analysis of the dynamics of iterative interference ...hcdc/library/pit paper (as published by...

Problems of Information Transmission, Vol. 40, No. 4, 2004, pp. 297–317. Translated from Problemy Peredachi Informatsii, No. 4, 2004, pp. 3–25.Original Russian Text Copyright c© 2004 by Burnashev, Schlegel, Krzymien, Shi.

INFORMATION THEORY

Analysis of the Dynamics of Iterative Interference

Cancellation in Iterative Decoding1

M. V. Burnashev?, C. B. Schlegel??, W. A. Krzymien??, and Z. Shi???

?Institute for Information Transmission Problems, RAS, [email protected]

??Department of Electrical and Computer Engineering, University of Alberta, [email protected] [email protected]

???Department of Electrical Engineering, University of Utah, [email protected]

Received May 18, 2004

Abstract—Characteristics of a successive cancellation scheme, widely used in iterative de-coding, is investigated. Comparison with another popular method, the minimum mean-squareerror (MMSE) method, is also provided.

1. INTRODUCTION

Iterative decoding in digital data transmission systems has become very popular with inventionof turbo coding [1]. Methods of iterative processing have quickly been realized to be applicable notonly to turbo coded signals but also to joint decoding of code-division multiple access (CDMA) sys-tems, intersymbol interference limited channels, multiple-input multiple-output (MIMO) systems,and others. In particular, iterative decoding of CDMA systems is proved to be a very powerfuldecoding method, which has the potential of realizing a significant portion of the capacity of sucha channel [2–8].

Key issues for iterative decoders are convergence conditions. Usually, it is not possible toinvestigate their convergence by purely analytical methods. For that reason, such methods aretypically combined with computer simulations. An example is density evolution analysis, firstapplied to low-density parity-check (LDPC) codes [9]. It should be noted that in many casesdensities involved are either exactly Gaussian, or they are well approximated by Gaussian densities.In these cases, much simpler analysis can be used instead [7, 10–12]. In particular, it has beenshown in [11] that simple iterative cancellation receivers (see Section 3) can achieve the capacity of amultiple access channel if powers (or rates) of participating users are chosen in a certain optimal way.

In this paper, which is a mathematical supplement to [11], dynamics of the behavior of iterativedecoders with iterative cancellation is studied by analytical methods. Some of the results derivedare used in [11] to describe the behavior of large-scale CDMA systems under various transmissionconditions (without mathematical proofs). Comparison to decoders using linear minimum mean-square error (MMSE) filters is also presented.

We consider the model in which the input, d = (d1, . . . , dK)T , and the output, y = (y1, . . . , yN )T ,channel vectors are related by the formula

y =K∑i=1

di√Pi ai + σn, (1)

1 Supported in part by the INTAS, Grant 00-738, and Russian Foundation for Basic Research, projectno. 03-01-00098.

0032-9460/04/4004-0297 c© 2004 MAIK “Nauka/Interperiodica”

298 BURNASHEV et al.

where d is a K × 1 column vector, and y, n, and all the ai are N × 1 column vectors. All entriesof the vectors are real. We assume the following:

(1) All components of n are independent N (0, 1) Gaussian random variables; noise intensity σis known;

(2) All vectors ai (perhaps, random) are known at the receiver;(3) {di} are independent random variables with P (di = 1) = P (di = −1) = 1/2, i = 1, . . . ,K;(4) {Pi} are some given positive constants (user powers).Clearly, (1) is a model with linearly modulated (by parameters {di}) signals {

√Pi ai} and

independent Gaussian noise. Precisely this scheme is as well used in multiuser detection.We are interested in the values P (dj = 1 |y) (for further use as soft decisions), as well as in

some good approximations for these values. For instance, note that

Y1(y) =P (d1 = 1 |y)P (d1 = −1 |y)

=

∑d2,...,dK

P (d1 = 1 |y, d2, . . . , dK)∑d2,...,dK

P (d1 = −1 |y, d2, . . . , dK)

=

∑d2,...,dK

exp

{−∥∥∥∥y −√P1 a1 −

K∑i=2

di√Pi ai

∥∥∥∥2/(2σ2)

}∑

d2,...,dK

exp

{−∥∥∥∥y +

√P1 a1 −

K∑i=2

di√Pi ai

∥∥∥∥2/(2σ2)

} .

Of course, the last expression is rather difficult to use for large K. For that reason, a numberof simple iterative procedures for evaluating (or approximating) Yj(y) have been proposed. Ourmain purpose is to investigate the popular successive cancellation scheme and to compare it withanother popular method, the MMSE detector. In this analysis we assume that both K and N arelarge enough so that at some points Gaussian approximation can be applied.

In Section 2, asymptotic (as K,N →∞) properties of the MMSE method (detector) are studied,while in Section 3 asymptotic properties of successive cancellation method are investigated, and, inparticular, its limiting (for a large number of iterations) effectiveness is found. Results derived areexpressed through the function g(b) = E[1−tanh(b2 +bξ)]2, b ≥ 0, where ξ ∼ N (0, 1). In Section 4,a series expansion for g(b) is presented, and rather accurate approximations for this function aregiven. In Section 5, using approximations from Section 4, properties of the successive cancellationmethod in the case of equal powers of all users are studied. Its convergence rate is also investigated.In Section 6, the case of unequal powers of users is investigated, and the optimal power profile isfound. In Section 7, the MMSE detector with unequal powers of users is considered. Some proofsare presented in the Appendix.

2. THE MMSE DETECTOR

Let us first consider a very popular method of direct application of Gaussian approximation inmodel (1). It is usually called the MMSE detector (method).

From the viewpoint of user 1, signals of other users represent additional noise, independent of

the primary noise. The random vectorK∑i=2

di√Pi ai has zero mean, and its correlation matrix is

E

(K∑i=2

di√Pi ai

)(K∑j=2

dj√Pj aj

)T=

K∑i=2

PiaiaTi = A1 diag1{Pi}AT

1 = W 1,

PROBLEMS OF INFORMATION TRANSMISSION Vol. 40 No. 4 2004

ANALYSIS OF THE DYNAMICS OF ITERATIVE INTERFERENCE CANCELLATION 299

where A1 = (a2, . . . ,aK) is an N×(K−1) matrix, diag1{Pi} is a (K−1)×(K−1) diagonal matrixwith entries {P2, . . . , PK}, and W 1 is an N ×N matrix. All matrices A1, diag1{Pi}, and W 1 areknown at the receiver.

If we assume that v =K∑i=2di√Pi ai is an N (0,W 1) Gaussian random vector, then model (1)

takes the form

y = d1

√P1 a1 + v + σn = d1

√P1 a1 + (W 1 + σ2IN )1/2n, (2)

where IN is the N×N identity matrix, and v is an N (0,W 1) Gaussian random vector independentof n.

The successive cancellation method, MMSE detector, and other similar methods are attemptsto somehow estimate the “noise” vector v in (2).

