Radar Equation 2.1 Prediction of Range Performance + The maximum radar range ar is given by ae form of expression. y heh eae Ree - ge)" ‘oe Ne om Sen (2.1.1) 7)" Prin Tats Gi dh e, Where, aie he P, is peak value of transmitted power G, is power gain of transmitting antenna. Li% @ is area of cross-section of target (m?) Prin is minimum detectable signal by receiving antenna. (W) * The radar equation shows that different parameters of radar system can be controlled except target cross-section area (c). When long range is desired, the transmitted power (P)) should be large, the transmitting gain must be large, also the receiver should be sensitive to weak signals. * However the above radar equation may not give predict actual performance of radar. In practice, the radar performance degrades because of following reasons. The minimum detectable signal is determined by receiver noise. Fluctuations and uncertainty in radar cross-section. Losses in various parts of radar system. Propagation effects due to earth’s surface and atmosphere. Encepl fete totgct Aacsscchion ae cL RCS, all he OMe loccqres Urls We . 2 1) Combes of KADAR Fens @ scanned with OKEN ScannerRadar Equation 2-2 Radar Systems e and target cross-section, the ¢ of receiver noise an ccs a decked probabilistically rather than single ange is de i SSE GRIEE rac range specication includes i : e. Sasi that radar will detect a target at a particular range. 4 Probability of making a false detection when no target is present. i the range of radar is a function of facts it can be concluded that the rang ae i bability of false alarm (Pj). The lity of detection (P,) and the probability Prediction of radar range is not accurate as there is uncertainty in various Parameters, Still radar range equation is an important tool for ') Assessing the performance of radar. 1) "Determining system tradeoffs for designing a new radar system. iti) Generating technical requirements for new radar procurement, 22 Minimum Detectable Signal The radar receiver's abil NB 4 Weak echo signal is decided by the noise present in the frequency spectrum, The weakest signal that a radar can detect is minim, is lum dete le si i A denoted by p stectable signal. The min mum detectable signal is { Deection ‘of radar gan ishit carn OF Padar sign '8 based’ on sstablishing threshold ley, = receiver. A target is said to be Present when the ger than the threshold, If the receiver output is below noise iN Teceiver, it is called as threshold detecti a re town it Fig. 3.94, This output een 2 € Versus tiy is i a time, it ig fluctuating because rand. _: ‘el for the Teceiver output is @ scanned with OKEN Scannerradar 22 Radar Equation a, ube Picce tee signal plus noise. When a large echo signal po erect it Is represented by ia patted ly, the output level does y A. If the threshold level is sett Not exceed threshold if noise only is present and it should exceed threshold when Strong echo is present. Lab a) Missed detection (failing to Tecognize a tar + The signal-to-noise ratio (SNR) j of a radar instead minimum detectable signal. 2.3 Receiver Noise and SNR + The noise affects the maximum radar range as it determines the minimum _ teceved power that radar can detect. Als, tie Tadar epee ee by decreasing minimum detectable power which depends on the sensitivity the receivers and hence on its noise figure. Noise figure F, is defined as Rae. _Input signal to noise ratio ~~ Output signal to noise ratio the definition of the noise figure, the noise power generated at the Of receiver can be calculated. Thus the power required at the input of ideal receiver having the same noise figure as the practical receiver F = S)input GIN) output Fea te Poo F Fe aie ig . ual @ scanned with OKEN ScannerRadar Equa, 2-4 a where, Pg ~ Input signal power Py = Output signal power Pa, = Input noise power Pao ~ Output noise power G, - Power gain of receiver “eg Pari - Noise power generated at the receiver input re therefore ai = (F=1) Poi = Ty BF where, KT) B = Noise input power of receiver K = Boltzmann's constant = 138x109 9/K Ty = Ambient temperature = 290 K B = Bandwidth of the receiver, Hz * Im the above equation it is assumed that antenna temperature is equal to the standard ambient temperature. The actual antenna temperature is important when a very low-noise amplifier is used. * Under threshold detection conditions, the minimum detectable signal is equal to the equivalent noise power at the input of receiver. . The ‘echoes from the target are repetitive, whereas noise impulses are random. Thus an integration of echoes takes place at the receiver, echo pulse can be obtained although their amplitude is no greater than that of noise impulses. When the target is stationary, the signal pulses keep on accumulating at the same spot so that the brightness at this spot of the Screen is maintained, whereas the noise impulses are of pure random in nature and are not additive. + If