--- title: "Digital exchange principles; non-blocking 3-stage" subject: "Telecommunications" chapter: 4 key: "semester_8/telecommunications:digital-exchange-nonblocking-3-stage" source: https://www.bctengineer.com/q/semester_8/telecommunications/digital-exchange-nonblocking-3-stage site: BCT Engineering --- # Digital exchange principles; non-blocking 3-stage Telecommunications · Chapter 4: Digital Switching - Asked in: 2071 Chaitra, 4+6 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-nonblocking-3-stage - Whole chapter: https://www.bctengineer.com/md/semester_8/telecommunications/chapter-4 ## Question What are principles of digital exchange? Describe non blocking switches with 3 stages switching matrix. ## Short answer Digital exchanges use 64 kbps PCM (E1 frames), time-slot interchange, stored program control, and BORSCHT interfaces. A three-stage non-blocking switch requires k ≥ 2n-1 (Clos); with n=√(N/2) crosspoints minimize; for N=128, 7680 crosspoints. ## Answer #### 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. ```text 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 $k \times n$ ```text 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). ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Which describes time division switching in an exchange? A. Each call has own hardware B. Hardware is shared in time; T switches change the slot and S switches change the highway C. Only space switches are used D. Switches use frequency division Answer: B. TSIs change slots, space switches change highways. (Key point: Digital exchange principle) 2. What is the strict non-blocking condition for a three-stage switch? A. k >= n B. k >= 2n - 1 C. k >= 2n + 1 D. k >= N/n Answer: B. The worst case uses 2(n-1) middle arrays already, so one more is needed: k >= 2n - 1. (Key point: Clos condition) 3. Crosspoint formula for a three-stage network? A. Nx = N^2 B. Nx = 2Nk + k(N/n)^2 C. Nx = Nk + n^2 D. Nx = 2nk + N^2 Answer: B. Outer stages give 2Nk and the middle stage k(N/n)^2. (Key point: Crosspoint count) 4. For N = 128, n = 8, k = 15, what is Nx? A. 16384 B. 3840 C. 7680 D. 1920 Answer: C. 2(128)(15) = 3840, and 15(16)^2 = 3840, total 7680. (Key point: Example) ## Related in Telecommunications - [Digital exchange and three-stage crosspoints](https://www.bctengineer.com/question.md?key=semester_8%2Ftelecommunications%3Adigital-exchange-crosspoints) — Telecommunications, chapter 4 - [Blocking and non-blocking switching](https://www.bctengineer.com/question.md?key=semester_8%2Ftelecommunications%3Ablocking-nonblocking-switching) — Telecommunications, chapter 4 - [Strictly non-blocking 3-stage condition](https://www.bctengineer.com/question.md?key=semester_8%2Ftelecommunications%3Astrictly-nonblocking-3-stage) — Telecommunications, chapter 4 - [S and T switches; 3-stage vs 2-stage](https://www.bctengineer.com/question.md?key=semester_8%2Ftelecommunications%3As-t-switches-3-stage) — Telecommunications, chapter 4 - [TST time slots for non-blocking operation](https://www.bctengineer.com/question.md?key=semester_8%2Ftelecommunications%3Atst-time-slots-numerical) — Telecommunications, chapter 4 - [Multistage switching and TST blocking](https://www.bctengineer.com/question.md?key=semester_8%2Ftelecommunications%3Amultistage-tst-blocking) — Telecommunications, chapter 4 ## Related in other subjects - [Tridiagonal matrix, river stage and water table](https://www.bctengineer.com/question.md?key=semester_8%2Fcomputational_techniques_in_civil_engineering%3Atridiagonal-matrix-river-aquifer) — Computational Techniques in Civil Engineering, chapter 7, asked 2× - [Analog and digital system block diagrams; linear and non-linear distortion](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aanalog-digital-block-linear-nonlinear-distortion) — Communication Systems, chapter 1 - [Stages of hydropower project study](https://www.bctengineer.com/question.md?key=semester_6%2Fhydro_power%3Ahydropower-study-stages) — Hydro Power, chapter 3, asked 3× - [Optimum filter; Remez exchange expression](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Aoptimum-filter-remez-expression) — Digital Signal Analysis and Processing, chapter 5, asked 5×