufficiently high. This general
conclusion is also valid for a frequency-selective channel, demonstrated in for example [16]. The
topic of low-resolution quantizers and for reading on the topic, see for example [35, 21, 36, 37].

18



2. System model

−1 0 1

−1

0

1

Re

Im

(a) 1 antenna.

−1 0 1

−1

0

1

Re
Im

(b) 8 antennas.

−1 0 1

−1

0

1

Re

Im

(c) 32 antennas.

−1 0 1

−1

0

1

Re

Im

(d) 128 antennas.

Figure 2.6: 32 OFDM symbols with 256 subcarriers transmitted over AWGN and quantized and aver-
aged at the receiver, SNR = 0 dB.

In Figure 2.6, we see the effect of adding antennas to the system. The constellation is gradually
becoming more and more defined and it is interesting to note that even with a modest number of
antennas, such as in Figure 2.6c, the constellation is clearly discernible. Given that the number
of antennas in Massive MIMO is likely to be on the order in 64 or larger [15], we can conclude
that one-bit quantizers could be a viable solution to reduce power consumption whilst retaining
support for higher order constellations.

19



2. System model

20



3
Analysis

Next, we will examine the effects of CFO and STO on a signal that is one-bit quantized. In order
to successfully perform an analysis of the quantized signal, we will need a tool to handle this kind
of nonlinear amplitude distortion. One such tool is Bussgang’s theorem, which will be introduced
and explained in Section 3.1. We will derive the closed-form expression for the received frequency-
domain signal in the presence of STO and CFO and then use Bussgang’s theorem to extend the
analysis to the one-bit case.

3.1 Bussgang’s theorem

A useful tools in the analysis of these kinds of system is Bussgang’s theorem, named after Julian J.
Bussgang who published it in 1952 [38]. In its original formulation, it states that for two Gaussian
signals, their cross-correlation will be the same up to a scaling before and after one of the signals has
undergone a nonlinear amplitude distortion. This situation is depicted in Figure 3.1a, where x(t)
and y(t) are two (generally) complex Gaussian signals. We let z(t) = D [y(t)] be some nonlinear
amplitude distortion and write

E [z(t)x∗(t)] = g E [y(t)x∗(t)] (3.1)

where g ∈ C is some scaling factor.

D

x(t)

y(t) z(t)

(a) The general statement.

Q
... ...

Q

y1[n] r1[n]

yB[n] rB[n]

(b) Our quantizer context.

Figure 3.1: A visualization of Bussgang’s theorem

In order to facilitate the analysis of the effect of one-bit quantization in a communication system,
we can frame Bussgang’s theorem as in Figure 3.1b. In each timing instant n, the B receive

21



3. Analysis

antennas are sampled and the values collected in the vector y[n] ∈ CB. Each entry yb[n] of the
vector y[n] is then put through the quantizer Q, characterized by (2.4). Collecting all outputs
rb[n] = Q(yb[n]) in the vector r[n] ∈ CB, we can write (3.1) in a vectorized form as

E
[
r[n]yH [n]

]
= G E

[
y[n]yH [n]

]
(3.2)

where the gain G is a B × B diagonal matrix. From (3.2), it follows that r[n] = Gy[n] + d[n],
where d[n] ∈ CB, if d[n] is uncorrelated with y[n], i.e. E

[
y[n]dH [n]

]
= 0. To verify, we can

examine a single entry rb[n] in r[n] and substitute on the left-hand side in (3.2). Then,

E [([G]b,b yb[n] + db[n])y∗b [n]] = [G]b,b E [yb[n]y∗b [n]]
E [[G]b,b yb[n]y∗b [n] + y∗b [n]db[n]] = [G]b,b E [yb[n]y∗b [n]]

[G]b,b E [yb[n]y∗b [n]] + E [y∗b [n]db[n]] = [G]b,b E [yb[n]y∗b [n]]

where the last step follows as the expectation E[·] is a linear operator. We see that for the above
to hold, the second term on the left-hand side must be equal to zero. Consequently, db[n] must be
uncorrelated with yb[n]. Consequently, we can indeed write

r[n] = Q(y[n]) = Gy[n] + d[n]. (3.3)

This formulation will become useful when we investigate the effects on the quantized signal in
the presence of STO and CFO. The term d[n] can be viewed as a distortion and captures the
adverse effects of the quantizer. Interstingly, while the power of the distortion caused by the
one-bit quantizers can be significant in relation to the signal power, illustrated in Figure 1.1d,
studies such as [39, 40, 16] has shown that 1) the performance loss is not necessarily as severe as
one might intuitively think and 2) only a few bits are required to make the gap within fractions
of a dB for low SNRs.

Next, we examine the gain G. A general expression where no particular constraints are placed
on the nonlinear amplitude distortion D(·) can be found in [41], but let us derive the gain in the
special case of a one-bit ADC. As mentioned previously G is a diagonal matrix and we will find
the expression for each diagonal entry [G]b,b. For notational clarity, we will here drop the index n
and consider a single point in time.

From Equation (3.2), we have

[G]b,b = E [rby∗b ]
E [yby∗b ]

. (3.4)

We can write the numerator of Equation (3.4) as

22



3. Analysis

E [rby∗b ] = E
[(

Re(rb) + jIm(rb)
)
·
(
Re(yb)− jIm(yb)

)]
= E [Re(rb) Re(yb)]− jE [Re(rb) Im(yb)] + jE [Im(rb) Re(yb)] + E [Im(rb) Im(yb)] .

Given that yb is circularly symmetric complex Gaussian variable, the real and imaginary part
are uncorrelated, i.e. E[Re(yb) Im(yb)] = 0. From (2.4), note that the real part of rb has no
relation to the imaginary part of yb, and equivalently for the relation between Im(rb) and Re(rb).
Consequently, as both the real and imaginary parts are zero-mean, we have

E [rby∗b ] = E [Re(zb) Re(yb)] + E [Im(zb) Im(yb)] .

