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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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Most popular questions from this chapter

The Birthday Problem. The birthday problem is: if there is a group of \(n\) people in a room, what is the probability that two or more of them have the same birthday? It is possible to determine the answer to this question by simulation. Write a function that calculates the probabulity that two or more of \(n\) people will have the same birthday, where \(n\) is a calling argument. (Hint: To do this, the function should create an array of size \(n\) and generate \(n\) birthdays in the range 1 to 365 randomly. It should then check to see if any of the \(n\) birthdays are identical. The function should perform this experiment at least 5000 times and calculate the fraction of those times in which two or more people had the same birthday.) Write a test program that calculates and prints out the probability that 2 or more of \(n\) people will have the same birthday for \(n=2,3, \ldots, 40\).

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?

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 \(V_{2}=\) \([0.5,3,2]\).

Write three MATLAB functions to caiculate the hyperbolic sine, cosine, and tangent functions. $$ \sinh (x)=\frac{e^{x}-e^{-x}}{2}, \cosh (x)=\frac{e^{x}+e^{-x}}{2}, \tanh (x)=\frac{e^{x}-e^{-x}}{e^{x}+e^{-x}} $$ Use your functions to plot the shapes of the hyperbolic sine, cosine, and tangent functions.

Derivative in the Presence of Noise. We will now explore the effects of input noise on the quality of a numerical derivative. First, generate an input vector containing 100 values of the function \(\sin x\) starting at \(x=0\), using a step size \(\Delta x\) of \(0.05\), just as you did in the previous problem. Next, use function randomo to generate a small amount of random noise with a maximum amplitude of \(\pm 0.02\), and add that random noise to the samples in your input vector (see Figure 5.8). Note that the peak amplitude of the noise is only \(2 \%\) of the peak amplitude of your signal, since the maximum value of \(\sin x\) is 1. Now take the derivative of the function using the derivative function that you developed in the last problem. How close to the theoretical value of the derivative did you come?
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