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Home , Decibel, Radian

APPENDIX 1

CONVERSION TO

Bels were named after . They were simply the of the ratio of out and power in. The bel was too small to be useful; so the quickly emerged. The definition of a decibel (dB) is contained in any radio engineer's handbook. It is

where P is power, V is , 0 is "out," and i is "in." In colloquial usage, dB are used to express any ratio. Decibels are just ten times the lOglO of that ratio. Some approximate dB conversions usually memorized by radar people are the following:

Ratio dB Ratio dB

I 0 20 13 1.25 I 100 20 1.6 2 10" IOn 2 3 ( 47T)3 33 3 5 Meters to nautical -33 4 6 Nmi to meters +33 5 7 Boltzmann's Constant (/oK) -228 6 8 Square degreeS in 47T 46 7 8.5 of light (meters/sec) 85 8 9 (feet/sec) 90 9 9.5 (miles/sec) 53 10 10

174 APPENDIX 2

THE ELECTROMAGNETIC SPECTRUM

Hertzian (3 kc - 3 X 106 MHz) t4+-----1.0------+-I·1

I HI\------"::...... ::::------ in MHz------~.1 1016 Cosmic rays I billionth of the electromagnetic spectrum

Radar bands -----~

VLF LF MF HF VHF UHF SHF EHF

.003 .03 .3 3 30 300 3000 30,000 t t Frequency (in megahertz) 3 X 105 3 X 106 I :Wavelength in meters I I 105 10,000 1000 100 1.0 .1 .01 .001 .0001 I I I I I

Note: radars are not included. They can range from about 107 MHz to 10/0 MHz.

175 APPENDIX 3

FOURIER SERIES AND TRANSFORMS*

Working in the field of transfer in the early nineteenth century, Jean B. S. Fourier found that virtually all functions of time, particularly repetitive ones, could be described in a series of and cosine waves of various and . His has been described as one of the most elegant developments in modern . Whatever its stature for the world, the benefits for the radar engineer are epic.

FOURIER SERIES

The statement for the Fourier series is that any may be broken down into the sum of and cosines of various amplitudes and frequencies. In mathematical notation

1 f(t) = "2 ao + al cos wt + a2 cos 2wt +

+ b l sin wt + b2 sin 2wt + .... (1) f(t) is a function of time here (it does not have to be), and the aj and bi are constants that are to be found in order to make the expression on the right equal to f(t); ~ ao is the de component of the function, if any.

The next step is to find a way to evaluate the aj and the bi . To do this, we use the orthogonality of the sine and cosine function, which is that the value of their cross products integrated from zero to 27T is zero. Furthermore, the sine and cosine each integrated from zero to 27T is zero. These integrations are easy to check by referring to a table of integrals; they are

(27T (27T Jo sin rnx dx = 0 Jo cos nx dx = 0 J:7T cos rnx cos nx dx = 0 (2)

* This presentation uses the approach taken by H. H. Skilling in Electrical Circuits (New York: Wiley, 1957), chaps. 14, 15.

176 Appendix III 177

Also

{27T {27T J0 sin mx sin nx dx = 0 Jo cos mx cox nx dx = 0 (3)

But

{27T (27T ) Jo (sin mx)2 dx = 1T Jo (cos mx)" dx = 1T (4)

Let us pick a coefficient to solve for, say, a2. Multiply through equation (1) by cos 2wt, then multiply both sides by d(wt) and integrate. We get

2IT . f21T I fo J(t) cos 2wt d(wt) = 0 2 ao cos 2wt d(wt) +

21T f21T fo al cos wt cos 2wt d(wt) + ... + 0 bl sin wt cos 2wt d(wt) +

We can see by inspection that most of these integrals are zero. On the right side of the equation, the first and terms are zero. But the third term equals a21T. In fact, all other terms written down and all implicit terms are zero, giving finally

21T fo J(t) cos 2wt d(wt) = a2 1T (5) and

1 f21T a2 = - J(t) cos 2wt d(wt) (6) 1T 0

Because we do know J(t), we can find a2, even if we have to do it by numerical integration on a home computer or programmable calculator. And because we did this in general, we can now generalize to

1 f21T an = - J(t) cos nwt d(wt) (7) 1T 0 and, after multiplying through by a sine function

1 f27T bn = - J(t) sin nwt d(wt) (8) 1T 0

Equation (1) is the Fourier series, but it can be rewritten for brevity as 178 Radar Principles for the Non-Specialist

1 x J(t) = "2 ao + ];1 (am COS mwt + bm sin mwt) (9)

Writing down a function and evaluating it is the essence of tedium, but by astute observation, the process can be shortened. Some of the waves we are interested in in radar are not too difficult. For example, a periodic square wave (Figure A3.1) has no even harmonics. It is just an infinite sum of odd harmonics with ever-increasing frequencies and ever-decreasing coefficients. The answer for a square wave is

4(. 1. 1. ) J (w) = -; sm wt + 3" sm 3wt + "5 sm 5wt + ... (10)

FOURIER TRANSFORMS

To arrive at the Fourier transforms, we must take a few preliminary steps. The first is to restate equation (1) in exponential form. We can do this by substituting for cosines and sines their identities, which are

We can now write the Fourier series as

(11)

f +1 on n_n I ;~ J U U U I I I I 0 'TT 2'TT 2 3 4 5 6

~ wt Hannonics of fundamental

FIGURE A3.1 A square wave and its Fourier spectrum Appendix III 179 or

The coefficients are related in this way

(12) and the coefficients can be found from the integral

I J,27T A = - Jet) e-jnwt d(wt) (13) n 27T 0 j(t) is the function of time to be expressed as a Fourier series and the integer n can now be positive, negative, or zero. The integral (13) is used just like equations (7) and (8) to obtain coefficients. It is more efficient because there is only a single integral, and it is much more readily integrated. Equation (13) can be derived from equations (7) and (8) through the use of Euler's theorem. By matching their infinite series, Euler showed that

ejO< = cos a + j sin a

Consider now a square wave, as in Figure A3.2, where we have made the width ofthe pulse equal the interpulse period divided by a constant. We solve for the coefficients with

