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    \begin{document}
    
    
    \maketitle
    
    

    
    \begin{titlepage}
    \begin{center}
        \vspace*{1cm}
        
        \textbf{Real Time Machine Learning Techniques for Wireless Techniques}
        
        \vspace{0.5cm}
        Thesis Subtitle
        
        \vspace{1.5cm}
        
        \textbf{Charles Meyers}
        
        \vfill
        
        \vspace{0.8cm}
        
        Applied Math\
        NYCCT\\
        5/23/2018
        
    \end{center}
\end{titlepage}

    \subsection{Back Ground Information}\label{back-ground-information}

\subsubsection{Signal Processing}\label{signal-processing}

To even begin how to model a signal at a given point of space and time,
we have to start with the model of the signal itself. All signals can be
modelled as the sum of a series of cosines and sines. In the simplest
case:

For a continuous variable \(E(t)\), we can model field strength as

\$ E(t) = \sum\_\{i=1\}\^{}N \textbar{}a\_i\textbar{}
\cos(2\emph{pi}f\_c t+\phi)\$

Where \(A_{Tx}\) is the amplitude at the transmitter, \(f_c\) is the
center frequency and \(\phi\) is a phase shift. As we can see from the
above equation, that only three parameters are within our control:
amplitude, frequency, and phase. With these three parameters, there are
many modulation schemes. The simplest case is show below in Figure 1.

Figure 1. - Different Modulation Techniques.

\(A_{Tx} \cos(2*\pi*f_c t+\phi)\)

where \(a_i\) is the average amplitude of the signal strength at the
receiver, given by the root-mean square estimation.

Modern wi-fi systems (since 802.11n) have deployed MIMO antenna arrays
that allow for a particular type of modulation that exploits the phase
and amplitude characteristics of signal, called Quadrature Amplitude
Modulation (QAM). This is where we must recall Euler's Identity

\(e^{i*\pi}+1=0\)

If we decompose this according to a basic trigonometric identity and let
\(x = 2*\pi*f_c t+\phi\), we see that

\(e^{i*\pi} = - (\sin^2(\omega)+\cos^2(x))\)

Without loss of generality, the above identity holds true for any
scaling factor \(E\). If we rewrite this in terms of phase and quadratur
components we find that

\(E(t) = I(t)*\cos(2*pi*f_c t+\phi)+Q(t)-\sin(2*pi*f_c t+\phi)\)

where \(Q\) is the phase of the signal and \(I\) is its amplitude.

Figure 2. - Phase-Amplitude Model.

Furthermore, since phase is a continuous variable, our sample space (as
a function of \(\theta\) can become arbitrarily small. However, there is
a trade-off here. As we decrease our \(\theta\) sampling window, we
increase our effective noise floor, as each pizza-shaped slice of our
measurement window contains the same information in less geometric
space. The effect of this is to transmit for bits per symbol. Note this
only applies to receiving systems with a single antenna.

\begin{longtable}[c]{@{}llll@{}}
\toprule
Modulation & Bits per Symbol & Symbol Rate & Minimum SNR\tabularnewline
\midrule
\endhead
BPSK & 2 & 1/2 bit rate & 3.5dB\tabularnewline
QPSK & 3 & 1/3 bit rate & 22.45\tabularnewline
8PSK & 4 & 1/3 bit rate & 26.96\tabularnewline
16QAM4 & 5 & 1/4 bit rate & 30.67\tabularnewline
32QAM & 5 & 1/5 bit rate & 32.49\tabularnewline
64QAM & 6 & 1/6 bit rate & 35.02\tabularnewline
\bottomrule
\end{longtable}

http://www.ni.com/tutorial/4805/en/
http://www.scielo.org.za/scielo.php?script=sci\_arttext\&pid=S0038-23532011000100012

