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Boeing B-52 Stratofortress (1955)

The B-52 Stratofortress is one of a selec t number of aircraft that hasbeen in service for more than half of the history of powered ight. Itis a long-range, subsonic, jet-powered strategic bomber with a crewof ve, designed and built in huge numbers by Boeing, and has beenoperated by the United States Airforce (USAF) since the 1950s. An upgradeprogram being conducted through 2015 will take its expected service into

the 2040s.

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77

Antenna of AustralianWedgetail AEW radar

The previous chapter introduced a nonmathematical way of viewing and manipulating radar signals. Here we provide some of the basic mathematical toolsneeded to support a more thorough yet still quitesimple description of radar systems. These tools are used tosynthesize and analyze signals and also to support signal-processing approaches used for target detection. They alsorelate directly to the phasor description of the last chapter, andboth the mathematical and nonmathematical approaches arefully compatible with each other. Further, this chapter showsthat phasors are also key to a mathematical understandingof radar enabling both visualization of phase and frequencyrelationships and their manipulation through complex numberrepresentation. First, the different types of signals used withinradar systems are formally dened.

However, if you prefer to grasp just the concepts underpinningradar without worrying about the underlying mathematics, feelfree to skip this chapter.

6.1 Signal Classication

Signals can be classied in a variety of ways. Perhaps the sim-plest is the periodic signal . (Most radar waveforms are peri-odic.) A periodic signal, f , repeats itself after a xed length oftime. This can be written as

f t f t T ( ) ( )= +

where T is the repetition period.

If a signal doesnt repeat itself after a xed amount of time itis said to be nonperiodic or aperiodic. Signals are generallyclassed as being energy signals or power signals .

An (analog) energy signal is dened as one where the totalenergy, E , dissipated between the star t and end of time (which

PreparatoryMath for Radar

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78 PART II: Essential Groundwork

will be taken to be the start and end of the signal) is nonzeroand nite:

E f t dt =

2 ( )

Radar signals are examples of energy signals as the energy canbe obtained simply by integrating the power over the durationof the pulse.

If the average power delivered by a signal from the beginningto the end of time is nonzero and nite, the signal is denedto be a power signal:

P T T f t dt T

T

= lim ( )12 2

One example of a power signal is direct current (DC). Powersignals are more common in communication systems. For peri-

odic signals the limits of integration can be set by the periodleading to an expression for the average power that can be written as

P T f t dt T

= 1 20

( )

Energy and power signals thus represent two very differentclasses. If a signal has nite energy its power will be zero andcannot therefore be a power signal. If a signal has nite powerits energy will be innite.

6.2 Complex NumbersComplex numbers are much less complex than their name sug-gests and provide a direct relationship to the nonmathematicalphasor description of signals. The sine and cosine functionsfrom which phasors were derived can be expressed in expo-nential as well as trigonometric forms. 1 For example:

A t A

A t A

sin ( )

cos ( )

=

=

e ej

e e

j -j

j -j

t t

t t

2

2+

The letter j in the exponential terms has the value 1.Because 1 cannot be evaluated, it is said to be an imaginarynumber. A variable (or number) having an imaginary part anda real part is called a complex variable (or number).

Often, sinusoidal functions are more easily manipulated inexponential form rather than in trigonometric form. At rstglance, the exponential terms e j t and e j t seem to have littlephysical meaning. However, the functions they represent canbe visualized quite easily with phasors. In a phasor diagram, e j

is a rotation in a counterclockwise direction and e j is a rotationin a clockwise direction.

1. The equivalence can be demonstrated by expanding thefunctions sin x, cos x, and e j x into power series with Maclaurinstheorem.

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CHAPTE R 6: Preparatory Math for Radar 79

The term e j t then is represented by a phasor of unit lengthrotating counterclockwise at a rate of radians per second.The term e j t is a phasor of unit length rotating in a clockwisedirection also at a rate of radians per second. This is illus-trated in Figure 6-1.

The sum e j t + e j t equals the sum of the projections of the twophasors onto the x axis and is shown in Figure 6-2. This sum is2cos t (remember to take one of the phasors and star t it from

the tip of the other to obtain the resultant).The difference e j t e j t equals the projection of the rstphasor onto the y axis minus the projection of the secondphasor onto the y axis as shown in Figure 6-3. This differenceis 2sin t (again take one of the phasors and start it from thetip of the other to obtain the resultant).

Using these basic relationships as building blocks and remem-bering the values of j raised to various powers,

j =

j

j j

j4

= =

= =

= = +

1

1 1 1

1 1

1 1 1

2

3

( )( )

one can easily visualize virtually any relationships involvingthe complex variable.

