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Universal Probability Density Function of Detected SNR
Radar echoes above an SNR threshold for a ground-based, atmospheric radar operating at 33.4 GHz
Details
The data presented here were recorded by a ground-based, 33.4 GHz radar operated at Yuma AZ in the spring of 2017. The data are of the form echo signal-to-noise (SNR) and corresponding echo range in meters. All echoes above a 2 dB threshold in the recording interval are presented. The measured echo SNR values are found to be well fitted by a Pareto distribution.
By detected signal to noise ratio (SNR) reference is made to radar echoes that are above a threshold that is large enough to insure that the echo is produced by reflection from a target and is not just a spike in the background noise level.
The SNR of a radar echo can be written in the simplified form y=F/ra where the symbol y denotes the SNR of the echo measured on a power scale, F is the figure of merit of the radar, r is the radial distance (or slant range) to the target and a is the decay rate constant. The cases a=1,2 and 4 respectively refer to the passive detection of an advancing cylindrical wave, the weather radar equation and the ordinary radar equation. If yT denotes the threshold SNR for the radar, then the maximum range at which the radar can make a detection is 
.
.If β denotes the constant density per unit volume of targets, then the number of detections produced by the radar at the two SNR thresholds yT and y>yT is 
and ND(y)=β ΩbeamF3/ay-3/a. If the radar is omnidirectional and the entire hemisphere is illuminated then Ωbeam=2π. The fraction of targets that are detected at SNR values y>yT is the ratio of these two quantities, i.e. (yT/y)3/a. The cumulative fraction of targets detected as a function of the SNR value y is G(y)=1-(yT/y)3/a for y≥yT and zero otherwise. This is the cummulative probability density function of a Pareto distribution with decay parameter α=3/a and threshold parameter yT.
and ND(y)=β ΩbeamF3/ay-3/a. If the radar is omnidirectional and the entire hemisphere is illuminated then Ωbeam=2π. The fraction of targets that are detected at SNR values y>yT is the ratio of these two quantities, i.e. (yT/y)3/a. The cumulative fraction of targets detected as a function of the SNR value y is G(y)=1-(yT/y)3/a for y≥yT and zero otherwise. This is the cummulative probability density function of a Pareto distribution with decay parameter α=3/a and threshold parameter yT.The range dependence in the underlying radar equation that governs the propagation physics is encoded in the parameter α=3/a. The median value of a Pareto distribution with threshold yT and decay constant α is 21/αyT. This implies that 1/2 of all detections will occur in the SNR range yT≤y≤21/αyT. For the case α=3/2 (weather radar equation with a=2), 1/2 of detections occur within 10log1022/3=2 dB of the threshold yT.
Examples
Basic Examples (2)
Retrieve the data as a TimeSeries object:
| In[1]:= | ![ResourceData[\!\(\*
TagBox["\"\<Universal Probability Density Function of Detected SNR\>\"",
#& ,
BoxID -> "ResourceTag-Universal Probability Density Function of Detected SNR-Input",
AutoDelete->True]\)]](https://www.wolframcloud.com/obj/resourcesystem/images/7a3/7a334e31-b67a-448a-a45e-8731ebd40ada/714411c0e192e5ea.png){kind=small} |
| Out[1]= |  |
Visualize the data:
| In[2]:= | ![ListPlot[ResourceData[\!\(\*
TagBox["\"\<Universal Probability Density Function of Detected SNR\>\"",
#& ,
BoxID -> "ResourceTag-Universal Probability Density Function of Detected SNR-Input",
AutoDelete->True]\)], Sequence[
PlotRange -> {{0, 50}, All}, Axes -> False, Frame -> True, FrameLabel -> {"SNR (dB)", "Slant Range (m)"}]]](https://www.wolframcloud.com/obj/resourcesystem/images/7a3/7a334e31-b67a-448a-a45e-8731ebd40ada/3e2f2422f595d094.png){kind=small} |
| Out[2]= |  |
Analysis (2)
Extract the SNR portion of the data and fit the values to a Pareto distribution. Create a Histogram of the data and compare it to the probability density function of the fitted Pareto distribution:
| In[3]:= | ![data = Keys[ResourceData[\!\(\*
TagBox["\"\<Universal Probability Density Function of Detected SNR\>\"",
#& ,
BoxID -> "ResourceTag-Universal Probability Density Function of Detected SNR-Input",
AutoDelete->True]\)]];
data = 10^(data/10);
estdist = EstimatedDistribution[data, ParetoDistribution[k, \[Alpha]]];
g1 = Histogram[data, {0, 15, 0.4}, Sequence[
"PDF", Axes -> False, Frame -> True, FrameLabel -> {"SNR (power)"},
PlotRange -> All]];
g2 = Plot[PDF[estdist, y], {y, 0, 15}, Sequence[
Axes -> False, Frame -> True, FrameLabel -> {"SNR (power)"}, PlotRange -> All]];
Show[g1, g2]](https://www.wolframcloud.com/obj/resourcesystem/images/7a3/7a334e31-b67a-448a-a45e-8731ebd40ada/6d33546645eb1df6.png) |
| Out[8]= |  |
The implied range decay coefficient is a=3/α where α is the fitted Pareto parameter. The value is almost exactly 3:
| In[9]:= | ![3/estdist[[2]]](https://www.wolframcloud.com/obj/resourcesystem/images/7a3/7a334e31-b67a-448a-a45e-8731ebd40ada/203b685f7e6f8af6.png){kind=small} |
| Out[9]= |  |
This implies 30log10r SNR range decay on a decibel scale in the data. This suggests that the range decay in the data is mid way between the weather radar equation (20log10r range decay) and the standard radar equation (40log10r range decay). An advantage of this technique is that it makes use of all the data and properly accounts for selection effects (thresholding).
Bibliographic Citation
Marshall Bradley, "Universal Probability Density Function of Detected SNR" from the Wolfram Data Repository (2026)