    \subsubsection{Design Constraints}\label{design-constraints}

I want to be able to build this path loss model using a large set of
numerical data. In order to do that, we must limit ourselves to
measuring things that are available on common radio circuits. For
model-building purposes, I set up tcpdump, iw, iperf, and ping to gather
network state data from the transmitter and receiver ends. I have
developed an android application using the Automate tool that gathers
sensor data from all three on-board radios (bluetooth, wifi, and cdma)
as well as the the 9 degree of freedom orientation chip that quantifies
location (in 3 dimensions), orientation (along each of 3 axes), and
magnetic field (in each of 3 orthogonal directions). In addition to
that, we get other common wifi packet data. In addition, the wi-fi
packet frames provide more information: * a frame control segment that
indicates whether the frame is a control, management, or data type *
address segments that include the MAC addresses of transmitter,
receiver, and the final destination at either end * sequence control
data that helps to reorder frames that arrive at phase-delayed times *
the actual body of the frame (data) * a frame sequence check (a 32 bit
checksum for error correction)

https://witestlab.poly.edu/blog/802-11-wireless-lan-2/

At the receiver end, I can verify the state of this data relative to the
transmit data, verify timing assumptions (to an imprecise degree),
gather RSSI levels and the bit error rate for a given modulation. At the
receiver end, I log the data continuosly for the sensor data. Every time
the Android OS has the ability (due to processor congestion and timing),
it measures the raw voltage levels on the 9-DoF chip. The network state
data, because it is user-space software, takes time to make its
measurements. Since this problem is outside of the scope of this
project, training data will only be taken from timestamped data that has
no missing features. It is possible that a time-series average of the
sensor data can replace the instantaneous measurement in the model.
However, more investigation is needed.

In addition to wifi data, I can collect the same information from a
bluetooth radio. However, instead of operating in the 2.4 or 5 Ghz
ranges familiar to wifi, it operates in the sub Ghz range around 900Mhz.
Because of the difference in these wavelenghts, the two fields operate
somewhat differently at a human scale. Bluetooth is much less sensitive
to fading and other non-line of sight measurements. The difference in
estimated fading margins between wifi and bluetooth can be used to get a
characteristic of the environment.

    \subsection{Previous Models}\label{previous-models}

\subsubsection{Friis Equation}\label{friis-equation}

In the simplest scenario, we can model the free space path loss. This
function of received power in terms of distance is known as Friis' Law:

\(P_{Rx}(d)=P_{Tx}G_{Tx}G_{Rx}(\frac{\lambda}{4*\pi*d})^2A_{Rx}\)

Where \(P\) is the power, \(G\) is the gain, and \(A\) is the area of
the receiving antenna, \(d\) is distance between the transmitters and
\(\lambda\) is the frequency. This equation only applies for systems
that are separated by at least one Rayleigh distance defined by

\(d_{R}=\frac{2{L_a}^2}{\lambda}\)

This is known as the far field. When dealing with link budgets, it is
best to use a logarithmic scale because signal levels will vary across
many orders of magnitude.

\(P_{Rx}(d)=P_{Tx}G_{Tx}G_{Rx}20\log(\frac{\lambda}{4*\pi*d})^2A_{Rx}\)

if and only if the powers and gains are consistently in dB or dBm.

    \subsubsection{Kirchoff Theory}\label{kirchoff-theory}

In the same way that Rayleigh distance defines the breaking point of
Friis system, Rayleigh roughness can be thought of as the

\paragraph{Perturbation Theory}\label{perturbation-theory}

    \subsubsection{Log-Distance Path Loss
Model}\label{log-distance-path-loss-model}

The next model, log-normal shadowing can be thought of as an extension
of the Friis model with the added inclusion of a random variable. If the
receiver is in the far field of the receiver (where \(d>d_R\)),
\(PL(d_0)\) is the path loss measured at a distance \(d_0\) from teh
transmitter, then the path loss when moving from distance \(d0\) to
\(d\) is given by the equation

