M-ary error probability; binary vs M-ary

Error probability of an M-ary (PAM) system

In an M-ary system each symbol carries $n = \log_2 M$ bits. Take M-ary baseband PAM: the receiver output sample (after the matched filter) takes one of $M$ equally spaced levels

$$a_k \in \{\pm A,\ \pm 3A,\ \dots,\ \pm(M-1)A\}$$

so adjacent levels are $2A$ apart. Assumptions: symbols equally likely, additive white Gaussian noise of PSD $N_0/2$, so the sampled noise $n$ is Gaussian with zero mean and variance $\sigma^2 = N_0/2$ (unit-energy pulse, matched filter). Decision thresholds are placed midway between levels: $0, \pm 2A, \pm 4A, \dots$

  -3A     -A      +A     +3A      (M = 4)
   x  |    x   |   x   |   x
     -2A       0      +2A        <- thresholds
 outer         inner        outer

Inner level (two neighbours): an error occurs if $|n| > A$:

$$P_{e,\text{inner}} = P(n>A) + P(n<-A) = 2Q\!\left(\frac{A}{\sigma}\right)$$

Outer level (one neighbour, e.g. $+(M-1)A$): an error occurs only if $n < -A$:

$$P_{e,\text{outer}} = Q\!\left(\frac{A}{\sigma}\right)$$

There are $M-2$ inner and $2$ outer levels, so the average symbol error probability is

$$\begin{aligned} P_e &= \frac{1}{M}\left[(M-2)\,2Q\!\left(\frac{A}{\sigma}\right) + 2\,Q\!\left(\frac{A}{\sigma}\right)\right] \\ &= 2\left(1-\frac{1}{M}\right)Q\!\left(\frac{A}{\sigma}\right) \end{aligned}$$

In terms of energy. The average symbol energy is

$$E_s = \frac{2}{M}\sum_{i=1}^{M/2}(2i-1)^2A^2 = \frac{(M^2-1)A^2}{3} \;\Rightarrow\; A^2 = \frac{3E_s}{M^2-1}$$

With $\sigma^2 = N_0/2$:

$$P_e = 2\left(1-\frac{1}{M}\right)Q\!\left(\sqrt{\frac{6E_s}{(M^2-1)N_0}}\right), \qquad E_s = E_b\log_2 M$$

Check, M = 2: $E_s = E_b$ and $P_e = Q\left(\sqrt{2E_b/N_0}\right)$, the polar binary result.

With Gray coding, bit error rate $\approx P_e/\log_2 M$.

Binary vs M-ary

So M-ary signalling saves bandwidth by a factor $\log_2 M$ but needs more signal power and a more complex system for the same error rate. It is used where bandwidth is scarce (telephone-line modems, digital radio).