Pseudo-random Puncturing: A Technique to Lower the Error Floor of Turbo
  Codes
Pseudo-random Puncturing: A Technique to Lower
the Error Floor of Turbo Codes
Ioannis Chatzigeorgiou, Miguel R. D. Rodrigues, Ian J. Wassell
Digital Technology Group, Computer Laboratory
University of Cambridge, United Kingdom
Email: {ic231, mrdr3, ijw24}@cam.ac.uk
Rolando Carrasco
School of EE&C Engineering
University of Newcastle, United Kingdom
Email: r.carrasco@ncl.ac.uk
Abstract— It has been observed that particular rate-1/2
partially systematic parallel concatenated convolutional codes
(PCCCs) can achieve a lower error floor than that of their
rate-1/3 parent codes. Nevertheless, good puncturing patterns
can only be identified by means of an exhaustive search,
whilst convergence towards low bit error probabilities can be
problematic when the systematic output of a rate-1/2 partially
systematic PCCC is heavily punctured. In this paper, we present
and study a family of rate-1/2 partially systematic PCCCs, which
we call pseudo-randomly punctured codes. We evaluate their bit
error rate performance and we show that they always yield
a lower error floor than that of their rate-1/3 parent codes.
Furthermore, we compare analytic results to simulations and
we demonstrate that their performance converges towards the
error floor region, owning to the moderate puncturing of their
systematic output. Consequently, we propose pseudo-random
puncturing as a means of improving the bandwidth efficiency
of a PCCC and simultaneously lowering its error floor.
I. INTRODUCTION
Although in certain applications, such as satellite
communications, link reliability is of essence and low
rate codes are used to support it, bandwidth occupancy is
more important in wireless communications and hence high
rate codes are preferred. A high rate convolutional code can
be obtained by periodic elimination, known as puncturing,
of particular codeword bits from the output of a parent low
rate convolutional encoder. Extensive analyses on punctured
convolutional codes have shown that their performance is
always inferior to the performance of their low rate parent
codes (e.g. see [1], [2]).
The performance of punctured parallel concatenated
convolutional codes (PCCCs), also known as punctured turbo
codes, has also been investigated. Design considerations have
been derived by analytical [3]–[5] as well as simulation-based
approaches [6]–[8], while upper bounds on the bit error
probability (BEP) were evaluated in [5], [9]. Punctured turbo
codes are usually classified as systematic, partially systematic
or non-systematic depending on whether all, some or none of
their systematic bits are transmitted [7]. Recent papers [7]–[9]
have demonstrated that partially systematic PCCCs yield lower
error floors than systematic PCCCs of the same rate.
In [10] we showed that rate-1/2 non-systematic PCCCs
can achieve error floors, which are lower even than those
of their rate-1/3 parent PCCCs. This interesting outcome is
valid when maximum-likelihood (ML) decoding is employed.
When suboptimal iterative decoding is used, the absence of
received systematic bits causes erroneous decisions, which
prohibit the iterative decoder from converging to the error
floor. Nevertheless, we demonstrated that rate-1/2 child codes,
whose BEP performance converges towards an error floor
which is lower than that of their rate-1/3 parent PCCC, can
still be found by means of an exhaustive search. During
this process, the union bound on the BEP of each rate-1/2
punctured PCCC is computed and compared to the union
bound of the rate-1/3 parent PCCC. Note that the union bound
coincides with the error floor of the code for high values of
Eb/N0 [11]. Punctured PCCCs that achieve a bound lower
than that of their rate-1/3 parent PCCC are selected.
Computation of the exact union bound on the BEP of
a punctured PCCC becomes intensive as the interleaver
size increases. In [12] we presented a simple technique
to approximate the union bound of a turbo code and we
demonstrated that this approximation is very accurate when
a large interleaver size is used. We used our technique to
identify a family of rate-1/2 partially systematic PCCCs, which
we called pseudo-randomly punctured PCCCs (PRP-PCCCs).
Although we did not explore their BEP performance in detail,
we observed that particular PRP-PCCC configurations could
achieve a lower error floor than that of their parent codes.
This paper builds upon the work carried out in [10] and [12].
