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\newlecturehead{CSG250}{Wireless Networks}{Fall 2007}{14 September 2007}{3}

\begin{center}
\Large{\bf Signal Encoding Techniques}
\end{center}

The notes in this document are based largely 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{Digital modulation: Digital data on analog signals}
There are two important factors that govern the effectiveness of a
digital modulation technique: {\em power efficiency}\/ and {\em
bandwidth efficiency}\/ (also called spectral efficiency).  The power
efficiency of a digital modulation technique is measured by the
$E_b/N_0$ ratio required at the receiver for a certain bit error rate.
The bandwidth efficiency is measured by the ratio of the information
rate to the bandwidth.  There is a natural tradeoff between the two
efficiency parameters.  A modulation technique with superior error
control has higher power efficiency, but lower bandwidth efficiency
since the different data symbols are encoded by signals separated in
the signal space.

In addition to the above two factors, digital modulation techniques
need to be simple and cheap to implement, especially in the context of
wireless communications since the vast majority of the wireless
devices (e.g., cellphones, PDAs, wireless sensor nodes) are heavily
resource-constrained.

\subsection{Amplitude-shift keying (ASK)}
Amplitude-Shift Keying modulation captures a bit by the amplitude of
the carrier signal.  Thus, if the carrier signal is $A \cos(2\pi f_c
t)$ and the digital data is given by $d(t)$, then in ASK
\[
s(t) = \left\{
\begin{array}{ll}
A \cos(2\pi f_c t) & \mbox{ } d(t) = 1\\
0 & \mbox{ } d(t) = 0
\end{array}
\right.
\]

\subsection{Frequency-shift keying (FSK)}
Frequency-shift keying encodes the digital data using multiple
frequencies around the center frequency $f_c$ of the carrier signal.
In Binary Frequency-Shift Keying (BFSK), we have
\[
s(t) = \left\{
\begin{array}{ll}
A \cos(2\pi f_1 t) & \mbox{ } d(t) = 1\\
A \cos(2\pi f_2 t) & \mbox{ } d(t) = 0,\\
\end{array}
\right.
\]
where $f_1, f_2 = f_c \pm f_d$.

In Multi-level Frequency Shift Keying (MFSK), we have 
\[
s_i(t) = A \cos(2\pi f_i t) \mbox{ } 1 \le i \le M,
\]
where $s_i(t)$ is the signal for the $i$th signal element, $f_i = f_c
+ (2i - 1 - M)f_d$, $f_c$ is the carrier frequency, $f_d$ is the
difference frequency, and $M = 2^L$ is the number of different signal
elements.

If $T$ seconds is the bit period of the transmission signal, then each
output signal element is held for $T_s = LT$ seconds.  The minimum
frequency separation required is $2f_d = 1/T_s$.  Consequently the
bandwidth required is $M/T_s$.

\subsection{Phase-shift keying (PSK)}
In Phase Shift Keying, the digital data is encoded by the phase of
the carrier signal.  In BPSK, we have
\[
s(t) = \left\{
\begin{array}{ll}
A \cos(2\pi f_c t) & \mbox{ } d(t) = 1\\
-A \cos(2\pi f_c t) & \mbox{ } d(t) = 0,\\
\end{array}
\right.
\]
Alternatively, if we have binary representation as $\{-1,1\}$ instead
of $\{0,1\}$, we obtain $s(t) = A d(t) \cos(2\pi f_c t)$.

In Differential Phase Shift Keying (DPSK), we encode a 1 by a change
of phase, and 0 by maintaining phase.  Just as in MFSK, we can use
multi-level signaling in PSK.  The QPSK modulation technique uses the
following encoding.
\[
s(t) = \left\{
\begin{array}{ll}
A \cos(2\pi f_c t + \pi/4) & \mbox{ } 11\\
A \cos(2\pi f_c t + 3\pi/4) & \mbox{ } 01\\
A \cos(2\pi f_c t - 3\pi/4) & \mbox{ } 00\\
A \cos(2\pi f_c t - \pi/4) & \mbox{ } 10\\
\end{array}
\right.
\]
The Quadrature Amplitude Modulation technique (QAM) is a combination
of ASK and PSK techniques.  It can also be seen as a logical extension
of QPSK.  The core idea is that it is possible to send two different
signals simultaneously on the same carrier frequency, one
phase-shifted by $\pi/2$ with respect to the other.

