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De Havilland Mosquito (1941)

The Mosquito was unusual in that its airframe was constructed almost

entirely of wood. It served multiple roles including tactical bomber,

night high-altitude bomber, day or night fighter, intruder, maritime strike

aircraft, and reconnaissance. Due to its wooden construction it had a high

maximum speed and was able to outrun almost all enemy aircraft. Its

wooden construction also gave it a low radar signature.

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Essential

Groundwork4. Radio Waves and Alternating

Current Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

5. A Nonmathematical Approach to Radar . . . . . . . 63

6. Preparatory Math for Radar . . . . . . . . . . . . . . . . . 77

PART
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Messerschmitt Me-262 (1944)

The Schwalbe(Swallow) was the first jet-powered aircraft to reach

operational status. Its advanced design allowed it to be used in a variety

of roles such as light bomber, reconnaissance, and night fighter. Its speed

and high rate of climb made it extremely difficult to counter. Multiple

B-1a trainers were converted into night fighters using the FuG 218 Neptun

radar, with Hirschgeweihantenna array.

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53

Cockpit of Beaufighter night fighter

Since radio waves and alternating current (AC) signals

are vital to all radar functions, any introduction to radar

logically begins with them. Indeed, many radar con-

cepts that at first glance may appear quite difficult are,

in fact, simple when viewed in the light of radio waves and AC

signals.

In this chapter the nature of radio waves is considered together

with their fundamental qualities.

4.1Nature of Radio Waves

Radio waves are perhaps best modeled as energy that has been

emitted into space. The energy exists partly in the form of an

electric field and partly in the form of a magnetic field. For this

reason, the waves are called electromagnetic.

Electric and Magnetic Fields. Whenever an electric current

flows, a magnetic field is produced. Familiar examples of both

of these field types can be identified. Electric fields are created

when a charge builds up between a cloud and the ground

and produces lightning (Fig. 4-1) or, on a much smaller scale,when a charge builds up on a comb on a particularly dry day,

enabling it to attract a scrap of paper. Magnetic fields encircle

the earth and cause compasses to react, surround a toy mag-

net or are produced when current flows through the coil in a

telephone earpiece and causes the diaphragm to vibrate and

produce sound waves.

The two types of fields are inextricably related. If an electric

field varies sinusoidal ly, so will the magnetic field it produces.

If a magnetic field varies sinusoidally, so will the electric

Radio Waves andAlternating CurrentSignals

Figure 4-1.A common example of an electric field is that which builds

up between a cloud and the ground.

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54 PART II: Essential Groundwork

field it produces. Whenever an electric charge accelerates

by changing speed or direction, a changing magnetic field is

produced and electromagnetic energy is radiated. Because of

thermal agitation of charged particles, everything around us

radiates electromagnetic energy, a tiny portion of which is at

radio frequencies. For an electric current to flow, whether in

a lightning bolt or in a telephone wire, an electric field must

exist. And whenever an electric current flows (Fig. 4-2), a

magnetic field is produced. The electromagnet is a common

example.

If the fields vary with time, the interrelationship extends even

further. Any change in a magnetic field, such as an increase or

decrease in magnitude produces an electric field. This relation-

ship can be observed in the operation of electric generators

and transformers. Similarly, although not so readily apparent,

the reverse is true and any change in an electric field produces

a magnetic field. A varying electric or magnetic field will cause

an electric current to flow in the form of an electromagnetic

wave.

In the second half of the nineteenth century, James Clerk

Maxwell conceived of the idea that a changing electric field

might produce a magnetic field. On the basis of this concept

(Fig. 4-3) and the already demonstrated characteristics of

electric and magnetic fields, he hypothesized the existence

of electromagnetic waves and described their behavior math-

ematically (Maxwells equations). Not until some 13 years

later was their existence actually demonstrated by Heinrich

Hertz.

Electromagnetic Radiation.The dynamic relationship between

the electric and magnetic fields gives rise to electromagnetic

waves. Because of this, whenever a charge, such as that car-

ried by an electron, accelerates, changes direction or rate of

motion, it will change the surrounding fields and electromag-

netic energy is radiated (Fig. 4-4). The change in the motion

of the charge causes a change in the surrounding magnetic

field that is produced by the particles motion. That change

creates a changing electric field a bit further out, which in

turn produces a changing magnetic field just beyond it, and

on, and on, and on.

