The present invention relates to a constant false alarm rate (CFAR) processor for use in a radar receiver and, more particularly, to an adaptive CFAR processor which can maintain a constant false alarm rate against any probability density distribution function of clutter signal and offers a remarkable increase in target detectability.

Among known techniques for removing undesirable signals other than target signals in a radar system, a moving target indicator (MTI) has found extensive use which is designed to cancel returns from hills, buildings and like objects fixed in location. The MTI canceller, however, fails to effectively operate against those types of clutter having velocity components, e. g. sea clutter, weather clutter and angel echoes.

Various propositions have heretofore been made for suppressing clutters other than the returns from the fixed objects. Typical of such propositions is the LOG/CFAR system (see J. Croney, "Clutter on Radar Display", Wireless Engineering, pp. 83-96, April 1956). The LOG/CFAR system has as its basis the assumption that the amplitudes of a clutter signal (referred to simply as clutter hereinafter) has a probability density function which is a Rayleigh function. The system suppresses the clutter and reduces it to about the receiver noise. level by sequential steps of logarithmically converting the clutter by a logarithmic converter which has a predetermined adequate characteristic constant, averaging the output of the logarithmic converter, subtracting the resulting average from a signal of interest, and subjecting the difference to an antilogarithmic conversion.

However, experiments revealed that clutter residues still exist despite the LOG/CFAR processing. This originates from the fact that not all the clutters have probability density functions which conform to the Rayleigh function to which the LOG/CFAR technique is applicable but rather, clutters generally have probability density functions which are expressed by the Weibull function except for limited cases. This was reported by D. C. Shleher in his paper entitled "Radar Detection in Weibull Clutter", IEEE Trans. , AES-12, No. 6, pp. 736-743, 1976. Apart from the Weibull function, clutters having the Log-Normal function or Rice function cannot be coped with by the LOG/CFAR system.

For the CFAR processing of clutter having the Weibull function (referred.to as Weibull clutter hereinafter),- a technique using equations disclosed in U. S. Patent No. 4,3 18, 101 to convert the Weibull clutter into the Rayleigh clutter may be advantageously practiced since an ordinary LOG/CFAR circuit suffices for the purpose. The subject of this technique is a clutter X of the Weibull function which is expressed as Then, using a scale parameter σ and a shape parameter k obtained from the clutter, a variable conversion of the clutter X is executed asThe resultant signal has a probability density function which is the Rayleigh function and given by(a function provided by substituting 2 for the k of equation (1)).

Such a technique is not fully acceptable because it is ineffective against clutters having the othertypes of functions (such as Log -Normal function and Rice function), that is, it finds application only to Weibull clutter.

V. G. Hansen proposed a technique for suppressing clutters having any type of function (any type of probability density function) in his paper entitled "Constant False Alarm Rate Processing in Search Radars" presented in International Conference on RADAR - PRESENT AND FUTURE held on October 23-25, 1973. Whatever the cumulative distribution function q_w (X) of clutter may be, the technique performs a variable transformation on the clutter as so that the probability density function of the resultant signal is transformed into an exponential function e-^Z.

It is true that the Hansen's technique can convert clutter into a signal having a predetermined exponential function regardless of the type of the probability density function of the clutter. This technique still involves some problems, however. The above- mentioned exponential function is a function derived from the Weibull function of equation (1) in which the k is replaced by 1 and, therefore, it is included in the Weibull function. Considering the relationship between the threshold value and the false alarm rate in CFAR processing Weibull clutter, it is known that the false alarm. rate for a common threshold value decreases with the increase in the shape parameter k. It is also known that the variance Var (Z) obtainable with a LOG/CFAR circuit decreases with the increase in the shape parameter k. For details, see the paper " Suppression of Weibull-Distributed Clutters Using a Cell-Averaging LOG/CFAR Receiver" by M. Sekine et al. , IEEE Trans. , AES-14, No. 5, pp. 823-826, September 1978, particularly Figure 3 on p. 825, and the aforementioned paper "Radar Detection in Weibull Clutter" by D. C. Schleher, Figures 2-6, and a paper "Suppression of Radar Clutter" by Sekine et al. , the Institute of Electronics And Communication Engineers of Japan, Trans. IECE, Vol. 62-B, No. 1, 1979, pp. 45-49, particularly Figures 3 and 4. Since the exponential function is the function given by substituting 1 for the shape parameter k in the Weibull function, the false alarm rate grows larger than in the case of Rayleigh function wherein k = 2. Moreover, a smaller shape parameter k needs a larger threshold value which would even cancel target signals of relatively low levels and depart from the function expected for a radar.

As discussed above, of the prior art systems, one relying on the variable conversion of equation (2) cannot process clutters other than Weibull clutter for a constant false alarm rate. The Hansen's system using the variable conversion of equation (4) can perform the CFAR processing regardless of the type of the probability density function of clutter. Still, this holds true only in the theoretical aspect; in practice, due to the shape parameter which is fixed at a value of 1, not only a small false alarm rate is unavailable but a large threshold value is necessary for the CFAR processing. Such a threshold value would cancel target signals together with clutters, resulting in a poor target detectability.

An object of the present invention is to provide an adaptive CFAR processor which can realize a constant false alarm rate regardless of the type of their probability density function or cumulative distribution function, only if the function is known.

Another object of the present invention is to provide an adaptive CFAR processor which can realize a constant false alarm rate easily with an ordinary.LOG/CFAR circuit regardless of the type of their probability density function or cumulative distribution function, only if the function is known.

Still another object of the present invention is to provide an adaptive CFAR processor which permits any desired threshold value to be selected for target detection in a LOG/CFAR circuit, therefore minimizes the false alarm rate and surely detects even target signals of relatively low levels.

Yet another object of the present invention is to provide an adaptive CFAR processor of the type described which simplifies the operating procedure or the construction.

In one embodiment of the present invention, a radar signal including clutter whose amplitude X is defined by any known cumulative distribution function F(X) or probability density function f (X) is subjected to a variable conversion expressed aswhere cr and k are arbitrarily determined constants. The resulting signal Y has σ and k as its scale parameter and shape parameter, respectively. The signal is processed by an ordinary CFAR circuit (e. g. LOG/CFAR circuit). Selecting a relatively large value for the constant k, which is open to choice as well as the constantσ , not only lowers the false alarm rate to a significant degree but markedly improves the ability for detecting a target signal buried in clutters.

Embodiments of the invention will now be described with reference to the accompanying drawings in which:-

• Figure 1 is a block diagram showing a basic construction of the adaptive constant false alarm processor of the present invention;
• Figure 2 is a detailed block diagram of a variable converting section 20 for Weibull clutter included in the construction of Figure 1;
• Figure 3 is a detailed block diagram of a variable convering section 20 for Log-Normal clutter;
• Figure 4 is a block diagram showing an example of a CFAR section 30 of Figure 1;
• Figure 5 is a graph representing a relationship between the shape parameter and the variance provided by the LOG/CFAR system; and
• Figure 6 is a graph demonstrating a relationship between the shape parameter and the false alarm rate.

Description will be first made on the principle of the present invention and then on a practical construction and arrangement which is based on said principle.

Suppose that X denotes a variable representing a clutter amplitude and that the cumulative distribution function F_x(X) of the variable X is known. The present invention proposes the following equation (5) or (6) in order to convert the variable X into a new variable Y which has a probability density function f_y(Y) with a shape parameter k and a scale parameter or which is the Weibull density function.

That the equation (6) can convert a signal X having any type of cumulative distribution function F_x(X) into an alternative variable Y whose probability density function is the Weibull function will be testified as follows.

Where the probability density function f_x(X) of the variable X is known and a relation Y = g (X) holds between the variables X and Y, the probability density function f_y(Y) of Y is generally given by

As well known, a probability density function f_x(X) and a cumulative distribution function F_x(X) have an interrelation which is expressed as

From equation (7), we obtain

From equation (6), we also obtain

Meanwhile, using equation (5), we have

Putting the relations represented by equations (8), (11) and (12) into equation (10) giveswhere σ > 0, k > 0 and Y > 0.

Based on probability density function characteristics, f_x(X) > 0 holds. Therefore,Putting equation (13) into equation (9), we have a probability density function fy(Y) of Y expressed aswhich is the Weibull density function having the shape parameter k and scale parameter σ.

What should be paid attention to in the above demonstration is the F (X) which has been treated as an entirely arbitrary function. Thus, the function F(X) may have any type of cumulative distribution characteristic. Another important feature is that the shape parameter k and scale parameter σ can also be of any desired value. Naturally, selecting 2 for the shape parameter k and leaving the scale parameter σ arbitrarily selectable means preparing a variable Y which has an arbitrary scale parameter σ and obeys the Rayleigh distribution.

