Iterative idma receivers with random and tree based interleavers

Journal of Information Engineering and Applications www.iiste.org
ISSN 2224-5758 (print) ISSN 2224-896X (online)
Vol 1, No.3, 2011

Iterative IDMA Receivers with Random and Tree Based
Interleavers
Kulbhushan Gupta 1 C.K. Shukla2 M. Shukla3*
3. Sam Higginbottom Inst. of Agriculture Tech. & Sciences (Deemed University), Allahabad, India
4. Sam Higginbottom Inst. of Agriculture Tech. & Sciences (Deemed University), Allahabad, India
5. Harcourt Butler Technological Institute, Kanpur, India
* E-mail of the corresponding author: manojkrshukla@gmail.com

Abstract
In recent days, on the horizon of wireless world, newly proposed multiple access scheme known as
Interleave-Division Multiple-Access (IDMA) has made its remarkable impact. Researchers all over world,
are making hard marks to establish the scheme to establish its claim as potential candidate for 4th generation
wireless communication systems. This paper is concerned with the performance enhancement of iterative
IDMA systems under coded & uncoded environment. The performance of an interleave division multiple
access (IDMA) system can be improved by the optimized power allocation techniques. Based on the
optimized power allocation technique we compare the performance of coded & uncoded IDMA system
with random interleaver & tree based interleaver. During the simulation, it has been observed that tree
based interleaver demonstrate the similar bit error rate (BER) performance to that of random interleaver
however on other fronts including bandwidth and memory requirement at transmitter and receiver ends, it
outperforms the random interleavers.
Keywords: Tree Based Interleaver, Random Interleaver, IDMA, linear programming, power allocation,
BER.

1. Introduction
The well established Direct-Sequence code-division multiple-access (DS-CDMA) which is also known as
simply CDMA has been adopted in second and third generation cellular mobile standards. The CDMA
scheme possesses many attractive features such as dynamic channel sharing, mitigation of cross-cell
interference, asynchronous transmission, ease of cell planning, and robustness against fading.
In CDMA system, many users share the same transmission media so that signals from different users are
superimposed, causing the multiple access interference (MAI) problems. At the receiver side, it is
necessary to separate the mixed signals. Multi-user detection (MUD) is a technique to improve
performance by jointly processing the signals forms all of the users.
Recently, it has been shown that multiuser detection together with unequal power control can enhance the
performance of code division multiple access (CDMA) systems. The complexity and computational cost is
the major problem of CDMA systems.
Interleave Division Multiple Access (IDMA) scheme in which interleavers are employed as the only means
of user separation. Interleave division multiple access (IDMA) is a special case of random waveform
CDMA, and the accompanying chip-by-chip (CBC) estimation algorithm is essentially a low cost iterative
soft cancellation technique [Ping L. & Lihai L. (2004)].
The chip-by-chip (CBC) MUD algorithm for IDMA is an iterative soft cancellation technique for treating
multiple access interference (MAI). Its computational cost is very low, being independent of the total
number of users when normalized to each user [Wang P., Ping L. & Liu L. (2006)].
With equal power control, it has been shown in [Liu L., Tong J. & Ping L. (2006)] that IDMA together with
CBC algorithm can achieve performance close to the theoretical limits. Unequal power allocation is a
technique to further enhance system performance. In [3], a linear programming technique is employed for
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power optimization of IDMA systems over additive white Gaussian noise (AWGN) multiple access
channels (MACs).
In this paper, we use the unequal optimized power allocation technique to the uncoded-IDMA system with
random interleavers & tree based interleavers. After that we compare their performance and reach to the
conclusion.
Section 2 contains an introduction of the IDMA communication system. In section 3, we discussed about
tree based interleaver. In section 4 we discussed the power allocation techniques in brief. Section 5
presents the simulation results of uncoded-IDMA with random interleaver and tree based interleaver.
Section 6 concludes the paper.

2. Iterative IDMA System

2.1. System Model
Consider an IDMA system with k users. At the transmitter side, the information bits for user k are first
encoded by an encoder (ENC) based on a FEC code and then interleaved and transmitted over a Gaussian
multiple access channel (MAC). The received signal can be written as;
K
r ( j ) = ∑ hk xk ( j ) + n ( j ) , j = 1,2,...J (1)
k =1
Where xk ( j ) ∈ {+1, −1} is the jth chip transmitted by user-k, hk the coefficient for user-k representing
{ }
the combined effect of power control and channel loss, and n ( j ) are samples of an AWGN process with
zero-mean and variance σ = N 0 / 2 [4].
2