If σ > 0, then W 1 +σ2IN is a strongly positive definite matrix, and the matrix (W 1 +σ2IN )1/2

is well defined. Now, preserving all information on d1, we can replace y with the vector

y1 = (W 1 + σ2IN )−1/2y = d1

√P1 b1 +n, b1 = (W 1 + σ2IN )−1/2a1,

from which we get(y1, b1)‖b1‖2

√P1

= d1 +1

‖b1‖√P1

ξ, ξ ∼ N (0, 1). (3)

Then, provided d1 = 1, from (3) we get

lnP (d1 = 1 |y)P (d1 = −1 |y)

= lnP (d1 = 1 | (y1, b1))P (d1 = −1 | (y1, b1))

= 2(y1, b1) = 2d1

√P1 ‖b1‖2 + 2‖b1‖ξ.

It is clear from (3) that 1/‖b1‖ plays the role of some effective noise intensity σeff . We areinterested in the value of E(P1/σ

2eff) = P1 E ‖b1‖2. In order to evaluate the expectation E ‖b1‖2 =

E(a1, (W 1 +σ2IN )−1a1

), we assume that all components of the vectors {ai} are independent iden-

tically distributed random variables with E aii = 0 and Ea2ii = 1/N . Then, taking the expectation

over a1, we have

E ‖b1‖2 = E(a1, (W 1 + σ2IN )−1a1

)=

1N

E tr[(W 1 + σ2IN )−1

]=

1N

EN∑i=1

1σ2 + λi

, (4)

where λ1, . . . , λN are eigenvalues of W 1 (all of them are nonnegative). Here we used the fact that,if for some x and λ we have W 1x = λx, then (W 1 + σ2IN )x = (σ2 + λ)x.

In the Appendix, we prove the following result.

Proposition 1. If P1 = . . . = PK and min{K,N} → ∞, then in model (3) with u = σ2/P1

we have

E ‖b1‖2 =1

2σ2

[√(α− 1 + u)2 + 4u− (α− 1 + u)

]=

2uσ2[√

(α− 1 + u)2 + 4u+ (α − 1 + u)] . (5)

Remark 1. The right-hand side of (5) is less than 1/σ2 for any α > 0.

Remark 2. Note that the same expression as in (5) will appear below, when considering thesuccessive cancellation method.



3. SUCCESSIVE CANCELLATION

Let us make soft decision for each symbol dj in model (1) by the successive cancellation method.This works as follows:

(1) First, multiplying y by each of the vectors aj (i.e., using matched filters), we get new outputs

y1,j = (y,aj) = dj√Pj ‖aj‖2 + n1,j, j = 1, . . . ,K,

n1,j =∑i6=j

di√Pi (ai,aj) + σ(n,aj).

Averaging over {di}, we have

En1,j = 0, En21,j =

∑i6=j

Pi(ai,aj)2 + σ2‖aj‖2 = σ21,j.

When we replace the observation y with the vector {y1,j, j = 1, . . . ,K}, we lose no informationabout {dj} (since we switch to a different sufficient statistics).

Now let us use the Gaussian approximation n1,j ∼ N (0, σ21,j). This is correct as long as K is

sufficiently large and the range of {Pi} is not too large (see the explanation in Section 6). Then wehave

lnP (dj = 1 |y1,j)P (dj = −1 |y1,j)

≈ 2y1,j√Pj ‖aj‖2σ2

1,j

= z1,j , P (dj = 1 |y1,j) ≈ez1,j

1 + ez1,j,

d1,j = E(dj |y1,j) =ez1,j − 11 + ez1,j

= tanh(z1,j

2

)= tanh

(y1,j

√Pj ‖aj‖2σ2

1,j

).

Denote y1 = ((y,a1), . . . , (y,aK))T .(2) At the second stage, we replace each vector y1,j by the vector

y2,j = y1,j −∑i6=j

d1,i

√Pi (ai,aj) = dj

√Pj ‖aj‖2 + n2,j, j = 1, . . . ,K,

n2,j =∑i6=j

(di − d1,i)√Pi (ai,aj) + σ(n,aj).

For any vectors {aj}, distributions of the variables (di − d1,i) and (di − d1,i)(dj − d1,j), i 6= j, aresymmetric with zero mean (due to the symmetry of {dj} and related variables). Then we have

En2,j = 0, En22,j =

∑i6=j

Pi E[(di − d1,i

)2(ai,aj)2]

+ σ2‖aj‖2 = σ22,j .

Continuing this process, at the mth stage we have

ym,j = dj√Pj + nm,j, j = 1, . . . ,K,

nm,j =∑i6=j

(di − dm−1,i

)√Pi (ai,aj) + σ(n,aj),

where

dm−1,j = tanh

(ym−1,j

√Pj

σ2m−1,j

)= tanh

((√Pj + nm−1,j

)√Pj

σ2m−1,j

).

Clearly, we have

Enm,j = 0, En2m,j =

∑i6=j

Pi E[(di − dm−1,i

)2(ai,aj)2]

+ σ2‖aj‖2.



In the sequel, we assume that all components of the vectors {ai} are independent identicallydistributed random variables with P (aij = 1/

√N) = P (aij = −1/

√N) = 1/2. Then ‖ai‖2 = 1,

and E(ai,aj)2 = 1/N for any i 6= j. Moreover, if ai = (ai,1, . . . , ai,N ), then, due to the symmetry,for all i 6= j we have

E[(di − dm−1,i

)2(ai,aj)2]

= E[(di − dm−1,i

)2 N∑k=1

a2i,ka

2j,k

]=

1N

E(di − dm−1,i

)2.

Therefore,

En2m,j =

1N

∑i6=j

Pi E(di − dm−1,i

)2 + σ2 = σ2m,j. (6)

We also have

dm−1,j = tanh

((√Pj + nm−1,j

)√Pj

σ2m−1,j

)≈ tanh

(Pj

σ2m−1,j

+√Pj

σm−1,jξ

),

where we have used the Gaussian approximation nm−1,j ∼ N (0, σ2m−1,j), and where ξ ∼ N (0, 1).

Then, denotingg(b) = E

[1− tanh(b2 + bξ)

]2, ξ ∼ N (0, 1), (7)

we get

E(di − dm−1,i

)2 = g

( √Pi

σm−1,i

),

and from (6) and (7) we have

σ2m,j = σ2 +

1N

∑i6=j

Pig

( √Pi

σm−1,i

), m = 1, 2, . . . . (8)

In equation (8) we formally set d0,i = 0 and σ0,i =∞ for all i.Clearly, we have σ1,j < σ0,j = ∞ for all j. Let us show that, in fact, for each j the sequence

{σm,j , m = 0, 1, . . .} is monotonically decreasing. Indeed, consider any j and assume that forsome m we have σ0,j > σ1,j ≥ . . . ≥ σm−1,j . Since g(b) is monotonically decreasing with b (seeProposition 2), from (8) we have

σ2m,j = σ2 +

1N

∑i6=j

Pig

( √Pi

σm−1,i

)≤ σ2 +

1N

∑i6=j

Pig

( √Pi

σm−2,i

)= σ2

m−1,j .

Then we conclude by the induction that the sequence {σm,j , m = 0, 1, . . .} is monotonically de-creasing and converges to some σ∞,j ≥ σ, which satisfies the system of equations

σ2∞,j = σ2 +

1N

K∑i=1

Pig

(√Pi

σ∞,i

)− PjNg

(√Pj

σ∞,j

), j = 1, . . . ,K. (9)

Note that the same arguments remain valid if we replace g(b) with any other monotone approxi-mation.