Previously mentioned U.S. Patent No. 4,318,101 proposes the following equation (16) to convert, for example, a Weibull distribution signal X having a shape parameter η and a scale parameter ν into a signal Y having the Rayleigh distribution.

The signal Y in accordance with this proposition has a probability density function fy(Y) which is given byand has the Rayleigh distribution.

The use of the variable conversion equation (6) of the present invention givesThis and 2 selected for the shape parameter k resultsUsing equation (15), the probability density function f_y(Y) of the variable Y is expressed aswhich obeys the Rayleigh distribution.

Thus, whereas equation (16) according to the prior art employs the scale parameter ν of X as a multiplier, equation (16) according to the present invention has a multiplier which is the freely selectable scale parameter σ. Where σ = 1 is selected, equation (19) giveswhich also works as an important variable conversion equation and, moreover, it cuts down the number of multiplying functions.

Figure 1 is a block diagram showing a basic construction of the adaptive CFAR processor proposed by the present invention. A return signal coming in through an antenna involves various clutters as well as a target signal as already discussed. The return signal reaching a receiver is mixed with a local oscillation signal by a mixer to be converted into an intermediate frequency (IF) signal. After amplification, the IF signal - is coupled to such a signal processing system as one shown in Figure 1. To facilitate digital processing in the following stages, an analog-to-digital (A/D) converter 10 transforms the input IF signal into a digital signal having an amplitude X (referred to as radar input signal hereinafter) in response to clock pulses CLK (not shown) of a predetermined frequency. A variable converting section 20 is supplied with a scale parameter S from a scale parameter σ setting unit 40 and a shape parameter k from a shape parameter k setting unit 50. Using these parameters σ and k and a known cumulative distribution function F (X) of the clutter contained in the radar signal, the variable converting section 20 processes the digital output of the A/D converter 10 according towhich has been indicated by equation (6). Then, the variable converting section 20 produces a converted signal Y. Should the cumulative distribution function F(X) be unknown and the probability density function known, equation (8) will be executed to obtain the cumulative distribution function F(X). Set and stored in the σ setting unit 40 and k setting unit 50 are the values which were arbitrarily selected. The signal Y is the Weibull function whose probability density function has a scale parameter Q; and a shape parameter k as described, so that it is fed to a CFAR section 30 for a constant false alarm rate processing which may employ a LOG/CFAR circuit, for example. If the k is 2, the constant false alarm rate processing can be performed with an ordinary LOG/CFAR circuit. If the k is not 2, such a processing is still attainable utilizing a technique which determines a threshold value for target detection in accordance with the k (see U. S. Patent No. 4, 242, 682 "Moving Target Indication Radar"). A probability density function of the clutter contained in the radar signal may be specified by preparing a circuitry preloaded with presumptive functions (Rayleigh, Weibull, Log-Normal, Rice) and selecting one of them which cancels the clutter more than the others. Such an implement facilitates cancellation of clutters which do not obey the Weibull distribution, as has.heretofore been impossible, while remarkably reducing the danger of cancelling a target signal together with the clutters.

Details of the variable converting section 20 for copying with Weibull clutter is illustrated in Figure 2.

As indicated in equation (1), the signal amplitude V of Weibull clutter has a probability density function f (V) which is given bywhere k' and σ' are parameters corresponding to k and σ, respectively.

The function f (V) therefore will give a cumulative distribution function F(V) when equation (8) is used. Thus, the variable conversion represented by equation (6) may be carried out using the resultant function F(V). However, to enlarge the dynamic range and permit the cumulative distribution function to be obtained with ease through a simple hardware design, the variable conversion will herein be performed using a signal X which is given by the logarithmic conversion of the signal V

It will be seen, as obvious from "Suppression of Weibull-Distributed Clutters Using a Cell-Averaging LOG/CFAR Receiver", IEEE, Vol. AES-14, that the averages <X> and <X²> of X and X², respectively, are expressed aswhere γ is the Euler's constant and approximately equals 0. 5772. The logarithmic amplifier is assumed to have a characteristic parameter 1.