Initially, we provide analytical expressions for the parameters
that influence the bit error performance of PCCCs. We then
evaluate those parameters and compute the union bound
approximations for both rate-1/3 parent PCCCs and rate-1/2
PRP-PCCCs. We demonstrate that the latter always exhibit
a lower error floor than the former, when large interleaver
sizes are considered. In order to verify our theoretical analysis,
we compare analytic results to simulations for specific PCCC
configurations. The paper concludes with a summary of the
main contributions.
II. PERFORMANCE EVALUATION OF PCCCS
Turbo codes, in the form of symmetric rate-1/3 PCCCs,
consist of two identical rate-1/2 recursive systematic
convolutional encoders separated by an interleaver of size N
[13]. The information bits are input to the first constituent
convolutional encoder, while an interleaved version of the
information bits are input to the second convolutional encoder.
http://arxiv.org/abs/0704.0361v1
The output of the turbo encoder consists of the systematic bits
of the first encoder, which are identical to the information bits,
the parity check bits of the first encoder and the parity check
bits of the second encoder.
The bit error probability Pb of a PCCC employing ML
soft decoding, on an additive white Gaussian noise (AWGN)
channel, is upper bounded as follows
Pb ≤ P
b (1)
where the union bound P u
is defined as
P ub =
P (w). (2)
Here, the sum runs over all possible values of input
information weight w, with P (w) being the contribution to
the union bound P u
of only those codeword sequences which
were generated by input sequences of a specific information
weight w. An individual contribution P (w) is given by [11],
P (w) =
Bw,dQ
2R · Eb
, (3)
where N is the interleaver size, R is the code rate of the turbo
encoder and Bw,d denotes the number of codeword sequences
having overall output weight d, which were generated by input
information sequences of weight w.
In [11] it was shown that the union bound on the BEP of
a PCCC using a uniform interleaver of size N coincides with
the average of the union bounds obtainable from the whole
class of deterministic interleavers of size N . For small values
of N , the union bound can be very loose compared with the
actual performance of turbo codes using specific deterministic
interleavers. However, for N≥1000, it has been observed that
randomly generated interleavers generally perform better than
deterministic interleaver designs [15]. Consequently, the union
bound provides a good indication of the actual bit error rate
performance of a PCCC operating in the error floor region,
when long interleavers are considered.
Derivation of all coefficients Bw,d becomes a
computationally intensive process as the interleaver size
increases, especially when punctured PCCCs are considered
[12]. However, the union bound can be approximated as
follows
P ub ≈ P (w=2), (4)
when long interleavers are used. This approximation is based
on a number of observations:
1) Codeword sequences, which were generated by input
sequences having the minimum possible information
weight, become the main contributors to the bit error rate
performance, as the size N of the interleaver increases
[12], [16].
2) Owning to the structure of the constituent encoders, the
minimum information weight of an input sequence is
always equal to two [16].
Therefore, P (w=2) is the dominant contribution to the union
bound over a broad range of bit error probabilities [12], [16]
and can be used to predict the error floor of turbo codes.
Throughout this paper, we use the union bound
approximation as the basis for the analytic performance
comparison of turbo codes. In particular, if P and P ′ are two
PCCCs using long interleavers of identical size, we say that
P yields a lower error floor than that of P ′ when their bound
approximations, PP(2) and PP
(2) respectively, satisfy
PP(2) < PP
(2). (5)
The above condition can be expanded using (3) as follows
2RP Eb
A. (6)
It was demonstrated in [16] that the free effective distance, df,
which conveys the minimum weight of a codeword sequence
for a weight-2 input information sequence, has a major impact
on the performance of a turbo code. Consequently, if dPf
and dP
f denote the free effective distances of P and P
respectively, condition (6) collapses to
2RP Eb
A, (7)
which only considers the first non-zero, that is the most
significant, term of each sum.
Function Q(ξ) is a monotonically decreasing function of
ξ, where ξ is a real number. Therefore, if ξ1 and ξ2 are real
numbers, with ξ1 > ξ2, we deduce that Q(ξ1) < Q(ξ2), and
vice versa, i.e,
Q(ξ1) < Q(ξ2) ⇔ ξ1 > ξ2. (8)
Consequently, inequality (7) reduces to
RPdPf > R
f , (9)
BP2,df ≤ B
. (10)
When the code rates are equal, the free effective distance of
turbo codes plays a role similar to that of the free distance
of convolutional codes, since the performance criterion (9) is
simplified to
dPf > d
f . (11)
Expressions (9) and (10) will be the basis for the comparison
of the BEP performance in the error floor region of two
PCCCs.