It follows that the sources of radiation are countless. As

a result of thermal agitation, electrons in all matter are incontinual random motion. Consequently, everything around

us radiates electromagnetic energy (Fig. 4-5). Most of the

energy is in the form of radiant heat (long wavelength infra-

red). But there is always a tiny fraction in the form of radio

waves. Radiant heat, light, and radio waves are, in fact, the

same thing: electromagnetic radiation. They differ only in

frequency.

In contrast to natural radiation, exciting a tuned circuit with a

strong electric current produces the waves radiated by a radar

system. The waves, therefore, all have substantially the same

Magnetic

Field

Current

Figure 4-2.Whenever an electric current flows, a magnetic field is

produced.

Magnetic

Field

ElectricField

Changing

Magnetic

Field

Changing

Electric

Field

Figure 4-3.Dynamic relationships giving rise to radio waves. If

an electric field varies sinusoidally, so will the magnetic field it

produces. If a magnetic field varies sinusoidally, so will the electric

field it produces.

Magnetic

Field

Figure 4-4.Whenever an electric charge accelerates, a changing

magnetic field is produced, and electromagnetic energy is radiated.

Figure 4-5.Because of thermal agitation, everything around us

radiates electromagnetic energy, a tiny portion of which is at radio

frequencies.

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CHAPT ER 4: Radio Waves and Alternating Current Signals 55

frequency and contain vastly more energy than the fraction of

the natural radiation with the same wavelength.

How Antennas Radiate Energy. For a picture of how radi-

ation takes place, consider a simple elemental antenna in

free space. For this purpose, there is no better model than

the dipole Hertz used in his original demonstration of radio

waves.This antenna consists of a thin straight conductor with flat

plates, like those of a capacitor, at either end (Fig. 4-6). An

alternating voltage applied at the center of the conductor causes

a current to surge back and forth between the plates. The cur-

rent produces a continuously changing magnetic field around

the conductor. At the same time, the positive and negative

charges that alternately build up on the plates as a result of

the current flowing in and out of them generate a continuously

changing electric field between the plates.

The fields are quite strong in the region immediately surround-

ing the antenna. As with the field of an electromagnet or thefield between the plates of a capacitor, most of the energy each

field contains returns to the antenna in the course of every

oscillation. However, a portion does not. The changing electric

field between the plates produces a changing magnetic field

just beyond it, which in turn creates a changing electric field

just beyond it, and so forth. Similarly, the changing magnetic

field surrounding the conductor produces a changing electric

field just beyond it, which creates a changing magnetic field

just beyond it, and so forth.

Within this mutual interchange of energy, the electric and mag-

netic fields propagate outward from the antenna. Like ripples

in a pond around a point where a stone has been thrown in

(Fig. 4-7), the fields move outward long after the current that

originally produced them has ceased (or in the case of the

stone has stopped moving). They and the energy they contain

have escaped.

Visualizing a Waves Field. Although the electric and mag-

netic fields cant be seen, they can both be visualized quite

easily. The electric field may be visualized as the force it

would exert on a tiny electrically charged particle suspended

in the waves path. The magnitude of the force corresponds

to the fields strength (E) and the direction of the force to the

fields direction.1As in Figure 4-8, the electric field is com-monly portrayed as a series of solid lines whose directions

indicate the fields direction and whose density (number of

lines per unit of area in a plane normal to the direction) indi-

cates the field strength.

The magnetic field may similarly be visualized as the force it

would exert on a tiny magnet suspended in the waves path.

The direction of the propagation of the wave is always per-

pendicular to the directions of both the electric and the mag-

netic fields. Again, the magnitude of the force corresponds to

the field strength (H) and the direction to the fields direction.

+++++++++

+++++++++

Figure 4-6.This illustration shows a simple dipole antenna such as

that used by Hertz to demonstrate radio waves.

Figure 4-7.Like ripples on a pond, radio waves move outward, long

after the disturbance that produced them has ceased.

+

Force

E

Figure 4-8.The electric field is best visualized as the force it

exerts on a charged particle.

1. The direction of travel of the electromagnetic wave isperpendicular to the directions of the electric and magnetic

fields that make up the electromagnetic wave.

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This field is portrayed in the same way as the electric field

except that the lines are dashed (Fig. 4-9).

