3.0 DETECTION THEORY. Figure 3-1 IF Receiver/Signal Processor Representation

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1 3.0 DEECON HEORY 3. NRODUCON n some of our raar range equation problems we looke at fining the etection range base on NRs of 3 an 0 B. We now want to evelop some of the theory that explains the use of these particular NR values. More specifically, we want to examine the concept of etection probability, P D. Our nee to stuy etection from a probabilistic perspective stems from the ct that the signals we eal with are noise-like. From our stuies of RC we foun that, in practice, the signal return looks ranom. n ct, werling has convince us that we shoul use statistical moels to represent target signals. Also, in aition to the target signal we foun that the signals in the raar contain a noise component, which also nees to be ealt with using the concepts of ranom variables, ranom processes an probabilities. o evelop the requisite equations for etection probability we nee to evelop a mathematical characterization of the target signal, the noise signal an the target-plusnoise signal at various points in the raar. From the above, we will use the concepts of ranom variables an ranom processes to characterize these quantities. We start with a characterization of noise an then progress to the target an target-plus-noise signals. 3. NOE N RECEVER We will characterize noise for the two most common types of receiver implementations. he first receiver configuration is illustrate in Figure 3- an is terme the F representation. n this representation, both the matche filter an the signal processor are implemente at some intermeiate frequency, or F. he secon receiver configuration is illustrate in Figure 3- an is terme the baseban representation. n this configuration, the signal processing is implemente at baseban. he F configuration is common in oler raars an the baseban representation is common in moern raars that use igital signal processing. Figure 3- F Receiver/ignal Processor Representation 3.. F Configuration n the F representation, the noise is represente by nf t NtcosF t φ t (3-) 0 M. C. Buge, Jr

2 where n t, N t an t F trig ientities we get φ are ranom processes. f we expan Equation (3-) using n tcos t n tsin t n t N t cos φ t cos t N t sin φ t sin t F F F where n t an t n t an t F F (3-) n are also ranom processes. n Equation (3-), we stipulate that n are joint, wie-sense stationary, zero-mean, equal variance, Gaussian ranom processes. hey are also such that the ranom variables n = n t are inepenent. he variances of n t an t tt he above statements mean that the ensity functions of given by t n n an t t n are both equal to. n t an t n are equal an n fn n f n e n. (3-3) We will now show that will further argue that the ranom variables inepenent. an N t is Rayleigh an φ t is uniform on, N N t an φ φ t are tt tt. We From probability an ranom variables if x an y are real ranom variables, r x y (3-4) tan y φ x, (3-5) where tan y x enotes the four-quarant arctangent, then the joint ensity of r an φ can be written in terms of the joint ensity of x an y as n our case f r, rf r cos, rsinu r rect rφ xy. (3-6) x n, y n, r N an φ φ. hus, we have N n n (3-7) an n φ tan n (3-8) E.g. Papoulis, A. Probability, Ranom Variables, an tochastic Processes McGraw-Hill 0 M. C. Buge, Jr

3 Now, since f N, Nf N cos, N sin U N rect Nφ n n. (3-9) n an n are inepenent, Gaussian an zero-mean with equal variance n n fn, n n q n fn n fn n e e. (3-0) n n e f we use this in Equation (3-9) with n Ncos an n Nsin we get N N cos N sin fnφ N, e U N rect. (3-) N N e U N rect From ranom variable theory, we can fin the marginal ensity from the joint ensity by integrating with respect to the variable we want to eliminate. hus, an N N, fn N fnφ N e U N (3-) fφ fnφ N, N rect. (3-3) his proves the assertion that prove that the ranom variables N t is Rayleigh an φ t is uniform on, N N t an t tt from Equations (3-), (3-) an (3-3) that, tt. o φ φ are inepenent we note fnφ N fn N fφ, (3-4) which means that N an φ are inepenent. ince we will nee it later, we want to fin the noise power out of the signal processor. ince n t is wie-sense stationary we can use Equation (3-) an write F n n cos n n cos F n sin F n n P E t E t t t sin t nf F F F E t t E t t E t t cos t sin t F F (3-5) 0 M. C. Buge, Jr 3

