Compound form of the K distribution

The clutter intensity has an exponential distribution of mean power \( x \)
\[ P(z \vert x ) = \frac {1} {x} \exp(-\frac{z}{x}) \] where x itself fluctuates with a Gamma distribution
\[ P(x) = \frac{b^\nu}{\Gamma(\nu)} x^{\nu - 1}\exp(-bx) \]
Here, \( \nu \) is the shape parameter, and \( \langle x \rangle = \frac{\nu}{b} \) is the sea clutter mean power.

Shape parameter dependence on sensor and environmental parameters

Empirical models for the K-distribution shape parameter have been developed using data collected using an I-band radar which provide an initial guide to the range of values to be expected.
\[ \log_{10} ( \nu ) = \frac{2}{3}\log_{10}(\phi^{0}_{gr}) + \frac{5}{8} \log_{10}(A_{c})-k_{pol}-\frac{\cos(2 \theta_{SW})}{3} \]

Probability of false alarm

Include the thermal noise in the mean power
\[ P(z \vert x ) = \frac {1} {x + p_n} \exp(-\frac{z}{x+ p_n}) \] If we assume that speckle decorrelates then the sum of N pulses \( \mu = \sum\limits_{i=1}^{N} z_{i} \) has the PDF
\[ P(\mu \vert x) = \frac{\mu^{N - 1}}{(x+p_{n})^{N}(N-1)!} \exp (\frac{-\mu}{x+ p_{n}}) \] The probability of false alarm, given \( x \), for a threshold \( Y \) is
\[ P_{FA} (Y \vert x) = \int_Y^\infty \frac{\mu^{N - 1}}{(x+p_{n})^{N}(N-1)!} \exp (\frac{-\mu}{x+ p_{n}}) d \mu = \\ \frac{1} {(N - 1)!} \Gamma(N, \frac{Y}{x + p_n}) \] The overall probability of false alarm after pulse-to-pulse integration is therefore
\[ P_{FA} (Y) = \frac{1} {(N - 1)!} \int_0^\infty \Gamma(N, \frac{Y}{x + p_n}) P(x) dx \] where \[ P(x) = \frac{b^\nu}{\Gamma(\nu)} x^{\nu - 1}\exp(-bx) \]
This is evaluated using numerical integration to produce radar detection performance predictions.

Probability of detection for a single pulse

The target plus noise and speckle is described by the Rice distribution
\[ P( z \vert A,x) = \frac{1}{x + p_n} \exp(- \frac{z + A^2}{x+p_n}) I_0( \frac {2A \sqrt{z}}{x + p_n}) \] Integrate for probability of detection given local mean
\[ P_D(Y \vert x) = \int_Y^\infty \frac{1}{x + p_n} \exp(-\frac{z + A^2}{x + p_n}) I_0(\frac{2A\sqrt{z}}{x+p_n}) dz \] Integrate result over the distribution of the local mean
\[ P_D(Y) = \int_0^\infty P_D(Y \vert x) P(x) dx \]

Probability of detection, pulse to pulse integration

The PDF of target plus noise and speckle has a multi-look Rice distribution given by
\[ P(\mu \vert s, N) = (\frac{\mu}{s})^{\frac{N-1}{2}} \exp(-(\mu + s)) I_{N-1}(2 \sqrt{\mu s}) \] where The probability of detection given the local mean is obtained by integration. The result is integrated over the distribution of the local mean
\[ P_D(Y) = \int_0^\infty P_D(Y \vert x) P(x) dx \]

Target fluctuations

Target plus noise and speckle, multi-look Rice distribution, only depends upon the sum of target powers, not the individual values. Therefore a Gamma distribution model for target fluctuations
\[ P(s \vert S, k) = \frac{s^{k-1}}{\Gamma(k)} (\frac{k}{S})^k e^{-\frac{ks}{S}} \] may be used to model correlated and uncorrelated targets including all Swerling models

Speckle correlation

There may be pulse to pulse correlation of speckle due to fixed frequency operation or limited frequency agility. If the noise is very low a reasonable approximation may be obtained by reducing the number of integrated pulses. However this is often not the case and the calculations become more complicated. < br> Put the clutter speckle (L independent pulses) with the target to form a Rice distribution \[ P(\beta \vert s, \alpha, L) = \frac{1}{\alpha} (\frac{\beta}{s})^{\frac{L-1}{2}} \exp(-\frac{s+\beta}{\alpha})I_{L-1}(\frac{2 \sqrt{s \beta}}{\alpha}) \] where Then treat this as the ‘target’ distribution in a new Rice distribution with the N noise pulses \[ P( \mu \vert \beta, N) = (\frac{\mu}{\beta}) ^{\frac{N-1}{2}} e^{-(\mu + \beta)} I_{N-1} (2 \sqrt{\mu \beta}) \] The marginal pdf is then given by \[ P( \mu \vert N, \alpha) = \int \int P(\mu \vert \beta; N) P(\beta \vert s; \alpha, L)P(s \vert S;k) d\beta ds \] This is integrated above the threshold to give the detection probability, given the local clutter mean \[ P_D(Y \vert N, \alpha) = \int \limits_{Y} ^{\infty} P(\mu \vert N, \alpha) d \mu \] Which is then integrated over the Gamma distribution of local mean This may seem an impossible numerical task, but using Shnidman's approach it simplifies down to what is effectively a single summation. \[ P_D(Y \vert x) = \int \limits_{0} ^{\infty} d\beta \int \limits_{0}^{\infty} ds \int \limits_{Y}^{\infty} d \mu P(\mu \vert \beta; N) P(\beta \vert s; \alpha, L) P(s \vert S; k) \\ = \sum \limits_{0} ^ { N -1} e^ {-Y} \frac{Y^m}{m!} + \sum \limits_{m = N} ^ { \infty} e^{-Y} \frac{Y ^ m}{m!} (1 - \sum \limits_{0}^{m - N} \langle e^{-\beta} \frac{\beta^i}{i!} \rangle) \] \[ \langle e^{-\beta} \frac{\beta^i}{i!} \rangle = \int e^{-\beta} \frac{\beta^i}{i!} P(\beta \vert s; \alpha, L)P(s \vert S,k) d\beta ds \] \[ \langle e^{-\beta} \frac{\beta^n}{n!} \rangle = \frac {(L-1+n)!}{n! (L-1)!} \times \\ \frac{\alpha^n}{(\alpha+1)^{L+n}} (\frac{k(\alpha + 1)}{k(\alpha + 1) + S})^k \Psi(n, L, k, \frac{S}{\alpha (k (\alpha+1)+S)}) \] The \( \Psi \) function can be evaluated efficiently from the recurrence relation
\[ (L+n) \Psi(n+1, L, k, z) =\\ (2n + L + z(n+k))\Psi(n, L, k, z) - n(z+1) \Psi(n - 1, L, k, z) \] where \( \Psi(0, L, k, z) = 1 \) and \( \Psi(1, L, k, z) = 1 + \frac{kz}{L} \)

Detection calculations conclusion

A method has been developed to evaluate detection performance in compound K distributed clutter with correlated speckle and noise The method incorporates Gamma distributed target fluctuations, which can be from pulse-to-pulse, scan-to-scan, or both. The method is sufficiently efficient that it can be used even for the special cases where other methods exist. The only parameter combination that has been found to cause difficulties is a high clutter to noise ratio. (This is currently overcome by an approximation of no noise).