III. DETERMINATION OF PARAMETERS THAT INFLUENCE
THE PERFORMANCE OF TURBO CODES
We will now determine the various parameters that affect
performance for two classes of turbo codes: conventional
rate-1/3 PCCCs and pseudo-randomly punctured rate-1/2
PCCCs. The turbo codes considered throughout this paper are
symmetric, i.e., the two constituent encoders are identical.
A. Rate-1/3 PCCCs
Criteria (9) and (10) require knowledge of the free effective
distance df and the coefficient B2,df of each PCCC. In the
remainder of the paper, we use the abbreviation “Par” to denote
a rate-1/3 parent PCCC. Its free effective distance dParf can
be expressed as the sum of the minimum weight dmin of the
codeword sequence generated by the first constituent encoder,
and the minimum weight zmin of the parity check sequence
generated by the second constituent encoder, when a sequence
of information weight w=2 in input to the PCCC
dParf = dmin + zmin. (12)
Taking into account that the turbo codes are symmetric and
the weight umin of the systematic output sequence is always 2
since w=2, we can write
dParf = (umin + zmin) + zmin = 2 + 2zmin. (13)
The number BPar
of codeword sequences, generated by
a turbo encoder using a uniform interleaver of size N , can
be associated with the number B2,dmin of codeword sequences
having weight dmin, generated by the first constituent encoder,
and the number B2,zmin of parity check sequences having
weight zmin, generated by the second constituent encoder, if we
elaborate on the expressions described in [11]. In particular,
we obtain
B2,dmin · B2,zmin
) , (14)
where B2,dmin and B2,zmin return the same value, since they
both consider the same trellis paths. Note that the first index
in the above notations refers to the input information weight,
which is two.
It was shown in [16] that good rate-1/3 PCCCs are obtained
when their feedback generator polynomial GR is chosen to be
primitive, whilst their feedforward generator polynomial GF
is different than GR. The period L of a primitive polynomial
is given by [17]
L = 2ν − 1, (15)
where ν is the order of the polynomial, or equivalently, the
memory size of each constituent code.
We demonstrated in [12] that when a primitive feedback
generator polynomial is used, the minimum weight zmin and
the coefficient B2,2,zmin can be expressed as
zmin = 2
ν−1 + 2,
B2,dmin = B2,zmin = N − L,
respectively. Consequently, expression (13) assumes the form
dParf = 6 + 2
ν , (17)
whilst, if we combine (14) and (16), the coefficient BPar
be expressed as a function of the intlerleaver size N and the
period L, as follows
2(N − L)2
N(N − 1)
. (18)
In the special case when the size N of the interleaver is an
integer multiple of the period L of the feedback generator
polynomial, i.e., N=µL, we can rewrite (18) as
2L(µ− 1)2
µ(µL− 1)
. (19)
B. Rate-1/2 Pseudo-randomly Punctured PCCCs
A high rate PCCC can be obtained by periodic elimination
of specific codeword bits from the output of a rate-1/3 parent
PCCC. A puncturing pattern P can be represented by a 3×M
matrix as follows:
p1,1 p1,2 . . . p1,M
p2,1 p2,2 . . . p2,M
p3,1 p3,2 . . . p3,M
 , (20)
where M is the puncturing period and pi,m ∈ {0, 1}, with
i=1, 2, 3 and m=1, . . . ,M . For pi,m=0 the corresponding
output bit is punctured, otherwise it is transmitted. The first
and second rows of the pattern are used to puncture the
systematic and parity check outputs, respectively, of the first
constituent encoder. The third row determines which parity
check bits from the output of the second constituent encoder
will be punctured.
Pseudo-random puncturing has been described in [12], in
detail. It is applied to rate-1/3 PCCCs, which use primitive
feedback generator polynomials, hence the polynomial period
L is also given by (15). The puncturing pattern can
be constructed once the parity check sequence y =
(y0, y1, . . . , yL) for an input sequence x = (1, 0, . . . , 0) of
length L+1, has been obtained at the output of the first
constituent encoder. As long as a trail of zeros follows the
first non-zero input bit, the component encoder behaves like a
pseudo-random generator, hence the parity check bits from y1
to yL form a pseudo-random sequence. We set the elements
of the second row of the puncturing pattern to be equal to
the bits of this pseudo-random sequence, but circularly shifted
rightwards by one, i.e., p2,m+1 = ym for m=1, . . . , L. Note
that in pseudo-random puncturing, the puncturing period M is
equal to the period L of the feedback polynomial, i.e., M=L.