For large N , the limiting values {σ∞,j} depend rather weakly on j. Indeed, since g(b) ≤ 1 andb2g(b) < 1 (see upper bound (12) for g(b)), we have Pjg(

√Pj/σ∞,j) < σ2

∞,j. Then for large N wemay neglect the last term on the right-hand side of (9). Then {σ∞,j} do not depend on j, and fortheir “common” value σ∞ (an upper bound for σ∞,j) we get the equation

σ2∞ = σ2 +

1N

K∑i=1

Pig

(√Piσ∞

). (10)



4. FUNCTION g(b) AND ITS APPROXIMATIONS

To investigate the solution of equation (10), we use some approximations for the function g(b) =E[1−tanh(b2 +bξ)

]2, b ≥ 0, where ξ ∼ N (0, 1). The following result represents the series expansionof and a rather tight upper bound for the function g(b) (for a proof, see the Appendix).

Proposition 2. (1) The function g(b), b ≥ 0, is monotone decreasing, and for any b > 0 wehave the relation

g(b) = 4e−b2/2

∞∑n=0

(−1)neb2(2n+1)2/2Q[b(2n + 1)], (11)

where Q(x) = (2π)−1/2∞∫xe−u

2/2 du. The absolute values of terms of sign-alternating series (11)

are monotone decreasing.(2) For any b ≥ 0, we have

g(b) ≤ min{

11 + b2

, πQ(b)}≤{

1/(1 + b2), b < 1,πQ(b), b ≥ 1.

(12)

Moreover, we have limb→∞

g(b)/(πQ(b)) = 1.

Remark 3. The error of the approximation

g(b) ≈{

1/(1 + b2), b < 1,πQ(b), b ≥ 1,

(13)

is less than 10% for all b ≥ 0.

Remark 4. A yet smaller error (of almost 1% for all b ≥ 0) is given by the approximation

g(b) ≈ 33 + 3b2 + 2b6

, b ≤ 0.6,

g(b) ≈ 8π(1 + b2)(π2 + 8b2)

Q(b), b > 0.6.(14)

5. SUCCESSIVE CANCELLATION FOR THE CASE OF EQUAL POWERS

If P1 = . . . = PK , then the values {σm,j} do not depend on j, and system (8) takes the form

σ2m = σ2 + αP1g

( √P1

σm−1

), α =

K − 1N

, m = 1, 2, . . . , (15)

with σ0 =∞ and σ21 = σ2 + αP1.

As above, the sequence {σm, m = 0, 1, . . .} is monotonically decreasing and converges to someσ∞ = σeff ≥ σ, which is the positive root of the equation

f(x) = σ2 + αP1g

(√P1

x

)− x2 = 0, x ≥ 0. (16)

Since f(∞) = −∞ and f(+0) = σ2 ≥ 0, there always exists at least one positive root to equa-tion (16) (if σ = 0, then a possible root is σ∞ = 0).



Now we apply approximation (13) to the function g(·) in (16). Since (13) includes two cases, weconsider them separately.

Case σ2∞ > P1. Applying the first approximation of (13), from (16) we get the equation

f(x) = σ2 +αP1x

2

P1 + x2− x2 = 0, x ≥ 0. (17)

If σ > 0, then equation (17) has the unique positive root

σ2∞ = σ2

eff =σ2 + (α− 1)P1 +

√(σ2 + (α− 1)P1)2 + 4σ2P1

2. (18)

We have σ2∞ > P1 if and only if 2σ2 + P1(α− 2) > 0. Note that

σ2∞ ≤ σ2 + (α− 1)P1 +

σ2P1

σ2 + (α− 1)P1,

and the right-hand side of this expression is an approximation for σ2∞ = σ2

eff for large α.If σ = 0, we have

σ2m =

αm(α− 1)P1

αm − 1, m = 1, 2, . . . , (19)

from which we get that, if α > 1, then σ2∞ = (α− 1)P1. If α ≤ 1, then σ∞ = 0. We have σ2

∞ > P1

if and only if α > 2. Equation (17) takes the form

f(x) = x2((α− 1)P1 − x2) = 0, x ≥ 0,

from which we can also find the same values σ2∞.

It is natural to ask how many iterations, m, we need to get σ2m ≈ σ2

∞. Since

σ2m+1 − σ2

∞ = αP1

[g

(√P1

σm

)− g

(√P1

σ∞

)]=

αP 21 (σ2

m − σ2∞)

(P1 + σ2m)(P1 + σ2

∞)≤ αP 2

1 (σ2m − σ2

∞)(P1 + σ2

∞)2,

with u = σ2/P1 we obtain

σ2m+1 − σ2

∞σ2m − σ2

∞≤ αP 2

1

(P1 + σ2∞)2

=4α[

u+ α+ 1 +√

(u+ α− 1)2 + 4u]2

≤ min{α,

1α,

1(1 + u)2

}, α > 0.

Therefore, if α is not too close to 1, then σ2m − σ2

∞ vanishes exponentially fast with m, and it issufficient to take a few iterations to get a good approximation for σ2

∞. For example, for α > 1 andσ2

1 = σ2 + αP1 we haveσ2m+1 − σ2

∞ ≤ (σ21 − σ2

∞)α−m, m ≥ 1.

Case σ2∞ ≤ P1. This case takes place when 2σ2 + P1(α − 2) ≤ 0. Then in (16) we apply the

second approximation of (13), and get for x = σ∞ the equation

f(x) = σ2 + απP1Q

(√P1

x

)− x2 = 0, x ≥ 0. (20)



Moreover, for any m such that σ2m ≤ P1 we also have

f(σm) = σ2 + παP1Q

(√P1

σm

)− σ2

m ≤ 0, (21)

Generally, equation (20) has no more than three positive roots (see below). Due to con-straint (21), the sequence {σm, m = 0, 1, . . .} with σ0 = ∞ converges to the maximal positiveroot xmax = xmax(σ, α) of equation (20). Clearly, xmax(σ, α) is a monotone increasing functionof σ, α.

For f(x), from (20) we have f(+0) = σ2 ≥ 0, f(∞) = −∞, and

2f ′(x) = x(√

2π αx−3P3/21 e−P1/(2x2) − 4

).

Since maxx≥0

{P

3/21 x−3e−P1/(2x2)

}= (3/e)3/2, there are two possible cases:

1. α ≤ α0, where

α0 =√

8π

(e

3

)3/2

≈ 1.376353.

Then f ′(x) ≤ 0, x ≥ 0, and there exists a unique root x1 of equation (20).2. α > α0. Then there exist two roots, 0 < x1 < x2 < ∞, of the equation f ′(x) = 0, x > 0.

Moreover, f ′(x) < 0 for 0 < x < x1 and x > x2, and f ′(x) > 0 for x1 < x < x2.If f(x1) > 0 or f(x2) < 0, then there exists a unique root of equation (20).If f(x1) < 0 and f(x2) > 0, then there exist three roots of equation (20). For instance, this is

the case if α = 1.99 and σ2/P1 = 0.01.Some of the results just obtained are summarized in the following statement.

Proposition 3. Let P1 = . . . = PK . Denote u = σ2/P1.(1) If 2u+α > 2, then the sequence {σ2

m, m = 0, 1, . . .} with σ0 =∞ is monotonically decreasingand converges to

σ2eff = σ2

∞ =σ2

2u

[√(α+ u− 1)2 + 4u+ α+ u− 1

]. (22)

(2) If 2u+α ≤ 2, then the sequence {σ2m, m = 0, 1, . . .} with σ0 =∞ is monotonically decreasing

and converges to x23, where x3 is the maximal positive root of equation (20).