Since the real and imaginary part of yb and zb are identically distributed, we have

E [rby∗b ] = 2 E [Re(rb) Re(yb)]

= 2
∫
yb,R

1√
2πσ2

yb,R

1√
2

sgn[yb,R] yb,R e−y
2
b,R/(2σ

2
yb,R

) dyb,R (3.5)

where we have written Re(y) as yb,R for readability. Denoting E [yby∗b ] as σ2
yb
, we can express the

distribution of the new variable yb,R as CN (0, σ2
yb
/2). Performing the substitution σ2

yb,R
= σ2

yb
/2

and then inserting (3.5) into (3.4) , we find the gain as

[G]b,b =
√

2
π

1
σ2
yb

√
σ2
yb

2
∫ ∞

0
yb,R e

−
y2
b,R

σ2
yb dyb,R

=
√

2
π

1
σ2
yb

√
σ2
yb

2
σ2
yb

2

=
√

2
π

1
σyb

. (3.6)

From (3.6), we see that the gain depends on the second-order statistics of the received signal y[n].
Consequently, the final expression for the gain G will vary with the assumed channel model and
for a frequency-flat channel model and symbol-rate sampling, we have, writing (2.3) in vector
notation,

23



3. Analysis

G =
√

2
π

(
diag

(
E
[
y[n]yH [n]

]))−1/2
(3.7)

=
√

2
π

diag

HE
[
x[n]xH [n]

]
︸ ︷︷ ︸

=IU

HH + E
[
w[n]wH [n]

]
︸ ︷︷ ︸

=IB ·N0



−1/2

=
√

2
π

(
diag

(
HHH + IB ·N0

))−1/2
(3.8)

where w[n] ∈ CB, IU and IB represents the identity matrix of size U and B, respectively and
diag(·) forms a new diagonal matrix from the main diagonal of a input matrix. The cross-terms
in (3.7) are zero as x[n] are w[n] are independent and the noise w[n] is zero-mean.

Lastly, note that the validity of Bussgang’s theorem hinges on the fact that the quantizer input,
y[n] ∈ CB, is a Gaussian signal. Using the OFDM signalling scheme, this condition can generally
be regarded as fulfilled.

3.2 Received signal due to STO and CFO

As mentioned in Section 2.3, the STO will impact the system differently depending on if the DFT
window is placed too late or too early. Additionally, the CFO can potentially place significant
limitations on the system performance. Our main goal is to find analytic expressions for how the
SINDR is affected due to these impairments, as well as the effect of one-bit quantization.

To be able to express the received signal at the bth antenna in a MU-MIMO uplink system
with B receive antennas and U UEs with STO and CFO, we begin by defining the set N =
{0, 1, . . . , N − 1}, i.e. all time-domain samples of an OFDM symbol. Next, we define the set S to
be some subset of N , i.e. S ∪ Sc = N . Given that the UEs are continuously transmitting OFDM
symbols, we use (2.3) and express the ith received symbol at the bth BS antenna as

y
(i)
b [n] =

U∑
u=1

hb,uej2πεun/N
(
x(i)
u [(n+ δu)N ] 1{Su}[n]

+ x(i−1)
u [n+N + P + δu] 1{Scu∧δu<−P}[n] + x(i+1)

u [n] 1{Scu∧δu>0}[n]
)

+ wb[n] , n = 0, 1, . . . , N − 1 (3.9)

where the (·)N defines the modulo operator, i.e. (n)N = n mod N . Further,

24



3. Analysis

Su =
{
{0, 1, . . . , N − δu − 1}, 0 < δu < N
{du, du + 1, . . . , N − 1}, −N < δu ≤ 0 (3.10)

is the set of samples corresponding to the ith transmitted symbol from the uth UE that are received
during the ith DFT window, and

du = max(−P − δu, 0). (3.11)

The parameter du captures the number of samples missed if δu < −P . As mentioned in Section 2.3,
as long as the starting point is taken within the cyclic prefix, no ISI or ICI will be incurred. To
keep track of this distinction, we use the parameter du. Lastly, as mentioned in 2.3, we will assume
that the frame detection preceding the DFT block has limited the STO for the ith symbol to ±N .
In fact, as any larger STO than ±N would mean that no samples of the ith symbol is present in
the ith DFT window, it not make much sense pursing an SINDR expression in that case.

As discussed in Section 2.3, the main causes of CFO are a mismatch between the oscillators in
the transmitter and receiver, as well as the Doppler effect. We can assume that a single clock is
driving all analog components in the receiver, but, naturally, this assumption would not be valid
for the transmitting side in a multi-user scenario. Further, we have no reason to assume that all
users move at roughly the same speed, meaning that all users have a unique Doppler shift relative
to the receiver. Consequently, we must separate the CFO for each user. Similarly, as we have no
reason to assume that the users are transmitting in a synchronized fashion equidistantly from the
receiver, we also must separate the STO for each user.

As noted in the previous section, the gain [G]b,b from (3.3) will depend on the second-order
statistics of the input, so in order to compute the analytic expression for the received signal, we
need to determine y(i)

b [n].

3.2.1 Infinite-precision case

In (3.9), we have used indicator functions to deal with STO larger than or less than zero. We will
treat these two cases separately, as they will cause either samples from the next symbol or from
the previous one to leak into the current block. In Figure 3.2, the two cases are shown. We will
begin with Figure 3.2b.

Case 0 < δu < N :

If the STO is positive, the DFT window will no longer include the complete symbol. We will
sample into the cyclic prefix of the next symbol and given a timing offset δu, we will miss the
δu − 1 first samples of the ith symbol and instead get first δu symbols of the following OFDM
symbol, i+ 1. Then, the DFT, defined by (2.2), of (3.9) reduces to

25



3. Analysis

D
ata CP . . .

δu

du

DFT window

(a) δu < 0.

CP
D

ata. . .

δu

DFT window

(b) δu > 0.

Figure 3.2: Illustration of δu > 0 and δu < 0.

ŷ
(i)
b [k] =

U∑
u=1

(
hb,u

1√
N

N−δu−1∑
n=0

ej2πεun/Nx(i)
u [(n+ δu)N ] e−j2πnk/N

+ hb,u
1√
N

N−1∑
n=N−δu

ej2πεun/N x(i+1)
u [n] e−j2πnk/N

)
︸ ︷︷ ︸

=îISI, right
b,u

[k]

+ 1√
N

N−1∑
n=0

wb[n] e−j2πnk/N . (3.12)

We note that the last term of (3.12) is the DFT of the thermal noise, ŵb[k] = DFT{wb[n]}. The
term îISI, right

b,u [k] is the ISI caused by the (i+ 1)th symbol during the ith DFT window.

With these definitions, we now have

ŷ
(i)
b [k] =

U∑
u=1

hb,u 1√
N

N−δu−1∑
n=0

ej2πεun/Nx(i)
u [(n+ δu)N ] e−j2πnk/N + îISI, right

b,u [k]
+ ŵb[k]. (3.13)

To continue, we write x(i)
u [n] in (3.13) in the frequency domain using (2.2). Simplifying the

exponents and rearranging the sums, we get

ŷ
(i)
b [k] =

U∑
u=1

hb,u
N−1∑
k′=0

x̂(i)
u [k′]ej2πδuk′/N 1

N

N−δu−1∑
n=0

ej2π(k′−k+εu)n/N

︸ ︷︷ ︸
=ψ[k,k′]

+îISI
b,u[k]

+ ŵb[k]. (3.14)

Now, we focus on ψ[k, k′]. Using the well-known formula for the geometric progression

26



3. Analysis

N−1∑
n=0

an = 1− aN
1− a (3.15)

we note that ψ[k, k′] bears a strong resemblance to the Euler formula for sine, i.e. sin(x) =
(exp(jx) − exp(−jx))/(2j). Multiplying the numerator and denominator with the appropriate
values, we get

ψ[k, k′] = ej πN (k′−k+εu)(N−δu−1) sin
(
π
N

(k′ − k + εu)(N − δu)
)

N · sin
(
π
N

(k′ − k + εu)
) . (3.16)

Now, consider the case when k′ = k, i.e. the subcarrier index of interest. Defining

g(α, β) =
sin

(
π
N

(N − α)β
)

N · sin(πβ
N

)
ej πN (N−α−1)β, (3.17)

note that when k′ = k, then, from (3.16), ψ[k, k] = g(δu, εu). This term captures the attenuation
caused by STO, as well as the attenuation and phase shift to due to CFO. Using (3.17), we can
now write (3.14) as

ŷ
(i)
b [k] =

U∑
u=1

(
hb,u g(δu, εu) ej2πδuk/N x̂(i)

u [k]

+ hb,u
N−1∑
k′=0
k′ 6=k

sin
(
π
N

(N − δu)(k′ − k + εu)
)

N · sin( π
N

(k′ − k + εu))
ej πN (N−δu−1)(k′−k+εu) ej2πδuk′/N x̂(i)

u [k′]

︸ ︷︷ ︸
=φ[k]

+ îISI, right
b,u [k]

)
+ ŵb[k]. (3.18)

Now, we perform the variable change k′′ = k − k′ and write φ[k] as

φ[k] =
N−1∑
k′′=1

sin
(
π
N

(N − δu)(εu − k′′)
)

N · sin
(
π
N

(εu − k′′)
) ej πN (N−δu−1)(εu−k′′) ej2 πN (k−k′′)δux̂(i)

u [(k − k′′)N ] . (3.19)

The first two factors of (3.19) is precisely g(δu, εu − k′′), so finally, (3.18) becomes

27



3. Analysis

ŷ
(i)
b [k] =

U∑
u=1

(
hb,u g(δu, εu) ej2πδuk/N x̂(i)

u [k]

+
N−1∑
k′′=1

hb,u g(δu, εu − k′′) ej2πδu(k−k′′)/N x̂(i)
u [(k − k′′)N ]︸ ︷︷ ︸

=îICI, right
b,u

[k]

+îISI, right
b,u [k]

)
+ ŵb[k]. (3.20)

Looking at the terms of (3.20), we see the expected ICI terms due to loss of subcarrier orthogonality
as well as ISI due to the next symbol.

Case −P < δu < 0:

Looking at Figure 3.2a, we see that in a flat-fading channel model, there will be no delay spread
caused by the preceding symbol that will distort the cyclic prefix of the current symbol. This
means that as long as −P ≤ δu ≤ 0, there will be no interference from the previous symbol. As
mentioned in Section 2.3, the only effect will be a rotation of the received constellation, as the
orthogonality between the subcarriers is not affected. If δu is in this interval, then du in (3.11) will
be zero and from (3.9), we have

y
(i)
b [n] =

U∑
u=1

hb,u ej2πεun/N x(i)
u [(n+ δu)N ] + wb[n]. (3.21)

Applying a DFT to (3.21) and proceeding in a similar manner as in the previous case, we arrive
at

ŷ
(i)
b [k] =

U∑
u=1

hb,u g(0, εu) ej2πδuk/N x̂(i)
u [k] + ŵb[k]. (3.22)

We see that the only effect from the STO is a rotation of the received symbol.

Case −N < δu < −P :

Here, we will miss the first du samples of symbol i and instead get the last du samples of symbol
i− 1. This means that we will get ISI. Applying the DFT to (3.9), we get

28



3. Analysis

ŷ
(i)
b [k] =

U∑
u=1

(
hb,u

1√
N

N−1∑
n=du

ej2πεun/Nx(i)
u [(n+ δu)N ] e−j2πnk/N

+ hb,u
1√
N

du−1∑
n=0

ej2πεun/N x(i−1)
u [n+N − du] e−j2πnk/N︸ ︷︷ ︸

=îISI, left
b,u

[k]

)

+ 1√
N

N−1∑
n=0

wb[n] e−j2πnk/N . (3.23)

Using the same steps in the case of 0 < δu < N , we arrive at

ŷ
(i)
b [k] =

U∑
u=1

hb,u
N−1∑
k′=0

x̂(i)
u [k′]ej2πδuk′/N 1

N

N−1∑
n=du

ej2π(k′−k+εu)n/N

︸ ︷︷ ︸
=ξ[k,k′]

+îISI, left
b,u [k]

+ ŵb[k]. (3.24)

Again, we focus at the innermost sum over n and write

ξ[k, k′] = 1
N

N−1∑
n=du

ej2π(k′−k+εu)n/N

= 1
N

N−1∑
n=0

ej2π(k′−k+εu)n/N −
du−1∑
n=0

ej2π(k′−k+εu)n/N


= 1
N

(
1− ej2π(k′−k+εu)