1 J7Tlk A = - e-jnJC dx n 27T -rrlk

T ~I +1 I-

T - k

0 'IT 'IT -'IT 0 'IT • X 2 2

FIGURE A3.2 Another square wave 180 Radar Principles for the Non-Specialist

e 0 /k de 1 1 sin (mr/k) .. I·k h f lor n = , Ao = 1 an lor n r 0, An = k n1T/ k ' glvmg a spectrum let at 0 Figure A3.3. The spectrum of Figure A3.3 has an envelope of the sin xix shape, which we know to be the of a single pulse and the larger the value of k, the closer the spectral lines are together and the more closely a is replicated. No matter how large we make k, however, this envelope is still caused by lines of discrete frequencies. How can those discrete frequencies exist? They exist because the time function they are describing goes on forever. If it starts and stops, there will of necessity be a continuum offrequen• cies, that is, the An will blend. There will be an An for every incremental distance along the frequency axis, not just at discrete points. The An will become a function offrequency, say, g(w), and the integer n will become an w on the other side of the equation. The limits of integration must be extended to find the stopping point of the time funrtiof'. These changes give us

g(w) = -1 IX f(t) e-}wt dt (14) 21T -x

for the frequency function and, by symmetry

f(t) = J:x g(w) e}wt dw (15)

These are called the Fourier integral equations or the Fourier transform pair. In this book, illumination functions are transformed into far-field antenna patterns and time functions into their spectra. In both these actions, equations (14) and (15) are applicable. Let us do an example with a single square pulse of height 1 and duration T

1 IT/2 . g(w) = - le-}wt dt (16) 21T -T/2

sinn1Tlk __-./ n1Tlk / \f-i k=4

/~2-'2 k

FIGURE A3.3 Another spectrum of a square wave Appendix III 181

T 1(1) .----If----...-- + 1

T T o +2: 2 T T T T T T

FIGURE A3.4 A single pulse and its Fourier transform

(17)

. T Sill w 1 . T T 2 = - Sill W - = - --- (18) TrW 2 2Tr T w- 2

The results are shown in Figure A3.4. Note that if e were substituted for T and 0 for w, the result would be for an antenna pattern (close to the boresight where sin o ~ 0). And g(w) can be converted back into f(t) rather easily by using the form g( w) takes in equation (17) to make up the integrand and remembering to integrate with dw from -00 to 00. APPENDIX 4

ANSWERS TO EXERCISES

Chapter 1: 1. 1200 km. 2. 2.5 x 10-22 joules/m2, 6.7 ms. 3. 181°K. Chapter 2: 1. .56 centimeters, 28 meters. 2. 400 elements, 100 m2. Chapter 3: 1. SIN = 13 dB. 2. -7 pulses. 3.

Required Threshold 4 3 2 o

Crossings

.6561 .9477 .9963 .9999 1.0

.0001 .0037 .0523 .3439 1.0

4. Pd - .98. 5. 24, less than 5. Chapter 4: 1. (a) 47TAIA2X_band (b) 50 percent of a. 2. 27Ta1A ~ I, at - 20 gigahertz. 3 ..6 degrees (peak to null on both sides of the ). 4. See Table 4.1. 5. -1500 m2. 6. -17 db. Chapter 5: 2. T = I millisecond, f3 = 5 MHz. 3. 500,000 feet. 4. T = 2 IJ-sec, T = I msec, .:1 ~ 3 msec. 5. (a) T = 2 IJ-sec, T = I msec, .:1 = 50 IJ-sec (PRF = 22,000) (b) 4.2 nmi. 6. (a) D = 20 meters (b) T = .016 sec., f3 = 10 MHz. 7. (a) 2000 (b) 2000 (c) 5 ft. (d) 5 kiloHz. Chapter 6: 1. -106 • 2. 0.2, 2.0. 3. 14.1 km, 44.6 km. 4. 1.6 Mw. Chapter 7: 1. (a)-I23 dB (b) -77 dB (c) -43 dbw (d) -120 dbw (e) Needed, at this and longer ranges, this and lower RCSs. 2. (a) PRF = 20,000, T = 3 ms, T = .2 IJ-sec (b) 4.2 nmi (c) Yes, but now PRF = 3000, T = 1150 sec, T = .2 IJ-sec, and unambiguous range = 28 mi. 3. Most clutter is at zero velocity. The rotating helicopter blades and tail propeller, and the vibration of the airframe cause the equiva-

182 Appendix IV 183

lent of clutter at varying , some quite high and, therefore, troublesome. 2V RA 2RA 4. (a) /lId = d at both ranges (b) T = Vd and Vd 2RA 4RA d. (c) (j2 and (j2 (d) 2 10 both cases. 5. Doppler spread is independent of range. liT is inversely propor• tional to range; n is proportional to range. Therefore, sensitivity considerations aside, resolution is independent of range. 6. -1.2 x 106 7. _.6°. About equal. 8. -40 inches. 9. -1.25 cm. Chapter 8: I. S-band, 2-3000 MHz. 2. -1,000,000 feet. 3. About 150 radars. 4. About 8. 5.5.4 miles. 6. -100 miles. APPENDIX 5

GLOSSARY

This glossary provides assistance in understanding terms used in this book. There may be more general or more specific or entirely different meanings for these terms when they are used elsewhere.

Adaptive array a steerable antenna designed to respond automatically and opti• mally to a variety of situations. Adaptive receiver a receiver with circuitry or perhaps programming that enables it to adjust to the characteristics of the incoming signal. AGe automatic control; keeping the excursions of the received signal within bounds with an automatic negative feedback loop. Ambiguous an angle where the characteristics of an antenna pattern are such that there is uncertainty about the angular location of a target (as with an interferometer). Ambiguous range the condition that the roundtrip time between pulses is insuffi• cient to accommodate all the targets that the radar will see, resulting in uncer• tainty about target location. Ambiguous velocity the situation that occurs when the doppler processing scheme of the radar is such that target velocities fold over, making those veloci• ties uncertain. 's law current passing through a conductor creates a magnetic field around the conductor; named after the man who discovered the phenomenon. the "height" of a trace on a display or record, usually referring to a measurement of voltage. Analog error signals indicators derived from a continuous process, that permit correction while the process is going on, as with a radar beam tracking a target. Analog signal processing performing various operations on signals while they are still in their received form, that is, continuous rather than quantized or digitized. Analog-to-digital (AID) converter electronic circuitry which samples an incom• ing signal and assigns a number (or a computer word) which describes each sample.