\(PL{d_0\rightarrow d}(dB) PL(d_0)+10n\log_{10}\frac{d}{d_0} + \chi_1\)

c where \(n\) is the path loss exponent, given by the table below and
\(\chi\) is a zero mean random normal distribution.

https://www.gaussianwaves.com/2013/09/log-distance-path-loss-or-log-normal-shadowing-model/

    \begin{longtable}[c]{@{}lll@{}}
\toprule
Environment & Path Loss Exponent min & Path Loss Exponent
Max\tabularnewline
\midrule
\endhead
Free Space & 2 & 2\tabularnewline
Urban area cellular radio & 2.7 & 3.5\tabularnewline
Shadowed urban cellular radio & 3 & 5\tabularnewline
Inside-LoS & 1.6 & 1.8\tabularnewline
Obstructed in building & 4 & 6\tabularnewline
Obstructed in Factory & 2 & 3\tabularnewline
\bottomrule
\end{longtable}

\textbf{\emph{Figure 1:}} Empirical Log-distance coefficients

    \subsubsection{Noise}\label{noise}

In most circumstances it is too cumbersome to describe all the sources
of noise. If it wasn't, then we'd solve Maxwell's Equations within the
system and be done with it. However there are many sources of
interference. Multipath propagation is a big limitation to predicting
the shape of a wireless network in an urban space as different rays from
the transmitter bounce around the room and reach the receiver at
different times. Because receivers cannot distinguish the 'true' signal
from the multi-path signal, the receiver just adds the components of
those multipaths up, creating interference. This interference can be
constructive or destructive. Small-scale fading occurs when a user is
moving-\/-when the user moves further away, they increase the phase and
possibly the measured voltage level. Additionally, the transmitter and
receiver each produce noise. The total power due to noise can be
described as

\(P = (\sum_{k=0}^{n}(x_I+x_Q))^2\)

Where \(x_I\) and \(x_Q\) are linearly independent vectors of amplitude,
\(I\), and phase, \(Q\). The signal to noise ratio can be expressed as

\(SNR = \frac{P_{signal}}{P_{noise}}\)

where P is the average power measured at the equivalent points in a
system and within the same system bandwidth. If we include interference,
the SNR becomes SINR or the signal to interference and noise ratio.

\(SINR = \frac{P_{signal}}{P_{noise}+P_{interference}}\)

Interference can also be created by objects in the environment.
Buildings and other obstacles to propagation can block regular
transmission, causing any signal to make it behind the building to be
greatly attenuated. This is called shadowing. Note that this not only
happens to the line-of-sight components of the wave front, but any multi
path ray! For this reason, buildings and other obstacles give rise to
large-scale fading by creating a diffraction pattern of the signal wave
in the shadow of the building.

These components can undergo many types of transformations. Reflection
is when a field reaches an object with very large dimensions compared to
the wavelength of the field. A wi-fi signal is reflected off of
buildings and the ground, creating patterns of de/coherence. Diffraction
occurs when a transmitter and receiver are obstructed by a surface,
'bending' the wave around the obstacle. When a field travels across a
surface with dimensions that are small compared to its wavelenght and
where the number of obstacles per volume is high. Refraction is a result
of the field traversing multiple mediums, much like how a pencil appears
to 'bend' when placed in a glass of water. For much higher frequencies
(starting at 15Ghz), atmospheric absorption is a problem, but for indoor
distances and low frequencies, the effects are negligible. {[}Source
Tum{]}

    \subsubsection{Log-Normal Shadowing Propagation Loss
Model}\label{log-normal-shadowing-propagation-loss-model}

The next model builds on the Log-Normal shadowing model by including a
random variable, \(chi\). This law can be expressed as:

\(PL_{d_0\rightarrow d}(dB)=PL(d_0)+10n\log_{10}\frac{d}{d_0}+\chi_2\)

Where \(\chi_2\) is a zero-mean Gaussian distributed random variable.
This variable is only used when there is a shadowing effect.
Equivalently, \(\chi_2=0\) when no shadowing effect is present.

