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63

NIMROD cockpit

Modern radar systems are coherent, meaning they

measure both amplitude and phase of echo sig-

nals. As will shortly be seen, the phase is measured

relative to a reference, usually the transmitted sig-

nal. Measuring the amplitude and phase provides a powerful

basis on which almost all advanced (and some not quite as

advanced) techniques are built. A powerful tool often used by

the radar engineer to represent the amplitude and phase of a

received echo is a graphic device called thephasor. Though no

more than an arrow, the phasor is key to nonmathematically

understanding many seemingly esoteric concepts encountered

in radar work such as the spectrum of a pulsed signal, thetime-bandwidth product, digital filtering, the formation of real

and synthetic antenna beams, and sidelobe reduction.

Unless you are already skilled in the use of phasors, dont yield

to the temptation to skip ahead to chapters about radar. Having

mastered the phasor, you will be able to unlock the secrets

of many intrinsically simple physical concepts that otherwise

you may find yourself struggling to understand. This is because

phasors represent the relationships between signals and can be

used to combine signals and describe the resultant. As well as

being easy to visualize, they have a rigorous mathematical basis

so results can be trusted both quantitatively and qualitatively.This chapter begins by briefly describing the phasor. To dem-

onstrate its application, phasors are then used to explain several

basic concepts that are essential to understanding material pre-

sented in later chapters. In addition, the decibel (dB) is intro-

duced. It is necessary to become familiar with the dB because

it is such a universal measure of many quantities used in radar.

5.1How a Phasor Represents a Signal

A phasor is nothing more than a rotating arrow (vector), yet

it can represent a sinusoidal signal completely (Fig. 5-1). The

A NonmathematicalApproach to Radar

A

Y

X

Phase

Figure 5 -1.A phasor rotates counterclockwise, making one

complete revolution for every cycle of the signal it represents.

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64 PART II: Essential Groundwork

arrow is scaled in length to the signals peak amplitude. It

rotates like the hand of a clock. Phase progression is rep-

resented as rotation and is positive in the counterclockwise

direction, making one complete revolution for every cycle of

the signal. The number of revolutions per second thus equals

the signals frequency.

The length of the projection of the arrow onto a vertical line

through the pivot point equals the amplitude times the sine of

the angle between the arrow and the horizontal axis (Fig. 5-2).

Consequently, if the signal is a sine wave, this projection cor-

responds to the signals instantaneous amplitude.

As the arrow rotates (Fig. 5-3), the projection lengthens until it

equals the arrows full length, shrinks to zero, then lengthens

in the opposite (negative) direction, and so on, exactly as the

instantaneous amplitude of the signal varies with time. If the

signal is a cosine wave, the projection on the horizontal axis

through the pivot corresponds to the instantaneous amplitude.

The 90-degree angle between the horizontal and vertical axes

shows that the cosine wave is a sine wave with a 90-degree

phase shift.

In the interest of simplicity, the arrow is drawn in a fixed posi-

tion. It can be thought of as illuminated by a strobe light thatflashes on at exactly the same point in every cycle. The strobe

point is the instant the arrow would have crossed the x axis

had the signal the arrow represents been in phase with a refer-

ence signal of the same frequency (Fig. 5-4). In other words,

the strobe light is the reference signal or, in radar parlance, the

local oscillator (LO) signal.

The angle the arrow makes with the x axis, therefore, corre-

sponds to the signals phaseand hence the name, phasor. If

the signal is in phase with the reference, the phasor will line

Asin kt

A

x

y

kt

Asin ktA

Time

kt

Figure 5-2.For a sine wave, projection of the phasor onto the y axis gives the signals instantaneous amplitude.

y

A

A

A

A

A

A

A

Time

+

Figure 5-3.As a phasor rotates, projection onto the y axis

lengthens to a maximum positive value, returns to zero, lengthens

to maximum negative value, and then returns to zero again.

A

y

x

Phase

Strobe

Light

Figure 5-4.A phasor can be thought of as illuminated by a

strobe light that flashes on at the same time as a reference phasor

would be crossing the x axis. The strobe provides the phase

reference.

