nd International Worshop on Materials Engineering and Coputer Sciences (IWMECS 015) Overlapping Signal Separation in DPX Spectru Based on EM Algorith Chuandang Liu 1, a, Luxi Lu 1, b 1 National Key Laboratory o Science and Technology on Blind Signal Processing. a liuchuandang0603@16.co, b yluluxi@gail.co Keywords: Gaa Mixture Model, EM Algorith, DPX. Abstract. Overlapping signal separation in spectru is a diicult proble. We use Gaa Mixture Model to orulate the distribution o signal power in each requency bin o digital phosphor technology (DPX) spectru. Then Expectation Maxiization (EM) Algorith is used to solve the odel. Siulation results show that when CIR is greater than.7db, paraeters estiation error rate o this algorith is less than 1e-5. Introduction The blind signal processing studies how to discover and analyze unnown, unlicensed and alicious signals. It attracts great interests o the spectru supervision departents around the world. To exploit spectru eiciently, supervision departents usually divide spectru into pieces and authorize organizations or individuals to use soe speciied pieces in speciied tie and speciied places. Thus, supervision departents will receive spectral overlapping signals. Beore udging which signal is illegal, they need to separate these overlapping signals. However, it is diicult or traditional ethods to separate overlapping signals because traditional ethods lie spectru ethod and tie-requency ethod entioned in [1] contain lots o averaging and soothing calculation. These calculations will decrease the power response o burst signals in spectru and even regard little signals as noise. Digital Phosphor Technology (DPX) can store the power response during a period o tie and display in a pseudo-color way []. Copared with traditional ethods, DPX only counts the appearance tie o signal power. Thus DPX spectru contains undistorted power inoration o signals. In this paper, we use Gaa Mixture Model to orulate the distribution o signal power in each requency bin o DPX spectru. Expectation Maxiization (EM) Algorith [3] is then adopted to calculate the odel paraeters which denote the power and the appearance tie o signals. Analysis o Received Signals in DPX Spectru Power Distribution o Received Signals in DPX. DPX counts tie-requency-power inoration obtained by Short-Tie Fourier Transor (STFT) and copresses it into a -D histogra whose coluns represent the trace o aplitude values and rows or points on the requency axis. Each grid in the -D histogra stores the appearance nuber o ties o signal whose power and requency correspond to the colun and row, respectively [4]. DPX is able to ephasize the signal o interests by pseudo-coloring ethod. However, there is no autoatic ethod to recognize overlapping signals and separate the in DPX. Fig. 1 DPX Histogra, DPX Display Illustration and DPX Iage o Signal 015. The authors - Published by Atlantis Press 464
Assue the received sapled signal is S rnsi nvn, n0,1,, (1) i1 where, si n denotes the i-th signal wave and n is the saple index. Each signal can be narrow-band bursts, wide-band bursts or continued signal. vn is noise and is usually not white in the broadband spectru. Ater the Discrete Fourier Transor (DFT) as (), the spectru is divided into ultiple narrowband requency bins. M1 S i, s,,,( 0,1,, 1; 0,1,, ) i v () H t r tm e H t H t M t 0 i1 where, / M, 0,1,, M 1 is the requency index, M is the length o DFT and t is the bloc index o DFT. Let the bandwidth o each requency bin be saller than the channel correlation bandwidth. Thereore, Hvt, in each bin ollows the coplex Gaussian distribution with zero-ean and -variance (the noise power). The easureents o our approach are the average power in requency. 