The digital data signal $d(t)$ is split into two signals, a signal
$d_o(t)$ consisting of the odd bits and a signal $d_e(t)$ consisting
of the even bits.  In QAM, the encoding is given by
\[
s(t) = d_o(t) \cos(2\pi f_c t) + d_e(t) \sin(2\pi f_c t).
\]

\subsection{Geometric representation of digital modulation techniques}
Suppose the digital data is expressed in terms of $M$ symbols.  So the
digital modulation uses $M$ signal elements given by the set $S =
\{s_i(t): 1 \le i \le M\}$.  Each signal can be viewed as a point in a
vector space.  Given a basis of this vector space, any point can be
expressed as a linear combination of the basis elements.  Thus, if the
basis elements are $\phi_1(t), \ldots \phi_N(t)$, then we can write
\[
s_i(t) = \sum_{j = 1}^{N} s_{ij} \phi_i(t).
\]
The basis elements satisfy the standard orthogonality condition
\[
\int_{-\infty}^{\infty} \phi_j(t) \phi_k(t) dt = 0, \mbox{ } j \neq k.
\]
Furthermore, the energy (per unit resistance) of each basis element
can be normalized to 1.
\[
\int_{-\infty}^{\infty} \phi^2_j(t) dt = 1.
\]
Let us now analyze the phase shift keying techniques using this
geometric representation.  In BPSK modulation, the basis consists of
only one element.  If $T_b$ is the time interval for a bit, then we have
\[
\phi_1(t) = \sqrt{\frac{2}{T_b}} \cos(2\pi f_c t), 0 \le t \le T_b,
\]
where $T_b$ is the time period per bit.  And we can write the two
signal elements, corresponding to the two bits, as
\[
s_1(t) = \sqrt{E_b} \phi_1(t) \mbox{,  } s_2(t) = -\sqrt{E_b}
\phi_1(t),
\]
where $E_b$ is energy per bit.  Thus, BPSK can be represented
geometrically by two points on a line, $(\sqrt{E_b})$ and
$(-\sqrt{E_b})$.

In QPSK, the four different signal elements can be expressed in terms
of two basis elements
\begin{eqnarray*}
\phi_1(t) & = & \sqrt{\frac{1}{T_b}} \cos(2\pi f_c t), 0 \le t \le 2T_b\\
\phi_2(t) & = & \sqrt{\frac{1}{T_b}} \sin(2\pi f_c t), 0 \le t \le 2T_b.
\end{eqnarray*}
We define each basis element for $2T_b$ seconds since each signal
element consists of 2 bits.  Now each of the four signals can be
expressed as
\[
s_i(t) = \pm \sqrt{E_b} \phi_1(t) \pm \sqrt{E_b} \phi_2(t).
\]
Thus QPSK can be expressed geometrically by four points on a 2-D plane
(in which each dimension represents a basis element) $(\sqrt{E_b},
\sqrt{E_b})$, $(-\sqrt{E_b}, \sqrt{E_b})$, $(\sqrt{E_b},
-\sqrt{E_b})$, and $(-\sqrt{E_b}, -\sqrt{E_b})$.

QAM can also be easily expressed geometrically by 16 points on a 2-D
plane.  Frequency-shift keying schemes can be represented using
higher-dimensional spaces by having one dimension for each of the
frequencies.

The geometric representation provides good intuition for the power
efficiency of a modulation technique.  Greater the minimum distance
between two signal points, higher the power efficiency since the more
immune is the encoding technique to noise.  Thus, QAM is more
bandwidth-efficient than QPSK but less power-efficient.  QPSK has the
same power-efficiency as BPSK but better bandwidth-efficiency.

In frequency-shift keying schemes, increasing the number of signal
elements increases the power efficiency at the expense of bandwidth
efficiency since the different signal elements remain well-separated
in the signal space (occupying orthogonal dimensions).  In phase-shift
keying schemes, increasing the number of signal elements decreases the
power efficiency at the expense of bandwidth efficiency.

\subsection{Bandwidth efficiency of digital modulation techniques}
The following are approximate relationships between the transmission
bandwidth $B_T$ of a modulation technique and the information rate
$R$.  For ASK, BPSK, and BFSK, we have
\[
B_T = (1 + r) R,
\]
where $0 < r < 1$ is a parameter related to the filtering technique
used in the modulation.

