PCM switching and three-stage crosspoints

Advantages of PCM switching over analog switching

• No accumulation of noise: digital signals are regenerated; quality does not depend on the number of switching stages.
• Time sharing of hardware: one crosspoint or memory location serves many calls, so fewer components are needed.
• Integration with PCM transmission: digital trunks are switched without A/D conversion at each exchange (integrated digital network).
• Low cost, size and power using VLSI; easy maintenance and SPC features.
• Data/ISDN services and fast common channel signaling.

Issues (problems) of PCM switching

• Synchronisation: all exchanges must run on a common clock, otherwise frame slips occur.
• Codec and hybrid per line: each analogue subscriber line needs BORSCHT functions, which costs more per line.
• Delay and echo: buffering adds delay (up to a frame per T stage), and 2-wire/4-wire hybrids cause echo that may need cancellers.
• Memory speed limits the size of a single time switch.
• Software complexity and dependence on processors.

Crosspoints in the three-stage switch

Reading of the data: 3 first-stage arrays, each with 4 input lines, so $N = 3 \times 4 = 12$, $n = 4$; 5 second-stage arrays, $k = 5$.

• Stage 1: 3 arrays of $4 \times 5$
• Stage 2: 5 arrays of $3 \times 3$
• Stage 3: 3 arrays of $5 \times 4$

 4 in [4x5] \             / [5x4] 4 out
 4 in [4x5] --> 5 x [3x3] --> [5x4] 4 out
 4 in [4x5] /    middle   \ [5x4] 4 out
 (12 inputs)                 (12 outputs)
 each outer array has 1 link to each
 of the 5 middle arrays

$$\begin{aligned} N_1 &= 3 \times (4 \times 5) = 60 \\ N_2 &= 5 \times (3 \times 3) = 45 \\ N_3 &= 3 \times (5 \times 4) = 60 \\ N_x &= 60 + 45 + 60 = 165 \end{aligned}$$

Check: $N_x = 2Nk + k(N/n)^2 = 2(12)(5) + 5(3)^2 = 120 + 45 = 165$.

Comment: a single-stage $12 \times 12$ switch needs only 144 crosspoints, so for such a small $N$ the three-stage form gives no saving; it pays off only for large $N$. Also $k = 5 < 2n - 1 = 7$, so the network is blocking.

Answer: total crosspoints = 165.