$ \begin{titlepage} \begin{center} \vspace*{1cm}
\textbf{Real Time Machine Learning Techniques for Wireless Techniques}
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Thesis Subtitle
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\textbf{Charles Meyers}
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\vspace{0.8cm}
Applied Math\
NYCCT\\
5/23/2018
\end{center}
\end{titlepage} $
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
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.
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.
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.
Modulation | Bits per Symbol | Symbol Rate | Minimum SNR |
---|---|---|---|
BPSK | 2 | 1/2 bit rate | 3.5dB |
QPSK | 3 | 1/3 bit rate | 22.45 |
8PSK | 4 | 1/3 bit rate | 26.96 |
16QAM4 | 5 | 1/4 bit rate | 30.67 |
32QAM | 5 | 1/5 bit rate | 32.49 |
64QAM | 6 | 1/6 bit rate | 35.02 |
http://www.ni.com/tutorial/4805/en/ http://www.scielo.org.za/scielo.php?script=sci_arttext&pid=S0038-23532011000100012
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:
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
https://www.gaussianwaves.com/2013/09/log-distance-path-loss-or-log-normal-shadowing-model/
Environment | Path Loss Exponent min | Path Loss Exponent Max |
---|---|---|
Free Space | 2 | 2 |
Urban area cellular radio | 2.7 | 3.5 |
Shadowed urban cellular radio | 3 | 5 |
Inside-LoS | 1.6 | 1.8 |
Obstructed in building | 4 | 6 |
Obstructed in Factory | 2 | 3 |
Figure 1: Empirical Log-distance coefficients
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
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.
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]
The next model builds on the Log-Normal shadowing model by including a random variable, $chi$. This law can be expressed as:
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.
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.
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.
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
$PL_{T−IPLM}(dB)=20 × log_{10}(f) + N_T × log_{10}(d)+10+\sum_{w=0}^{w=k}+FAF-20$
NT for different number of obstacles
Channel 1 | Channel 7 | Channel 11 | ||||
---|---|---|---|---|---|---|
No. Obstacles | $N_T$ | No. Obstacles | $N_T$ | No. Obstacles | $N_T$ | |
1 | 31.1 | 1 | 32.9 | 1 | 29.3 | |
2 | 30.1 | 2 | 28.5 | 2 | 28.4 | |
3 | 31.8 | 3 | 26.7 | 3 | 27 | |
4 | 31.2 | 4 | 29.1 | 4 | 28 | |
5 | 31.3 | 5 | 27.4 | 5 | 28.4 |
Floor Wise Attenuation Factor (FAF)
Scenario | FAF (dB) |
---|---|
-2 floors | 36 |
-1 floors | 21 |
0 floors | 0 |
+1 floors | 21 |
+2 floors | 33 |
+3 floors | 40 |
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
## Purely Statistical Models
$x = cdf(r_{min})\approx \frac{r^2_{min}}{2\sigma^2}$
$BER_{Doppler} = \frac{1}{2} \pi^2(v_{max}T_B)^2 $
$BER = K (\frac{S_{\tau}}{T_B})^2$
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.
Model | Required Features | ||||||
---|---|---|---|---|---|---|---|
Friis | Receive Power | Transmit Power | Transmitter Gain | Receiver Gain | distance | frequency |
Machine Learning Tools can signinficantly reduce the cost of measurement.