components of a PCM SO1 separated by the baud rate are correlated. The spectral self-coherence property is shared by niost communication signals as a result ...
Application of the SCORE Algorithm and SCORE Extensions to Sorting in the Rank-L Spectral Self-coherence Environment Stephan V. Schell' Dept. of Electrical Engineering and Computer Science University of California, Davis, CA 95616
restore the spectral self-coherence of the SO1 to the receiver output signal. The SCORE algorithms were originally developed to maximize the spectral self-coherence of the signal output of an adaptive array. The resulting solution was shown to extract a spectrally self-coherent SO1 from arbitrary interference provided the interference does not exhibit spectral self-coherence at the frequency separation of interest. This paper extends this investigation to the rank-L spectral self-coherence environment where L signals are spectrally self-coherent at the frequency separation of interest (e.g. in communication nets). It is shown that alternate or sub-optimal solutions to the cross-SCORE optimization problem exist that can extract multiple SOIs from rank L spectral self-coherence environments. For example, each of the alternate solutions to the optimization problem corresponds to extracting one of several PCM SOIs having the same baud rate and impinging from different directions on the receiver. A brief development of the original cross-SCORE algorithm and of the modified SCORE algorithm is presented, followed by analysis and experimental verification of the ability of these algorithms to blindly adapt the receiver to extract multiple spectrally self-coherent SOIs, i.e., to sort the SOIs on the basis of their spectral self-coherence. The theory of spectral correlation (spectral self-coherence) [5,6] is used to prove perfect extraction of each spectrally self-coherent SO1 in the L-signal noiseless environment ( L 5 number of sensors) and near-optimum extraction in noisy environments. Furthermore, the analysis shows the modified SCORE algorithm is more robust than the cross-SCORE algorithm in that it can extract signals from many environments in which the original algorithm fails. In particular, it is shown that the modified algorithm can sort signals on the basis of differing complex-valued spectral self-coherence (e.g., different timing phase of the signals in a communications net), whereas the cross-SCORE algorithm can only sort on the basis of differing spectral self-coherence strength (magnitude). These analytical results are verified by computer simulation for a narrow-band antenna array excited by multiple PCM SOIs having the same baud rate and by FDh4-FM and TV interference.
Abstract
The Self Coherence ILEstoral (SCORE) algorithm has been shown to blindly extract a single signal of interest with spectral correlation at a known value of frequency separation (e.g., baud rate) from an arbitrary number of interferers without correlation at that frequency separation. This paper investigates the ability of the SCORE approach to extract cyclostationary signals from a rank L spectral se[f-coherence environment where L signals exhibit spectral correlation at the same frequency separation. It is shown analytically and by computer simulation that the SCORE algorithm can separate multiple signals with spectral self-coherence at the same value of frequency separation, provided that those signals have different self-coherence strengths. An extension of the SCOICE algorithm is also developed that can separate signals if their complex-valued spectral self-coherence functions are different, e.g., in a communications net where the signals have different timing phases. It is shown that the SINR of the separated signals approaches the maximum attainable SINR.
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Brian G. Agee AGI Consulting 3000 Cowell Blvd. Davis, CA 95616
Introduction
The recently discovered Self Coherence REstoral (SCORE) approach has been shown [2,3,4] to blindly extract signals of interest (SOIs) from co-channel interference in environments in which only the SOIs exhibit spectral self-coherence at the frequency separation of interest, for example, where spectral components of a PCM SO1 separated by the baud rate are correlated. The spectral self-coherence property is shared by niost communication signals as a result of periodic switching, gating, or mixing operations at the transmitter [5,6,7]. For example, spectral self-coherence is induced at multiples of the baud rate in PCM signals, including BPSK, QPSIC, and QAM, and at twice the carrier frequency in DSB-AM and VSB-Ah4 signals. The spectral self-coherence of a received signal is degraded by additive interference and noise which does not exhibit spectral self-coherence a t the frequency separation of interest (e.g., if a PCM SO1 is corrupted by a PChiI interfrrer having a different baud rate). In this light, the SCORE algorithm can be viewed as a property restoral algoiithni that adapts the receiver to remove interference so as to 'This work was supported in part by ESL, Inc. and the Calif. State MICRO Program. 22nd Asilomar Conference. @lSSS hlaple Press.