6.3 Fourier Series

The Fourier series and the Fourier transform (see Section 6.4)are named after Jean Baptiste Joseph Fourier (17681830), aFrench mathematician and physicist. It can be demonstratedboth mathematically and graphically that any continuous,

e j t =

1

t

e j t =

1

t

Figure 6-1. Phasors of unit length are pictured here rotating counter-clockwise, e j t , and clockwise, e j t , at a rate of radians per second.

e j t + e j t = t

t

2 cos t

Figure 6-2. This graph illustrates the sum of the projections oftwo phasors.

e j t e j t =

1 1

t

t

t t

2 sin t

Figure 6-3. This shows the projection of the rst phasor onto they axis minus the projection of the second phasor onto the y axis. This difference is 2 sin t .

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periodically repeated waveform, such as a pulsed signal, canbe represented by a series of sine waves of specic ampli-tudes and phases whose frequencies are integer multiples ofthe repetition frequency of the wave shape. The repetition fre-quency is called the fundamental ; the multiples of it are calledharmonics. The mathematical expression for this collection of

waves is the Fourier series.

Mathematical Description of a Fourier Series. The mathemati-cal description begins with any well-behaved periodic functionof time, f (t + T ), as illustrated in Figure 6-4. T is the period ofrepetition (and is assumed to continue from the beginning tothe end of time).

The function of time, f (t + T ), is represented by the sum of aconstant, a 0, plus a series of sine terms whose frequencies areinteger multiples of the repetition frequency, f r (= 1/T ). Thesesine terms comprise the various harmonic components of theperiodic signal. For example,

f t A A t A t ( ) ( ) ( )= + + + +0 0 01 1 2 2sin sin 2

(First Harmonic) (Secon

dd Harmonic)

sin 3 sin 4

(Third Harmon

3 3 4 o 4+ + + + A t A t ( ) ( ) 0

iic) (Fourth Harmonic)

where 0 = 2 f r , f r = 1/ T , and 1, 2, 3, 4 are the phases ofthe harmonics.

The phase angles can be eliminated by resolving the terms intoin-phase and quadrature components as illustrated in Figure 6-5.

f t a a t b t ( ) = 0 1First Harmonic

+ +1 0 0cos ( ) (sin ( )

( )

++ +

+

a t b t

a

2 0 0

3

2 2cos ( ) sin ( )

c

( )

2

SecondHarmonic

oos ( ) sin ( )( )

3 30 0 t b t + 3ThirdHarmonic

The complete series can be written compactly as the summa-tion of n terms for which n has values of 1, 2, 3

f t a a n t b n t nn

n( ) = +0 0 0+=

1

cos sin

This is the general form of the Fourier trigonometric series.The coefcients a n and b n are the Fourier sine and cosine coef-cients representing the amplitudes of the various componentscomprising the signal. The constant term a 0 represents the

Time

T

+

Figure 6-4. This periodic pulse train has period T (=1/f r ).

a 1

1

b1 A1

a 2 a 3

b 3 2 3

b2 A2

A3

f t a a t b t a( )= 0 1First Harmonic

+ + +1 0 0cos( ) sin( )( )

22 0 0 32 2cos( ) sin( ) co( )

t b t a+ +2Second Harmonic

ss( ) sin( )

( )

3 30 0 t b t + 3 Third Harmonic

Figure 6-5. This equation resolves the phasors into I and Q components.

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constant or DC component of the signal. Although the sumof the series is strictly a periodic variation, it can be used todescribe an arbitrary signal over an interval of length T .

Graphical Description of the Fourier Series. The concept under-lying the Fourier series is illustrated graphically for a square

wave in Figure 6-6.

The sum in Figure 6-6 is obtained by adding the fundamental,third, and fth harmonics. The reinforcement and cancellationof the different components creates a sum signal that beginsto approximate a square wave. By adding more and more har-monics the approximation to the square wave gets better andbetter. Note that the shape of the composite wave depends asmuch on the phases of the harmonics as on their amplitudes.To produce a rectangular wave, the phases must be such thatall harmonics go through a positive or negative maximum atthe same time as the fundamental.

A wave of a more general rectangular shape is illustrated in

Figure 6-7. Theoretically, to produce a perfectly rectangular wave an innite number of harmonics would be required. Actually, the amplitudes of the higher order harmonics are

Fundamental

Third harmonic

Fifth harmonic

Sum

Figure 6-6. This square wave is produced by adding two harmonicsto the fundamental. Because positive and negative excursions areof equal duration, amplitude of even harmonics is zero.

0.0

Figure 6-7. This rectangular wave is produced by adding four harmonics to the fundamental. Shape of composite wave is determined by relativeamplitudes and phases of harmonics. For shape to be rectangular, all harmonics must go through a positive or negative maximum at the same timeas the fundamental.

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relatively small so reasonably rectangular wave shapes can beproduced with a limited number of harmonics. A recognizablyrectangular wave was produced in Figure 6-6 by adding onlytwo harmonics to the fundamental. And a still more rectangu-lar wave has been produced in Figure 6-7 by adding only fourharmonics.

The more harmonics included, the more rectangular the wave will be and the less pronounced the ripple.

In Figure 6-8, the ripple has been reduced to negligible propor-tions, except at the sharp corners, by including 100 harmonics.

To create a train of pulses (i.e., a waveform whose amplitudealternates between zero and, say, one), such as A + B, with aseries of sine waves, a zero-frequency or DC component mustbe added.