184 Glossary 185

Angle resolution the capability of an antenna to separate two objects in angle. Angle tracking to follow a target in angle, by whatever means. Angular error the amount by which an antenna or other angle measuring system fails to indicate the exact angle of a target. Angular rate the rapidity with which a target's changes in angle. Antenna pattern the "intensity" of the field at locations around the antenna at long distances from it; usually established by taking a series of measurements. Antijam gain or margin the ratio by which a particular waveform or other tech• nique is able to mitigate the effects of jamming. Aperture literally an "opening"; its size determines the amount of electromag• netic energy intercepted; thus, any antenna is called an aperture. Arecibo the location in Puerto Rico of a thousand-foot -aperture ionospheric and astronomic radar. Array factor the pattern of an entire steerable array. Aspect angle the angle made by some arbitrary characteristic of the target, say, its axis, with the axis of the radar antenna beam. Auroral ionosphere the ionosphere in the vicinity of the earth's magnetic poles when excited by magnetic storms. the angle from a fixed point, say, due north, in the plane of the earth's surface.

Barrage jamming disrupting and interfering signals that are calculated to contin• uously envelope all the frequencies being radiated. Beam steering moving an antenna pattern about the sky, usually by electronic means. Beamwidth the lateral dimension (in angle) of the principal lobe of an antenna pattern. Bi-static radar a radar whose transmit and receive antennas are substantially separated. Blanking circuit an electronic scheme by which particular range or angle loca• tions in the radar coverage are wiped out. Blass array an electronically steered array consisting of stacked beams that are turned on and off with a matrix; named after its inventor. Blind ranges ranges where a filter that suppresses clutter at one range also suppresses signals at other ranges. 186 Radar Principles for the Non-Specialist

Blind speeds at which a filter that suppresses clutter at one velocity also suppresses signals at other velocities. Boltzmann's constant the number by which temperature (in degrees ) is related to energy (in joules) per of bandwidth; it is 1.38 x 10-23 . Boresight the forward direction on the axis of a parabolic dish antenna; loosely, the direction any antenna is pointing. Butler array or matrix an electronically steered array with the characteristics that there are as many beams formed as there are elements, they are orthogonal and the output is the Fourier transform of the input; named after its inventor.

Carrier frequency the rate of oscillation of the radio waves that carry a signal through space. Cassegrain feed an antenna feed patterned after the optical telescope feed of the same name; the feed is located at the center of a parabolic dish, reflecting onto the dish via a hyperboloid at its focus. CFAR constant false alarm rate; keeping level constant by normalizing to a noise sample taken near the target. Chaff small, light pieces of material that have high radar cross section. a characteristic of radio waves whose electric vector ro• tates 360 degrees during each radio frequency cycle. Clutter unwanted and interfering radar returns from objects other than targets. Clutter coefficient the ratio of the return received from clutter to what would have been received from a perfectly conducting of the same physical . Clutter fence a screen around a radar site that prevents low-angle clutter from being illuminated. Coherence the preservation of fixed relationships over time in the con• duct of radar operations. Collision frequency the rate at which the free electrons in an ionized medium collide with ions or atoms. Comb filters an array of filters, arranged like the teeth of a comb, whose re• sponse frequencies are close together. Cone sphere the shape created by a sphere capping a truncated cone; similar in shape to an ice cream cone. Conical scan a tracking technique in which an antenna feed or the antenna itself makes a small, , sequentially comparing returns in order to obtain more accurate angle information about the target. Glossary 187

Convolution multiplying the overlapping portions of two functions continuously as one is moved across the other. Comer reflector (dihedral & trihedral) radar targets designed to have high retro• directive radar cross-section. The dihedral has two faces, the trihedral three. Cross range a measurement orthogonal to the axis of an antenna, differentiated from derived from an angle measurement.

Data link a communications channel for information, usually digitized and wide• band. Dead zone the region near a radar from which returns are not received because the receivers are turned off while the transmitter is radiating. Decibel ten bels, a bel being the logarithm to the base ten of the ratio of output power to input power. Difference beam the remainder or residue that results when the voltage from a signal in one beam is subtracted from that in an adjoining beam. Diffraction grating a matrix of narrow slits that breaks light up into fringes, caused by the constructive and destructive interference of light waves. Diffuse to break up and distribute on reflection, as with an incident electromag• netic wave. Digital signal processing performing various operations on signals after they have been converted to digital form, as differentiated from analog signal pro• cessing. Directional coupler a switch that connects one electric circuit with another in such a way that energy moves easily in one direction but not in the other. Distributions various probability density functions of mathematical statistics (exponential, gamma, Gaussian, normal, and Rayleigh) that find applications in radar theory. Doppler ambiguity a condition in which velocity data folds over so that there is uncertainty about the true doppler frequency. Doppler sidelobes the residue of a filter's that appears in adjacent filters. Doppler spread the band of frequencies within which doppler returns might occur. Downrange away from the radar along the axis of its antenna.

Electric vector the direction and of the voltage measured in an elec• tromagnetic field. 188 Radar Principles for the Non-Specialist

Electron density the number of free electrons per unit volume of an ionized gas or plasma. Electronic countermeasures (ECM) using electronic techniques to disrupt radar or communications; counter-countermeasures (ECCM) are activities to counter ECM. Element factor the scanning coverage of which a steerable array antenna is capable, determined by the pattern of its elements. Envelope the shape, amplitude, or modulation of a signal after the radio fre• quency carrier has been removed; the postdetection content of the signal. Envelope integration building up the signal by summing two or more signal envelopes; postdetection integration. Erf(x) a form of the integral of the standard normal distribution in which the standard deviation has been made narrower by the square root of two. ERP of a radar or jammer (peak power x transmit ). Ether an imagined medium by which, it was thought until the nineteenth cen• tury, light waves were able to propagate in space.

