Abstract-In this paper we develop an automatic modulation classification algorithm utilizing the statistical moments of the signal phase and use it to classify the ...
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 40, NO. 5, MAY 1992
Signal Classification Using Statistical Moments Samir S. Soliman, Senior Member, IEEE, a n d Shue-Zen Hsue
Abstract-In this paper we develop an automatic modulation classification algorithm utilizing the statistical moments of the signal phase and use it to classify the modulation type of general M-ary PSK signals. We show that for M-ary PSK signals, the nth moment (n even) of the phase of the signal is a monotonic increasing function of M. Based on this property, we formulate a general hypothesis test, develop a decision rule and derive an analytic expression for the probability of misclassification. Two examples are given to demonstrate the performance of the algorithm. At low carrier-to-noise ratio (CNR), we establish that the eighth moment is adequate to classify BPSK signals with reasonable performance. Finally, we compare the performance of the suggested algorithm to the quasi log-likelihood ratio (qLLRC), square-law (SLC), and phase-based (PBC) classifiers. The suggested algorithm is outperformed by the qLLRC algorithm at low CNR but is comparable to the SLC and is better than the PBC algorithms. The qLLRC algorithm is only valid at CNR < 0 dB and can be used only to discriminate between BPSK and QPSK signals whereas, the moments algorithm is more general.
I. INTRODUCTION
A
N automatic modulation classifier is a system that automatically identifies the modulation type of the received radio signal given that the signal exists and its parameters lie in a known range. That is, modulation classification is an intermediate step between signal detection (interception) and data demodulation (information extraction). Modulation type classifiers play an important role in some communication applications such as signal confirmation, interference identification, monitoring, spectrum management, surveillance, electronic warfare, military threat analysis, and electronic counter-counter measure. For example, threat analysis can be accomplished by comparing the signal characteristics of intercepted emitters against a catalog of characteristics or sorting parameters. One of the most important sorting parameters is the modulation type. In general, modulation classification can be approached either from a decision-theoretic or a statistical pattern recognition framework. The decision-theoretic approach is valid when the number of modulation candidates are finite and is based on composite hypotheses testing. Thus, the resulting classifier is optimum in the sense that it minimizes the average cost function. The complexity of the decision-theoretic based classifier depends on the number of unknown parameters that are associated with the received waveform and the classifier Paper approved by the Editor for Data Communications and Modulation of the IEEE Communications Society. Manuscript received Decemher 7, 1989; revised August 30, 1990. S . S . Soliman is with Qualcomm Inc., San Diego, CA 92121. S.-Z. Hsue is with the Department of Electrical Engineering, Southern Methodist University, Dallas, TX 75275. IEEE Log Number 9108001.
Signal space Fig. 1,
Observation space
Feature space
Decision space
Pattern recognition model for the classification problem.
performance is expressed in terms of the probability of making a correct decision. In the pattern recognition approach, the classifier is composed of two subsystems. The first is called the feature extraction subsystem and its role is to extract useful information from the raw data and can be viewed as a mapping of the set of incoming signals into a chosen feature space. The second is a pattern recognizer subsystem whose function is to indicate the membership of the modulation type. A model of a modulation classifier based on statistical pattern recognition framework is shown in Fig. 1. In general, the model is well defined and the resulting algorithms are easy to implement, but the problem lies in defining a good set of features and providing a rigorous mathematical proof or justification that these algorithms are optimum in a statistical sense. In most cases, pattern recognition algorithms rely on defining and extracting a set of features that, intuitively, look as an optimum set. Some of the ad-hoc statistical features that has been used are: mean, harmonic mean, log geometric mean, energy, entropy, peak count, percent occupying bandwidth, etc. The performance of pattern recognition based classifier is measured in terms of its confusion matrix. Scanning the open literature, we find that there are few articles published in the area of automatic modulation classification. The following is an overview of some of these modulation classification algorithms. Kim et al. [I] follow the