Coded Code-Division Multiple Access (CDMA) The combination of forward error control (FEC) coding with code-division multiple-access (CDMA).
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1 Coded Code-Division Multiple Access (CDMA) The combination of forward error control (FEC) coding with code-division multiple-access (CDMA). Optimal Processing: Maximum Likehood (ML) decoding applied to the time-varying trellis with a number of states which is exponential in the product of the number of users in the system and the contraint length of the codes used []. Linear Pre-Processing: Linear filtering operations are used to preprocess the received signal and generate appropriate FEC decoder metrics. Such preprocessors can be the decorrelator or the MMSE filter, but the projection filter is best suited for FEC codes and asynchronous random CDMA [, 5, 6]. No iterative processing is required. Iterative Processing: CDMA has a time-varying trellis respresentation similar to a convolutional code, and FEC coded CDMA systems are interpreted as the concatenation of a bank of FEC codes with a CDMA channel encoder. The Turbo decoding principle is applied at the receiver, where the CDMA multiuser detectors and a bank of single-user error control APP decoders form a closed loop system and exchange soft updated information. I Interference Cancellation: subtraction of interference estimates from the received matched-filter signals [4], lowest complexity O(K). II Minimum Mean-Square Error Filtering: linear operation on the interference-cancelled singals to minimize the mean-square error between the transmitted signals and the filter outputs [9], polynomial complexity O(K 3 ). III A Posteriori Probability Decoding: CDMA decoding is performed with an APP channel decoder [3], but this leasds to prohibitive complexity O( K ).
2 EE 795: Statistical Communication Theory Serial Concatenated Coded CDMA Under some circumstances, in particular on channels with low signal-tonoise ratio, powerful error control codes, which are themselves serially concatenated, are required. u Error Control Encoder Outer Encoder Π Inner Encoder d π Spreading Function a (t) x (t) τ u Outer Encoder Π Inner Encoder d π a (t) τ x (t) + x(t) a K (t) u K Outer Encoder Π K Inner Encoder d K π K τ K x K (t) Assuming the equal powers of all users, the signal from the kth user is L x k (t) = d k (j)a j,k (t jt τ k ) j= L: number of encoded symbols in a frame; u k : information bits for user k; d k : serially concatenated encoded symbols for user k; a j,k (t): supported on the interval [,T], is the energy-normalized spreading waveform for user k during symbol j; τ k : <T time delay of user k.
3 EE 795: Statistical Communication Theory 3 Serial Concatenated CDMA Model Using a nondistorting channel model, the received CDMA signal embedded in additive white Gaussian noise (AWGN) is y(t) = K x k (t)+n(t) k= where n(t) is the zero mean white Gaussin noise with two-sided spectral density σ. Assuming perfect timing and phase knowledge yielded through normal procedures, the received signal sampled by chip matched filter is in the discrete form y = Ad + n A: (L +)N by LK matrix whose jth column is a j/k,k =[ in+τk /T c, a T k,i, (L i)n k] T, with l being a length-l all-zero row vector; d: [d (),...,d K (),d (),...,d K (L)] T is a length-(l +)N vector of encoded symbols; n: length-(l +)N vector of white noise samples with variance σ.
4 EE 795: Statistical Communication Theory 4 Iterative Receiver y CDMA Canceller Π Π Error Control Decoder Outer Outer π π Inner Inner Π Π Λ(d ) Λ(d ) Π K Outer π K Inner Π K Λ(d K ) tanh() tanh() tanh() Error Control Decoder: Normal iterative decoders of serial concatenated codes for single users with soft output; CDMA Canceller: interference resolution function which generates reliability outputs of the encoded symbols of different data stream. The most sophisticated case is an APP decoder for the CDMA channel, which generates log-lokelihood ) ratios (LLR) of the encoded symbols d, λ(d k ) = log. Due to complexity, ( Pr (d k = y) P r (d k = y) we are interested in linear interference suppression algorithms, e.g., interference cancellation, MMSE filter, etc.
