APPENDIX 1 CONVERSION TO DECIBELS Bels were named after Alexander Graham Bell. They were simply the logarithm of the ratio of power out and power in. The bel was too small to be useful; so the decibel quickly emerged. The definition of a decibel (dB) is contained in any radio engineer's handbook. It is where P is power, V is voltage, 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 miles -33 4 6 Nmi to meters +33 5 7 Boltzmann's Constant (joules/oK) -228 6 8 Square degreeS in 47T steradians 46 7 8.5 Velocity 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 waves (3 kc - 3 X 106 MHz) t4+-----1.0------+-I·1 I HI\------"::......::::------Frequency 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: Laser 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 heat 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 sine and cosine waves of various frequencies and amplitudes. His work has been described as one of the most elegant developments in modern mathematics. 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 wave may be broken down into the sum of sines 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 Engineering 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 second 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 Fourier transform of a single pulse and the larger the value of k, the closer the spectral lines are together and the more closely a continuous function 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, antenna illumination functions are transformed into far-field antenna patterns and waveform 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).