Its value, B, equals the amplitude of the negative loops of therectangular wave, with sign reversed (Fig. 6-9).

Exponential Form of the Fourier Series. The Fourier series canalso be expressed in exponential form using complex num-bers. To derive the exponential form of the Fourier series, thesines and cosines are replaced by the complex numbers.

a n t a

e e

b n t b

j e e

nn jn t jn t

nn jn t

cos( )

sin( )

0

0

2

2

0 0

0

= +

=

j jn t 0

This leads directly to

a n t b n t X e X e n n n jn t n jn t cos( ) sin( )

0 0 0 0+ = +

where

X a jb X a jb n n n n n n= = +12

12( ); ( )

The general form of this exponential series describing a peri-odic signal is therefore

f t X X e X e

X e X e

X e

j t j t

j t j t

n jn

( ) = + + +

+ + +

=

0 1 22

1 22

0 0

0 0

0

t t

n

n

=

=

The coefcients X n are complex and specify the amplitudesand phases of the harmonic components describing thesignal f (t ).

6.4 The Fourier Transform

The Fourier transform is a mathematical construct that allows atime domain signal to be represented in the frequency domainand vice versa.

By switching between the time and the frequency domains,signals that can appear very scrambled in one domain may

Rectangularwave

DC component

Pulse train

(+)

(+)

()

(+)

0

0

A

B

B

A + B

() () ()

(+)

(+)

(+) (+)

(+)

0

Figure 6-9. To produce a train of rectangular pulses from arectangular wave, a DC component must be added.

Figure 6-8. This rectangular wave shape is produced by combining100 harmonics. Note the reduction in ripple.

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CHAPT ER 6: Preparatory Math for Radar 83

have a very clear structure in the other. The Fourier transformenables signals to be viewed in both the time and frequencydomains. Further, in switching from time to frequency theFourier transform is directly related to Doppler motion andhence is a basic tool in the detection of moving targets. TheFourier transform has a wide variety of other roles and isused in radar (and many other) topics as diverse as anten-nas (Chapter 8) and synthetic aperture radar (SAR) processing(Section VII).

Time and Frequency Domains. A graph (or equation) relatingthe amplitude of a signal to time represents the signal in thetime domain (Fig. 6-10).

A graph (or equation) relating the amplitude and phase of thesignal to frequency (the signals spectrum ) represents the signalin the frequency domain (Fig. 6-11).

A signal can be represented completely in either domain.

Switching between Domains. The representation of a signalin one domain can readily be transformed into the equivalentrepresentation in the other domain through the Fourier trans-form process (Fig. 6-12).

The mathematical expression for transforming from thetime domain to the frequency domain is called the Fouriertransform .

The mathematical expression for transforming from the fre-quency domain to the time domain is called the inverse

Fourier transform (Figure 6-13). Together, the two transformsare called a transform pair . Thus, Figures 6-12 and 6-13 are a

Fourier pair. In this way a sine wave in the time domain wouldtransform into a single line (at the frequency of the sine wave)in the Fourier or frequency domain, two very different repre-sentations of the same signal.

Calculating the Transforms. To calculate the Fourier transform,an expression for the signal, f (t ), is inserted into

F f t t ( ) =

+

( ) e dtj

and the integration is then performed.

Similarly, to calculate the inverse transform, an expression forthe signal, F ( ), is inserted into

f t F ( ) = ( ) e d+j t1

2

+

Again, the integration is carried out. The variable is fre-quency in radians per second ( = 2 f ), and e j t is the expo-nential form of the expression, cos t j sin t .

However, with the advent of digital signal processing in realradar systems, this is done using a discrete Fourier transform .

Time Domain

Time

A m p l i t u d e

0 (+)()

Figure 6-10. This shows a time domain, square wave pulse.

Frequency Domain

Frequency A m p l i t u d e

0 (+)

(+)

()()

Figure 6-11. This shows a frequency domain (i.e., a Fouriertransformed) representation of the time domain signal ofFigure 6-10.

Fourier Transform

Frequency Domain

Frequency

Time Domain

Time() (+)0 () (+)0

Figure 6-12. This is a Fourier transform of a time domain signal.

InverseFourier Transform

Frequency Domain

A m p l i t u d e

Time Domain

Time() (+)0 () (+)0

Figure 6 -13. This illustrates the inverse Fourier transform of afrequency domain signal.

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The continuous time forms of the Fourier transform and inverseFourier transform become

F k f n e k j N nk

n

N

[ ] [ ] , , , ,( / )= =

=

20

1

1 2 3 4

where f [n ] is value of the discrete time domain signal at t = n , X [k ] is the value of the discrete frequency domain signal atw = k , and N is the sampling interval.