the target is moving and it moves significantly between successive scans, a system called moving-target-indicator is used. Therefore minimum signal Pian Should be atleast equal to. Pyin = KT B(F-1) @ scanned with OKEN ScannerRadar Systems 2-5 Radar Equation ee ee The radar range equation is Substituting Pain in radar range equation (4x)? KT) BF-1) + The radar range can be predicted fairly accurately E ly using above equation. Still there are other factors which affect the radar range and for very accurate calculations these factors have also to be considered. These factors include : a) System losses b) Receiver non-linearities ©) Antenna imperfections d) Anomalous propagation ¢) Interference by nearby noise sources ) Operators error Since Gre -the 2 And Ae= nA ae (] re xD? xD G = 4xx Sey -005(2Y Also Ty = 290° and K = 138x 10-9 J/°K * The expression for maximum radar range is PD's Va toe [ar where, P, - Peak transmitted power, watts D - Diameter at antenna, metres 9 - Effective cross-sectional area of target, metre” B - Bandwidth of receiver, Hz af @ scanned with OKEN Scanner a.i patente - er A - Wavelength, m F - Noise figu ym> Example 2.3.1: Calculate the minimum receivable signal in a radar receiver operating in standard ambient temperature, which has an IF bandwidth of 2 MHz and a noise figure of 12 4B. Solution : B = 2 MHz = 2x 10° Hz F = 12dBs 12 F = Antilog 55 F = 15.84 Minimum receivable signal is given by, Pmin = K Ty BF - 1) K = 138x109 J/* T = 20% Poe 10° 3 x 290 x 2 x 10° [15.84 - 1] Prin = 118.85 x 10” watts «Ans. i. mp Example 2.3.2: A low power radar has a receiver whose noise figure is 5.3 aB. If the antenna diameter is 1.5 m, the IF bandwidth is 500 kHz, the operating frequency is 8.1 GHz and the radar set is supposed to be capable of detecting targets of 5 m° cross-sectional area at a maximum distance of 15 km. What must be the peak transmitted power ? Solution : F=53 dB D=15m B = 500 kHz = 500x 10° Hz f = 81 GHz = 81x10? Hz o = 5m Rmax = 15 km PR = to find Since F = 53 dBs @ scanned with OKEN ScannerRadar Systems 2-7 Radar Equation = 5.3 = Antilog 2 Antilog 3 F = 3.388 And c i 3x108 = oxi = 0.037 Ano a Maximum range of radar and transmitted power are related by equation note eet a [3 ~ | where Ruy is in km PD‘ BA(F-1) BA(F-1) (15 ¢ Paes) (15 Dis (#8 p, = 300x108 x(0.037)?(3.388-1) (15 oy 154 x5 48 0.6158 watts Ans, ay ump Example 2.3.3: A radar system operates at 6 GHz 3 MW power output. If the antenna diameter is 5 m and the receiver bandwidth is 1.5 MHz and has a 12 dB noise figure, what is the maximum detection range for 1 m? target. Solution : F = 6 GHz P, = 3MW=3xx 10° watt D = 5 metres 1.5 MHz = 15x 10° Hz F = 12dB o=1m c Eta 4 mai @ scanned with OKEN Scanner@ scanned with OKEN ScannerRadar Systems ro Hadar Enysation Radar range equation is given by, L [t Ge mu iar Tus} 20 ee Ri. AGio eaiieaille | (49)? Prain ’ G? = REA? Pin Pio R449)? Prag JE Gy = [RE Pain Pio 1 3x10° x(0.24)? x1 Gy = 1643.76 "Also for paraboloid effective area is given by, Se 1643.76 x (0.24)? . b xx | A, = 753 The effective area A, is related with physical area A by relation Ae = nA 753 = 065% A a A = 1159 4 a ‘Area of paraboloid A = xD? Fp us ba D = 38tmere Ans. @ scanned with OKEN Scanner@ scanned with OKEN Scanner| ae ON ——— ee 8 ee pia eon ; Radar Equation f signal energ: ton scienes, Hal beed energy during tectification process. The Ot Post-detection integration is, given b (9), ration is given by, © © nN) (244) same integrat ) « For Brated SNR, post-detection integration requires more pulses een ore spsuming SNR per pulse is same in both cases. When a edetecian ni integrated (small SNR per pulse), the difference between predetection and post detection is significant «The radar equation with n pulse integrated is given by, : RGA | GaKn aN, Rina (245) where, (i), is signal-to-noise ratio of each pulse. 25 Radar Cross-Section (RCS) of Targets * The amount of power reflected by the target depends on many factors including the size, shape, material (metal, plastic, wood or water) and edges (sharp or rounded) of the target, as well as the frequency of the incident radar signal and the angle between the radar system and the target. * The radar cross-section 6 is the property of a scattering object which represents the magnitude of the echo signal returned to the radar by the target. * The radar cross-section depends on the characteristic dimensions of the object compared to the radar wavelength. When the wavelength is large compared me to the object’s dimensions, scattering is said to be in Rayleigh region. The radar cross-section in the Rayleigh region is proportional to the fourth power of the frequency. * When the wavelength is small compared to the object's dimensions is called the optical region. * In between the Rayleigh region and optical regions is the resonance regions, where the radar wavelength is comparable to the objects dimensions (approximately 1 to 10 wavelengths). it @ scanned with OKEN Scanner@ scanned with OKEN ScannerRadar Systems 45 sig b, = Pinoy i "phe or For perfectly conducting sphere where, jn iS spherical Bessel function of the first kind. Nh? is spherical Hankel function of second kind, nis summation counter. 