# Communication Systems/Microwave Systems

Microwaves are high frequency radio waves (\> 1 GHz) and they propagate
much like any other electromagnetic phenomenon in free space. Because
they are electromagnetic waves and not electrons, they do not propagate
well down copper cables. instead, the waves are guided down hollow
conductors.

## Waveguides

Two boundary conditions must be met in a waveguide for a wave to travel
down the guide:

:   • The electric field must terminate normally on the conductor (the
    tangential component of the electric field must be zero).
:   • The magnetic field must lie entirely tangent along the wall
    surface. (the normal component of the magnetic field must be zero).

Consequently, TEM waves cannot be conducted in a waveguide. (Recall that
in a TEM wave, the electric and magnetic fields are at right angles to
each other and the direction of propagation.)

The most common wave used in waveguides is the TE~10~, meaning
transverse electric. The subscript TE~ab~ denotes the number of half
cycles which appear in the *a* and *b* dimensions of the waveguide.

:   ![](Waveguide.gif "fig:Waveguide.gif")

There are many other waveguide modes. A TE~10~ wave may be formed when
two TEM wave intersect:

:   ![](TE10a.gif "fig:TE10a.gif")

```{=html}
<!-- -->
```

:   ![](TE10b.gif "fig:TE10b.gif")

```{=html}
<!-- -->
```

:

    :   The velocity of a TEM wave in air is: $c=\lambda f$

```{=html}
<!-- -->
```

:

    :   The phase velocity (the apparent wave velocity along the guide
        wall) of a TE wave: $v_{p}=\lambda _{g}f$

```{=html}
<!-- -->
```

:

    :   Therefor the phase or guide velocity is:
        $v_{p}=\frac{\lambda _{g}}{\lambda }c$

```{=html}
<!-- -->
```

:

    :   From the right angle triangle BCD, we obtain:
        $\cos \alpha =\frac{\lambda }{2a}$

```{=html}
<!-- -->
```

:

    :   From the right angle triangle ABD we obtain:
        $\sin \alpha =\frac{\lambda }{\lambda _{g}}$

Since

$$\text{sin}^{2}\alpha \text{ + cos}^{2}\alpha \text{ = 1}$$

:

    :   We obtain:
        $$\left( \frac{\lambda }{\lambda _{g}} \right)^{2}+\left( \frac{\lambda }{2a} \right)^{2}=1$$

```{=html}
<!-- -->
```

:

    :   or
        $$\frac{1}{\lambda _{g}^{2}}=\frac{1}{\lambda ^{2}}-\frac{1}{\left( 2a \right)^{2}}$$

```{=html}
<!-- -->
```

:

    :   where
        $$\lambda _{g}=$$ guide wavelength for the TE~10~ mode

        $$\lambda =$$ free space wavelength of the TEM wave

        $$a=$$ broad dimension of the waveguide

When $\lambda =2a$, the guide wavelength becomes infinite. This
corresponds to a TEM wave bouncing from side to side in the guide with
no velocity component along the guide. This is cutoff wavelength
$\left( \lambda _{c} \right)$and represents the lowest frequency that
can be propagated.

**Phase & Group Velocity**

:   ![](TE10c.gif "fig:TE10c.gif")

The equations above can be manipulated to obtain:

$$\lambda _{g}=\frac{\lambda }{\sqrt{1-\left( \frac{\lambda }{\lambda _{c}} \right)^{2}}}$$

This means that the phase velocity is always greater than the speed of
light. The TEM components zigzag through the guide at the speed of
light, but they convey power at the group velocity. The group velocity
is always less than the speed of light.

**Analogy**

Nothing physical can exceed the speed of light, but it is possible for
some aspect of a wave to exceed light speed. For example, the splash
that occurs when a wave washes up on the beach can travel faster than
the wave, if it strikes at some angle other than 90^0^.

:   ![](Microwave_Systems_splash.gif "fig:Microwave_Systems_splash.gif")

```{=html}
<!-- -->
```

:   **Example**
:   A rectangular waveguide measuring 0.9 by 0.45 inches is fed with a
    10 GHz carrier. Determine if a TE~10~ wave will propagate, and if
    so, determine its guide wavelength, group velocity and phase
    velocity.

$$a=0.9\ \text{in}\text{.}\ =2.28\ cm$$

$$\lambda _{c}=2\times 2.28=4.58\ cm$$

$$\lambda =\frac{3\times 10^{8}}{10^{10}}=3\ cm$$

:   Since $\lambda <\lambda _{c}$, a TE~10~ wave will propagate.

$$\lambda _{g}=\frac{\lambda }{\sqrt{1-\left( \frac{\lambda }{\lambda _{c}} \right)^{2}}}=\frac{3}{\sqrt{1-\left( \frac{3}{4.58} \right)^{2}}}=3.97\ cm$$

$$v_{p}=c\frac{\lambda _{g}}{\lambda }=3\times 10^{8}\frac{3.97}{3}=4.4\times 10^{8}\ m/\sec$$

$$v_{g}=c\frac{\lambda }{\lambda _{g}}=3\times 10^{8}\frac{3}{3.97}=2.27\times 10^{8}\ m/\sec$$

### Wave Impedance

The ratio of the transverse electric and magnetic fields constitutes an
impedance. This should not be considered the same as a resistance,
because it does not dissipate power.

The TE~10~ wave consists of two equal TEM waves where the electric field
is transverse along the *b* dimension but the magnetic field is along
the a dimension is described by: $H_{a}=H\sin \alpha$. Consequently, the
wave impedance is:

$$Z_{w}=\frac{E_{b}}{H_{a}}=\frac{E}{H\sin \alpha }$$

:   For a TEM wave: $\frac{E}{H}=120\pi$

Therefore:

$$Z_{w}=\frac{120\pi }{\sin \alpha }=120\pi \frac{\lambda _{g}}{\lambda }\quad \Omega$$

:   or

$$\frac{120\pi }{\sqrt{1-\left( \frac{\lambda }{\lambda _{c}} \right)^{2}}}$$
for TE waves.

$$120\pi \sqrt{1-\left( \frac{\lambda }{\lambda _{c}} \right)^{2}}$$ for
TM waves.

A TE wave has a characteristic impedance $\Omega >120\pi$. As the cutoff
frequency is approached, the impedance approaches infinity. These modes
are generated by means of a vertical probe antenna.

A TM wave has a characteristic impedance $\Omega <120\pi$. As the cutoff
frequency is approached, the impedance approaches zero. These modes are
generated by means of a horizontal probe antenna.

A waveguide with a dielectric other than air will have its cutoff
wavelength increased by the square root of the dielectric constant.

Waveguide attenuation characteristics are quite complex and are
generally derived empirically. Attenuation varies with interior wall
coating, guide dimensions, and operating frequency Attenuation can occur
because the induced wall currents are electrons interacting with the
guide.

There is a limit to the amount of power which can be conveyed in a
guide. If this limit is exceeded, electrical arcing will occur and
severe attenuation results. The maximum power handling capability of an
air filled rectangular waveguide operating in the TE mode is:

$$P_{\max }=6.63\times 10^{-4}E_{\max }^{2}ab\frac{\lambda }{\lambda _{g}}$$

:   where *E~max~* = maximum voltage gradient in v/cm

### Standing Waves

Standing waves act much like the plucked string on a guitar. The signal
magnitude changes but its location does not. One of the most useful
places to create a standing wave is at a radiating antenna, but it may
also be useful to create one in a guide.

:

    :   ![](Standing_waves.gif "fig:Standing_waves.gif")

**Wall Currents**

Traveling waves are time invariant. The pattern appears to keep its
shape as it moves down the guide. However, as they propagate down the
line, they induce currents in the guide walls, at right angles to the
magnetic field.

:

    :   ![](Wall_currents.gif "fig:Wall_currents.gif")

Slots that do not noticeably interrupt wall current, do not radiate.

### Microwave Components

**Coupling Energy into a Waveguide**

Probe or capacitive coupling acts like a 1/4$\lambda$ Marconi antenna.
The probe, which might be the center conductor of a coaxial cable, is
positioned 1/4 $\lambda$ from the end of the guide. Since a total
inversion occurs when the wave reflects, it adds in phase with the
coupled signal.

:

    :   ![](Microwave_probe.gif "fig:Microwave_probe.gif")

The probe can be tapered to handle wideband signals. TE modes are
generated if the probe is placed in the wide dimension of the guide,
while TM modes are produced when the probe is placed in the narrow
sidewall.

Loop or magnetic coupling creates magnetic fields that are launched from
the antenna. The loop may be placed anywhere the magnetic field exists.

:

    :   ![](Loop_coupling.gif "fig:Loop_coupling.gif")

Aperture or slot coupling occurs when a slot is cut in the maximum of
the electric or magnetic field area

**Bends & Twists**

:

    :   ![](Bends_twists.gif "fig:Bends_twists.gif")

The twist section is used to change between horizontal and vertical
polarization.

**Tees**

:

    :   ![](Tees.gif "fig:Tees.gif")

Shunt tee - The signal at port C is the vector sum of ports A and B if
they are used as inputs. If port C is used as the input, the power is
evenly split between A and B.

Series Tee - Using the top port as the input, the power will b evenly
split between A and B however, they will be anti phase.

Hybrid Tee - If A and B are used as the inputs, port C will be the
vector sum and port D will be the vector difference. If C is the input,
the energy will be evenly split between A and B. This device is often
used to connect a transmitter and receiver to the same antenna. The
antenna is at port B, the transmitter at C, the receiver at D, and a
matched load at A.

**Tuners**

:

    :   ![](Tuners.gif "fig:Tuners.gif")

If the reactive stub is \< 1/4$\lambda$ in the waveguide, it acts like a
capacitor. If it is \> 1/4$\lambda$ in the waveguide, it acts like an
inductor. If it is 1/4$\lambda$ in the waveguide, it acts like an LC
resonant circuit where the Q is proportional to the post diameter.

Tuning is sometimes done to minimize the amount of signal reflected from
a waveguide component. This has a significant effect on the reflection
coefficient and VSWR (voltage standing wave ratio). The effect of these
things can be plotted and observed on a Smith Chart.

**Terminators**

All of the energy travelling down a waveguide must be absorbed in order
to prevent reflections. This is accomplished by placing a resistor in
the center of the guide where the electric field intensity is the
greatest.

:

    :   ![](Resistive_terminator.gif "fig:Resistive_terminator.gif")

A short circuit is used if reflections are required.

A variable attenuator can be constructed by repositioning the absorbing
material in the electric field.

**Directional Couplers**

:

    :   ![](Directional_coupler.gif "fig:Directional_coupler.gif")

### Microwave Cavities

The dimensions of a microwave cavity must be some multiple of 1/2 l, and
the Q may be as high as 100,000.

$$Q=\frac{\text{cavity}\ \text{volume}}{\text{interior}\ \text{area}}$$

Energy is coupled into the cavity by means of a probe, loop, slot, or
electron beam.

