Talker and listener echo in four-wire circuit

Assumptions (standard four-wire model): $L$ = overall loss of the four-wire circuit between its two-wire terminals (hybrid to hybrid), $B$ = balance return loss of each hybrid, stability margin $S = L + B$ (half the loss round the four-wire loop). Each subscriber is joined to a hybrid by a two-wire circuit of 6 dB loss and 1 ms delay.

Four-wire loss

$$L = S - B = 3 - 3 = 0\ \text{dB}$$

 Talker─[2W: 6dB,1ms]─H1══ 4-wire (L) ══H2─[2W]─Listener
           ^                               │ reflects (B)
           └──────── talker echo <─────────┘

Talker echo (talker's speech reflected at the far hybrid and heard by the talker):

$$\begin{aligned} A_{TE} &= 6 + L + B + L + 6 \\ &= 6 + 0 + 3 + 0 + 6 = 15\ \text{dB} \end{aligned}$$ $$\begin{aligned} \text{Delay}_{TE} &= 20\ \text{ms (4-wire round trip)} + 2 \times 1\ \text{ms (own 2-wire both ways)} \\ &= 22\ \text{ms} \end{aligned}$$

Listener echo (signal reflected at the far hybrid, then again at the near hybrid, reaching the listener late). Compared with the direct signal it travels once more round the four-wire loop:

$$A_{LE} = 2L + 2B = 2S = 2 \times 3 = 6\ \text{dB below the direct signal}$$

(absolute: $6 + 3L + 2B + 6 = 18$ dB from talker to listener, against 12 dB for the direct signal), delayed by 20 ms relative to it.

Answer: talker echo attenuation = 15 dB, talker echo delay = 22 ms, listener echo attenuation = 6 dB (relative to the received speech, i.e. 18 dB overall).