Remark 5. Note that expression (22) for σ2∞ coincides with the similar expression 1/E ‖b1‖2

from (5) for the MMSE detector.

Let us elaborate a little on this remark. Note first that the optimal decoder for model (1) is aNeyman–Pearson test based on the likelihood ratio Y1(y) from Section 1 (which is difficult to use).The test investigated in Section 2 (MMSE detector) is based on the Gaussian approximation fora part of the output vector y. Of course, generally, such a detector is not optimal; in particular,successive cancellation can give better results.

Second, it follows from Propositions 1 and 3 and from the proof of Proposition 3 that, for thecase of equal powers, the performance of the MMSE detector (e.g., its variance) coincides withthe performance of successive cancellation if we replace g(b) with the function 1/(1 + b2). Then,due to the inequality g(b) ≤ 1/(1 + b2), b > 0, the successive cancellation performance cannotbe worse than that of the MMSE detector. If 2u + α − 2 > 0, then the difference between theseperformances is rather small (since the approximation g(b) ≈ 1/(1 + b2) is very accurate for b < 1).If 2u + α − 2 < 0, it becomes better to use the second of upper bounds (12). In this case, thesuccessive cancellation performance can be much better than that of the MMSE detector.



6. SUCCESSIVE CANCELLATION FOR UNEQUAL POWERSAND OPTIMAL POWER PROFILE

Assume that P1 ≤ P2 ≤ . . . ≤ PK . Then there are two possible cases: σ2∞ > PK and σ2

∞ ≤ PK .If σ2

∞ > PK , then, applying the first approximation of (13), from (10) we get the equation

σ2∞ = σ2 +

1N

K∑i=1

Piσ2∞

σ2∞ + Pi

= σ2 +1N

K∑i=1

Pi −1N

K∑i=1

P 2i

σ2∞ + Pi

. (23)

To investigate equation (23), we consider the function

f(x) = σ2 +1N

K∑i=1

Pix

x+ Pi− x = σ2 +

1N

K∑i=1

Pi −1N

K∑i=1

P 2i

x+ Pi− x.

Then σ2∞ is the unique root of the equation f(x) = 0 since f ′′(x) < 0, f(0) = σ2 > 0, and

f(∞) = −∞. The condition σ2∞ > PK = max

iPi is equivalent to f(PK) > 0.

If σ2∞ ≤ PK , then let K1 ≤ K be such that PK1−1 < σ2

∞ ≤ PK1. Then, applying approxima-tions (13), from (10) we get the equation

σ2∞ = σ2 +

1N

K1−1∑i=1

Piσ2∞

σ2∞ + Pi

+π

N

K∑i=K1

PiQ

(√Piσ∞

). (24)

Up to now, we investigated soft decoding of each separate symbol {dj}, and there was no codingin our considerations so far. To apply these results to long codes, we make the frequently usedassumption [2, 5, 7, 10] that the variance transfer curve of such a code has the form

σ2d,j =

{1 if σ2

∞,j > τPj ,

0 if σ2∞,j ≤ τPj ,

(25)

where 1/τ is the signal-to-noise ratio threshold of the code, and Pj is the power of the jth user.In (25) it is also assumed that each user employs the same code (otherwise, τ would depend on j).It is known that approximation (25) is applicable to many codes.

Assume that P1 < P2 < . . . < PK . Then relation (25) means that, if σ∞,j ≤ τPj , then for sucha code we have the following:

(1) With very high probability, the symbol dj is decoded correctly;(2) When decoding the next lower symbol (say, dj−1), we may assume that we exactly know all

the preceding symbols {di, i = j, j + 1, . . . ,K}.Then, for successful decoding of all symbols {dj}, we must have

τP1 ≥ σ2,

τPj ≥ σ2∞,j = σ2 +

1N

j−1∑i=1

Pig

(√Pi

σ∞,j

), j = 2, . . . ,K.

(26)

The minimal possible {Pj} satisfying system (26) are of the form Pj = caj−1, j = 1, 2, . . . ,K,with some c > 0 and a > 1 (see below). Recall that in our analysis of the successive cancellationmethod we approximated

∑i6=j

(di − d1,i

)√Pi (ai,aj) by a Gaussian random variable. If {Pj} grow

exponentially fast with j, then the validity of such approximation needs to be established. More



precisely, we have to show that the Lindeberg condition [17, Section VIII.4, Theorem 2] is fulfilled(i.e., we have to prove asymptotic negligibility of each random variable in the sum considered).

A simple way to avoid this technical problem is to have not one user but a large number of users ineach group of equal powers. Then the Lindeberg condition is clearly fulfilled, while asymptoticallywe lose nothing in optimality (see below). To this end, we partition all the K users into J groupsof powers P1 < P2 < . . . < PJ with K1 users, K1 = K/J � 1, in each group j. Denote alsoα1 = K1/N = α/J . When decoding group j, we may assume that all higher groups j + 1, . . . , Jhave already been decoded. Since we assume that K1 � 1, we can apply the results obtainedearlier.

Now, for the minimum possible {Pj}, instead of system (26) we get the system of equations(σ∞,j = σj)

σ2j = σ2 +

α1

τ

j∑i=1

σ2i g

(σi

σj√τ

), τPj = σ2

j , j = 1, . . . , J. (27)

To solve system (27), we separately consider the cases τ > 1 and τ ≤ 1.

Case τ > 1. Since σi/(σj√τ) ≤ 1, i = 1, . . . , j, we use the approximation g(b) ≈ 1/(1 + b2)

from (13). Then for j = 2, . . . , J we get

σ2j = σ2 +

α1σ2j

1 + τ+ α1σ

2j

j−1∑i=1

σ2i

σ2i + τσ2

j

, σ21 = σ2

(1− α1

1 + τ

)−1

. (28)

To investigate solutions of system (28), for fixed σ1, . . . , σj−1 consider the function

fj(x) = x− σ2 − α1x

1 + τ− α1x

j−1∑i=1

σ2i

σ2i + τx

= fj−1(x)−α1σ

2j−1x

σ2j−1 + τx

. (29)

Clearly, the root σ2j of system (28) is a root of the equation fj(x) = 0. Since fj(0) < 0,

fj(∞) = ∞, and f ′′j (x) > 0, there exits a unique root of fj(x) = 0, and therefore there exists aunique solution of system (28).

The solution of (28) is monotonically increasing; i.e., σ1 < σ2 < . . . < σJ . Indeed, sincefj−1(σ2

j−1) = 0, due to relation (29) we have fj(σ2j−1) < 0. Then for the root of fj(x) = 0 we must

have x = σ2j > σ2

j−1.Note also that, if for some x0 we have fj(x0) > 0, then x0 > σ2

j . This allows us to get someupper bounds for the root σ2

j .

Let us look for an upper bound on σ2j of the form σ2

j ≤ cσ2aj−1 with some c > 0 and a > 1.Then we should have

caj−1 ≥ 1 +cα1a

j−1

1 + τ+ cα1a

j−1j−2∑i=0

ai

τaj−1 + ai, j = 1, . . . , J.

Note thatj−2∑i=0

ai

τaj−1 + ai≤

j−1∫0

ax

τaj−1 + axdx =

1ln a

ln(1 + τ)aj−1

1 + τaj−1.

Therefore, it is sufficient to have

caj−1 ≥ 1 +cα1a

j−1

1 + τ+cα1a

j−1

ln aln

(1 + τ)aj−1

1 + τaj−1, j = 1, . . . , J,



or, equivalently,g(caj) ≥ 0, j = 0, . . . , J − 1,

whereg(x) = 1− 1

x− α1

1 + τ− α1

ln aln

(1 + τ)xc+ τx

.

We haveg′(x) =

1x2− cα1

x(c+ τx) ln a,

g′′(x) = − 1x3

+τ

x2(c+ τx)− (c+ 2τx)g′(x)

x(c+ τx).

Note that, if for some x0 > 0 we have g′(x0) = 0, then g′′(x0) < 0, and therefore x0 correspondsto a local maximum of g(x).

Hence, it is sufficient to have the inequalities g(c) ≥ 0 and g(∞) ≥ 0 fulfilled, which areequivalent to the conditions

c ≥ c1 = (1− α1

1 + τ)−1, a ≥ a1 = (

1 + τ

τ)c1α1 . (30)

Therefore, we haveσ2j ≤ c1σ2aj−1

1 , j = 1, . . . , J.

Then, for τ > 1 and J large, for the average power per user P we have (τPj = σ2j )

P =1J

J∑j=1

Pj =1τJ

J∑j=1

σ2j ≤

c1σ2

τJ

J∑j=1

aj−11 =

c1σ2

τJ(a1 − 1)(aJ1 − 1)

≤ σ2

ατ ln(1 + 1/τ)

[(1 + τ

τ

)c1α− 1

](31)

sincec1

(a1 − 1)≤ c1

ln a1=

1α1 ln(1 + 1/τ)

.

Remark 6. In fact, it is sufficient to have the inequalities g(c) ≥ 0 and g(caJ−1) ≥ 0 fulfilled.This gives the same value c = c1, but the value aJ = z for large J is determined by the inequality

(z − 1) ln z ≥ αz ln(1 + τ)z1 + τz

,

which gives a slightly smaller result for aJ .

Case τ ≤ 1. Using the approximation g(b) ≈ min{1/(1 + b2), πQ(b)}, from (27) we have

σ2j = σ2 + α1σ

2j

k−1∑i=1

σ2i

σ2j τ + σ2

i

+πα1

τ

j∑i=k

σ2iQ

(σi

σj√τ

), j = 2, . . . , J,

where k = k(j) is such that σk−1/(σj√τ) < 1 and σk/(σj

√τ) ≥ 1.

Again, we look for an upper bound on σ2j of the form σ2

j ≤ cσ2aj−1 with some c > 0 and a > 1.Then we should have

caj−1 ≥ 1 + cα1aj−1

k−1∑i=1

ai−1

τaj−1 + ai−1+πcα1

τ

j∑i=k

ai−1Q

(ai/2

aj/2√τ

).



Note that

k−1∑i=1

ai−1

τaj−1 + ai−1≤

k−1∫0

ax

τaj−1 + axdx =

1ln a

lnτaj−1 + ak−1

τaj−1 + 1≈ 1

ln aln

2τaj−1

τaj−1 + 1

and

j∑i=k

ai−1Q(ai/2

aj/2√τ

) ≈ 1a

j∫k−1

axQ(ax/2

aj/2√τ

) dx ≈ 2τaj−1

ln a

1/√τ∫

1

zQ(z) dz

=τaj−1

√2π ln a

[e−1/2 − 1√

τe−1/(2τ) +

√2π(

1τ− 1

)Q

(1√τ

)]

since (see the Appendix)

2√

2πa∫

1

zQ(z) dz = e−1/2 − ae−a2/2 +√

2π(a2 − 1)Q(a). (32)

Therefore, we should haveg(caj) ≥ 0, j = 0, . . . , J − 1,

where

g(x) = 1− 1x− α1b(τ)

ln a+

α1

ln alnτx+ c

τx,

b(τ) = ln 2 +π√2π

[e−1/2 − 1√

τe−1/(2τ) +

√2π(

1τ− 1

)Q

(1√τ

)].

As before, it is sufficient to have g(c) ≥ 0 and g(∞) ≥ 0, which is equivalent to the conditions

c ≥ c2 =[1 +

α

J ln aln

1 + τ

τ− αb(τ)J ln a

]−1

=b(τ)

ln(1 + 1/τ), a ≥ a2 = eαb(τ)/J . (33)

Therefore, for τ ≤ 1 we haveσ2j ≤ c2σ2aj−1

2 , j = 1, . . . , J.

Then, for τ ≤ 1 and J large, for the average power per user P we have

P =1J

J∑j=1

Pj ≤c2σ

2

τJ

J∑j=1

aj−12 =

c2σ2(aJ2 − 1)

τJ(a2 − 1)≤ σ2(eαb(τ) − 1)ατ ln(1 + 1/τ)

. (34)

Note that for small τ we have

b(τ) = ln 2 +√π

2e+ o(1) ≈ 1.45 + o(1), τ → 0.

We present some of the obtained results in the following statement.

Proposition 4. The optimal power profile is of the form Pj ≤ ciσ2aj−1i /τ , j = 1, . . . , J , where

i = 1 if τ > 1, and i = 2 if τ ≤ 1, with c1, c2, a1, and a2 defined in (30) and (33) respectively.For its average power per user P , upper bounds (31) and (34) are valid.



For comparison, consider the case where all the K users have the same power P1. Then, forτ > 1 and u1 = σ2/P1, we have

u1 = τ

[1− α

1 + τ

]+.

All the constraints are only fulfilled provided that 1 + τ − α > 0. Then for P1 we get

P1 =σ2

τ(1− α/(1 + τ)).

Note that, if τ > 1 and 1 + τ − α > 0, then (see (31))

1(1− α/(1 + τ))

>1

α ln(1 + 1/τ)

[(1 + τ

τ

)α− 1

].

To prove this inequality, note that the function

f(τ, α) = α ln(

1 +1τ

)−(

1− α

1 + τ

) [(1 + τ

τ

)α− 1

]satisfies f ′′α2(τ, α) > 0 and f ′α(τ, 0) = 0. Then we have

minα≥0

f(τ, α) = f(τ, 0) = 0,

and therefore the case of all powers equal requires larger energy per user.

7. UNEQUAL POWERS: THE MMSE DETECTOR

Consider the MMSE detector for model (1) where power levels {Pi} are allowed to be different.

Denote P = K−1K∑i=1

Pi. As before, we have

(yi, bi)‖bi‖2

√Pi

= di +1

‖bi‖√Piξ, ξ ∼ N (0, 1), i = 1, . . . ,K,

b1 = (W 1 + σ2IN )−1/2a1, W 1 = A1 diag1{Pi}AT1 ,

where A1, diag1{Pi}, and W 1 are defined in Section 2.We should have Pi‖bi‖2 ≥ 1/τ with high probability for all i. In particular, we must have

Pi E ‖bi‖2 ≥ 1/τ for all i.Consider the user with the smallest power, say, P1. To evaluate E ‖b1‖2, it is sufficient to

consider the case where all the remaining powers {Pi, i = 2, . . . ,K} are chosen as independentidentically distributed random variables with an arbitrary distribution function H(x) such thatH(P1 − ε) = 0 for any ε > 0 (see [18,19]).

