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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 49, NO. 5, MAY 2001 793

Interleaver Properties and Their Applications to theTrellis Complexity Analysis of Turbo Codes

Roberto Garello, Member, IEEE, Guido Montorsi, Member, IEEE, Sergio Benedetto, Fellow, IEEE, andGiovanni Cancellieri

AbstractIn the first part of this paper, the basic theory of in-terleavers is revisited in a semi-tutorial manner, and extended toencompass noncausal interleavers. The parameters that charac-terize the interleaver behavior (like delay, latency, and period) areclearly defined. The inputoutput interleaver code is introducedand its complexity studied. Connections among various interleaverparameters are explored. The classes of convolutionaland blockin-terleavers are considered, and their practical implementation dis-cussed. In the second part, the trellis complexity of turbo codes istied to the complexity of the constituent interleaver. A procedureof complexity reduction by coordinate permutation is presented,together with some examples of its application.

Index TermsConcatenated codes, interleavers, trellis com-plexity, turbo codes.

I. INTRODUCTION

I NTERLEAVERS are simple devices that permute (usuallybinary) sequences: they are widely used for improving errorcorrection capabilities of coding schemes over bursty channels.

Their basic theory has received a relatively limited attention in

the past, apart from some classical papers [1], [2]. Since the in-

troduction of turbo codes [3], where interleavers play a funda-

mental role, researchers have dedicated many efforts to inter-

leaver design. However, misunderstanding of basic interleaver

theory often causes confusion in the turbo code literature.In this paper, interleaver theory is revisited. Our intent is

twofold: first, we want to establish a clear mathematical frame-

work which encompasses old definitions and results on causal

interleavers. Second, we want to extend this theory to noncausal

interleavers, which can be useful for some application like, for

example, turbo codes.

We begin by a proper definition of the key quantities

that characterize an interleaver, like its minimum/maximum

delay, its characteristic latency, and its period. Then, interleaver

equivalence and deinterleavers are carefully studied. Interleaver

trellis complexity is analyzed by the introduction of a proper

input/output interleaver code. Connections between interleaver

Paper approved by M. Fossorier, the Editor for Coding and CommunicationTheory of the IEEE Communications Society. Manuscript received August 1,1997; revised July 6, 1999 and April 13, 2000. This work was supported in partby Qualcomm, Inc. and Agenzia Spaziale Italiana. This paper was presentedin part at the Sixth Communication Mini-Conference, Phoenix, AZ, November1997, and at the 2000 International Symposium on Information Theory, Sor-rento, Italy, June 2000.

R. Garello and G. Cancellieri are with the Dipartimento di Elettronica edAutomatica, Universit di Ancona, 60131 Ancona, Italy.

G. Montorsi and S. Benedetto are with the Dipartimento di Elettronica, Po-litecnico di Torino, 10129 Torino, Italy.

Publisher Item Identifier S 0090-6778(01)04090-9.

quantities are then explored, especially those concerning

latency, a key parameter for applications.

Finally, the class of convolutional interleavers is considered,

and their practical implementation discussed. Block inter-

leavers, which are the basis of most turbo code schemes, are

introduced as a special case.

A turbo code is obtained by the parallel concatenation of

simple constituent convolutional codes connected through an

interleaver. Turbo codes, although decoded through an iterative

suboptimum decoding algorithm, yield performance extremely

close to the Shannon limit [3]. In spite of the fact that severalresearch issues are still open, turbo codes are obtaining a large

success and their introduction in many international standards

is in progress. For decoding purposes, the trellises of the con-

stituent convolutional encoders are almost always terminated,

so that the turbo code can be considered as a block code.

Every linear block code can be represented by a minimal

trellis, which, in turn, can be used for soft-decision decoding

with the Viterbi or the BCJR [4] algorithms. For decoding pur-

poses, the complexity of a given trellis is usually expressed in

terms of complexity parameters like the maximum state com-

plexity (logarithm of the maximum number of states), the max-

imum branch complexity (logarithm of the maximum number

of branches), and the average branch symbol complexity (av-erage number of branch symbols per information bit). More-

over, fora given block code, itis well knownthatall com-

plexity measures strongly depend on the ordering of the time

axis, which in turn involves in principle all possible permuta-

tions of its segments.