Conventional tubes have constant velocity beams and vary the number of
electrons emitted to modulate the beam intensity. Microwave tubes emit a
constant stream of electrons and vary their velocity to modulate the
beam intensity.

**Two Cavity Klystron**

An electron beam is created at the cathode and passed by the buncher
cavity. The RF energy coupled into the cavity causes the beam velocity
to vary, thus causing the electrons to bunch. Energy is coupled to the
catcher cavity when the beam passes its port. The net result is that the
RF energy may be amplified by a factor of 1000.

:

    :   ![](Two_cavity_klystron.gif "fig:Two_cavity_klystron.gif")

The klystron acts as a high Q narrowband amplifier. It may have more
bunching cavities to increase the gain or the cavities may be staggered
tuned to increase the bandwidth.

**Reflex Klystron**

The reflex klystron has a single cavity that acts as both buncher and
catcher. The beam is passed by the cavity, bunched, repelled and passed
by the cavity a second time, and the absorbed in the sidewalls. It is
necessary to turn on the repeller voltage prior to turning on the anode
supply, otherwise the tube will self-destruct.

The reflex klystron is used as a low power, low efficiency, variable
frequency oscillator. The operating frequency is adjusted by changing
the cavity volume.

**Magnetron**

The magnetron is a high power microwave oscillator that has found
widespread use in radar, radio beacon, and microwave oven applications.

The cylindrical cathode is surrounded by the anode, which has several
resonant cavities. The entire assembly is placed in an intense right
magnetic field running parallel to the cathode axis. This field causes
an electron emitted from the cathode to follow a curved path on its way
to the anode. However, the electron passes cavities on the route and so
looses energy, causing it to fall back towards the cathode. The cathode
repels the falling electron forcing it back towards the anode. This
process continues ad infinitum. Electron bunching occurs each time the
beam passes a cavity.

:

    :   ![](Magnetron.gif "fig:Magnetron.gif")

The critical magnetic field is that which allows the electron to just
graze the anode. If the field intensity falls below this value,
oscillation will cease. The dominant or π mode occurs when the polarity
on adjacent poles is 180o apart. This causes the phase focusing effect
where electrons are bunched in adjacent cavities. The beam shift around
the interior is some integer multiple of the wavelength.

To prevent the bunched electrons to skip some cavities thus creating
spurious modes, every second pole is strapped together. The output is
taken from one of the cavities via a probe antenna. The magnetron can be
mechanically tuned by about 5%.

The continuous power output can be as high as 25 kW, but with low duty
cycle pulsing \[.001\] the peak output can be in the mega watt range.

**TWT**

Traveling wave tubes have a higher bandwidth but lower Q than klystrons.
The RF signal traveling a helical path around the beam causes the beam
to bunch. This in turn reinforces the RF signal causing amplification.

It can be used as a low power \[30 mW\], low noise amplifier with about
a bandwidth of about one octave. Medium power devices operate at about
25 watts. High-powered devices can be pulsed at about 100 kW.

TWTs are used as broadband amplifiers and repeaters in TV, radar and
satellite applications. It may be frequency or amplitude modulated.

### Microwave Semiconductors

**Gunn Diodes**

The Gunn diode is not a diode in the truest sense because it does not
contain a semiconductor junction. It is simply a piece of N-type gallium
arsenide.

Normally the mobility of electrons increases as the electric field
increases in a semiconductor. In this device, there is a range where the
mobility of the electrons actually decreases as the electric field
increases. Essentially the current falls as the voltage rises. This
means that for a narrow range, the device exhibits a negative
resistance. This is a very unstable situation and the device has a
tendency to oscillate when biased in this region. The actual frequency
of oscillation is dependent on the physical size of the semiconductor
and occurs when the electron transit time is equal to one oscillation
period.

**Pin Diodes**

PIN diodes are used as high frequency switches. The have very little
junction capacitance and a very low forward voltage drop.

### Microwave Measurements

Reflections in a waveguide are undesirable. They send power back to the
source, which may ultimately damage the signal source, and they create
standing waves.

**Standing Waves**

If waves are allowed to reflect from a microwave device, the incident
and reflected waves will interact to create a standing wave. This is
somewhat similar to the vibration of a stringed instrument.

The reflection coefficient is defined as the ratio of the reflected and
incident voltage.

$$\Gamma =\frac{E_{r}}{E_{i}}$$

The ratio of maximum to minimum voltage of a standing wave is known as
the VSWR (voltage standing wave ratio).

$$VSWR=\frac{\left| E_{i} \right|+\left| E_{r} \right|}{\left| E_{i} \right|-\left| E_{r} \right|}=\frac{E_{\max }}{E_{\min }}=\frac{1+\left| \Gamma  \right|}{1-\left| \Gamma  \right|}$$

To eliminate the constant use of absolute magnitude bars, the reflection
coefficient is often written as:

$$\rho =\left| \Gamma  \right|$$

And since voltage is generally assumed to be the measured parameter:

$$SWR=\frac{1+\rho }{1-\rho }$$ or $\rho =\frac{SWR-1}{SWR+1}$

The physical distance between two consecutive minimums corresponds to
one-half the guide wavelength $\frac{\lambda _{2}}{2}$

The VSWR can take on any value between 1 and ∞.

If VSWR = ∞, total reflection occurs. Ideally for a matched load, VSWR =
1 and there are no reflections. In practice, it is generally not cost
effective to try to reduce the VSWR below 1.1.

**Smith Chart**

All values on the Smith chart are normalized to the characteristic
impedance of the line.

$$Z_{norm}=\frac{Z}{Z_{0}}$$

Impedances are described by circles on a Smith chart.

:

    :   ![](Smith_chart_loci.gif "fig:Smith_chart_loci.gif")

This peculiar plotting method has a number of benefits. For one thing,
the center of the chart corresponds to 1+1j. Consequently, it can be
used to characterize any normalized transmission environment.

Furthermore, the SWR appears as a circle in the chart.

:

    :   ![](SWR_circle.gif "fig:SWR_circle.gif")

The normalized impedance is located at the intersection of SWR circle
and reflection coefficient phase angle.

**Review Questions**

Quick Quiz

:   1\. The magnetic field in a waveguide must terminate \[normally,
    tangentially\] on the conductor.
:   2\. The \[phase, guide\] velocity in a waveguide can exceed the
    speed of light.
:   3\. In a waveguide the electric field must terminate \[normally,
    tangentially\] on the conductor.
:   4\. TM waves can be generated by placing a probe antenna in the
    \[narrow, broad\] side of a waveguide.
:   5\. A \[shunt, series, hybrid\] tee is typically used to connect a
    transmitter and receiver to the same antenna.
:   6\. When capacitively coupling energy into a waveguide, the probe is
    placed \[1/4, 1/2, 1\] $\lambda$ away from the end plate.
:   7\. A reactive stub acts like a capacitor when it is inserted
    \[more, less\] than 1/4 $\lambda$ into the guide.
:   8\. A TE wave has an impedance \[greater, less\] than 120 π ohms.
:   9\. \[TE, TM, TEM\] waves cannot propagate in a waveguide.
:   10\. The subscript in TEab denotes the number of \[quarter, half,
    complete\] cycles which appear in the a and b dimensions of the
    waveguide.
:   11\. The dominant waveguide mode is \[TE01, TE10, TE11\].
:   12\. The speed at which energy travels down a waveguide is known as
    the \[group, phase\] velocity.
:   13\. The cutoff wavelength represents the \[highest, lowest\]
    frequency that can propagate down a waveguide.
:   14\. The \[shunt, series, hybrid\] tee evenly splits power between
    two branches, but the outputs are phase inverted with respect to
    each other.
:   15\. The \[shunt, series, hybrid\] tee is used to couple a
    transmitter and receiver to the same antenna.
:   16\. The \[magnetron, TWT\] is a microwave oscillator.
:   17\. The \[reflex klystron, TWT\] is a microwave amplifier.
:   18\. The magnetron \[is, is not\] a tube.
:   19\. The klystron can be used as either an amplifier or an
    oscillator. \[True, False\]
:   20\. The microwave tubes modulate the electron beam \[magnitude,
    velocity\].
:   21\. TWTs use buncher cavities to modulate the beam. \[True, False\]

Analytical Questions

:   1\. A rectangular waveguide has the following characteristics:

Internal dimensions: 10.16 x 22.86 mm Feed: 10.5 GHz gunn diode

`       Determine if:`

:   a\) If a TE~20~ wave will propagate
:   b\) If a TE~10~ wave will propagate
:   c\) The dominant mode guide wavelength
:   d\) The dominant mode group velocity
:   e\) The dominant mode phase velocity
:   f\) The dominant mode wave impedance
:   g\) Explain how you would measure the guide wavelength.

Composition Questions

:   1\. Sketch the cross section of a two-hole directional coupler and
    discuss its operation.
:   2\. Sketch the cross section of a magnetron and discuss its
    operation.
:   3\. Sketch the TE10 mode in a rectangular waveguide, and show the
    position of all of the radiating and non-radiating slots.


# Communication Systems/Fiber Optic Systems

Long-haul trunks were the first application of optical fiber to gain
universal acceptance. Three alternate methods are coax (copper),
terrestrial microwave and satellite. While microwave towers can be used
to bridge relatively small bodies of water, cables or satellites are
needed to span oceans.

Telegraph cables were first deployed in the mid-1800s and the first
successful trans-Atlantic cable was laid in 1858. For 100 years copper
submarine cables were the principal means of communication between North
America and Europe. In the 1960s satellites gained ascendancy but today
fiber cable dominates.

Fiber cable has some significant advantages over satellite technology:

:   • Fiber systems can be repaired, geo-stationary satellites cannot
:   • Fiber systems have a slightly longer life expectancy
:   • Fiber systems can be upgraded while in service
:   • Fiber deployment has a lower risk than launching a satellite
:   • Fiber propagation delay is significantly lower than satellites

## Fiber Deployment (ground based)

**Telecom Canada**

The current Trans-Canada fiber system stretches some 4100 miles and
includes under water links to PEI and Newfoundland. The cable is buried
at a minimum depth of 5 feet in most places. The cable is buried 2 feet
in rocky areas, but over major rivers such as the Thompson, it is 6 feet
below the riverbed.

Eight fibers are reserved for Trans-Canada traffic, with additional
fibers are used for toll traffic from the participating operating
companies.

The cable has a Kevlar strength member surrounded by a star shaped
polyethylene core, which can support up to 5 fiber tubes. Each fiber
tube can contain up to 6 fibers. The cable has multiple layers of
protection, but in avalanche prone areas, it was also encased in an 8
inch steel conduit.

## Submarine Fiber

## Optical Phenominon

### Wave-Particle Duality

In 1678 Huygens showed that reflection and refraction could be explained
by wave theory. While not rejecting the wave theory entirely, Einstein
in 1905 suggested that light could be thought of as a small energy
packet called a photon. This idea although not actually original, seemed
to explain the Compton effect later discovered in 1921.

