Page 540 - Fundamentals of Radar Signal Processing

c. Regardless of the LR found in part (b), suppose the rule is

chosen as the detection test. Sketch the resulting curves for P vs. T
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and P vs. T as T varies from –1.5 to +1.5. Label all important
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values.

Problems 3 to 6 are a related group exploring how changes in the values
of m, , and T affect detection and false alarm performance in the real
constant-in-WGN example.

3. Consider detection of a real-valued constant in zero-mean real-valued
Gaussian noise. Let the noise variance , the number of samples N = 1,

and the constant m = 4. What is the SNR χ for this case? Sketch the
distributions p(y | H ) and p(y | H ); label appropriate numerical values on
1
0
the axes. Write the likelihood ratio and log-likelihood ratio for this
problem. Simplify the expressions.

4. Continuing with the same parameters given in the previous problem,
suppose P = 0.01 (1 percent) is required. What is the required value of
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the threshold T? What is the resulting value of P ? Lookup tables or
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MATLAB can be used to calculate the values of functions such as erf(·),
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–1
erfc(·), erf (·), or erfc (·) that may be needed.
–1
5. Suppose m in Prob. 3 is increased so as to double the SNR; and N = 1
still. What is the new value of m? Sketch and label the distributions p(y |
H ) and p(y | H ) with this new value of m. If the same threshold T found in
0
1
Prob. 4 is retained, does P change and, if so, what is the new value? If
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that same threshold T is retained, does P change and, if so, what is the
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new value?

6. Go back to the case of m = 4, but now reduce the noise variance to . N
= 1 still. What is the SNR χ now? Sketch and label the distributions p(y|H )
0
and p(y|H ) with this value of m and . If the threshold value used in
1
Probs. 4 and 5 is still retained, what is the value of P ? What is the value
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of P ?
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7. Consider detection of a constant in complex Gaussian noise. Let the total
noise variance , the number of samples N = 1, and the constant m = 4.
What is the SNR c for this case? Suppose P = 0.01 (1 percent) is
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required. What is the required value of the threshold T? What is the
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resulting value of P ? Lookup tables or MATLAB can be used to
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–1
–1
calculate the values of functions such as erf(·), erfc(·), erf (·), or erfc (·)
that may be needed.
8. Compute the threshold T and probability of detection P for the case of a
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constant in zero-mean complex Gaussian noise, but now with unknown