i am having trouble understanding the union bound concept, which in my case is used to get an upper bound on the probability of error in a given system.
Consider having 3 matrices: $M1, M2, M3$
The PEP (pairwise error probability) is the probability of mistaking $Ma$ for $Mb$ which is $P(M_a\rightarrow M_b)$.
So to get an upper bound on the total probability of error, [Proakis, 2001, Sec. 5.2] states that the union bound could be used:
$ P_{tot} \le \frac{1}{|\mathcal{M}|} \sum_{M_a \in \mathcal{M}} \sum_{M_b \in \mathcal{M},\;b\neq a } P(M_a\rightarrow M_b)$
where $|\mathcal{M}|$ is the cardinality of the group of matrices used, in our case $|\mathcal{M}|=3$
Let us assume that all the PEP's of the above statement are equal to 1, by my caclulations, this would lead to the $P_{tot}\le \frac{6}{3}$.
which i find not to be logical since the total error probability should not surpass 1.
Am i making a mistake in my calculations ? or is there more to the union bound that i don't get?
Thank you
Nothing logically wrong to get trivial upper bounds.
However, one thing to note that we can't really have all $PEP$ to be equal to one. That would $P(M_1 \to M_2)=1$ and $P(M_1 \to M_3)=1$, we can't have the probability of decodign $M_1$ to be $M_2$ and $M_3$ to be both $1$.
In fact we should have $P(M_1 \to M_2) + P(M_1 \to M_3) \le 1$.