Today light is generally thought of as an electromagnetic wave when it
is propagating. However, when it interacts with matter by means of
emission or absorption, it is preferable to think in terms of photons.
Photons have energy, but no rest mass. If a photon stops, it ceases to
exist as a particle, and is transformed into some other form of energy,
such as heat.

A rather hybrid way of thinking of light, is as a wave packet. Perhaps
someday, a better model will be developed.

:

    :   ![](Wave_packet.gif "fig:Wave_packet.gif")

Light in the everyday world is incoherent. That is to say that the wave
packets arrive in a chaotic, random fashion. On the other hand, coherent
light as generated from a laser has the wave packets synchronized or in
phase.

The perceived color of light is a function of its wavelength. The
characteristic wavelength in the packet is related to its velocity and
frequency:

$$v=\lambda f$$

:   where:

$$v=$$ velocity

$$\lambda =$$ wavelength

$$f=$$ frequency

It has been experimentally determined that the velocity of light in a
vacuum, is about 299,793,000 meters per second. Fiber optic systems
operate at a frequency of approximately 3 x 10^8^ GHz, with the most
common transmission bands at wavelengths of 0.8 - 0.9 μm and 1.2 - 1.4
μm. The amount of energy in light can be determined by quantum theory
and is proportional to frequency and given by:

$$E=hf=\frac{hc}{\lambda }$$

:   where:

$$E=$$ Energy in Joules

$$h=6.625\times 10^{-34}\frac{J}{\sec }=$$ Planck\'s constant

$$f=$$ frequency

$$c=3\times 10^{8}\frac{m}{\sec }=$$ velocity of light

$$\lambda =$$ wavelength

The radiated spectral energy in watts per unit area is given by:

If we examine light at the macroscopic level, we observe that waves
radiate spherically from their source. This is readily observable with
incoherent sources and less pronounced with coherent sources. At a large
distance, the wavefront flattens out into a plane wave. A ray path shows
the direction that the wave or photon is traveling. In the case of a
point source, this might resemble:

:

    :   ![](Ray_path.gif "fig:Ray_path.gif")

When light strikes an object, it can be reflected, refracted, or
absorbed. A ray of light striking the boundary between two dissimilar
transparent materials is usually split into reflected and refracted rays
with generally very little absorption.

Different wavelengths of light travel at the same velocity in a vacuum,
but this is not true in other mediums. This change in velocity leads to
refraction and color separation. If the speed is slowed in a medium, the
ray is redirected towards the normal. As light leaves a slower medium
and enters a faster one, it accelerates and is redirected away from the
normal.

:

    :   ![](Reflection_refraction.gif "fig:Reflection_refraction.gif")

By convention, the angles involved in reflection and refraction are
measured with respect to the normal at the point of contact. All three
components are in the same plane \[there are some exceptions\]. In the
above illustration, the plane of incidence is the paper.

**Displacement and Polarization**

Energy displacement in waves can be either longitudinal or transverse
with respect to the direction of travel. Longitudinal waves such as
sound and seismic waves, are displaced in the direction of travel.
Transverse waves such as radio and water waves are displaced at right
angles to the direction of propagation.

Light travels as a TEM wave, having electric and magnetic fields
perpendicular to the direction of travel. If there is no particular
reason for the electrical fields of light to favor any particular
orientation, the field of the waves are directed (polarized) at any
angle perpendicular to the direction of propagation. This is the case
with sunlight.

The combined effect of many simultaneous waves is the vector sum of all
of their components. This is known as the superposition principle. This
principle holds true with all forms of waves, including light. This
raises an interesting question: Since the polarization and phase of
natural light is totally random, why don't the electric field components
of all the waves simply average out to zero and the light cancel itself
out? The reason is simple: they do average out, but, since this
averaging is not exact, but is a random one, the laws of statistics
hold. So, when we sum N random fields which oscillate, say, from minus
to plus one, we expect that the sum is of the order of the square root
of N, which can be enormously less than N, but is not zero. And, since
the energy (the intensity) of light is proportional to the square of the
electric field, we have that the mean energy of the sum is just N times
the individual energies, which is precisely what we expect.

A cancellation or enhancement effect do occur, but to observe it the
light must be coherent. Traditionally coherent light was created by
passing light from some distant source through a small hole. This
effectively selects light coming from only a very small part of the
originating source thus causing it to be more uniform. Today, it is
easier to use a laser, which is not only very coher ent, but also
essentially monochromatic. Interference patterns of light and dark areas
can be observed when coherent waves intersect.

Unpolarized light can be polarized by absorption, scattering and
reflection.

### Reflection

There are two types of reflection: diffuse and specular.

**Diffuse Reflection**

This is the most common form of reflection. If the illuminated object is
not microscopically homogeneous in its structure, the light is scattered
in all directions by multiple reflections or diffusions by its
inhomogeneities (both internal and on the surface). Most materials are
inhomogeneous: minerals are generally polycrystalline, organic matter is
generally made by cells or fibers, so the light sent to our eyes by most
objects is diffusely reflected. Because of this phenomenon, we can see
other objects.

:

    :   ![](Diffuse_reflection.gif "fig:Diffuse_reflection.gif")

Sunlight contains all of the perceived colors in such a way that their
combined values look colorless. When it strikes an object, some
wavelengths are absorbed, while others are reflected. The perceived
color is the vector sum of all the reflected waves.

**Specular Reflection**

Light striking a smooth surface is reflected at the same angle with
which it strikes, but suffers some loss. Instead of seeing the
reflecting surface, one sees an image of where the light originates.

The reflected light can have the appearance of coming from either in
front of or behind the reflecting surface. If the image appears to be
behind the surface, as in the case of a mirror, it is referred to as a
virtual image.

:

    :   ![](Virtual_image.gif "fig:Virtual_image.gif")

If the image is seen on a screen placed in front of the reflecting
surface, it is called a real image. This principle is used in reflecting
telescopes. Real images are most often generated by refraction and are
used in cameras and projectors.

:

    :   ![](Real_image.gif "fig:Real_image.gif")

**Variable Reflective Coatings**

About 4% of light striking a smooth air-glass boundary is reflected.
This can be reduced by applying non-reflecting coatings. Photographers
and fiber optics designers are interested in non-glare coatings, since
reflection reduces the amount of light entering and exiting the glass.

By depositing a thin high refractive index film over a thin low
refractive index film, the amount of reflection can be increased to
about 50%. This principle is used to make a beam splitter, a device of
great value in optical measuring equipment and color TV cameras.

**External Reflection**

A phase reversal occurs when light is reflected by a more optically
dense medium. This also happens when radio waves strike a metallic
surface. The electric field component is short-circuited. Since the
incident and reflected fields cancel at the point of incidence, the
reflected waves are equal in amplitude but opposite in phase to the
incident wave.

The relationship between angle of incidence and amount of reflection is
very complex. At normal incidence, a reflecting surface reflects all
components equally well. However, at other angles, objects prefer to
reflect light having the electric field component perpendicular to the
plane of incidence. At the extreme case, Brewster's angle, only
perpendicular components are reflected, and complete polarization
occurs. Brewster's angle is defined by:

$$\tan \left( \theta _{P} \right)=\frac{\eta _{1}}{\eta _{2}}$$

At Brewster's angle, the reflected and refracted rays are at right
angles to each other.

:

    :   ![](Brewsters_angle.gif "fig:Brewsters_angle.gif")

When noncoherent light strikes glass, about 15% of the perpendicular
components are reflected, and none of the others. The refracted ray
consists of the remaining 85% of the perpendicular component as well as
all other orientations. The polarized reflection can be enhanced by
stacking many thin glass plates together.

**Internal Reflection**

If light strikes an optically less dense medium, the reflected wave is
not phase reversed. When the angle of incidence is greater than the
critical angle, total internal reflection occurs.

However, a refracted beam of sorts does exist. It is sometimes
designated as frustrated total reflection or more commonly evanescent
wave. This wave does not dissipate power, and extends only a few
wavelengths into the faster medium. It decays exponentially according to
the relationship:

$$e^{\alpha x}$$

:   where

$$\alpha =k_{0}\sqrt{\eta _{1}^{2}\sin ^{2}\Theta _{i}-\eta _{2}^{2}}$$

$$k_{0}=$$ free space propagation factor

$$\Theta _{i}=$$ angle of incidence

$$x=$$ distance into the faster medium

### Refraction

Refracted rays create diverse effects ranging from the rainbow to the
apparent bending of sticks protruding into water. From experimentation,
it was observed that the ratio of the sines of the incident and
refracted rays is constant:

$$\frac{\sin \left( \theta _{0} \right)}{\sin \left( \theta _{1} \right)}=\text{ constant}$$

This constant is actually a function of the optical wavelength.
Therefor, white light is separated into its various color components
when refracted.

If the originating medium is a vacuum, this ratio is known as the
refractive index. In practice, the refractive index for air is 1.000293
for yellow light. Therefor, for terrestrial measurements, air is
regarded as a vacuum, with a refractive index of 1.

:

    :   ![](White_light_refraction.gif "fig:White_light_refraction.gif")

**Snell\'s Law**

The relationship between the incident and refracted angles is given by
Snell's Law:

$$\eta _0\sin \left( \theta _{0} \right)=\eta _{1}\sin \left( \theta _{1} \right)$$

When a light ray passes through a glass block, it is refracted twice. If
the block has parallel faces, the ray will resume it original angle of
attack but is slightly displaced.

The refractive index of a material is the ratio of the speed of light in
a vacuum to the speed of light in the specific material.

:

    :   ![](Refraction_in_glass.gif "fig:Refraction_in_glass.gif")

If the faces are not parallel, such as in a prism, the white light is
separated into its various spectral components.

:

    :   ![](Prism_refraction.gif "fig:Prism_refraction.gif")

In some cases these simple rules of reflection and refraction are not
valid. Reflection for example, assumes a smooth boundary, and Snell's
law of refraction assumes an isotopic material. The velocity of light
varies with direction in anisotopic materials.

Snell's Law predicts the following refraction response in glass:

:

    :   ![](Borosilicate_Glass.gif "fig:Borosilicate_Glass.gif")

The reflected intensity depends on the angle of incidence and on the
polarization of incident light. At a high angle of incidence, for
example, the reflectance can be very high. At a certain angle
(Brewster\'s angle) reflected light is completely polarized. The
Fresnel equations give
quantitatively the intensity of reflected light as a function of the
angle of incidence and of its polarization (and, of course, of
refraction index).


# Communication Systems/Baseband and Broadband Signals

It is important to know the difference between a baseband signal, and a
broad band signal. In the Fourier Domain, a baseband signal is a signal
that occupies the frequency range from 0 Hz up to a certain cutoff. It
is called the baseband because it occupies the **base**, or the lowest
range of the spectrum.

In contrast, a broadband signal is a signal which does not occupy the
lowest range, but instead a higher range, 1 MHz to 3 MHz, for example. A
wire may have only one baseband signal, but it may hold any number of
broadband signals, because they can occur anywhere in the spectrum.

