# A short survey on QPSK Costas loop mathematical models

## TL;DR

This survey reviews mathematical models of the QPSK Costas loop, highlighting the challenges of nonlinear analysis and discrepancies between simplified models and real circuit behavior.

## Contribution

It provides a comprehensive overview of existing mathematical models and discusses their limitations in accurately representing the QPSK Costas loop.

## Key findings

- Simplified models often differ from actual circuit behavior.
- Nonlinear analysis of the Costas loop remains a complex challenge.
- The survey identifies key issues in current modeling approaches.

## Abstract

The Costas loop is a modification of the phase-locked loop circuit, which demodulates data and recovers carrier from the input signal. The Costas loop is essentially a nonlinear control system and its nonlinear analysis is a challenging task. Thus, simplified mathematical models and their numerical simulation are widely used for its analysis. At the same time for phase-locked loop circuits there are known various examples where the results of such simplified analysis are differ substantially from the real behavior of the circuit. In this survey the corresponding problems are demonstrated and discussed for the QPSK Costas loop.