1− ej2π(k′−k+εu)/N −
1− ej2π(k′−k+εu)du/N

1− ej2π(k′−k+εu)/N

)

= 1
N
· ej2π(k′−k+εu)du/N − ej2π(k′−k+εu)

1− e2π(k′−k+εu)/N . (3.25)

Multiplying (3.25) with the appropriate factors, we can again use Euler’s formula for sine to find

ξ[k, k′] =
sin

(
π
N

(k′ − k + εu)(N − du)
)

N · sin
(
π
N

(k′ − k + εu)
) ejπ(k′−k+εu)(N−du−1)/N e−j2π(k′−k+εu)du/N . (3.26)

For the case k′ = k, (3.26) simplifies to

ξ[k, k] = g(du, εu)e−j2πεudu/N . (3.27)

29



3. Analysis

Using the same change of variable k′′ = k − k′ as before, we can finally write (3.23) as

ŷ
(i)
b [k] =

U∑
u=1

(
hb,u g(du, εu) ej2πδuk/N e−j2πεuδu/N x̂(i)

u [k]

+
N−1∑
k′′=1

hb,u g(du, εu − k′′) ej2πδu(k−k′′)/N e−j2π(k′′+εu)du/N x̂(i)
u [(k − k′′)N ]︸ ︷︷ ︸

=îICI, left
b,u

[k]

+îISI, left
b,u [k]

)
+ ŵb[k]. (3.28)

Comparing Equation (3.20) and (3.28), we note that they are highly similar. Again, we get ISI
due to the misaligned window, as well as ICI since the subcarrier orthogonality is not preserved.
This is to be expected, as the effect in either direction largely amounts to the same thing.

To verify our calculations, we examine the expression when either δu or εu is set to zero. Beginning
with no CFO, we get

g(δu, 0) = N − δu
N

using l’Hospital’s rule. For g(δu,−k′′), we get

g(δu,−k′′) = 1
N
· 1− e−j2πk′′ej2πδuk′′/N

ejπk′′/N − e−jπk′′/N ejπk′′/N

= 1
N
· 1− ej2πδuk′′/N

1− ej2πk′′/N

where we used that ej2πk′′ = 1, since k′′ ∈ Z.

With these results, (3.20) reduces to

ŷ
(i)
b [k] =

U∑
u=1

(
N − δu
N

hb,uej2πδuk/N hb,u x̂
(i)
u [k]

+ 1
N

N−1∑
k′′=1

hb,u
1− ej2πk′′δu/N
1− ej2πk′′/N ej2πδu(k−k′′)/N x̂(i)

u [(k − k′′)N ]

+ îISI, right
b,u [k]

)
+ ŵb[k]. (3.29)

This is the same result stated in for example [42, cf. (1)]. Similarly, (3.28) reduces to [42, cf. (3)]
if εu = 0.

30



3. Analysis

Setting δu (or du) to zero, we get,

g(0, εu) = sin(πεu)
N · sin(πεu

N
) ejπεu(N−1)/N

Using this, we can reduce (3.20) and (3.28) to

ŷ
(i)
b [k] =

U∑
u=1

( sin(πεu)
N · sin(πεu

N
) ejπεu(N−1)/N x̂(i)

u [k]

+
N−1∑
k′′=1

hb,u
sin(π(εu − k′′))

N · sin( π
N

(εu − k′′))
ejπ(εu−k′′)(N−1)/N x̂(i)

u [(k − k′′)N ]
)

+ ŵb[k]. (3.30)

The expression (3.30) can be compared to for example [32], where a similar expression was found
for the case hb,u = 1.

3.2.2 One-bit quantization

Now that we have found the expressions for yb[n] and ŷb[k], we are ready to continue with the
one-bit quantizer.

From (3.6), we see that we need to determine the power of the received signal on each antenna,
σ2
yb
. Equivalently, we can examine the power of ŷb[k] = DFT{yb[n]}, given the DFT definition

in (2.2). Moreover, in this particular setting, it is trivial to see that

r̂
(i)
b [k] = DFT{r(i)

b [n]}
= [G]b,bŷ(i)

b [k] + d̂b[k] (3.31)

where d̂b[k] = DFT{db[n]} and [G]b,b is given by (3.6). From (3.20) and (3.28), we see that the
signal ŷ(i)

b [k] on antenna b after the DFT has the following form

ŷ
(i)
b [k] = ŝb,u[k] + îISI

b,u[k] + îICI
b,u [k] +

U∑
u′=1
u′ 6=u

ŝb,u′ [k] + îISI
b,u′ [k] + îICI

b,u′ [k]

︸ ︷︷ ︸
îMUI
b,u

[k]

+ŵb[k] (3.32)

31



3. Analysis

where

îISI
b,u[k] =

î
ISI, right
b,u [k], 0 < δu < N

îISI, left
b,u [k], −N < δu < −P

(3.33)

and

îICI
b,u [k] =

î
ICI, right
b,u [k], 0 < δu < N

îICI, left
b,u [k], −N < δu < −P.

(3.34)

Furthermore, ŝb,u[k] defines the part of the received signal from the uth UE, i.e.

ŝb,u[k] = hb,u γ(δu, εu) x̂(i)
u [k] Ψ[δu] (3.35)

where

γ(δu, εu) =
g(δu, εu), 0 < δu < N

g(du, εu), −N ≤ δu ≤ 0
(3.36)

and

Ψ[δu] =
ej2πδuk/N , −P ≤ δu < N

ej2πδuk/N e−j2πεuδu/N , −N < δu < −P.
(3.37)

In (3.32), îISI
b,u[k] and îICI

b,u [k] represents the interference, stemming from the uth UE on the bth
antenna, caused by STO and CFO. In addition to these interferences, there is also MU interference,
denoted with îMUI

b,u [k], as well as thermal noise, ŵb[k].