## Wideband vs Narrowband

in form of frequency modulation. wideband fm has been defined as that in
which the modulation index normally exceeds unity.

## Frequency Spectrum

A graphical representation of the various frequency components on a
given transmission medium is called a frequency spectrum.


# Communication Systems/Time-Division Multiplexing

Consider a situation where there are multiple signals which would all
like to use the same wire (or medium). For instance, a telephone company
wants multiple signals on the same wire at the same time. It certainly
would save a great deal of space and money by doing this, not to mention
time by not having to install new wires. How would they be able to do
this? One simple answer is known as **Time-Division Multiplexing**.

## Time Division Multiplexing

**Time-Division Multiplexing** (TDM) is a convenient method for
combining various digital signals onto a single transmission media such
as wires, fiber optics or even radio. These signals may be interleaved
at the bit, byte, or some other level. The resulting pattern may be
transmitted directly, as in digital carrier systems, or passed through a
modem to allow the data to pass over an analog network. Digital data is
generally organized into frames for transmission and individual users
assigned a time slot, during which frames may be sent. If a user
requires a higher data rate than that provided by a single channel,
multiple time slots can be assigned.

Digital transmission schemes in North America and Europe have developed
along two slightly different paths, leading to considerable
incompatibility between the networks found on the two continents.

BRA (basic rate access) is a single digitized voice channel, the basic
unit of digital multiplexing.

:

    :   ![](BRA_comparison.gif "fig:BRA_comparison.gif")

```{=html}
<!-- -->
```

:

    :   ![](digital_transmission_systems.gif "fig:digital_transmission_systems.gif")

### North American TDM

The various transmission rates are not integral numbers of the basic
rate. This is because additional framing and synchronization bits are
required at every multiplexing level.

:

    :   ![](north_american_TDM.gif "fig:north_american_TDM.gif")

In North America, the basic digital channel format is known as DS-0.
These are grouped into frames of 24 channels each. A concatenation of 24
channels and a start bit is called a frame. Groups of 12 frames are
called multiframes or superframes. These vary the start bit to aid in
synchronizing the link and add signaling bits to pass control messages.

:

    :   ![](DS_framing.gif "fig:DS_framing.gif")

#### S Bit Synchronization

The S bit is used to identify the start of a DS-1 frame. There are 8
thousand S bits per second. They have an encoded pattern, to aid in
locating channel position within the frame.

:

    :   ![](s_bits.gif "fig:s_bits.gif")

This forms a regular pattern of 1 0 1 0 1 0 for the odd frames and 0 0 1
1 1 0 for the even frames. Additional synchronization information is
encoded in the DS-1 frame when it is used for digital data applications,
so lock is more readily acquired and maintained.

For data customers, channel 24 is reserved as a special sync byte, and
bit 8 of the other channels is used to indicate if the remaining 7 bits
are user data or system control information. Under such conditions, the
customer has an effective channel capacity of 56 Kbps.

To meet the needs of low speed customers, an additional bit is robbed to
support sub-rate multiplexer synchronization, leaving 6 x 8 Kbps = 48
Kbps available. Each DS-0 can be utilized as:

:

    :   • 5 x 9.6 Kbps channels or
    :   • 10 x 4.8 Kbps channels or
    :   • 20 x 2.48 Kbps channels.

In the DS-2 format, 4 DS-1 links are interleaved, 12 bits at a time. An
additional 136 Kbps is added for framing and control functions resulting
in a total bit rate of 6.312 Mbps.

#### Signaling

Signaling provides control and routing information. Two bits, called the
A and B bits, are taken from each channel in frames 6 and 12 in the
multiframe. The A bit is the least significant bit in each channel in
frame 6, and the B bit is the least significant bit in each channel in
frame 12. This provides a signaling rate of 666 2/3 bps per channel.

The quality of voice transmission is not noticeably affected when 2% of
the signal is robbed for signaling. For data, it may be a different
story. If the data is encoded in an analog format such as FSK or PSK,
then robbing bits is of no consequence, but if the data is already in
digital form, then robbing bits results in unacceptable error rates. It
is for this reason that in North America, a 64 Kbps clear channel cannot
readily be switched through the PSTN. This means that data customers are
limited to 56 Kbps clear channels. This simple condition has a profound
effect on the development of new services such as ISDN. In most
facilities, the A and B bits represent the status of the telephone hook
switch, and correspond to the M lead on the E&M interface of the calling
party.

#### ESF

CCITT has modified the North American digital hierarchy for the
deployment of ISDN, by means of recommendation G.704. ESF consists of 24
DS-0 channels in a frame, but groups them into a 24-frame multiframe
instead of the usual 12-frame multiframe.

The S bit is renamed the F bit, but only 1/4 of them are used for
synchronization. This is possible because of improvements in frame
search techniques and allows more signaling states to be defined.

Bit robbing is still used for signaling over an ESF link, but with the
advent of ISDN, it will not be permitted. Instead, channel 24 is used to
support a D channel.

:

    :   ![](ESF.gif "fig:ESF.gif")

#### Typical T1 CPE Application

The large telecom carriers are not the only ones who deploy high-speed
TDM facilities. In many cases, heavy users of voice or data services can
reduce their transmission costs by concentrating their numerous low
speed lines on to a high speed facility.

There are many types of T1 multiplexers available today. Some are
relatively simple devices, while others allow for channel concatenation,
thus supporting a wide range of data rates. The ability to support
multiple DS-0s allows for easy facilitation of such protocols as the
video teleconferencing standard, Px64.

:

    :   ![](T1_mux.gif "fig:T1_mux.gif")

#### Multiplexers

Multiplexing units are often designated by the generic term M~*ab*~
where*a* is input DS level and *b* is the output DS level. Thus, an M13
multiplexer combines 28 DS--1s into a single DS--3 and an M23
multiplexer combines 7 DS--2s into a single DS--3.

:

    :   ![](M13_mux.gif "fig:M13_mux.gif")

#### ZBTSI

ZBTSI (zero byte time slot interchange) is used on DS--4 links. Four
DS-1 frames are loaded into a register, and renumbered 1--96. If there
are any empty slots \[all zeros\], the first framing bit is inverted and
all blank slots are relocated to the front of the frame. Channel 1 is
then loaded with a 7-bit number corresponding to the original position
of the first empty slot. Bit 8 used to indicate whether the following
channel contains user information or another address for an empty slot.

If there is a second vacancy, bit 8 in the previous channel is set, and
the empty slot address is placed in channel 2. This process continues
until all empty positions are filled.

The decoding process at the receiver is done in reverse. Borrowing 1 in
4 framing bits for this system is not enough to cause loss of
synchronization and provides a 64 Kbps clear channel to the end-user.

### European TDM Carriers

European systems were developed along slightly different principles. The
64 Kbps channel is still the basic unit, but signaling is not included
in each channel. Instead, common channel signaling is used. In a level 1
carrier, channels 0 and 16 are reserved for signaling and control. This
subtle difference means that European systems did not experience the
toll fraud and 56 k bottlenecks common to North American systems, and
they experience a much larger penetration of ISDN services.

:

    :   ![](European_TDM.gif "fig:European_TDM.gif")

```{=html}
<!-- -->
```

:

    :   ![](E1_frame.gif "fig:E1_frame.gif")

## Zero Substitutions

In order to prevent transmission systems from loosing lock on the data
stream, it is necessary to avoid long strings of zeros. One of the most
effective ways of doing this is to replace the zeros with a
predetermined code. This substitution must be done in such a way that
the receiver can identify it and strip it off before passing the data
stream to the client.

AMI provides a simple means of detecting substitutions. In the normal
course of events, alternate marks are inverted. Therefor, deliberately
inducing a bipolarvariation at the transmitter can alert the receiver of
a substitution. However, a single violation is indistinguishable from a
transmission error. Consequently, some additional condition must also
occur.

There are two common methods to create a second condition:

:   · Create a second bipolar violation in the opposite direction,
    within a specified time. This has the effect of keeping the average
    signal level at zero.
:   · Count the number of marks from the last substitution to predict
    the next type of violation

### B6ZS

B6ZS (binary six zero substitution) is used on T2 AMI transmission
links.

Synchronization can be maintained by replacing strings of zeros with
bipolar violations. Since alternate marks have alternate polarity, two
consecutive pulses of the same polarity constitute a violation.
Therefore, violations can be substituted for strings of zeros, and the
receiver can determine where substitutions were made.

Since the last mark may have been either positive (+) or negative (-),
there are two types of substitutions:

:

    :   ![](B6ZS_substitutions.gif "fig:B6ZS_substitutions.gif")

These substitutions force two consecutive violations. A single bit error
does not create this condition.

:

    :   ![](B6ZS_example.gif "fig:B6ZS_example.gif")

### B8ZS

This scheme uses the same substitution as B6ZS.

:

    :   ![](B8ZS_substitutions.gif "fig:B8ZS_substitutions.gif")

### B3ZS

B3ZS is more involved than B6ZS, and is used on DS--3 carrier systems.
The substitution is not only dependent on the polarity of the last mark,
but also on the number of marks since the last substitution.

:

    :   ![](B3ZS_substitutions.gif "fig:B3ZS_substitutions.gif")

```{=html}
<!-- -->
```

:

    :   ![](B3ZS_example.gif "fig:B3ZS_example.gif")

### HDB3

HDB3 (high density binary 3) introduces bipolar violations when four
consecutive zeros occur. It can therefore also be called B4ZS. The
second and thirds zeros are left unchanged, but the fourth zero is given
the same polarity as the last mark. The first zero may be modified to a
one to make sure that successive violations are of alternate polarity.

:

    :   ![](HDB3_substitutions.gif "fig:HDB3_substitutions.gif")

HDB3 is used in Europe. Violation, or V pulses are injected after three
consecutive zeros. The fourth zero is given the same polarity as the
last mark. In the event of long strings of zeros occurring, a succession
of single polarity pulses would occur, and a dc offset would build-up.

To prevent this, the first zero in a group of 4, may be modified to a 1.
This B or balancing pulse assures that successive violations are of
alternate polarity.

:

    :   ![](HDB3_waveform.gif "fig:HDB3_waveform.gif")

### Block Code Substitution

These schemes operate on bytes rather than a bit at a time. Some
transmit the signal as binary levels, but most use multi-level pulses.
Some authors categorize these as line codes.

A binary block code has the designation *n*B*m*B, where *n* input bits
are encoded into *m* output bits. The most common of these is the 3B4B
code.

:

    :   ![](3B4B_coding.gif "fig:3B4B_coding.gif")

In Europe 4B3T, which encodes 4 binary bits into 3 ternary levels, has
been selected as the BRA for ISDN. In North America, 2B1Q which encodes
2 binary bits into 1 quaternary level has been selected for BRA.

:

    :   ![](2B1Q_coding.gif "fig:2B1Q_coding.gif")

Some block codes do not generate multilevel pulses. For example, 24B1P
or 24B25B simply adds a P or parity bit to a 24 bit block.