False-alarm probability the likelihood that noise alone will cross a threshold and be erroneously accepted as a signal. False-alarm rate the frequency with which noise alone crosses a threshold and is erroneously accepted as a signal. Faraday's law a time-varying magnetic field will induce a voltage in a circuit immersed in that field; also known as the induction law; named after its dis• coverer. Faraday rotation the turning of the electric vector of an electromagnetic wave as it passes through an ionized medium, an that occurs due to Faraday's law. Far field the region sufficiently far from an antenna that the phases of wavelets arriving from the antenna edges will be negligibly different from those arriving from the center. Fast Fourier transform (FFT) an algorithm for efficiently calculating the fre• quency content of a digitized time function. Fat beams antenna patterns whose main lobes occupy a relatively large . Feedhom the expanding end of a waveguide that acts as a launcher of electric waves. Glossary 189

FM chirp a frequency-modulated signal that smoothly changes frequency up• ward or downward during its transmission; if it were a wave, it would be heard as a chirp. FM ramp the ramp like shape of the plot of a waveform whose frequency is changing either upward or downward during transmission. Fourier transform a mathematical operation that changes a function to reveal its characteristics in a different dimension; named after its inventor. Fraunhoffer diffraction the far-field patterns that appear when light is propa• gated through a narrow slit or grating. Frequency ambiguity uncertainty about the true location of a target because of foldover in the doppler processor. Frequency diversity a characteristic of radars that can radiate at anyone of a large number of frequencies; a design to reduce jamming vulnerability. Frequency domain the dimension in which a function is evaluated for its spectral content. Frequency scanning a technique by which an array radar surveys a solid angle by changing its carrier frequency.

Gain the focusing power of an antenna as measured by the ratio of the angular area of a sphere to the angular area of the antenna beam, or the power of a processor to build up the signal-noise ratio by applying various techniques. Galactic noise unwanted and interfering electromagnetic radiation that enters the radar from the cosmos. Gaussian refers to the normal distribution; phenomena whose events are nor• mally distributed are "Gaussian." Geometric region the region where the scattering of electromagnetic waves is from objects whose characteristic dimension is much larger than a wavelength. Geosynchronous synchronized with the turning of the earth; satellites in orbits whose period is 24 and whose inclination is zero degrees are geosyn• chronous. Gigahertz billions of cycles per second. Grating lobes patterns in which energy appears at almost equal amplitUdes in several locations, as with diffraction through multiple slits or interferometry. Grazing angle the angle an antenna beam makes with the surface of the earth. Gun-barrel analogy the inference that the trajectories of radar targets can be compared with those of bullets emerging from a gun barrel. 190 Radar PrincipLes for the Non-Specialist

Half-wave a radiating or receiving element consisting of a straight con• ducting wire one-half as long as the wavelength of the associated radio waves. Hamming weighting tailoring the amplitude of an antenna illumination function or a waveform so that it is the shape of a cosine on a pedestal; named after its designer. Haystack radar a very large, high-precision dish radar atop Haystack Hill in Massachusetts. Horizontal polarization the condition of a radio wave whose electric vector is in the plane of the earth's surface. Huggins beam steering a method of moving a radar beam about the sky by adding and then subtracting the correct phases from the carrier frequency; named after its inventor. Hybrid a device that employs a mixture of two or more electronic technologies, such as mixing tubes and solid-state devices, analog and digital processing, and waveguide and electronic circuits. Hypothesis test a method of decision making in mathematical statistics in which a threshold is set at a predetermined level, fixing the probability of incorrectly accepting or rejecting the hypothesis.

Dlumination function the voltage or power pattern with which an antenna is excited. Incoherent integration the adding together of the envelopes or modulation of signals without regard for the phase of the carrier frequency; postdetection integration. Incremental sources imaginary small radiators of electromagnetic energy used as a convenience for developing theory. Inertial guidance a completely autonomous system of navigation that measures and deduces other quantities from those measurements and origi• nal position. Interferometer a system that uses phase differences (constructive and destruc• tive interference) to determine angular position to high accuracy. Interpulse period the interval between radar pulses, pulse repetition interval, the reciprocal of the pulse repetition frequency. Inverse synthetic aperture the use of the deterministic rotation of an object in a radar beam to derive differential doppler information and thereby resolve the object in angle; the "inverse" of having the beam pass across the target. Ionospheric radars radars that irradiate the ionosphere so that scientific informa• tion can be derived from the incoherent backscatter received. Glossary 191

Ionospheric sounders systems that probe the ionosphere by transmitting rapidly varying frequencies toward it; returns are evaluated to obtain estimates of electron density as a function of height. Isotropic radiator an imaginary source of electromagnetic energy that radiates equal amplitudes and phases in all directions.

Jitter small, rapid, perhaps random fluctuations about an intended point or lo• cation.

Kalman filter a well-known tracking algorithm (named after its creator) which weights measurements according to their quality to optimize results. Keplerian motion movement dictated only by the of gravity, such as the movement of planets and satellites. Kilohertz thousands of cycles per second. Klystrons high-power of radio frequency energy, capable of coherent operation.

Laser radar a radar at optical, infrared or ultraviolet frequencies. Lidar for "light detection and ranging"; a laser radar; sometimes, Ladar. Linear aperture an antenna consisting of elements arranged in a line. Line feed a feed whose elements are arranged in a line; they may also be phased, as in a line feed that removes spherical aberration. LPIR low probability of intercept radar; a radar with waveform and antenna designed to minimize the power radiated in both spatial and spectral domains.

Main beam that part of an antenna pattern that contains the major portion of energy. Matched filter a filter in a radar receiver whose spectral response is matched to the spectral content of the transmitted signal. Megahertz millions of cycles per second. an angle measure of one one-thousandth of a , that is, .0573 degrees. MMIC monolithic microwave integrated circuit; refers to an all solid-state array element integrated on a single piece of substrate. Modulation the impression upon the radio frequency carrier of the signal fluctu• ations. 192 Radar Principles for the Non-Specialist

Monopulse tracking deriving out of a single pulse all the information necessary to obtain high angular accuracy. Moving target indication (MTI) the use of the doppler content of the radar returns to achieve cancellation of clutter at zero velocity (or some other specific velocity). MTI cancelers the electronic circuits that accomplish the clutter cancellation for MTI.

Newtonian trajectories flight paths that have no forces acting upon them except the forces of gravity. Noise unwanted, sometimes random, electromagnetic energy that mixes with and interferes with the wanted signal energy. Normalize to refer data to a convenient or common reference point and apply a standard interval from that point. North filter a filter that gives an optimum response for signal against Gaussian noise; named after the person who first analyzed it. Nutate to nod or wobble slightly. Nutation a slow or small rotation superimposed on a more rapid or larger one.