As follows from the above,Accordingly, k' is given byPutting k' into equation (24) givesThen, σ' is obtained by

Meanwhile, the probability density function f (X) of the signal X is given byTherefore, the cumulative distribution function F(X) is expressed as

Substituting y for e^x, we haveAccordingly,Since equation (33) is the same as the cumulative distribution function for In y of the Weibull probability density function, we obtainRe-substituting In y by x, the cumulative distribution function F(X) of X is expressed as

Thus,

Turning back to Figure 2, the Weibull clutter signal V is processed by a known logarithmic converter 211 into a signal X and then coupled to an averaging circuit 212. The averaging circuit 212 comprises a known shift register 212a having a predetermined number of shift stages (in this embodiment, (2N + 1) stages where N is a natural number), an adder 212b and a divider 212c. If desired, the shift register 212a may be replaced by a delay tapped line. The digital output of the A/D converter 10 shown in Figure 1 is sequentially coupled to and shifted in the shift register 212a at the predetermined sampling intervals at the A/D converter 10. The adder 212b sums the signals stored in the respective stages of the shift register 212a except for the central (N + 1)th stage. The divider 212c divides a summation output of the adder 212b by 2N to produce an average signal <X>. It will be understood that, if the signal in the central stage of the shift register 212a ( signal of interest) where not excepted in averaging the radar signal, the signal of interest would also be suppressed in the clutter suppressive processing to degrade the S/N ratio in case where a target signal is superposed on clutter. This is also true for the various embodiments which will follow.

The output of the logarithmic converter 211 is also fed to a square calculator 215 and therefrom to an averaging circuit 216. An output signal <x²> of the averaging circuit 216 indicating an average value is coupled to a subtractor 217. The averaging circuit 216, like the averaging circuit 212, comprises a shift register 216a, an adder 216b and a divider 216c. Again, the input to the adder 216b lacks the radar signal in the central stage of the shift register 216a. The subtractor 217 provides a difference between a square <x²> of an average < x) prepared by a square calculator 214 and the square average <x²>. The difference is supplied to a k' extractor 218. The k' extractor 218 calculates k' from the output <x²> - <x>² of the subtractor 217 using equation (27) and produces an output indicative of the k'. A divider 219 is adapted to produce a signal indicating a ratio of Euler's constant γ to the k' signal, I /k'.

A subtractor 213 subtracts the γ/k' from the average < x> to produce an output <x> - as is shown in equation (28). This substractor output is applied to an exponential converter 221 and thereby exponentially converted into a signal which indicates the scale parameter σ' shown in equation (29). The signal x of interest in the central stage of the shift register 212a is coupled to a sign converter 220 which inverts the input signal into a -x signal. The -x signal from the sign converter 220 is fed to a divider 222 which then produces a signal indicating - . A power multiplier 223 raises the - , signal to k'-th power which is the output of the k' extractor 218. The output ( - x σ')^k' of the power multiplier 223 is supplied to an exponential converter 224 which then produces a signal indicating exp ( - )^k'. Since exp ( - )^k' is [1.0 - F(X)] as will be apparent from equation (36), the _exp ( - )^k' signal is fed to a logarithmic amplifier 225 and thereby processed into a signal indicating In 1. 0 - F(X) The output of the logarithmic amplifier 225 is applied to a power multiplier 226 which raises the input to 1/k-th power using an output signal of a k setting unit 50 indicative of an arbitrarily selected value of k. The output of the power multiplier 226 is multiplied at a multiplier 227 by a signal output of a σ setting unit 40 indicative of an arbitrarily selected value of σ As a result, the multiplier 227 produces a signal which has undergone the conversion indicated by equation (6). This signal is a Weibull signal having arbitrarily selected parameters σ; and k and, thus, it is processed by the CFAR section 30 for a constant false alarm rate.

Referring to Figure 3, there is shown a detailed block diagram of the variable converting section 20 for Log-Normal clutter.

As well known in the art, the signal amplitude V of clutter is generally understood to "obey the Log-Normal distribution" when its probability density function is expressed as where β and M are the parameters which specify the distribution. While in the present invention it suffices to obtain the cumulative distribution function F(V) from the probability density function f(V) and then perform the variable conversion of equation (6), the signal V will be logarithmically converted according to an equation.This will enlarge the available dynamic range and simplify the construction for the variable conversion, as has been discussed in conjunction with the embodiment of Figure 2.

Thus, the probability density function f (X) of the signal X is given byFrom equation (39), we have a cumulative distribution function F(X) expressed aswhere erf is an error function, and

As will be readily understood, equation (39) represents a normal distribution whose standard deviation is P and expected value (average value) is lnM. Therefore, lnM and β are given.From equations (42) and (43), we can obtain ln M and β and, therefore, F(X) of equation (40).