Similarly, the discrete inverse Fourier transform is given by

f n N F k e n j N kn

n

N

[ ] [ ] , , , , ,( / )= ==

1 0 1 2 3 420

1

An important difference between the continuous and dis-crete time domain variants is due to signals being periodicallysampled in time (i.e., digitized). This leads to the spectrumrepeating at the inverse of the sampling period (Fig. 6-14). The

discrete Fourier transform is usually implemented in a formknown as a fast Fourier transform (FFT), whose computationstake advantage of some redundancy in the Fourier transformresulting in a very considerable speed up. The good news isthat mathematical software typically used to process radar sig-nals will compute the FFT automatically after simply enteringthe signal vector. However, some care is required to ensurecorrect results, and the interested reader should delve moredeeply than space permits here.

Value of the Concept. The concept of time and frequencydomains and the transformation between them is immensely

useful. In radar, it is pretty well indispensable. Translating fromone representation to the other is crucial to modern signalprocessing understanding and design. Range resolution andrange measurement may be readily generated only in the timedomain. Doppler resolution, Doppler range-rate measurement,and certain aspects of high-resolution ground mapping, on theother hand, may be readily generated only in the frequencydomain. The Fourier transform is used in almost all radar sys-tems and, in modern systems, usually in discrete form.

0

0 0

FT

DFT

n

t

x (t )

x (n)

B /

B

s s /2 s /2 s 2 s B B

B

X ( j )

1/ T

X *( j )

Figure 6-14. This graph demonstrates the Fourier transform of a sampled pulse.

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6.5 Statistics and Probability

The world in which we live is statistical in nature, and it shouldcome as no surprise to nd that this is also true for radar.Indeed, although radar systems may transmit nicely designedregular signals, the same cannot be said of the echoes that arecaptured by the receiver. In the simplest terms these either can

be random (thermal) noise or can have the same characteristicsas noise (e.g., due to scattering from a randomly rough sur-face). This means that simple detection of targets by radar hasto be done against a background of random noise (as mightbe present in the radar receiver) or by scattering from a roughsurface. Hence, probabilistic descriptions characterizing radarperformance are routinely used, and this means understandingsome basic aspect of statistics. Chapter 12 describes noise andits role in target detection.

Realizations of a Random Process. As a practical example of arandom process, consider a group of students in a lecture. The

students were asked to ip coins and to record the sequence ofheads and tails they observed. Figure 6-15 shows an outcomeof such an experiment in which 20 students participated, eachdoing 24 coin ips. An upward stem and dot means the coinip was a head; a downward dot means the coin ip was a tail.

It is clear that two students had biased coins since student 2ipped all heads and student 14 ipped all tails. The other stu-dents had fair coins with roughly equal numbers of heads andtails. The terminology relating to the experiments is as follows:

The ipping of one coin is a random event .

A sequence of 24 coin ips from a student is a realization .

The sequences of 24 coin ips from all 20 students constitutean experiment that sampled a random process .

The collection of all possible realizations in an experiment iscalled an ensemble . Here, 20 sequences or realizations fromthe ensemble are explored.

Time Average and Ensemble Average. It is important to under-stand, because of its random nature, the same experiment withthe same students and the same coins done on another occasion

would give different results. Each students sequence of coinips would be different. However, the statistical properties suchas the average would be similar. This is also true of radar detec-tion against a background of random noise. Numerical averagescan be calculated by assigning a value of +1 for a head and 1 fora tail. Then the averages may be calculated in one of two ways:

1. If the coins are fair, then each student would ip a roughlyequal number of heads and tails. Thus, the average ofeach students coin ipping sequence should be zero (orclose to zero). Taking averages in this way is equivalent toaveraging the rows in the experiment shown in the previ-ous gure. It can be seen that, apart from students 2 and14 who had biased coins the row averages are all quite

0 5 10 15 20 25

2019181716151413121110 9 8 7

6 5 4 3 2 1 0.08

1.000.330.080.080.00

0.250.170.000.080.420.080.171.000.000.250.330.000.170.33

0.00 0.10 0.30 0.20

S t u

d e n t

Coin Flip Number

Ensemble Average

T i m e

A v e r a g e

Figure 6 -15. These results show 20 students each ipping a coin24 times.

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close to zero. These averages are called time averages or sequence averages .

2. The averages may also be done column-wise (i.e., averagesacross all the students). They are then called ensemble aver-ages . There is not room to put all the ensemble averages onthe plot, and just a few are shown. For instance, the averageacross all the students results for the 10th coin ip was 0.1.

All the ensemble averages are close to zero even though there were some biased coins. That is because the biased coins ofstudent 2 (all heads) and student 14 (all tails) cancel each otherout. However, even an unmatched biased coin has little effectbecause in the ensemble average its results are diluted by thoseof the other fair coins.

Estimates of Averages and True Averages. It is possible to dis-cover the true averages using only the whole ensemble. Theexperiment of 20 students doing 24 coin ips is just a smallsample of the ensemble (an innite number f students doing

an innite number of coin ips), and therefore calculation ofaverages from the experiment gives estimates only of the trueaverages (which is why the estimates of the coin ips are closeto zero but not necessarily actually zero).

