Hacettepe University Department of Electrical and Electronics Engineering An Adaptive Blind Channel Shortening Algorithm for MCM Systems Blind, Adaptive Channel Shortening Equalizer Algorithm which provides Shortened Channel State Information (BACS-SI) Cenk Toker & Gökhan Altın Hacettepe University, Ankara, Turkey 1/
Outline Introduction Channel Shortening in the Literature MMSE SAM Blind, Adaptive Channel Shortening Equaliser Algorithm which provides Shortened Channel State Information (BACS-SI) Definition of the Cost Function, Cost Surface Application of the Genetic Algorithms Simulations and Results Conclusions /
Introduction Multicarrier Modulation (MCM) can successfully combat InterSymbol Interference (ISI), but requires Cyclic Prefix (CP). v samples N samples CP s y m b o l ( i ) CP s y m b o l ( i+1) copy copy CP prevents ISI/ICI. CP decreases throughput by N/(N+v). 3/
MCM(DMT) System TEQ aims at shortening the channel to a fixed length, rather than full equalisation, FEQ cancels the phase rotations caused by the channel. Data bits Serial to Parallel S/P m 0 bit m 1 bit QAM QAM m N/-1 bit QAM X 0 X 1 X N/-1 X * N/-1 X * 1 X * 0 IFFT x 0 x 1 x N-1 Parallel to Serial P/S * Add CP Output bits P/S QAM QAM QAM 1/B 0 X 0 X 1 1/B 1 FEQ x x 1/B N/-1 X N/-1 x Y 0 Y 1 Y N/-1 Y * N/-1 Y * 1 FFT Remove CP TEQ 4/ Y * 0 y 0 y 1 y N-1 S/P DMT Channel + TX-RX Filters noise +
Channel Shortening Channel Shortening Methods in the Literature, - MMSE -MSSNR -MBR, etc. requires Channel State Information (CSI) requires Channel Estimation requires Training Sequence (reduces bit rate) SOLUTION Blind Channel Shortening (does not require training) Blind Channel Shortening Methods in the Literature: MERRY (Martin, et al., 0) SAM (Balakrishnan, et al. 03) SLAM (Nawaz, et al. 04), etc. 5/
MMSE n x n n r n Equaliser y n e Channel (TEQ) + n + h w - ŷ n Delay Target Impulse Response (TIR) b MMSE aims at minimising the variance of the difference between the signals at the output of - the Equaliser (TEQ) and - the Target Impulse Response (TIR). 6/
SAM* x n Channel h n n + r n Equaliser (TEQ) w y n c=h*w (shortened channel) Adaptive Algorithm Main idea: - If the length of the shortened channel is v+1 taps, - then the length of its autocorrelation R cc will also be v+1 taps. don t care suppress suppress -(v+1) v+1 SAM l= v+ 1 * Balakrishnan, et al., Blind, Adaptive Channel Shortening Sum-Squared Autocorrelation Minimization (SAM), IEEE Trans. Signal Process., Dec.003 7/ J = n c 1 R cc ( l)
MMSE vs. SAM MMSE Does NOT require communication CSI (i.e. blind equalisation) X CAN provide shortened channel CSI SAM X BACS-SI 8/
Blind Equalisation: Delay BACS-SI Blind, Adaptive Channel Shortening Equaliser which provides Shortened Channel State Information (BACS-SI) n(n) Equaliser R x(n) Channel r(n) yy e(n) + (TEQ) + h - w No access to - the channel coefficients h, or - the input bits x(n) is possible. Only access to - the received signal r(n) Target Impulse Response (TIR) b - the statistics of x(n), i.e. E{x(n)}=0, var{x(n)}=1 9/ R yy ^^
BACS-SI Equality of the autocorrelations of the outputs of the upper and lower branches means the autocorrelations of the equalised channel (c=h*w), and the TIR, b, (of length v+1) will be equal. don t care in SAM, cared by TIR in BACS suppress suppress -(v+1) v+1 J BACS = n 1 c l= 0 R yy ( l) R yy ˆ ˆ ( l) J SAM = n c 1 l= v+ 1 R yy ( l) 10/
BACS-SI Cost surface of BACS-SI is multimodal: J J BACS BACS = = E{( R n 1 c l= 0 yy ( R ( l) R cc yy ˆˆ ( l) + σ ( l)) n 0 R ww } ( l) R bb ( l)) = n 1 c l= 0 ( R cc ( l) R bb ( l)) - h=[1 0.3 0.] T, b=[1 0.5] T - equaliser: w=[w 1 w w 3 ] T θ = φ = arctan arctan ( ) w + w w 1 ( w w ) / 1 3 11/
BACS-SI Proposition: Taking the conjugate/reciprocal of any combination of zeros of a sequence w.r.t. the unit circle in the z-plane does NOT alter the autocorrelation of that sequence. a=[0.5088-0.4097 0.36-0.6768 0.0018 0.0936-0.0009] b=[0.0679-0.0183 0.483 0.3070-0.4138 0.7043-0.0068] 1/