The procedure described above i:s practicable with a hardware construction shown in Figure 3. Referring to Figure 3, the digital signal (radar signal) from the A/D converter 10 is coupled to a known logarithmic converter 231 which may be constituted by a read only memory (ROM). The resulting output X of the logarithmic converter 231 is given by equation (38). The signal X is supplied to an averaging circuit 232 in which, as in the embodiment of Figure 2, the average (expected value lnM) < x> of 2N data of (2N + 1) digital data excepting the central (N + 1)th data is obtained. The averaging circuit 232 comprises a shift register (or a tapped delay line) 232a having (2N + 1) shift stages, for example, an adder 232b for summing signals frcm the respective stages of the shift register 232a except the central stage, and a divider 232c for dividing an output of the adder 232b by 2N. A sub tractor 233 subtracts the signal indicating the average 1nM from the signal output X (signal data of interest) of the (N + 1)th stage of the shift register 232a, thereby producing a signal (X - 1nM). Meanwhile, the output X of the logarithmic converter 231 is also fed to a square calculator 235 and therefrom to an averaging circuit 236. The averaging circuit 236, like the averaging circuit 232, comprises a shift register 236a, an adder 236b and a divider 236c. The averaging circuit 236 provides a square average <x²> of the data in the respective stages of the shift register 236a except one in the (N + 1)th stage. Supplied with the square average <x²) , a subtractor 237 subtracts an output of a square calculator 234 from the square average < x²>. The difference from the subtractor 237 is coupled to a square root calculator 238 whose output (indicating β ) is in turn coupled to a divider 239. The divider 239 divides the output (x - 1nM) of the subtractor 233 by the output β of the square root calculator 238, the resulting being applied to an error function (erf) generator 240. The error function generator 240 is in the form of a read only memory (ROM) which calculates in advance erf with as a variable and has these values as its addresses and stored values. Thus, the error function generator 240 will produce a signal indicating erf which corresponds to an input . An adder 241 adds a signal indicating a numerical value 0.5 to an output erf of the erf generator 240, thereby providing a cumulative distribution function F(X) indicated by equation (40). A subtractor 242 provides a difference between the output F(X) of the adder 241 and asignal indicative of a numerical value 1.0. The difference 1.0-F(X) is coupled from the subtractor 242 to a logarithmic converter 225 which then produces a signal indicating In 1.0 - F(X)}. A power multiplier 226 calculates [In {1.0 - F(X)}] 1/k using the shape parameter supplied from the k setting unit 50 of Figure 1. Finally a multiplier 227 multiplies the output of the power multiplier 226 by the signal output of the σ setting unit 40 which indicates the scale parameter σ. The output σ[ln{1.0 - F(X) 1) 1/k of the power multiplier 226 is supplied to the CFAR section 30. As already stated, the signal inputted to the CFAR section 30 has undergone a variable conversion of equation (6) and, accordingly, the probability density distribution function of the converted signal is the Weibull function having a scale parameter σ and a shape parameter k. It will be clear in Figure 3 that the same operation proceeds even if the adder 241 is omitted and the numerical value coupled to the subtractor 242 is 0.5 instead of 1. 0.

While the A/D converter 10 has been shown and described in each of the foregoing embodiments as being located just ahead of the the variable converting section 20, it will be apparent that the logarithmic converters 211, 231 in Figures 2 and 3 may be comprised of logarithmic amplifiers to locate the A/D converter 10 just past of them. Further, since the value in the variable converting section 20 is also open to choice, selecting 1 for said value σ will permit the multiplier 227 in Figure 2, or 3 to be omitted.

Referring to Figure 4, a LOG/CFAR circuit which is a typical example of the CFAR section 30 is illustrated. Before entering details of the LOG/CFAR circuit, the principles of a LOG/CFAR processing system will be briefly described. Let Y denote a signal which has been subjected to a variable conversion to have Weibull distribution with a specific shape parameter k and a scale parameter σ. By the LOG/CFAR processing, the signal Y is logarithmically convertedAn average <Z> of the Z is calculated and then a difference between Z and < Z>. The difference is defined asSince Wis the result of subtraction from clutter an average value of the clutter, it represents a fluctuation component of the clutter. Further, W is antilogarithmically converted according toWhere a, b, c and d are constants, if ad = 1. 0, a variance Var(N) of the N is given by Here, f denotes a gamma function and γ = 0.5772 (Euler's constant).