4 n Equation (3-5), the term on the thir line is zero because t n = n are inepenent an zero-mean. tt t n n an t t 3.. Baseban Configuration Figure 3- Baseban Receiver/ignal Processor Representation n the baseban configuration of Figure 3- we represent the noise at the signal processor output as a complex ranom process of the form where nb t n t jn t (3-6) n t an t n are joint, wie-sense stationary, zero-mean, equal variance, Gaussian ranom processes. hey are also such that the ranom variables an n = n t are inepenent. he variances of n t an t tt. he constant of n n t t t n are both equal to is inclue to provie consistency between the noises in the baseban an F receiver configurations. he power in the properties of n t an t n ) B t n is given by (making use of PnB En B tnb t E n t jn t n t jn t En t En t. (3-7) P nf We note that we can write n t in polar form as where t B N jφt n B t e (3-8) 0 M. C. Buge, Jr 4

5 an t t t N n n (3-9) φ t t t n tan. (3-0) n t will be note that the efinitions n t. n t, N t an t φ are consistent between the F an baseban representations. his means that both representations are equivalent in terms of the statistical properties of the noise. We will reach the same conclusion for the signal. he ramifications of this are that the etection an lse alarm performance of both types of receiver/signal processor configurations will be the same. hus, the future etection an lse alarm probability equations that we erive will be applicable to either receiver configuration. t shoul be note that, if the receiver you are analyzing is not of one of the two forms inicate above, the ensuing etection an lse alarm probability equations may not be applicable to it. he most notable exception to the two representations above is the case where the receiver uses only the or channel in baseban processing. While this is not a common receiver configuration, it is sometimes use. n this case, one woul nee to erive a ifferent set of etection an lse alarm probability equations that woul be specifically applicable to the configuration. 3.3 GNAL N RECEVER 3.3. ntrouction an Backgroun We now want to turn our attention to eveloping a representation of the signals at the output of the signal processor. Consistent with the noise case, we want to consier both F an baseban receiver configurations. hus, for our analyses we will use Figures t t t t t t n t with 3- an 3- but replace n with s, N with, φ with θ, s t an n t with s t. We will nee to evelop three signal representations: one for W0/W5 targets, one for W/W targets an one for W3/W4 targets. We have alreay acknowlege that the W through W4 target RC moels are ranom process moels. o be consistent with this, an consistent with what happens in an actual raar, we will also use a ranom process moel for the W0/W5 target RC. ince the target RC moels are ranom processes we must also represent the target voltage signals in the raar (henceforth terme the target signal) as ranom processes. o that en, the F representation of the target signal is where s t t cos t θ t s t cos t s t sin t (3-) F F F F 0 M. C. Buge, Jr 5

6 an cos s t t θ t (3-) sin s t t θ t. (3-3) he baseban signal moel is sb t s t js t. (3-4) t will be note that both of the signal moels are consistent with the noise voltage moel of the previous sections. inepenent. Consistent with the noise moel, we assume that t an t tt θ θ are At this point we nee to evelop separate signal moels for the ifferent types of targets because the signal amplitue fluctuations, t, of each are governe by ifferent moels. tt 3.3. ignal Moel for W0/W5 argets For the W0/W5 target case we assume that the target RC is constant. his means that the target power, an thus the target signal amplitue, will be constant. With this, we let t. (3-5) he F signal moel becomes sf t cos Ft θ cosθcosft sin θsinft. (3-6) s cos ts sin t F F We introuce the ranom variable θ to force s t F to be a ranom process. We specifically choose θ uniform on, 3. his means that s an s are also ranom We have mae a large number of assumptions concerning the statistical properties of the signal an noise. A natural question is: Are the assumptions reasonable? he best answer to this question is that we esign raars so that the assumptions are satisfie. n particular, we eneavor to make the receiver an signal processor linear. Because of this an the central limit theorem, we can reasonably assume that n t an n t are Gaussian. Further, if we enforce reasonable constraints on the banwith of receiver components we can reasonably assume the inepenence requirements are vali. he stationarity requirements are easily satisfie if we assume that the receiver gains on t change with time. We enforce the zero-mean assumption by using banpass filters to eliminate DC components. For signals, we won t nee the Gaussian requirement. However, we will nee the stationarity, zero-mean an other requirements. hese are usually satisfie for signals base on the same assumptions as for noise, an by the requirement that the t is uniform on target RC is a wie sense stationary process, an that, an is wie-sense stationary. Both of the latter assumptions are vali for practical raars an targets. 0 M. C. Buge, Jr 6

