E1 digital hierarchy; SQNR in uniformly quantized PCM

E1 digital hierarchy

E1 is the ITU-T (CEPT) primary PCM-TDM multiplex used in Europe and most of Asia, including Nepal.

E1 frame

• Voice (300–3400 Hz) sampled at 8 kHz, A-law companded, 8 bits/sample: 64 kbps per channel.
• Frame duration 125 µs, 32 time slots × 8 bits = 256 bits.
• TS0: frame synchronisation/alignment; TS16: signalling (multiframe of 16 frames); TS1–15 and TS17–31: 30 voice channels.
• Rate $= 256 \times 8000 = 2.048$ Mbps.

|TS0|TS1 ... TS15|TS16|TS17 ... TS31|  125 us
sync   voice 1-15  sig   voice 16-30

Higher orders (PDH) multiplex 4 tributaries each, plus framing and justification bits:

(The North American counterpart is T1: 24 channels, 1.544 Mbps.)

SQNR of uniformly quantized PCM

Message range $\pm m_p$, $L = 2^n$ levels.

1. Step size: $\Delta = \dfrac{2m_p}{L}$.
2. Error $q$ uniform in $(-\Delta/2, \Delta/2)$, pdf $1/\Delta$:

$$N_q = \int_{-\Delta/2}^{\Delta/2}\frac{q^2}{\Delta}\,dq = \frac{\Delta^2}{12} = \frac{m_p^2}{3L^2}$$

3. Signal power $S = \overline{m^2}$:

$$\text{SQNR} = \frac{S}{N_q} = 3L^2\frac{\overline{m^2}}{m_p^2}$$

4. For a full-load sinusoid, $\overline{m^2} = m_p^2/2$:

$$\begin{aligned} \text{SQNR} &= \frac{3}{2}L^2 = 1.5 \times 2^{2n} \\ \text{SQNR}_{dB} &= 1.76 + 6.02n\ \text{dB} \end{aligned}$$

So SQNR increases by about 6 dB for each added bit. Example: for E1 channels with 8-bit uniform coding, $1.76 + 48.16 = 49.9$ dB for a full-scale tone (companding keeps it high for weak signals too).