Orthogonal polarization orientation of the of electromagnetic radia• tion (including light) at right to a reference electric field. Oscillators electric circuits that produce sinusoidal waves (waves that rise and fall smoothly and harmonically). Over-the-horizon radar radar that uses ionospheric reflection to detect targets beyond the horizon, operating in the frequency band that supports the phenom• enon: the HF (high frequency) band.

Parabolic antenna a device that radiates and focuses electromagnetic energy by use of the shape of the curve of a parabola. Paraboloid a surface made up by revolving a parabola about its axis; the shape of a . Parameter an arbitrary constant that may take on or be assigned various values. Passive ECM the use of such things as chaff and decoys to defeat radar or communications passively. Pedestal the structure that supports a radar antenna. Glossary 193

Pencil beam electromagnetic energy focused to a narrow angle in two dimen• sions, as with a searchlight beam. Phased array an antenna that forms a beam by assigning phases to a number of separate radiating elements. Phase locked held at a constant phase relationship by being tied electrically to a reference oscillator, usually a stable local oscillator. Phase shifter electronic circuits that shift phase in discrete, predetermined steps. Plan view the perspective from above. Plasma frequency the rate of vibration of the electrons in an ionized medium. Polarize to align the electric vectors of electromagnetic radiation. Potted physically fixed in position by being embedded in a resin or other seal. Power-aperture the product of a radar's average power and the physical aper- ture of its antenna. Power the power present in the frequency constituents of a function (usually a plot of these quantities). Propagation the outward spreading of electromagnetic waves. Pseudorandom code a series of random quantities (numbers or levels) whose randomness is unproven; or, random quantities which are replicated when generated for later auto-correlation. Pulse-burst waveform a pattern of transmitted energy that consists of a train of pulses. Pulse compression a technique by which more bandwidth is inserted into a pulse than its duration would imply it could contain. Pulse doppler a radar or a waveform that uses a series of pulses that are pro• cessed for their velocity content.

Quanta small, discrete packages of energy. Quantum mechanics a theory of physics that treats the interactions of radiation and matter; its name derives from the observation that these interactions take place only in discrete packages.

Radar an instrument for radio detection and ranging. Radar altimeter a radar that measures altitude. 194 Radar Principles for the Non-Specialist

Radar cross section a measure of the amount of electromagnetic energy a radar target intercepts and scatters back toward the radar. Radar signature identifying features or patterns in a target's radar cross section. Radial velocity the component of velocity on the radial toward or away from a point, for example, along the line of sight from a radar. Radian form to express an angle in . Radian frequency to express frequency in radians per second rather than in cycles or degrees per second. Random variable in statistics, a function defined over a sample space. Range error the inaccuracy in a range measurement. Range rate the velocity derived from two or more range measurements. Range sidelobes the residues of a pulse compression waveform that spill over into and contaminate adjacent range cells. Range tracking following a target in range. Rayleigh region a region where electromagnetic energy is scattering from targets that are smaller than a wavelength; named after Lord Rayleigh, who calculated the magnitude of that scattering. Rectification processing of electric waves that swing both positive and negative into waves that swing only positive. Refraction the bending of electromagnetic waves that takes place as the medium varies over the propagation path. Resolution bin or cell the extent of the region (in angle, range, or velocity) filled by the return from a single-point target. Resonance region the region where the wavelength of the scattered electromag• netic energy is of the same order as the characteristic dimension of the target. Root-mean-square (RMS) the square root of the average of the sum of the squares of a series of values.

Scanning moving a radar beam around the sky to cover a prescribed region. Scattering the reflection of electromagnetic energy from a target. Scintillation rapid variations in the level of scattering from a target. Semi-isotropic radiation emission from a point source of equal levels of energy in all directions within a hemisphere. Servodrive the power provided to equipment by a technique that uses system output to determine partially what the input will be. Glossary 195

Servoloops the electric circuits that sample system output and refer it back to the input. Servomechanism an automatic device that uses feedback to control systems, usually by inserting at the input control signals derived from samples of the output. Sidelobe the unwanted, out-of-place residue of an antenna pattern or waveform. Sidelobe jamming the act of sending interfering and disrupting signals into the antenna sidelobes. Side-looking radar a radar which points at an angle substantially off the velocity of its carrier, hence, another name for a synthetic aperture radar. Signal-processing gain the improvement in signal-to-noise ratio that results when various processing techniques are applied. Signal-to-noise ratio the ratio of the RMS signal power to the RMS noise power at the output of a radar receiver. Sinusoid sine or cosine plots. Skip distance the range at which a radio wave propagated from the earth toward the ionosphere returns to the earth's surface. Solid angle an area in angle, a number of square degrees or steradians. Spark gap a mechanism by which electromagnetic energy is radiated by building up the field intensity across a gap until the intervening medium breaks down and a spark occurs. Specular returns radar reflections of high amplitude and short duration, like flashes from a mirror. Spherical aberration in a wavefront, resulting when it is reflected off a spherical surface rather than a parabola. Spherics bursts of electromagnetic interference caused by disturbances in the atmosphere. Squint angle the angle off the velocity of its carrier that a synthetic aperture radar may be pointed. STC sensitivity time control; attenuating the radar return signal exponentially as a function of time to keep near-in returns from saturating the receiver. the solid angle subtended by an area on the surface of a sphere equal to its squared. Subarray factor the coverage boundaries of a group of elements in a steerable array. Subclutter visibility the capability of a radar processor to suppress clutter. 196 Radar Principles for the Non-Specialist

Sum beam the adding together of two slightly displaced antenna beams to form a single beam that is the sum of the two, as with monopulse tracking. Surveillance providing coverage of or keeping watch over. Synthetic aperture radar (SAR) a system that uses movement of an antenna beam across an area to synthesize a very large aperture and provide very good a!1gle resolution. Synthetic display an uncluttered presentation obtained by distilling essential in• formation from noisy data and rejecting the latter.