4.2Characteristics of Radio Waves

A radio wave has several fundamental qualities: speed, direc-

tion, polarization, intensity, wavelength, frequency, and phase.

Speed.In a vacuum, radio waves travel at constant speed, thespeed of light, represented by the letter c. In the earths atmo-

sphere they travel a tiny bit slower. Moreover, their speed var-

ies slightly not only with the composition of the atmosphere

but also with temperature and pressure.

The variation, however, is so extremely small that most practi-

cal purposes radio waves can be assumed to travel at a con-

stant speed, the same as that in a vacuum. This speed is very

nearly equal to 3 108m/sec. This is the value usually used in

radar computations.

Direction.The direction in which a wave travels, that is, the

direction of propagation (Fig. 4-10), is always perpendicularto the directions of both the electric and the magnetic fields.

These are always such that the direction of propagation is

away from the radiator.

When a wave strikes a (retro) reflecting object, the direction

of one or the other of the fields is reversed, thereby revers-

ing the direction of propagation. As will be made clear in the

blue panel on the next page, which field reverses depends uon

the electrical characteristics of the object.

Polarization.This is the term used to express the orientation

of the waves fields. By convention, it is taken as the direction

of the electric field (the direction of the force exerted on anelectrically charged particle). In free space, outside the imme-

diate vicinity of the radiator the magnetic field is perpendicular

to the electrical field (Fig. 4-11) and the direction of propaga-

tion is perpendicular to both.

When the electric field is vertical (with respect to the ground),

the wave is termed vertically polarized. When the electric field

is horizontal, the wave is termed horizontally polarized.

If the radiating element emitting the wave is a length of

thin conductor, the electric field in the direction of maxi-

mum radiation will be parallel to the conductor. If the con-

ductor is positioned such that it is vertical, the element isvertically polarized (Fig. 4-12). Conversely, if the conductor

E

H

Direction of

propagation

Figure 4-10.The direction of propagation is always perpendicular

to the directions of both the electric and the magnetic fields.

E

H

Figure 4-11.In free space, a waves magnetic field is always

perpendicular to its electric field. Direction of travel is

perpendicular to both.

Figure 4-12.If the radiating element is vertical, the element is said to be

vertically polarized.

N

H Force

S

Figure 4-9.The magnetic field is best visualized as the force it

would exert on a tiny magnet.

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CHAPTE R 4: Radio Waves and Alternating Current Signals 57

is positioned so that it is horizontal, the element is horizon-

tally polarized.

A receiving antenna placed in the path of a wave can extract

the maximum amount of energy from it if the polarization (ori-

entation) of the antenna and the polarization of the wave are

the same. If the polarizations are not the same, the extracted

energy is reduced in proportion to the cosine of the angle

between them.

When a wave is reflected, the polarization of the reflected

wave depends not only on the polarization of the incident

wave but also on the structure of the reflecting object. In this

way the polarization of radar echoes can, in fact, provide use-

ful information about the object being illuminated.For the sake of simplicity, the discussion here has been lim-

ited to linearly polarized waves (waves whose polarization

is the same throughout their length). In some applications,

it is desirable to transmit waves whose polarization rotates

through 360 in every wavelength (Fig. 4-13). This is called

circular polarization. It may be achieved by simultaneously

transmitting horizontally and vertically polarized waves that

are 90 out of phase. In the most general case, polariza-

tion is elliptical (i.e., anything other than 90 out of phase).

Circular and linear polarizations are special cases of elliptical

polarization.

The Speed of Light and Radio Waves

THESPEEDOFLIGHTINANONMAGNETICMEDIUM, SUCHASTHEATMOSPHERE, IS

c =

meters/s*299 7925 106

1 2

.

( )/

e

where e is a characteristic, called the dielectric constant,

of the medium through which the radiation is propagat-

ing. The dielectric constant for air is roughly 1.000536 at sea

level. However, this speed is very nearly equal to 3 108 m/s

and hence is still the value used in the vast majority of radar

computations.

Space: e= 1

Air: e= 1.000536

Speed in the Atmosphere. The dielectric constant of the

atmosphere varies slightly with the composition, tempera-

ture, and pressure of the atmosphere. The variation is such

that the speed of light is slightly higher at higher altitudes.

