Introduction

Binary integration (BI), also known as M-of-N processing, is a technique for combining multiple threshold detection test outcomes to form a single, final detection test outcome. The goal is to achieve specified probabilities of detection ($P_D$) and false alarm ($P_{FA}$) with a lower single-sample signal-to-noise ratio (SNR) than would be required for a single threshold test based on a single measurement.

The paper develops a procedure for computing the binary integration gain ($G_{BI}$), which is the factor by which the required single-measurement SNR needed to achieve a given $P_D$ and $P_{FA}$ is decreased when using BI, compared to using a single measurement.

Procedure for Computing Binary Integration Gain

a. Given desired values of M, N, $P_D$, and $P_{FA}$: Find the reference SNR $\chi$ required to achieve $P_D$ and $P_{FA}$ using a single measurement and threshold test. This can be done using Shnidman's SNR equation.

b. Solve the following equation for the value of $P_{D1}$ (single-trial detection probability) required to achieve the specified $P_D$:

$$ \sum_{r=M}^N \binom{N}{r} P_{D1}^r (1-P_{D1})^{N-r} = P_D. $$

c. Repeat step (b) to find the value of $P_{FA1}$ (single-trial false alarm probability) required to achieve the specified $P_{FA}$. (replace $P_D$ with $P_{FA}$) d. Find the SNR $\chi_1$ required to achieve the single-trial probabilities $P_{FA1}$ and $P_{D1}$ on a single measurement and threshold test. e. The binary integration gain $G_{BI}$ is the ratio $\chi/\chi_1$.

MATLAB code for the complete procedure:

function G_BI_dB = binary_integration_gain(M, N, P_D, P_FA)
    % Find the reference SNR χ required to achieve P_D and P_FA using a single measurement and threshold test
    chi = shnidman(P_D, P_FA, 1, 0);

    % Solve for the value of P_D1 required to achieve the specified P_D
    fun = @(p) (MofN_probability(p, M, N) - P_D);
    P_D1 = fzero(fun, [0 1]);

    % Solve for the value of P_FA1 required to achieve the specified P_FA
    fun = @(p) (MofN_probability(p, M, N) - P_FA);
    P_FA1 = fzero(fun, [0 1]);

    % Find the SNR χ_1 required to achieve the single-trial probabilities P_FA1 and P_D1
    if P_D1 <= 0.1, P_D1 = 0.1; end
    chi_1 = shnidman(P_D1, P_FA1, 1, 0);

    % Compute the binary integration gain G_BI
    G_BI_dB = chi - chi_1;
end

function prob = MofN_probability(p, M, N)
    prob = 0;
    for r = M:N
        prob = prob + nchoosek(N, r) * p^r * (1-p)^(N-r);
    end
end

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References

[1] M. A. Richards, “Binary Integration Gain” Oct. 13, 2016

[2] D. A. Shnidman, "Binary integration for Swerling target fluctuations," in IEEE Transactions on Aerospace and Electronic Systems, vol. 34, no. 3, pp. 1043-1053, July 1998.