## Benefits of TDM

TDM is all about cost: fewer wires and simpler receivers are used to
transmit data from multiple sources to multiple destinations. TDM also
uses less bandwidth than Frequency-Division Multiplexing (FDM) signals,
unless the bitrate is increased, which will subsequently increase the
necessary bandwidth of the transmission.

## Synchronous TDM

Synchronous TDM is a system where the transmitter and the receiver both
know exactly which signal is being sent. Consider the following diagram:

`Signal A ---> |---| |A|B|C|A|B|C|   |------| ---> Signal A`\
`Signal B ---> |TDM| --------------> |De-TDM| ---> Signal B`\
`Signal C ---> |---|                 |------| ---> Signal C`

In this system, starting at time-slice 0, every third time-slice is
reserved for Signal A; starting at time-slice 1, every third time-slice
is reserved for Signal B; and starting at time-slice 2, every third
time-slice is reserved for Signal C. In this situation, the receiver
(De-TDM) needs only to switch after the signal on each time-slice is
received.

------------------------------------------------------------------------

The data flow of each input connection is divided into units where each
input occupies one input time slot. Each input connection has a time
slot alloted in the output irrespective of the fact whether it is
sending data or not.

`      A -----|A3|A2|A1|---> |---| .............|C3|B3|A3|C2|B2|A2|C1|B1|A1|   |------| --->  A`\
`             |  |  |  |                        |        |        |`\
`      B -----|B3|B2|B1|---> |MUX| -------------|--------|--------|----------> |De-MUX| --->  B`\
`             |  |  |  |                        |        |        |`\
`      C -----|C3|C2|C1|---> |---|              |        |        |            |------| --->  C`\
`                   <-->                                 <-------->`\
`                  Bit Interval                       Frame (x seconds)`

Sync TDM is inefficient when one or more input lines have no data to
send. Thus, it is used with lines with high data traffic.

------------------------------------------------------------------------

Sampling rate is same for all signals. Maximum sampling rate = twice the
maximum frequency all the signals.

## Statistical TDM

Synchronous TDM is beneficial because the receiver and transmitter can
both cost very little. However, consider the most well-known network:
the Internet. In the Internet, a given computer might have a data rate
of 1kbps when hardly anything is happening, but might have a data rate
of 100kbps when downloading a large file from a fast server. How are the
time-slices divided in this instance? If every time slice is made big
enough to hold 100Kbps, when the computer isn\'t downloading any data,
all of that time and electricity will be wasted. If every time-slice is
only big enough for the minimum case, the time required to download
bigger files will be greatly increased.

The solution to this problem is called **Statistical TDM**, and is the
solution that the Internet currently uses. In Statistical TDM, each data
item, known as the payload (we used time-slices to describe these
earlier), is appended with a certain amount of information about who
sent it, and who is supposed to receive it (the header). The combination
of a payload and a header is called a **packet**. Packets are like
envelopes in the traditional \"snail mail\" system: Each packet contains
a destination address and a return address as well as some enclosed
data. Because of this, we know where each packet was sent from and where
it is going.

The downside to statistical TDM is that the sender needs to be smart
enough to write a header, and the receiver needs to be smart enough to
read the header and (if the packet is to be forwarded,) send the packet
toward its destination.

### Link Utilization

Statistical multiplexing attempts to maximize the use of a communication
path. The study of this is often called *queuing theory*. A queue is
simply a line of customers or packets waiting to be served. Under most
circumstances, the arrival rate is unpredictable and therefor follows a
random or Poisson distribution pattern, whereas the service time is
constant.

The utilization or fraction of time actually used by a packet
multiplexing system to process packets is given by:

:

    :   ![](Utilization.gif "fig:Utilization.gif")

The queue length or average number of items waiting to be served is
given by:

$$q = \frac{{\rho ^2 }}{{2\left( {1 - \rho } \right)}} + \rho$$

:

    :   ![](Queue_length.gif "fig:Queue_length.gif")

```{=html}
<!-- -->
```

:

    :   **Example**

```{=html}
<!-- -->
```

:

    :   A T1 link has been divided into a number of 9.6 Kbps channels
        and has a combined user data rate of 1.152 Mbps. Access to this
        channel is offered to 100 customers, each requiring 9.6 Kbps
        data 20% of the time. If the user arrival time is strictly
        random find the T1 link utilization.

```{=html}
<!-- -->
```

:

    :   **Solution**

```{=html}
<!-- -->
```

:

    :   The utilization or fraction of time used by the system to
        process packets is given by:

$$\rho  = \frac{{\alpha NR}}{M} = \frac{{0.2 \times 100 \times 9.6 \times 10^3 }}{{1.152 \times 10^6 }} = 0.167$$

:

    :   A 24 channel system dedicated to DATA, can place five 9.6 Kbps
        customers in each of 23 channels, for a total of 115 customers.
        In the above statistical link, 100 customers created an
        *average* utilization of 0.167 and were easily fitted, with room
        to spare if they transmit on the average 20% of the time. If
        however, the customer usage were not randomly distributed, then
        the above analysis would have to be modified.

This example shows the potential for statistical multiplexing. If
channels were assigned on a demand basis (only when the customer had
something to send), a single T1 may be able to support hundreds of low
volume users.

A utilization above 0.8 is undesirable in a statistical system, since
the slightest variation in customer requests for service would lead to
buffer overflow. Service providers carefully monitor delay and
utilization and assign customers to maximize utilization and minimize
cost.

## Packets

Packets will be discussed in greater detail once we start talking about
digital networks (specifically the Internet). Packet headers not only
contain address information, but may also include a number of different
fields that will display information about the packet. Many headers
contain error-checking information (checksum, Cyclic Redundancy Check)
that enables the receiver to check if the packet has had any errors due
to interference, such as electrical noise.

## Duty Cycles

Duty cycle is defined as \" the time that is effectively used to send or
receive the data, expressed as a percentage of total period of time.\"
The more the duty cycle , the more effective transmission or reception.

We can define the pulse width, τ, as being the time that a bit occupies
from within its total alloted bit-time *T~b~*. If we have a duty cycle
of *D*, we can define the pulse width as:

$$\tau = DT_b$$

Where:

$$0 < \tau \le T_b$$

The pulse width is equal to the bit time if we are using a 100% duty
cycle.


# Communication Systems/Frequency-Division Multiplexing

## Introduction

It turns out that many wires have a much higher bandwidth than is needed
for the signals that they are currently carrying. Analog Telephone
transmissions, for instance, require only 3 000 Hz of bandwidth to
transmit human voice signals. Over short distances, however,
twisted-pair telephone wire has an available bandwidth of nearly
100000 Hz!

There are several terrestrial radio based communications systems
deployed today. They include:

-   Cellular radio
-   Mobile radio
-   Digital microwave radio

Mobile radio service was first introduced in the St. Louis in 1946. This
system was essentially a radio dispatching system with an operator who
was able to patch the caller to the PSTN via a switchboard. Later, an
improved mobile telephone system, IMTS, allowed customers to dial their
own calls without the need for an operator. This in turn developed into
the cellular radio networks we see today.

The long haul PSTNs and packet data networks use a wide variety of
transmission media including

-   Terrestrial microwave
-   Satellite microwave
-   Fiber optics
-   Coaxial cable

In this section, we will be concerned with terrestrial microwave
systems. Originally, microwave links used FDM exclusively as the access
technique, but recent developments are changing analog systems to
digital where TDM is more appropriate.

### Fixed Access Assignment

Three basic methods can be used to combine customers on to fixed channel
radio links:

-   FDMA - (Frequency division multiple access) analog or digital
-   TDMA - (Time division multiple access) three conversation paths are
    time division multiplexed in 6.7 mSec time slots on a single
    carrier.
-   CDMA - (Code division multiple access) this uses spread spectrum
    techniques to increase the subscriber density. The transmitter hops
    through a pseudo-random sequence of frequencies. The receiver is
    given the sequence list and is able to follow the transmitter. As
    more customers are added to the system, the signal to noise will
    gradually degrade. This is in contrast to AMPS where customers are
    denied access once all of the frequencies are assigned code division
    multiple access \[digital only\]

:

    :   ![](Fixed_assgnment_access.gif "fig:Fixed_assgnment_access.gif")

## What is FDM?

**Frequency Division Multiplexing** (FDM) allows engineers to utilize
the extra space in each wire to carry more than one signal. By
frequency-shifting some signals by a certain amount, engineers can shift
the spectrum of that signal up into the unused band on that wire. In
this way, multiple signals can be carried on the same wire, without
having to divy up time-slices as in Time-Division Multiplexing
schemes.In analog transmission, signals are commonly multiplexed using
frequency-division multiplexing (FDM), in which the carrier bandwidth is
divided into subchannels of different frequency widths, each carrying a
signal at the same time in parallel

Traditional terrestrial microwave and satellite links employ FDM.
Although FDM in telecommunications is being reduced, several systems
will continue to use this technique, namely: broadcast & cable TV, and
commercial & cellular radio.

### Analog Carrier Systems

The standard telephony voice band \[300 -- 3400 Hz\] is heterodyned and
stacked on high frequency carriers by single sideband amplitude
modulation. This is the most bandwidth efficient scheme possible.

:

    :   ![](Single_voice_channel.gif "fig:Single_voice_channel.gif")

The analog voice channels are pre-grouped into threes and heterodyned on
carriers at 12, 16, and 20 kHz. The resulting upper sidebands of four
such pregroups are then heterodyned on carriers at 84, 96, 108, and
120 kHz to form a 12-channel group.

Since the lower sideband is selected in the second mixing stage, the
channel sequence is reversed and a frequency inversion occurs within
each channel.

:

    :   ![](12_channel_group.gif "fig:12_channel_group.gif")

This process can continue until the available bandwidth on the coaxial
cable or microwave link is exhausted.

:

    :   ![](Analog_carrier_systems.gif "fig:Analog_carrier_systems.gif")

In the North American system, there are:

-   12 channels per group
-   5 groups per supergroup
-   10 super groups per mastergroup
-   6 master groups per jumbogroup

In the European CCITT system, there are:

-   12 channels per group
-   5 groups per supergroup
-   5 super groups per mastergroup
-   3 master groups per supermastergroup

There are other FDM schemes including:

-   L600 - 600 voice channels 60--2788 kHz
-   U600 - 600 voice channels 564--3084 kHz
-   L3 - 1860 voice channels 312--8284 kHz, comprised of 3 mastergroups
    and a supergroup
-   L4 - 3600 voice channels, comprised of six U600s

## Benefits of FDM

FDM allows engineers to transmit multiple data streams simultaneously
over the same channel, at the expense of bandwidth. To that extent, FDM
provides a trade-off: faster data for less bandwidth. Also, to
demultiplex an FDM signal requires a series of bandpass filters to
isolate each individual signal. Bandpass filters are relatively
complicated and expensive, therefore the receivers in an FDM system are
generally expensive.

