折叠Kolmogorov-Smirnov调制分类器：提高鲁棒性，降低复杂度

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"这篇研究论文提出了一种基于折叠的Kolmogorov-Smirnov调制分类器，用于在电子战和干扰消除等应用场景中识别调制格式。该方法通过对接收到的信号进行折叠预处理，利用不同调制格式的对称特性来提升识别效果。" Kolmogorov-Smirnov调制分类器是一种新颖的信号处理工具，它在处理调制识别问题时，借鉴了统计学中的Kolmogorov-Smirnov检验（K-S检验）。K-S检验通常用于比较两个分布是否一致，而在此文中，这一概念被巧妙地应用于分类不同类型的调制格式。折叠操作是该分类器的核心，它可以将接收到的信号映射到一个或多个对称轴上，这些轴反映了各种调制方式的特性。在介绍部分，作者强调了调制分类的重要性，尤其是在军事和民用领域，如电子战和频谱监控，这些都需要在未知调制格式的情况下进行信号解调。传统的基于最优似然的分类器虽然性能优异，但在面对噪声不确定性和计算复杂性时可能存在挑战。而提出的折叠Kolmogorov-Smirnov分类器则旨在解决这些问题。仿真结果显示，新分类器的性能接近最优似然分类器，这表明其在识别精度上具有竞争力。更重要的是，通过改善对噪声不确定性的鲁棒性，该分类器在实际应用中可能更为稳定。此外，由于减少了计算需求，其运行速度也有所提升，这对于实时系统尤其关键。在论文的主体部分，作者可能详细介绍了折叠操作的具体实现，包括如何构建对称轴、如何计算K-S统计量以及如何设置阈值进行分类决策。他们可能还对比了新方法与其他分类方法的性能，提供了仿真条件和实验结果，以证明其优势。这篇研究论文提出了一个创新的调制识别方案，它利用了折叠和K-S检验的特性，有效地平衡了识别准确性和计算效率，对于未来在复杂通信环境中的应用具有潜在价值。

IEEE SIGNAL PROCESSING LETTERS, VOL. 23, NO. 7, JULY 2016 1003

Fold-based Kolmogorov–Smirnov

Modulation Classiﬁer

Fanggang Wang, Octavia A. Dobre, Chung Chan, and Jingwen Zhang

Abstract—Modulation classiﬁcation is crucial in applications

such as electronic warfare and interference cancellation. In this

letter, a novel feature-based Kolmogorov–Smirnov classiﬁer is pro-

posed for the identiﬁcation of the modulation formats. The received

signal is ﬁrst preprocessed with a folding operation that helps iden-

tify the modulation formats based on their different axes of symme-

try. Simulation results show that the performance of the proposed

classiﬁer is close to that of the optimal likelihood-based classiﬁer,

while its robustness to noise uncertainty is improved and its com-

putational complexity is reduced compared to that of the optimal

likelihood-based classiﬁer.

I. INTRODUCTION

ODULATION classiﬁcation represents a step performed

before demodulation when the modulation format is un-

known to the receiver, and plays a key role in various military

and civilian applications, such as electronic warfare, spectrum

surveillance, software-deﬁned and cognitive radios, and inter-

ference cancellation [1]–[5]. Likelihood and feature-based clas-

siﬁers represent two main classes of modulation classiﬁcation

techniques, being surveyed in [1]–[5]. The maximum likeli-

hood classiﬁer [6], [7] and the cumulant-based classiﬁer [8]

are seminal works for these two classes, respectively. Recently,

research on modulation classiﬁcation has focused on: 1) the

investigation of the classiﬁcation problems in non-ideal sce-

narios, e.g., in [9]–[11]; and 2) ﬁnding powerful features and

tools to improvethe performance of the feature-based classiﬁers,

e.g., in [12] and [13]. Speciﬁcally, in [9], a likelihood-based

classiﬁer is proposed for classiﬁcation in time-correlated non-

Gaussian channels, which involves a whitening ﬁlter. In [10],

the cumulant-based classiﬁer is extended to multipath channels.