The first row of the pattern is set to be the complement of the
second row, thus p1,m=1− p2,m. In order to achieve a code
rate of 1/2, we do not puncture the parity check output of the
second constituent encoder, hence all the elements of the third
row are set to one, i.e., p3,m=1.
As an example, let us consider a rate-1/3 PCCC with
generator polynomials (GF , GR)=(5, 7)8 in octal form. The
memory size of each constituent encoder is ν = 2, thus the
period of GR is found to be L = 2
2−1 = 3. Consequently,
we set the input sequence to (1, 0, 0, 0) and we obtain the
parity check sequence (1, 1, 1, 0) at the output of the first
constituent encoder. The block of the last L=3 parity check
bits, i.e., (1, 1, 0), forms a pseudo-random sequence. If we
circularly shift the bits of this pseudo-random sequence to the
right by one and map them to the elements of the second row
of the puncturing pattern, we obtain [0 1 1]. Eventually the
puncturing pattern, based on which the rate-1/2 PRP-PCCC is
generated from the rate-1/3 parent PCCC, assumes the form
1 0 0
0 1 1
1 1 1
 . (21)
We emphasize that the puncturing pattern depends on the
generator polynomials of the rate-1/3 parent PCCC, hence
different polynomials yield different puncturing patterns.
Furthermore, a rate-1/2 PRP-PCCC can be obtained only if the
parent PCCC uses primitive feedback generator polynomials.
We have previously determined [12] the minimum weight
d′min of the codeword sequence generated by the first
constituent encoder, when a sequence of information weight
w = 2 in input to the rate-1/2 PRP-PCCC. In particular, we
found that
d′min = 2
ν−2 + 2. (22)
The parity check sequence generated by the second constituent
encoder is not punctured, thus its minimum weight is also
given by (16). Therefore, we can compute the free effective
distance dPRPf of a rate-1/2 PRP-PCCC as follows
dPRPf = d
min + zmin
= (2ν−2 + 2) + 2ν−1 + 2
= 4 + 3(2ν−2).
Every time a particular column m of the puncturing
pattern is active during the N time steps of the coding
process, codeword sequences having minimum weight d′min
are generated. Their exact number, Am, can be computed
using the expressions in [12]. In particular, we find that for
M =L the number of minimum-weight codeword sequences
Am, generated when column m is active, is given by
⌊N/M⌋ − 1, if (N mod M)<m
⌊N/M⌋ , otherwise,
where (ξ1 mod ξ2) denotes the remainder of division of ξ1
by ξ2, and ⌊ξ⌋ denotes the integer part of ξ. In order to
facilitate our analysis, we assume that the interleaver size N is
an integer multiple of the puncturing period M , i.e., N=µM ,
where µ is a positive integer. Hence, (24) collapses to
Am = µ− 1, (25)
since (N mod M) is always zero and m>0.
It has been demonstrated in [12] that minimum-weight
codeword sequences can be obtained only when the active
column m is in the range 2 ≤ m ≤ M ; every time one
of these M−1 columns of the puncturing pattern is active,
Am minimum-weight codeword sequences are generated.
Consequently, the total number of codeword sequences having
weight d′min assumes the value
B2,d′
= (M − 1)Am, (26)
or, equivalently
B2,d′min = (L− 1)(µ− 1), (27)
where M has been replaced by L, since they are equal
quantities and they can be used interchangeably.
Similarly to the second constituent encoder of the rate-1/3
parent PCCC, the second constituent encoder of the rate-1/2
PRP-PCCC also generates a total of B2,zmin sequences having
weight zmin, since its parity check output is not punctured.
Consequently, the coefficient BPRP
or a rate-1/2 PRP-PCCC
can be expressed as
BPRP2,df =
B2,d′min · B2,zmin
[(L− 1)(µ− 1)] · (N − L)
2(L− 1)(µ− 1)2
µ(µL− 1)
invoking (14), which can be used when PCCCs employing
uniform interleavers of size N are considered.