In equation (47), the variance Var (N) depends on the parameter k selected, but it will remain constant regardless of the distribution characteristics once the parameter k is determined. It follows that a constant false alarm rate is achievable if a threshold value higher than the variance Var(N) is selected.

It has been customary in LOG/CFAR processing to use the value of equation (47) which corresponds to the Rayleigh distribution where k = 2. In contrast, the threshold value in accordance with the present invention is determined to be optimum in accordance with the parameter k which can have any desired value. This offers a more desirable CFAR system featuring an improved target detectability.

Turning back to Figure 4, the Weibull clutter signal supplied from the variable converting section 20 is subjected to a logarithmic conversion by a logarithmic converter 311 the output of which is coupled to an averaging circuit. This averaging circuit is constructed in the same way as the averaging circuits shown in Figures 2 and 3 and has a shift register 312 furnished with (2N + 1) stages, an adder 313 and a divider 314. A subtractor 315 subtracts an output of the divider 314 from the data stored in the central (N + I)th stage of the shift register 312. An antilogarithmic converter 316 antilogarithmically converts the resulting difference from the subtractor 315. The thus converted signal of the clutter has a variance of a constant value, which is dictated by the shape parameter k as previously discussed. Hence, in a threshold setting unit 318 a threshold value is determined on the basis of the shape parameter k supplied from the k setting unit 50. A comparator 317 produces a target detection signal when the output of the antilogarithmic converter 316 is larger than the threshold level, that is, when a target is present.

A characteristic feature of the present invention resides, as has been repeatedly mentioned, in the selectivity of the shape parameter k and the threshold value which is determined in accordance with the value of the shape parameter k. This not only promotes CFAR processing for clutter regardless of the type of its probability density function but, since the threshold value can be lowered to match a situation, realizes sure detection of low level target signals.

As previously states, it is known that the variance decreases with the increase in shape parameter. An example based on "Suppression of Weibull Clutters Using a Cell-Averaging LOG/CFAR Receiver" will be described with reference to Figure 5. In Figure 5, the abscissa indicates the shape parameter k and the ordinate the value given by dividing the variance by the square of a characteristic constant c of an antilogarithmic converter. It will be seen from the graph that the larger the shape parameter k, the smaller the value Var(Z)/c^z becomes and, therefore, a smaller threshold value can be selected. Consequently, though the signal level of a true target may be relatively low, it can be safely detected if the parameter k has a relatively large value. In the graph for instance, a function (exponential function) when k = 1 in a Weibull function has Var(Z)/c² whose value is, though out of the graph and indistinct, about 10 considering the characteristic curve; the threshold value must be set to a relatively large value which corresponds to such value Var(Z)/c². Then, the probability of detection of a low level target signal at the comparator 317 would be significantly lowered. Since the case concerned has the Rayleigh function k = 2, the value of Var(Z)/c² is determined from the graph to be about 0. 5 which is smaller by one digit than the case of K = 2 and, thus, proportionally increases the target detectability. It will be clear, however, that a farther increase in the shape parameter k would set up an additional increase in the target detectability. If the shape parameter k which is open to choice in accordance with the invention is selected to be 8, the Var(Z)/c² will be about 0.02 which allows the threshold value to be far smaller compared to the case with a Rayleigh function, with the resultant remarkable increase in the target detectability.

The present invention with the freely selectable shape parameter k achieves a drastic improvement in the aspect of false alarm as well. This will be appreciated also from Figure 6 which graphically demonstrates a relationship between the false alarm rate P_fa obtained by the above procedure and the shape parameter k.

The false alarm rate P_fa is the probability with which clutter is erroneously determined as a target when a threshold value T is determined, and it is given bywhere Z denotes the clutter level and γ the Euler's constant (γ = 0. 5772). The value c is the characteristic constant of an antilogarithmic converter and is herein assumed to be 1 for simplicity. Then, equation (48) can be modified asThus, a threshold value T which makes P_fa 10^-6 when k = 2 is 7.527. Putting this value of T into equation (49), we obtainWhen therefore a predetermined value of shape parameter k is put into equation (50), there can be set up a relation which indicates the dependency of the fals'e alarm rate P_fa on the shape parameter k.

The relationship shown in Figure 6 was determined by the above procedure. Figure 6 shows that, as the shape parameter k increases, the false alarm rate decreases in a manner of exponential function. For instance, whereas the false alarm rate Pfa is 10^-6 when k = 2, it is 10-⁸ when k = 2. 24 and this constitutes a significant improvement over the prior art LOG/CFAR in which k = 2.