5 EE 795: Statistical Communication Theory 5 Interference Cancellation ( λ(dj (i)) ) Soft bit estimation: dj (i) =E[d j (i) λ(d j (i))] = tanh is the soft estimate of the coded symbol of user j at time i, which was generated during the previous iteration; Interference substraction: the estimated interference is reconstructed and subtracted from the CDMA channel observations ỹ = {}}{ L L K Pk d k (i)a i,k + Pk (d j (i) d j (i))a i,j +n i= i= j= (j k) I k Correlation: Correlated with spreading sequence for user k z k (i) =a T i,kỹ = P k d k (i)+ L l= K Pj h kj (i, l)(d j (l) d j (l)) + n k,i j= (j k) where h kj (i, l) is the correlation between user k at time i and user j at time l. Asymptotic VTR : Cancellation leads back to single user FEC decoder systems. The input variance to the FEC decoders is a linear function of the system load and cancelled soft symbol variance. σk,ic = σ + K N Pσ d K,N = σ + αp σd σ k,ic = f VTR (σ d) ( α = K ) N the output variance of the interference supression detector can be view as a function of its input variance, which is termed variance transfer function (VTR).
6 EE 795: Statistical Communication Theory 6 Minimum-Mean Square Error (MMSE) Receiver MMSE filtering is a concept from estimation theory transplanted to the field of (multiuser) detection. This filter minimizes the expected squared error: ω k = arg min E ( d k ω T z ) ω R N where z =[z,z,...,z K ] T contains the interference cancellation outputs. The MMSE detector for user k is now given by d k,mmse = ω T k z = P k +P k a T k V a k a k V y where [ a k is the spreading sequence for user k, and V = A T k D k A k, D k = ] diag E[P (d d ),,P k (d k d k ),P k+ (d k+ d k+ ),,P K (d K d K ) ] and A k =[a,, a k, a k+,, a K ]. Asymptotic Signal-to-Interference Ratio (SIR) SIR mmse = I(σ dp, P k, SIR mmse ) = P k σ + αe [I(σd P, P k, SIR mmse )] σd PP k P k + σd P SIR mmse Inference MMSE filter outperforms interference cancellation: σ k,mmse = P k SIR mmse = σ + αe [ I(σ dp, P k, SIR mmse ) ] <σ k,ic Iteriave versus non-iterative processing: SIR mmse σ d = }{{} non iterative SIR mmse }{{} iterative
7 EE 795: Statistical Communication Theory 7 Asymptotic MMSE VTR Function General solution does not exist for SNR mmse, however, if the powers are equal for all users SIR mmse = σ + σd (α )P σ4 + σd 4 σd + (α ) P +σd (α +)σ P σ σd σ The VTR plots show the normalized output variance σ k P (= SIR mmse )asa function of input symbol variance σd for MMSE and interference cancellation..5 normalized output variance σ k.5.5 α = : Interference cancellation : MMSE cancellation E s / N = db E s / N residual bit variance σ d Observations: MMSE is superior to interference cancellation, and the gap between the two decreases as bit variance σ d ; Asymptotic VTR of MMSE is non-linear for low SNR, and linear for high SNR.
8 EE 795: Statistical Communication Theory 8 Limiting Performance Theorem: As E b N symbol variance σd, the normalized variance σ k P k for the MMSE filter, with α. α IC = α mmse = K mmse = K IC + N is linear w.r.t. the Given the same FEC coding, N extra users can be accommodated by MMSE than interference cancellation in the limiting case. Example: 6 5 (4) 4 (3) K / N 3 () () () SCC, Simple Cancellation () SCC, MMSE Cancellation (3) /3 PC code, Simple cancelltion (4) /3 PC code, MMSE Cancellation E b / N (db)
9 EE 795: Statistical Communication Theory 9 VTR for FEC Codes No closed-form variance transfer functions are known for FEC APP decoders. VTR curves must be evaluated by simulation (6) (4) 3 () () (3) (4) ().5 σ k () σ k.5 () SCC () SCC (3) SCC 3.5 (4) SCC 4 (5) SCC 5 (6) SCC σd (3) (5).5.5 () rate /3 repetition code () rate /3 CC (v=) (3) rate /3 CC (v=3) (4) rate /3 CC (v=4) σd Arrows in the figure point to the variance levels at which the convergence may work for the specific codes and small loads. Serial concatenated codes are able to function in higher channel noise than the simple FEC codes. The slope of FEC code VTR curves limit the maximum system load for the given multiuser detectors in the case E b N. Consequently, single FEC codes are more efficient in high signal-to-noise ratios.
