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--- kb:signal_detection [2021-05-12 22:01] – created jaeyoung
+++ kb:signal_detection [2024-04-30 04:03] (current) – external edit 127.0.0.1
@@ Line 15: / Line 15: @@
 Given that $W[n]$ is Gaussian, this can be rewritten as:
-$$ \frac{\prod_{n = 0}^{L - 1} \left( \frac{1}{\sigma \sqrt{2 \pi}}e^{-\frac{(r[n] - s[n])^2}{2 \sigma^2}} \right)}{\prod_{n = 0}^{L - 1} \left( \frac{1}{\sigma \sqrt{2 \pi}}e^{-\frac{(r[n])^2}{2 \sigma^2}} \right)} $$
+$$ \frac{\prod_{n = 0}^{L - 1} \left( \frac{1}{\sigma \sqrt{2 \pi}}e^{-\frac{(r[n] - s[n])^2}{2 \sigma^2}} \right)}{\prod_{n = 0}^{L - 1} \left( \frac{1}{\sigma \sqrt{2 \pi}}e^{-\frac{(r[n])^2}{2 \sigma^2}} \right)} \overbrace{\gt}^{'H_1'} \underbrace{\lt}_{'H_0'} \frac{p_0}{p_1} $$
+After some simplifications, we get:
+$$ g = \sum_{n = 0}^{L - 1} r[n] s[n] \overbrace{\gt}^{'H_1'} \underbrace{\lt}_{'H_0'} \sigma^2 \ln \eta + \frac{\varepsilon}{2} = \gamma $$
+where $\eta = \frac{p_0}{p_1}$ and $\varepsilon = \sum_{n = 0}^{L - 1} s^2[n]$ (Energy)
+Let $G$ be the random variable of which $g$ is a realized value. Similarly, $R[n]$ is the random process of which $r[n]$ is a realized instance. Then,
+$$ G = \sum_{n = 0}^{L - 1} R[n]s[n] $$
+The distributions of $G$ are:
+$$ H_0: G \sim \mathcal{N}(0, \sigma^2 \varepsilon) $$
+$$ H_1: G \sim \mathcal{N}(\varepsilon, \sigma^2 \varepsilon) $$
+Note that the variance is the same in both cases.
+===== Matched filter =====
+A matched filter is used to detect a known signal $s[n]$ in white Gaussian noise.
+The filter is the time reverse of the signal:
+$$ h[n] = s[-n] $$
+In the frequency domain:
+$$ H(e^{j\Omega}) = S(e^{-j\Omega}) = |S(e^{j\Omega})| e^{-j\angle S(e^{j\Omega})} $$
+Consider filtering a noisy signal $r[n]$ with the matched filter $h[n]$:
+$$ g[n] = (h \ast r)[n] = (\overleftarrow{s} \ast r)[n] $$
+In the ideal case where $r[n] = s[n]$, the output is deteministic autocorrelation:
+$$ g[n] = (h \ast r)[n] = (s \ast \overleftarrow{s})[n] = \bar{R}_{ss}[n] $$
+The matched filter maximizes the spread between the $H_0$ and $H_1$ cases.
+Compare the $g[n]$ with the threshold $\gamma = \sigma_W^2 \ln \frac{p_0}{p_1} + \frac{\varepsilon}{2}$. If $g[n] > \gamma$, declare $'H_1'$. Otherwise, declare $'H_0'$.
+===== Probability of error =====
+The conditional probability of false alarm is:
+$$ P_{FA} = Q\left(\frac{\gamma}{\sigma\sqrt{\varepsilon}}\right) $$
+$$ P_M = 1 - Q\left(\frac{\gamma - \varepsilon}{\sigma\sqrt{\varepsilon}}\right) $$
+Total probability is:
+$$ P_e = p_0 P_{FA} + p_1 P_M $$