1 tn N 1 (3) N ttrn H t, is r r, Ht, Rt where tr 0,1,, is the tie index o easureents, noise only during one easureent tie, according to the eature o coplex Gaussian distribution, N is the average length. When Rt r, ollows Gaa distribution. The probability density unction (PDF) o r, N 1 x exp x, x0 x, N N 0, x 0 Rt is where, / N contains the power inoration o signals or noise, and is Gaa Function. Sybols in digital odulated signal approxiately have independent unior distribution. According to Central Liit Theore, the greater N is, the closer signal s power su distribution approaches to Gaussian distribution, and the closer Rt r, approaches to Gaa distribution. Actually, received signals in every requency bin are the cobination o pure signals and noise, and noise in one requency bin can be regarded as an independent wideband signal. For siplicity, we use signal to express received signal or noise. Gaa Mixture Model in DPX Frequency Bin. There ay be several signals in one requency bin and the distribution o every signal s power ollows Gaa Distribution. So we deine a Gaa Mixture Model to orulate the power distribution in each requency bin. K P( y θ) ( y ) (5) 1 where, K denotes the nuber o dierent Gaa distributions (equals to the nuber o noise and signals). ( y ) is the PDF o Gaa Distribution with paraeter. y ( y1, y,, y N ) is an observation set o signal power in requency bin, θ ( 1,,, K ; 1,,, K ) is paraeter vector, and is the proportion o ( y ) and equals to the ratio that appearance nuber o ties o ( y ) divide the total appearance nuber o ties in requency bin, subect to Our wor in this paper is to calculate vector θ in each requency bin. K 1 (4) 1 (6) 465
EM Algorith Solution o Gaa Mixture Model EM Algorith was proposed by Depster et al. in 1977. It is an iterative ethod to solve the Maxiu Lielihood Estiation (MLE) proble o probability odel paraeters with hidden variables. It is proved in [3] and [5] that the iterative sequence o EM Algorith converges to the stable point o logarithic lielihood unction sequence. Gaa Mixture Model in (5) ulills the conditions to use EM Algorith. Let be the hidden variables o observation y, 1, i y is ro ( x ) (7) 0, otherwise The coplete hidden variable vector to generate observation lielihood unction is where N n, n K 1 y is 1 K γ (,,, ), whose 1 K N K N N N exp n y y n P( yγ, θ ) ( y ) (8) 1 1 1 1 N N. 1 Derive the Q-unction o P( yγ, θ ) K N N () i Q( θθ, ) ( E ) log ( E ) log ( N)+ Nlog + N-1y - y (9) 1 1 1 where, i denotes the iteration index, and E denotes the expectation o hidden variable. Then we use Bayes Forula to copute E in (9). where, Py ( =1, ) ( =1 ) ( ) θ P θ y 1,,, N E =, K K 1,,, K Py ( =1, ) P( =1 ) ( y ) θ θ =1 1 is the integration interval width o ( x ) in logarithic horizontal coordinate. Last, we need to derive the MLE o Q-unction. Let the partial derivatives o (9) with respect to and equal to 0 (subect to (6)), thus we have N N N ˆ = ˆ ˆ ˆ ˆ N y, N (11) 1 1 1 We can get a suicient good solution o θ by iterating (10) and (11) until stopping conditions ulilled. Stopping criterion or iteration can be a dierence threshold o paraeter vector θ between adacent iterations. Processing Flow o EM Algorith Potential signal aount (paraeter K in (5)) in each requency bin o DPX spectru is an iportant input o EM algorith. We ind that the proportions o existing signals are approxiately the real value and the spare proportions are approxiately zero, i K is greater than the real nuber o signals. Table 1 shows the calculation results o one requency bin in Fig. 5 with an increasing K. Actually there are two signals in the requency bin in total. When K is greater than, spare proportions are approxiately equal to 0. Table 0shows the EM Algorith results o soe requency bin in Fig. 5 when K 5. denotes the signal nuber o Gaa Distribution. All are between 0.1 and 0.3, but the values o are all around -66.74dB and vary less than 1%. Thus, these ive results should be regarded as one signal. In act, there is only one signal in that requency bin. (10) 466
