Iterative Multiuser Detection Benjamin Vigoda 6.975 EECS Graduate - - PowerPoint PPT Presentation

Iterative Multiuser Detection Benjamin Vigoda 6.975 EECS Graduate Seminar in Communications

Multiuser Detection

• multiuser detection uplink channel from a handset to a base
• station. Base station must demodulate/decode K − 1 inter-

fering handset signals.

• interference cancellation down-link channel from the base

station to the handset. Handset must separate the signal intended for it from others useful supplementary text: Sergio, Verdu “Multiuser Detection”

Gaussian Channel p(y|xi) = exp

• − 1

2σ2

• [y(t) − xi(t)]2dt
• (1)

x transmitted signal and y is the received signal. Hypothesize two possible transmitted signals xi and xj, which likelihood is greater (maximum)? p(y|xi) <> p(y|xj), (2) − 1 2σ2

• [y(t) − xi(t)]2dt

<> − 1 2σ2

• [y(t) − xj(t)]2dt,

(3)

• [y(t) − xi(t)]2dt

<>

• [y(t) − xj(t)]2dt,

(4)

• y(t)xi(t)dt − 1

2

• xi(t)2dt

<>

• y(t)xj(t)dt − 1

2

• xj(t)2dt.

(5)

Matched Filter In the single user CDMA channel y(t) = Axs(t) + σn(t), t ∈ [0, T], (6) signature sequence s, transmitted symbol b ∈ −1, 1, h(t) = as(t) ˆ x = sgn

• h(t)y(t)dt − 1

2

• xi(t)2dt
• (7)

sufficient statistic:

• y(t)xi(t)dt.

(8)

Multiuser channel, matrix form Y =

K

• i=1

Xisi + W (9) IF

• bank of matched filters, 1 per user
• users perfectly synchronized, bit and chip
• signature sequences sk linearly independent

Then = single user performance (optimum)

Nonorthogonal Signature Sequences But maintaining strict orthogonality involves synchronization → Hard because of real-valued multi-path time delays. Nonorthogonal is good anyway:

• # users is looser → graceful degradation of channel sharing
• Reliability depends on # simultaneous users not # potential

users

• Trade reception quality for increased capacity

Nonorthogonal Signature Sequences Matched filter is no longer optimal (near-far problem But can make new receievrs to exploit structure of the multi- access interference (MAI) to:

• Increased spectral efficiency
• Decreased output power
• Robustness against imbalances in the received powers

Decorrelator

• Linear like matched filter (MF)
• Uses information from all users unlike MF

Inverts the channel → Leaves received signal without interference → But increases the noise.

Decorrelator Received signal: Y = SX + W, (10) X data bits, S signature sequence matrix, W noise.

S for single-user, bit and chip synchronous channel:

                       

s1,1 s1,2 . . . s1,N s2,1 s2,2 . . . s2,N ... sTu,1 sTu,2 . . . sTu,N

                       

(11)

S for multi-user, bit and chip synchronous channel:

                  

s1

1,1

. . . sK

1,1

s1

1,2

. . . sK

1,2

. . . . . . s1

1,N

. . . sK

1,N

s1

2,1

. . . sK

2,1

s1

2,2

. . . sK

2,2

. . . . . . s1

2,N

. . . sK

2,N

...

                  

(12)

Decorrelator Bank of matched filters: multiplying STY: R = STSX + STW (13) Decorrelator, also multiply by (STS)−1: U = (STS)−1R = X + (STS)−1STW (14) Decorrelator = (STS)−1ST.

Decorrelator No knowledge of the received power necessary Solves near-far problem: Performance is independent of the power of interfering users

Optimum Multiuser Detector (Nonlinear)

• NOT computationally feasible
• upper bound on performance
• starting point for reduced complexity decoders

Optimum Multiuser Detector (Nonlinear)

• required to know:

– signature waveform – timing (synchronization) – amplitude of each user – noise level

Two User Synchronous Optimum Multiuser Detector (is feasible) Individual minimum probability of error for user 1: MAP value

• f b1 ∈ −1, +1

P[b1|y(t), 0 ≤ t ≤ T] (15) Joint (Both Users) Minimum Probability of Error: Select the pair (b1, b2) that jointly maximizes APP: P[(b1, b2)|y(t), 0 ≤ t ≤ T]. (16) If transmitted data are equiprobable → joint MAP = ML

Individual optimum similar to joint optimum Received signal: y(t) = A1x1s1(t) + A2b2s2(t) + σn(t), t ∈ [0, T], (17) Joint optimum decisions for two users are given by ˆ b1 = sgn

• A1y1 + 1

2|A2y2 − A1A2ρ| − 1 2|A2y2 + A1A2ρ|

• ,

(18) A1, A2 are amplitudes of users, ρij = R is signature sequence crosscorrelation matrix ρ =

T

0 s1(t)s2(t)dt.

(19)

Joint optimum similar to Individual Optimum ˆ b1 = sgn

  y1 − σ2

2A1 log cosh

A2y2+A1A2ρ

σ2

• cosh

A2y2−A1A2ρ

σ2

• 

  ,

(20) absolute value function is replaced by cosh. for large SNR, individual optimum decision converges to joint

• ptimum, cosh → | · |

Iterative Multiuser Detection Suboptimal (but lower complexity) non-linear detectors:

• Multistage receivers
• Decision feedback equalizers (DFE)
• Xie et al. trellis-based suboptimal MLSE (much better than

MF)

• Iterative decoders (Turbo/Factor graph inspired)

With randomly generated spreading codes:

• (and many users) → synchronous or asynchronous average

performance is the same

• It is theoretically possible to achieve single-user performance

Prior work combining convolutional decoding and CDMA decoding Giallorenzi et al.: Optimal MLSE with convolutional error cor- rection coding (ECC) They jointly estimate CDMA and ECC Complexity exponential in K (# users) and # of states in ECC. Find a way to factorize

Convolutional Coded Synchronous Multiuser Channel

Encoder bt(K) st(K) dt(K) Encoder bt(1) st(1) dt(1) . . . AWGN et Matched Filter 1 Matched Filter K Iterative Receiver/ Decoder yt(1) yt(K) bt(1) ^ bt(K) ^

The channel output et = Atdt + nt, (21)

where dt = (d(1)

t

, . . . , d(K)

t

)T ∈ {+1, −1}K (22) is the data vector. At = (s1

t , . . . , sK t ) ∈ {−1/

√ N, . . . , +1/ √ N}N×K (23) is the bank of spreading codes, one spreading code for each user.

Matched filter (MF) output yt = AT

t Atdt + AT t nt

= Htdt + zt (24) where Ht = AT

t At is the crosscorrelation matrix of the spreading

sequences, zt and nt are the correlated and uncorrelated noise vectors, respectively.

Decomposition of Iterative Multiuser Receiver with Chan- nel Coding

Encoder (K users) Multiuser Channel Likelihood Calculation Metric Generator K Single User Decoders bt dt yt p(yt | dt) p(yt | dt(K)) Pr(dt(K) =d | y(K)) bt ^

The Algorithm:

• 1. Matched filter channel output → conditional channel proba-

bilities p(yt|dt), (multivariate Gaussian conditional probabili- ties).

• 2. The metric generator then calculates the marginal probabil-

ities p(yt|d(k)

t

) for the kth decoder.

• 3. The single user soft-in/soft-out FEC decoders then generate

the a posteriori coded bit probabilities Pr{d(k)

t

= d|y(k)} for user k for coded block size 0 to L − 1.

• 4. The a posteriori coded bit probabilities are then used as a

priori information for the metric generator on the next itera- tion.

• 5. Output from the single user’s decoder can be taken as bit

estimates after a suitable number of iterations.

Algorithm Complexity Joint detection of DS/CDMA channel and FEC code → O(2K+Kν) When partition the receiver: separate FEC decoder and DS/CDMA channel decoder → O(2K + 2ν)

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