Companding; T1 hierarchy of TDM-PCM telephony

Companding

Companding = compressing + expanding. Before uniform quantisation, the signal passes through a compressor that boosts weak amplitudes and compresses strong ones. At the receiver, an expander applies the inverse curve. The result works like non-uniform quantisation: small step sizes for weak signals and large steps for loud ones.

This gives a nearly constant SQNR over a wide range of speech levels, using only 8 bits.

 Tx: m(t) -> [Compressor] -> [Uniform quantizer
                              + encoder] ~~> channel
 Rx: channel ~~> [Decoder] -> [Expander] -> m(t)

Standard laws: µ-law ($\mu = 255$, North America/T1) and A-law ($A = 87.6$, Europe/E1).

$$y = \frac{\ln(1 + \mu |x|)}{\ln(1 + \mu)} \operatorname{sgn}(x), \quad |x| \le 1$$

T1 hierarchy of TDM-PCM telephony

T1 (DS1) multiplexes 24 PCM voice channels:

|<--------- one frame = 125 us = 193 bits --------->|
+---+--------+--------+--------+-------+--------+
| F |  Ch 1  |  Ch 2  |  Ch 3  |  ...  | Ch 24  |
+---+--------+--------+--------+-------+--------+
 1 bit  8 bits  8 bits  8 bits           8 bits
 F = framing bit (pattern spread over 12 frames)

$$\begin{aligned} \text{Bits per frame} &= 24 \times 8 + 1 = 193\ \text{bits} \\ \text{Frames per second} &= f_s = 8000\ \text{frames/s} \\ R_b &= 193 \times 8000 = 1\,544\,000\ \text{b/s} = 1.544\ \text{Mb/s} \end{aligned}$$

Higher levels are formed by multiplexing T1 streams and adding framing and stuffing bits:

 24 voice ch x 64 kb/s
        |
   [ DS1 / T1 ]  1.544 Mb/s    24 ch
        | x4
   [ DS2 / T2 ]  6.312 Mb/s    96 ch
        | x7
   [ DS3 / T3 ]  44.736 Mb/s   672 ch
        | x6
   [ DS4 / T4 ]  274.176 Mb/s  4032 ch