    \subsubsection{Two Ray Ground Propagation Loss
Model}\label{two-ray-ground-propagation-loss-model}

Thus far, we have not needed to think of our signal and other multipath
components as being complex vectors. When we start trying to predict the
way these multiple paths interact analytically, phase shifting due to
refraction, fading, and other multipath effects can be modelled by using
a two path tracing technique. The figure below illustrates this model.

Figure 1. - Two Ray Path Loss Model.

    \subsubsection{ITU Propagation Loss
Model}\label{itu-propagation-loss-model}

The next model is a semi-empirical model, based on the same Friis
equatoin, with added parameters for noise due to fading between floors
in a building.\\

\(PL_{d_0\rightarrow d}(dB)=PL(d_0)+20n\log_{f}+N(\log_{10}(d)+P_f(n)-28\)

Where \(P_f\)(n) is a power loss function due to the number of floors
through which the field propagates. In addition, the \(\chi\) value of
previous models has been set to -28 in the ITU model, taken from
standardized urban measurements. Source: ITU.pdf

    \subsubsection{Motley-Keenan Model}\label{motley-keenan-model}

The Motley-Keenan Model follows the same logic.\\
\(PL_{d_0\rightarrow d}(dB)=PL(d_0)+20\log_{10}\frac{d}{d_0}+\sum a_K\)\\
\(PL_{LoS}(d)[dB]=20log_{10}\frac{(4πd_0)}{λ}+10_{n_{LoS}}log_{10}(d)+\chi_{\sigma}\)\\
\(PL_{NLoS}(d)[dB]=20log_{10}\frac{(4πd_0)}{λ}+10_{n_{NLoS}}log_{10}(d)+\chi_{\sigma}\)
\(PL_{LoS}(d)[dB]=P_{LoS}*PL_{LoS}(d)[dB]+(1-P_{LoS})*PL_{NLoS}(d)[dB]\)

Source: m-k 3

    \subsubsection{Tata Modified ITU Model}\label{tata-modified-itu-model}

\(PL_{T−IPLM}(dB)=20 × log_{10}(f) + N_T × log_{10}(d)+10+\sum_{w=0}^{w=k}+FAF-20\)

NT for different number of obstacles

\begin{longtable}[c]{@{}llllll@{}}
\toprule
Channel 1 & & Channel 7 & & Channel 11 &\tabularnewline
\midrule
\endhead
No. Obstacles & \(N_T\) & No. Obstacles & \(N_T\) & No. Obstacles &
\(N_T\)\tabularnewline
1 & 31.1 & 1 & 32.9 & 1 & 29.3\tabularnewline
2 & 30.1 & 2 & 28.5 & 2 & 28.4\tabularnewline
3 & 31.8 & 3 & 26.7 & 3 & 27\tabularnewline
4 & 31.2 & 4 & 29.1 & 4 & 28\tabularnewline
5 & 31.3 & 5 & 27.4 & 5 & 28.4\tabularnewline
\bottomrule
\end{longtable}

    Floor Wise Attenuation Factor (FAF)

\begin{longtable}[c]{@{}ll@{}}
\toprule
Scenario & FAF (dB)\tabularnewline
\midrule
\endhead
-2 floors & 36\tabularnewline
-1 floors & 21\tabularnewline
0 floors & 0\tabularnewline
+1 floors & 21\tabularnewline
+2 floors & 33\tabularnewline
+3 floors & 40\tabularnewline
\bottomrule
\end{longtable}

    Longley-Rice Model:

Frequency\\
Distance\\
Antenna Heights\\
Polarization\\
Terrain irregularity (\(\Delta h\))\\
Electrical Ground constants\\
Refractivity\\
Climate\\
Siting\\
Reliability and confidence level

w' (t,ℓ,s) = W0 + yS(s) + δL(s) yL(ℓ) + δT(s) yT(t),\\
https://www.ntia.doc.gov/files/ntia/publications/ntia\_82-100\_20121129145031\_555510.pdf

