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# Consider detection of a real-valued constant in zero-mean real-valued Gaussian noise. Let the noise variance b 2 = 2, the number of samples N = 1, and the constant m = 4. What is the SNR for this...

1. Consider detection of a real-valued constant in zero-mean real-valued Gaussian noise. Let the noise variance b 2 = 2, the number of samples N = 1, and the constant m = 4. What is the SNR for this case? Sketch approximately the distributions p(y|H0) and p(y|H1); be sure to label appropriate numerical values on your axes. Write the likelihood ratio and log-likelihood ratio for this problem. Simplify your expressions.
2. Continuing with the same parameters given in problem #1, suppose we want to have PFA = XXXXXXXXXX%). What is the required value of the threshold T? What is the resulting value of PD? You can use lookup tables or MATLAB to calculate the values of functions such as erf(), errc(), erf-1(), or erfc-1() that you may need.
3. Consider detection of a constant in complex Gaussian noise. Let the total noise variance b 2 = 2, the number of samples N = 1, and the constant m = 4. What is the SNR c for this case? Suppose we want to have PFA = XXXXXXXXXX%). What is the required value of the threshold T? What is the resulting value of PD? You can use lookup tables or MATLAB to calculate the values of functions such as erf(), errc(), erf-1(), or erfc-1() that you may need.
4. Repeat the computation of the threshold T and probability of detection PD for the case of a constant in zero-mean complex Gaussian noise, but now with unknown phase. Use m = 4, b2 = 2 and N = 1 again. You will need to evaluate the Marcum Q function QM; you can use MATLAB marcumq (provided in Communications Toolbox or Signal Processing Toolbox; if you have neither, you can download the matlab function from the class website).
5. Use Albersheim’s equation to estimate the single-sample SNR c1 required to achieve PFA = 0.01 and PD equal to the same value you obtained in problem #4. How does the compare to the actual SNR in problem #4?
6. Consider 3-out-of-5 (M=3, N=5) binary integration. Determine the required values of the single-trail probabilities PD and PFA such that the cumulative probabilities are PCFA = 10-8 and PCD = 0.99. You can use a small-probability approximation to solve for PFA, but finding PD will require some numerical trial-and-error; your PD should be accurate to 2 decimal places. (Hint: the correct answer lies in the range .)
7. Suppose we want to do a detection test on a single noncoherently detected sample of a nonfluctuating target in complex Gaussian noise with power b2 = 1. We are using a square-law detector. Assuming we know this interference power level exactly, what ideal value of threshold T is required to obtain PFA = 10-4? If the SNR is c = 10 dB, what is PD? You will have to use MATLAB to evaluate the Marcum Q function QM.
8. Now assume that we do not know the interference level a priori, so we use a cell-averaging CFAR to do the detection test. Choose N = 30 reference cells. What will be the threshold multiplier a such that the average false alarm probability remains at 10-4? It turns out that if the signal-to-noise ratio is c = 10 dB, the value of PD using the ideal threshold is XXXXXXXXXXwith know interference power). Assuming the SNR is still c = 10 dB, what will be the average detection probability using the CA-CFAR?

## Solution

David answered on Dec 31 2021
Homework #1
1. Consider detection of a real-valued constant in zero-mean real-valued Gaussian noise. Let the noise variance 2 =
2, the number of samples N = 1, and the constant m = 4. What is the SNR for this case? Sketch approximately the
distributions p(y|H0) and p(y|H1); be sure to label appropriate numerical values on your axes. Write the likelihood
atio and log-likelihood ratio for this problem. Simplify your expressions.
Solution
Given that 2 = 2, the number of samples N = 1, and the constant m = 4
So SNR = Nm2/2
SNR = 1 x 42 / 2 = 16/2 = 8 so,
Sketch of p(y|H0) and p(y|H1)
Likelihood Ratio
Log Likelihood Ratio
On simlifying we get
And finally we have
Likelihood ratio in simple form
Log Likelihood ratio in simple form
2. Continuing with the same parameters given in problem #1, suppose we want to have PFA = 0.01 (1%). What is the
equired value of the threshold T? What is the resulting value of PD? You can use lookup tables or MATLAB to
calculate the values of functions such as erf(), e
c(), erf-1(), or erfc-1() that you may need.
Solution
Given that 2 = 2, the number of samples N = 1, and the constant m = 4
For calculating threshold we have
Hence we have