The receiver consists of an elementary signal estimator (ESE) and k a posteriori probability (APP)
decoders (DECs), operating iteratively.
For each user k, we rewrite (1) as;
r ( j ) = hk xk ( j ) + ξ k ( j ) (2)
where,

ξ k ( j ) = ∑ hk ′ xk ′ ( j ) + n ( j ) (3)
k ′≠ k

2.2 The CBC Algorithm
(
E ( xk ( j ) ) = tanh eDEC ( xk ( j ) ) 2 , ∀k , j ) (4)
(
Var ( xk ( j ) ) = 1 − E ( xk ( j ) ) , ∀k , j )
2
(5)
E (ξ k ( j ) ) = ∑ hk ′ xk ′ ( j ) , ∀k , j (6)
Var (ξ k ( j ) ) = ∑ hkk′ ′≠Var ( xk ′ ( j ) ) + σ 2 , ∀k , j
2k
(7)
(
eESE = {2hk / Var (ξk ( j ) )} r( j) − E (ξkk′≠ kj ) ) , ∀k, j
( ) (8)
After (8), the APP decoding in the DECs is performed to generate the LLRs
{ }
eDEC ( xk ( j ) ) , ∀k , j .Then go back to (4) for the next iteration.

3. Tree Based Interleaving Mechanism
In iterative IDMA system each user has a specific interleaver {π k } having length equal to chip length ‘J’.
A considerable amount of memory will be required to store the indexes for these interleavers. The tree
based interleaver is basically aimed to minimize the computational complexity and memory requirement
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that occurs in power interleaver & random interleavers respectively [Shukla M., Srivastava V.K. & Tiwari
S. (2009)].
In a tree based interleaver generation, two randomly generated interleaver are chosen. Let π 1 and π 2 be
the two random interleavers. The combinations of these two interleavers in a particular fashion as shown in
the figure 2 are used as interleaving masks for the users [Shukla M., Srivastava V.K. & Tiwari S. (2009)].
The allocations of the interleaving masks follow the binary tree format. The interleaving masking diagram
is shown upon 14 users for simplicity. It is clearly shown through the figure that, for obtaining the
interleaving sequence of the 14th user, it need only 2 cycles, as in [Shukla M., Srivastava V.K. & Tiwari S.
(2009)].
π 14 = π 2 (π 2 (π 2 ) )

4. Power Allocation Strategy for Users
This section is concerned with the power optimization technique for IDMA systems over AWGN channels.

4.1 Problem Formulation:

{ } ( )
For simplicity, we assume that, for all users, the same FEC code is used and the same BER performance is
(l ) (l )
required. Let γ k be the average SNIR for the outputs of the ESE after the l-th iteration & f k γ k
(l )
be the average variance of the outputs of DECk driven by an input sequence with SNIR γ k [Shukla
M., Srivastava V.K. & Tiwari S. (2006), Shukla M., Srivastava V.K. & Tiwari S. (2009)].
(l )
We can approximately track γ k using the following recursion;
2
( l +1) Pk hk
γk = , ∀k , l = 0,1,....( L − 1)
( )
(10)
∑P h fi γ i ( ) + σ 2
2 l
i i
i≠k
where L is the maximum number of iterations.
For AWGN channels, we have,
hk = 1, ∀κ
2

Then recursion (10) reduces to,
Pk
γ k (l +1) = , ∀k , l = 0,1,....( L − 1)
( )
(11)
∑ Pi fi γ i(l ) + σ 2
i≠k
Our objective now is to minimize total power ∑ Pk while achieving the required
(l )
performance γ k ≥ Γ, ∀k , for specified Γ [2]. This power optimization problem can be formulated as
k

follows.

4.2 Power optimization over AWGN channels:
Find the distribution { Pk } that minimizes,
Φ = ∑ Pk (12)
subject to, k
γ k (l ) ≥ Γ, ∀k
{ }
(13)
(l ) (0)
where γ k are obtained through the following SNIR evolution process with initialization γ k =0,
for all k.
Pk
γ k (l +1) = , ∀k , l = 0,1,....( L − 1)
( )
(14)
∑ Pi fi γ i( ) + σ 2
i≠k
l

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Problem (14) is generally nonlinear and non-convex. The problem considered above is to find the
minimum sum-power solution within the feasible region [Shukla M., Srivastava V.K. & Tiwari S. (2009)].
We define the total interference after the l-th iteration of recursion (11) as,
I ( ) = ∑ Pk f k γ k ( ) + σ 2
l

k =1
( ) l
(15)

Then (11) can be rewritten as
Pk
γ k (l +1) = , ∀k , l = 0,1,....( L − 1)
( )
(16)
(l )
− Pk f k γ k ( )
l
I

