--- title: "Error probability of M-ary system" subject: "Communication Systems" chapter: 9 key: "semester_6/communication_systems:mary-error-probability" source: https://www.bctengineer.com/q/semester_6/communication_systems/mary-error-probability site: BCT Engineering --- # Error probability of M-ary system Communication Systems · Chapter 9: Noise in Communication Systems - Asked in: 2075 Asoj (CS II), 8 marks - Video: [ESE 471 A General Formula for Prob Symbol Error in M ary PAM](https://www.youtube.com/watch?v=6o-uUbFNicQ) — Neal Patwari - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_6/communication_systems/mary-error-probability - Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-9 ## Question Derive expression for evaluating error probability of M-ary system. ## Short answer For M-ary PAM with equally spaced levels and thresholds midway, inner-level error is erfc(A/√2σ) and outer-level error half that. Averaging yields symbol error P_e=(1-1/M)erfc(A/√2σ). Substituting A²=3P_av/(M²-1) gives P_e=(1-1/M)erfc(√[3P_av/(2(M²-1)σ²)]). ## Answer In an **M-ary system** each symbol represents $k=\log_2M$ bits. For baseband M-ary PAM the receiver samples the filtered signal and decides which of the $M$ levels was sent using $M-1$ thresholds. **M-ary baseband PAM.** Each symbol takes one of $M$ equally likely levels spaced $2A$ apart, symmetric about zero: $$ a_i=\pm A,\ \pm3A,\ \dots,\ \pm(M-1)A $$ The received sample is $y=a_i+n$, $n\sim\mathcal{N}(0,\sigma^2)$. Thresholds are placed midway between adjacent levels (at $0,\pm2A,\pm4A,\dots$). ```text levels: -3A -A +A +3A (M = 4) -------*--|---*--|---*--|---*-------> y -2A 0 +2A thresholds outer levels: error on one side only inner levels: error on both sides ``` - **Inner level** (there are $M-2$): error if $\lvert n\rvert>A$: $P_{inner}=2\cdot\frac12\operatorname{erfc}\left(\frac{A}{\sqrt2\sigma}\right)=\operatorname{erfc}\left(\frac{A}{\sqrt2\sigma}\right)$ - **Outer level** (there are 2): error only if noise pushes it inward by more than $A$: $P_{outer}=\frac12\operatorname{erfc}\left(\frac{A}{\sqrt2\sigma}\right)$ Average symbol error probability: $$ \begin{aligned} P_e &= \frac{1}{M}\left[(M-2)\operatorname{erfc}\left(\frac{A}{\sqrt2\sigma}\right)+2\cdot\frac12\operatorname{erfc}\left(\frac{A}{\sqrt2\sigma}\right)\right] \\ &= \left(1-\frac1M\right)\operatorname{erfc}\left(\frac{A}{\sqrt2\,\sigma}\right)=2\left(1-\frac1M\right)Q\left(\frac{A}{\sigma}\right) \end{aligned} $$ **In terms of signal power.** Average symbol power $$ P_{av}=\frac{2}{M}\sum_{i=1}^{M/2}(2i-1)^2A^2=\frac{(M^2-1)A^2}{3}\ \Rightarrow\ A^2=\frac{3P_{av}}{M^2-1} $$ so $$ P_e=\left(1-\frac1M\right)\operatorname{erfc}\left(\sqrt{\frac{3}{2(M^2-1)}\cdot\frac{P_{av}}{\sigma^2}}\right) $$ For $M=2$ this reduces to the binary polar result $P_e=\frac12\operatorname{erfc}\left(\frac{A}{\sqrt2\sigma}\right)$. **Observations:** - For fixed $A/\sigma$, $P_e$ rises slowly with $M$ (factor $1-1/M$), but for fixed average power the spacing shrinks as $\sqrt{3/(M^2-1)}$, so $P_e$ rises sharply. - With Gray coding, bit error probability $\approx P_e/\log_2M$. - M-ary signalling reduces bandwidth by $\log_2M$ but needs more power: a bandwidth–power trade-off. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. For M-level PAM, how does an inner level err? A. Only if noise exceeds +A B. Never, because thresholds are far away C. If |n| > A on either side, giving erfc(A/(sqrt2 sigma)) D. Only if noise exceeds 2A Answer: C. Inner levels have a threshold on each side at distance A, so each side contributes (1/2) erfc. (Key point: Inner level errors) 2. How many outer levels are there and what is their error probability? A. 2 outer levels, each (1/2) erfc(A/(sqrt2 sigma)) B. M-2 outer levels, erfc C. 2 outer levels, each erfc D. M outer levels, each (1/2) erfc Answer: A. Outer levels have only one inward threshold, so only one side contributes. (Key point: Outer level errors) 3. What is the average symbol error probability? A. (1/M) erfc(A/(sqrt2 sigma)) B. (1 - 1/M) erfc(A/(sqrt2 sigma)) C. (1 + 1/M) erfc(A/(sqrt2 sigma)) D. (1/2) erfc(A/(sqrt2 sigma)) Answer: B. (1/M)[(M-2) erfc + 2(1/2)erfc] = ((M-1)/M) erfc. (Key point: Average symbol error probability) 4. What is the average symbol power of M-ary PAM with levels +/-A, +/-3A, ...? A. (M^2 - 1) A^2 / 3 B. M A^2 C. (M - 1) A^2 D. (M^2 + 1) A^2 / 3 Answer: A. Averaging squared odd multiples of A gives (M^2-1)A^2/3, so A^2 = 3P_av/(M^2-1). (Key point: Average power) ## Close variants set in other exams - [Binary baseband error probability; comparison with M-ary](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aerror-prob-binary-vs-mary) — Communication Systems, chapter 9, asked 3× ## Related in Communication Systems - [Error probability of binary ASK extended to M-ary](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aask-error-prob-mary) — Communication Systems, chapter 9 - [M-ary error probability; binary vs M-ary](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Amary-error-compare-binary) — Communication Systems, chapter 9 - [Binary PAM error probability extended to M-ary](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Abinary-pam-mary-error) — Communication Systems, chapter 9 - [Error probability in binary baseband system](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aerror-probability-binary-baseband) — Communication Systems, chapter 9, asked 2× - [Error probability in a binary communication system](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aerror-probability-binary-system) — Communication Systems, chapter 9 - [Binary error probability; threshold effect in FM and mitigation](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aerror-prob-threshold-effect) — Communication Systems, chapter 9, asked 2× ## Related in other subjects - [Derive expression for tile drain spacing](https://www.bctengineer.com/question.md?key=semester_6%2Firrigation_and_drainage_engineering%3Atile-drain-spacing-derivation) — Irrigation and Drainage Engineering, chapter 9, asked 8× - [Derive development length expression](https://www.bctengineer.com/question.md?key=semester_7%2Fdesign_of_rcc_structure%3Aderive-development-length-expression) — Design of RCC Structure, chapter 6, asked 6× - [Derive PET expression by Penman's method](https://www.bctengineer.com/question.md?key=semester_5%2Fengineering_hydrology%3Aderive-penman-pet) — Engineering Hydrology, chapter 3 - [Analytical resection, derive coordinate expression](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Aanalytical-resection-derive-three-known) — Surveying II, chapter 5