# Communication Systems/Binary Modulation Schemes

This page discusses the binary modulation schemes and \"keying\".

## What is \"Keying?\"

Square waves, sinc waves, and raised-cosine rolloff waves are all well
and good, but all of them have drawbacks. If we use an optimal, matched
filter, we can eliminate the effect of jitter, so frankly, why would we
consider square waves at all? Without jitter as a concern, it makes no
sense to correct for jitter, or even take it into consideration.
However, since the matched filter needs to look at individual symbols,
the transmitted signal can\'t suffer from any intersymbol interference
either. Therefore, we aren\'t using the sinc pulse.

Since the raised-cosine roll-off wave suffers from both these problems
(in smaller amounts, however), we don\'t want to use that pulse either.

So the question is, what other types of pulses can we send?

It turns out that if we use some of the techniques we have developed
using analog signal modulation, and implement a sinusoidal carrier wave,
we can create a signal with no inter-symbol interference, very low
bandwidth, and no worries about jitter. Just like analog modulation,
there are 3 aspects of the carrier wave that we can change: the
amplitude, the frequency, and the phase angle. Instead of
\"modulation\", we call these techniques **keying** techniques, because
they are operating on a binary-number basis.

There is one important point to note before continuing with this
discussion: Binary signals **are not** periodic signals. Therefore, we
cannot expect that a binary signal is going to have a discrete spectra
like a periodic squarewave will have. For this reason, the spectral
components of binary data are continuous spectra.

## Amplitude Shift Keying

In an ASK system, we are changing the amplitude of the sine wave to
transmit digitial data. We have the following cases:

-   Binary 1: $A_1 \sin(f_c t)$
-   Binary 0: $A_0 \sin(f_c t)$

The simplest modulation scheme sets A0 = 0V (turning the transmitter
off), and setting A1 = +5V (any random non-zero number turns the
transmitter on). This special case of ASK is called OOK (On-Off keying).
Morse code uses OOK.

Another common special case of ASK sets A1 to some positive number, and
A0 to the corresponding negative number A0 = -A1. We will mention this
case again later.

In ASK, we have the following equation:

$$a(t) \sin(\omega t)$$

by the principal of duality, multiplication in the time domain becomes
convolution in the frequency domain, and vice-versa. Therefore, our
frequency spectrum will have the following equation:

$$A(j\omega) * \delta(t - \omega)$$

where the impulse function is the fourier-transform of the sinusoid,
centered at the frequency of the wave. the value for A is going to be a
sinc wave, with a width dependent on the bitrate. We remember from the
Signals and Systems book that
convolution of a signal with an impulse is that signal centered where
the impulse was centered. Therefore, we know now that the frequency
domain shape of this curve is a sinc wave centered at the carrier
frequency.

## Frequency Shift Keying

In **Frequency Shift Keying** (FSK), we can logically assume that the
parameter that we will be changing is the frequency of the sine wave.
FSK is unique among the different keying methods in that data is never
transmitted *at* the carrier frequency, but is instead transmitted at a
certain offset from the carrier frequency. If we have a carrier
frequency of $f_c$, and a frequency offset of $\Delta f$, we can
transmit binary values as such:

-   Binary 1: $A \sin ((f_c + \Delta f)t)$
-   Binary 0: $A \sin ((f_c - \Delta f)t)$

Similar to ASK, we have FSK, which uses 2 different frequencies to
transmit data. For now we will call them $\omega1, \omega2$. Using the
same logic that we used above, the fourier representations of these
waves will be (respectively):

$$A_1(j\omega) * \delta(t - \omega1)$$

$$A_0(j\omega) * \delta(t - \omega2)$$

With one sinc wave centered at the first frequency, and one sinc wave
centered at the second frequency. Notice that A1 and A0 are the
half-square waves associated with the 1s and the 0s, respectively. These
will be described later.

### Error Rate

The BER of coherent QPSK in the presence of gaussian and Rayleigh noise
is as follows:

:   {\| border=1

\|- \| Gaussian Noise \|\| Rayleigh Fading \|- \|
$\frac{1}{2}\operatorname{erfc} \left( \sqrt{\frac{E_b}{N_0}} \right)$
\| $\frac{1}{2}\left( 1 - \sqrt{\frac{\gamma_0}{2 + \gamma_0}} \right)$
\|}

## Phase Shift Keying

PSK systems are slightly different then ASK and FSK systems, and because
of this difference, we can exploit an interesting little trick of
trigonometry. PSK is when we vary the phase angle of the wave to
transmit different bits. For instance:

-   Binary 1: $A \sin(f_c t + \phi_1)$
-   Binary 0: $A \sin(f_c t + \phi_0)$

If we evenly space them out around the unit-circle, we can give
ourselves the following nice values:

-   Binary 1: $A \sin(f_c t + 0)$
-   Binary 0: $A \sin(f_c t + \pi)$

Now, according to trigonometry, we have the following identity:

$\sin(f_c t + \pi) = -\sin(f_c t)$

So in general, our equations for each signal (s) is given by:

-   $s_1(t) = A\sin(f_c t)$
-   $s_0(t) = -A\sin(f_c t)$

Which looks awfully like an ASK signal. Therefore, we can show that the
spectrum of a PSK signal is the same as the spectrum of an ASK signal.

There are two
commonally
used forms of Phase Shift keying Modulation:

Binary Phase Shift Keying (BPSK)

Quadrature Phase Shift Keying (QPSK)

Binary Phase Shift keying is set out above.

### QPSK

Quadrature Phase Shift Keying utilises the fact that a cosine wave is in
quadrature to a sine wave, allowing 2 bits to be simultaneously
represented.

-   Binary 11: $A \sin(f_c t + 0) + \cos(f_c+\pi/2)$
-   Binary 10: $A \sin(f_c t + 0) + \cos(f_c -\pi/2)$
-   Binary 01: $A \sin(f_c t + \pi) + + \cos(f_c+\pi/2)$
-   Binary 00: $A \sin(f_c t + \pi) + + \cos(f_c - \pi/2)$

QPSK has the advantage over BPSK of requiring half the transmission band
width for the same data rate, and error probability.

### Error Rate

The BER of coherent BPSK in the presence of gaussian and Rayleigh noise
is as follows:

:   {\| border=1

\|- \| Gaussian Noise \|\| Rayleigh Fading \|- \|
$\frac{1}{2}\operatorname{erfc} \left( \sqrt{\frac{E_b}{N_0}} \right)$
\| $\frac{1}{2}\left( 1 - \sqrt{\frac{\gamma_0}{1 + \gamma_0}} \right)$
\|}

## Binary Transmitters

## Binary Receivers
