ter such as carrier frequency and symbol rate are available,. Polydoros and Kim [6] derived an approximate likelihood ratio modulation classifier to separate ...
IDENTIFICATION OF DIGITAL MODULATION TYPES USING THE WAVELET TRANSFORM Liang Hong
K.C. Ho
Department of Electrical Engineering, University of Missouri-Columbia, Columbia MO 65211, USA ABSTRACT Automatic identification of the digital modulation type of a signal has found applications in many areas, including electronic warfare, surveillance and threat analysis. This paper studies the use of wavelet transform to distinguish QAM signal, PSK signal and FSK signal. The approach is to use wavelet transform to extract the transient characteristics in a digital modulation signal, and apply the distinct pattern in wavelet transform domain for simple identification. The relevant statistics for optimum threshold selection are derived under the condition that the input noise is additive white Gaussian. The performance of the identification scheme is investigated through simulations. When CNR is greater than 5 dB, the percentage of correct identification is about 97% with 50 observation symbols. 1. INTRODUCTION Automatic identification of the digital modulation type of a signal is a rapidly evolving area [1]. It has found wide applications in many areas, including electronic warfare, surveillance and threat analysis. A variety of techniques have been proposed to distinguish quadrature amplitude modulation(QAM) signal, phase shift keying(PSK) signal and frequency shift keying(FSK) signal. Hsue and Soliman [2] proposed to use the variance of the zero-crossing interval sequence as well as the frequency and phase difference histogram for identification. Beidas and Weber [3] proposed a classifier based on the time domain higher-order correlations. Hero et al. [4] proposed a method using eigendecomposition of moment matrices. Huo and Donoho [5] used the counts of signals falling into different parts of the signal plane to identify the digital modulation types. These methods are either computationally intensive or require a high carrier-to-noise ratio (CNR). Under the assumption that some communication parameter such as carrier frequency and symbol rate are available, Polydoros and Kim [6] derived an approximate likelihood ratio modulation classifier to separate BPSK and QPSK. The classifier works well even at low CNR. Another approach for digital modulation types identification is to use wavelet transform (WT) [7]. Different digi-
tal modulation signals contain different transients in amplitude, frequency or phase. The WT has capability to extract transient information and thereby allowing simple methods to perform modulation identification. Lin and Kuo [8] applied Morlet wavelet to detect the phase changes, and used the likelihood function based on the total number of detected phase changes as a feature to classify M-ary PSK signal. Ho et al. [9], on the other hand, proposed a method to identify PSK and FSK signals using the Haar WT without the need of any communication parameter of a modulated signal. In ideal case, the Haar WT magnitude(jHW T j) of a PSK signal is a constant and that of a FSK signal is a multistep function. Hence the variance of jHWT j of an input signal is used as a feature to classify the two signals. This paper extends the approach of [9] to include QAM signal in the signal set for identification. Compared to PSK and FSK signals, one distinction in QAM signal is that it does not have a constant amplitude. The jHWT j of a QAM signal is a multi-step function similar to that of a FSK signal because of the change in amplitude as symbol changes. On the other hand, if we normalize a QAM signal to unity amplitude, it will behave like a PSK signal and its jHW T j will be a constant. Amplitude normalization has little effect on the jHWT j of PSK and FSK signals since both of them have constant amplitude. Hence we can distinguish QAM, PSK and FSK signals by computing the variances of jHWT j with and without applying amplitude normalization. We shall elaborate in detail the methodology for digital modulation identification using WT in the next section. Section 3 gives the simulation results. Finally, a conclusion is made in Section 4. 2. MODULATION IDENTIFICATION BY WT Let the received waveform r(t), 0 t T be described as r(t) = s(t)+ n(t), where n(t) is a complex white Gaussian noise. The signal s(t) can be represented in complex form as
s(t) = s~(t)ej (!c t+c ) (1) where !c is the carrier frequency and c is the carrier phase.