7 variables (rather than ranom processes). presence of the t term. F F t s is a ranom process because of the he ensity functions of s an s are the same an are given by s f s f s rect s s s. (3-7) We cannot assert that the ranom variables s an s are inepenent because we have no means of showing that f s, s f s f s he signal power is given by ss s s. s cosθcos sin θsin P E t E t t sf F F F θ F θ E cos cos t E sin sin Ft E cosθsin θ cos t sin t n the above we can write θ θ F E cos cos f cos rect imilarly, we get an E E sin F. (3-8). (3-9) cos θ (3-30) cos θsin θ 0. (3-3) ubstituting Equations (3-9), (3-30) an (3-3) into Equation (3-8) results in PsF cos Ft sin Ft 0 cosftsin Ft. (3-3) From Equation (3-4) the baseban signal moel is sb t sb s js cosθ jsin θ. (3-33) he signal power is 3 his moel is actually very consistent with what happens in the actual raar. pecifically, the phase of the signal is ranom. 0 M. C. Buge, Jr 7

8 P EssE cos jsin cos jsin P sb B B θ θ θ θ. (3-34) sf ignal Moel for W/W argets For the W/W target case we have alreay state that the target RC is governe by the ensity function AV fσ e U. (3-35) AV ince the power is a irect function of the RC (from the raar range equations), the signal power at the signal processor output has a ensity function that is the same form as Equation (3-35). hat is where pp fp p e U p (3-36) P P GGR P 3 4 AV (3-37) 4 RL From ranom variable theory it can be shown that the signal amplitue, governe by the ensity function t, is P f e U. (3-38) P Which is recognize as a Rayleigh ensity function. his, combine with the ct that t θ θ t θ t in Equation (3-) is uniform, an the assumption that an are inepenent, leas to the interesting observation that the signal moel for a W/W target is of the same form as the noise moel. hat is, the F signal moel for a W/W target is of the form where s t t cos t θ t s t cos t s t sin t (3-39) F F F F t is Rayleigh an θ t is uniform on, s t an t noise stuy we arrive at the conclusion that ensity functions Furthermore, tt tt. f we aapt the results from our s are Gaussian with the s P fs s f s e s. (3-40) P s s t an t t t he signal power is given by s s are inepenent. tt 0 M. C. Buge, Jr 8

9 s s cos s s cos F s sin F s s cos sin P E t E t t t sin t sf F F F E t t E t t E t t t t F F nvoking the inepenence of t. (3-4) s s t an s s t an the ct that t t t s are zero mean an have equal variances of P leas to the conclusion that tt s an P sf P. (3-4) he baseban representation of the signal is t jθt sb t s t js t e (3-43) where the various terms are as efine above. he power in the baseban signal representation can be written as as expecte. PsB Es B tsb t E s t js t s t js t Es t Es t P (3-44) ignal Moel for W3/W4 argets For the W3/W4 target case we have alreay state that the target RC is governe by the ensity function 4 AV AV fσ e U. (3-45) ince the power is a irect function of the RC (from the raar range equation), the signal power at the signal processor output has a ensity function that is the same form as Equation (3-45). hat is where 4 p P pp fp p e U p (3-46) P GGR P 3 4 AV (3-47) 4 RL From ranom variable theory it can be shown that the signal amplitue, governe by the ensity function t, is 0 M. C. Buge, Jr 9

