\documentclass[11pt]{article}
\usepackage{handout2e}
\begin{document}
\input{lecmacros}
%\input{macros}
\input{courseMacros}
\thispagestyle{empty}
\setlength{\parindent}{0pt}
\setlength{\parskip}{1.8ex}

\newlecturehead{CSG250}{Wireless Networks}{Spring 2007}{16 January 2007}{2}

\begin{center}
\Large{\bf Antennas and Propagation}
\end{center}

The notes in this document are based almost entirely on Chapter~5 of
the textbook~\cite{stallings:wirelessBook}.  Rappaport's text is also
a good reference for wireless signal
propagation~\cite{rappaport:wirelessBook}.
\section{Antennas} 
\label{sec:antenna}
An antenna is an electrical conductor or a system of conductors that
radiates/collects (transmits or receives) electromagnetic energy
into/from space.  An idealized isotropic antenna radiates equally in
all directions.  The directivity of an antenna is captured by its beam
width; it is the angle within which power radiated is at least half of
that in the most preferred direction (that is, half of the maximum
power).

Antenna gain is the power output, in a particular direction, compared
to that produced in any direction by an idealized omnidirectional
antenna.  If $f$ is the carrier frequency and $A_e$ is the effective
area of the antenna, then the antenna gain is given by
\[
G = \frac{4\pi A_e f^2}{c^2}, 
\]
where $c$ is the speed of light.  The effective area of an antenna
depends on the size and shape of the antenna.  The effective area of
an idealized isotropic antenna is 1, by definition.

\section{Propagation modes}
Wireless transmissions propagate in three modes: {\em ground-wave},
{\em sky-wave}, and {\em line-of-sight}.  Ground wave propagation
follows the contour of the earth, while sky wave propagation uses
reflection by both earth and ionosphere.  Finally line of sight
propagation requires the transmitting and receiving antennas to be
within line of sight of each other.  Which of these propagation modes
dominates depends on the frequency of the underlying signal.

Examples of ground wave and sky wave communication are AM radio and
international broadcasts such as BBC.  Above 30 MHz, neither ground
wave nor sky wave propagation operates and the communication is
through line of sight.

If $h$ is the height of a transmitting (resp., receiving) antenna in
meters, then the distance to the receiver (resp., transmitter) for
line-of-sight transmission should be at most
\[
d = 3.57\sqrt{h} \mbox{ kms}.
\]
This can be proved using elementary geometry.  Since microwaves are
bent or refracted by the atmosphere, the ``effective'' line of sight
is, in fact, larger than the true line of sight.  We introduce an
adjustment factor $K$ to capture the refraction effect and obtain
\[
d = 3.57\sqrt{Kh} \mbox{ kms}.
\]
If the two antenna have heights $h_1$ and $h_2$ meters, then the
distance between them for LOS propagation should be at most
\[
3.57\sqrt{K}(\sqrt{h_1} + \sqrt{h_2}) \mbox{ kms}.
\]
\section{Transmission limitations}
There are four classes of limitations that affect electromagnetic wave
transmissions.  

\BfPara{Attenuation}
The strength of signal falls with distance over transmission medium.
The extent of attenuation is a function of distance, transmission
medium, as well as the frequency of the underlying transmission.  Even
in free space, with no other impairment, the transmitted signal
attenuates over distance simply because the signal is being spread
over a larger and larger area.  For an ideal isotropic antenna, the
ratio between the transmitted power $P_t$ and the received power $P_r$
for a separation distance of $d$ is given by 
\[
\frac{P_r}{P_t} = \frac{(4\pi d)^2}{\lambda^2} = \frac{(4\pi f
d)^2}{c^2},
\]
where $\lambda$, $f$, and $c$ are the wavelength, frequency, and the
speed of the signal.  

One can express the above ratio in terms of antenna gains and
effective areas of the antennas by applying the appropriate formulae.

\BfPara{Distortion}
Since signals at different frequencies attenuate to different extents,
as the above formula indicates, a signal comprising of components over
a range of frequencies gets distorted; i.e., the shape of the received
signal changes.  A standard method of resolving this problem (and
recovering the original shape) is to amplify higher frequencies and
thus equalize attenuation over a band of frequencies.

\BfPara{Dispersion} Dispersion is the phenomenon of spreading of a
burst of electromagnetic energy during propagation.  It is especially
prevalent in wireline transmissions such as an optical fiber.  Bursts
of data sent in rapid succession tend to merge due to dispersion.  The
longer the length of the wire, the more severe is the effect of
dispersion.  The effect of dispersion is to limit the product $R
\times L$, where $R$ is the data rate and $L$ is distance.