For MPSK, we have
\[
B_T = \left(\frac{1+r}{\log_2 M}\right) R,
\]
and for MFSK, we have
\[
B_T = \left(\frac{(1+r)M}{\log_2 M}\right) R.
\]

\section{Analog modulation: Analog data on analog signals}
There are two principles reasons to modulate analog data over analog
signals.  First, the frequency spectrum of the underlying data may be
different than that needed for transmission.  In the case of wireless
transmissions, especially, it is improbable to transmit baseband
signals without modulation.  The other reason is that analog
modulation permits frequency division multiplexing since we can
transmit data of different users over different bands by appropriately
modulating them.
\subsection{Amplitude modulation}
In Amplitude Modulation (AM), the analog data is encoded in the
amplitude of the signal.  Specifically, if the analog data is given by
$m(t)$, then the encoded signal is given by
\[
s(t) = (1 + n_a m(t)) \cos(2\pi f_c t),
\]
where $n_a$ is the ratio of the amplitude of the signal to that of the
data and is called the amplitude modulation index.  The frequency
spectrum of this signal is twice as wide as the bandwidth of $m(t)$.
To see this, suppose $m(t) = 2\cos(2\pi f t)$.  Then $s(t)$ can be
written as
\[
s(t) = cos(2\pi f_c t) + \frac{n_a}{2} \left(\cos(2\pi(f_c + f) t) +
\cos(2\pi(f_c - f)t\right).
\]
As the above equation shows, any frequency component at frequency $f$
in the data $m(t)$ shows up at frequencies $f_c + f$ and $f_c - f$ in
the encoded signal.  This is also easily seen in the figures given in
the textbook.  Such an AM encoding is referred to as Double
Side-Banded and requires a transmission bandwidth $B_T$ that is twice
the bandwidth $B$ of the original signal.  Single-side banded AM
encoding filters out one of the bands, and has $B_T = B$.

\subsection{Angle modulation}
Modulating the analog signal by varying its phase or frequency can be
viewed as two variants of {\em angle modulation}.  If the analog data
is given by $m(t)$, then an angle modulation encoded signal is given
by
\[
s(t) = A \cos(2\pi f_c t + \phi(t),
\]
where $\phi(t) = n_p m(t)$ for phase modulation and $\phi'(t) = n_f
m(t)$ for frequency modulation.  Here $n_p$ refers to a parameter
called the phase modulation index, $n_f$ is the frequency modulation
index and $\phi'(t)$ is the derivative of $\phi(t)$ wrt $t$.

One property of angle modulation is that it has constant power
regardless of how the analog data varies.  This makes it less
susceptible to noise.  The transmission bandwidth required is given by
$B_T = 2(\beta + 1) B$, where $\beta = n_p A$ for phase modulation and
$\beta = n_f A/(2\pi B)$ for frequency modulation.

\section{Digitization: Analog data on digital signals}
There are two popular ways of digitizing analog data.  {\em Pulse code
modulation}\/ (PCM) is a process in which the analog data is sampled
at a rate given by Nyquist's sampling theorem (that is, twice the
bandwidth of the analog data).  If we store each sample to full
precision, then we can recover the entire analog data (assuming it has
a finite bandwidth).  This is, however, unrealistic.  In PCM, we
divide the range of the analog data to a number of intervals (a
process called quantization), and replace a sample by the nearest (or
closest higher/lower) quantized level.

The number of quantized levels determines the resolution of
digitization.  More the number of levels, the more precise (or higher
resolution) the digitization process.  That is, the error between the
analog data value at any time, and the quantized value that we replace
it by is small.  The quantization step in PCM may be either uniform
(all intervals are identical size) or non-uniform in which the
intervals are smaller at lower analog data values (resulting in more
uniform relative precision).

An alternative digitization technique is {\em Delta Modulation}, in
which rather than encoding the actual analog data values, we encode
the variation in data.  We select a step size $s$ and modulation rate
$r$, and every $1/r$ seconds increase (resp., decrease) the digital
signal value by $s$ if the analog data value increases (resp.,
decreases) in $1/r$ seconds.  Clearly, the resolution of delta
modulation depends on the choice of $r$ and $s$; higher the $r$,
better the resolution.  The effect of $s$ is more subtle and depends
on how the analog data varies.  For instance, if $s$ is too small, the
digital signal may be underestimating (resp., overestimating) the
analog data when it is increasing (resp., decreasing).

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