10 3 P P f e U. (3-48) Unfortunately, this is about as r as we can carry the signal moel evelopment for the W3/W4 case. We can invoke the previous statements an write an s t t cos t θ t s t cos t s t sin t (3-49) F F F F t jθt sb t s t js t e. (3-50) However, we on t know the form of has proven very laborious an elusive. s t an s t We can fin the power in the signal from s cos θ P E t E t t t sf F F PsB Es B tsb t E t P We will nee to eal with the inability to characterize the characterization of signal-plus-noise.. Furthermore, eriving its form. (3-5) s t an t s when we consier 3.4 GNAL-PLU-NOE N RECEVER 3.4. General Formulation Now that we have characterizations for the signal an noise we want to evelop characterizations for the sum of signal an noise. hat is, we want to evelop the appropriate ensity functions for t t t v s n. (3-5) f we are using the F representation we woul write t t t tcosft θt NtcosFt φt V tcos t ψt v s n F F F F an if we are using the baseban representation we woul write, (3-53) 0 M. C. Buge, Jr 0

11 v t jvt V t ψ t v t s t n t j s t n t B e. (3-54) n either representation, the primary variable of interest is the magnitue of the signalplus-noise voltage, V t, since this is the quantity use in computing etection probability. We will compute the other quantities as neee, an as we are able. We will begin the evelopment with the easiest case, which is the W/W case, an progress through the W0/W5 case to the most ifficult, which is the W3/W4 case ignal-plus-noise Moel for W/W argets For the W/W case we foun that the real an imaginary parts of both the signal an noise were zero-mean, Gaussian ranom processes. ince Gaussian ranom processes are relatively easy to work with we will use the baseban representation to erive the ensity function of t t t v t will also be Gaussian. ince Finally, since sum of the variances of V. ince s an n are Gaussian, s t an n t are zero-mean, t s t an n t are inepenent, the variance of t s t an n t. hat is v v will also be zero-mean. v will equal to the P. (3-55) With this we get v P f v e v. (3-56) P By similar reasoning we get v P f v v f v e v. (3-57) P ince s s t, n n t, s s t an t t t t t inepenent, v t v t an t t tt tt tt n n are mutually tt v v are inepenent. his, with the above an our previous iscussions of noise an the W/W signal moel, leas to the observation that t V t is V is Rayleigh. hus the ensity of V V P fv V e U V. (3-58) P 0 M. C. Buge, Jr

12 3.4.3 ignal-plus-noise Moel for W0/W5 argets ince s t an s t are not Gaussian for the W0/W5 case when we a them to n t an n t the resulting v t an v t will not be Gaussian. his means that irectly manipulating v t an v t to obtain the ensity function of V t will be ifficult. herefore, we take a ifferent tack an invoke some properties of joint an marginal ensity functions. pecifically, we use We then use,,, f V f V θ f. (3-59) Vψθ Vψ θ,, fv V fvψθ V (3-60) to get the ensity function of V t. his proceure involves some teious math but it is math that can be foun in many books on ranom variable theory. o execute the erivation we start with the F representation an write vf t cos Ft θ NtcosFt φ t (3-6) where we have mae use of Equation (3-6). f we expan Equation (3-6) an group terms we get v t cos θ n t cos t sin θ n t sin t. (3-6) F F F Accoring to the conitional ensity of Equation (3-59) we want to consier Equation (3-6) for the specific value of θ. f we o this we get tcos t v tsin t tcos Ft ψ t v t cos n t cos t sin n t sin t F θ F F v V With this we note that F F cos n t an sin t. (3-63) n are Gaussian ranom variables with means of cos an sin. hey also have the same variance of. Further more, since t t cos n an n n an n n are inepenent t t tt sin n are also inepenent. With this we can write v cos sin v fv v, v v θ e. (3-64) f we invoke the iscussions relate to Equations (3-4), (3-5) an (3-6), we can write 0 M. C. Buge, Jr