3.2.3 ZF equalization

Inserting (3.32) into (3.31), and inserting (3.31) into (2.5), we get that the estimated frequency-
domain symbol on from the uth UE during the ith DFT window, which can be written as

x̂est,(i)
u [k] = aTuGhuγ(δu, εu) x̂(i)

u [k]Ψ[δu]
+ aTuGîISI

u [k] + aTuGîICI
u [k] + aTuGîMUI

u [k]
+ aTu d̂[k] + aTuGŵ[k] (3.38)

32



3. Analysis

where au = [au,1, au,2, . . . , au,B]T is the uth row of the ZF equalization matrix A defined in (2.6)
and hu = [h1,u, h2,u, . . . , hB,u]T is the uth column of the channel matrix H. We have fur-
ther defined îISI

u [k] =
[̂
iISI
1,u[k], îISI

2,u[k], . . . , îISI
B,u[k]

]T
, îICI

u [k] =
[̂
iICI
1,u [k], îICI

2,u [k], . . . , îICI
B,u[k]

]T
, îMUI

u [k] =[̂
iMUI
1,u [k], îMUI

2,u [k], . . . , îMUI
B,u [k]

]T
, d̂[k] =

[
d̂1[k], d̂2[k], . . . , d̂B[k]

]T
, and ŵ[k] = [ŵ1[k], ŵ2[k], . . . , ŵB[k]]T .

From (3.38), we find that the SINDR at the uth UE can be written as

SINDRu = |γ(δu, εu)|2|aTuGhu|2
I ISI
u + I ICI

u + IMUI
u + aTuE

[
d̂[k]d̂H [k]

]
a∗u +N0aTuGGHa∗u

(3.39)

since the interference sources, namely the ISI, ICI, and MU interference, quantization distortion,
and thermal noise are uncorrelated. The numerator in (3.39) is the power of the received signal
from the uth UE, i.e. the first term in (3.38). For the terms in the denominator, we will now
examine them individually.

Power of IISI
u :

To compute the power in the ISI term, we need to distinguish between two cases. If 0 < δ < N ,
the DFT window includes δu samples of the next symbol. If −N < δu < −P , du, the DFT
window includes du samples of the previous symbol. In either case, the power of this term will
be a fraction of the power of a full symbol, directly tied to the STO. The power of a full symbol
simply |aTuGhu|2, so we find

I ISI
u = E

[
|̂iISI
u [k]|2

]
=
{

δu
N
|aTuGhu|2, δu > 0

du
N
|aTuGhu|2, δu ≤ 0. (3.40)

Power of IICI
u :

Starting from (3.20),

E
[
|aTuGîICI, right

u [k]|2
]

= |aTuGhu|2
N−1∑
k′=1

N−1∑
k′′=1

g(δu, εu − k′) g∗(δu, εu − k′′)

· E
[
x̂(i)
u [(k − k′)N ]x̂∗,(i)u [(k − k′′)N ]

]
︸ ︷︷ ︸
=

1, k′ = k′′

0, k′ 6= k′′
, since symbol-sampling

= |aTuGhu|2
N−1∑
k′=1

[
sin( π

N
(N − δu)(εu − k′))

N · sin( π
N

(εu − k′))

]2

= |aTuGhu|2
N−1∑
k′=0

[
sin( π

N
(N − δu)(εu − k′))

N · sin( π
N

(εu − k′))

]2

−
[

sin( π
N

(N − δu)εu)
N · sin(πεu

N
)

]2
 .

(3.41)

33



3. Analysis

In (3.41), note that the second term is precisely g|(δu, εu)|2. For the first term, we find

N−1∑
k′=0

[
sin( π

N
(N − δu)(εu − k′))

N · sin( π
N

(εu − k′))

]2

= N − δu
N

. (3.42)

Hence,

E
[
|aTuGîICI, right

u [k]|2
]

= |aTuGhu|2
[
N − δu
N

− |g(δu, εu)|2
]
. (3.43)

Starting from (3.28) instead, we obtain

E
[
|aTuGîICI, left

u [k]|2
]

= |aTuGhu|2
[
N − du
N

− |g(du, εu)|2
]
. (3.44)

Examining (3.43) and (3.44), we note that we can use γ(δu, εu) defined in (3.36) to write

I ICI
u = E

[
|aTuGîICI

u [k]|2
]

= |aTuGhu|2
[
γ(δu, 0)− |γ(δu, εu)|2

]
. (3.45)

Power of IMUI
u :

As the transmitted signal is normalized, the multi-user interference will only depend on the power
in the channel, i.e.

IMUI
u = E

[
|aTuGîMUI

u |2
]

=
U∑

u′=1
u′ 6=u

|aTuGhu|2. (3.46)

Power of quantization distortion:

To find the power of the quantization distortion, we look at (3.3). This describes the quantizer
input-output relationship. Rearranging, we find

E
[
d[k]dH [k]

]
= E

[
r[k]rH [k]

]
︸ ︷︷ ︸

=Crr

−GE
[
y[k]yH [k]

]
︸ ︷︷ ︸

=Cyy

G (3.47)

where Crr and Cyy are used to denote the covariance matrix of the output and inputs to the
quantizer, respectively. The covariance of the input in time-domain was given in (3.8). Computing

34



3. Analysis

the covariance matrix Crr is in the special case of one-bit quantizers fairly straightforward, as we
can resort to the arcsine law [43]. Using this law, we find

Crr = 2
π

(
arcsin

(
Dy
−1/2Re(Cyy)Dy

−1/2
)

+ j arcsin
(
Dy
−1/2Im(Cyy)Dy

−1/2
))

(3.48)

where Dy refers to the diagonal elements of Cyy, i.e. Dy = diag (Cyy). Consequently, we can
write the power of the distortion term as

E
[
d[k]dH [k]

]
= Crr −GCyyG. (3.49)

In (3.39), we have the quantization distortion in the frequency-domain. However, due to Parseval’s
theorem, we can examine the power in either the time- or frequency-domain if the DFT operation
is defined as in (2.2). Hence, we can use (3.49) to express the power of the quantization distortion.

Lastly, the power of the noise is found as

E
[
(aTuGŵ[k])(aTuGŵ[k])H

]
= aTuGE

[
ŵ[k]ŵ[k]H

]
Ga∗u

= N0aTuGGHa∗u (3.50)

where the second equality follows as the noise is white and zero mean.

Inserting the computed power terms (3.40), (3.45) and (3.46) into (3.39), we find a closed-form
expression for the SINDR for a given STO and CFO.