10 EE 795: Statistical Communication Theory Variance Exchange The multiuser detection and the error control decoders have complementary VTR functions, and the turbo decoding can be viewed as a variance exchange (VEG) process between these two blocks. Example: The left-hand plot shows the VEG for serially coded CDMA in low SNR situations, MMSE receiver outperforms multiuser interference cancellation (MUIC) by a factor of 3 at E b N =.6 db; The right-hand plot shows convolutionally coded CDMA is superior to serially coded CDMA in high SNRs..4 () SCC VTR () MMSE (K/N=, E b /N =.6 db). (3) MMSE (K/N=, E b /N =.6 db) (4) IC (K/N=.5, E /N =.6 db) b (4) () () rate /3 CC (v=) () SCC (3) MUIC (K/N=3.5) (4)MUIC (K/N=.7).8 σk.6.4. () (3) E b /N small σk.5.5 () () (3) (4) E b /N o σ d σ d
11 EE 795: Statistical Communication Theory Iteration Examples The decoding behavior of the turbo receiver can be tracked and visualized as trajactories between the VTR curves of the multiuser detection and error control decoders. Low SNR, low system load. 3.5 σk 3.5 σk 3.5 SCC K =5,N =5 3.5 Rate 3 CC K =5,N =5 E b N = db E b N = 5dB σ d σ d high SNR, high system load 3.5 σk σ k SCC K =3,N =.5 Rate 3 CC K =3,N = E b N = db E b N = db σ d σ d
12 EE 795: Statistical Communication Theory Simulation Examples BER of SCC and rate 3 CC in both low SNR and high SNR BER - SCC, 6 iterations Rate 3 CC, 4 iterations Note that the CC system is noise limited. Hence the shallow error curve compared to the turbo system. - Both system are fully loaded K = N E b N [db] BER Rate 3 CC iterations SCC 7 iteations The system using the simple CC code has a lod of α = 3, while the SCC system achieves only α =.3. The better system has less decoding complexity E b N [db]
13 EE 795: Statistical Communication Theory 3 Spectral Efficiency Capacity of random CDMA and asymptotic performance of coded CDMA systems using codes of rate R = 3, and iterative detection. AWGN capacity optimal processing () Sum Capacity [bits/dimension] (8) (7) (4) (3) (5) (6) () MMSE noniterative MF noniterative. () IC (rate /3, repetition code) () MMSE (rate /3, repetition code (3) IC (rate /3 CC, v=) (4) MMSE (rate /3 CC, v=) (5) IC (SCC ) (6) MMSE (SCC ) (7) IC (SCC ) (8) MMSE (SCC ) Average E /N [db] b
14 EE 795: Statistical Communication Theory 4 References [] P.D. Alexander, L. Rasmussen, and C. Schlegel, A class of linear receivers for coded CDMA, IEEE Trans. Commun., Vol. 45, No. 5, pp. 65 6, May 997. [] T.R. Giallorenzi and S.G. Wilson, Multiuser ML Sequence Estimator for Convolutional Code Asynchronous DS-CDMA systems, IEEE Trans. Commun., vol. COM-44, pp , Aug [3] M. Moher, An Iterative Multiuser Decoder for Near-Capacity Communications, IEEE Trans. Commun., vol. 47, pp , July 998. [4] M.C. Reed, C. Schlegel, P.D. Alexander, and J.A. Asenstorfer, Iterative Multiuser Detection for CDMA with FEC: Near-Single-User Performance, IEEE Trans. Commun., vol. 46, no,, pp, , Dec [5] C. Schlegel, S. Roy, and P.D. Alexander, Coded asynchronous CDMA and its efficient detection, IEEE Trans. Inform. Theory, Vol. 44, No. 6, November 998. [6] C. Schlegel, P.D. Alexander, S. Roy, and Z. Xiang, Multi-user projection receivers, IEEE J. Select. Areas Commun., Vol. 4, No. 8, pp. 6 68, October 996. [7] Z. Shi and C. Schlegel, Joint Iterative Decoding of Serially Concatenated Error Control Coded CDMA, IEEE Journal on Selected Areas in Communications, pp , August. [8] Z. Shi and C. Schlegel, Asymptotic Efficiency of FEC Coded Random CDMA using Joint Multi-user Dectection, IEEE Trans. on Inform. Theory, submitted, January. [9] X. Wang and H.V. Poor, Iterative (Turbo) Soft Interference Cancellation and Decoding for Coded CDMA, IEEE Trans. Commun., vol. 47, no. 7, pp. 46 6, July 999.
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