## Examples of FDM

As an example of an FDM system, Commercial broadcast radio (AM and FM
radio) simultaneously transmits multiple signals or \"stations\" over
the airwaves. These stations each get their own frequency band to use,
and a radio can be tuned to receive each different station. Another good
example is cable television, which simultaneously transmits every
channel, and the TV \"tunes in\" to which channel it wants to watch.

## Orthogonal FDM

Orthogonal Frequency Division Multiplexing (OFDM) is a more modern
variant of FDM that uses orthogonal sub-carriers to transmit data that
does not overlap in the frequency spectrum and is able to be separated
out using frequency methods. OFDM has a similar data rate to traditional
FDM systems, but has a higher resilience to disruptive channel
conditions such as noise and channel fading.


# Communication Systems/What is Modulation?

Modulation is a term that is going to be used very frequently in this
book. So much in fact, that we could almost have renamed this book
\"Principles of Modulation\", without having to delete too many
chapters. So, the logical question arises: What exactly is modulation?

## Definition

**Modulation** is a process of mixing a signal with a sinusoid to
produce a new signal. This new signal, conceivably, will have certain
benefits over an un-modulated signal. Mixing of low frequency signal
with high frequency carrier signal is called modulation.

$$f(t) = A \sin(\omega t + \phi)$$

we can see that this sinusoid has 3 parameters that can be altered, to
affect the shape of the graph. The first term, A, is called the
magnitude, or amplitude of the sinusoid. The next term, $\omega$ is
known as the frequency, and the last term, $\phi$ is known as the phase
angle. All 3 parameters can be altered to transmit data.

The sinusoidal signal that is used in the modulation is known as the
**carrier signal**, or simply \"the carrier\". The signal that is used
in modulating the carrier signal(or sinusoidal signal) is known as the
\"data signal\" or the \"message signal\". It is important to notice
that a simple sinusoidal carrier contains no information of its own.

In other words we can say that modulation is used because some data
signals are not always suitable for direct transmission, but the
modulated signal may be more suitable.

## Types of Modulation

There are 3 basic types of modulation: Amplitude modulation, Frequency
modulation, and Phase modulation.

amplitude modulation
:   a type of modulation where the amplitude of the carrier signal is
    modulated (changed) in proportion to the message signal while the
    frequency and phase are kept constant.

frequency modulation
:   a type of modulation where the frequency of the carrier signal is
    modulated (changed) in proportion to the message signal while the
    amplitude and phase are kept constant.

phase modulation
:   a type of modulation where the phase of the carrier signal is varied
    accordance to the low frequency of the message signal is known as
    phase modulation.

## Why Use Modulation?

Why use modulation at all? To answer this question, let\'s consider a
channel that essentially acts like a bandpass filter: both the lowest
frequency components and the highest frequency components are attenuated
or unusable in some way, with transmission only being practical over
some intermediate frequency range. If we can\'t send low-frequency
signals, then we need to shift our signal up the frequency ladder.
Modulation allows us to send a signal over a bandpass frequency range.
If every signal gets its own frequency range, then we can transmit
multiple signals *simultaneously* over a single channel, all using
different frequency ranges.

Another reason to modulate a signal is to allow the use of a smaller
antenna. A baseband (low frequency) signal would need a huge antenna
because in order to be efficient, the antenna needs to be about 1/10th
the length of the wavelength. Modulation shifts the baseband signal up
to a much higher frequency, which has much smaller wavelengths and
allows the use of a much smaller antenna.

## Examples

Think about your car radio. There are more than a dozen (or so) channels
on the radio at any time, each with a given frequency: 100.1 MHz,
102.5 MHz etc\... Each channel gets a certain range (usually about
0.22 MHz), and the entire bayot gets transmitted over that range.
Modulation makes it all possible, because it allows us to send voice and
music (which are essential baseband signals) over a bandpass (or
\"Broadband\") channel.

## Non-sinusoidal modulation

A sine wave at one frequency can be separated from a sine wave at
another frequency (or a cosine wave at the same frequency) because the
two signals are \"orthogonal\".

There are other sets of signals, such that every signal in the set is
orthogonal to every other signal in the set.

A simple orthogonal set is time multiplexed division (TDM) \-- only one
transmitter is active at any one time.

Other more complicated sets of orthogonal waveforms---Walsh codes and
various pseudo-noise codes such as Gold codes and maximum length
sequences---are also used in some communication systems.

The process of combining these waveforms with data signals is sometimes
called \"modulation\", because it is so very similar to the way
modulation combines sine waves with data signals.

## Further reading

-   Data Coding Theory/Spectrum
    Spreading
-   Wikipedia:Walsh code
-   Wikipedia:Gold code
-   Wikipedia:pseudonoise code
-   Wikipedia:maximum length
    sequence


# Communication Systems/Analog vs. Digital

There is lots of talk nowadays about buzzwords such as \"Analog\" and
\"Digital\". Certainly, engineers who are interested in creating a new
communication system should understand the difference. Which is better,
analog or digital? What is the difference? What are the pros and cons of
each? This chapter will look at the answers to some of these questions.

## What are They?

What exactly is an analog signal, and what is a digital signal?

Analog : Analog signals are continuous in both time and value. Analog signals are used in many systems, although the use of analog signals has declined with the advent of cheap digital signals. All natural signals are Analog in nature or analog signal is that signal which amplitude on Y axis change with time on X axis\...

```{=html}
<!-- -->
```

Digital : Digital signals are discrete in time and value. Digital signals are signals that are represented by binary numbers, \"1\" or \"0\". The 1 and 0 values can correspond to different discrete voltage values, and any signal that *doesn\'t quite fit* into the scheme just gets rounded off.

or digital signal is that signal which have certain or fixed value on Y
axis change with time on X axis\...

Digital signals are sampled, quantized & encoded version of continuous
time signals which they represent. In addition, some techniques also
make the signal undergo encryption to make the system more tolerant to
the channel.

## What are the Pros and Cons?

Each paradigm has its own benefits and problems

Analog : Analog systems are less tolerant to noise, make good use of bandwidth, and are easy to manipulate mathematically. However, analog signals require hardware receivers and transmitters that are designed to perfectly fit the particular transmission.\
Digital : Digital signals are more tolerant to noise, but digital signals can be completely corrupted in the presence of excess noise. In digital signals, noise could cause a 1 to be interpreted as a 0 and vice versa, which makes the received data different than the original data. Imagine if the army transmitted a position coordinate to a missile digitally, and a single bit was received in error. This single bit error could cause a missile to miss its target by miles. Luckily, there are systems in place to prevent this sort of scenario, such as checksums and CRCs, which tell the receiver when a bit has been corrupted and ask the transmitter to resend the data. The primary benefit of digital signals is that they can be handled by simple, standardized receivers and transmitters, and the signal can be then dealt with in software (which is comparatively cheap to change).

```{=html}
<!-- -->
```

Discrete Digital and Analogue:

Discrete data has a fixed set of possible values.\
Digital data is a type of Discrete data where the fixed value can either
be 1 or 0.\
Analogue data can take on any real value.

## Sampling and Reconstruction

The process of converting from analog data to digital data is called
\"sampling\". The process of recreating an analog signal from a digital
one is called \"reconstruction\". This book will not talk about either
of these subjects in much depth beyond this, although other books on the
topic of EE might, such as A-level Physics (Advancing
Physics)/Digitisation/Digitisation "wikilink").

## Further reading

-   Electronics/Digital to Analog & Analog to Digital
    Converters


# Communication Systems/Communication Mediums

Signals need a channel to follow, so that they can move from place to
place. These **Communication Mediums**, or \"channels\" are things like
wires and antennae that transmit the signal from one location to
another. Some of the most common channels are listed below:

## Twisted Pair Wire

**Twisted Pair** is a transmission medium that uses two conductors that
are twisted together to form a pair. The concept for the twist of the
conductors is to prevent interference. Ideally, each conductor of the
pair basically receives the same amount of interference, positive and
negative, effectively cancelling the effect of the interference.
Typically, most inside cabling has four pairs with each pair having a
different twist rate. The different twist rates help to further reduce
the chance of crosstalk by making the pairs appear electrically
different in reference to each other. If the pairs all had the same
twist rate, they would be electrically identical in reference to each
other causing crosstalk, which is also referred to as capacitive
coupling. Twisted pair wire is commonly used in telephone and data
cables with variations of categories and twist rates.

Other variants of Twisted Pair are the **Shielded Twisted Pair** cables.
The shielded types operate very similar to the non-shielded variety,
except that Shielded Twisted Pair also has a layer of metal foil or mesh
shielding around all the pairs or each individual pair to further shield
the pairs from electromagnetic interference. Shielded twisted pair is
typically deployed in situations where the cabling is subjected to
higher than normal levels of interference.

## Coaxial Cable

Another common type of wire is **Coaxial Cable**. Coaxial cable (or
simply, \"coax\") is a type of cable with a single data line, surrounded
by various layers of padding and shielding. The most common coax cable,
common television cable, has a layer of wire mesh surrounding the padded
core, that absorbs a large amount of EM interference, and helps to
ensure a relatively clean signal is transmitted and received. Coax cable
has a much higher bandwidth than a twisted pair, but coax is also
significantly more expensive than an equal length of twisted pair wire.
Coax cable frequently has an available bandwidth in excess of hundreds
of megahertz (in comparison with the hundreds of kilohertz available on
twisted pair wires).

Originally, Coax cable was used as the backbone of the telephone network
because a single coaxial cable could hold hundreds of simultaneous phone
calls by a method known as \"Frequency Division Multiplexing\"
(discussed in a later chapter). Recently however, Fiber Optic cables
have replaced Coaxial Cable as the backbone of the telephone network
because Fiber Optic channels can hold many more simultaneous phone
conversations (thousands at a time), and are less susceptible to
interference, crosstalk, and noise then Coaxial Cable.

## Fiber Optics

**Fiber Optic** cables are thin strands of glass that carry pulses of
light (frequently infrared light) across long distances. Fiber Optic
channels are usually immune to common RF interference, and can transmit
incredibly high amounts of data very quickly. There are 2 general types
of fiber optic cable: single frequency cable, and multi-frequency cable.
single frequency cable carries only a single frequency of laser light,
and because of this there is no self-interference on the line.
Single-frequency fiber optic cables can attain incredible bandwidths of
many gigahertz. Multi-Frequency fiber optics cables allow a
Frequency-Division Multiplexed series of signals to each inhabit a given
frequency range. However, interference between the different signals can
decrease the range over which reliable data can be transmitted.

## Wireless Transmission

In **wireless transmission systems**, signals are propagated as
Electro-Magnetic waves through free space. Wireless signals are
transmitted by a transmitter, and received by a receiver. Wireless
systems are inexpensive because no wires need to be installed to
transmit the signal, but wireless transmissions are susceptible not only
to EM interference, but also to physical interference. A large building
in a city, for instance can interfere with cell-phone reception, and a
large mountain could block AM radio transmissions. Also, WiFi internet
users may have noticed that their wireless internet signals don\'t
travel through walls very well.