The estimates get worse as the experiment gets smaller. Forinstance, the extreme case is an experiment where one studentipped one coin just one time and got a tail, for example. Theestimated average and ensemble average based on that oneresult is 1 even though, for a fair coin, the true average is zero.Thus, in general the estimated averages get closer to the trueaverages as the experiment gets bigger.

The E-Notation. Ensemble averages are denoted by E, which isknown as the expectation operator . When dealing with random

variables it is important to remember that E is an ensembleaverage, not a time average.

Let the m -th result from the n -th student be denoted by f n [m].Using this notation f 15[20]= H because student 15 got a headon his or her 20th coin ip. Let the sequence of results fromthe n -th student be { f n [m]}. For example, student 1 had thesequence { f n [m]}= {HTHTHH } (or { f n [m]}= {1,1,1,1,1,1 } ifheads are assigned a value of 1 and tails a value of 1).

The ensemble average is estimated from a nite number of

realizations. For instance, the average value over 20 studentsof the 20th coin ip (column 10 in Fig. 6-15) is given by the fol-lowing expression with N = 20 and m = 10. The index n is thestudent counter, and m is the coin-ip counter:

S m N f m

S f

E nn

N

E nn

[ ] [ ]

[ ] [ ]

=

=

=

=

1

10 1

20 10

1

1

20

Note that the numerical calculation given here is an esti-mate of the true ensemble average for the m -th coin ip

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S m E f m E [ ] ( [ ]) . The true ensemble average for the m -th coinip is found as N becomes very large so that statistical uctua-tions are minimized:

E f m n N f mn

n

N

( [ ]) lim

[ ]=

=

1

1

In the coin-ipping experiment, the ensemble average givesinformation about the average properties of all the coins. If allthe coins are fair, then the ensemble average is zero for everym . Even if a number of coins are biased toward either heads ortails, the effects of averaging over many coins means that theeffects of biased coins tend to cancel each other out.

Calculation of a Time Average. A time or sequence average canbe calculated from each students coin-ip sequence. The timeaverage is taken over M coin ips and the index in the summa-tion is m , the coin-ip number. The time average for the n -thstudent is given by

S M f mnT n

m

M

==

11

[ ]

If the number of coin ips is large, then the value of S nT con- verges toward the true time average. Time averages are usuallydenoted by a bar or a hat, such as f n or

f n . Thus,

f M M f mn n

m

M

=

=

lim[ ]

1

1

The time average contains information about the properties of

the n -th coin (or possibly the skill of the n -th student in ip-ping the coin). If the n -th coin is fair, then f n = 0 . If the coinis biased toward heads then f n > 0, and if it biased towardtails then f n < 0.

Variance and Mean Square Deviation. Variance and meansquare deviation give a measure of the width about the meanof a distribution. Variance is a term used to describe the vari-ability of an ensemble and therefore applies to the columns ofan experiment:

Var f E f E f ( ) (( ( )) )=

2

The quantity f E ( f ) is termed mean-centered data ; that is,the ensemble mean or expected value has been subtracted.

Thus, ( f E ( f ))2 is the square of the deviations or variancefrom the mean. For instance, if heads are denoted by +1 andtails by 1 and if E ( f ) = 0, then ( f E ( f ))2 is always +1; there-fore, the expected value of the variance is +1. That is, the val-ues are expected to uctuate by +1 and 1 (which is all that canhappen in the coin-ipping experiment).

Thus, for the coin-ipping experiment, Var ( f ) = E (( f E ( f ))2) = 1.

The mean square deviation (MSD) is used to estimate vari-ability in a random sequence (i.e., within one realization) and

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therefore applies to the rows of the coin-ipping experiment.The expression is

MSD f M M f m f m

M

( ) lim

[ ]= ( )

=

1

1

2

The MSD of an unbiased coin is +1. For the biased coin, asused by student number 2, the MSD is zero. All the f [m] were+1 (Heads); thus, the time average is +1 and all the f [m] f

values are zero, which leads to a value of zero for the MSD ofthe biased coin.

Numerical Estimates for Variance and Mean Square Values. The following expression shows how to estimate a variancefrom experimental data as long as E ( f ) is known. For exam-ple, in the coin-ipping experiment, if the coins are unbiasedit is possible to say in advance that E ( f ) = 0. Then the varianceis computed using

Var f N f E f n

n

N

( ) ( ( )) =

1 21

If an estimate of E ( f ) rst has to be calculated from the sameset of data then the computation of the variance is done using

Var f N f N f n n

n

N

n

N

( )

==

11

1

11

2

where the unknown E ( f ) has been approximated using

E f N f n

n

N

( ) =

11

In the case where E ( f ) has been derived approximately fromthe data, statistical theory shows that the estimate of Var ( f )is a little larger than in the case where E ( f ) is known before-hand, which explains the presence of N 1 rather than N in thedenominator of the expression for Var ( f ). Similar expressionsapply to the mean square deviations. If f is known before-hand, then

MSD f M f m f

m

M

( ) ( [ ] )

=

1 2

1

If is f estimated from the data, then

MSD f M f m M f m

m

M

m

M

( ) [ ] [ ]=

==

11

1

1

2

1

Ergodic and Stationary Sequences. An ergodic sequencearises from a very special type of random process in whichthe ensemble and time averages are the same. A coin-ippingexperiment using only fair coins would be ergodic. The ben-et of ergodicity from a signal processing viewpoint is thatproperties of the ensemble can be determined from a singlerealization. In the coin-ipping experiment, it means that an

experimenter would be able to assume that 100 coins each

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ipped once give the same average result as one coin ipped100 times. This is a reasonable assumption if the coin is fair.