In [11], a method relying on cyclic cumulant is proposed, which

is robust to carrier phase and frequency offsets. In [12], the

concept of using the empirical cumulative distribution function

of the received signal for modulation classiﬁcation is ﬁrst pro-

posed, which is further exploited in [13]. This technique has the

Manuscript received November 04, 2015; revised May 17, 2016; accepted

May 22, 2016. Date of publication May 24, 2016; date of current version June

22, 2016. This work was supported in part by the National Natural Science

Foundation under Grant 61571034 and under Grant U1334202, the Fundamen-

tal Research Funds for the Central Universities under Grant 2015JBM112,

the State Key Laboratory of Rail Trafﬁc Control and Safety under Grant

RCS2016ZT013, and the Natural Sciences and Engineering Research Coun-

cil of Canada (NSERC) under Grant 327285. The associate editor coordinating

the review of this manuscript and approving it for publication was Prof. Peter

K. Willett.

F. Wang and J. Zhang are with the Beijing Jiaotong University, Beijing

100044, China (e-mail: wangfg@bjtu.edu.cn; 12111009@bjtu.edu.cn).

O. A. Dobre is with the Memorial University, NL A1B 3X5, Canada (e-mail:

odobre@mun.ca).

C. Chan is with the Chinese University of Hong Kong, Hong Kong (e-mail:

cchan@inc.cuhk.edu.hk).

Color versions of one or more of the ﬁgures in this paper are available online

at http://ieeexplore.ieee.org.

Digital Object Identiﬁer 10.1109/LSP.2016.2572666

advantage of low computational complexity and robustness to

nonideal scenarios when compared with the optimal likelihood-

based approach; however, there exists an obvious gap in the

classiﬁcation performance.

In this letter, we propose a feature-based Kolmogorov–

Smirnov (K-S) classiﬁer that further improves the classiﬁcation

performance of those in [12] and [13], reducing the perfor-

mance gap from the optimal approach. To illustrate the idea

of the proposed classiﬁer, suppose that the signal of interest is

real-valued and modulated by either binary phase-shift keying

(BPSK) or amplitude-shift keying (ASK) with four points in

the signal constellation, i.e., 4-ASK. The signal constellation

of BPSK consists of two points, while that of 4-ASK consists

of four points, and the samples obtained from the noisy signal

are scattered around the points in the signal constellation. Let

us consider the case of low noise power. Given that the number

of samples is large enough, they would concentrate around the

signal points and it would then be easy to identify the mod-

ulation formats by counting the number of “hot spots” where

the samples concentrate. However, if there are not enough sam-

ples, it becomes difﬁcult to identify and count these hot spots.

Furthermore, the problem becomes more difﬁcult for higher

order modulation formats. Hence, the goal is to ﬁnd a suit-

able feature that identiﬁes different modulation formats with

a small number of samples. We propose such a feature based

on the following folding operation. For BPSK, by folding the

negative axis towards the positive axis, the two signal constel-

lation points merge into a single point. All the samples now

concentrate around the positive signal point, and the number

of samples around this point basically doubles. For 4-ASK,

the folding operation has the same effect, merging four signal

points into two. Then, through a second folding operation, the

two signal points merge into one.

As a result, the samples

concentrate around a single point whose location depends on

the modulation format. Then, classiﬁcation can be performed

by observing how these samples concentrate using the K-S

test [12].

The rest of this letter is organized as follows: the system

model and the K-S test are presented in Section II; the pro-

posed classiﬁers for quadrature amplitude modulation (QAM)

and PSK are introduced in Sections III and IV, respectively,

while discussions are provided in Section V; simulation re-

sults are shown in Section VI and conclusions are drawn in

Section VII.

II. S

YSTEM DESCRIPTION

Consider the following discrete-time signal model

= x

+ w

,n=1,...,N (1)

A better folding scheme is described in Section III; here we provide folding

of the constellation points as an example for ease of understanding.

See http://www.ieee.org/publications

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