The dielectric constant of the atmosphere also varies to some

extent with wavelength. As a result, the speeds of light and

radio waves are not quite the same, and the speed of radio

waves is slightly different in different parts of the radio fre-

quency spectrum.

FREE SPACE: e=1

AIR: e=1.000536

*From Maxwells equation, c =()1/2, where =om, and =oe. But, (00)1/2 =299.7925 106and, in a nonmagnetic medium, the perme-

ability m=1.

0

/8

/4

Figure 4-13.Pictured here is polarization of a circularly polarized

wave at points separated by 1/8 wavelength. Wave is produced by

combining two equal-amplitude waves that are 90 out of phase.

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Reflection, Refraction, and Diffraction

ANYOFTHREEMECHANISMSMAYCAUSEARADIOWAVETOCHANGEDIRECTIONS:reflection (which makes radar possible), refraction, anddiffraction or any combination of the three.

Reflection from a Conductive Surface.When a wave strikes a

conducting surface, its electric field causes a current to flow that

in turn creates a wave whose energy is reradiated, forming the

process of reflection. From a flat surface (i.e., any irregularities

are small compared with a wavelength), reflection is mirror-

like and is called specular. From an irregular (i.e., any irregu-

larities are of the order of a wavelength or larger) or complex

surface (e.g., that of trees or an aircraft), reflection is diffuse,

and radiation is scattered in all directions. This is termed diffuse

scattering.

E Time 1 Time 2 Time 3

ReflectedEnergy

IncidentEnergy Incident

Energy

Specular Reflection Scattering

1 1

Reflection from a Nonconductive Surface. When a wave

enters a nonconducting medium (e.g., Plexiglas) with a dif-

ferent dielectric constant from the medium through whichthe wave has been propagating (e.g., air), some of the waves

energy is reflected (just as from a conducting surface). The

reason is that the dielectric constant, e, of the medium deter-

mines the division of energy between the waves electric and

magnetic fields. (In a vacuum, where e= 1, the energy is

divided equally between the two fields.) To adjust the balance

to the new dielectric constant, some of the incident energy

must be rejected, which occurs through the reflection.

1

2

Refraction. If the angle of incidence, 1, is greater than zero,

when a wave enters a region of different dielectric constant

the energy passing through is deflected in a phenomenon

called refraction. The deflection increases with the angle of inci-

dence and the difference of the two dielectric constants, that

is, with the difference between the speeds in the two media.

Assuming that material 1has a higher dielectric constant than

2, the wavefront will travels faster in 2 than in 1. Thus, the

portion of the wavefront reaching 2 first will travel start to

travel at a higher speed in the new material. The portion of the

wavefront yet to reach 2continues at its previous speed until italso reaches 2. This progressive change in speed of the wave-

front causes it to propagate in 2at a larger angle 2. The ratio

of the velocities in the two media is called the refractive index.

1

RefractedEnergyReflected

Energy

2

1

1 2

Atmospheric Refraction. A form of refraction occurs in the

atmosphere. Because of the increase in the speed of light

(decrease in e) with altitude, the path of a horizontally propa-

gating wave gradually bends toward the earth. This phenom-

enon enables us to see the sun for a short time after it has set.

It similarly enables a radar system to see over the horizon.

Diffraction.A wave spreads around objects whose size is com-

parable to a wavelength and bends around the edges of larger

obstructions. For a given size of obstruction, the longer the

wavelength, the more significant the effect. That is why radiobroadcast stations (operating at wavelengths of a few hundred

meters) can be heard in the shadows of buildings and moun-

tains, whereas TV stations operating at wavelengths of only a

few meters cannot.

Particle Diffraction Edge Diffraction

This phenomenon, called diffraction, stems from the fact that

the energy at each point in a wave is passed on just as if a radia-

tor actually existed at that point. The wave as a whole propa-

gates in a given direction only because the radiation from

all points in every wavefront reinforces in that direction and

cancels in others. If the wavefronts are broken by an obstruc-

tion, cancellation at the edge of the wave is incomplete, which

causes the part of the wavefront nearest to the obstruction to

propagate differently.