Tapering varying the density of the elements in an array or the power radiated by the array elements in order to obtain a tailored aperture illumination to control the array far-field pattern. Thermal noise unwanted signals generated by the heat inherent in the operation of a radar system, a major factor in the first stage of radio frequency amplifi• cation. Threshold a level established for decision making as to whether or not a signal is present. Time-delay networks circuits in array radars that point the antenna beam in various directions by progressively delaying the radiation of the signal from the array elements. Time domain viewing a multidimensional function in its time dimension. Time sidelobes the spreading residues in range of pulse compression wave• forms, also called range sidelobes. Tracking following selected targets over time, whether in range, angle, or ve• locity. Tracking gate a region of special attention around a target being tracked, usually with appropriate logic to keep the gate moving with the target. Track-while-scan the radar operation in which targets are followed by the radar's routine scanning function, differentiated from an operation where the radar changes its routine to do tracking. Transponders equipment that generates and radiates energy as a result ofreceiv• ing a signal; it may reradiate an enhanced version of the signal received or send new information. Truncated cone a conical shape that has been cut off at right angles to the axis of the cone at some arbitrary point. Glossary 197

Unambiguous range the range associated with the time between radar pulses, that is, the maximum distance at which the roundtrip to a target can be com• pleted before the next pulse is sent. Unambiguous velocity a waveform design feature that provides that the veloci• ties of the targets of interest will not fold over in the signal processor.

Variance a statistical quantity indicating the spread of a distribution about its mean. Velocity ambiguities foldover of velocities in the signal processor, requiring additional processing to determine the actual velocities of the targets. Velocity spectrum the frequencies generated by a target moving at a given velocity. Vertical polarization by convention, an electromagnetic wave whose electric vector is perpendicular to the earth's surface. Vertical return that part of an airborne radar signal that returns from directly beneath the aircraft where the angle of incidence is 90 degrees. Video integration adding up signals after they have been through the second detector when only the envelope of the original signal remains.

Waveforms various shapes of radar pulses or groups of pulses designed to ac• complish particular objectives. Weighting changing the shapes of pulses or the envelopes of groups of pulses to tailor their sidelobes, usually by rounding the ends with a slowly varying func• tion, such as a cosine. Woodward ambiguity function the surface that results when responses to wave• forms are mapped in both velocity and time; named after the person who led in analyzing ambiguity functions. INDEX

Acceleration, 91-92 over-the-horizon, 127-128 accuracy, 92 parabolic, 14-16 resolution, 91 pattern derivation, 18-21 Accuracy, 58, 87, 92 radar altimeter, 130 , 92 receive, 6, 9, 37-38, 128 angle, 58 sidelobes, 23-26, 37 range, 87-88 transmit, 6, 16-17, 127 Aegis system, 26 Anti-jam gain, 114 ALCOR radar, 11 Aperture, 7-10, 23, 133, 185 Alpha error, 50 circular, 23 Ambiguity, 22, 98-101,136-137,142,151- size, 10 152 Area MTI, 135 functions, 101, 102 Arecibo radar, 133-135 in interferometers, 22 line feed, 134 mitigation, 101 range, 135 range and doppler, 98-101,142,151-152 Array radars, 26-40 in SAR radars, 151-152 Arrays, 26-40, 127 Ampere's law, 162 Blass, 33 Angle accuracy, 58 Butler, 32 Angle tracking, 55-58 corporate fed, 38-39 Antennas, 6-7, 9, 13-40 design, 38-40 area, 7, 9 element spacing, 33-34 array. See Arrays factors, 38-40 beam width , 17,21-22 gain, 35-36 circular, 23 grating lobes, 35 definition, 13 sidelobes, 36-38 far-field, 16, 148, 164-165 space fed, 38-39 focusing, 14-15 steering, 26-34 gain, 6-8, 14 subarrays, 38-40 half-power points, 17 thinning, 36-38 incremental sources, 16-17 wrap-up factor, 29 optical analog, 14 A-scope, 86, 104

199 200 Index

Attenuation, 128, 161-162 Clutter, 74-77, 128, 143-145 in ionosphere, 128, 162 for airborne pulse-doppler, 143-145 in troposphere, 164-165 cancelers, 138-141 Automatic gain control (AGC), 104 fences, 76 AWACS radar, 142 -limited radar, 77 mapping, 145 in MTI, 135-141 Backscatter, OTH, 128-129 and OTH-B radars, 128 Backscatter, radar. See Radar cross• velocity, 138-141 section as target, 76 Ballistic missile early warning system Clutter RCS, 75-76, 143-145 (BMEWS), 11, 135 ground, 75, 143-145 Bandpass filter, 85 reflection coefficient, 75-76 Bandwidth, 8, 85-86, 92-94, 154 roughness, 75 defined, 85 sea, 75 filter, 8, 85 wavelength dependence, 75-76 in laser radar, 154 wind,75 ratio, 93-94 Cobra Dane radar, 26 Barrage jamming, 112-115 Coincidence filter, 100 Bayes formula, 61 Coherence, 51,101,128,136-141 Beam-shape loss, 23 Coherent MTI, 136-141 Beam steering, 27-34 Comb filters, 61 frequency, 29 Conical scan, 55-56 Huggins, 32 Constant false alarm rate (CFAR), 104 phase, 28-33 Continuous wave systems, 126-133 phase-shifter, 29-32 in altimeters, 126-130 time-delay, 27-28 in OTH-B radar, 130-133 Beamwidth, 17, 21, 23, 25 Cosine illumination, 24-25 Bessel function, 23 Cosine weighting, 24-25, 97 Beta error, 50 Costs, 10, 11 Blanking, 76, 144 Cross-section. See Radar cross-section Blind speeds, 137 Boltzmann's constant, 6, 156 Dead zone, 5, 144 and Lidars, 156 Decoys, 123 Brightness, 154-155 DeForest, Lee, 3 B-scope, 104-105 Decibel (table), 145 Detection, 41-54 Cancelers, 116-119, 138-141 m-out-of-n, 52-54 clutter, 138-141 probability of, 48, 50-51 mainlobe, 118 techniques, 47-54 sidelobe, 116-119 threshold, 47, 51, 53 Center-of- tracking, 61, 132 Difference beam, 56-57 Chaff radar cross-section, 73-74, 121-122 Difference frequency, 93 average, 74 Digital processing, 107-108, 129-130 of cloud, 74 Diode phase shifters, 29, 129-130 maximum, 74 Displays, 4, 86, 103-105 simple theory, 73-74 A-scope, 86, 104 Chain Home radars, 3 B-scope, 104-105 Circular aperture, 23 E-scope, 105 Circular polarization, 71 PPI scope, 104 Index 201

