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In several cases (e.g., thermal noise, weather echoes, …), the incoming signal to a radar receiver can be assumed to be Rayleigh distributed. When estimating the mean power from the inherently fluctuating Rayleigh signals, it is necessary to average either the echo power intensities or the echo logarithmic levels. Until now, it has been accepted that averaging the echo intensities provides smaller variance values, for the same number of independent samples. This has been known for decades as the implicit consequence of two works that were presented in the open literature. The present note deals with the deriving of analytical expressions of the variance of the two typical estimators of mean values of echo power, based on echo intensities and echo logarithmic levels. The derived expressions explicitly show that the variance associated to an average of the echo intensities is lower than that associated to an average of logarithmic levels. Consequently, it is better to average echo intensities rather than logarithms. With the availability of digital IF receivers, which facilitate the averaging of echo power, the result has a practical value. As a practical example, the variance obtained from two sets of noise samples, is compared with that predicted with the analytical expression derived in this note (

When interpreting a fluctuating echo from a randomly distributed target, the usual problem is to estimate the long-term mean echo power, in order to obtain an estimate of the scatterers contained in the volume sampled by the radar pulse (e.g., Doviak and Zrnic [

The derived formulas can also be useful to estimate the variance of radar receiver thermal noise, given the number of averaged samples. Prior to the envelop detector (rectification), which means energy detection in the electronic circuit, receiver thermal noise is expected to have a Gaussian distribution with a zero mean. After rectification, noise has an exponential one-sided probability distribution that fluctuates around a mean, which is the root-mean-squared value of the unrectified fluctuations. In other words, the results by Marshall and Hitschfeld [

It has been shown (Kerr and Goldstein [_{0} is a parameter that characterizes the exponential decay rate and which is coincident with the mean and standard deviation of the distribution; _{m} has been chosen to be equal to 1 mW and a decimal logarithmic scale is used. Consequently, the logarithmic power level L is expressed in dBm, namely _{m}), where _{m} = 1 mW and [

The probability density function of the logarithmic power level L is (see Wallace [_{0}, is coincident with the mean power E{_{0}, once it is expressed on a decimal logarithmic scale and multiplied by 10; in other words, the most probable value, _{0}, is the mean power expressed in dBm:
_{0}=10 × Log(E{_{m}_{0}/_{m}

As shown in the sketch presented on page 966 in Marshall and Hitschfeld [_{L}_{0}, is 2.51 dB larger than the mean logarithmic power level, regardless of the mean power value _{0}. While the standard deviation that characterizes the probability distribution function _{P}_{0}, the standard deviation of the probability distribution function _{L}_{L}

On the basis of the two kinds of samples, namely intensity, _{i}_{j}_{i}_{ML} indicate that this estimator is the maximum-likelihood one. The estimator based on the logarithmic level samples, _{j}_{0} is estimated in dBm.

The uncertainty in the estimates of the mean echo power, starting from intensity or logarithmic level samples, is quantified by determining which of the two estimators has the smaller variance. Since the two estimators are for different (although directly related) parameters, the dBm value of the estimator in Equation (4) is computed:

The performance index 2 σ/E is computed as a criterion for comparing the estimators in Equations (5) and (6), where E is the average value of the estimator and σ is its standard deviation. This means that the performance index characterizes the relative uncertainty of the estimator and is related to the confidence intervals normalised to the mean value. It is shown, in the _{L}_{0}_{m}

The standard deviation of the estimators in Equations (5) and (6) are

Equation (7) tends to Equation (8) for large values of _{ML}_{L}_{ML}_{L}

Expected standard deviation from an ensemble of

As a practical example of Rayleigh-distributed (in amplitude, hence exponential-distributed in power) signals, radar noise measurements have been analyzed at the output of a radar receiver. The principal contribution at weather radar frequencies is found to be receiver thermal noise. Prior to the envelop detector (rectification), which means energy detection in the electronic circuit, receiver thermal noise is expected to have a Gaussian distribution with a zero mean. After rectification, the noise has an exponential one-sided probability distribution, fluctuating around a mean that is the root-mean-squared value of the unrectified fluctuations. Both presented data sets have been derived from the output of a low-cost and easily achievable civil marine radar. The main features of the system are described in Table 1 of the paper by Gabella ^{0.5} = 0.16 dB, as can easily be derived from Equation (10). However, this is not the case, since it is known that intensities were averaged (algebraic average). Consequently, in order to predict the expected standard deviation, we would better use Equation (8), which in fact gives 0.13 dB. This value of standard deviation is much closer to those which were derived using the two civil marine radar experimental data sets.

When the incoming signal envelope to a radar receiver is Rayleigh-distributed, how is the variance reduced by the geometric or algebraic average of

In order of appearance in the text:

Total number of available samples.

Equivalent number of independent samples.

Echo power.

_{0}

Mean echo power.

Step function, defined as 1 if

Power logarithmic Level in dBm (that is, the reference value is set to 1 mW)

Natural logarithm of

_{0}

Level of the mean echo power.

Base-10 logarithm of

_{i}

Echo power sample.

_{i}

Level of the echo power sample (dBm).

_{ML}

Maximum likelihood estimate of the parameter β.

Mean value of an estimator.

Standard deviation of an estimator.

_{ML}

Mean value of the Log-transformed ML estimator.

_{ML}

Standard deviation of the Log-transformed ML estimator.

_{L}

Mean value of the level samples based estimator

_{L}

Standard deviation of the level samples based estimator.

Euler’s constant.

Riemann’s zeta function.

The author is grateful to Jürg Joss for his valuable help in estimating the standard deviation of the two noise measurement sets, which consist of 10,416 samples, with a 0.4 dB quantization. He also would like to thank Maurizio Sartori, Dennis Vollbracht and Olivier Progin for interesting discussions regarding fluctuating radar echoes from thermal noise.

The author declares no conflict of interest.

We obtain the standard deviation of the estimator {10 × Log(_{0}_{m})

We know that the random variable in Equation (4) has the following distribution of probability (see Wallace [

Hence the standard deviation of the right therm in Equation (6) will be

We have:
^{x}

Since:
^{x}

From Equations (A7), (A9) and (A10) we conclude that the estimator in Equation (6) has the following mean and variance: