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Asymptotically Optimal Power Allocation and Code Selection for Iterative Joint Detection of CDMA Christian Schlegel

Zhenning Shi

Department of Electrical Engineering University of Alberta Edmonton, AB, CANADA Email: [email protected]

Department of Electrical Engineering University of Utah Salt Lake City, Utah Email: [email protected]

Marat Burnashev Institute for Problems of Information Transmission Moskow, RUSSIA Email: [email protected]

2004 International Zurich Seminar on Digital Communications ¨ Zurich, Switzerland, February 2004 ¨ supported in part by iCORE Alberta, NSF Grant 9732962, and the Russian Fund for Fundamental Research

Iterative “Turbo” Detection of Coded CDMA Systems

Conventional Correlation Detection Create parallel single-user channels using filtering matched to spreading sey quences. Linear Joint Detection Various filter techniques, most prominently MMSE filtering. The Turbo Principle is applied to the serial concatenation of an outer error control code with the inner CDMA channel.

2 Canceller Noise Variance σeff

Interference ˆ k +n rk = Sd − Sd Canceller

The CDMA “Channel” Joint access of K users with individual spreading sequences.

ω1

π1−1

Soft-Output Error Control Decoder

π1

ωK

−1 πK

Soft-Output Error Control Decoder

πK

λ(d1)

λ(dK )

σd2 dˆ1,j dˆK,j

tanh() tanh()

λ(d1,j ) λ(dK,j )

Early Work on Iterative CDMA Detection [ReScAlAs98] M.C. Reed, C.B. Schlegel, P.D. Alexander, and J.A. Asenstorfer, “Iterative multiuser detection for CDMA with near single user performance,” IEEE Trans. Commun., Vol. 46, No. 12, December 1998. [Moh98] M. Moher, “An iterative multiuser decoder for near-capacity communications,” IEEE Trans. Commun., vol. 46, pp. 870–880, July 1998. [AleGra98] P.D. Alexander, A.J. Grant, and M.C. Reed, “Iterative detection on code-division multiple-access with error control coding,” European Transactions on Telecommunications, vol. 9, no. 5, pp. 419–426, Sept.-Oct. 1998. [WanPoo99] X. Wang and V. Poor, “Iterative (turbo) soft interference cancellation and decoding for coded CDMA,” IEEE Trans. Commun, vol. 47, pp. 1046–61, July 1999. Asymptotically Optimal Power Allocation and Code Selection for Iterative Joint Detection of CDMA IZS’04, Zurich — Christian Schlegel, Zhenning Shi, and Marat Burnashev ¨

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Iterative Joint Detection — A Dynamical System Interpretation The Turbo Principle implies a dynamical system model with the signal variance as parameter: 2 σeff

Resolution

y

CDMA Interference

y1 y2

yK

(1)

wi,1t

π1−1

APP Dec

π1

Λ(di ) Dynamical Model: (2)

wi,2t

π2−1

APP Dec

π2

Λ(di ) fIC (σd2 )

(K)

wi,K t

−1 πK

APP Dec

πK

σd2

2 σeff

Λ(di ) 2 ) g (σeff

σd2

tanh() tanh() tanh()

Convergence of this system can be studied by examining the evolution of the values of the variances. The PDFs at the output of the filters are Gaussian according to the CLT. [BouCai02]

J. Boutros and G. Caire, “Iterative multiuser joint decoding: unified framework and asymptotic analysis”, IEEE Trans. Inform. Theory, Vol. 48, No. 7, pp. 1772–1793, July 2002.

Asymptotically Optimal Power Allocation and Code Selection for Iterative Joint Detection of CDMA IZS’04, Zurich — Christian Schlegel, Zhenning Shi, and Marat Burnashev ¨

2

Iterative “Turbo” Detection – Fundamental Observations CDMA Turbo Detection: exhibits a “Turbo Cliff” and an “Error Floor”. g (σd2 ) Canceller Noise Variance

3.5 3

Bit Error Rate

1

Rate 13 CC (ν = 2) K = 45 N = 15 −1 dual-stage filter 10

2.5

1 iteration 5 iterations 10 iterations

imitation

nce L Interfere

Single User

2

15 iterations

1.5

10−3

20 iterations 30 iterations

2 ) fIC (σeff

1

Noise Limitation

0.5

10−4

β=3

0 0

0.2

0.4

0.6

0.8

1

10−5 2.5 3 3.5 4 Note: Analysis and discussion of large-scale cancellation systems: FEC Decoder Variance

[ShiSch01] [ShiSch03]

4.5

5

5.5

Eb N0

Z. Shi and C. Schlegel, “Joint iterative decoding of serially concatenated error control coded CDMA”, IEEE J. Select. Areas Commun., Vol. 19, No. 8, August 2001. Z. Shi and C. Schlegel, Asymptotic Iterative Multi-User Detection of Coded Random CDMA, submitted to IEEE Trans. Commun., March 2003.


