Digital exchange principles; non-blocking 3-stage

Principles of a digital exchange

1. Digital (PCM) representation: speech is sampled at 8 kHz, quantised and coded into 8 bits (64 kbps per channel). 32 channels form a 2.048 Mbps E1 frame of 125 µs.
2. Time division switching: switching hardware is shared in time. Time switches (TSIs) change the time slot of a sample; space switches change the highway. They are combined as TST, STS, TSST, etc.
3. Stored program control: a processor runs software for call processing, routing, charging and maintenance.
4. Line interface (BORSCHT): battery feed, over-voltage protection, ringing, supervision, coding, hybrid and testing for each analogue line.
5. Digital trunks and CCS: inter-exchange trunks are PCM links, and signaling is usually SS7.
6. Synchronisation: all exchanges work on a common clock to avoid slips.

 lines-->[Line units]-->[ T-S-T network ]-->[Trunks]
                ^              ^              ^
                +------[ SPC processor ]------+

Non-blocking switch with a three-stage matrix

A single $N \times N$ matrix is non-blocking but needs $N^2$ crosspoints. A three-stage network uses fewer.

Structure: $N$ inputs in groups of $n$:

• Stage 1: $N/n$ arrays of $n \times k$
• Stage 2: $k$ arrays of $(N/n) \times (N/n)$
• Stage 3: $N/n$ arrays of $k \times n$

 n in [n x k] \               / [k x n] n out
 n in [n x k] --> k arrays  --> [k x n] n out
  ...         /  (N/n x N/n)  \    ...

$$N_x = 2Nk + k\left(\frac{N}{n}\right)^2$$

Clos condition for strict non-blocking: a new call from input array A to output array B is worst-hit when the other $n-1$ inputs of A use $n-1$ middle arrays and the other $n-1$ outputs of B use $n-1$ different middle arrays. One more is needed:

$$k \ge 2(n - 1) + 1 = 2n - 1$$

Minimum crosspoints: with $k = 2n - 1$, $N_x$ is minimum at $n = \sqrt{N/2}$, giving

$$N_{x,min} = 4N\left(\sqrt{2N} - 1\right)$$

Example: $N = 128$: $n = 8$, $k = 15$, $N/n = 16$:

$$N_x = 2(128)(15) + 15(16)^2 = 3840 + 3840 = 7680$$

A single-stage switch would need $128^2 = 16\,384$ crosspoints.

Digital equivalent: in a TST switch the internal time slots act like middle arrays, so it is non-blocking when $l \ge 2c - 1$ (e.g. $c = 32$, $l = 63$, so 64 internal slots are used).