There are 2 types of antennas that are used in wireless communications,
**isotropic**, and **directional**.

### Isotropic

People should be familiar with isotropic antennas because they are
everywhere: in your car, on your radio, etc\... Isotropic antennas are
omni-directional in the sense that they transmit data out equally (or
nearly equally) in all directions. These antennas are excellent for
systems (such as FM radio transmission) that need to transmit data to
multiple receivers in multiple directions. Also, Isotropic antennas are
good for systems in which the direction of the receiver, relative to the
transmitter is not known (such as cellular phone systems).

### Directional

Directional antennas focus their transmission power in a single narrow
direction range. Some examples of directional antennas are satellite
dishes, and wave-guides. The downfall of the directional antennas is
that they need to be pointed directly at the receiver all the time to
maintain transmission power. This is useful when the receiver and the
transmitter are not moving (such as in communicating with a
geo-synchronous satellite).


# Communication Systems/Coherent Receivers

## Receiver Design

It turns out that if we know what kind of signal to expect, we can
better receive those signals. This should be intuitive, because it is
hard to find something if we don\'t know what precisely we are looking
for. How is a receiver supposed to know what is data and what is noise,
if it doesnt know what data looks like?

Coherent transmissions are transmissions where the receiver knows what
type of data is being sent. Coherency implies a strict timing mechanism,
because even a data signal may look like noise if you look at the wrong
part of it. In contrast, noncoherent receivers don\'t know exactly what
they are looking for, and therefore noncoherent communication systems
need to be far more complex (both in terms of hardware and mathematical
models) to operate properly.

This section will talk about coherent receivers, first discussing the
\"Simple Receiver\" case, and then going into theory about what the
optimal case is. Once we know mathematically what an optimal receiver
should be, we then discuss two actual implementations of the optimal
receiver.

It should be noted that the remainder of this book will discuss optimal
receivers. After all, why would a communication\'s engineer use anything
that is less than the best?

## The Simple Receiver

A simple receiver is just that: simple. A general simple receiver will
consist of a low-pass filter (to remove excess high-frequency noise),
and then a sampler, that will select values at certain points in the
wave, and interpolate those values to form a smooth output curve. In
place of a sampler (for purely analog systems), a general envelope
filter can also be used, especially in AM systems. In other systems,
different tricks can be used to demodulate an input signal, and acquire
the data. However simple receivers, while cheap, are not the best choice
for a receiver. Occasionally they are employed because of their price,
but where performance is an issue, a better alternative receiver should
be used.

## The Optimal Receiver

Engineers are able to mathematically predict the structure of the
optimal receiver. Read that sentence again: Engineers are able to
design, analyze, and build the best possible receiver, for any given
signal. This is an important development for several reasons. First, it
means that no more research should go into finding a better receiver.
The best receiver has already been found, after all. Second, it means
any communications system will not be hampered (much) by the receiver.

### Derivation

here we will attempt to show how the coherent receiver is derived.

### Matched Receiver

The matched receiver is the logical conclusion of the optimal receiver
calculation. The matched receiver convolves the signal with itself, and
then tests the output. Here is a diagram:

`s(t)----->(Convolve with r(t))----->`

This looks simple enough, except that convolution modules are often
expensive. An alternative to this approach is to use a correlation
receiver.

### Correlation Receiver

The correlation receiver is similar to the matched receiver, instead
with a simple switch: The multiplication happens first, and the
integration happens second.

Here is a general diagram:

`           r(t)`\
`            |`\
`            v`\
`s(t) ----->(X)----->(Integrator)--->`

In a digital system, the integrator would then be followed by a
threshold detector, while in an analog receiver, it might be followed by
another detector, like an envelope detector.

## Conclusion

To do the best job of receiving a signal we need to know the form of the
signal that we are sending. After all we can\'t design a receiver until
after we\'ve decided how the signal will be sent. This method poses some
problems in that the receiver must be able to line up the received
signal with the given reference signal to work the magic: if the
received signal and the reference signal are out of sync with each
other, either as a function of an error in phase or an error in
frequency, then the optimal receiver will not work.

## Further reading


# Communication Systems/Analog Modulation Introduction

## Analog Modulation Overview

Let\'s take a look at a generalized sinewave:

$$x\left( t \right) = A\sin \left( {\omega t + \theta } \right)$$ It
consists of three components namely; amplitude, frequency and phase.
Each of which can be decomposed to provide finer detail:

$$x(t) = A s(t) \sin ( 2 \pi [f_c + kf_m(t)] t + \alpha \phi(t) )$$

## Types of Analog Modulation

We can see 3 parameters that can be changed in this sine wave to send
information:

-   $A s(t)$. This term is called the \"Amplitude\", and changing it is
    called \"Amplitude Modulation\" (AM)
-   $kf_m(t)$ This term is called the \"Frequency Shift\", and changing
    it is called \"Frequency Modulation\"
-   $\alpha \phi (t)$. this term is called the \"Phase angle\", and
    changing it is called \"Phase Modulation\".
-   The terms frequency and phase modulation are often combined into a
    more general group called \"Angle Modulation\".

## The Breakdown

Each term consists of a coefficient (called a \"scaling factor\"), and a
function of time that corresponds to the information that we want to
send. The scaling factor out front, A, is also used as the transmission
power coefficient. When a radio station wants their signal to be
stronger (regardless of whether it is AM, FM, or PM), they \"crank-up\"
the power of A, and send more power out onto the airwaves.

## How we Will Cover the Material

We are going to go into separate chapters for each different type of
modulation. This book will attempt to discuss some of the mathematical
models and techniques used with different modulation techniques. It will
also discuss some practical information about how to construct a
transmitter/receiver, and how to use each modulation technique
effectively.


# Communication Systems/FM and PM Generalization

## Concept

We can see from our initial overviews that FM and PM modulation schemes
have a lot in common. Both of them are altering the angle of the carrier
sinusoid according to some function. It turns out that we can go so far
as to generalize the two together into a single modulation scheme known
as **angle modulation**. Note that we will never abbreviate \"angle
modulation\" with the letters \"AM\", because Amplitude modulation is
completely different from angle modulation.

## Instantaneous Phase

Let us now look at some things that FM and PM have of common:

$$s_{FM} = A\cos (2 \pi [f_c + ks(t)]t + \phi)$$

$$s_{PM} = A\cos (2 \pi f_c t + \alpha s(t))$$

What we want to analyze is the *argument of the sinusoid*, and we will
call it Psi. Let us show the Psi for the bare carrier, the FM case, and
the PM case:

$$\Psi_{carrier}(t) = 2 \pi f_c t + \phi$$

$$\Psi_{FM}(t) = 2 \pi[f_c + ks(t)] t + \phi$$

$$\Psi_{PM}(t) = 2 \pi f_c t + \alpha s(t)$$

$$s(t) = A\cos(\Psi(t))$$

This Psi value is called the *Instantaneous phase* of the sinusoid.

## Instantaneous Frequency

Using the Instantaneous phase value, we can find the *Instantaneous
frequency* of the wave with the following formula:

$$f(t) = \frac{d\Psi(t)}{dt}$$

We can also express the instantaneous phase in terms of the
instantaneous frequency:

$$\Psi(t) = \int_{-\infty}^t f(\lambda)d\lambda$$

Where the Greek letter \"lambda\" is simply a dummy variable used for
integration. Using these relationships, we can begin to study FM and PM
signals further.

## Determining FM or PM

If we are given the equation for the instantaneous phase of a particular
angle modulated transmission, is it possible to determine if the
transmission is using FM or PM? it turns out that it is possible to
determine which is which, by following 2 simple rules:

1.  In PM, instantaneous phase is a linear function.
2.  In FM, instantaneous frequency minus carrier frequency is a linear
    function.

For a refresher course on Linearity, there is a chapter on the subject
in the Signals and Systems
book
worth re-reading.

## Bandwidth

In a PM system, we know that the value $\alpha s(t)$ can never go
outside the bounds of $(-\pi, \pi]$. Since sinusoidal functions
oscillate between \[-1, 1\], we can use them as a general PM generating
function. Now, we can combine FM and PM signals into a general equation,
called *angle modulation*:

$$v(t) = A \sin ( 2 \pi f_c t + \beta \sin (2 \pi f_m t))$$

If we want to analyze the spectral components of this equation, we will
need to take the Fourier transform of this. But, we can\'t integrate a
sinusoid of a sinusoid, much less find the transform of it. So, what do
we do?

It turns out (and the derivation will be omitted here, for now) that we
can express this equation as an infinite sum, as such:

$$v(t) = A \sum_{n = -\infty}^{\infty}J_n(\beta) \sin [2 \pi(nf_m + f_c)t]$$

But, what is the term $J_n(\beta)$? J is the *Bessel function*, which is
a function that exists only as an open integral (it is impossible to
write it in closed form). Fortunately for us, there are extensive tables
tabulating Bessle function values.

## The Bessel Function

The definition of the Bessel function is the following equation:

$$J_n(\beta) = \frac{1}{2\pi} \int_{-\pi}^\pi e^{j[\beta sin\theta - n\theta]} d\theta$$

The bessel function is a function of 2 variables, N and $\beta$.

Bessel Functions have the following properties:

-   If n is even:

$$J_{-n}(\beta) = J_n(\beta)$$

-   If n is odd:

$$J_{-n}(\beta) = -J_n(\beta)$$

-   $J_{n-1} + J_{n+1} = \frac{2n}{\beta}J_n(\beta)$.

The bessel function is a relatively advanced mathematical tool, and we
will not analyze it further in this book.

## Carson\'s Rule

If we have our generalized function:

$$v(t) = A\sin ( 2 \pi f_c t + \beta \sin (2 \pi f_m t))$$

We can find the bandwidth BW of the signal using the following formula:

$$BW=2(\beta + 1)f_m = 2(\Delta f + f_m)$$

where $\Delta f$ is the maximum frequency deviation, of the transmitted
signal, from the carrier frequency. It is important to note that
Carson\'s rule is only an approximation (albeit one that is used in
industry frequently).

## Demodulation: First Step

Now, it is important to note that FM and PM signals both do the same
first 2 steps during demodulation:

1.  Pass the signal through a **limiter**, to remove amplitude peaks
2.  Pass the signal through a bandpass filter to remove low and high
    frequency noise (as much as possible, without filtering out the
    signal).

Once we perform these two steps, we no longer have white noise, because
we\'ve passed the noise through a filter. Now, we say the noise is
**colored**.

here is a basic diagram of our demodulator, so far:

`      channel`\
`s(t) ---------> r(t) --->|Limiter|--->|Bandpass Filter|---->z(t)`

Where z(t) is the output of the bandpass filter.