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p
where Si = A2i + Bi2 is the amplitude of the ith symbol and i = tan,1(Bi =Ai) is the phase of the ith symbol. Taking the magnitude of (6) to eliminate the unknown carrier phase gives
XN (Ai + jBi)uT (t , iT );
For QAM signal,
s~QAM (t) =
i=1
Ai ; Bi 2 f2m , 1 , M; m = 1; 2; :::;M g: For PSK signal,
p
s~PSK (t) = S
(2)
When the Haar wavelet covers a symbol change, the Haar WT of a QAM signal is
i
'i 2 f 2M (m , 1); m = 1; 2; :::;M g: s~FSK (t) =
(3)
X S ej ! t uT (t , iT );
For FSK signal,
p
N
(
i=1
i
+
i
)
that the three signals have the same symbol duration. It can be seen from (2)-(4) that symbol changes will give rise to transients in the modulated signals. The transients are created independently in the changes of amplitude, phase and frequency respectively. WT can characterize these transients effectively and allowing simple method for identification of the three signals. The continuous WT of a signal s(t) is defined as [7]
Z s(t) (t)dt a Z p1 s(t) ( t , )dt:
=
a
a
(5)
where a is the scale, is the translation and the superscript * denotes complex conjugate. The function (t) is the mother wavelet and the baby wavelet a (t) comes from time-scaling and translation of the mother wavelet. The choice of a mother wavelet depends on its application. Some widely used wavelets are Morlet, Haar and Shannon [7]. Due to its simple form and ease of computation, we shall consider the Haar wavelet only. The following analysis derives the Haar WT of QAM, PSK and FSK signals. In ideal case, when the Haar wavelet is within a symbol time, the Haar WT of a QAM signal is
Z (A + jB )ej ! t dt i i , Z , (A + jB )ej ! t dt)
CW T (a; ) = p1a (
0
a2
(
a2
i
i
(
c
( + )+
c( +
)+
c
)
c)
0 p 4 Si sin2 (! a )ej (!c +c + i ) : = j pa! c4
c
(6)
Z d (A + jB )ej ! t
CWT (a; ) = p1a (
+
!i 2 f!1; !2; :::; !M g; i 2 (0; 2): (4) In (2), (3) and (4), S is the signal power, N is the number of observed symbols, T is the symbol duration and uT (t) is the standard unit pulse of duration T . In this study, we assume
CWT (a; ) =
(7)
c
XN ej' uT (t , iT ); i=1
4pSi sin2 ( !c a ) jCWTQAM (a; )j = p a! 4
Z
Z
0
d
a2
, a2
i
i
(
c( +
c ) dt
)+
(Ai+1 + jBi+1 )ej (!c (t+ )+c ) dt
, (Ai+1 + jBi+1 )ej (!c (t+ )+c ) dt) 0 1 = j pa! ej (!c +c + i ) [Si (ej!c d , e,j!c a2 ) c +Si+1 ej (2 , ej!c d , ej!c a2 )]:
(8)
where Si and Si+1 are the amplitudes of the ith and (i + 1)th symbols, i and i+1 are the phases of ith and (i + 1)th symbols, = i+1 , i is the phase change and d is the time instance when the symbol changes relative to the center of the wavelet. It is assumed to be negative in (8). For positive d, we can derive a similar formula as (8). It is observed from (6)-(8) that when the wavelet is within a symbol period, the jHWT j of a QAM signal without noise is constant independent of the translation . Since the amplitude of the QAM signal Si is variable, the jHW T j pattern of a QAM signal is a multi-step function. There will be distinct peaks in the jHWT j resulted from phase changes at the times where the wavelet covers a symbol change. The jHWT j of PSK signal and FSK signal have been derived from [9] when the Haar wavelet is within a symbol time: p
jCWTPSK (a; )j = p4 a!S sin2 ( !4c a ) c
(9)
p 4 jCWTFSK (a; )j = pa(! S+ ! ) sin2[ (!c +4 !i)a ] c i
(10)
Note that !i is fixed within a symbol. It is clear from (9) and (10) that in ideal case, the jHW T j of PSK signal is a constant, while the jHWT j of FSK signal is a multi-step function since the frequency is a variable. Similar to QAM signal, there will be distinct peaks in the jHWT j of the two signals when the Haar wavelet covers a symbol change. Figure 1 shows the jHWT j of QAM, PSK and FSK signals.
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arctan
s~(t) as (11)
i=1
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XN ej' uT (t , iT ) i XN sFSK (t) = ej ! t uT (t , iT ) sPSK (t) =
i
(12)
(13)
=1
(
i=1
i
+
i)
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7000
Figure 2: jHWT j of QAM, PSK and FSK signals with amplitude normalization.
From (2)-(4), we have the amplitude normalized modulation signals:
sQAM (t) =
0
n
Figure 1: jHWT j of QAM, PSK and FSK signals. We define amplitude normalization of a signal follows:
1
(14)
Amplitude normalization does not affect the modulation type of the PSK signal and FSK signal since they are constant amplitude modulation. On the other hand, the amplitude variations in QAM signal disappear after amplitude normalization. sQAM (t) has the same form as sPSK (t). As a result, a QAM signal with amplitude normalization will have a constant jHWT j with peaks due to phase change. However, the characteristics of the jHWT j of amplitude normalized PSK signal and FSK signal will remain the same. Figure 2 shows the jHWT j of the three signals after amplitude normalization. To differentiate QAM, PSK and FSK signals, we must find a common feature and select a criterion based on differences. If we ignore the peaks, comparing Figure 1 and
Figure 2 reveals that the jHWT j of a QAM signal is a multistep function. It becomes a dc when amplitude normalization is applied to the signal. Again, ignoring the peaks the jHWT j of a PSK signal is a constant and that of a FSK signal is multi-step function, no matter their amplitudes are normalized to unity or not. The variance of a constant is zero while that of a multi-step function is larger than zero. Hence we can distinguish QAM signal, PSK signal and FSK signal by computing the variances of the jHWT j with and without amplitude normalization, after removing the peaks by median filtering. Figure 3 is the WT modulation identifier for QAM, PSK and FSK signals. The identifier consists of two branches and a decision block. One branch is without amplitude normalization and the other is with amplitude normalization. The identifier first finds the jHWT j of an input signal. After removing the peaks of jHWT j by a median filter, the identifier computes the variance of the median filter outputs. The decision block compares the two variances with two thresholds to decide the modulation type of an input signal. If the variance produced by the branch without amplitude normalization is larger than the threshold while the other variance is smaller than the threshold, the input is classified as a QAM signal. If both variances are lower than the thresholds, the input is classified as a PSK signal. If both variances are higher than the thresholds, the input is classified as a FSK signal. Under the condition that the input noise is white Gaussian, the relevant statistics for optimum threshold selection can be evaluated. First, we analyze the theoretical variance
0-7803-5538-5/99/$10.00 (c) 1999 IEEE
|HWT|
Median filter
Decision Block
Compute + variance
Input Normalize amplitude to unity
QAM
PSK FSK
>0
0
