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the users and is explained in next section. Also, a decorrelating decision-feedback (DF) detector for synchronous and asynchronous CDMA channels have been proposed in the literature [3], [4]. Performance gains of DF detectors are more significant than decorrelator, especially for relatively weak users [3], [4]. B. Spatial- Temporal Decorrelating (STD) Detector Integrated spatial-temporal processing of the re- ceived signal has been shown to provide significant performance improvement over the decorrelator [5]. In most cases the multiple users are distributed in space in such a way that they are intercepted at the detector from var- ious directions. By exploiting the signals' spatial distribution (direction matrix) and the temporal properties (crosscorrelation matrix R) followed by linear decorrela- tion (decorrelator) a uniformly superior performance has been achieved. The signal at the output of the matched filter is given as y = MWb + n. (3) The above equation is similar to Eqn.1 except the matrix filter M. Where M here is the spatial-temporal crosscorrelation matrix. The next stage of processing is similar to the decorrelator and the output of the matrix filter in this case is y = M-'y = Wb + 6, (4) where n is a Gaussian noise vector with the autocorrela- tion matrix M(n) = (r2M-l. Comparative results with other suboptimum detectors have shown superior performance gains of STD detectors [5]. The superior performance of STD detectors has been the motivation of this work, in which spatial filtering is combined with decision- feedback (STDF). The STDF detector is derived in next section and its performance is compared with the other suboptimum detectors. 11. SPATIAL-TEMPORAL DECORRELATING DECISION-FEEDBACK (STDF) DETECTOR It has already been shown that for STD the complexity of the detector grows exponentially as the number of users increase [5]. Hence, the same complexity can be ex- pected for optimum STDF detectors. Here only a suboptimum STDF is considered. The suboptimum STDF detector is derived by exploiting the spatial-temporal crosscorrelation matrix M given in Eqn.3. The matrix M can be written as the Hadamard or Schur product of matrices A and R. M = R O (A~A), (5) where o denotes the Hadamard product of matrices [6]. A is the directions matrix comprising the direction vec- torsal,aa,...,ak,i.eA= [ a1 a2 ... aK ],andthe 2- v- 3+ : Spatid Pl, '3 MF2 (cjl ' 1 Filter I I :: FLll @J- w% I I : CL :: :: I I 2Kl I .. n 1- - - I I I 12 @L : I K I , direction vector ak = [ 1 ejsxl ... ejokp 1'. The direction vector ak expresses the phases and gains of the P sensors relative to a reference sensor in the direction of arrival of the wavefront of user k. AH is the conjugate transpose of A. Analysis shows that R is a positivedefinite matrix. Also, it can be easily shown that AHA is always a positive definite matrix, that implies matrix M will also be a positive definite matrix. If M is a positive definite matrix, then it can be decomposed as M = L ~L, (6) where LH is an upper triangular matrix and L is a lower triangular matrix. The above method of matrix decomposition is known as the Cholesky decomposition [7]. The matrices LH and L correspond to causal and stable matrix filters. If the filters are represented as a spectrum (frequency domain representation), then the Cholesky decomposition can be viewed as a spectral factorization the- orem [8]. In STDF, the sampled output y of the matched filter is passed through the feedforward filter (LH)-l, and the resulting output vector of the matrix filter (LH)-' is given as y = (L*)-'y = LWb + n, (7) where n is a white Gaussian noise matrix with the di- agonal elements being the variance u'. Therefore, the feedforward filter (LH)-' is nothing but a whitening filter. The model given in Eqn.7 is a white noise model of the CDMA channel. Also the expression given in Eqn.7 makes analysis simpler. The kth component of the vector 203 Authorized licensed use limited to: Ryerson University Library. Downloaded on July 6, 2009 at 16:08 from IEEE Xplore. Restrictions apply. bK
24 y can be written as Now the SNR of kth user can be given as From the above equation it can be seen that the kth user term has a desired term, an interference term, and a white noise t,erm. The above model gives rise to the decision- feedback technique used in STDF. The users signals in STDF are arranged in decreasing order of energy (E1 2 & 2 . . . 2 &). Where, user 1 will be the strongest user and the I
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Signal Analysis Research (SAR) Grou
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2006.05 Support Vector Machines Bas
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A Novel Way of Lossless Compression
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CONTENT BASED AUDIO CLASSIFICATION
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