## Filtered Noise

To denote the new, filtered noise, and new filtered signal, we have the
following equation:

$$z(t) = \gamma A \cos (\Psi(t)) + n_0(t)$$

Where we call the additive noise $n_0(t)$ because it has been filtered,
and is not white noise anymore. $n_0(t)$ is known as **narrow band
noise**, and can be denoted as such:

$$n_0(t) = \mathbf{W}(t) \cos(2 \pi f_c t) + \mathbf{Z}(t) \sin(2 \pi f_c t)$$

Now, once we have it in this form, we can use a trigonometric identity
to make this equation more simple:

$$n_0(t) = \mathbf{R}(t) \cos(2 \pi f_c t + \mathbf{\Theta}(t))$$

Where

$$\mathbf{R}(t) = \sqrt{\mathbf{W}(t)^2 + \mathbf{Z}(t)^2}$$

$$\mathbf{\Theta}(t) = tan^{-1} (\mathbf{Z}(t)/\mathbf{W(t)})$$

Here, the new noise parameter R(t) is a **rayleigh** random variable,
and is discussed in the next chapter.

## Noise Analysis

R(t) is a noise function that affects the amplitude of our received
signal. However, our receiver passes the signal through a limiter, which
will remove amplitude fluctuations from our signal. For this reason,
R(t) doesnt affect our signal, and can be safely ignored for now. This
means that the only random variable that is affecting our signal is the
variable $\mathbf{\Theta}(t)$, \"Theta\". Theta is a uniform random
variable, with values between pi and -pi. Values outside this range
\"Wrap around\" because phase is circular.


# Communication Systems/AM Receivers

## AM Receivers

The most common receivers in use today are the super heterodyne type.
They consist of:

::\*Antenna

::\*RF amplifier

::\*Local Oscillator and Mixer

::\*IF Section

::\*Detector and Amplifier

The need for these subsystems can be seen when one considers the much
simpler and inadequate TRF or tuned radio frequency amplifier.

### TRF Amplifier

It is possible to design an RF amplifier to accept only a narrow range
of frequencies, such as one radio station on the AM band.

:

    :   ![](TRF_amplifier.gif "fig:TRF_amplifier.gif")

By adjusting the center frequency of the tuned circuit, all other input
signals can be excluded.

:

    :   ![](Channel_selection.gif "fig:Channel_selection.gif")

The AM band ranges from about 500 kHz to 1600 kHz. Each station requires
10 kHz of this spectrum, although the baseband signal is only 5 kHz.

Recall that for a tuned circuit: $Q = \frac{{f_c }}{B}$. The center or
resonant frequency in an RLC network is most often adjusted by varying
the capacitor value. However, the Q remains approximately constant as
the center frequency is adjusted. This suggests that as the bandwidth
varies as the circuit is tuned.

:

    :   For example, the Q required at the lower end of the AM band to
        select only one radio station would be approximately:

$$Q = \frac{{f_c }}{B} = \frac{{500\;kHz}}{{10\;kHz}} = 50$$

:

    :   As the tuned circuit is adjusted to the higher end of the AM
        band, the resulting bandwidth is:

$$B = \frac{{f_c }}{Q} = \frac{{1600\;kHz}}{{50}} = 30\;kHz$$

A bandwidth this high could conceivably pass three adjacent stations,
thus making meaningful reception impossible.

To prevent this, the incoming RF signal is heterodyned to a fixed IF or
intermediate frequency and passed through a constant bandwidth circuit.

### Superheterodyne Receiver

:

    :   ![](Heterodyne.gif "fig:Heterodyne.gif")

The RF amplifier boosts the RF signal into the mixer. It has broad
tuning and amplifies not just one RF station, but many of them
simultaneously. It also amplifies any input noise and even contributes
some of its own.

The other mixer input is a high frequency sine wave created by a local
oscillator. In AM receivers, it is always 455 kHz above the desired
station carrier frequency. An ideal mixer will combine the incoming
carrier with the local oscillator to create sum and difference
frequencies. .

A real mixer combines two signals and creates a host of new frequencies:

:

    :   • A dc level
    :   • The original two frequencies
    :   • The sum and difference of the two input frequencies
    :   • Harmonics of the two input frequencies
    :   • Sums and differences of all of the harmonics

Since the RF amplifier passes several radio stations at once, the mixer
output can be very complex. However, the only signal of real interest is
the difference between the desired station carrier frequency and the
local oscillator frequency. This difference frequency, also called the
IF (intermediate frequency) will alway be 455 kHz. By passing this
through a 10 kHz BPF (band pass filter) centered at 455 kHz, the bulk of
the unwanted signals can be eliminated.

#### Local Oscillator Frequency

Since the mixer generates sum and difference frequencies, it is possible
to generate the 455 kHz IF signal if the local oscillator is either
above or below the IF. The inevitable question is which is preferable.

:

    :   **Case I The local Oscillator is above the IF.** This would
        require that the oscillator tune from (500 + 455) kHz to
        (1600 + 455) kHz or approximately 1 to 2 MHz. It is normally the
        capacitor in a tuned RLC circuit, which is varied to adjust the
        center frequency while the inductor is left fixed.

```{=html}
<!-- -->
```

:

    :   Since $f_c  = \frac{1}{{2\pi \sqrt {LC} }}$

```{=html}
<!-- -->
```

:

    :   solving for *C* we obtain
        $C = \frac{1}{{L\left( {2\pi f_c } \right)^2 }}$

```{=html}
<!-- -->
```

:

    :   When the tuning frequency is a maximum, the tuning capacitor is
        a minimum and vice versa. Since we know the range of frequencies
        to be created, we can deduce the range of capacitance required.

$$\frac{{C_{\max } }}{{C_{\min } }} = \frac{{L\left( {2\pi f_{\max } } \right)^2 }}{{L\left( {2\pi f_{\min } } \right)^2 }} = \left( {\frac{{2\;MHz}}{{1\;MHz}}} \right)^2  = 4$$

Making a capacitor with a 4:1 value change is well within the realm of
possibility.

:

    :   **Case II The local Oscillator is below the IF**. This would
        require that the oscillator tune from (500 - 455) kHz to
        (1600 - 455) kHz or approximately 45 kHz to 1145 kHz, in which
        case:

$$\frac{{C_{\max } }}{{C_{\min } }} = \left( {\frac{{1145\;kHz}}{{45\;kHz}}} \right)^2  \approx 648$$

It is not practical to make a tunable capacitor with this type of range.
Therefore the local oscillator in a standard AM receiver is above the
radio band.

#### Image Frequency

Just as there are two oscillator frequencies, which can create the same
IF, two different station frequencies can create the IF. The undesired
station frequency is known as the image frequency.

;;;![](Image_frequency.gif "Image_frequency.gif")

If any circuit in the radio front end exhibits non-linearities, there is
a possibility that other combinations may create the intermediate
frequency. Once the image frequency is in the mixer, there is no way to
remove it since it is now heterodyned into the same IF band as the
desired station.

### AM Detection

There are two basic types of AM detection, coherent and non-coherent. Of
these two, the non-coherent is the simpler method.

::\*Non-coherent detection does not rely on regenerating the carrier
signal. The information or modulation envelope can be removed or
detected by a diode followed by an audio filter.

::\*Coherent detection relies on regenerating the carrier and mixing it
with the AM signal. This creates sum and difference frequencies. The
difference frequency corresponds to the original modulation signal.

Both of these detection techniques have certain drawbacks. Consequently,
most radio receivers use a combination of both.

#### Envelope Detector

:

    :   ![](Envelope_detector.gif "fig:Envelope_detector.gif")

An envelope detector is simply a half wave rectifier followed by a low
pass filter. In the case of commercial AM radio receivers, the detector
is placed after the IF section. The carrier at this point is 455 kHz
while the maximum envelope frequency is only 5 kHz. Since the ripple
component is nearly 100 times the frequency of the highest baseband
signal and does not pass through any subsequent audio amplifiers.

:

    :   An AM signal where the carrier frequency is only 10 times the
        envelope frequency would have considerable ripple:

```{=html}
<!-- -->
```

:

    :   ![](Exagerated_am_ripple.gif "fig:Exagerated_am_ripple.gif"){width="300"}

#### Synchronous Detector

In a synchronous or coherent detector, the incoming AM signal is mixed
with the original carrier frequency.

:

    :   ![](Synchronous_detector.gif "fig:Synchronous_detector.gif")

If you think this looks suspiciously like a mixer, you are absolutely
right! A synchronous detector is one where the difference frequency
between the two inputs is zero Hz. Of in other words, the two input
frequencies are the same. Let\'s check the math.

Recall that the AM input is mathematically defined by:

$$e_{am}  = \underbrace {\sin \omega _c t}_{{\rm{Carrier}}} + \underbrace {\frac{m}{2}\sin \left( {\omega _c  - \omega _m } \right)t}_{{\rm{Lower}}\;{\rm{Sideband}}} - \underbrace {\frac{m}{2}\sin \left( {\omega _c  + \omega _m } \right)t}_{{\rm{Upper}}\;{\rm{Sideband}}}$$

:

    :   At the multiplier output, we obtain:

$${\rm{mixer}}\;{\rm{out  =  }}e_{am}  \times \sin \omega _c t = \underbrace { - \frac{m}{2}\sin \omega _m t}_{{\rm{Original}}\;{\rm{Modulation}}\;{\rm{Signal}}}\underbrace { - \frac{1}{2}\sin 2\omega _c t - \frac{m}{4}\sin \left( {2\omega _c  - \omega _m } \right)t + \frac{m}{4}\sin \left( {2\omega _c  + \omega _m } \right)t}_{{\rm{AM}}\;{\rm{signal}}\;{\rm{centered}}\;{\rm{at}}\;{\rm{2}}\;{\rm{times}}\;{\rm{the}}\;{\rm{carrier}}\;{\rm{frequency}}}$$

:

    :   The high frequency component can be filtered off leaving only
        the original modulation signal.

```{=html}
<!-- -->
```

:

    :   This technique has one serious drawback. The problem is how to
        create the exact carrier frequency. If the frequency is not
        exact, the entire baseband signal will be shifted by the
        difference. A shift of only 50 Hz will make the human voice
        unrecognizable. It is possible to use a PLL (phase locked loop),
        but making one tunable for the entire AM band is not trivial.

As a result, most radio receivers use an oscillator to create a fixed
intermediate frequency. This is then followed by an envelope detector or
a fixed frequency PLL.

#### Squaring Detector

The squaring detector is also a synchronous or coherent detector. It
avoids the problem of having to recreate the carrier by simply squaring
the input signal. It essentially uses the AM signal itself as a sort of
wideband carrier.

:

    :   ![](Squaring_detector.gif "fig:Squaring_detector.gif")

```{=html}
<!-- -->
```

:

    :   The output of the multiplier is the square of the input AM
        signal:

$$\left( {e_{am} } \right)^2  = \left( {\sin \omega _c t + \frac{m}{2}\sin \left( {\omega _c  - \omega _m } \right)t - \frac{m}{2}\sin \left( {\omega _c  + \omega _m } \right)t} \right)^2$$

Since the input is being multiplied by the ${\sin \omega _c t}$
component, one of the resulting difference terms is the original
modulation signal. The principle difficulty with this approach is trying
to create a linear, high frequency multiplier.