Distribution, 44-48, 137-140 Exponential distribution, 44-45, 137-138 erf(x), 44-48 mean and variance of, 44 exponential, 137-140 normal, 46-47, 98 Faraday, Michael, 2, 162, 164 Rayleigh, 44 Far-field, 16, 18, 148, 164-165 Doppler, 89-92, 98, 128, 141-144, 147, 149 Far-field equation, 148, 165 acceleration, 91 Federal Aviation Agency (FAA), 54, 126 accuracy, 90 Feed, 16,24,71,133-134 ambiguity, 98,100,102,132-133,142- Cassegrain, 16 143 Ferrite phase shifter, 29 cross-talk, 95 Filter, 6, 61-63, 85, 103, 105 derivation, 89-90 bandpass, 85 processing, 106-108 bandwidth, 6, 85 vs. range rate, 91 bank, 61, 106 relativistic, 90 cut-off, 85 resolution, 90, 98, 128 digital, 107-108 sidelobes. 101 Kalman, 61-63 tracking, 61 matched, 6, 85, 103 Duty cycle. 4, 128 North,85 First-time-around range, 4 FM chirp, 92-94 E-2C, 142 FM/CW waveform, 128-129, 132 E-3A, 142 range errors, 94-95, 132-133 Einstein, Albert, 153 time sidelobes, 96-98 Electromagnetic waves, 2-3 FM ramp, 92-94, 128-129, 132 spectrum of, 2, 175 Focusing, 14, 149-150 velocity of, 2 Four-bit phase shifters, 30, 32 Electronic counter-countermeasures Fourier series, 176-178 (ECCM), 114, 116-119, 121, 123 Fourier transform, 25, 37, 84, 180 Electronic countermeasures (ECM), 111- fast, 32-33, 108 124 Frequency, 2, 4 active, 111-121 radian, 4 chaff, 121-122 Frequency steering, 29 decoys, 123 jacks, 122-123 Gain, 6-8, 14,22-23,36,40,51-52,94 passive, 121-123 antenna, 6-8, 22-23 retrorefiectors, 122-123 array, 35-36 Element gain, 35-36 defined, 6 Element spacing, 33-34 element, 34-35 Energy, 2, 6-8, 46, 52 integration, 51-52 electromagnetic, 2 signal processing, 94 signal, 6-8, 46, 52 Glossary, 184-197 Envelope integration, 51, 105 Grating lobes, 34 Erf(x), 47-48 Ground clutter, 75, 143-145 Error voltage, 54 Ground maps, 147-152 Errors, 30-31, 59-60, 94-95,136,145-146 Gun-barrel analogy, 59-60 angle, 30-31, 59-60 range, 59-60, 94-95 Hamming weighting, 98 E-scope, 105 Haystack radar, 169, 172 Ether, 2 Hertz, Heinrich, 2 202 Index

High frequency (HF) spectrum, 126-127, extended target, 157 175 range equation, 154-155 Horizon, radar, 160-161 receiver noise, 155-156 Hypothesis testing, 50 resolution, 154 signal-to-noise ratio, 155-156 Illumination functions, 18,20-21,24,37 speckle, 156 Incoherent integration, 51 theory, 153-154 Incoherent MTI, 135 uses and limitations, 154 Incremental source, 16, 147 wavelengths, 154, 175 Integration, 51-52, 105-106 Leading-edge tracking, 60-61 coherent, 51-52 Length measurement, 78, 80, 158 envelope, 51, 105 Levels, sidelobe, 23-26, 96-98 human operators, 106 Lindar. See Laser radars incoherent, 51 Light, velocity of, 2 post-detection, 105 Lincoln Laboratory, 11, 169 pre-detection, 105 Line feed, 134 by recirculating delay lines, 105-106 Losses, 7, 23, 71-73 SIN effects, 52 beam-shape, 23 time, 128, 147-148 polarization, 71-73 video, 51, 105 system, 7 Interferometer, 22 Low-probability-of-intercept radar (LPIR), beamwidth, 22 119-120 Ionosphere, 126-129, 162-164 , 162 Magnetic fields, 2, 163-164 polarization rotation, 163-164 Earth's, 163-164 reflection, 128 Matched filter, 7, 85, 106 refraction, 128, 163 Mathematical statistics, 44-50, 87-88 sounders, 162 Maximum likelihood estimate, 87-88 Ionospheric radars, 133-135 Maxwell, James Clerk, 2, 66 Isotropic radiator, 5-6, 35, 66 Measurement, 57-58, 78, 86-87, 89-92, and radar cross-section, 66 158 angle, 57-58 Jacks, 123 doppler, 89-92 Jamming, 112-116 length, 78, 80, 158 equations, 114-115 range, 86-87 mainlobe, 112-115 Medium, propagation, 161-164 repeater, 121 Monopulse tracking, 56-58 screening range, 114 M-out-of-n detection, 52-54 sidelobe, 115-116 Moving target indication (MTI), 135-141 signal-to-jam ratio (S/J), 112, 114-115 blind speeds, 137 Jittered PRF, 100-101 coherent, 136-141 figure of merit, 137-140 Kalman filter, 61-63 non-coherent, 135-136 Keplerian motion, 78-79, 191 Naval Research Laboratory (NRL), 3 Ladar. See Laser radars Navigation, 170-171 Laser radars, 153-157 Navstar GPS, 170 bandwidth, 154 Newtonian trajectory, 59 brightness, 154-155 Noise, 6, 44-50, 52, 85, 155-156 cross-section, 156-157 figure, 6 Index 203