In terms of the Stieltjes transform mW1(z) (see the Appendix), we have E ‖b1‖2 = mW1(−σ2);moreover, m = mW1(−σ2) satisfies the equation

m =

σ2 + α

∞∫−∞

x dH(x)1 + xm

−1

=

σ2 + α

∞∫P1

x dH(x)1 + xm

−1

.

Since x/(1 + xm) ≥ P1/(1 + P1m) for x ≥ P1, we obtain

m ≤[σ2 +

αP1

1 + P1m

]−1

.



Then, denoting u = σ2/P1, we have

P1 E ‖b1‖2 = P1m ≤2√

(α+ u− 1)2 + 4u+ α+ u− 1.

The maximum of the right-hand side of the last expression is attained for the minimal possible u, i.e.,u = u0 = σ2/P . This means that for the MMSE detector it is best of all to have P1 = . . . = PK = P .This yields the following result.

Proposition 5. For the MMSE detector, it is best of all to use equal powers. In particular,the performance of the MMSE detector is not better than that of successive cancellation with equalpowers.

8. CONCLUSION

In the paper, dynamics of iterative interference cancellation has been studied. The dynamicsgovern the behavior of iterative receivers for numerous applications such as code-division multiple-access communications (CDMA) or multiple antenna transmission systems. In particular, it hasbeen shown that, under equal powers of users, performance characteristics of such iterating softdecisions compare well to those of the MMSE detector. In fact, they are even better if the channelnoise is not too high. Note also that the performance of iterative interference cancellation is betterfor unequal power distributions, whereas that of the MMSE detector is worse. Altogether, thisshows the inherent advantage of iterative signal cancellation. Of course, in order to apply thisinterference cancellation method, some form of error control coding needs to be applied, usinglarge enough codes so that the independence assumptions made is our analysis are satisfied.

For an unequal distribution of powers, the optimal power profile has been calculated (it turnedout to be exponential). It was shown to provide superior performance with respect to the equalpower profile.

The exact formula and good approximations for the expectation related with soft symbol de-cisions have been found. This allows us to use fast computer evaluations instead of cumbersomecomputer simulations.

We conclude that iterative interference cancellation is a powerful and simple receiver techniquewhich, in contrast to linear filtering methods, benefits from an optimized distribution. Moreover,it outperforms these linear methods even in equal power situations (as arise in multiple antennatransmission systems or power-controlled CDMA).

APPENDIX

Proof of Proposition 1. First, we recall two auxiliary results, which we need below.1. Let {ai, i = 1, . . . ,K} be N × 1 column vectors, and let A = (a1, . . . ,aK). For n =

max{K,N} and m = min{K,N}, introduce the matrix

W =

{AAT if N < K,

ATA if N ≥ K.

Then W is an m×m nonnegative definite matrix.Assume that all entries {ai,j} of A are independent identically distributed random variables

with E ai,j = 0 and E a2i,j = 1. Denote by FA(x) the empirical distribution of eigenvalues of an

m×m Hermitian matrix A:

FA(x) =the number of eigenvalues of A less than x

m.



It is known [18,19] that, if n/m = τ ≥ 1 and m→∞, then

dF1mW (x)dx

−→{√

(ν+ − x)(x− ν−)/(2πx) for x ∈ [ν−, ν+],0 otherwise,

(A.1)

where ν− = (√τ − 1)2 and ν+ = (

√τ + 1)2.

2. We also need the formula1∫

0

√x(1− x)

(a+ x)(b+ x)dx =

(√b(b+ 1)−

√a(a+ 1)

)π

(b− a)− π. (A.2)

To derive it, it is sufficient to evaluate the integral (upon the substitution x = sin2 z)

1∫0

√x(1− x)(a+ x)

dx =π

2+ aπ − 2a(a+ 1)

π/2∫0

dz

a+ sin2 z

=π

2+ aπ − 2a(a+ 1)

π∫0

dv

2a+ 1− cos v=π

2+ aπ − π

√a(a+ 1)

since [20, formula 3.613.1]π∫

0

dv

1 + b cos v=

π√1− b2

.

Now let us give two proofs of formula (5).

Proof 1. In order to evaluate the expectation on the right-hand side of (4), denote n =max{K,N} and m = min{K,N}. The matrix W 1 has no more than m nonzero eigenvalues,say, λ1, . . . , λm. Therefore,

N E ‖b1‖2 = EN∑i=1

(σ2 + λi)−1 =(N −m)

σ2+ E

m∑i=1

1σ2 + λi

.

If P1 = . . . = PK = P , then W 1 has the form W 1 = PW /N , where W = AAT , and A is anN × (K − 1) matrix consisting of independent identically distributed random variables with zeromean and unit variance. Let FB(x) be the empirical distribution of eigenvalues of a matrix B.Since FW (x) = F cW (cx), c > 0, we have

Em∑i=1

1σ2 + λi

= E∞∫0

m

(σ2 + x)dFW1(x) = E

∞∫0

m

(σ2 + Pmx/N)dF

1mW (x).

If n/m = τ and m → ∞, then, using formula (A.1) with ν− = (√τ − 1)2, ν+ = (

√τ + 1)2, and

m(ν+ − ν−) = 4√NK and using formula (A.2), we get

E∞∫0

m

(σ2 + Pmx/N)dF

1mW (x) =

m

2π

ν+∫ν−

√(ν+ − x)(x− ν−)

(σ2 + Pmx/N)xdx+ o(1)

=m(ν+ − ν−)2

2π

1∫0

√z(1 − z)

(σ2 + Pmz/N)(ν− + (ν+ − ν−)z)dz + o(1)

=N

2πP

1∫0

√z(1− z)

(a+ z)(b + z)dz + o(1) =

N

2P

[√b(b+ 1)−

√a(a+ 1)

(b− a)− 1

]+ o(1),



a =ν−

ν+ − ν−=

(√N −

√K)2

4√NK

, b =σ2N + Pmν−Pm(ν+ − ν−)

= a+σ2

4P

√N

K.

Now with α = K/N and u = σ2/P we have

a =(√α− 1)2

4√α

, b = a+u

4√α,

√a(a+ 1) =

|N −K|4√NK

=|α− 1|4√α,√

b(b+ 1) =1

4√α

√(α− 1)2 + 2(1 + α)u+ u2.

Therefore, we get

E ‖b1‖2 =(1− α)+

σ2+

12P

[√b(b+ 1)−

√a(a+ 1)

b− a − 1

]

=(1− α)+

σ2+

12σ2

[√(α− 1)2 + 2(1 + α)u+ u2 − |α− 1| − u

]=

12σ2

[√(α− 1 + u)2 + 4u− (α− 1 + u)

],

from which formula (5) follows.

Proof 2. If we introduce the Stieltjes transform

m(z) = mW1(z) =∞∫−∞

1x− z dF

W1(x),

then E ‖b1‖2 = mW1(−σ2). It is known [18,19] that m(z) satisfies the equation

m(z) = −

z − α ∞∫−∞

x dH(x)1 + xm(z)

−1

, α =K

N.