Ergodicity applies to all the statistical properties of the process:

1. The expected value is equal to the time average value.2. The variance is equal to the mean square deviation.3. The probability distribution function (PDF) of the random

process is the same as the distribution of values in therealization.

In Figure 6-16, the upper panel is a single realization generatedfrom a random noise-like process having a Gaussian distribu-tion (which in this case was known). The bars on the graph of

p( x ) in the lower panel of were determined from the data inthe upper panel, while the smooth curve is the Gaussian dis-tribution for a random process. Because the process is ergodicthe statistical distribution of the data from the single realizationcan be used to estimate the distribution for the process thatgenerated the data.

A sequence is stationary if its ensemble properties E ( f ) andVar ( f ), and more generally its probability distribution func-tion, do not change with time. Otherwise, it is a nonstationary sequence or process. A nonstationary process cannot be ergo-dic. This does occur with radar data and ultimately requires

very sophisticated processing, which is outside the scope ofthis book.

Nonstationary processes also often occur in many forms. Oneexample is the daily temperatures in the Northern Hemispherefor the months January to March. This is because they havea rising trend. The average temperature on January 1 overan ensemble of years such as 20102020 will be below theensemble average for March 31 (averaged over the same years)because the temperature usually rises between January andMarch. Also note that both these values would be differentfrom the temperature averaged over the period January toMarch (i.e., the time average) in any given year.

Probability Distributions. Noise in radar systems, such as ther-mal noise in receivers and in some cases echoes from distrib-uted targets (e.g., the sea or vegetation) can be described bythe Gaussian probability distribution function. For this reasonit features very centrally when describing the performance

of radar systems (as we shall see in a number of the follow-ing chapters). Further, it is the statistical nature of noise (andof target echoes) that results in radar detection performancebeing described in statistical terms. The mathematical expres-sion describing the probability of a quantity that ts a knownGaussian distribution, such as random noise, is

p f

f ( ) exp

( )=

12 2

2

2

where and are parameters that dene the distribution. 2 isthe variance of the quantity, is the standard deviation, and

is the mean value .

0 100 200 300 4004

2

0

2

4

x - v

a l u e

Sample Number

4 2 0 2 40

0.2

0.4

p ( x )

x -value

Figure 6-16. A random noise-like sequence has a Gaussiandistribution of probabilities. The bars on the p( x ) graph are fromthe data in the upper panel, while the smooth curve is the Gaussiandistribution for a random process.

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A second and related PDF is the Rayleigh distribution.Remember that most modern radar systems downconvert anduse in-phase and quadrature channels to obtain values of bothamplitude and phase. Noise will be present on both the I andthe Q channels and will be Gaussian with a zero mean and a

variance of 2. The resultant amplitude is ( ) I Q 2 2+ and hasa Rayleigh distribution. The mathematical expression deninga Rayleigh PDF is

p f f f

( ) exp= 2

2

22

where is the standard deviation, and 2 is the variance. InChapter 12 the Rayleigh distribution is used to compute thelikelihood of noise being observed as a falsely detected target.This is represented as the probability of a false alarm. It is alsopart of the calculation that determines the likelihood of detect-ing a true target, the probability of detection.

6.6 Convolution, Cross-Correlation, andAutocorrelation

Convolution and its close cousin correlation are invaluabletools in the analysis of signals and are used routinely in manyaspects of radar signal processing. Convolution is a basic ingre-dient in lter design and is used specically in the matchedlter (Chapter 16), a method of maximizing the signal-to-noiseratio. It is also employed in the closely related topic of pulsecompression (Chapter 16), a method of generating high rangeresolution. The cross-correlation function examines the like-ness between two signals, and the autocorrelation function

evaluates the self-similarity of a given signal to see if and howit contains components that repeat.

Convolution. The discrete form of the convolution of two sam-pled, time domain signals, f [n ] and g [n ], can be written as

R l f n g n l l K fg n

[ ] [ ] [ ], , , ,= = =

0 1 2

where l is the lag and represents a discrete sliding of one signalpast the other as the value of l changes, and K is the nite timeextent of the signal being convolved . Note: this is the convolu-tion of signal f [n ] with signal g [n ]. We can also write similar

expression for the convolution of signal g [n ] with signal f [n ] bysimply reversing the order. For example,

R l g n f n l l K gf n

[ ] [ ] [ ], , , ,= = =

0 1 2

This equation takes one of the signals, ips it around, andthen slides it across the other as the products of the overlappingparts (determined by a value of l ) are integrated (summed).This ipping is a neat way of representing the direction of sig-nals. For example, pulse compression, a technique to achievehigh range resolution (Chapter 16), involves convolving the echo

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with a replica of the transmitted signal. Of course the echo trav-els in the opposite direction and the ipping accounts for this.