decision-theoretic approach to discriminate between BPSK and QPSK signals. In their development, they use the low CNR approximation and relay on simulation to substantiate their results. Their classifier is less robust in the sense that signal parameters such as carrier frequency, initial phase, symbol rate and CNR are all assumed to be available to the classifier. Mammone et al. [2] follow the pattern recognition approach and use the phase derivative of an analytic signal as a feature to discriminate between CW, BPSK and QPSK signals. Through simulation they demonstrate that the classifier performs well for CNR above 35 dB. Although the classifier requires high CNR, it is robust in the sense that carrier frequency, initial phase, CNR and symbol rate are assumed unavailable to the classifier. In [3] Hipp develops
0090-6778/92$03.00 0 1992 IEEE
yo9
SOLIMAN AND HSUE: SIGNAL CLASSIFICATION USING STATISTICAL MOMENTS
a classification algorithm using statistical pattern recognition techniques. The algorithm uses statistical moments of both the demodulated signal and the signal spectrum as the modulation identifying parameters. The algorithm is tested on numerically simulated signals. Chan et al. (41 use the ratio of the variance to the mean squared of the squared envelope of the received signal as a feature to discriminate between SSR, AM, DSB and constant-envelope modulated signals. Liedtke [5] provides a systematic approach to classify digitally modulated signals such as ASK, PSK and FSK signals. Callaghan et al. [6] provide algorithms to classify different modulation types such as CW, AM, FM, SSB, FSK, ASK by using both zero-crossing sampling and voltage sampling. Hagiwara er al. [7] use a histogram made of the distribution of the envelope of the input signal as discrimination feature to classify QPSK and MSK signals. Hsue and Soliman [9]-[ll] use zero-crossing techniques to classify PSK and FSK signals. In this paper, we develop a moments-based algorithm to classify general M-ary PSK signals, A4 = 2", a = 0, 1 , 2 . . . ((U= 0 corresponds to CW waveform). The paper is organized as follows. In Section 11, we introduce the problem of modulation classification and derive expressions for the probability density functions and moments of the phase components. In Section 111, we formulate the modulation classification problem as a hypothesis testing problem and derive a suboptimal decision rule. We then analyze the performance of the proposed moments-based modulation classifier for a finite observation interval. In Section IV, we present two examples to demonstrate the numerical results and compare the performance of the proposed classifier and other known classifiers. Finally, we present the conclusions and summary in Section V. 11. DEVELOPMENT
The received signal ~ ( tis) composed of two uncorrelated components, a signal component s ( t ) and an additive white Gaussian noise w(t), that is
The signal component s ( t ) is assumed to be one of the modulation types: CW or MPSK, and ~ ( tis) modeled as white Gaussian noise with zero-mean and power spectral density ne.. The signal s ( t ) contains the phase information that can be extracted by means of IQ-technique or by using discrete Hilbert transform [12]. The extracted phase can be represented as
&(?) = Q , ( 7 )
+4 7 )
- 3c,
< / < x.
-7r
< O n ( / ) I 7r
A. Probability Denyity Functions
For a CW signal ((2 = O), the probability density function (pdf) of the phase 4" is given in [13] as
.Q[-
ros(4o)/fi]
- 7r
< 4 n I 7r (3)
where ?c.
(4)
and A is the amplitude of the transmitted CW signal. Due to the complex nature of (3), one well-known asymptotic expression of (3) under the assumption a i + 0 is the Tikhonov probability density function [14]
where lo[.] is the zero-order modified Bessel function of the first kind. The approximation in (6) is considered good for values of y > 6 dB and is considered fair for values of y around 0 dB. As y increases, the Tikhonov pdf can be further approximated by a Gaussian pdf, that is,
(7) Note that (3) is the true phase distribution, whereas (6) and (7) are asymptotic approximations of (3). In the following, we restrict the analysis to the Tikhonov pdf because of its simplicity and quite acceptable accuracy. With equally likely M phases, the pdf of 8, can be written as
where r l k ( a ) is the phase of kth phase states and can be expressed as
IY
= 0 . . . . , log2 M .
From (2), the pdf f b ( ? j ; n ) of Tikhonov functions
.
4n is a sum
(9 of the noncentral
2"
(2)
e,(/)
where is the sampled phase component of the transmitted MPSK signal s ( t ) , and ~ ( z )is the random phase attributed to the receiver noise w(t) and any other measurement errors. Without loss of generality, {4n(7)} and {d,(z)} are assumed independent and identically distributed with zero-means. In the following, w e derive and discuss the probability density functions and the moments of the phase component cbn (7').