It is a folk theorem among researchers in the field that

Maximum Likelihood decoding of turbo codes presents a

computational complexity increasing exponentially with the

interleaver length, so that it becomes prohibitively complex

when the turbo code employs a large interleaver, because of the

huge number of states involved. Thus, simpler, suboptimum

iterative decoding strategies have been proposed and suc-

cessfully applied. The previous, qualitative folk theorem maycontain a reasonable amount of truth. The trellis complexity of

turbo codes, however, has never been properly studied. In this

paper, it is analyzed and related to that of the constituent codes

and interleaver. By the introduction of the uniform interleaver,

the evaluation of the average complexity of a turbo code of a

given length can also be performed.

Strong connections exist between the true trellis com-

plexity parameters (minimized by coordinate permutation) and

the free distance of the code , that characterizes the

error probability performance at high signal-to-noise ratios. As

00906778/01$10.00 2001 IEEE

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GARELLO et al.: INTERLEAVER PROPERTIES AND THEIR APPLICATIONS TO THE TRELLIS COMPLEXITY ANALYSIS OF TURBO CODES 795

Fig. 1. Minimal trellises for the codes of (a) Example 1 and (b) Example 2.

Example 2: Given the code of Example 1, the following per-

mutation:

leads to . The gener-

ator matrix

has the LR property. It follows

We have .

The minimal trellis of is depicted in Fig. 1(b).

In general, the problem of finding the best permutation for

complexity minimization is NP-complete [25], [26].

D. Sectionalization

So far, we have considered an atomic representation of ,

i.e., a bit at every time . For clearness, in the fol-

lowing we will denote by , and , the complexity

parameters obtained when a sectionalization occurs such that agroup of bits corresponds to any time . The super-

script will be omitted only when obvious from the context.

As pointed out in [16], trellis sectionalization varies the state

and branchcomplexities,up topathological cases like .

When , the definition of the average branch symbol com-

plexity becomes

is not affected by sectionalization; moreover, it is

strictly tied to the computational complexity of the decoding

algorithms [22]. This suggests that is perhaps the mostappropriate parameter for complexity considerations. Optimal

sectionalization of the code trellis to reduce its complexity has

been studied in [29]. Several authors also allow for sectional-

ization into varying numbers of bits per trellis section, as

in [29].

III. INTERLEAVER THEORY: PAST WORK ON THE SUBJECT

The theory of interleavers was established in the two classical

papers [1] and [2]. In [1] causal interleavers were deeply ana-

lyzed: several definitions and results, equivalent to those pre-

sented in this paper, will be referred in the following. More-

over, many significant results on the spreading properties of in-terleavers (a subject that we have not considered in this paper)

and on their implementation were presented.

Other results were presented in [2] (see also [30]): Forney

invented periodic convolutional interleavers that are employed

in many applications and international standards.

Two considerations are important. First, classical interleaver

theory only concerned causal devices. As a consequence, some

quantities defined in our paper (like, for example, equivalence

classes of interleavers, or the interleaver code) were not strictly

necessary, and not introduced in [1] and [2]. Second, other re-

sults on causal interleavers were not explicitly presented in [1]

and [2], and are poorly documented or scattered elsewhere in

the literature.As a matter of fact, we think that most of the results on causal

interleavers presented in this paper are to be considered as a

re-interpretation of previous ones, even when an explicit refer-

ence is missing. Among the main contribution of this paper are

the general mathematical framework and all results concerning

noncausal interleavers, a subject which was not studied in the

past.

Recently, we partially presented our results in a preliminary

form in [31]. Similar definitions for causal interleavers can be

found in [32] together with some results about the interleaver

spread. Some other results concerning the cycle decomposition

of an interleaver and its implementation are presented in [33].

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796 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 49, NO. 5, MAY 2001

Fig. 2. The delay function and some parameters of the interleaver of Example 3.

IV. INTERLEAVERS: BASIC DEFINITIONS

We begin to revisit interleaver theory by introducing some

basic definitions. They can be considered an extension of thedefinitions introduced in [1] for causal interleavers.

An interleaver is a device characterized by a fixed per-

mutation of the infinite time axis . maps bi-in-

finite input sequences into permuted

output sequences , with

. Although irrelevant for the interleaver proper-

ties, for simplicity we will assume in the following that is the

binary alphabet .

A. Delays and Period

We define the delay function as

The interleaver action can then be described as

We also define the following.