13 f V, Vf V cos, V sin U V Vψ θ v rect v θ. (3-65) f we substitute from Equation (3-64) we get V V cos cos V sin sin f V, θ e U V rect Vψ. (3-66) We can manipulate the exponent to yiel V V V cos f V, θ e U V rect Vψ (3-67) Finally we can use f rect θ (3-68) along with Equation (3-59) to write V V V cos fvψθ V,, e U V rect rect. (3-69) For the next step we nee to integrate f V,, erive the esire marginal ensity, f V V Vψθ with respect to an to. hat is (after a little manipulation) V V fv V e U V. (3-70) V cos e rect rect We want to first consier the integral with respect to. hat is, V cos V cos, V e rect e (3-7) We recognize that the integran is perioic with a perio of an that the integral is performe over a perio. his means that we can evaluate the integral over any perio. pecifically, we will choose the perio from to. With this we get V cos, V e. (3-7) f we make the change of variables the integral becomes V cos V, V e 0 (3-73) 0 0 M. C. Buge, Jr 3

14 where 0 x is a moifie Bessel function of the first kin. f we substitute Equation (3-73) into Equation (3-70) the latter becomes V V V fv V e U V 0 rect V V V 0 e U V where the last step erives from the ct that the integral with respect to is equal to one. Equation (3-74) is the esire result, which is the ensity function of V t. (3-74) ignal-plus-noise Moel for W3/W4 argets As with the W0/W5 case, case. hus, when we a them to not be Gaussian. his means that irectly manipulating t s t an t n t an n t the resulting v t an t v t an t s are not Gaussian for the W3/W4 v will v to obtain the ensity function V will be ifficult. Base on our experience with the W0/W5 case, we will again use the joint/conitional ensity approach. We note that the F signal-plus-noise voltage is given by cos Vtcos t ψt cos F v t t t θ t N t t φ t F F F n this case we will nee to fin the joint ensity of perform the appropriate integration to get the marginal ensity of specifically, we will fin an Further, since,,,,,,. (3-75) V t, t, ψ t an t V t. More θ an f V f V θ f (3-76) Vψθ Vψ θ,,, fv V fvψθ V. (3-77) We can raw on our work from the W0/W5 case to write V V V cos f V,, θ e U V rect Vψ. (3-78) t an t θ are, by efinition, inepenent, we can write 3 P, f f f e U rect θ θ P. (3-79) 0 M. C. Buge, Jr 4

15 f we substitute Equations (3-78) an (3-79) into Equation (3-76) we get V V V cos fvψθ V,,, e U V rect. (3-80) 3 P e U rect P From Equation (3-77) we can write where an 3 V V f V V e e, V U U V (3-8) P (3-8) P, V e rect rect V cos. (3-83) We recognize Equation (3-83) as the same ouble integral of Equation (3-70). hus, using the iscussions relate to Equation (3-73) we get an V, V, V 0 3 V V fv V e e 0 U U V P V 3 V V V e e 0 U V P 0 o complete the calculation of f V where 3 s s e 0 ss 0 V we must compute the integral (3-84). (3-85) (3-86) V. (3-87) t turns out that Maple was able to compute the integral as 4 e 4. (3-88) 0 M. C. Buge, Jr 5

16 With this f V V becomes V 4 V 4 fv V e e U V (3-89) P which, after manipulation can be written as V PV V P f V V e U V. (3-90) P P Now that we have complete the characterization of noise, signal an signal-plusnoise we are reay to attack the etection problem. 3.5 DEECON PROBABLY 3.5. ntrouction A functional block iagram of the etection process is illustrate in Figure 3-3. t consists of an amplitue etector an a threshol evice. he amplitue etector etermines the magnitue of the signal coming from the signal processor an the threshol evice is a binary ecision evice that outputs a etection eclaration if the signal magnitue is above some threshol, or a no-etection eclaration if the signal magnitue is below the threshol. Figure 3-3 Block Diagram of the Detector an hreshol Device 3.5. Amplitue Detector ypes he amplitue etector can be a square-law etector or a linear etector. Both variants are illustrate functionally in Figure 3-4 for the F implementation an the baseban implementation. n the F implementation, the etector consist, functionally, of a ioe followe by a low-pass filter. f the circuit is esigne such that it uses small voltage levels, the ioe will be operating in its low signal region an will result in a square-law etector. f the circuit is esigne such that it uses large voltage levels the ioe will be operating in its large signal region an will result in a linear etector. For the baseban case, the igital harware (which we assume in the baseban signal processing case) will actually form the square of the magnitue of the complex signal out of the signal processor by squaring the real an imaginary components of the 0 M. C. Buge, Jr 6

