---
title: "M-ary error probability; binary vs M-ary"
subject: "Communication Systems"
chapter: 9
key: "semester_6/communication_systems:mary-error-compare-binary"
source: https://www.bctengineer.com/q/semester_6/communication_systems/mary-error-compare-binary
site: BCT Engineering
---

# M-ary error probability; binary vs M-ary

Communication Systems · Chapter 9: Noise in Communication Systems

- Asked in: 2071 Shrawan (CS II), 5+3 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-compare-binary
- Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-9

## Question

Derive the expression of error probability in case of M-ary system. Compare binary and M-ary scheme in terms of bandwidth efficiency and system complexity.

## Short answer

For M-ary PAM, symbol error probability is P_e = 2(1-1/M) Q(√(6E_s/((M^2-1)N_0))) with E_s = E_b log2 M. M-ary signaling improves bandwidth efficiency by a factor of log2 M but requires higher power and more complex hardware (M-1 thresholds) for the same error rate.

## Answer

#### Error probability of an M-ary (PAM) system

In an M-ary system each symbol carries $n = \log_2 M$ bits. Take **M-ary baseband PAM**: the receiver output sample (after the matched filter) takes one of $M$ equally spaced levels

$$
a_k \in \{\pm A,\ \pm 3A,\ \dots,\ \pm(M-1)A\}
$$

so adjacent levels are $2A$ apart. Assumptions: symbols equally likely, additive white Gaussian noise of PSD $N_0/2$, so the sampled noise $n$ is Gaussian with zero mean and variance $\sigma^2 = N_0/2$ (unit-energy pulse, matched filter). Decision thresholds are placed midway between levels: $0, \pm 2A, \pm 4A, \dots$

```text
  -3A     -A      +A     +3A      (M = 4)
   x  |    x   |   x   |   x
     -2A       0      +2A        <- thresholds
 outer         inner        outer
```

**Inner level** (two neighbours): an error occurs if $|n| > A$:

$$
P_{e,\text{inner}} = P(n>A) + P(n<-A) = 2Q\!\left(\frac{A}{\sigma}\right)
$$

**Outer level** (one neighbour, e.g. $+(M-1)A$): an error occurs only if $n < -A$:

$$
P_{e,\text{outer}} = Q\!\left(\frac{A}{\sigma}\right)
$$

There are $M-2$ inner and $2$ outer levels, so the average symbol error probability is

$$
\begin{aligned}
P_e &= \frac{1}{M}\left[(M-2)\,2Q\!\left(\frac{A}{\sigma}\right) + 2\,Q\!\left(\frac{A}{\sigma}\right)\right] \\
&= 2\left(1-\frac{1}{M}\right)Q\!\left(\frac{A}{\sigma}\right)
\end{aligned}
$$

**In terms of energy.** The average symbol energy is

$$
E_s = \frac{2}{M}\sum_{i=1}^{M/2}(2i-1)^2A^2 = \frac{(M^2-1)A^2}{3}
\;\Rightarrow\; A^2 = \frac{3E_s}{M^2-1}
$$

With $\sigma^2 = N_0/2$:

$$
P_e = 2\left(1-\frac{1}{M}\right)Q\!\left(\sqrt{\frac{6E_s}{(M^2-1)N_0}}\right), \qquad E_s = E_b\log_2 M
$$

**Check, M = 2:** $E_s = E_b$ and $P_e = Q\left(\sqrt{2E_b/N_0}\right)$, the polar binary result.

With Gray coding, bit error rate $\approx P_e/\log_2 M$.

#### Binary vs M-ary

| Point | Binary | M-ary |
|---|---|---|
| Bits per symbol | 1 | $\log_2 M$ |
| Symbol rate for bit rate $R_b$ | $R_b$ | $R_b/\log_2 M$ |
| Bandwidth needed | $\approx R_b/2$ (Nyquist) | $\approx R_b/(2\log_2 M)$ |
| Bandwidth efficiency | 2 bit/s/Hz (ideal) | $2\log_2 M$ bit/s/Hz |
| Power for same $P_e$ | lower | higher (levels closer) |
| Noise immunity | better | poorer |
| Transmitter/receiver | simple (one threshold) | complex ($M-1$ thresholds, AGC, precise level control) |

So M-ary signalling **saves bandwidth** by a factor $\log_2 M$ but needs **more signal power and a more complex system** for the same error rate. It is used where bandwidth is scarce (telephone-line modems, digital radio).

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. In M-ary PAM with equally spaced levels, why do inner levels have twice the error probability of outer levels?
   A. Inner levels have higher energy
   B. Inner levels have two neighbours, so noise can push the sample across either threshold
   C. Outer levels use a larger threshold
   D. Inner levels are sent more often
   Answer: B. An inner level errs if |n| > A giving 2Q(A/sigma); an outer level errs on one side only, giving Q(A/sigma). (Key point: Inner vs outer level errors)

2. The average symbol error probability of M-ary PAM is
   A. Q(A/sigma)
   B. 2(1 - 1/M) Q(A/sigma)
   C. (1/M) Q(A/sigma)
   D. 2M Q(A/sigma)
   Answer: B. Averaging (M-2) inner levels at 2Q and 2 outer levels at Q over M levels gives 2(1 - 1/M)Q(A/sigma). (Key point: Average symbol error probability)

3. Compared to binary signalling at the same bit rate, M-ary signalling needs a bandwidth smaller by a factor of
   A. M
   B. log2 M
   C. M^2
   D. M - 1
   Answer: B. Each symbol carries log2 M bits, so symbol rate and required bandwidth fall by log2 M. (Key point: Bandwidth efficiency of M-ary)

4. What is the price paid for the bandwidth saving of M-ary over binary?
   A. Lower data rate
   B. More power for the same error rate and a more complex system
   C. Less noise immunity but simpler receiver
   D. Larger symbol rate
   Answer: B. Levels are closer so more power is needed for the same P_e, and the receiver needs M-1 thresholds, AGC and precise level control. (Key point: Cost of M-ary)

## Related in Communication Systems

- [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×
- [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
- [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
- [AM vs FM efficiency and complexity; coherent ASK error](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aam-fm-bw-efficiency-ask-error) — Communication Systems, chapter 9, asked 2×
- [Error probability of M-ary system](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Amary-error-probability) — Communication Systems, chapter 9, asked 2×
- [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×

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