The maximum delay

The minimum delay

The characteristic latency

We say that is periodic with period , if there exists a

positive integer such that for all

i.e.,

In this paper, we will only consider periodic interleavers, be-

cause nonperiodic permutations are not used in practice. The

period is usually referred to, in the turbo code literature,

as the interleaver length or size, and is a crucial parameter in

determining the code performance. Often, it is also directly re-

lated to the latency introduced in the transmission chain; this

is not strictly correct, as we will see soon that the minimum

delay introduced by the interleaverdeinterleaver pair is instead

the characteristic latency . The ambiguity stems from the

common suboptimal implementation of a block interleaver (see

Section VII-B).

Example 3: Consider this interleaver and its action on the

binary sequences , as shown at the bottom of the page.

has period , minimum and maximum delay

, and characteristic latency .

Its delay function is depicted in Fig. 2.

In the following, we will describe an interleaver of period

only by its values in the fundamental interval

. For the interleaver of Example 3 we

have .

B. Interleaver Equivalence

Formal definition of equivalence classes of interleavers, notintroduced in classical theory, will prove useful in our frame-

work.

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GARELLO et al.: INTERLEAVER PROPERTIES AND THEIR APPLICATIONS TO THE TRELLIS COMPLEXITY ANALYSIS OF TURBO CODES 801

Fig. 5. Structure of a convolutional interleaver.

where is a finite basic permutation of length

,a nd is t he th e lement o f a shiftvector of elements

taking values in .

A visual implementation of the convolutional interleaver de-

scribed in (1) is shown in Fig. 5, where the reader can recognize

the similarities with the familiar structure of convolutional in-

terleavers described by Ramsey and Forney in [1] and [2]. The

input stream is permuted according to the basic permutation .The th output of the permutation register is sent to a delay line

with a delay , whose output is read and sent out. Notice

that in the realization of Fig. 5 the permutation is always non-

causal, except for the case of the identity permutation. As a con-

sequence, it is in general not physically realizable. In the next

subsection, we will describe a minimal realization of any peri-

odic causal interleaver.

Example 12: A simple class of interleavers stems from the

choice of the identity basic permutation

As an example, choosing , and

yields the equation shown at the bottom of the

page.

1) Implementation of Convolutional Interleavers with Min-

imal Memory Requirement: We have already proved in Corol-

lary 2 that the minimal memory requirement for the implemen-

tation of a causal interleaver is the constraint length , which

in turn is equal to the average of the delay profile. An important

property that relates the shift vector to the constraint length

of causal interleavers is given in the following lemma.

Lemma 5: The constraint length of a causal interleaver ofperiod is related to the shift vector through

Fig. 6. Minimal memory implementation of an interleaver.

Proof: For a causal interleaver, we proved in Corollary 3

that the average delay is equal to the constraint length, so that

From (1) we then have

In this section we describe an implementation with minimal

memory requirements of the encoder for a generic periodic

causal interleaver. The encoder is realized through a single

memory with size equal to the constraint length . The input

and output bits at a given time are read and stored with a single

Read-Modify-Write operation acting on a single memory cell

whose address is derived as follows (see Fig. 6).

At time we read from a memory cell, and write

into it. As a consequence, the successive contents of

that memory cell are

Thus, the permutation induces a partition

of the set of integers with the following equivalence relation-

ship:

and the constraint length is also the cardinality of this parti-

tion.

With the previous considerations in mind, we can devise the

following algorithm to compute the addresses to be used in

the minimal memory interleaver of Fig. 6.

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806 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 49, NO. 5, MAY 2001

branch symbol complexity decreases from 253 625 branch sym-bols per information bits to 77 625 branch symbols per informa-tion bits.

Example 21: Consider a turbo code with the same con-

stituent convolutional encoders of Example 18, and rectangular

interleavers with different length , but equal column

number . Let us consider ,

and . In Fig. 10 we report the stateprofile of the two turbo codes, evaluated directly and through

the permutation . The result is striking: the two curves

obtained directly show a dependence of the maximum state

complexity on , and, in particular, a value of equal to

18 for and to 130 for . Instead, the two curves

obtained after applying yield a significant complexity

reduction, and, more important, a maximum state complexity

that is independent from , and given by . Using

the algorithm described in [37], we have computed the free

distance of the two turbo codes, and verified that it is also

independent from ; this suggests that a simple increase of

the interleaver length may not be beneficial, and, instead, that

the interleaver in a PCCC construction should be chosen aimingat maximizing the true trellis complexities parameters.