IV. PERFORMANCE COMPARISON OF ANALYTIC TO
SIMULATION RESULTS
Having evaluated the parameters that influence the
performance of the PCCCs under investigation, we are now in
the position to explore whether a rate-1/2 PRP-PCCC exhibits
a lower bound approximation than that of its rate-1/3 parent
PCCC. We observe that dPRPf can be expressed in terms of
dParf , if we subtract (17) from (23)
dPRPf = d
f − (2 + 2
ν−2). (29)
Coefficient BPRP
can also be represented in terms of BPar
we divide (28) by (19)
BPRP2,df =
BPar2,df . (30)
According to (9) and (10), if both conditions
dPRPf >
dParf (31)
< BPar
are satisfied, a rate-1/2 PRP-PCCC yields a lower bound
approximation than that of its rate-1/3 parent code. We deduce
from (30) that BPRP
is always less than BPar
, thus the second
condition holds true. The first condition assumes the following
form, if we substitute dPRPf with its equivalent, based on (29),
dParf > 6 + 3(2
ν−2). (33)
Nevertheless, we have shown in (17) that the free effective
distance of the parent PCCC is given by dParf =6+ 2
ν , which
can be rewritten as dParf = 6 + 4(2
ν−2). Therefore, dParf is
always greater than 6 + 3(2ν−2), and hence, both conditions
are satisfied.
The outcome of this investigation reveals that rate-1/2
PRP-PCCCs using long interleavers are always expected to
0 1 2 3 4 5 6
 (dB)
Simulation, Rate−1/3 PCCC (ν=2)
Bound Approx., Rate−1/3 PCCC (ν=2)
Simulation, Rate−1/2 PRP−PCCC (ν=2)
Bound Approx., Rate−1/2 PRP−PCCC (ν=2)
Simulation, Rate−1/3 PCCC (ν=3)
Bound Approx., Rate−1/3 PCCC (ν=3)
Simulation, Rate−1/2 PRP−PCCC (ν=3)
Bound Approx., Rate−1/2 PRP−PCCC (ν=3)
Fig. 1. Comparison of bound approximations to simulation results. The
exact log-MAP algorithm is applied over 8 iterations and an interleaver size
of 1, 000 bits is used.
yield a lower bound approximation, or equivalently a lower
error floor, than that of their rate-1/3 parent codes.
Fig.1 compares bound approximations to simulation results
for rate-1/3 parent PCCCs and rate-1/2 PRP-PCCCs of
memory size ν =2 and ν =3, over the AWGN channel. For
ν=2, the generator polynomials of the PCCCs are taken to be
(GF , GR)=(5, 7)8, whilst for ν=3, the PCCCs are described
by (GF , GR) = (17, 15)8. The component decoders employ
the conventional exact log-MAP algorithm [18]. A moderate
interleaver size of 1, 000 bits has been chosen, so as to allow
the bit error rate performance of the PCCCs to approach the
corresponding bound approximations at BEPs in the region of
10−6 to 10−7.
As expected, Fig.1 confirms that for high values of Eb/N0,
the BEP of each rate-1/2 PRP-PCCC is indeed lower than
that of the corresponding rate-1/3 parent code, whilst after 8
iterations the performance curves of all turbo codes approach
the respective bound approximation curves.
V. CONCLUSION
In previous work [9], [10], [12] we introduced techniques
to evaluate the performance of punctured PCCCs and we
observed that, in some cases, the error floor could be lowered
by reducing the rate of a PCCC from 1/3 to 1/2. Nevertheless,
good puncturing patterns were identified by means of an
exhaustive search, whilst convergence towards low bit error
probabilities of those rate-1/2 PCCCs whose systematic output
was heavily punctured, had to be investigated.
In this paper, we established that rate-1/2 pseudo-randomly
punctured PCCCs, which form a subset of rate-1/2 partially
systematic PCCCs, not only approach the error floor region
for an increasing number of iterations but always yield a
lower error floor than that of their rate-1/3 parent codes.
Consequently, pseudo-random puncturing can be used to
reduce the rate of a PCCC from 1/3 to 1/2 and at the same
time achieve a coding gain at low bit error probabilities.
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	Introduction
	Performance Evaluation of PCCCs
	Determination of Parameters that Influence the Performance of Turbo Codes
	Rate-1/3 PCCCs
	Rate-1/2 Pseudo-randomly Punctured PCCCs
	Performance Comparison of Analytic to Simulation Results
	Conclusion
	References