Fig. 2 demonstrates the pdf's of at CNR = 7 dB for CW, BPSK, QPSK, and 8PSK, respectively. The number of peaks in the figures indicates the number of signal phase states. As the
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 40, NO. 5, MAY 1992
910
By expanding exp[2y (.os( y (12) becomes
a.IT1 x
0
-A
--x
( a ) ) ]into a Fourier series,
n
0
( P H A S E ANGLE )
(b) B P S K
1 5
1 cw
1
20
5
--x
n odd
where I k [ . ] is the modified Bessel function of the first kind and order k . By changing the variable L = y - ql(cy) and using binomial expansion, (13) can be written as
-
U0 5
0
K
&
7r 71
0 0
-77
7r
( P H A S E ANGLE ) (c)
QPSK
-77
0
A
m n ( a )= n+1
2^
+ 2"7rIo[2Yl
cc
2=1 k = l
( P H A S E ANGLE
(d) B P S K
Fig. 2. pdf of CW and MPSK signals at CNR = 7 dB. (a) CW; ( b ) BPSK; (c) QPSK; (d) IIPSK.
CNR decreases, the peaks smear off and finally the pdf reaches that of a uniformly distributed random variable. Also, as shown in Fig. 2, fq(y; cy) for large a is flatter than those with small a. Generally, f#(y: a ) will approach 1 / 2 as ~ CNR + -cc dB or (y: m. --f
B. Ensemble Moments
In the absence of the noise, the nth moment of
4n is
In the presence of noise, the nth moment of q5rl is
m n ( a )=
i
y'"f,(y:
a ) dy
where LqJ denotes the largest integer that is 5 q . It can be shown that even moments of 4o is a monotonic increasing function with respect to N (see Appendix A), that is m,,(cy
+ 1) > m,(tr).
(15)
-7r
Fig. 3 shows the moments of MPSK signals at CNR = 3 dB. From the figure, it may be intuitively predicted that even moments m,(tr) can be used as a feature to classify CW and general MPSK signals. Higher order moments may be needed to discriminate between signals with large M .
~
91 1
SOLIMAN AND HSUE: SIGNAL CLASSIFICATION USING STATISTICAL MOMENTS
Input signal r ( t )
18 CNR= 3 d B
1 o4
n=10
A --
I
Extraction
n
A
n=0
-
1o3
d
Moments
v
d
g
Number o f samples L
n=6 0-
Computation
lo;!
1
3
?-
Id U-8-
n=2
8
Threshold Comparison
1oo
Set o f thresholds T,
Decision
16'
CW
BPSK
QPSK
0PSK
16PSK
32PSK
MODULATION TYPE Fig. 4.
Block diagram of the MPSK classifier using moments
Fig. 3. Moments of MPSK signals at CNR = 3 dB.
111. CLASSIFIERPERFORMANCE
C. Measured Moments
The suggested algorithm uses estimates of the ensemble moments. Convenient estimates are the unbiased sample averages of even powers of the extracted phase, namely,
&(a) =
l
L Cmi)
(16)
To evaluate the performance of the suggested algorithm, the problem of classifying CW and general MPSK signals may be formulated as a hypothesis testing problem based on the densities of phase moments. pdf's of measured even moments have been shown to be Gaussian densities with mean p n ( a ) and variance c: ( a )defined in (I 7) and (18), respectively. The hypothesis testing problem is then formulated as
2=1
where L is the number of samples observed in a finite interval. The central limit theorem can be invoked to show that the pdf of &(cy) approaches a Gaussian density as L increases. nn( a ) )where p n ( a ) and Let the pdf of fin ( a )be N ( p n ( 0). n i ( a ) are P n ( @ )= E { f i , n ( a ) )=
L xl CE{#ai)) = mn(a)
(17)
2=1
and
Ha : pdf of f n n ( a )= N ( p n ( a ) , a , ( n ) ) .
+
+
1 hypotheses and ( a 1)* alternatives that There are a may occur each time the experiment is conducted. The Bayes criterion assigns a cost to each of these alternatives, assumes a set of a priori probabilities Po,Pl!. . . , Pa, and minimizes the average cost. If any error is of equal importance, then the optimum processor is the maximum a posteriori probability computer. A simpler test is
< Tl
7^nn(a)
That is, the random variables ri2n(a) have means that are monotonic increasing functions of a and therefore can be used as a discriminating feature to classify MPSK signals. Fig. 4 is a block diagram that details the steps involved in the classification procedure.