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Optimal Power Distribution Profiles “Strong” Codes: J groups of users with partial loads αj .

τ is the idealized SNR threshold. Optimal Power Levels: which maximize the slope of the load curve: j σ2 Y 1 Pj = 1−j τ τ − αj j 0 =1

1≤j≤J

the optimized signal-to-noise ratio is

Eb N0

opt

1 = 2Rα

"

τ τ − α/J

J

#

10 J = 10 Sum Capacity [bits/dimension]

The variance transfer function is idealized (upper-bounded) for strong codes by 2 2 ≥ τP σm 1 if σm j 2 = σd = g 2 0 if σm < τ Pj Pj

Achievable Spectral Efficiencies: with Rate R = 1/3 codes

J =5 J =1

0.1

−1 −2

0

2

4


6

8

10

12

14 16 Eb/N0 4

Optimal Error Control Coding The average variance transfer curve of any code approaches a straight line as J → ∞. Strong Codes

10 9

K1 = 22, K2 = 18, K3 = 16 P1: P2: P3 = 1: 2: 4 N = 15, Eb/N0 = 13.45 dB

8 7

8 7

6 5

6 5

2 σeff

2 σeff

10 9

4 3

4 3

2

2

1

1

0

0

0.2

0.4

0.6

0.8

As J → ∞, and R → 0, the capacity of an AWGN channel is achieved:

Eb N0

= lim

22C

−1 ; 2C

C = Rα

1 σd2

0

0

Weak Codes K1 = K2 = K3 = 20 P1: P2: P3 = 1: 2: 4 N = 15, Eb/N0 = 8.68 dB

0.2

0.4

0.6

0.8

1

We are working on a proof of this conjecture: Using any code in conjunction an with optimal power allocation can achieve the AWGN channel capacity.


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Equal Power Users J = 1 We are working on a proof of Conjecture 2:

This is in marked contrast to what is known for linear filters, in particular MMSE filters:

Using iterative turbo detection, the achievable total spectral efficiency of the system is minimized by the equal power assignment J = 1.

The total spectral efficiency of a CDMA system using an MMSE filter to separate the different users is maximized by the equal power assignment.

3.5

Observation “Weak” codes with smoother variance transfer functions support higher system loads.

2 is the variance of the output informa• σllr tion bit log-likelihood ratio.

Iterative decoding with weak codes may be incomplete.

3 2.5 2 2 σm

We need to consider the resulting Binary-Input Gaussian Channel, leading to the system capacity: P C = αRCB 2 (α, C ) σllr

Rate R = 1/3 Repetition Code

1.5 1

2 , σ2) Intersect Point: (σ∞ d

0.5 0

0

0.2

0. 4


0.6

0. 8

1 σd2 6

Repetition Codes for Equal Power CDMA Observation: Among a large number of codes analyzed, they provide the highest spectral efficiency. Calculation of extrinsic LLRs is simple: X 2 X 0 (E) yi λ (di) = λ(yi) = 2 σ m 0 0

AWGN Capacity (2) (1) (3)

1

i 6=i

i 6=i

Capacity bits/dimension 10

(4) (5)

The LLRs and extrinsic LLRs are exactly Gaussian distributed. Low rate codes perform best (Rate R = 1/L): Lemma: The system spectral efficiency of iteratively decoded CDMA using repetition codes is maximized for L → ∞.

0.1 -2

Using the following inequality:

2

E 1 − tanh b + bξ

2

≤ min

n

(1) Iterative Rate 1/3, repetition code (2) Iterative Rate 1/10, repetition code (3) MMSE, load = 1 (4) MMSE, load = 2 (5) MMSE, load = 5

0

2

4

6

we find:  1 , πQ(b) 1 + b2

o

2 σ∞ ≤

1 0 σ2 α − 1+ + 2 L− 1

8

s


10 12 14 16 Eb/N0

α0 − 1+

σ2 L− 1

2 +

 4σ 2  L− 1

7

Conclusions and some Speculations • An iterative cancellation detector using “weak” FEC codes should be viewed as an iterative nonlinear filter. • As such, using R = 1/L repetition codes, the iterative filter with load α and equal powers has a performance lowerbounded by that of an MMSE filter with load α/(L − 1). • Note that both information rate loads are identical • For large signal-to-noise ratios, the iterative cancellation filter strictly outperforms the MMSE filter.

Conjecture 3: For unequal power levels, the performance of the iterative filter is strictly larger that that of an MMSE filter. Conjecture 4: The iterative cancellation filter using repetition codes can achieve the maximum capacity of the CDMA channel using optimum power levels.


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