Table 1 Proportion Calculation Results o Soe Frequency Bin in Fig. 5 K 1 3 4 1 1.0000 - - - 0.7177 0.83 - - 3 0.0000 0.7177 0.83-4 0.0000 0.0000 0.7177 0.83 Table EM Algorith Results o Soe Frequency Bin in Fig. 5, K 5 (db) 1 0.1416-66.734 0.1670-66.737 3 0.1960-66.741 4 0.90-66.744 5 0.663-66.747 Then we propose a selection strategy o K as ollows: 1) set a threshold to ilter the results with sall signal proportions; ) quantiy signal power and erge the signals with close power; 3) process all requency bins with step 1) and step ) and ar the results with pseudo-color. The entire processing low is shown in Fig.. Fig. Paraeters Estiation Algorith Flow Algorith Perorance Analysis and Siulation We use Monte-Carlo Siulation to calculate signal separation error rate. To avoid the occasional error, we record the nuber o Monte-Carlo siulation ties when signal separation error occurs 100 ties. In every Monte-Carlo siulation we generate 1000 power saples in a requency bin. The error criterion o signal separation is deined that the dierence between calculated power value and the real power value is greater than 0.5dB or calculated proportion diers ro the real ones ore than 0.05. Let 40 N, 0.5 and CIR increase ro 0.5dB to 4dB. Siulation results in Fig. 3 shows N that the error rate decreases along with CIR. When CIR is greater than.7db, error rate is less than 1e-5. The reason is that the distances between signals distribution axius depend on CIR. The higher the CIR is, the ore distinct the signal is ro the intererence. 467
Let N 0.5, CIR =3dB, 4dB, 5dB, 6dB, and N increase ro to 40. Siulation results in Fig. 4 deonstrates that the error rate decreases along with N. According to the Central Liit Theore, signal power distribution approaches closer to Gaa Distribution along with the increase o N, so the estiation will be ore accurate when N is higher. Fig. 5 shows the DPX iage o siulation signal. The length o DFT in DPX is 51. Siulation signals contain: 1) one -67dB continued BPSK signal with 6.5MHz carrier requency and 4.4MHz bandwidth; ) one -79dB burst BPSK signal with 16.3MHz carrier requency and 8.75MHz bandwidth; 3) a set o requency hopping BPSK signals whose carrier requency set is {5.5, 8., 10.9, 13.6, 16.3}MHz, bandwidth o each hop is 1MHz and power is -7dB; and 4) one 8.1dB coplex white Gaussian noise. Siulation signals separation result is showed in Fig. 6. 10 0 N =40, N =0.5 10-1 Error Rate 10-10 -3 10-4 10-5 0 0.5 1 1.5.5 3 3.5 4 CIR(dB) Fig. 3 Paraeters Estiation Error Rate Curve 10 0 10-1 CIR=3dB CIR=4dB CIR=5dB CIR=6dB Error Rate 10-10 -3 10-4 0 5 10 15 0 5 30 35 40 N Fig. 4 Paraeters Estiation Error Rate Curve 468
-65 Power(dB) -70-75 -80-85 50 100 150 00 50 Frequency Bin Fig. 5 DPX Iage o Siulation Signal -66-68 -70 Power(dB) -7-74 -76-78 -80-8 -84-86 0 50 100 150 00 50 Frequency Bin Fig. 6 Siulation Signal Classiication and Visualization Conclusion In this paper, we ocus on the overlapping signal separation. We use Gaa Mixture Model to orulate the power distribution o ixed signals in each requency bin o DPX spectru. Then EM Algorith is adopted to solve this odel. Our siulation results deonstrate that the error rate o the algorith is less than 1e-5 when CIR is greater than.7db. Further wor will be directed to a precise signal classiication and recognition. Reerences [1] Tetronix, Fundaentals o Real-Tie Spectru Analysis. (009) [] Tetronix, DPX Acquisition Technology or Spectru Analyzers Fundaentals Prier. (009) [3] P. Depster, N. M. Laird and D. B. Rubin, Maxiu Lielihood ro Incoplete Data via E-M Algorith, Journal o the Royal Statistical Society, 39 (1977). [4] S. Guo, J. Su, G. Sun, Z. Zhao, and Z. Chen, Design o DPX Syste or Real-Tie Spectru Analysis and Signal Detection, IEICE Electronics Express, 10 (013) 1-8. [5] G. McLachlan and T. Krishnan, The EM Algorith and Extensions, New Yor: John Wiley & Sons, 1996. 469