35



3. Analysis

36



4
Synchronization in OFDM systems

From the analysis in Section 3, it is clear that the effects of both STO and CFO can potentially
be quite severe in the context of OFDM. The inherent sensitivity to these errors is something that
has been known for a long time. For example in [44], it was demonstrated that OFDM is several
orders of magnitude more sensitive to frequency offsets than single-carrier modulation. This is
not particularly surprising, as orthogonality between the subchannels is instantly destroyed if the
received signal is demodulated with the wrong frequencies. Consequently, it is critical to devise
strategies to estimate these offsets as accurately as possible.

Early papers resembling the methods of today in this field, such as [45, 46, 47], demonstrated
different approaches to both temporally locate the signal, as well as estimating its carrier frequency.
A ground-breaking paper in the area of synchronization for OFDM was presented by in 1997
Schmidl and Cox in [48], even if the fundamental ideas are presented already in 1996 [49]. They
expanded on the method in [46] and their work found its way into several standards, such as
the wireless local area network (WLAN) standard 802.11a [50]. This method will be examined
in detail and an extension developed in [51], as well as a method relying on the cyclic prefix
presented in [52, 53] and, lastly, a more modern method reminiscent of how synchronization is
achieved in the current generation of mobile systems. Before delving into the different methods of
synchronization, we will take a brief look at the history of synchronization itself.

4.1 Background

At the most fundamental level, timing synchronization requires the localization of a specific se-
quence within a window of samples. For example, in a system transmitting raw bits, you could
devise a system where every payload is preceded by a known sequence. If we could accurately
locate the position of the synchronization sequence, we would also have located the payload. Even
though a system transmitting raw bits might seem far from the OFDM system considered in this
thesis, it is a good starting point in order to say something general about synchronization.

The natural questions to ask in this setting would be how to go about finding a known sequence
within a window and, next, whether the design of the synchronization sequence has any influence
on our ability to locate it. Going further, it could also be questioned whether the synchronization

37



4. Synchronization in OFDM systems

sequence needs to be explicitly known at the receiver.

Regarding how to determine the frequency offset of a signal, it hinges on accurate estimates of the
temporal position of a sequence. Say for example that a number of known symbols are transmitted
and then correctly located on the receiving side. By comparing the phase differences of the symbols
in the known sequence on the transmitting and receiving sides, we are able to extract information
about the frequency offset. As the timing estimation is key to extract information of a possible
frequency offset, the following discussion will focus on the timing aspect.

Optimally locating a sequence

To the first question, the standard method was (and is) to find the peak in the correlation spectrum
produced when correlating the window of samples with the known sequence. Whether or not this
is optimal and how the nature of the samples in which the sequence is embedded might have an
influence, was not definitely settled until [54]. The set-up considered is depicted in Figure 4.1a,
where the top part shows the signal x ∈ XN ,X = {−1, 1} to be transmitted over an AWGN
channel.

0 20 40 60 80 100

−1

−0.5

0

0.5

1

x

p

0 20 40 60 80 100

−2

0

2

Samples

, y

(a) The context in [54], SNR = 0 dB.

0 10 20 30 40 50 60 70
−10

0

10

S1

S2

arg maxd S1

δ

0 10 20 30 40 50 60 70

−20

−10

0

d

S1 − S2

arg maxd (S1 − S2 )

δ

(b) The output of the correlation and the
optimum rule.

Figure 4.1: The optimum rule for detecting a known sequence under the influence of AWGN.

In x, a known sequence p of length L < N is embedded, beginning at an unknown location δ.
The sequence x is then subjected to noise, forming y = x + w, where w ∼ N (0, N0/2). This is
depicted in the bottom part of 4.1a and the task is to precisely locate known sequence. Starting
from these assumption, Massey derived the optimal rule for the estimation of the position δest as

38



4. Synchronization in OFDM systems

δest = arg max
d


L−1∑
m=0

p[m]y[m+ d]︸ ︷︷ ︸
S1[d]

−
L−1∑
m=0

f(y[m+ d])︸ ︷︷ ︸
S2[d]

 (4.1)

We see that the first term S1 of (4.1) is indeed a correlation term, in line with the expectations
at the time. In addition to the correlation term, there is also a second term S2 present. This a
correction term that accounts for the power in the received signal y. With the assumption that a
sequence of ±1s given by x is transmitted, S2 is depicted in the top part of Figure 4.1b, together
with S1. We see that if we only maximize S1, which will be referred to as the correlation rule,
it will produce an erroneous decision regarding the true position of the synchronization sequence,
as the peak at δest = 35 is greater than the true position δ = 63. However, when adding the
correction term, as in the bottom part of Figure 4.1b, the correct location is indeed found.

In Figure 4.2a, the probability of δest 6= δ is depicted. We see that in the SNR region near 0 dB,
Massey’s optimum rule defined by (4.1) provides an approximate 3 dB gain over the correlation
rule and around twice that a few dBs higher. Figure 4.2a also depicts a high SNR approximation
for the correction term, which has a very similar performance as the optimum rule. In this setting,
the optimal Sopt

2 has the form

Sopt
2 [d] = N0

2
√

2

L−1∑
m=0

ln
(
cosh

(√
2/N0 y[m+ d]

))
(4.2)

The computational complexity of (4.2) in apparent, involving both the ln(·) and cosh(·) functions.
The high SNR approximation, on the other hand, takes the form

Sapprox.
2 [d] =

L−1∑
m=0

|y[m+ d]| (4.3)

Consequently, it requires only L(N − L) additions.

It is quite fascinating that the high SNR approximation of the correction term seems to yield
comparable performance to when using the optimum Sopt

2 . This was noted already in Massey’s
original paper, and conclusively confirmed after extensive simulations in [55]. Going further on
the topic of simplifications of the optimal rule, it has been argued in for example [56], that as
the received signal in a real system is always constrained to some fixed interval, owing to the
AGC/ADC, the correction term will vary very little. Hence, it can in most practical cases be
disregarded. As we will see in for example Section 4.3, a variant of the correlation rule without
calculating any correction term is used to find the STO.