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CHAPTE R 5: A Nonmathematical Approach to Radar 65

up with the x axis (Fig. 5-5). If the signal is 90 out of phase

(i.e., in quadrature) with the reference, i.e., is in quadrature

with it, the phasor will line up with the y axis. For a signal that

which leads the reference by 90, the phasor will point up; for

a signal that lags behind the reference by 90, the phasor will

point down.

Generally, the rate of rotation of a phasor is represented bythe Greek omega, . While the value of can be expressed

in many different units (e.g., in revolutions per second or

degrees per second), it is most commonly expressed in

radians per second. A radian is an angle that, if drawn from

the center of a circle, is subtended by an arc the length of

the radius. Since the circumference of a circle is 2times the

radius, the rate of rotation of a phasor in radians per second

is 2times the number of revolutions per second, or the fre-

quency (Fig. 5-6). Thus,

= 2 f

wherefis the frequency of the signal, in Hz.

The real power of phasors lies in their ability to represent

the relationships between two or more signals clearly and

concisely. Phasors may be manipulated to portray the addi-

tion of signals of the same frequency but different phases, the

addition of signals of different frequencies, and the resolution

of signals into in-phase and quadrature components (a key

part of modern radar systems). Several common but impor-

tant aspects of radar operationincluding target scintillation,

frequency translation, image frequencies, and the creation

of sidebandscan illustrate the kind of insights that may begained from phasors.

5.2Combining Signals of Different Phase

To see how radio waves of the same frequency but different

phases combine, consider drawing two phasors from the same

pivot point. Sliding one laterally, one is added to the tip of

the other. A third phasor from the pivot point to the tip of the

second arrow can then be drawn. This phasor, which rotates

counterclockwise in unison with the others, represents their

sum (Fig. 5-7).

The sum can also be obtained without moving the second

phasor by constructing a parallelogram with two adjacent sides

made up of the phasors to be added. The sum is a phasor

drawn from the pivot point to the opposite corner of the paral-

lelogram (Fig. 5-8). The value of such a seemingly simple rep-

resentation of the sum of two signals can be used to explain

target scintillation.

Scintillation. Consider a situation where the reflections of

a radars transmitted waves are received primarily from two

R

= 2f

t

Figure 5-6.Rate of rotation, , is generally expressed in radians/

second. Since there are 2radians in a circle, =2f.

B

B

B

AA A

A + B

Figure 5-7.To add phasorsAand B, simply slide to the tip

ofA. The sum is a phasor drawn from the origin to the tip of B.

B

A

A + B

Figure 5-8.Phasors can also be added by constructing a

parallelogram and drawing an arrow from the pivot to the opposite

corner.

In phase

with reference

In quadrature

with reference

(Leading)

(Lagging)

Figure 5-5.If the signal a phasor represents is in phase with the

reference (strobe light), the phasor will line up with the x axis. If

signal is in quadrature, the phasor will line up with the y axis.

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66 PART II: Essential Groundwork

parts of a target (Fig. 5-9). The fields of the reflected waves

will merge. To see what the resulting wave will be like under

various conditions, the waves are represented by phasors.

To begin with, assume that the targets orientation is such that

the distances from the radar to the two parts of the target are

almost the same (or differ by roughly a whole multiple of a

wavelength). Therefore, the two waves are nearly in phase. Asillustrated by the left-hand diagram in Figure 5-10, the ampli-

tude of the resulting wave very nearly equals the sum of the

amplitudes of the individual waves.

Next, assume that the orientation of the target changes ever

so slightly, as it might in normal flight, but enough so that the

reflected waves are roughly 180 out of phase. The waves now

largely cancel (right-hand diagram in Fig. 5-10).

Clearly, if the phase difference is somewhere between these

extremes, the waves neither add nor cancel completely, and

their sum has some intermediate value. Thus, the sum may

vary wildly from one moment to the next. Recognizing, of

course, that appreciable returns may be reflected from many

different parts of a target, this wildly varying sum explains

why a targets echoes scintillate. This also explains why the

maximum detection range of a target is predicted in statisti-

cal terms.

What happens to the rest of the reflected energy when the

waves dont add up completely? It doesnt disappear. The

waves just add up more constructively in directions different

to that of the radar receiver.