17 signal processor output an then aing them. he result of this operation will be a square-law etector. n some instances the etector also performs a square root to form the magnitue. Figure 3-4 F an Baseban Detectors Linear an quare Law n either the F or baseban representation the output of the square-law etector will be t V t when N when only noise is present at the signal processor output an signal-plus-noise is present at the signal processor output. For the linear etector the output will be t V t N when only noise is present at the signal processor output an when signal-plus-noise is present at the signal processor output Detection Logic ince both N t an t V are ranom processes we must use concepts from ranom processes theory to characterize the performance of the etection logic. n particular, we will use probabilities to characterize the performance of the etection logic. ince we have two signal conitions (noise only an signal-plus-noise) an two outcomes from the threshol check we have four possible events to consier:. signal-plus-noise threshol etection. signal-plus-noise < threshol misse etection 3. noise threshol lse alarm 4. noise < threshol no lse alarm 0 M. C. Buge, Jr 7

18 Of the above, the two esire events are an 4. hat is, we want to etect targets when they are present an we on t want to etect noise when targets are not present. ince events an are relate an events 3 an 4 are relate we only fin probabilities associate with events an 3. We term the probability of the first event occurring the etection probability an the probability of the thir even occurring the lse alarm probability. n equation form an where P - etection probability P V target present (3-9) P - lse alarm probability P N target not present. (3-9) t V V is the signal-plus-noise voltage evaluate at a specific time an t tt tt N N is the noise voltage evaluate at a specific time. he above efinition carries some subtle implications. First, when one fins etection probability it is tacitly assume that the target return is present at the time the output of the threshol evice is checke. Likewise, when one fins lse alarm probability it is tacitly assume that the target return is not present at the time the output of the threshol evice is checke. n practical applications it is more appropriate to say: At the time the output of the threshol evice is checke the probability that there will be a threshol crossing is equal to P if the signal contains a target signal an P if the signal oes not contain a target signal. n typical applications the output of the threshol evice will be checke at times separate by a pulse with an will result in many checks per PR. t will be note that the above probabilities are conitional probabilities. n normal practice we on t explicitly use the conitioning an write an P P V (3-93) P P N (3-94) an recognize that we shoul use signal-plus-noise when we assume the target is present an noise only when we assume that the target is not present. he above assumes that the etector preceing the threshol evice is a linear etector. f the etector is a square law etector the appropriate equations woul be an P P V (3-95) P P N. (3-96) 0 M. C. Buge, Jr 8

19 3.5.4 Calculation of P an P From probability theory we can write an P f v v V or P fv v v (3-97) P f n n N or n the above is the threshol voltage level an normalize power. P fn n n (3-98) is the threshol expresse as o avoi having to use two sets of P an P equations we will igress to show that we can compute them using either of the integrals of Equations (3-97) an (3-98). f we write t can be shown 4 that if x y an y 0 then fx x xfy x. (3-99) P fv v v (3-00) we can use Equation (3-99) to write P fv v v vf v v. (3-0) V f we make the change of variables x v we can write P f v v f x x V V. (3-0) imilar results apply to P an inicate that one can use either form to compute etection an lse alarm probability. f we examine the equations for P an P we note that both are integrals over the same limits. his integration is illustrate graphically in Figure 3-5. t will be note that P an P are areas uner their respective ensity functions to the right of the threshol value. hus, increasing the threshol ecreases the probabilities an ecreasing the threshol increases the probabilities. his is not exactly what we want. eally, we want to select the threshol so that we have P 0 an P. However this is not 4 Papoulis, A. Probability, Ranom Variables, an tochastic Processes McGraw-Hill 0 M. C. Buge, Jr 9