B. Linear Programming Approach:

In problem (12), the constraints (13) & (14) are nonlinear with respect to the optimization variables { Pk } ,
which makes the problem complicated. We therefore consider a linearization technique to solve this
problem.
We quantize the power levels { Pk } into (M+1) discrete levels : { P(m), m=0,1,……M} with P(m-1) <
P(m), m=1,2,…….M, and partition the k users into (M+1) groups according to their power levels. Letting
λ (m) denote the number of users assigned with power level P(m) [Shukla M., Srivastava V.K. & Tiwari S.
(2009)].
Then total power (12) is rewritten as
φ = ∑ λ (m)P(m) (17)
m
and the number of users k can be represented as,
k = ∑ λ ( m) (18)
m
γ ( l ) ( m)
be the SNIR of the ESE outputs for the users in the group with power P(m) after the l-th iteration
and assume that user-k has power P(m). Then (15) & (16) can be rewritten as,

I ( ) = ∑ λ ( m) P ( m) f γ
l

m
( (l )
(m) +σ 2 ) (19)

( l +1) P (m)
γ (m) = , ∀k , l = 0,1,....( L − 1) (20)
I (l )
− P ( m) f γ ( (l )
( m) )
Since {I ( ) } is a monotonically decreasing sequence, i.e.,
l

I(
l +1)
≤ I ( ) , l=0, 1,….. (L-1)
l
(21)
Substituting (21) into (20) and introducing a delay factor δ ( 0 < δ < 1) to control the convergence speed
of the iterative detection, we get,

∑ λ (m)P(m) f (γ ( ) (m) ) + σ
m
l 2
≤ (1 − δ ) I ( )
l
(22)

l=0, 1,………… (L-1)
Now we take {λ ( m)} as optimization variables. In this case both the target function (17) and the
constraints (18) & (22) appear linear with respect to {λ ( m)} . However γ
( l +1)
(m) in (20) is dependent
on {λ ( m)} . Now, γ
( l +1)
(m) can be treated as pre-calculated parameter when linear programming is
applied.
Using an approximation,

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( l +1) P ( m)
γ ( m ) ≈ γ ∗ ( m) = (23)
I (l )
* denotes approximation.
We can rewrite (22) as
 P ( m) 
∑ λ (m)P(m) f
m

 I 
 + σ ≤ (1 − δ ) I , I min ≤ I ≤ I max
2
(24)

where I max & I min specify the total interference at the beginning and end of the iterative detection and the
(l )
iteration index l for I is omitted. We evaluate (24) at a set of quantized values of I:{I(n),n=0,1,..N}. By
{ } { }
pre-calculating f m , n = f ( P ( m) I ( n ) ) and using f m ,n as constants, we obtain a group of linear
constraints with respect to {λ ( m)} [2].
In summary, we have
φ = ∑ λ (m)P(m) (25)
m
subject to
∑ λ ( m) = k (26)
λ (m) ≥ 0, m = 0,1,....M
m (27)
 P ( m) 
∑ λ (m)P(m) f
m

 I 
 + σ ≤ (1 − δ ) I , I min ≤ I ≤ I max
2 (28)

Let y (m) denotes the total power of these λ (m) users. As such,
y(m)= λ (m) P(m) (29)
Substituting (29) into (25),(27) & (28) we get
φ = ∑ y ( m) (30)
subject to, m

 P ( m) 
∑ y ( m) f
m

 I 
 + σ ≤ (1 − δ ) I , I min ≤ I ≤ I max
2
(31)

y (m) ≥ 0, m = 0,1,....M . (32)
In practice in [1], both P and I should be quantized. We need to determine the searching
ranges Pmin , Pmax , I min & I max . Let us quantize P and I as
P(m) = α m Pmin , m = 0,1,....M (33)
I (n) = β n I min , n = 0,1,....N (34)
with P(0)= Pmin >0, P(M)= Pmax ,I(0)= I min >0, I(N)= I max , α >1 and β >1.
Then (31) becomes,
∑ y ( m ) f (α
m
m
β − nγ ) + σ 2 ≤ (1 − δ ) β n I min (35)

Since γ = Pmin I min (36)
The minimum SNR is obtained at the end of the iterative decoding process. Here γ can be determined by
the desired BER. For more details about the relation between γ and BER see in [1]. Since
y ( m) y (m)α − m
k = ∑ λ (m) = ∑ =∑ (37)
m m P(m) m γ I min
We get,
∑ y(m)α
m
−m
− kγ I min = 0 (38)