\BfPara{Noise}
The most pervasive form of noise is thermal noise, which is often
modeled using an additive Gaussian model.  Thermal noise is due to
thermal agitation of electrons and is uniformly distributed across the
frequency spectrum.  The thermal noise to be found in 1 Hz of
bandwidth is 
\[
N_0 = kT \mbox{ W/Hz}, 
\] 
where $T$ is the absolute temperature of the medium (conductor) and
$k$ is Boltzmann's constant ($1.3803 \times 10^{-23}$ Joules/Kelvin).
The amount of thermal noise in $B$ Hz of bandwidth is simply $N_0B$.
Other forms of noise include intermodulation noise (caused by signals
produced at frequencies that are sums or differences of carrier
frequencies), crosstalk (interference between two signals), and
impulse noise (irregular pulses of high energy caused by external
electromagnetic disturbances).  While an impulse noise may not have a
significant impact on analog data, it has a noticeable effect on
digital data, causing burst errors.

The effect of noise on the achievable information rate is best
captured by Shannon's capacity theorem which we derived last week,
which relates the capacity to the signal-to-noise ratio.  A more
convenient form of this equation is in terms a different ratio,
referred to as the $E_b/N_0$ term.  Here $E_b$ refers to the energy
per bit of the signal and $N_0$ is the noise power density introduced
above.  In terms of the signal power $P$ and the information rate $R$,
we can write $E_b$ as $P/R$.  Noting that $N = N_0B$, we obtain
$E_b/N_0 = (P/N)(B/R)$, where $B$ is the bandwidth.  Thus, the
$E_b/N_0$ value needed to achieve the capacity of a channel must
satisfy the following equation.
\[
C = B \log_2 \left(1 +  \frac{E_b}{N_0} \frac{C}{B}\right) \Rightarrow
\frac{E_b}{N+0} = \frac{B}{C}(2^{C/B} - 1).
\]
For signal modulation techniques, the bit-error rate is often
expressed as a function of the $E_b/N_0$ ratio.
\section{Fading}
\label{sec:fading}
Fading refers to the variation of the signal strength with respect to
time/distance and is widely prevalent in wireless transmissions.  The
most common causes of fading in the wireless environment are multipath
propagation and mobility (of objects as well as the communicating
devices).

\BfPara{Multipath propagation} In wireless media, signals propagate
using three principles: reflection, scattering, and diffraction.
Reflection occurs when the signal encounters a large solid surface,
whose size is much larger than the wavelength of the signal, e.g., a
solid wall.  Diffraction occurs when the signal encounters an edge or
a corner, whose size is larger than the wavelength of the signal,
e.g., an edge of a wall.  Finally, scattering occurs when the signal
encounters small objects of size smaller than the wavelength of the
signal.  One consequence of multipath propagation is that multiple
copies of a signal propagation along multiple different paths, and
arrive at any point at different times.  So the signal received at a
point is not only affected by the inherent noise, distortion,
attenuation, and dispersion in the channel but also the interaction of
signals propagated along multiple paths.

\BfPara{Delay spread} Suppose we transmit a probing pulse from a
location and measure the received signal at the recepient location as
a function of time.  The signal power of the received signal spreads
over time due to multipath propagation.  The {\em delay spread}\/ is
determined by the density function of the resulting spread of the
delay over time.  Average delay spread and root mean square delay
spread are two parameters that can be calculated.

\BfPara{Doppler spread} This is a measure of spectral broadening
caused by the time rate of change of the mobile radio channel.  It is
caused by either relative motion between the mobile and base station
or by movement of objects in the channel.  When a pure sinusoid of
frequency $f$ is transmitted the received signal spectrum, the Doppler
spectrum, is the range $[f - f_d, f | f_d]$, where $f_d$ is referred
to as the Doppler spread, given by
\[
f_d = \frac{v}{\lambda} \cos \theta,
\]
where $v$ is the relative velocity of the mobile and $\theta$ is the
angle formed by the direction of the motion and that of the signal,
and $\lambda$ is the wavelength of the carrier.

The relative size of the delay spread, as compared to the symbol
period, has an impact on whether the effect of fading is uniform over
all frequencies or varies across frequencies in the spectrum of the
channel.  {\em Flat fading}\/ occurs when the bandwidth of the signal
is less than the bandwidth of the channel, or delay spread is less
than the symbol period.  {\em Frequency selective fading}\/ occurs
when the bandwidth of the signal is greater than the bandwidth of the
the channel, or the delay spread is greater than symbol period.

There are several fading models that capture different multipath
propagation characteristics.  {\em Rayleigh fading}\/ models a
worst-case scenario in which no path dominates (in fact, a LOS path
may not even exist).  {\em Rician fading}\/ assumes that one of the
paths, usually LOS, dominates all the other paths.

When the velocity of the mobile is high, the Doppler spread is high,
and the the resulting channel variations are faster than that of the
baseband signal; this is referred to as {\em fast fading}.  When
channel variations are slower than the baseband signal variations,
then the resulting fading is referred to as {\em slow fading}.
\bibliographystyle{alpha}
\bibliography{localrefs}

\end{document}