Figure 6-17 shows an example of convolution. Here two rect-angular wave functions of differing magnitudes are convolved

with one another, and the slide and integrate mechanismdescribed by the function are indicated. First, the graph of the

rectangular function on the right is ipped about the verticalaxis and repositioned on the left. It is then slid from left to rightacross the other rectangular function. The sliding and integra-tion result in a triangular output. Note how its integral is at amaximum when the two rectangular functions exactly overlap(i.e., the lag l is zero). Further, the range of the lag value is dic-tated by the duration of the signal (for a given sampling rate)and does not have to be extended to plus and minus innity;that is, there is no point convolving the zero values outside ofthe signal with more zero values.

If f [n ] and g [n ] were expressed in the frequency domain, thenthey could be multiplied together and an inverse Fourier trans-form could be applied to obtain the same result.

Correlation. Almost identical to convolution, correlation dif-fers only in that there is no ipping of one of the signals.Correlation can be used either to examine how alike two sig-nals, f [n ] and g [n ], are or to examine especially their periodicproperties. The cross-correlation function is written as

Rxy l x n y n l l K

n

[ ] [ ] [ ], , , ,= + = =

0 1 2

Figure 6-18 shows the cross-correlation of two random noise-

like signals, which indicates that there is no similarity betweenthe two signals (as expected for random noise). If the two sig-nals were periodic with the same period, such as sine wavesof the same frequency, then we would get a highly structuredand strongly correlated output since the two signals come inand out of phase as a function of the lag value.

Autocorrelation. This is the correlation of a signal f [n ] with itself. It is used to reveal properties of a signal such as whether or not it is periodic and also to different iate a cho-sen signal from others and from sources of interference suchas noise. The autocorrelation of f [n ] comes from substituting

f [n ] for g [n ] in the expression for cross-correlation and isthus given by

R l f n f n l l K ff

n

[ ] [ ] [ ], , , ,= + = =

0 1 2

Figure 6-19 shows the autocorrelation of a random noise-likesignal. Here we see that it is completely correlated with a lagl of zero. This is true for all signals and is a direct result ofthe fact that every element of the sum in the autocorrelationfunction has the same value. However, with only a lag valueof one ( l = 1) the correlation of the signal plunges and hovers,

50 40 30 20 10 0 10 20 30 40 500.4

0.2

0

0.2

0.4

0.6

0.8

1Auto-Correlation of a Random Signal

Figure 6-19. This graph shows the autocorrelation of a randomsignal.

100 80 60 40 20 0 20 40 60 80 1000.2

0.15

0.1

0.05

0

0.05

0.1

0.15

Figure 6-18. This graph shows the cross-correlation of two randomnoise-like signals.

g(n)

n n n

f(n)

g(n)*f(n)

Figure 6-17. This illustration shows the convolution of two

rectangular functions.

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noise-like, around a zero mean. This shows that there are noperiodic or repeating components in a random noise-like signal.

Autocovariance. This property of a random process indicatesthe degree of similarity of a signal in one part of a time series

with that taken at a later part in the same time series. It is cal-culated using the expectation

ff l E f n f n l ( ) ( [ ] [ ])= +

In this expression f [n ] is a mean-centered random variable;that is, the expected value of E( f ) = 0, or the mean value hasbeen subtracted from f . The equation states that autocovariance ff is a function of lag l and is dened as the expected value of

f at time n multiplied by f at a later time n + l . If f [n + 1] is cor-related with f [n ] then ff (l ) would be nonzero, whereas if thereis no correlation then ff (l ) is zero. The autocovariance can beestimated from a single data sequence of length N . Both f [n ]and f [n + 1] must have their means removed before they canbe used in the following expression:

ff n

N

l N f n f n l ( ) [ ] [ ] +

=

1 11

1

where n is the sample number.

Insight into Autocovariance. When l = 0, the autocovarianceis equal to Var ( f ), the expression has a maximum value (as

f [n ] = f [n ]), and the answer is simply saying that the two sig-nals are identical. As l increases, the value of Var ( f ) falls; therate at which it falls as a function of l provides informationabout the self similarity of the signal (i.e., periodic nature) ofthe signal. For example, a special case arises when f [n ] formsan oscillating signal. Suppose there were exactly 20 samplesper cycle of oscillation:

f f f f f f

[ ] [ ] [ ] ...

[ ] [ ] [ ] ...

1 21 41

2 22 42

= =

= =

In other words, apart from random noise present in the signal, f [n ] is identical to f [n + 20]. Thus, the autocorrelation will alsobe large when l = 20. Moreover, if the peaks of the oscillatingsignal are at f [1], f [21], f [41] then there are valleys (minima)at f [11], f [31], f [41] Thus, the autocovariance will be largeand negative when l = 10.