We can replace (31) by (35). We can also include (38) as part of the LP constraints. In this way
I min becomes an optimization variable and Pmin can be determined from I min by (36) [1].
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We still need to determine M and N in (33) & (34). In general, we just use a sufficient large M, since the
LP will automatically determine the maximum power level used. We start with a relatively small N.
If after the LP,
I max = ∑ y (m) ×1 +σ 2 > I ( N ) (39)
m
then the quantization range in (34) is not sufficient and we increase N and repeat the LP until
I max ≤ I ( N ) .
5. Simulation Results
The power profiles for 32, 48, 64 users are given below in the Table 1.
As we see that the performance improves with increased data lengths. We can also see that a data length of
512 is sufficient to achieve relatively good performance. Performance enhancement is marginal with
information block longer than 512.
−4
If user count is k ≤ 32, then fewer than 20 iterations are sufficient (measured at 10 ) and more iterations
do not bring about significant performance improvement. However, when user count is K ≥ 48, more
iteration can be beneficial [Shukla M., Srivastava V.K. & Tiwari S. (2008), Shukla M., Srivastava V.K. &
Tiwari S. (2006)].

6. Conclusion
As we see from the simulation results that in unequal power allocation technique with Tree based
interleaver based uncoded and coded IDMA system gives the better results in comparison to random
interleaver based IDMA system.
So using Tree based interleaver in IDMA system with unequal optimized power allocation technique will
enhance the performance in comparison to random interleaver and also reduce the memory cost &
computational complexity.

References
Ping L. & Lihai L. (2004), “Analysis and Design of IDMA Systems Based on SNR Evolution and Power
Allocation”, Proceedings of Vehicular Technology Conference VTC 2004-Fall, IEEE, 1068-1072.
Wang P., Ping L. & Liu L. (2006), “Power Allocation for Multiple Access Systems with Practical Coding
and Iterative Multi-User Detection”, Proceedings of International Conference ICC`06, 11, IEEE, 4971-
4976.
Liu L., Tong J. & Ping L. (2006), “Analysis and Optimization of CDMA Systems with Chip-Level
Interleavers”, Journal Selected Areas in Communication, 24, IEEE, 141-150.
Ping L., Wu Y. & Leung W. (2006),”Interleave-Division Multiple-Access”, Transaction on Wireless
Communication, 5 (4), IEEE. 938-947.
Shukla M., Srivastava V.K. & Tiwari S. (2009), “Analysis and Design of Optimum Interleaver for Iterative
Receivers in IDMA Scheme”, Journal of Wireless Communication and Mobile Computing, 9(10), Wiley
1312-1317.
Shukla M., Srivastava V.K. & Tiwari S. (2006), “Interleave Division Multiple Access for Wireless
Communication”, Proceedings of International Conference on Next Generation Communication Systems:
A Perspective’, “ICONGENCOM 06”, J.K. Institute, Allahabad, India, 150-154.
Shukla M., Srivastava V.K. & Tiwari S. (2008), “Analysis and Design of Tree Based Interleaver for
Multiuser Receivers in IDMA Scheme”, Proceedings of International Conference on Networks “ICON
2008”, Delhi, India, IEEE, 1-4.

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Vol 1, No.3, 2011

Shukla M., Shukla A., Srivastava V.K., & Tiwari S. (2009), “Different Designing Factors for IDMA
Systems”, Proceedings of International Conference on Computer, Communication, and Control and
Information Technology “C3 IT 2009”, Calcutta, India, Academy Publishers, 748-756.
Shukla M., Srivastava V.K. & Tiwari S. (2006), “A Novel Interleaver for Interleave Division Multiple
Access Scheme”, Proceedings of International Conference on Information and Communication Techniques
“ICCT 07”, Dehradun, India, 843-846.
Shukla M., Srivastava V.K. & Tiwari S. (2009), “Analysis of Optimum Interleaver for Iterative Receivers in
IDMA Scheme”, Proceedings of International Conference on Computing and Networking “ICDCN 2009”,
Springer, 400-407.
Shukla M., Srivastava V.K. & Tiwari S. (2009), “Performance Analysis of Tree Based Interleaver with
IDMA Systems using Optimum Power Allocation Algorithm”, Proceedings of International Conference on
Emerging Trends in Engineering & Technology “ICETET-09”, IEEE, 1173-1177.

Table 1
Power Profile For 32, 48 & 64 Users
(power level(dB)) × (user number)
K=32 0 × 25, 5.3811 × 7
K=48 0 × 26, 7.4509 × 8, 10.3484 × 8, 10.7623 × 6
K=64 0 × 25, 7.8645 × 7, 8.2784 × 7, 13.2455 × 5
13.6594 × 7, 18.6266 × 13

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Figure 1. (a) The transmitter for user-k. (b) A part of the receiver related to user-k.