(19)
T, < f k n ( a )< T,+1
say CW signaling. say MPSK signal with M = 2". (20)
Using the fact that the density functions are Gaussian, it can be shown that
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 40, NO. 5, MAY 1Y92
Using the nth moment, the quality of the classifier is measured in terms of two probabilities of misclassification. These probabilities are defined as:
Pe(n. 0) = Prob(misc1assificationlHo) = JNP (”
dY
TI
and
In general, the probability of misclassification Pe(n. a ) is a function of n, CNR and L. We expected that increasing n will provide us with more classification power. Unfortunately, this was not true. As we increase n, the variance of & ( a ) increases, and hence the probability of misclassification. There is always an optimum value for n for each specific value of L. IV. NUMERICAL RESULTS
In this section we present two examples to demonstrate the performance of the suggested algorithm. In the first example, the classifier is designed to discriminate between BPSK and QPSK signals. In the second example, the classifier is designed to recognize one of the following modulations: CW, BPSK, QPSK, and 8PSK. A . Example 1
The problem can be modeled as
The probabilities of misclassification Pe(n, 1) and Pe(n, 2 ) for L = 1024 are given in Figs. 5 and 6, respectively. For CNR < -5 dB, the probability of misclassification ~ 0 . 5 and , the order of moments does not play any part in determining the performance of the classifier. For these values of CNR the means of the two distributions are too close, and intuitively, it
Fig. 5.
Probability of misclassification of BPSK signals in Example 1.
becomes hard on the classifier to distinguish between BPSK and QPSK. As we increase CNR, the performance of the classifier improves as n increases up to n = 8. Increasing n beyond 8 will degrade the performance. This behavior is again attributed to the behavior of the variance as a function of n. As n increases, the variances of the two distributions increase at a faster rate than the means of the two distributions. B. Example2
In this example we try to classify the received waveform as one of the following modulations: CW, BPSK, QPSK, and 8PSK. The problem can be modeled as
Ho : pdf of r?771(0)= N ( / i 7 L ( 0a,(O)) ). H i : pdf of h T 1 (=l )l V ( / ~ ~ ~ ( l ) . ~ ~ ( l ) ) H2 : pdf of T ? I ~ ~=( ~N) ( p n ( 2 ).,(a)) , H3 : pdf of h T L ( 3=) N ( / L , , ( ~o )r,1 ( 3 ) ) . The probabilities of misclassification Pe(n, i), z = 0, 1, 2, 3 for L = 1024 are given in Figs. 7, 8, 9, and 10, respectively. In Fig. 7, a classifier using the second moment has the best performance in terms of correctly classifying CW signals. This
SOLIMAN AND HSUE: SIGNAL CLASSIFICATION USING STATISTICAL MOMENTS
913
1oo
L= 1024
Fig. 6. Probability of misclassification of QPSK signals in Example 1
phenomena can be explained by noticing that the nth moment always emphasize the part of the distribution near T or -7r. For the case of CW, this part is mainly due to the phase noise (the distribution of CW is centered around the origin). Increasing n, will enhance the noise component and hence, degrade the performance. In Fig. 8, the best choice of n for identifying BPSK signal is n = 8, whereas Figs. 9 and 10, classifiers using moments of order n > 8 have better performance. These results can be explained also using Fig. 3. Note that for n = 2 , BPSK, QPSK, and 8PSK have almost the same second moment, while C W has a different second moment. With CNR = 3 dB, QPSK and 8PSK have almost the same 10th moment, this means higher order moments are needed to discriminate between these two type of modulations.
Fig. 7. Probability of misclassification of CW signals in Example 2.
1oo
L= 1024
C. Classifier Comparison
At this stage we want to compare the moments-based classifier with other known classifiers. Three kinds of modulation classifiers were reported by Kim et al. [l]. These are the decision theoretic or the quasi log-likelihood ratio (qLLRC), square-law (SLC) and the phase-based (PBC) classifiers. The qLLRC is a local optimal detector that is obtained by approximating the likelihood ratio. The SLC is a harmonic detector which detects the energy of the second harmonic of the carrier frequency. The PBC uses the phase difference histogram made of the distribution of the difference between two consecutive symbol phases as a feature [5]. To have a fair performance comparison, we adopt the same assumptions that are used in [l], namely, signal parameters
CNR (dB) Fig. 8. Probability of misclassification of BPSK signals in Example 2
such as carrier frequency, initial phase, symbol rate (l/Ts) and CNR are assumed to be available to all of the modulation
IEEE TKANSACTIONS ON COMMUNICATIONS. VOL. 40, NO. 5, MAY 1992
914
1
/ 0 95
0 90
H z
2
085
2 V
5 3
080
075
V w
0: 070
V
8
'Ea tl
065
./
/
B
060
/
/
0 55
0 50
-8
-6
-4
-2
0
( ya in dB) Fig. 9. Probability of misclassification of QPSK signals in Example 2.