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Spectral Self-coherence
A signal waveform s ( t ) is said to be spectrally self-coherent ut frequency separation a [6] if s ( t ) is correlated with s ( t ) frequency-shifted by a for some lag 7,i.e., if the spectral selfcoherence function &(.) is not identically zero,
where
3.1
Algorithms and Analysis Cross-SCORE
In light of the previous discussion of spectral self-coherence, an intuitive approach to the blind signal extraction problem is to choose the weight vector w that maximizes the spectral self-coherence of the processor output y ( t ) = w H x ( t ) ,since this minimizes the contribution of signals that are not spectrally self-coherent (e.g., stationary noise and interference, PCM signals at different baud rates, etc.). Investigation of methods for adapting the vector c in (5) has led to the cross-SCORE algorithm which maximizes the spectral cross-coherence between the processor output y ( t ) = w H x ( t )and the reference beamformer output r ( t )= c H x ( t ) . The algorithm can be mathematically stated as
(.)M denotes infinite time-averaging. The function
R:s(.r)is referred to as the cyclic autocorrelation function. The spectral self-coherence function and cyclic correlation function are developed in detail in the theory of spectral correlation [5,6], where it is shown that cyclostationay a.nd almost-cyclostationary waveforms exhibit spectral selfcoherence at discrete multiples of the time periodicities of the waveform statistics. This class of waveforms includes most communication signals; for instance, each PCM signal exhibits spectral self-coherence at multiples of its baud rate. The spectral self-coherence concept is particularly useful in interference environments because non-spectrally-selfcoherent interference has zero cyclic correlation. For example, consider a received signal vector x ( t ) which is the sum of the SO1 s ( t ) and an uncorrelated interference field i(t),
In [3] it is shown that (6) is accomplished for c given by c = R;iXxH(~)w,
(7)
where w is the dominant mode of the cross-SCORE eigenequation RG,(.)R;$XxH(~)w = XRXxw. (8)
x ( t ) = a s ( t )+ i ( t ) . (2) If s ( t ) is the only component of x(t) spectrally self-coherent at a, (i.e., if this is a rant-i spectral self-coherence environment), then the cyclic autocorrelation of x ( t ) can be expressed a scaled version of the crosscorrelation between x ( t ) and s ( t ) ,
In the rank-I spectral self-coherence environment only the dominant mode of (8) performs signal extraction. However, in the rank-L spectral self-coherence environment the eigenvectors corresponding to the smaller eigenvalues also perform useful signal extraction. In the following analysis this behavior is shown analytically in the noiseless rank-l environment. Given L independent signals impinging on an M-element antenna array, L 5 M, the received signal vector x ( t ) is given by L
x(t)=
ais;(t) = As(t),
(9)
i=l
yielding the following relations: Post-multiplying (4) by any vector c not orthogonal to a yields a vector proportional to R,,(O),
= AR,",(.)A~ Rxx(0) = ARss(0)AH,
%,(TI
Thus, a scaled version of the crosscorrelation R,,(O) needed for the conventional MMSE processor w = RG:R,,(O) is obtained without knowing the signal waveform s ( t ) . Methods for exploiting this property for a fixed c led to the leastsquares SCORE algorithm reported in [2,3]. As shown there, the Icast-squares SCORE algorithm is applicable only to the rank-1 spectral self-coherence environment where a single signal cxhibits spectral self-coherence at the a of interest, and it suffers from slow convergence. This restricted applicability motivates the development of techniques to adapt the vector c to reject other signals (including other spectrally self-coherent signals) and to speed convergence.
(10) (11)
where R,",(T)and Rs,(0) are the diagonal cyclic- and conventional-autocorrelation matrices of the transmitted signals, respectively. If A has full column rank L then (6) can be written in terms of R&(T)and h S ( O ) as:
where f and g are relative strengths of the signals s i ( t ) within each beamformer output,
f = AHw, g = AHc,
(13)
that is, where w H x ( t )= f H s ( t )and c H x ( t )= g H s ( t ) . The resulting solution f satisfies the following simple eigenequa-
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tion, diag
(I P : , ~ , ( T ) ~ } ~
t=l
where the eigenvalues X1 5 lP:,s,(T)/2
... 5
f =
Xf,
(14)
eigenequation (14), the following mathematical progression yields such an algorithm:
XL take on values A, =
.
If the eigenvalues are distinct (each signal has a different spectral self-coherence magnitude), then the elements of each resulting eigenvector f, for non-zero P:,~,( T ) are all zero except for the zth element which is one, i.e.,
f, = [O,. . . ,o, 1,0,. . .
,oy.
(15)
Comparison with the definition (13) reveals that
ayw, = 0 for all 1
# 2,
and arw, = 1,
(16)
implying that each cross-SCORE eigensolution perfectly extracts one signal, that is, that the cross-SCORE algorithm is able to sort through the environment on the basis of the spectral self-coherence magnitude of each transmitted signal. A complete analysis of the general noisy environment has not yet been completed. However, it has been shown [l] that the noiseless results will be a close approximation if the generalized SINR, G , AHRi’AF&,, is small (llGsll
