Lec 27: Probability of Error for PSK

Lec 27: Probability of Error for PSK

by Prof. Ribhu

Formal & Physical Sciences Engineering & Technology TJKCommunications engineering
🎙 Prof. Ribhu 👥 226K 📅 July 31, 2026 ⏱ 28 min 👁 18 📄 lecture 🧭 2026-08-02
Available in: English (current) Français

Keywords

M-PSKprobability of errorconstellationmaximum likelihoodcircularly symmetric noise

Summary

This lecture, part of the NPTEL course ‘Analog and Digital Communications II’, focuses on deriving the probability of error for M-ary Phase Shift Keying (M-PSK). The instructor begins by reviewing the constellation diagram for M-PSK, where symbols are equally spaced on a circle. He then sets up the maximum likelihood detection rule, noting that due to symmetry, the error probability is the same for all symbols, so he can focus on symbol S1. The received signal is modeled as a complex Gaussian random variable with circularly symmetric noise. The decision region for S1 is defined by the angle of the received signal, which must lie between -π/M and π/M. The probability of error is expressed as an integral over this angular region, involving the Rician distribution for the magnitude. The instructor points out that this integral cannot be evaluated in closed form and requires numerical methods. He also mentions that a similar issue arises for QAM, which will be discussed in the next lecture. The lecture concludes by setting the stage for further exploration of error probability in digital modulation schemes.

181 words

Critical Evaluation

The lecture provides a rigorous derivation of the probability of error for M-PSK, a fundamental topic in digital communications. The instructor, Prof. Ribhu, demonstrates a strong command of the subject, systematically building from the constellation diagram to the final integral expression. The argumentation is logical and well-structured, with clear explanations of the symmetry arguments that simplify the analysis. The use of complex Gaussian noise and the circular symmetry property is appropriate and correctly applied. The derivation of the probability density function for the received signal’s magnitude is stated without full derivation, but the instructor acknowledges this and refers to standard results from probability theory, which is acceptable for a lecture of this level. The main limitation is that the final integral is not evaluated in closed form, and the instructor does not provide numerical results or approximations, which might leave the student without a concrete sense of the error probability values. However, this is a common issue in such derivations, and the instructor explicitly mentions that numerical methods are required. The lecture is well-paced and suitable for an advanced undergraduate or graduate audience. The title accurately reflects the content, and the lecture fulfills its promise. Overall, this is a high-quality educational resource, though it lacks practical examples or simulations that could enhance understanding.

213 words

Title / Content Match

The title accurately reflects the content, which focuses on deriving the probability of error for M-PSK modulation.

Quality & Reliability

8/10

Lecture by a professor from IIT Guwahati, part of a formal NPTEL course. The content is mathematically rigorous, with derivations and references to standard results. However, the video is a lecture, not peer-reviewed, and some steps are stated without full derivation.

Key Moments

Markers derived by PSI from the transcript: the creator did not define chapters.

Cited Sources

Concurring Sources

  • Digital Communications by John G. Proakis — Standard textbook covering error probability for M-PSK.
  • Principles of Digital Communication by Robert G. Gallager — Another authoritative source on digital modulation and error analysis.

Contribution & Novelties

This lecture provides a clear and systematic derivation of the probability of error for M-PSK, highlighting the role of symmetry and the intractability of the resulting integral. It bridges the gap between theoretical concepts and practical computation, emphasizing the need for numerical methods.

Pour aller plus loin :

  • Phase-shift keying — Overview of PSK modulation techniques.
  • Rician fading — The magnitude distribution of a complex Gaussian with non-zero mean, relevant to the derived PDF.
  • Numerical integration — Methods like Simpson’s rule mentioned in the lecture for evaluating the error probability integral.

91 words

Radar Profile

The radar profile shows high scores in technical level and information quality, reflecting the advanced mathematical content and rigorous derivation. The quantity of information is also high, but the overall reliability is slightly lower due to the lack of peer review and the reliance on stated results without full derivation.

Reliability 8/10