IX. CONCLUSION

In this paper, we have defined the main parameters that char-

acterize an interleaver, by extending classical concepts to non-

causal interleavers, too. After introducing the input/output in-

terleaver code, we have evaluated its complexity, and explored

the connections between the various parameters. We have then

focused on causal interleavers and on block interleavers, that are

both important for applications. These concepts have been ap-

plied to the evaluation of the trellis complexity of turbo codes.

We have also faced the problem of finding a time axis permuta-tion that reduces the complexity of turbo codes.

ACKNOWLEDGMENT

The authors wish to thank the Editor and the reviewers for

their valuable comments that helped to refocus and improve the

original manuscript. They are also grateful to F. Chiaraluce and

G. D. Forney, Jr., for their appropriate comments.

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[37] R. Garello, S. Benedetto, and P. Pierleoni, Computing the free distanceof turbo codes and serially concatenated codes with interleavers: Algo-rithms and applications, J. Select. Areas Commun., to be published.

Roberto Garello (M90) was born in Torino, Italy,on December 18, 1965. He received the Laurea inIngegneria Elettronicadegree (summa cum laude) in1990, and the Ph.D. degree in electronic engineeringin 1994, both from Politecnico di Torino.

In 1993, he spent six months visiting the Signaland Information Processing Laboratory, ETH,Zurich, Switzerland, and the Laboratory for Infor-mation and Decision Systems, MIT, Boston, MA.From October 1994 to August 1997, he was a DesignEngineer at Marconi Communications, Genova,

working on high-rate all-digital modems and signal processing applications.From September 1997 to September 1998, he was with the Dipartimento diElettronica of Politecnico di Torino working on the DSP implementation of aturbo decoder. Since November 1998, he has been an Associate Professor at theDipartimento di Elettronica ed Automatica of Universit di Ancona, Italy. Hismain interests include teaching, coding theory, and spread-spectrum systems.

Guido Montorsi (S93M95) was born in Turin,

Italy, on January 1, 1965. He received the Laureadegree in ingegneria elettronica in 1990 fromPolitecnico di Torino, Turin, Italy, with a mastersthesis concerning the study and design of codingschemes for HDTV, developed at the RAI ResearchCenter, Turin. In 1992, he spent the year as VisitingScholar in the Department of Electrical Engineeringat Rensselaer Polytechnic Institute, Troy, NY. In1994, he received the Ph.D. degree in telecommu-nications from the Dipartimento di Elettronica of

Politecnico di Torino.Since December 1997,he hasbeen an AssistantProfessor at thePolitecnicodi

Torino. His current interests include the area of channel coding, particularly onthe analysis and design of concatenated coding schemes and study of interativedecoding strategies.

Sergio Benedetto (SM90F97) received theLaurea in ingegneria elettronica (summa cum laude)from Politecnico di Torino, Italy, in 1969.

From 1970 to1979,he waswiththe Istituto di Elet-tronica e Telecomunicazioni, first as a Research En-gineer, then as an Associate Professor. In 1980, hewas made a Professor in Radio Communications atthe Universit di Bari. In 1981, he rejoined to Po-litecnico di Torino as a Professor of Data Transmis-

sion Theory in the Dipartimento di Elettronica. From1980 to 1981, he spent nine months at the SystemScience Department of University of California, Los Angeles, as a Visiting Pro-fessor, and three months at the University of Canterbury, New Zealand, as anErskine Fellow. He has co-authored two books in signal theory and probabilityand random variables (in Italian), and the books Digital Transmission Theory(Englewood Cliffs, NJ: Prentice-Hall, 1987) and Optical Fiber Communica-tions Systems (Boston, MA: Artech House, 1996), as well as about 200 papersfor leading engineering journals and conferences. Active in the field of digitaltransmission systems since 1970, his current interests are in the field of opticalfiber communications systems, performance evaluation and simulation of dig-ital communication systems, trellis-coded modulation and concatenated codingschemes.

Dr. Benedetto is currently the Area Editor for Signal Design, Modulation andDetection of the IEEE TRANSACTIONS ON COMMUNICATIONS.

Giovanni Cancellieri was born in Florence, Italy,in 1952. He graduated from the University ofBologna, Bologna, Italy, with a degree in electronicengineering and in physics.

He developed research in telecommunication sys-tems, with particular attention to networks and fibreoptics components. In 1980, he joined the Universityof Ancona, where he is currently Full Professor ofTelecommunications. He is co-author of more than100 papers and nine books. He is a consultant for theItalian Ministry of Research for funding projects of

advanced products and large enterprises.