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Chapter 5: Problem 26

Probability of Detection \(\left(P_{2}\right)\) versus Probability of False Alarm \(\left(P_{\text {a }}\right)\). The signal strength returned by a radar target usually fluctuates over time. The target will be detected if its signal strength exceeds the detection threshold for any given look. The probability that the target will be detected can be calculated as $$ P_{i=}=\frac{\text { Number of Target Detections }}{\text { Total Number of Looks }} $$ Suppose that a specific radar looks repeatedly in a given direction. On cach look, the range between \(10 \mathrm{~km}\) and \(20 \mathrm{~km}\) is divided into 100 independent range samples (called range gates). One of these range gates contains a target whose amplitude has a normal distribution with a mean amplitude of 7 volts and a standard deviation of I volt. All 100 of the range gates contain system noise with a mean amplitude of 2 volts and a Rayleigh distribution. Determine both the probability of target detection \(P_{d}\) and the probability of a false alarm \(P_{1,}\) on any given look for detection thresholds of \(8.0,8.5,9.0,9.5,10.0,10.5,11.0,11.5\), and \(12.0 \mathrm{~dB}\). What threshold would you use for detection in this radar? (Hint: Perform the experiment many times for each threshold and average the results to determine valid probabilities.)

Short Answer

Expert verified
Based on the given information about the radar system and its noise, one could estimate the probabilities of target detection (\(P_d\)) and false alarm (\(P_a\)) by conducting multiple experiments for each of the given detection thresholds. The key to this process is to generate random samples from both the target and noise distributions, assessing whether the sum of the target amplitude and noise amplitude exceeds the threshold. To calculate these probabilities, one could use the formulas: \(P_d = \text{{Number of Target Detections}} / \text{{Total Number of Looks}}\) and similarly for \(P_a\). In order to recommend an optimal threshold, one could plot \(P_d\) and \(P_a\) for each detection threshold to identify a suitable value that satisfies a balance between maximising the probability of detecting a target and minimising the probability of a false alarm. This balance is often a trade-off based on specific system requirements and acceptable levels of false alarms. Once the threshold is determined, the radar system should be able to perform detection within the desired performance parameters.

Step by step solution

01

Convert Decibel Values to Voltages

First, we can convert the given thresholds from decibels to voltages. To convert from decibels to voltage, we can use the equation: \[V = 10^{\frac{dB}{20}}\] where V is the voltage and dB is the decibel value. Apply this equation to each of the given detection thresholds.
02

Perform Experiments for Each Threshold

Next, for each threshold value, perform a large number of experiments by generating random samples from the given target and noise distributions. Use Python or any other programming language to do this. For each experiment, the detection threshold can be compared with the sum of the target amplitude and the noise amplitude. If this sum is greater than the threshold, the target will be detected. Keep track of the number of detections and false alarms for each threshold.
03

Compute Probabilities

After completing the experiments for each threshold, we can now compute the probabilities of detection (P_d) and false alarm (P_a). For each threshold, use the following formula: \[P_d = \frac{\text{Number of Target Detections}}{\text{Total Number of Looks}}\] Similarly, compute the probability of a false alarm (P_a) using the number of false alarms for each threshold.
04

Determine the Optimal Threshold

Now that we have the probabilities for detection and false alarm for each detection threshold, we need to determine an optimal threshold for the radar system. To do this, plot P_d and P_a for each threshold. The goal here is to find a balance between maximizing the probability of detecting a target (P_d) and minimizing the probability of a false alarm (P_a). An optimal threshold is the one that maximizes the probability of detecting a target while keeping the probability of a false alarm to a tolerable level as per system requirements. Generally speaking, choose the threshold that strikes an appropriate balance between P_d and P_a. Finally, with the chosen threshold, the radar system can perform detection with an acceptable performance of target detection and false alarm probabilities.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Radar Signal Processing
Radar signal processing is a technique used to analyze the data received by radar systems to identify targets and measure their properties. It involves processing the signals captured by radar to make accurate detections and measurements. Radar, which stands for Radio Detection and Ranging, works by emitting radio waves and analyzing the waves that bounce back from objects. This analysis allows the radar system to determine various features of the target, such as its distance and speed.

In radar signal processing, one of the main challenges is distinguishing the actual target signal from the noise. Noise can come from various sources, including electronic interference and environmental factors. It can mask or distort the signal returned from the target. Therefore, sophisticated processing techniques are required to enhance the target signal and reduce the effect of noise. A pivotal part of this is understanding the probability of detection, which describes how likely the system is to identify and confirm the presence of a target. Being familiar with this and other metrics helps in optimizing radar performance and achieving reliable detection outcomes.
Noise Distribution
In radar systems, noise plays a critical role as it can affect the accuracy of target detection. Noise in radar refers to the random variations in the signal that can obscure the presence of a target. The noise distribution represents how noise values are spread or distributed over different amplitudes.

For radar systems like the one described in the exercise, it is typical to encounter noise with a Rayleigh distribution. A Rayleigh distribution is characterized by having a single mode or peak, which occurs at a value greater than zero, and it is used to model the distribution of noise amplitudes in environments where received signals scatter around obstacles like buildings or vegetation.