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Intensity. The rate at which a radio wave carries energy

through space, intensity is defined as the amount of energy

flowing per second through a unit of area in a plane normal to

the direction of propagation (Fig. 4-14).2

The intensity is directly related to the strengths of the elec-

tric and magnetic fields. Its instantaneous value equals the

product of the strengths of the two fields times the sine ofthe angle between them. As previously noted, in free space

outside the immediate vicinity of the antenna, that angle is

90; thus, the intensity is simply the product of the two field

strengths (EH).

Generally, what is of interest is not the instantaneous value of

the intensity but the average value. If an antenna is interposed

at some point in a waves path, multiplying the waves average

intensity at that point by the area of the antenna gives the amount

of energy per second intercepted by the antenna (Fig. 4-15).

In an electrical circuit, the term used for the rate of flow of

energy is power. Consequently, in considering the transmis-sion and reception of radio waves, the term power density

is often used for the waves average intensity. The two terms

are equivalent. The power of the received signal is the power

density of the intercepted wave times the area of the antenna.

Wavelength. If a linearly polarized radio wave were frozen

in time and its two fields were viewed over some distance in

space, two things would be observed. First, the strength of the

fields varies cyclically in the direction of the waves travel. It

builds up gradually from zero to its maximum value, returns

gradually to zero, builds up to its maximum value again, and

so on. The fields in the planes of two successive maxima areshown in Figure 4-16. Second, each time the intensity goes

through zero, the directions of both fields are reversed.

UnitofArea

1

1

Flowof

Energy

Figure 4-14.Intensity of a wave is the amount of energy flowing

per second through a unit of area normal to the direction of

propagation.

Antenna

Figure 4 -15.Power of received signal equals power density of

intercepted wave times area of antenna. (Power density is another

term for intensity.)

E

E

H

H

Figure 4-16.The fields of a radio wave, at points of maximum intensity,

frozen in space. When intensities go through zero, directions of fields

reverse.

2. Other terms for this rate are energy flux and power flow.

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The intensity of the fields is plotted versus distance along the

direction of travel in Figure 4-17. It is negative when the direc-

tions of the forces exerted by the fields are reversed. The shape

is the same as a plot of the sine of an angle versus the angles

size (strictly speaking, the wave should be infinitely long).

Because of this, radio waves are referred to as sinusoidal, or

sine waves.3

Referring again to Figure 4-17, the distance between successive

crests, or troughs, is the wavelength, which is usually repre-

sented by a lowercase Greek lambda, , and is expressed in

meters, centimeters, or millimeters depending on its length.

Frequency.The frequency of a radio wave is directly related to

the wavelength. To see the relationship, visualize a radio wave

traveling past a fixed point in space. The intensity of the elec-

tric and magnetic fields at this point increases and decreases

cyclically as the wave goes by.

Placing a receiving antenna in the waves path and the volt-

age developed across the antenna terminals is observed on anoscilloscope will reveal that it has the same shape (amplitude

versus time) as the earlier plot of the intensity of the fields

versus distance along the direction of travel (Fig. 4-17). The

number of cycles this signal completes per second is the waves

frequency.

Frequency is usually represented by a lowercase f and is

expressed in Hertz (Hz) in honor of Heinrich Hertz: 1 Hz is one

cycle per second; 1000 Hz is 1 kilohertz (kHz); 1000 000 Hz is

1 megahertz (MHz); 1000 MHz is 1 gigahertz (GHz); and

1000 GHz is a terahertz (THz).

Since a radio wave travels at a constant speed in a given

medium, its frequency is inversely proportional to its wave-

length. The shorter the wavelength, the more closely spaced

the crests and the greater the number of them that will pass a

given point in a given period of timehence the greater the

frequency (Fig. 4-18).

The constant of proportionality between frequency and wave-

length is, of course, the waves speed. Expressed mathematically,

f c=

where f=frequency, c=speed of the wave (3 108m/sec),

and =wavelength. With this formula, the frequency corre-

sponding to any wavelength can be quickly found. A wave

having a wavelength of 3 cm, for example, has a frequency of

10,000 MHz or 10 GHz.

Knowing the frequency, the wavelength can be found simply

by inverting the formula:

= c

f

Wavelength

Distance

(

)Intensity(+)

Figure 4-17.This graph shows the variation in intensity of fields in

direction of travel. Distance between crests is wavelength.

f1

f2= 3f1

Distance

Distance

Fixed Observation Point

Speed

Figure 4-18.Since a radio wave travels at a constant speed, the

shorter the wavelength, the higher the frequency.

