Suppose $D$ is a string of length $L$ where its values are uniformly distributed in $[-a,a]$. Also, all values of $D$ are independent (i.i.d).

$X$ is a noisy version of $D$ in this way: $X=D+N$ where $N$ is a white Gaussian noise with mean and variance of $0$ and $\sigma_N^2$.

$P$ is a sub-string of $D$ where its length is $l$ $(l<L)$.

To detect $P$ in $X$, I use the cross-correlation of $X$ and $P$: $$ C_n = \sum_{i=1}^l X_{n+i-1}P_i \space\space , \space\space 1<n<L-l+1 $$ and a threshold of $T=K\sigma_{C_n}$ where $K$ is a constant value and $\sigma_{C_n}$ is the standard deviation of the correlation signal $C_n$.

A false negative occurs If no points of the correlation signal reaches the threshold.

Now the question is: how can I theoretically calculate the false negative probability? It tried to use Q-function, but I couldn't achieve the answer.


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