20 possible an we therefore usually choose the threshol as some sort of traeoff between P an P. n ct, what we actually o is choose the threshol to achieve a certain P an fin other means of increasing P. f we refer to Equation (3-) the only parameter that affects n fn is the noise power,. While we have some control over this via noise figure an effective noise banwith, executing this control can be very expensive. On the other han, fv v is epenent upon both P an. hus, this gives us some egree of control. n ct, what we usually try to o is affect both fn n an fv v by increasing P an ecreasing. he net result of this is that we try to maximize NR. Figure 3-5 Probability Density Functions for Noise an ignal-plus-noise False Alarm Probability f we use Equation (3-) in Equation (3-98) we get n n P fn nn e n e. (3-03) 0 M. C. Buge, Jr 0

21 n this equation we efine NR (3-04) as the threshol-to-noise ratio. As inicate earlier, we usually select a esire from this erive the require NR as ln P P an, NR. (3-05) Detection Probability We can compute etection probability for the three target classes by substituting Equations (3-58), (3-74) an (3-90) into Equation (3-0). he results for W0/W5 targets is where P erf NR NR NR NR NR NR e NR NR 4 NR 4 NR 6NR. (3-06) P NR (3-07) is the signal-to-noise ratio that one woul compute from the raar range equation an erf x u (3-08) x e u is one form of the error function. 5 0 he etection probability equations for the W/W case an for the W3/W4 case are, respectively an NR NR P e (3-09) P NRNR NR e NR NR. (3-0) n Equations (3-09) an (3-0) NR is the signal-to-noise ratio compute from the raar range equation. 5 his Equation (3-06) shoul only be use for cases where NR is larger than NR. 0 M. C. Buge, Jr

22 3.5.5 P Behavior vs. arget ype Figure 3-6 contains plots of P versus NR for the three target types an 6 P 0, a typical value. t is interesting to note the P behavior for the three target types. n general, the W0/W5 target provies the largest P for a given NR, the W/W target provies the lowest P an the W3/W4 is somewhere between the other two. With some thought this makes sense. For the W0/W5 target moel the only thing affecting a threshol crossing is the noise (since the RC of the target is constant). For the W/W the target RC can fluctuate consierably, thus both noise an RC fluctuation affects the threshol crossing. he stanar assumption for the W3/W4 moel is that it consists of a preominant (presumably constant RC) scatterer an several smaller scatterers. hus, the threshol crossing for the W3/W4 target is affecte somewhat by RC fluctuation, but not to the extent of the W/W target. 6 t is interesting to note that the NR require for P 0.5, with P 0, on a W/W target is about 3 B. his same NR gives a P 0.9 on a W0/W5 target. o obtain a P 0.9 on a W/W target requires a NR of about B. hese numbers are the origin of the 3 B an 0 B NR numbers we use in our raar range equation stuies. 0 M. C. Buge, Jr

23 Figure P vs. NR for hree arget ypes an P DEERMNAON OF FALE ALARM PROBABLY One of the parameters in the etection probability equations is threshol-to-noise ratio, NR. As inicate in Equation (3-05), NR ln P, where P is the lse alarm probability. False alarm probability is set by system requirements. n a raar, lse alarms result in waste raar resources (energy, timeline an harware) in that every time a lse alarm occurs the raar must expen resources etermining that it i, in ct, occur. ai another way, every time the output of the amplitue etector excees the threshol,, a etection is recore. he raar ata processor oes not know, a priori, whether the etection is a target etection or the result 0 M. C. Buge, Jr 3