Another interesting fact regarding the optimum rule is that it becomes the correlation rule in the
case of one-bit quantization, depicted in Figure 4.2b. Here, we form r = Q(y) = ±1/

√
(2). In

39



4. Synchronization in OFDM systems

−10 −5 0 5 10
10−3

10−2

10−1

100

SNR [dB]

Pr
ob

ab
ili

ty
of

δes
t

6=
δ

[%
]

Correlation rule

Optimum rule

High SNR approx. rule

(a) The probability of δest 6= δ.

−10 −5 0 5 10
10−3

10−2

10−1

100

SNR [dB]

Pr
ob

ab
ili

ty
of

δes
t

6=
δ

[%
]

Correlation rule

Optimum rule

(b) The error probability with one-bit
quantization.

Figure 4.2: The probabilities of an erroneous decision regarding the true position δ of the synchronization
sequence. Results obtained via simulation the detection of a length 13 Barker sequence over the AWGN
channel 105 times.

this particular setting, it can be understood from the fact that as cosh(·) from (4.2) is an even
function and as r = ±1, Sopt

2 will be the same for all values of d. An explanation on a more
intuitive level is to consider the purpose of the correction term. It is intended to take the data
surrounding the synchronization sequence into account weaken peaks in the correlation spectrum
due to particularly powerful input. In the case of a one-bit quantizer, the power of the input to
the correlator is constant, so there is no need to consider this effect.

While the first example given by Massey is the one depicted above, it was also shown that if the
surrounding data is instead distributed as zero-mean Gaussian random variables, the optimum
rule has precisely the same form as (4.1). Further, Massey demonstrated the optimum rule in
an AWGN channel with essentially BPSK signaling and interestingly, subsequent studies in more
complex settings have arrived at similar conclusions. In [57], it was shown that the optimum rule
for location a synchronization sequence with an M -ary signalling scheme over the AWGN channel
consisted of essentially the same components as (4.1) from [54]. They were also able to again
demonstrate that the high SNR approximation performs very close to the optimum rule. Going
further, AWGN channels with ISI was considered in [58], flat fading channels in [59] and frequency-
selective channels in [60], to name some of the papers in this field. Similar conclusions was also
reached by [53] and [46] under different sets of assumptions, where the maximum likelihood (ML)
rules again and again was showed to involve a correlation in one way or another. All of these
results have decisively demonstrated that the core principle of that finding the maximum in a
correlation spectrum is central to timing synchronization.

40



4. Synchronization in OFDM systems

Design of synchronization sequences

Regarding the design of synchronization sequence, it is intuitively clear that there are at least some
designs which are outright unusable. Given that the task is to locate a known sequence somewhere
within a received signal, we can generally say that the sequence must be unique. This means that
if our synchronization sequence where to appear by chance somewhere else in the window, we will
not be able to uniquely determine its position. However, if the sequence does appear twice in the
received signal and the distance between the repetitions is known, we can use that information to
temporally locate the data. We will soon return to this idea and for now focus on finding a single,
known sequence within a window of received samples.

To avoid an instance where our synchronization sequence appears more than once by chance rather
than by design, the sequence must be of sufficient length so that the probability of it showing up as
a result of a stochastic process is sufficiently small. Further, as we established that the correlator
is the optimum metric to maximize, we would like the sequence to have a peak at zero lag and,
preferably, zero at all other timing instants. Due to the implementation specifics, we can place
additional constraints of the sequence, such as the PAPR. Fundamentally, however, its correlation
properties are paramount. Other important aspects of a synchronization sequence could be, as
listed in [61], minimal overhead as well as rapid and low-complexity detection.

Early work in this field came from R.H. Barker, who developed the well-known Barker sequences [61].
These were originally formulated with a specific condition on the autocorrelation properties and
the original solution was later slightly relaxed into what is now referred to a Barker sequences [62].
A sequence sB[n] ∈ {−1, 1} of length L is a Barker sequence if its circular autocorrelation Rc is

|Rc
sB [m]| = {0, 1} , ∀ m 6= 0 (4.4)

where Rc
s is defined as [63]

Rc
s(si, sj) =

N−j−1∑
i=0

si s
∗
i+j +

N−1∑
i=N−j

si s
∗
i+j−N , j = 0, 1, . . . N (4.5)

If i = j, then (4.5) reduces to ∑N−1
i=0 sis

∗
i .

Only eight of these sequences have been found, the longest of length L = 13. It has not been
formally proven that no longer sequences can be found, but it exists an overwhelming number of
indicators supporting that claim and via computer simulations, no sequence of length 13 ≤ L <
1022 have been found [64]. From a practical point of view, it means that no usable sequence longer
than 13 exists, as 1022 bits reserved for synchronization is would certainly be infeasible. Barker
sequences have had a number of uses over the years, for example to increasing the length of a
radar signal via pulse compression [65].

Another type of sequence with specific autocorrelation properties is the Maximal Length Sequence,

41



4. Synchronization in OFDM systems

or m-sequence for short. It was originally developed by Solomon Golomb and has been extensively
used since its inception in the 1950s [66]. They can be defined with the aid of linear feedback shift
registers, where certain configurations will yield periodic sequences with the maximum possible
period. In Figure 4.3, one such configuration is shown. The state vector r of this particular
setting will, given that the initial state is not the zero vector, cycle through every possible five-bit
combination except for the all-zero state exactly once before repeating itself. This means that the
output vector s will be periodic with 25 − 1 = 31 and in general, an linear shift register of degree
N will produce a m-sequence of period 2N − 1 [67]. Note further that different initial values of
the shift registers, apart from the zero vector, all produce cyclically shifted versions of the same
sequence.

r4[n] r3[n] r2[n] r1[n] r0[n]
s[n]

Figure 4.3: The generator for the m-sequence used in LTE [68], where the initial state is set to r =
[0 0 0 0 1].

M-sequences have a number of interesting properties and the key characteristic yielding them
especially useful for synchronization, relates to their autocorrelation properties. Feeding the binary
vector s to a binary phase shift keying (BPSK) modulator, we produce the vector s̃, i.e. s̃[n] =
1− 2s[n], n ∈ {0, 1, . . . , N}. Then, the autocorrelation Cs̃ is given as [69]

Cs̃[m] =
N−1∑
n=0

s̃[n] s̃[(n+m)N ] = −1 , ∀ m 6= 0 (4.6)

Note that (4.6) is the same circular autocorrelation defined in (4.5), however expressed more
compactly with the circular shift notation (·)N .