BACS-SI All minima are related to each other by simple zero flipping operations, If we know a single minimum, we can directly find all minima. Only half of the minima perform shortening, Each minimum corresponds to a particular system delay, Each minimum results in different system performance (e.g. bit rate) How to find a minimum? Stochastic gradient descent algorithm Blind operation no prior information for initialisation arbitrary initial point. Variable to update: - TEQ coefficients, w - TIR coefficients, b 13/
BACS-SI C.Toker 14/ BACS-SI Composite variable: Single step size, µ Update equation of the iterative algorithm: Iterate until a stopping criterion is satisfied, e.g. Max. no. iterations, to find the optimum TEQ, w, and TIR, b, coefficients. = = + ) ( ) ( ) ( ) ( 1 1 n J n J n J n J n n b w f f f f μ = b w f
BACS-SI For proper operation of the MCM system, FEQ coefficients needed. We need the Fourier Trans. of the shortened channel, c, for FEQ. No direct access to the shortened channel, c. Substitute TIR, b, instead, In MMSE Shortened CSI and TIR are (ideally) identical In BACS, their autocorrelations are identical, but themselves may not be. 15/
Genetic Algorithms How to find matching shortened CSI, c, and TIR, b? We propose an approach based on genetic algorithms. Each minimum of J BACS results in a different (w,b) pair, Each minimum is connected to others through zero flipping operations for the zeros of both w and b. Problem is binary in nature (inside or outside the unit circle). Genetic algorithm parameters: Each relative position is represented by a gene (inside: 0 or outside: 1) Collection of all w and b genes for a particular minimum of J BACS : a chromosome: 16/
Fitness function: Genetic Algorithms We can use pilot tones that are already there (for other puposes) For example, ADSL, 64th tone is reserved for timing recovery, If TEQ & FEQ are running properly, FEQ output at 64th tone is R( 64) = 1 To check whether the shortened channel and TIR are the same for a particular solution (chromosome), we can check k K R( k) 1 Initial population, 18 randomly generated chromosomes, Mutation rate 30%, i.e., 30% of all genes change state (1 0, 0 1), Stop algorithm after 750 iterations. 17/
Genetic Algorithms magnitude 0.3 0. 0.1 0 before GA shortened channel IR TIR magnitude autocorrelation -0.1 0 5 10 15 0 5 30 35 0.3 0. 0.1 0 samples after GA -0.1 0 5 10 15 0 5 30 35 0.3 0. 0.1 0 samples autocorrelations of shortened channel IR and TIR before/after GA shortened channel IR TIR shortened channel IR before GA shortened channel IR after GA TIR before GA TIR after GA -0.1 0 10 0 30 40 50 60 70 samples 18/
SIMULATIONS and RESULTS FFT size 51, TEQ length 16, CP length 3, TIR length 33, and channel CSA test loop 1 Fixed Step Size BACS-SI SAM Adaptive Step Size BACS-SI SAM İterations 19/
SIMULATIONS and RESULTS Channel Impulse Response Bit Rate Orig. Channel BACS-SI SAM 3.5 4 x 106 3 magnitude bits per second.5 1.5 taps 1 BACS MFB 0.5 MSSNR SAM 0 0 500 1000 1500 000 number of iterations 0/
CONCLUSIONS Most of the channel shortening equaliser proposals in the literature assume perfect channel state information. This requires channel estimation utilising training sequences which do not convey information, hence reducing throughput. Blind channel equalisation does not require channel estimation, hence training sequences. The blind channel shortening equaliser in the literature (MERRY, SAM, SLAM, etc.) do not directly provide shortened channel impulse response which is vital for the FEQ, hence the proper operation of an MCM system. The proposed BACS-SI algorithm can provide this information without any additional effort. 1/
Thank You. /