Fig. 11. Performance comparison.
performance comparison between the proposed classifier using the 8th moment and the above three classifiers, for L = 100 symbols. The performance curves of qLLRC, SLC, and PBC which were experimentally generated are directly chosen from [ l , Fig. 41 and were obtained using simulation. The probability of correct classification for the moments-based algorithm in Fig. 10 is obtained analytically and is defined as
Prob(H1IH1)
Pc =
+ Prob(Hz1H.L) 2
and the symbol-CNR ys is defined as
ys = YTS where T, is the symbol duration. Although the performance of the qLLRC is better than the suggested algorithm, the capability of the qLLRC algorithm is limited since it is only valid for low CNR (CNR < 0 dB) and can only be used to discriminate between BPSK and QPSK signals only. On the other hand, the suggested algorithm is better than the SLC classifier for CNR > 0 dB and outperforms the PBC for all CNR's. V. CONCLUSION CNR (dB) Fig. 10. Probability of misclassitication of RPSK signals in Example 2.
classifiers to be compared. Note that certain classifiers may not require all of the above assumptions. Fig. 11 demonstrates the
An automatic modulation classifier using statistical moments has been developed to classify CW and general MPSK signals with M = 2". It is shown that the nth moment is a monotonic increasing function of a . In general there is an optimum value for II beyond which the performance of the classifier starts to degrade. This phenomena is attributed to the
SOLIMAN AND HSUE: SIGNAL CLASSIFICATION USING STATISTICAL MOMENTS
915
way the variance of the sample moments behaves as a function and of n. A comparison is made between the performance of the suggested classifier and the three classifiers reported in [l]. C= We demonstrated that using a moments classifier with only the eighth moment is comparable to the SLC and is better than the PBC. More moments are needed to classify MPSK signals with A4 2 8. Furthermore, performance improvement is expected if we increase the number of samples L.
2"
(1. 7r
(Y
k=l
+ 8)"W ) [ 2 Y COS(Y
Theorem: The even moments rrLTL(a)of the phase of MPSK signals are monotonic increasing functions of a where a = 0 , l . . log, M . Proof: From (12),
+
/ (w
- io)" exp[2ycos(y - % ( a ) )d]Y
-7r
= C = 0.
( ;)Y'fi"-'
+ /j)'L + (Y - /I)" - &/"I 1=O
' '
2"+'
> 0, and B
11
[(Y
7/k(c-u))l dy
-a+ j
In the following, we will show that A Note that
APPENDIX A
-
7r
+
m,L(o 1 ) =
1=l
2y"
-7r
exp[2ycos(y - 7 1 i ( a
+ I ) ) ] dy . [l
+ (-1)"-1]
11-2
=2
where
fi
+
5{ J
1
2"+27r10[2y]
k=l
(y++)'l
-7r-3
. exp[2ycos(y - 7 / k ( Q ) ) ] dy +
T
-x+3
. (y - p)"cXp[2ycos(y The difference between ,rrL,(a as ,rrLIL( ( 2
where
A=
+ 1)
51
k=l
m n( a ) =
[(y
+
}
I
5{ 7
(y+p)"~xp[2ycos(?/-7/k(a))]dy
k=l
-7r-
I
-
2a+27r10[2y]
-
/j)'L
-
+
[ A B - C]
2yrL]
3
(-7J
-
p)'Lexp[2ycos(y - 7/rc(a))] dy
-7r
-
f 2"
5 { .~p + +/ (y
k=l
Similarly, it can be shown that C = 0.
x+,J
7r
-
I
(QED)
REFERENCES
p)" exp[2ycos(y - 7 I k ( n ) ) 1 (iy
-7r-,j
(y
1
exp[2ycos(;y - ~ / 2 < > - k + l ( c ~ ) ) ]dy ] = 0.
-7r
. exp[2ycos(y - * r / k ( a ) ) dy ]
B=
=
-7r+A
+ (y
/j)l2
- T / k ( ( Y ) ) ]dy
+ 1) and m,,(a) can be written 1
-
>0
and ..ence A > 0. By changing variables and using the property q k ( a ) = - 7 ] 2 ~ - , + + 1 ( N ) , B can be written as
= 7r/a0+'. The last equation can be written as
7n,,(a 1) =
(;)$8'"-'
/I)'' exp[2y cos(?/
- , r / k ( c t ) ) ] r,ty
[ l ] K. Kim and A. Polydoros, "Digital modulation classification: The BPSK and QPSK case," in Proc. MILCOM'88, San Diego, CA, vol. 2, Oct. 1988, pp. 431-436. [2] R. J. Mammone, R. J. Rothaker, and C. I. Podilchuk, "Estimation of carrier frequency, modulated type and bit rate of an unknown modulation signal," in Proc. ICC'K7, Seattle, WA, vol. 2, June 1987, pp. 1006-1012.