Grasping how noise behaves is crucial because it influences the radar's performance criteria, such as the Probability of Detection (P_d) and Probability of False Alarm (P_a). If the noise level is high, it becomes more challenging to distinguish actual signals from noise, leading to potential errors in signal identification. Ultimately, analyzing noise distribution helps radar engineers to set appropriate thresholds for detection, ensuring that the system can make reliable detections despite the presence of noise.
Threshold Optimization
Threshold optimization is an important aspect of radar operations. It involves selecting a threshold value that determines whether a signal is considered a target detection or not. The threshold is crucial because it directly impacts the radar's Probability of Detection (P_d) and Probability of False Alarm (P_a).

By adjusting the detection threshold, radar operators can control the balance between P_d and P_a. A lower threshold might increase P_d 鈭� indicating more targets are detected. However, it also raises the chance of false alarms since noise might exceed the lower threshold. Conversely, a higher threshold reduces false alarms but might cause missed detections as weak target signals are ignored.

The exercise involves converting detection thresholds from decibels to voltages, performing experiments, and then computing probabilities of detection and false alarms for each threshold. The optimal threshold is the one where P_d is maximized with P_a kept within acceptable limits. This balance is essential for effective radar performance, ensuring targets are detected without overwhelming the system with false positives. This process of determination allows for making informed decisions about the threshold that meets operational requirements.

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Most popular questions from this chapter

Derivative of a Function. The derivative of a continuous function \(f(x)\) is defined by the equation $$ \frac{d}{d x} f(x)=\lim _{\Delta x=0} \frac{f(x+\Delta x)-f(x)}{\Delta x} $$ In a sampled function, this definition becomes $$ f^{\prime}\left(x_{i}\right)=\frac{f\left(x_{i+1}\right)-f\left(x_{i}\right)}{\Delta x} $$ where \(\Delta x=x_{i+1}-x_{i}\) Assume that a vector vect contains nsamp samples of a function taken at a spacing of \(d x\) per sample. Write a function that will calculate the derivative of this vector from Equation (5.12). The function should check to make sure that \(\mathrm{dx}\) is greater than zero to prevent divide-by-zero errors in the function. To check your function, you should generate a data set whose derivative is known, and compare the result of the function with the known correct answer. A good choice for a test function is \(\sin x\). From elemen\(\operatorname{tary}\) calculus, we know that \(\frac{d}{d x}(\sin x)-\cos x\). Generate an input vector containing 100 values of the function \(\sin x\) starting at \(x=0\) and using a step size \(\Delta x\) of \(0.05\). Take the derivative of the vector with your function, and then compare the resulting answers to the known correct answer. How close did your function come to calculating the correct value for the derivative?

Use the Help Browser to look up information about the standard MATLAB function sortrows and compare the performance of sortrows with the sort-with-carry function created in the previous exercise. To do this, create two copies of a \(1000 \times 2\) element array containing random values, and sort column 1 of each array while carrying along column 2 using both functions. Determine the execution times of each sort function using tic and toe. How does the speed of your function compare with the speed of the standard function sorerows?

What is the difference between a script file and a function?

Cross Product. Write a function to calculate the cross product of two vectors \(\mathbf{V}_{1}\) and \(\mathbf{V}_{2}\) \(\mathbf{V}_{1} \times \mathbf{V}_{2}=\left(V_{y 1} V_{x 2}-V_{v 2} V_{21}\right) \mathbf{i}+\left(V_{21} V_{x 2}-V_{22} V_{x 1}\right) \mathbf{j}+\left(V_{41} V_{v 2}-V_{x 2} V_{v 1}\right) \mathbf{k}\) where \(\mathbf{V}_{1}=V_{x 1} \mathbf{i}+V_{x 1} \mathbf{j}+V_{z 1} \mathbf{k}\) and \(\mathbf{V}_{2}=V_{x 2} \mathbf{i}+{y_{12}} \mathbf{j}+y_{x 2} \mathbf{k}\). Note that this function will return a real array as its result. Use the function to calculate the cross product of the two vectors \(V_{1}=[-2,4,0.5]\) and \(V_{2}=\) \([0.5,3,2]\).

Sort with Carry. It is often useful to sort an array are1 into ascending order, while simultancously carrying along a second array arr2. In such a sort, each time an element of array arrl is exchanged with another element of arr1, the corresponding elements of array arr2 are also swapped. When the sort is over, the elements of array arri are in ascending order, whereas the elements of array arr 2 that were associated with particular elements of array arrl are still associated with them. For example, suppose we have the following two arrays. \(\begin{array}{ccc}\text { Element } & \text { arr1 } & \frac{\operatorname{arc} 2}{1 .} \\ 1 . & 6 . & 0 . \\ 2 . & 1 . & 10 . \\\ 3 . & 2 . & 10 .\end{array}\) After sorting array arrl while carrying along array arr2, the contents of the two arrays will be \(\begin{array}{ccc}\text { Element } & \text { arr1 } & \text { are2 } \\ 1 . & 1 . & 0 . \\ 2 . & 2 . & 10 . \\ 3 . & 6 . & 1 .\end{array}\) Write a function to sort one real array into ascending order while carrying along a second one. Test the function with the following two 9 -element arrays. $$ \begin{aligned} &a=[-1, \quad 12, \quad-6, \quad 17, \quad-23,0,5, \quad 1,-1]: \\ &b=[-31,101,36,-17, \quad 0,10,-8,-1,-1]= \end{aligned} $$
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