In our case, the distribution H(x) is concentrated at x = P . Therefore, m = mW1(−σ2) = E ‖b1‖2satisfies the equation

m =[σ2 +

αP

1 + Pm

]−1

,

from which we get formula (5).

Proof of Proposition 2. Since 1− tanhx = 2/(1 + e2x), we have

E[1− tanh(b2 + bξ)

]2= 4 E

[(1 + e2bξ+2b2)−2

]. (A.3)

Let us get a series representation for the expectation on the right-hand side of (A.3). Differentiating

the series 1/(1+x) =∞∑n=0

(−1)nxn, which uniformly converges for |x| ≤ 1− δ, we get another series,

also uniformly converging for |x| ≤ 1− δ:

(1 + x)−2 =∞∑n=0

(−1)n(n+ 1)xn, |x| < 1. (A.4)

We split the expectation on the right-hand side of (A.3) into two parts: |ξ+b| ≥ ε and |ξ+b| ≤ ε,with some (sufficiently small) ε > 0.



For a random variable η and a set A, denote E[η;A] = E[ηI{A}]. We also use the simple formulas

E[eaξ; ξ ≤ b

]= E

[eaξI{ξ≤b}

]= ea

2/2Q(a− b), E[eaξ ; ξ ≥ b

]= ea

2/2Q(b− a). (A.5)

Note that, if ξ + b ≤ −ε, then e2b2+2bξ < 1. Then from (A.4) and (A.5) we have

E[(

1 + e2b2+2bξ)−2

; ξ + b ≤ −ε]

=∞∑n=0

(−1)n(n+ 1)e2b2n(n+1)Q[b(2n+ 1) + ε

].

For ξ + b ≥ ε we use a similar series:

(1 + x)−2 =∞∑n=0

(−1)n(n+ 1)x−(n+2), |x| > 1.

Then we have

E[(

1 + e2b2+2bξ)−2

; ξ + b ≥ ε]

=∞∑n=0

(−1)n(n+ 1)e2b2(n+1)(n+2)Q[b(2n + 3) + ε

]=∞∑n=0

(−1)n+1ne2b2n(n+1)Q[b(2n + 1) + ε

].

Therefore, for any ε > 0 we get

E[(

1 + e2b2+2bξ)−2

; |ξ + b| ≥ ε]

=∞∑n=0

(−1)ne2b2n(n+1)Q[b(2n+ 1) + ε

]. (A.6)

The absolute values of terms of this series are monotonically decreasing (see below).It remains to evaluate the expectation

E[(

1 + e2b2+2bξ)−2

; |ξ + b| ≤ ε]

=14

E[[

1 +(e2b2+2bξ − 1

)/2]−2

; |ξ + b| ≤ ε].

Assuming e2bε < 3 and using the formula

E[eaξ; |ξ + b| ≤ ε

]= ea

2/2[Q(a+ b− ε)−Q(a+ b+ ε)],

we have

E[(

1 + e2b2+2bξ)−2

; |ξ + b| ≤ ε]

=14

∞∑n=0

(n + 1)2−n E[(

1− e2b2+2bξ)n

; |ξ + b| ≤ ε]

=14

∞∑n=0

(n + 1)2−nn∑i=0

(−1)i(n

i

)E[e2ib2+2ibξ; |ξ + b| ≤ ε

]=

14

∞∑n=0

(n + 1)2−nn∑i=0

(−1)i(n

i

)e2b2i(i+1)[Q(b(2i + 1)− ε)−Q(b(2i + 1) + ε)

]=

14

∞∑i=0

(−1)ie2b2i(i+1)[Q(b(2i+ 1)− ε)−Q(b(2i+ 1) + ε)] ∞∑n=i

(n+ 1)2−n(n

i

)

=∞∑i=0

(−1)i(i+ 1)e2b2i(i+1)[Q(b(2i+ 1)− ε)−Q(b(2i + 1) + ε)],



where we used the formula

∞∑n=i

(n+ 1)2−n(n

i

)= (i+ 1)

∞∑n=i

2−n(n+ 1i+ 1

)

= (i+ 1)∞∑k=0

2−k−i(k + i+ 1

k

)= (i+ 1)2−i(1− 2−1)−(i+2) = 4(i+ 1)

since

(1− x)−(i+2) =∞∑k=0

(k + i+ 1

k

)xk, |x| < 1.

Therefore, for any b > 0 and any ε > 0 such that e2bε < 3, we have the representation

E

[1

(1 + e2b2+2bξ)2

]=∞∑n=0

(−1)ne2b2n(n+1)Q[b(2n + 1) + ε

]+∞∑n=0

(−1)n(n+ 1)e2b2n(n+1)[Q(b(2n + 1)− ε)−Q(b(2n + 1) + ε)]. (A.7)

The absolute values of terms of the first sign-alternating series on the right-hand side of (A.7)are monotonically decreasing. Formula (11) follows from (A.7) by taking the limit as ε→ 0.

Now let us prove the second upper bound in (12). Combining terms with n = 2k and n = 2k+1for k = 0, 1, . . . , from (11) we have

g(b) = 4[Q(b)− e4b2Q(3b)

]+

4e−b2/2

√2π

∞∑k=1

ck,

whereck =

√2π{eb

2(4k+1)2/2Q[b(4k + 1)]− eb2(4k+3)2/2Q[b(4k + 3)]}.

Since√

2π ea2/2Q(a) =

∞∫0

ue−u2/2

√a2 + u2

du =1a−∞∫0

ue−u2/2

(a2 + u2)3/2du,

we haveck ≤

2b(4k + 1)(4k + 3)

, k = 0, 1, . . . .

Then, taking into account that∞∑k=0

1(4k + 1)(4k + 3)

=π

8,

we have

g(b) ≤ 4[Q(b)− e4b2Q(3b)

]+e−b

2/2

b√

2π

(π − 8

3

). (A.8)

Combining analytical and numerical methods, we can check that the right-hand side of (A.8) doesnot exceed πQ(b) for any b > 0.201. Since πQ(0.201) ≈ 1.32 and g(b) ≤ 1, b ≥ 0, we get the upperbound g(b) ≤ πQ(b), b ≥ 0.

To get the second approximation of (14), we take one more term in the expansion for ck anduse a similar expansion for Q(3b).



This approach does not work for small b. For such b, we use the Taylor formula(1− tanh(bξ + b2)

)2 = 1− 2bξ + (ξ2 − 2)b2 + (2ξ3/3 + 2ξ)b3 + (1− 2ξ4/3 + 2ξ2)b4

+ (2ξ − 8ξ3/3− 4ξ5/15)b5 + (2/3 − 4ξ2 − 4ξ4/3 + 17ξ6/45)b6 +O(b8), b→ 0.

Although the remainder term O(b8) depends on ξ, we have P (|ξ| >√

2c ln(1/b)) < bc for any c > 0.Then we may take expectations of both sides and get, for sufficiently small b,

g(b) = 1− b2 + b4 − 53b6 +O

(b8 ln(1/b)

)≤ 1

1 + b2, b→ 0.

Note also that 1/(1 + b2) > πQ(b) for b > 1. Finally, we can numerically check that the inequalityg(b) ≤ 1/(1 + b2) remains valid for all b ≥ 0. Together with the inequality g(b) ≤ πQ(b), b ≥ 0,it proves upper bound (12).