lidar, 155-156 Propagation, 161-164 in integration, 52 Pulse burst , 98-101 optimum filter, 85 ambiguities, 99-102 power, 44-45, 85 ambiguity mitigation, 100-10 I radar, 6, 44, 85 sidelobes, 101 receiver, 6, 44, 85 time and frequency domains, 98-99 system, 6 Pulse, 83-86, 88 thermal, 6, 44 Pulse compression, 92-96 Normal distribution. 44 FM chirp, 92-95 North, D.O., 46, 85 other techniques, 95-96 filter, 85 Pulse-doppler radars, 141-146 Nulls, sidelobe, 17-18,23-24 airborne, 143-146 Pulse repetition frequency (PRF), 4, 98- Optics region, 69-70 99, 138, 140, 142 Orbits, satellite, 169-170 Pulse repetition interval (PRI), 4, 98 Over-the-horizon (OTH) radar. 126-130 antennas, 127-128 Radar anti-jam performance, 129 altimeters, 130-133 skip distance, 127 average power, 7 technology, 129 astronomy, 133-135 waveforms, 128-129 block diagram, 4 Chain Home, 3 Parabolic antennas, 14-16 costs, 10-11 characteristics, 15-16 cross-section. See Radar cross-section Parabolic reflectors, 14-16 definition, I PAVE PAWS radars, 26 history, 3 Phase, 16 horizon, 160-161 Phase difference, 17, 19 ionospheric, 133-135 Phase shifters, 29-32 laser, 153-157 diode, 30-32 limitations, 168-173 ferrite, 29 noise, 9 four-bit, 30, 32 , 6 three-bit, 30-32 over-the-horizon (OTH), 126-130 two-bit, 29-30 peak power, 6 Planck's constant, 155 pervasiveness, 125-126 Plane wave, 15 potential, 168-173 Plan position indicator (PP!) scope, 104 pulse-doppler, 141-146 Polarization, 71-73, 163-164 range equation, 5-10, 128, 131 circular, 71 rules of thumb, 166-167 horizontal, 71 scanning, 8-11 losses, 72-73 signatures, 77-79, 152-153, 171-172 and rain penetration, 72 synthetic aperture, 147-153 rotation, 163-164 systems application, 125-159 vertical, 72 tracking, II and weather radars, 72 Radar cross-section (RCS), 6, 65-80, 128- Power, 6-9 m Probability of detection, 46-47, 49, 52-54 of chaff, 73-74, 121-122 Probability of false alarm, 45-46, 49, 52- of clutter, 75-76 54 of complex objects, 66, 68 Processing. See Signal processing of comer reflectors, 66-67, 122-123 204 Index

of a cylinder, 68-69 RMS (root-mean-squared), defined, 3 defined, 66 Rules of thumb, 166-167 of diffuse targets, 74-77, 130-131 images, 77-78,152-153,171-172 SAL Treaty, 11 of large curved bodies, 68 Satellite object identification (SOl), 77-80, of a long wire, 69 152-153 of nosetips, 70 Satellite orbits, 169-170 of objects small with respect to wave- Satellite surveillance, 169-170 length, 69, 129 Science, 172-173 patterns, 78 Sea clutter, 75 and radar altimeters, 130-131 Sensitivity time control (STC), 104 in Rayleigh region, 69, 129 cancelers, 116-119 of reentry vehicles, 70 Sidelobes, 17,21,23-26,36-38,96-98, relationships, 79 101,132 in resonance region, 69-70 angle, 17,21, 23-26, 36-38 of simple bodies, 68-71 approximations, 25-26 of a sphere, 66-70 jamming, 115-116 and synthetic aperture radars, 146, 153 levels, 24, 26, 36-37, 97-98 terrain modeling, 74-76 maxima, 24, 37 volume measurement, 69 nulls, 17-18,22-23 Radian frequency, 4 pulse burst waveform, 101 Radio waves, 2 time, 96, 98, 101 Range, 3 Signal, 6-7, 46-47, 50-51 ambiguous, 98-99, 132, 142-143 power, 46, 51 equation, 5-10,131,138,154-155 Signal-plus-noise, 46-47 gate, 105 Signal processing, 82-108 tracking, 58-60 digital, 107-108, 126-127, 129, 135 unambiguous, 98-99, 142-143 I & Q channels, 108 Range measurement, 86-87 topics, 102-108 accuracy, 87-88 Signal processing gain, 114 resolution, 88-89, 92-93 Signal-to-clutter ratio, 77 Rayleigh, Lord, 44, 69, 88, 129 Signal-to-jam ratio, 112, 114-115 criterion, 88 Signal-to-noise ratio, 7-10, 46-50, 86, 119, distribution, 44 155-156 region ReS, 69, 129 in laser radars, 155-156 Rectification, 44 Signatures, 77-79, 152-153, 171-172 Reentry vehicle ReS, 70 Simple pulse, 83-86 Reflection coefficient, 75 Fourier transform of, 84 Relativistic doppler, 89-90 with noise, 85-86 Resolution, 17,21, 23, 50, 88-90, 92-93, power spectrum of, 84-85 98 Sin x/x, 20, 23, 37, 84, 96, 132, 181 3-dB criterion, 88 Snell's law, 15 angle, 17,21,23 Solid angle, 6, 14 bins, 50 Spark gap generators, 2-3 doppler, 90, 98 Speckle, 156 range, 88-89, 92-93, 98 Sphere ReS, 66-70 Rayleigh criterion, 88 Spherical abberation, 133-134 for unequal targets, 90 Spotlight mapping, 150-151 Resonance region, 69-70 Squint angle, 150 Retroreflectors, 122-123 Sum beam, 56 Index 205

Surveillance, 7, 63, 168-170 leading edge, 60-61 satellite, 63, 169-170 monopulse, 56-58 Synthetic aperture radar, 147-153 range, 58-60 ambiguities of, 151 Track-while-scan, 54 angle resolution of, 148-149 Transmit antenna, 6, 15-16,36-40 focusing, 149-150 Tropospheric attenuation, 164-165 inverse, 152-153 limitations, 151-152 University of Michigan, 149 non-focusing, 147 platform motion, 152 processing load, 152 Vacuum tubes, 3 and satellite signatures, 153 Velocity ambiguity, 98, 100, 142, 151-152 spotlight mapping, 150-151 Velocity of light, 2 squint angle, 150 Verification pulse, 50 System , 6 Video , 103-104 Video integration, 51 Thermal noise, 44 Three-bit phase shifter, 30-32 Waveforms, 82-102, 128-129, 132 Thresholds, 46-49, 50-51, 54 FM chirp, 92-94 Time-bandwidth product, 93-94 FM/CW, 128-129, 132 Time sidelobes, 96-98, 132 pulse compression, 92-96 in FM chirp, 96 pulse burst, 98-101 in FM/CW, 132 pulse-doppler, 141-143 weighting functions for, 96-98 simple pulse, 83-86 Track-and-scan, 54 Wavelength, 4 Tracking, 11, 54-63 Waves, 2-3, 15 algorithms, 58-59, 61-63 radio, 2-3 angle, 55-58 plane, 15 conical scan, 55-56 Weighting functions, 24-25, 36-38, 96-98 doppler, 61 , 85 Kalman filter, 61-63 Woodward ambiguity function, 101-102