5.5  Estimating Probability of Error for PAM

The goal of this section is to obtain an expression for the symbol error probability $P_e$ for PAM on AWGN channels using ML. The symbols are assumed to have a uniform prior distribution and the optimal MAP criterion coincides with the ML.

The PAM constellation is assumed to have the same distance $d$ among the neighbor symbols and all symbols are equiprobable. Listing 2.8 gives an example where $d\; =\; 2$. In general, the PAM symbols are: $\pm \frac{d}{2},\pm \frac{3d}{2},\pm \frac{5d}{2},\dots ,\pm \frac{(M-1)d}{2}$. In the case of PAM, the 2 symbols at the extrema of the constellation have only one neighbor, while the others have two. Hence, for a M-PAM with the symbols uniformly distributed with probability $P(m)\; =\; 1∕M,\forall m$,

where $M\; =2^b$. It is useful to rewrite Eq. (5.6) in terms of SNR. Assuming matched filtering with $E_p=\; 1$, it has been shown that

where the subscript nMF recalls that this SNR is obtained at the output of a matched filter when the (normalized) shaping pulse has unitary energy.

For PAM, $E_c$ depends only on $d$ and $M$. Hence, using Eq. (2.10), to express $d$ in terms of $E_c$ and $M$ leads to

and, for PAM, one can write

such that for PAM:

Note that, for sufficiently high $M$ (say $M\; \ge \; 32$ or, equivalently, $b\; \ge \; 5$)

Estimating the bit error probability $P_b$ is more evolved and the approximation of Eq. (2.4) is often adopted.

5.5.1  The union bound

Sometimes it is not necessary to have a precise value for $P_e$ but only a bound on its maximum value in a given setup.

The union bound states that the probability of symbol error for the ML detector on AWGN with an $M$-point constellation is

min$2\sigma ),$$where$d_min$istheminimumdistancebetweenanypairofpointsoftheconstellation,i.e.$$d_{}$

min = $\underset{i\ne j}{min}||m_i-m_j||,\forall i,j.$$$