This reasoning highlights an important result concerning auto-covariance. An oscillating signals autocovariance is oscillatory

with the same period. Moreover, since the calculation for auto-covariance includes an average over N l samples, the auto-covariance is much less noisy than the original signal. Theautocovariance reveals the degree of periodicity in a signal. Thiscan be completely masked if observed in the time domain only.

Covariance estimation is a technique increasingly used todetermine whether or not a range cell contains clutter or atarget in advanced concepts such as space-time adaptive pro-cessing (STAP).

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Cross-Covariance. The cross-covariance of stationary zero-mean signals { f [n ]} and { g [n ]} is

fg l E f n g n l E f n l g n[ ] ( [ ] [ ]) ( [ ] [ ])= + =

The notation is fg , meaning the cross-covariance between{ f [n ]} and later values { g [n ]}. By contrast, the following is the

cross-covariance between { g [n ]} and later values in { f [n ]}:

gf l E g n f n l E g n l f n[ ] ( [ ] [ ]) ( [ ] [ ])= + =

The lag can also be negative. Here, because the lag is negative(l ), the covariance is between { f [n ]} and earlier values in { g [n ]}:

fg l E f n g n[ ] ( [ ] [ ]) = + 1

The cross-covariance function is not symmetrical:

fg fg l l [ ] [ ]

For instance, if { f [n ]} is an input sequence into a causal 2 digital

lter whose output is { g [n ]}, then the input sequence { f [n ]} isexpected to correlate with later values in the { g [n ]} sequence insome way. Thus, fg (l ) would not in general be zero. But if thelter is causal there would be no relationship between { f (n)}and earlier values in the { g [n ]} sequence and so gf (l ) wouldbe zero. In this way the cross-covariance provides informa-tion about the similarity of two signals similar to the way theautocovariance provides information about the self-similarity

within a single signal.

The Wiener-Khinchin Theorem. This theorem shows that theautocovariance function provides a method for estimating a

signals power spectrum (Chapter 19)directly from the dis-crete Fourier transform:

S k N F k ff [ ] | [ ] |

=1 2

where k is an integer.

The Wiener-Khinchin theorem states that this estimate may alsobe obtained from the Fourier transform of the autocovariance:

S k l e ff ff

j N

l

N

[ ] ( )=

=

2 1

0

1

where l is an integer indicating the number of lags.Conversely, as might be expected, the autocovariance can bedetermined from the inverse Fourier transform of an N l channel power spectrum:

ff ff

j l k N

k

N

l N S k e [ ] ( )=

=

1 20

1

6.7 Summary

This chapter introduced a number of mathematical conceptsand constructs that are widely used in the description of radar

2. A system is causal if the output at any time depends only on

values of the input at the present time or in the past. Thatis, it cannot have an output until it has an input and cannotanticipate future inputs. A vehicle is a causal system because itcannot anticipate the drivers future actions (at least not yet).

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performance and in the evaluation and manipulation of radarsignals.

Radar signals are examples of energy signals. Complex numbersare a mathematical way of representing phasors and make auseful connection with our nonmathematical way of describingradar signals. Periodic signals are described by the Fourier series

that is made up of a number of harmonically related frequencies.The Fourier transforms provides a means for moving betweenfrequency and time domain representations of radar signals.

Noise in a radar system is described by a Gaussian probabilitydistribution function when observed as a zero mean voltagein the I and the Q channels. The overall detected amplitude

( ) I Q 2 2+ is described by Rayleigh distribution function.

Convolution, correlation, and autocorrelation provide a power-ful means of realizing matched lters, compressing pulses forhigh range resolution, and forming synthetic aperture, amonga host of others, and are also useful tools for determining the

properties of radar signals. The Fourier transform of the auto-covariance provides an alternative means of estimating thepower spectrum of a radar signal.

Further Reading

R. N. Bracewell, The Fourier Transform and Its Applications ,3rd ed., McGraw-Hill, 1999.

D. P. Bertsekas and J. N. Tsitsiklis, Introduction to Probability ,2nd ed., Athena Scientic, 2008.

M. L. Meade and C. R. Dillon, Signals and Systems , Chapmanand Hall, 2009.

H. Hsu, Schaums Outline of Probability, Random Variables,and Random Processes , 2nd ed., McGraw-Hill, 2010.

J. F. James, A Students Guide to Fourier Transforms; with Applications in Physics and Engineering , CambridgeUniversity Press, 2011.

H. Hsu, Schaums Outline of Signals and Systems , 3rd ed.,McGraw-Hill, 2013.

Test your understanding1. What is the amplitude and phase represented by the complex number 1 + j ?

2. Sketch the Fourier transform pairs for the following time domain signals: (a) an impulse; (b) asquare pulse; and (c) a Gaussian pulse

3. Explain what is meant by the mean and variance of thermal noise.

4. Sketch the autocorrelations functions of (a) a sine wave and (b) thermal noise.

5. Sketch the cross-correlation function of two independent realizations of thermal noise.

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