Short note: M-ary baseband communication system

In an M-ary baseband system, the transmitter does not send one bit per pulse. It groups $k = \log_2 M$ bits into one symbol and sends one of $M$ distinct pulse amplitudes (M-ary PAM) for each symbol. Binary transmission is the special case $M = 2$.

Working

• The bit stream is split into groups of $k$ bits, e.g. $M = 4$ gives 2 bits per symbol.
• Each group is mapped (usually with Gray code) to one of the levels $\pm 1, \pm 3, \dots, \pm (M-1)$ times $A$.
• The receiver samples each symbol and compares it with $M - 1$ thresholds to decide the level, then maps the level back to bits.

 4-ary PAM (Gray coded)
 +3A  --  10
 +1A  --  11
 -1A  --  01
 -3A  --  00
 bits 10 01 11 00 -> levels +3A -1A +1A -3A

Key relations

$$R_s = \frac{R_b}{\log_2 M}, \qquad B_{min} = \frac{R_s}{2} = \frac{R_b}{2 \log_2 M}$$

So for the same bit rate, the needed bandwidth falls by a factor of $\log_2 M$. The bandwidth efficiency rises to $2\log_2 M$ b/s/Hz (Nyquist).

Advantages

• Higher data rate in a band-limited channel (e.g. telephone lines, 2B1Q in ISDN).
• Lower symbol rate, so less ISI and slower symbol timing.

Disadvantages

• For the same peak power, the levels are closer together, so the noise margin is smaller and the error probability rises. Keeping the same error rate needs more transmit power (roughly 6 dB more for each doubling of $M$ at large $M$).
• The receiver needs more decision thresholds and accurate gain control.

M-ary signalling therefore trades power for bandwidth.