24 of noise (i.e. a lse alarm). herefore, the raar must verify each etection. his usually requires transmission of another pulse an another threshol check (an expeniture of time an energy). Further, until the etection is verifie, it must be carrie in the computer as a vali target etection (an expeniture of harware). n orer to minimize waste raar resources we wish to minimize the probability of a lse alarm. ai another way, we want to minimize P. However, we can t set P to an arbitrarily small value because this will increase NR an reuce etection probability, P (see Equations (3-06), (3-09) an (3-0)). As a result we set P to provie an acceptable number of lse alarms within a given time perio. his last statement provies the criterion normally use to compute P. pecifically, one states that P is chosen to provie an average of one lse alarm within a time perio that is terme the lse alarm time,. is usually set by some criterion that is riven by raar resource limitations. he classical metho of etermining P is base strictly on timing. his can be explaine with the help of Figure 3-7 which contains a plot of noise at the output of the amplitue etector. he horizontal line labele hreshol, represents the etection threshol voltage level. t will be note that the noise voltage is above the threshol for three time intervals of length t, t an t 3. Further, the spacings between threshol crossings are an. ince a threshol crossing constitutes a lse alarm one can say that over the interval lse alarms occur for a perio of t. Likewise, over the interval lse alarms occur for a perio of t, an so forth. f we were to average all of the t k we woul have the average time that the noise is above the threshol, t k. Likewise, if we were to average all of the k we woul have the average time between lse alarms; i.e. the lse alarm time, to, i.e.. o get the lse alarm probability we woul take the ratio of t k P t k. (3-) Figure 3-7 llustration of False Alarm ime 0 M. C. Buge, Jr 4

25 While is reasonably easy to specify, the specification of t k is not obvious. he stanar assumption is to set t k to the range resolution expresse as time, an unmoulate pulse, R is the pulse with. For a moulate pulse, R is the reciprocal of the moulation banwith. R. For t has been the author s experience that the above metho of etermining P not very accurate. While it woul be possible to place the requisite number of caveats on Equation (3-) to make it accurate, with moern raars this is not necessary. he previously escribe metho of etermining P was base on the assumption that etections were recore via harware operating on a continuous-time signal. n moern raars, etection is base on examining signals that have been converte to the iscrete-time omain by sampling or by an analog-to-igital converter. his makes etermination of P easier, an more intuitively appealing, in that one can eal with iscrete events. With moern raars one computes the number of lse alarm chances, N, within the esire lse alarm time,, an computes the probability of lse alarm from P. (3-) N o compute N one nees to know certain things about the operation of the raar. We will outline some thoughts along this line. n a typical raar, the return signal from each pulse is sample with a perio equal to the range resolution, R, of the pulse. As inicate above, this woul be equal to the pulse with for an unmoulate pulse an the reciprocal of the moulation banwith for a moulate pulse. hese range samples are usually taken over the instrumente range,. n a search raar, might be only slightly less than the PR,. However, for a track raar,, may be significantly less than. With the above, we can compute the number of range samples per PR as N R. (3-3) R Each of the range samples provies a chance that a lse alarm will occur. n a time perio of the raar will transmit N pulse (3-4) pulses. hus, the number of range samples (an thus chances for lse alarm) that one has over the time perio of is N N N. (3-5) R pulse 0 M. C. Buge, Jr 5

26 n some raars, the signal processor consists of several ( N Dop ), parallel Doppler channels. his means that it will also contain N Dop amplitue etectors. Each amplitue etector will generate N R range samples per PR. hus, in this case, the total number of range samples in the time perio woul be N NRN pulsen Dop. (3-6) n either case, the lse alarm probability woul be give by Equation (3-) Example o illustrate the above, we consier a simple example. We have a search raar that has a PR of 400 s. t uses a 50 s pulse with linear frequency moulation (LFM) where the LFM banwith is MHz. With this we get R s. We assume that the raar starts its range samples one pulse-with after the transmit pulse an stops taking range samples one pulse-with before the succeeing transmit pulse. From this we get 300 s. he signal processor is not a multi-channel Doppler processor. he raar has a search scan time of s an we esire no more than one lse alarm every two scans. PR we get From From the last sentence above we get s. f we combine this with the N pulse (3-7) an R we get N R his results in an N P 300 s 300. (3-8) s R 6 NRN pulse (3-9) N (3-0) 0 M. C. Buge, Jr 6