2 & o 4 Bos Mie or Resonance region 0.05 o4 02 05 1 Circumference a= 2) ==> Fig. 2.5.1 @ scanned with OKEN Scanner@ scanned with OKEN Scanner28 o> Fig. 2.5.2 cross-section of cone sphere is a very low and is considéred to be of 7 missile. A large cross-section occurs when a radar views the cone ¢ ws ’ rage transmitter power is defined over the duration of total transmission d. The transmitted waveform is a train of rectangular pulses of width t pulse repetition period Tp. fs pe repetion fequency «tie at Thy od A UP at i @ scanned with OKEN Scanner@ scanned with OKEN ScannerRadar Systems 2.47 8 (x | = SNR required as if direction were based on only single Pulse f, = Pulse repetition frequency (Hz) 2.7 PRF and Range Ambiguities «PRE is uy determined by maximum unambiguous range. The PRF corresponding to maximum, unambiguous range is given by, = 2Runambiguous —aenbigwoe where, cis velocity of Propagation. «Echo signals that are received later than pulse repetition period are known as second time around echoes or multiple time around echoes. These echoes may cause error and confusion. Also it can mask unambiguous target echoes at shorter ranges. Pulse doppler radars have usually problem of range ambiguities because of prf. The problem of multiple time around echoes occurs when the range is greater than 2 Runambiguous: These echoes cannot be determined from constant prf-———"6* * Consider the targets located at three different positions A, B and C as shown in Fig. 27.1. Ais located within unambiguous range interval Rusanbiguose Bis located at a distance > Ruranbiguow but less than 2 wanbiguou Cis located at a distance > 2. Rusanbiguou but less than 3 Reranbiguour * Fig. 27.1 (a) shows target B is a second time around echo and target C is _ multiple time around echo. Fig. 27.1 (b) shows radar display when these _ three pulse repetition intervals are superimposed. The ambiguous echoes B and C looks very similar to unambiguous range echo of A. Out of these three echoes only the range of A is correct but it is hard to determine that ‘other two are not the apparent ranges, The ambiguous range echoes are recognized by changing the prf of the -tadar. On changing the prf, the unambiguous echo ( 2.0 dB (CFAR) Automatic integrator 1.5 to 2.0 dB 1.0 to 2.0 dB Upto 1 dB 2.8.4 Collapsing Loss * When additional noise samples are integrated with signal + noise pulses, this added noise causes degradation called collapsing loss. The collapsing loss is given by L. i(m+n) Le where, ‘ Licmen) is integration loss for m+n pulses. Li(a) is integration loss for n pulses. — 'm is noise pulses. nis signal - to - noise pulses. Le¢mn) @ scanned with OKEN Scanneryp Example 28.4: A square law det 30 noise pulses. If integration lose Toss due to noise pulses (10) is 1,7 solution : Given leclor integrates 10 signal plus noise pulses along with for signal + noise pulses (40) 18 3.5 dB and integration 4B. Calculate collapsing loss of the radar antenna. Signal plus noise pulses (m) = 10 Noise pulses (n) = 30 Integration loss L. = 35 4B (men) = Integration loss Li.) = 1.7 4B Collapsing loss is ee, i(m+n) Lion Lomeny = o a Lemen) = Q Effects ‘The propagation effects of radar wave has significant impact on losses. Major of propagation on radar performance are under mentioned. Reflections from earth's surface "i Refraction Propagation in atmospheric ducts Attenuation in clear atmosphere. Te propgation efits are mot computed under astm ls but under 2 Radar range equation is an important todl for i) Assessing radar performance ii) Designing radar system iii) Generating technical requirements a @ scanned with OKEN ScannerF Sdection of proper threshold is necessary ) Missed detection . ‘of pulses returned after hitting target is given by, 3. Number of pu! eal n= = - . of summing all the radar echoes pulses is the process 4. The integration of oa of pied before second detector is known as predetection 5. The integra j integration or coherent integration. 6 The integration of pulses after second detector is known as post detection or coherent integration. 7. Integration efficiency for post detection integration e,. « On *) “ 2G), 8. Radar cross-section (RCS) is given by 2 is 2 LErl ° ae peat 9. RCS for sphere is given by, A o= 2) Sirens (en ho) P lnat 10. Average transmitted power is given by Rom Py = 11. Duty cycle of radar is dependent on i) Type of waveform if) Pulse wit Mi pPviee coomeceniony iv) Radar range @ scanned with OKEN ScannerSS verge transmitted power? How itafects radar range ? cribe briefly range ambiguities in radar system. various system losses in any radar. death ca @ scanned with OKEN Scanner