3. A radio wave will have a pure sinusoidal shape if it is

continuous and its peak amplitude, frequency, and phaseare constant (i.e., the wave is unmodulated).

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Period.Another measure of frequency is period, T, the length

of time a wave or signal takes to complete one cycle (Fig.

4-19). If the frequency is known, the period (in seconds) can

be obtained by taking the inverse of the number of cycles per

second:

Period= seconds)1

f (

For example, if the frequency is 1 MHz (i.e., the wave or signal

completes one million cycles every second), it will complete

one cycle in one-millionth of a second. Its period is one-

millionth of a second, or 1 microsecond.

Phase.This concept is essential to understanding many aspects

of radar operation. It is the degree to which the individual

cycles of a wave or signal coincide with those of a reference of

the same frequency (Fig. 4-20).

Phase is commonly defined in terms of the points in time at

which the amplitude of a signal goes through zero in a positivedirection. The signals phase, then, is the amount that these

zero crossings lead or lag the corresponding points in the ref-

erence signal. This amount can be expressed in several ways.

Perhaps the simplest is as a fraction of a wavelength or cycle.

However, phase is generally expressed in degrees with 360

corresponding to a complete cycle. If, for instance, a wave is

lagging a quarter of a wavelength behind the reference, its

phase is 360 1/4 =90. As will be seen later, when the phase

of a target echo is repeatedly changing, it indicates the speed

of motion of the target or parts of the target.

4.3Summary

Radio waves are radiated whenever an electric charge accel-

erates whether due to thermal agitation in matter or a current

surging back and forth through a conductor. Their energy is

contained partly in an electric field and partly in a magnetic

field. The fields may be visualized in terms of the magnitude

and direction of the forces they would exert on an electri-

cally charged particle and a tiny magnet, suspended in the

waves path.

The polarization of the wave is the direction of the electric

field. The direction of propagation is always perpendicular tothe directions of both fields. In free space at a distance of sev-

eral wavelengths from the radiator, the magnetic field is per-

pendicular to the electric field, and the rate of flow of energy

equals the product of the magnitudes of the two fields. In an

unmodulated signal, the intensity of the fields varies sinusoi-

dally as the wave passes by. The distance between successive

crests is the wavelength.

If a receiving antenna is placed in the path of a wave, an

AC voltage proportional to the electric field will appear across

its terminals. The number of cycles this signal completes per

Period

T

Time

Amplitude

Figure 4 -19.Period is length of time a signal takes to complete

one cycle.

Reference Signal

Phase

Figure 4-20.Phase is the degree to which the cycles of a wave

or signal coincide with those of a reference signal of the same

frequency.

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second is the waves frequency. The length of time the signal

takes to complete one cycle is its period. Phase is the fraction

of a cycle by which a signal leads or lags a reference signal

of the same frequency. It is commonly expressed in degrees.

Some relationships to keep in mind are as follows:

Speed o f radio waves = m/s

= 300 m/ s

3 108

Wave length = Frequency300 106

Period = Frequency

1

Further Reading

S. E. Schwarz,Electromagnetics for Engineers, Oxford University

Press, 1995.

K. Lonngren, S. Savov, and R. Jost, Fundamentals ofElectromagnetics with MATLAB, 2nd ed., SciTech-IET, 2007.

J. W. Nilsson and S. Reidel, Sinusoidal Steady-State Analysis,

chapter 9 inElectric Circuits, Prentice-Hall, 2011.

F. T. Ulaby, E. Michielssen, and U. Ravaioli, Fundamentals of

Applied Electromagnetics, 6th ed., Pearson, 2014.

Test your understanding

1. How does an antenna radiate energy?

2. What is diffraction?

3. Explain what is meant bypolarization.

4. At what speed do electromagnetic waves travel in free space? How might this differ if the radiowaves propagate in the earths atmosphere?

5. Sketch graphs to show the wavelengthandfrequencyof a radio wave. Take care to correctlylabel the axes of the graphs.

6. If a radar system operates at a transmission frequency of 3 GHz, what is the wavelength

(the velocity of light can be taken to be 3 108m/s)?

7. For a radar system transmitting at an operating frequency of 3 GHz, what is the period?

8. Describe three mechanisms that cause a radio wave to change direction.

9. Describe what is meant by the phase of a signal.

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