П1(П1(П1)) 7
3

П1(П1)
П1(П1(П2)) 11

1

П1(П2(П1)) 9
П1
5

П1(П2)
П1(П2(П2)) 13

П2(П1(П1)) 8
4

П2(П1)
2

П2(П1(П2)) 12
П2

П2(П2(П1)) 10
6

П2(П2)

П2(П2(П2)) 14

Figure 2. Interleaving mask allocation for the proposed Tree based interleaving scheme

0
withoutcoding random interleaver 32 users
10

-1
10

-2
10
Bit Error Rate

-3 datalength=128
10
datalength=256
datalength=512
-4 datalength=1024
10

-5
10

-6
10
2 4 6 8 10 12 14 16
Eb/No

Figure 3 (a). 32 users Random Interleaver

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0
without coding tree based interleaver 32 users
10

-1
10

-2
10
Bit Error Rate

-3 datalength=128
10
datalength=256
datalength=512
-4
10 datalength=1024

-5
10

-6
10
2 4 6 8 10 12 14 16
Eb/No
Figure 3 (b). 32 users Tree Based Interleaver

0
without coding 64 users random interleaver
10

-1
10

-2
10
Bit Error Rate

-3 datalength=128
10
datalength=256
datalength=512
-4 datalength=1024
10

-5
10

-6
10
0 5 10 15 20 25
Eb/No

Figure 4 (a). 64 users Random Interleaver

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0
without coding tree based interleaver 64 users
10

-1
10
Bit Error Rate

-2
10 datalength=512
datalength=1024
datalength=256
datalength=128
-3
10

-4
10
2 4 6 8 10 12 14 16 18 20 22
Eb/No

Figure 4 (b). 64 users Tree Based Interleaver

Comparison of with and without coding 16 users random interleaver
-1
10

datalength=512
datalength=2048
-2
10
Bit Error Rate

-3
10

-4
10

-5
10
2 4 6 8 10 12
Eb/No
Figure 5 (a). 16 users Random Interleaver with and without coding

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-1
16 users with and without coding tree based interleaver
10
datalength=512
datalength=2048
-2
10

-3
Bit Error Rate

10

-4
10

-5
10

-6
10
2 4 6 8 10 12
Eb/No
Figure 5 (b). 16 users Tree-Based Interleaver with and without Coding

Kulbhushan Gupta is engaged as Assistant Professor in B.B.S. College of Engineering, Allahabad, India
and is perusing Ph.D. form Sam Higginbottom Inst. of Agriculture Tech. & Sciences (Deemed University),
Allahabad, India. His area of interest is Ad-Hoc Neworks, Multiple Access Scheme and Modulation
Schemes.

C.K. Shukla is associated with Sam Higginbottom Inst. of Agriculture Tech. & Sciences (Deemed
University), Allahabad, India as Professor in Electronics & Communication Engineering Department. At
present he is working in the area of Ad-Hoc Networks, routing algorithms, and Mobile Computing
mechanisms. He has published several papers in national and international conferences of repute apart from
various journal publications. He is also member of various professional bodies and is active in research
areas. He may be contacted at http://shiats.edu.in/faculty/Colleges/dept/staff/facProfile.asp?txtOpt=1519 .

M. Shukla is associate professor of Electronics Engineering Department in Harcourt Butler Technological
Institute, Kanpur, India. As an engineer-scientist-turned-faculty, Dr. Shukla’s research agenda has been
extensive, ranging from information systems development to case-driven strategic technological issues. His
current research interests include network security, neural network and fuzzy applications in electronics
engineering, design and analysis of interleavers for multiple access schemes in wireless and mobile networks,
channel coding, QOS issues, VHDL coding of communication models, and next generation networks. He has
more than eighteen years of teaching and research experience in the area of circuit design and
Communication Engineering. He has supervised a number of M. Tech. theses. He has worked as reviewer for
several conferences and journals of repute. He has published more than fifteen research papers in different
journals and conferences. Dr. Shukla’s papers have appeared in journals such as Wiley Transactions on
Wireless Communication and Mobile Computing, International Journal of Applied Engineering Research, The

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IUP Journal of Telecommunications, and Springer Lecture Notes on Computer Science, among others. He is a
member of the Institute of Electrical and Electronics Engineers (IEEE), Institution of Electronic and
Telecommunication Engineers (IETE), International Society of Electronics & Electrical Engineers (ISEEE),
and the Indian Society for Technical Education (ISTE). He may be ‘virtually’ reached at
http://www.hbti.ac.in/Dept/ET/manoj.htm and http://www.manojkrshukla.weebly.com.

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