A key difference between the m-sequence and the Barker sequence, is that there is not any upper
bound on the length of the m-sequence. This means that the difference between the main peak
of the correlation spectrum and the side peaks can be made arbitrarily large. In Figure 4.4, this
difference is shown. The peak of both spectra is equal to the length their respective sequence, so
as the m-sequence can be designed to have any length, we can make the difference between the
peaks significantly larger than for the longest possible Barker sequence.

A key aspect of the m-sequence is that within one period, it appears almost completely as white
noise. The definition of white noise states that its autocorrelation at any locations other than
the zero lag should be identically zero, and from Figure 4.4b, it is clear that the m-sequence
approximates this to a stunning degree. The fact that an m-sequences is a completely deterministic
sequence that appears fully random, is perhaps the major reason that they have found such
widespread use. This is also why they are sometimes called pseudonoise (PN) sequences. They
are for example used as a modulating sequence in code-division multiple access (CDMA) systems,
as well as for synchronization purposes in LTE [70, 68].

42



4. Synchronization in OFDM systems

0 2 4 6 8 10 12 14

−1

−0.5

0

0.5

1

0 5 10 15 20 25 30

−1

−0.5

0

0.5

1

n

(a) The sequences.

−8 −6 −4 −2 0 2 4 6 8

0

5

10

15

−15 −10 −5 0 5 10 15

0

10

20

30

m

(b) The correlation spectra.

Figure 4.4: A length 13 Barker sequence (top) and a length 31 m-sequence (bottom) and their respective
correlation spectra.

Lastly, we mention the topic of the cross-correlation of two different m-sequences. This is an
important aspect in CDMA, where each user requires a unique sequence to modulate its data with
and we would, preferably, like the cross-correlation of two different m-sequences to be close to
zero, meaning that a m-sequence from one user appears as white noise when correlated with any
other m-sequence used in the system. This leads us to the topic of Gold sequences, which we will
not look into further, but note that it is possible to construct a set of sequences such that their
respective cross-correlation is bounded. The construction of this set is based on the m-sequences
and Gold sequences have found significant use in various contexts [66, 69].

We have now covered some the relevant background with regards to binary sequences. The concepts
developed so far can be extended to cover sequences consisting of a larger alphabet, such as
those made from a string of complex numbers. We will return to this topic when we discuss the
synchronization procedure in LTE, as it is not applicable for any of the other methods that we
will investigate.

Data-aided and non-data aided synchronization

The discussion so far has fundamentally been about our ability to detect a known sequence and
the design of such a sequence. Going further, we could also question whether the synchronization
sequence actually needs to be explicitly known at the receiver. This was hinted to earlier and for
example, say that each payload was preceded by a repetition of some unknown sequence but of
known length. In this setting, we could simply look for a place in the window where a given number
samples of are repeated once or more times to accurately determine the start of the payload. With
this approach, the received signal only needs to contain some predetermined structure, rather

43



4. Synchronization in OFDM systems

than a specific sequence. Broadly speaking, we can categorize synchronization methods into two
major groups called data-aided (DA), where the sequence is explicitly known or non-data aided
(NDA) methods, where some structural property is used. We will discuss methods of both types
in following sections.

4.2 Cyclic prefix-based synchronization

As implied by title of this section, these methods relies on the cyclic prefix to find symbol timing
and frequency offsets. Clearly, this is only relevant to systems using a CP, which OFDM happens
to do. As mentioned in Section 2.1, we normally prepend each OFDM symbol with a prefix of
length P made up of the last P samples the symbol. Consequently, an OFDM symbol itself already
has repetitive structure, which can be used for synchronization. As the exact content of the cyclic
prefix is unknown to the receiver, we would call this a NDA method.

An early paper in this field came in 1995 [71, 52], treating mainly the timing offset estimation.
The same authors developed the concept further to cover both the frequency and timing offset
estimates in a well-known paper from 1997 [53]. Additional developments can also be found in [72],
where the idea is extend to synchronization in multiuser OFDM and in [73], where pilots used in
channel estimation are exploited in tandem with the cyclic prefix to increase the accuracy of the
offset estimation.

We will now present an outline of the method and its performance. Assume we transmit OFDM
symbols with N subcarriers, each prepended with a cyclic prefix of length P over an AWGN
channel. At the receiver, a window of length 2N + P is observed, as depicted in Figure 4.5. The
received signal is affected by some unknown normalized frequency offset ε0 and the correct location
of the window is defined to be in the start of the cyclic prefix and parameter measuring the shift
from this location is called µ. This means that the STO δ0 as defined in Chapter 3 relates to µ as
δ0 = µ− P .

OFDM block:
CP Data CP Data CP Data

1 µ 2N + P

Figure 4.5: The window of received samples considered in [53].

We collect the received samples y[n] in the vector y = {y[1], y[2], . . . , y[2N + P ]}. The log-
likelihood function L(µ, ε) of p(y|µ, ε) can then be shown to be [52, 53]

L(µ, ε) = |γ(µ)| cos (2πε0 + ∠γ(µ))− ρΦ(µ) (4.7)

44



4. Synchronization in OFDM systems

where

γ(m) =
m+P−1∑
n=m

y[n]y∗[n+N ] (4.8)

Φ(m) = 1
2

m+P−1∑
n=m

|y[n]|2 + |y[n+N ]|2 (4.9)

ρ =

∣∣∣∣∣∣∣∣
E
{
y[n]y∗[n+N ]

}
√
E
{
|y[n]|2

}
E
{
|y[n+N ]|2

}
∣∣∣∣∣∣∣∣ = SNR

SNR + 1 (4.10)

Maximizing (4.7) w.r.t. to ε0 yields the following maximum likelihood (ML) estimate for ε

εest
ML = − 1

2π∠γ(µ) (4.11)

assuming that |ε0| < 1/2. This is valid as long it can be assumed that a coarse synchronization
in the acquisition step has already confined the CFO within this interval. With other methods of
synchronization, this restriction can be lifted. In