    \begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}5}]:} \PY{c+c1}{\PYZsh{}\PYZsh{} Purely Statistical Models}
\end{Verbatim}


    \subsubsection{Rayleigh Fading}\label{rayleigh-fading}

\(x = cdf(r_{min})\approx \frac{r^2_{min}}{2\sigma^2}\)

    \subsubsection{Rician Fading}\label{rician-fading}

    \subsubsection{Doppler Adjustments}\label{doppler-adjustments}

\$BER\_\{Doppler\} = \frac{1}{2} \pi\textsuperscript{2(v\_\{max\}T\_B)}2
\$

\(BER = K (\frac{S_{\tau}}{T_B})^2\)

    \subsubsection{Machine Learning}\label{machine-learning}

    \subsection{Problem}\label{problem}

Standard wireless models are great at modelling unbostructed
line-of-sight connections. However, they fail to accurately model the
network states at peak noise, in complicated urban environments both
indoors and outdoors. In addition, these models live entirely in the
proprietary world of the IEEE website. Implementing these models in
Python would go a long way to modelling the bigger problem. This is the
first step. Since some quantities required for these models are unknown,
a gridsearch algorithm will be used to optimize these algorithms.

Since these models require different sets of parameters, what features
our dataset needs is an open question. The first method will be using a
k-random-forest algorithm to see what features are relevant to the
signal strength between two nodes.

The next method will take the most useful features and use them to build
a linear (or maybe quadratic) regressor that predicts signal to noise
ratio between two nodes.

Then, each of these models (the set of 802.11 standard models and my
from-scratch regressor) will be compared using a Xi-Square test.
Differences between my 'simplified' model and the more complex 802.11
models will be compared and analyzed with respect to the cost of finding
the additional features.

    \begin{longtable}[c]{@{}lllllll@{}}
\toprule
\begin{minipage}[b]{0.05\columnwidth}\raggedright\strut
Model
\strut\end{minipage} &
\begin{minipage}[b]{0.16\columnwidth}\raggedright\strut
Required Features
\strut\end{minipage} &
\begin{minipage}[b]{0.16\columnwidth}\raggedright\strut
\strut\end{minipage} &
\begin{minipage}[b]{0.14\columnwidth}\raggedright\strut
\strut\end{minipage} &
\begin{minipage}[b]{0.13\columnwidth}\raggedright\strut
\strut\end{minipage} &
\begin{minipage}[b]{0.08\columnwidth}\raggedright\strut
\strut\end{minipage} &
\begin{minipage}[b]{0.09\columnwidth}\raggedright\strut
\strut\end{minipage}\tabularnewline
\midrule
\endhead
\begin{minipage}[t]{0.05\columnwidth}\raggedright\strut
Friis
\strut\end{minipage} &
\begin{minipage}[t]{0.16\columnwidth}\raggedright\strut
Receive Power
\strut\end{minipage} &
\begin{minipage}[t]{0.16\columnwidth}\raggedright\strut
Transmit Power
\strut\end{minipage} &
\begin{minipage}[t]{0.14\columnwidth}\raggedright\strut
Transmitter Gain
\strut\end{minipage} &
\begin{minipage}[t]{0.13\columnwidth}\raggedright\strut
Receiver Gain
\strut\end{minipage} &
\begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
distance
\strut\end{minipage} &
\begin{minipage}[t]{0.09\columnwidth}\raggedright\strut
frequency
\strut\end{minipage}\tabularnewline
\bottomrule
\end{longtable}

    \subsection{Hypothesis}\label{hypothesis}

Machine Learning Tools can signinficantly reduce the cost of
measurement.


    % Add a bibliography block to the postdoc
    
    
    
    \end{document}