Error probability of M-ary system

In an M-ary system each symbol represents $k=\log_2M$ bits. For baseband M-ary PAM the receiver samples the filtered signal and decides which of the $M$ levels was sent using $M-1$ thresholds.

M-ary baseband PAM. Each symbol takes one of $M$ equally likely levels spaced $2A$ apart, symmetric about zero:

$$a_i=\pm A,\ \pm3A,\ \dots,\ \pm(M-1)A$$

The received sample is $y=a_i+n$, $n\sim\mathcal{N}(0,\sigma^2)$. Thresholds are placed midway between adjacent levels (at $0,\pm2A,\pm4A,\dots$).

  levels:  -3A    -A     +A    +3A        (M = 4)
    -------*--|---*--|---*--|---*-------> y
             -2A     0     +2A       thresholds
  outer levels: error on one side only
  inner levels: error on both sides

• Inner level (there are $M-2$): error if $\lvert n\rvert>A$: $P_{inner}=2\cdot\frac12\operatorname{erfc}\left(\frac{A}{\sqrt2\sigma}\right)=\operatorname{erfc}\left(\frac{A}{\sqrt2\sigma}\right)$
• Outer level (there are 2): error only if noise pushes it inward by more than $A$: $P_{outer}=\frac12\operatorname{erfc}\left(\frac{A}{\sqrt2\sigma}\right)$

Average symbol error probability:

$$\begin{aligned} P_e &= \frac{1}{M}\left[(M-2)\operatorname{erfc}\left(\frac{A}{\sqrt2\sigma}\right)+2\cdot\frac12\operatorname{erfc}\left(\frac{A}{\sqrt2\sigma}\right)\right] \\ &= \left(1-\frac1M\right)\operatorname{erfc}\left(\frac{A}{\sqrt2\,\sigma}\right)=2\left(1-\frac1M\right)Q\left(\frac{A}{\sigma}\right) \end{aligned}$$

In terms of signal power. Average symbol power

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

so

$$P_e=\left(1-\frac1M\right)\operatorname{erfc}\left(\sqrt{\frac{3}{2(M^2-1)}\cdot\frac{P_{av}}{\sigma^2}}\right)$$

For $M=2$ this reduces to the binary polar result $P_e=\frac12\operatorname{erfc}\left(\frac{A}{\sqrt2\sigma}\right)$.

Observations:

• For fixed $A/\sigma$, $P_e$ rises slowly with $M$ (factor $1-1/M$), but for fixed average power the spacing shrinks as $\sqrt{3/(M^2-1)}$, so $P_e$ rises sharply.
• With Gray coding, bit error probability $\approx P_e/\log_2M$.
• M-ary signalling reduces bandwidth by $\log_2M$ but needs more power: a bandwidth–power trade-off.