Performance Model for Square Law Detectors Followed by Accumulators and an ORing Device

A mathematical model is developed to characterize the performance of square law detectors followed by accumulators and an ORing device. The performance of a Gaussian signal in Gaussian noise is determined for a specified number of channels ORed. The ORing loss is computed as a function of the number of samples in the accumulator preceding the ORing device for various fixed input signal-to-noise ratios (SNR) to the square law detector. It is concluded that the ORing loss decreases with an increase in the number of samples in the accumulator preceding the ORing device.

the inverse of the noise variance at each point Such functions can easily be constructed by a Gram-Schmidt or modified Gram-Schmidt technique [ 5 ] provided the noise variance at each point is known.Such a method of forcing orthonormality is not necessarily a hindrance.The periodic sampling of functions of a continuous variable which are orthonormal on an interval does not generally yield sequences which are orthonormal on the same discretized interval, and hence one is often compelled in such instances to use a Gram-Schmidt technique to generate a set of orthonormal sequences.In such instances, the sequences should be constructed to satisfy (24) rather than (2) if the noise variances are not constant on the interval.One obvious disadvantage of variances that are not constant is that algorithms such as the fast Fourier transform cannot be used to rapidly determine unbiased coefficient estimates, since the discrete sine and cosine functions would not be orthonormal on the interval with respect to the weights.
If the noise associated with the data is not Gaussian, as, for example, the time sequence of Poisson distributed counts from a radioactive source, then minimization of the weighted square error is not equivalent to maximization of the likelihood function.In such cases, the approximation is sometimes still made after carrying out a suitable transformation of the noise variance at each point [ 4 ] .Alternatively, one may minimize a different quantity, such as a non-Euclidian norm.The danger in this approach is that maximization of the likelihood function may not correspond to the minimization of a single quantity, as in the case of the least squares approximation.
In any event, one should be aware of the limitations in bias and accuracy when estimating coefficients in an orthonormal expansion, in order to assure that the results from such techniques are meaningful.

INTRODUCTION
Many signal processing systems produce a large quantity of data that are not feasible to display.It is desired to reduce the amount of data displayed, and thereby reduce the amount of hardware needed, One technique that provides a reduction of data is ORing, a process wherein one picks that single channel with the most energy.
The subsequent analysis characterizes the performance of square law detectors followed by accumulators and an ORing device.Some relevant past work on this topic is given in t11-[31.

MATHEMATICAL ANALYSIS
The system of interest is shown in Fig. 1.The input to the ORing device X1, X,, * * * , X , contains either N channels of noise or N -1 channels of noise, and one channel of signal plus noise.The random variables X I , X , , * . ., X , are statistically independent, and they are identically independently distributed (iid) for the noise-only channels.The ,accumulators following the square law detectors are electrical integrators that sum M statistically independent samples.
It is assumed that the input to the square law detector is an envelope of a Gaussian signal in Gaussian noise, both having zero mean.The input signal-to-noise ratio (SNR) to the square law detector (SNRID) is given by where u : = signal power at the input to the square law detector and u i = noise power at the input to the square law detector.
The input SNR to the ORing device (SNRIOR) for Gaussian input statistics is where dIoR = d@(u:/ui) =input SNR to the ORing device in nondecibel form and M = number of samples in the accumulator.
Finally, the SNR at the output of the ORing device (SNRooR) is given by where p y 1 = mean of the signal plus noise after ORing, ~Y O = mean of the noise after ORing, and uy0 = standard deviation of the noise after ORing.
In order to evaluate ( 3 ) , it is necessary to determine pya, p y l , and uyo.The output of the ORing device is mathematically defined as Y = max ( X , , X * , * * ., X N ) .
For the case in which one of the Xi's is signal plus noise, the cumulative distribution of Y , F1 ( y ) is defined as

,-ca
The probability density function f l ( y ) for the signal plus noise case is defined as The mean values p y 0 and pyl in (3) for the signal-absent and signal-present cases are

(BN -A&)"' .pl(Y)+P;-'(Y)Pl(Y)I 0 . (10)
It is now desired to evaluate the performance of the system The standard deviation for the noise-only case (Tyg in (3) is in Fig, 1.The integration over M statistically independent samples produces a gain.However, this gain will be reduced ( l by an amount equal to the ORing loss.The ORing loss relative to the input to the ORing device (SNRloSOR) is The only thing remaining is to determine the cumulative dis--20 log CN ( ~I O R ) -AN ] (dB).( 16) The ORing loss relative to the detector input (SNR1,,,) is Equations ( 12)-( 15) were programmed on a digital computer.The input SNR to the square law detector was varied from -17 to -1 dB.The SNR at the output of the ORing device was determined for a specified number of chhnnel ORed, N, and a specified number of samples in the accumulator, M .Equation (1 7) was then used to compute the ORing loss.

CONCLUSIONS
The analytical results for the ORing loss are presented in Figs.2-4.The following conclusions can be drawn from an examination of these figures: 1) the greater the number of channels ORed, the greater the ORing loss, and 2) the more samples in the accumulator preceding the ORing device, the lower the ORing loss.