---
title: "Error probability of binary PAM"
subject: "Communication Systems"
chapter: 9
key: "semester_6/communication_systems:binary-pam-error-probability"
source: https://www.bctengineer.com/q/semester_6/communication_systems/binary-pam-error-probability
site: BCT Engineering
---

# Error probability of binary PAM

Communication Systems · Chapter 9: Noise in Communication Systems

- Asked in: 2072 Kartik (CS II), 5 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/binary-pam-error-probability
- Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-9

## Question

Derive the expression of error probability for binary PAM signal.

## Short answer

For binary polar PAM with matched filtering and equal prior probabilities, the error probability is Pe = 0.5 erfc(√(Eb/N0)). The received sample ±A + n is compared to a zero threshold; the noise variance is σ² = N0/2.

## Answer

In **binary (polar) PAM** bit 1 is sent as a pulse of amplitude $+A$ and bit 0 as $-A$. White Gaussian noise is added; the receiver samples once per bit and decides with threshold $\lambda=0$.

The sample is $y=\pm A+n$, $n\sim\mathcal{N}(0,\sigma^2)$:

$$
f(y\mid1)=\frac{1}{\sqrt{2\pi}\sigma}e^{-(y-A)^2/2\sigma^2},\qquad f(y\mid0)=\frac{1}{\sqrt{2\pi}\sigma}e^{-(y+A)^2/2\sigma^2}
$$

```text
    f(y|0)       f(y|1)
     .--.    |    .--.
    /    \   |   /    \
 --/------\..|../------\---> y
  -A         0        +A
```

**Error when 0 is sent** ($y>0$): let $u=\frac{y+A}{\sqrt2\sigma}$

$$
\begin{aligned}
P_{e0} &= \int_0^{\infty}\frac{1}{\sqrt{2\pi}\sigma}e^{-(y+A)^2/2\sigma^2}dy=\frac{1}{\sqrt\pi}\int_{A/\sqrt2\sigma}^{\infty}e^{-u^2}du \\
&= \frac12\operatorname{erfc}\left(\frac{A}{\sqrt2\,\sigma}\right)
\end{aligned}
$$

By symmetry $P_{e1}=P_{e0}$. With equal priors:

$$
P_e=\frac12\operatorname{erfc}\left(\frac{A}{\sqrt2\,\sigma}\right)=Q\left(\frac{A}{\sigma}\right)
$$

With a matched filter, $\frac{A^2}{\sigma^2}=\frac{2E_b}{N_0}$, so $P_e=\frac12\operatorname{erfc}\sqrt{E_b/N_0}$. For **unipolar** PAM ($A$, 0, threshold $A/2$): $P_e=\frac12\operatorname{erfc}\left(\frac{A}{2\sqrt2\sigma}\right)$, which needs twice the average power for the same $P_e$.

## Check yourself

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

1. In polar binary PAM (+A for 1, -A for 0) with Gaussian noise of variance σ^2, the optimum threshold with equal priors is:
   A. A
   B. 0, midway between -A and +A
   C. A/2
   D. -A
   Answer: B. The two pdfs are Gaussians centred at ±A with equal variance, so the equal-error threshold is 0. (Key point: Decision rule and noise model)

2. The error probability of polar binary PAM is:
   A. (1/2) erfc( A / (√2 σ) ) = Q(A/σ)
   B. (1/2) erfc( A / (2√2 σ) )
   C. Q(2A/σ)
   D. (1/2) exp(-A/σ)
   Answer: A. P_e0 = integral over y > 0 of the Gaussian centred at -A, giving (1/2) erfc(A/(√2σ)); by symmetry P_e1 is the same. (Key point: Error probability expression)

3. For unipolar PAM (A and 0, threshold A/2) the same P_e needs, compared to polar PAM:
   A. Half the average power
   B. Equal power
   C. Twice the average power
   D. Four times the peak amplitude
   Answer: C. Unipolar error argument is A/(2√2σ), half the polar one, so it needs twice the average power for the same P_e. (Key point: Unipolar vs polar)

## Related in Communication Systems

- [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
- [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×
- [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×
- [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 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

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