To get the first approximation of (14), we use the formula

g(b) = 1− b2 + b4 − 53b6 +O

(b8 ln(1/b)

)≈ 1

1 + b2 + 2b6/3.

Monotonicity of g(b). Since series (11) does not converge absolutely, formally we may notdifferentiate it. But we may differentiate series (A.7) and then take the limit as ε→ 0. As a result,for b > 0 we get

g′(b) = 4∞∑n=0

(−1)n[4n(n+ 1)be2n(n+1)b2Q[b(2n + 1)]− (2n + 1)√

2πe−b

2/2].

Using for Q(x) the representation

√2πQ(x)ex

2/2 =1x− 1x3

+3x5− 15c(x)

x7, 0 < c(x) < 1, x > 0,

we get

g′(b) = −4e−b2/2

b2√

2π

∞∑n=0

(−1)n[

(1 + b2)(2n+ 1)

− (3 + b2)b2(2n + 1)3

+3

b2(2n+ 1)5+

60cnn(n+ 1)b4(2n + 1)7

]

= − e−b2/2

b2√

2π

[(1 + b2)π − π3

8+π3(5π2 − 48)

128b2+

241θb4

], |θ| < 1,

where some known formulas [20] for series sums were used.Evaluating the right-hand side of the last expression, we conclude that g′(b) < 0 for b ≥ 2.2.

The inequality g′(b) < 0 for b < 2.2 can be checked numerically. 4Monotonicity of series (A.6). We should show that for any b > 0, ε > 0, and n ≥ 1 we have

the inequalitye2b2n(n+1)Q[b(2n + 1) + ε]e2b2n(n−1)Q[b(2n − 1) + ε]

< 1.

The left-hand side of the inequality is

e[b(2n+1)+ε]2/2−2bεQ[b(2n+ 1) + ε]e[b(2n−1)+ε]2/2Q[b(2n − 1) + ε]

<e[b(2n+1)+ε]2/2Q[b(2n+ 1) + ε]e[b(2n−1)+ε]2/2Q[b(2n− 1) + ε]

< 1

since the function ex2/2Q(x) monotonically decreases for x ≥ 0. 4



Proof of formula (32). We have

2√

2πa∫

1

zQ(z) dz = 2a∫

1

z

∞∫z

e−u2/2 du dz

= 2a∫

1

e−u2/2

u∫1

z dz du+ 2∞∫a

e−u2/2

a∫1

z dz du

=a∫

1

e−u2/2(u2 − 1) du + (a2 − 1)

∞∫a

e−u2/2 du

= e−1/2 − ae−a2/2 +√

2π(a2 − 1)Q(a).

REFERENCES

1. Berrou, C. and Glavieux, A., Near Optimum Error-Correcting Coding and Decoding: Turbo-Codes,IEEE Trans. Commun., 1996, vol. 44, no. 10, pp. 1261–1271.

2. Alexander, P.D., Grant, A.J., and Reed, M.C., Iterative Detection in Code-Division Multiple-Access withError Control Coding, European Trans. Telecommun., Special Issue on CDMA Techniques for WirelessCommun. Syst., 1998, vol. 9, no. 5, pp. 419–426.

3. Reed, M.C., Schlegel, C.B., Alexander, P.D., and Asenstorfer, J.A., Iterative Multiuser Detectionfor CDMA with FEC: Near-Single-User Performance, IEEE Trans. Commun., 1998, vol. 46, no. 12,pp. 1693–1699.

4. Moher, M., An Iterative Multiuser Decoder for Near-Capacity Communications, IEEE Trans. Commun.,1998, vol. 46, no. 7, pp. 870–880.

5. Alexander, P.D., Reed, M.C., Asenstorfer, J.A., and Schlegel, C.B., Iterative Multiuser InterferenceReduction: Turbo CDMA, IEEE Trans. Commun., 1999, vol. 47, no. 7, pp. 1008–1014.

6. Wang, X. and Poor, H.V., Iterative (Turbo) Soft Interference Cancellation and Decoding for CodedCDMA, IEEE Trans. Commun. 1999, vol. 47, no. 7, pp. 1046–1061.

7. Shi, Z. and Schlegel, C.B., Joint Iterative Decoding of Serially Concatenated Error Control CodedCDMA, IEEE J. Select. Areas Commun., 2001, vol. 19, no. 8, pp. 1646–1653.

8. Schlegel, C.B. and Perez, L.C., Trellis and Turbo Coding, Piscataway: IEEE; Hoboken: Wiley, 2004.

9. Richardson, T. and Urbanke, R., The Capacity of Low-Density Parity-Check Codes under Message-Passing Decoding, IEEE Trans. Inf. Theory, 2001, vol. 47, no. 2, pp. 599–618.

10. Boutros, J. and Caire, G., Iterative Multiuser Joint Decoding: Unified Framework and AsymptoticAnalysis, IEEE Trans. Inf. Theory, 2002, vol. 48, no. 7, pp. 1772–1793.

11. Schlegel, C.B., Shi, Z., and Burnashev, M.V., Asymptotically Optimal Power Allocation and CodeSelection for Iterative Joint Detection of Coded Random CDMA, IEEE Trans. Inf. Theory, submitted.

12. Caire, G., Muller, R., and Tanaka, G., Iterative Multiuser Joint Decoding: Optimal Power Allocationand Low-Complexity Implementation, IEEE Trans. Inf. Theory, submitted.

13. Grant, A. and Alexander, P., Random Sequence Multisets for Synchronous Code-Division Multiple-Access Channels, IEEE Trans. Inf. Theory, 1998, vol. 44, no. 7, pp. 2832–2836.

14. Narayanan, K.R. and Stuber, G.L., A Serial Concatenation Approach to Iterative Demodulation andDecoding, IEEE Trans. Commun., 1999, vol. 47, no. 7, pp. 956–961.

15. Shi, Z. and Schlegel, C.B., Asymptotic Iterative Multi-User Detection of Coded Random CDMA, IEEETrans. Commun., submitted.



16. Xie, Z., Short, R.T., and Rushforth, C.K., A Family of Suboptimum Detectors for Coherent MultiuserCommunications, IEEE J. Select. Areas Commun., 1990, vol. 8, no. 4, pp. 683–690.

17. Feller, W., An Introduction to Probability Theory and Its Applications, vol. 2, New York: Wiley, 1971,2nd ed. Translated under the title Vvedenie v teoriyu veroyatnostej i ee prilozheniya, Moscow: Mir, 1984.

18. Marchenko, V.A. and Pastur, L.A., The Distribution of Eigenvalues in Certain Sets of Random Matrices,Mat. Sb., 1967, vol. 72(114), no. 4, pp. 507–536.

19. Bai, Z.D., Methodologies in Spectral Analysis of Large Dimensional Random Matrices, A Review, Statist.Sinica, 1999, vol. 9, no. 3, pp. 611–677.

20. Gradshtein, I.S. and Ryzhik, I.M., Tablitsy integralov, summ, ryadov i proizvedenii, Moscow: Nauka,1971, 5th ed. Translated under the title Table of Integrals, Series, and Products, New York: Academic,corr. and enl. ed., 1980.


analysis of the dynamics of iterative interference ...hcdc/library/pit paper (as published by...

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