5.3Combining Signals of Different Frequency

The application of phasors is not limited to signals of the

same frequency. They can also be used to illustrate what hap-

pens when two or more signals of different frequency are

added together or when the amplitude or phase of a signal of

one frequency is varied (i.e., modulated) at a lower frequency.

To see how two signals of slightly different frequency combine,

consider drawing a series of phasor diagrams, each showing

the relationship between the signals at a progressively later

instant in time. If instants are chosen so they are synchronized

with the counterclockwise rotation of one of the phasors (i.e.,

adjusting the frequency of the imaginary strobe light so it is

the same as the frequency of one of the phasors), that phasor

will occupy the same position in every diagram (phasor Ain

Fig. 5-11).

The second phasor will occupy progressively different posi-

tions. The difference from diagram to diagram corresponds to

the difference between the two frequencies.

If the difference is positive and the second frequency is higher,

the second phasor will rotate counterclockwise relative to the

Time 1 Time 2 Time 3 Time 4

A

B

A

B

A

B

B

A

Figure 5-11.How signals of different frequencies combine. If the

strobe light is synchronized with the rotation of phasor A, it will

appear to remain stationary and phasor Bwill rotate relative to it.

d1 d

2

Figure 5-9.In this situation, a radar receives return primarily from

two points on a target. Distances to the points are d1and d2.

2

Sum

Sum

1

2

1

d1 d

2~ ~d1

d2

2

Figure 5-10.If distances d1and d2to the two points on the target

are roughly equal, the combined return will be large, but if the

distances differ by roughly half a wavelength, the combined return

will be small as they will sum in near anti-phase.

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CHAPT ER 5: A Nonmathematical Approach to Radar 67

first (Fig. 5-12). If the difference is negative and the second fre-

quency is now lower, the second phasor will rotate clockwise

relative to the first.

As the phasors slip into and out of phase, the amplitude oftheir sum fluctuates (or is modulated) at a rate equal to the

difference between the two frequencies. The phase of the sum

also is modulated at this rate. It falls behind during one-half

of the differencefrequency cycle and slides ahead during the

other half. As the phase changes, the rate of rotation of the sum

phasor changes: the frequency of the signal is also modulated.

By representing signals of different frequencies in this way, many

important aspects of a radars operation can easily be illustrated

graphically using image frequencies or creating sidebands.

Frequency Translation.Since the amplitude of the sum of two

phasors fluctuates at a rate equal to the difference between the

rates of rotation of the phasors, a signal can be readily shifted

down in frequency by any desired amount. Adding one signal

to another at a suitably different frequency does this, and then

the amplitude fluctuation is extracted. Figure 5-13 shows how

this is carried out in a radar receiver. The frequencies of the

local oscillator (fLO) and intermediate frequency (fIF) are two

very important design parameters for any radar.

In the early stage of vir tually every radio or radar receiver, the

received signal is translated to a lower intermediate frequency,

or IF (Fig. 5-13). Translation is accomplished by mixing the

signal with the output of a local oscillator, whose frequency isoffset from the signals frequency by an amount equal to the

desired intermediate frequency (fIF).

In one mixing technique, the signal, fsis simply added to the

LO output, as in Figure 5-14, and the fluctuation in the ampli-

tude of the sum is extracted (detected). In another mixing tech-

nique, the amplitude of the received signal itself is modulated

by the LO output. Amplitude modulation produces image fre-

quencies or sidebands. In this case, the frequency of one of

the sidebands is the difference between the frequencies of the

received signal and LO signalfIF.1

fs

fLO

Frequency, fIF= fs fLO

Time

Fluctuation in amplitude of sum

Fluctuation

in phase

Figure 5 -14.If the LO signal is stronger than the received signal,

then the fluctuation in amplitude of the sum is virtually identical to

the received signal except for being shifted to fIF.

ReceivedSignal, fs

ReceivedSignal, fIF

(fIF= fs fLO)

Mixer

ExtractAmplitude

Fluctuation*

LocalOscillator

* By passing the sum through anonlinear circuit and its outputthrough a bandpass filter.

+

fLO

Figure 5-13.A received signal may be translated to a lower

frequency fIFby adding it to an LO signal and extracting the

amplitude modulation of the sum.

fB>fA

AB

AB

fB