--- title: "Digital exchange and three-stage crosspoints" subject: "Telecommunications" chapter: 4 key: "semester_8/telecommunications:digital-exchange-crosspoints" source: https://www.bctengineer.com/q/semester_8/telecommunications/digital-exchange-crosspoints site: BCT Engineering --- # Digital exchange and three-stage crosspoints Telecommunications · Chapter 4: Digital Switching - Asked in: 2080 Bhadra, 4+4 marks - Video: [Switching Engineering - Cross-points and Number of Switches - Calculations](https://www.youtube.com/watch?v=Bbm3lKI7O5Y) — KINEMARC - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_8/telecommunications/digital-exchange-crosspoints - Whole chapter: https://www.bctengineer.com/md/semester_8/telecommunications/chapter-4 ## Question Explain the working principle of Digital Telephone Exchange. Calculate how many cross point used in three stage switching. ## Short answer A digital exchange switches PCM samples through time and space stages under stored program control, with line cards performing BORSCHT functions. For a three-stage space switch, crosspoints = 2Nk + k(N/n)^2. With N=100, n=10, k=2n−1=19, this equals 2(100)(19) + 19(10)^2 = 3800+1900 = 5700, 43% less than 10000 in a single stage. ## Answer #### Working principle of a digital telephone exchange A **digital exchange** switches speech as PCM samples (8-bit words, 8000 per second) using time-division techniques, under **stored program control (SPC)**. ```text Subscriber +-------+ +----------+ +-------+ lines ---->| Line |->| Digital |->| Trunk |--> other (analog) | units |<-| switching|<-| units | exch. |BORSCHT| | (TST) | |(E1) | +-------+ +----------+ +-------+ ^ ^ ^ +-------------------------------+ | Control processor (SPC) | | + signaling | +-------------------------------+ ``` 1. **Line unit:** each analogue line has a line card doing the **BORSCHT** functions: Battery feed, Over-voltage protection, Ringing, Supervision, Coding (codec: A/D and D/A), Hybrid (2-wire to 4-wire), Testing. 2. **Concentration and multiplexing:** coded samples from many lines are multiplexed onto 32-channel (2.048 Mbps) PCM highways. 3. **Digital switching network:** time switches (TSIs) and space switches (usually in TST form) move each sample from its input slot and highway to the required output slot and highway. 4. **Control:** the processor detects off-hook, collects digits, finds a free path, writes the control memories of the T and S stages, sends ringing and tones, supervises and records charging data. 5. **Trunk and signaling units:** connect to other exchanges over E1 links; signaling is by CAS or CCS (SS7). #### Crosspoints in a three-stage switch Take $N$ inputs and $N$ outputs, divided into groups of $n$ lines, with $k$ middle-stage arrays. - First stage: $N/n$ arrays, each $n \times k$ - Second stage: $k$ arrays, each $(N/n) \times (N/n)$ - Third stage: $N/n$ arrays, each $k \times n$ $$ \begin{aligned} N_x &= \frac{N}{n}(nk) + k\left(\frac{N}{n}\right)^2 + \frac{N}{n}(kn) \\ &= 2Nk + k\left(\frac{N}{n}\right)^2 \end{aligned} $$ For a strictly non-blocking (Clos) network, $k = 2n - 1$: $$ N_x = (2n-1)\left[2N + \left(\frac{N}{n}\right)^2\right] $$ **Example:** $N = 100$, $n = 10$, $k = 2n-1 = 19$: $$ \begin{aligned} N_x &= 2(100)(19) + 19\left(\frac{100}{10}\right)^2 \\ &= 3800 + 1900 = 5700 \end{aligned} $$ A single-stage $100 \times 100$ switch needs $N^2 = 10\,000$ crosspoints, so the three-stage design saves 43% and is still non-blocking. **Answer:** $N_x = 2Nk + k(N/n)^2$; for $N = 100$, $n = 10$, $k = 19$, $N_x = 5700$ crosspoints. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What does the C in BORSCHT stand for and do? A. Coding: codec doing A/D and D/A conversion B. Charging of the call C. Concentration of lines D. Clocking Answer: A. Coding is the codec on each line card that converts analogue speech to PCM and back. (Key point: BORSCHT) 2. For N lines, groups of n and k middle arrays, the total crosspoints in a three-stage switch are A. Nk + k(N/n) B. 2Nk + k(N/n)^2 C. 2Nk + k(N/n) D. N^2/n Answer: B. Stages 1 and 3 give N/n arrays of n x k each (Nk each), and stage 2 gives k arrays of (N/n)^2. (Key point: Crosspoint formula) 3. For N = 100, n = 10 with strictly non-blocking k, what are k and the crosspoints? A. k = 10, 3000 B. k = 19, 5700 C. k = 20, 10000 D. k = 19, 3800 Answer: B. k = 2n - 1 = 19; Nx = 2(100)(19) + 19(10)^2 = 3800 + 1900 = 5700. (Key point: Clos condition and result) 4. What does the control processor of the exchange do when a call arrives? A. Only supplies battery B. Collects digits, finds a free path and writes the control memories of the switching stages C. Converts E1 to T1 D. Only stores billing Answer: B. It detects off-hook, collects digits, finds a free path, and sets up the T and S stage control memories. (Key point: Function of control processor) ## Related in Telecommunications - [Working principle of a digital telephone exchange](https://www.bctengineer.com/question.md?key=semester_8%2Ftelecommunications%3Adigital-telephone-exchange-principle) — Telecommunications, chapter 4, asked 4× - [Crosspoints and three-stage switching](https://www.bctengineer.com/question.md?key=semester_8%2Ftelecommunications%3Acrosspoint-switching-array) — Telecommunications, chapter 4 - [Switching functions and three-stage crosspoints](https://www.bctengineer.com/question.md?key=semester_8%2Ftelecommunications%3Aswitching-functions-crosspoints-4x4) — Telecommunications, chapter 4, asked 2× - [PCM switching and three-stage crosspoints](https://www.bctengineer.com/question.md?key=semester_8%2Ftelecommunications%3Apcm-switching-crosspoints-4x5) — Telecommunications, chapter 4 - [Crosspoints in three-stage switch (4 inputs, 5 arrays)](https://www.bctengineer.com/question.md?key=semester_8%2Ftelecommunications%3Acrosspoints-4-input-5-second) — Telecommunications, chapter 4, asked 2× - [Telephone exchange and switching hierarchy](https://www.bctengineer.com/question.md?key=semester_8%2Ftelecommunications%3Aexchange-switching-hierarchy) — Telecommunications, chapter 1, asked 3×