ILEE lRANSACrlONS ON COMMUNICATIONS, VOL 40, NO 5, MAY 1YY2
Y 16
[3] J. E. Hipp, “Modulation classification based on statistical moments,” in Proc. MlLCOM’86, Monterey, CA, 1986, pp. 20.2.1-20.2.6. (41 Y.T. Chan, L.G. Gadbois and P. Yansouni, “Identification of the modulation type of a signal,” in Proc. IEEE Int. Conf Acoust., Speech, Signal Processing, Tampa, FL, Mar. 1985, pp. 838-841. [SI F. F. Leidtke, “Computer simulation of an automatic classification procedure for digitally modulated communication signals with unknown parameters,” Signal Processing, vol. 6, no. 4, pp. 31 1-323, Aug. 1984. [6] T. G. Callaghan, J. L. Perry, and J. K. Tjho, “Sampling and algorithms aid modulation recognition,” Microwaves RF, pp. 117- 121, Sept. 1985. [7] M. Hagiwara and M. Nakagawa, “Automatic estimation of an input signal type,” in Proc. GLOBECOM’87, Tokyo, Japan, 1987, pp. 254-258. [ 8 ] P. M. Petrovit, Z. B. Krsmanovit, and N. K. Remenski, “Automatic realtime HF signal classification,” in Proc. MELECON ‘87, Mediterrian Electrotech. Conf. and 34th Cong. Electron. Joint Conf., Italy, Mar. 1987, pp. 175-178. (91 S. Hsue and Samir S. Soliman, “Automatic modulation recognition of digitally modulated signals,” in Proc. MILCOM’89. Boston, MA, Oct. 1989, pp. 37.4.1-37.4.5. [IO] -, “Automatic modulation recognition of using zero-crossing,” IEE Proc. -Part F Radar Signal Processing, vol. 137, no. 6, Dec. 1990. [ l l ] Y. Yang and S . Soliman, “Statistical moments based classifier for MPSK signals,” IEEE GLOBECOM ’91, Phoenix, AZ, Dec. 1991. 1121 A. V. Okatov and Y. E. Chernvshov. “Comoutation of the components of an analytic signal using a digital Hilbert filter,” Telecommun Radio E n g , vol 39, no 11, pp 109-111, Nov 1984 [13] W R Bennett, “Methods of solving noise problems,” Ptoc IRE, vol 44, no 5 , pp. 609-638, May 1956 [14] H Leib dnd S Pasupathy, “The phase of a vector perturbed by ~aUSSidn noise and differentially coherent receivers,” IEEE Trans Inform Theory, vol. 34, pp. 1491-1501, Nov. 1988 L
Samir S. Soliman (S’80-M’83-5M’88) was born in Cdiro. Egypt He received the B Sc degrees in electricdl engineering and in applied mdthemdtics, both with honors, from Ain Shdms Univeristy, Cdiro, in 1974 dnd 1977, respectively, dnd the M S and Ph D degrees in electricdl engineering from the University of Southern Cdlifornid, Los Angeles, i n 1980 and 1983, respectively His resedrch interests include synchronization dnd coding over fading dispersive chdnnels, detection dnd estlmdtion over non-Gdus\idn channels, electronic countermedsures, rdddr systems design, dnd personal communications networks dnd seivices From August 1983 to May 1990 he was with the Department of Electricdi Engineering, Southern Methodist University, Dallas, TX He spent the Summer of 1990 on ubbatical dt Bell-Northern Research, conducting resedrch on indoor wireless communicdtions He is currently with Qudko” Inc, San Diego, CA He is the principdl author of the book, Continuous and D u t t e t e Signulr and Sy&nzs (Prent~ce-Hall,Englewood Cliffs, NJ, 1990) Dr Soliman is a member of Signid XI dnd Eta Kdppd Nu
,
Shue-Zen Hsue wa\ born in Taipei, TaiWdn, on Februdry 1, 1953 He received the B S. and M S degrees from the National Chiao-Tung University, Hsin Chu, TdiWdn, in 1975 and 1979, respectively He received the Ph D degree from the Southern Methodist University, Texds, in 1990, all degree\ in electricdl engineering Since 1979, he hds been employed by the Chung Shan Institute of Science and Technology, Taiwan, ds an assistant researcher Currently, he is an associate researcher in the institute
