Signal Processing - March 2017 - 102
A shifted-repetition code is a rate
channel introduces a burst of length
In contrast to RLC codes,
B = 20 in the interval t ! [i, i + 19]. The
R
=
1/2 code, where each source packet is
the parity-check packets
infinite-memory code will force the
repeated once after a delay of T packets-
in a shifted-repetition
decoder to use the next 20 parity checks
i.e., we can express x [i] = (s [i], s [i - T ]) .
code do not involve a
in the interval [i + 20, i + 39] to recover
In contrast to RLC codes, the parity-check
linear combination of the
packets in a shifted-repetition code do
the erased source sequence. Any additionsource packets.
not involve a linear combination of the
al losses in this period will cause longer
source packets. We replace (4) with simply
delays. The code with memory m = 5
will behave very differently. It will skip parity checks in the
p [i] = s [i - 5]. We note the following properties:
interval [i + 20, i + 24], which are the only received parities
1) Single isolated loss: When there is a single isolated loss,
the corresponding source packet can be recovered with a
that depend on the burst interval. Thereafter, any parity
delay of T = 5 packets. For example, if x [0] is lost, then
checks can be used to recover from any future losses. Thus,
due to delay constraints, the code with memory m = 5 is
the source packet s [0] is recovered when its repeated copy
more desirable in the event of such burst losses.
at time T = 5 is received.
It should be noted that the construction in (4) applies to
2) Two isolated losses: In general, the shifted-repetition code
any arbitrary rate R. There is nothing special about R = 1/2,
cannot recover from two or more isolated losses. As an
example, if the erasures happen at time t = 0 and t = 5 ,
except the simple construction (5). The following result shows
the burst- and isolated-error correction properties of RLC
then the source packet s [0] cannot be recovered. Thus, the
codes [13] for an arbitrary rate R.
delay for this case is 3.
3) Burst-erasure channel: The shifted-repetition code can
correct a burst of length B = 5 with a delay of T = 5.
Theorem 1 (error-correction properties of RLC codes
Suppose
that the erasure burst spans the interval [0, 4] .
at a given maximum delay)
Then
(
n
,
k
,
m
)
R
=
(
k
/
n
)
and
s
[
0
] is recovered at time t = 5 from p [5] = s [0] .
Consider an
RLC code with rate
memory m $ T. Such a code can recover from a burst-erasure
Likewise, each s [ j] for j = 0, f, 4 is recovered at time
channel with a maximum burst length B, or from an isolatedt = j + 5 in a sequential manner.
It is clear that a shifted-repetition code with delay T
erasure channel with a maximum of N erasures, with a
maximum delay of T, provided that
will recover any burst of length B # T . This is clearly the
maximum burst length that can be recovered by any code.
However, the rate of the code is fixed at R = 1/2. MaxiB # (1 - R) (T + 1),
(6)
mally short (MS) codes [24], [25] are a generalization
N # (1 - R) (T + 1) .
(7)
of the shifted-repetition code that achieve optimal burst
correction. For a given rate R and delay T, they achieve
RLC codes have the same threshold for burst-error and isoB = min ^1, (1 - R) /R h T. We review a variation of the origilated-error correction. To explain this, recall that RLC
nal construction in the "General Code Constructions" seccodes perform simultaneous recovery of the source packets
tion. It should be noted that the value of B is larger than that
in the event of an erasure burst. They treat each parity check
of RLC codes in Theorem 1. Unfortunately, like the shiftedas providing an equation involving the source symbols and
repetition codes, these codes are sensitive to the isolatedrecover all the erased symbols simultaneously when suffierasure channel with N $ 2 . We will see that this can lead
ciently many parity checks are received. This is illustrated
to a significant degradation over statistical channels, such
in Figure 6(b). They are not able to recover in an
as the Gilbert-Elliott channel. Nevertheless, the MS codes
opportunistic fashion earlier source packets whose deadconstitute an important building block for the more robust
lines occur earlier. In the following four sections, we discodes discussed in the sequel.
cuss the class of streaming codes that can achieve such a
sequential recovery and thus provide improved performance
Shifted-RLC codes
over burst-erasure channels.
Shifted-RLC codes combine concepts of shifted-repetition
code with RLC code. They achieve a longer burst-error correction threshold than RLC codes in Theorem 1, but smaller
Shifted-repetition code
than the shifted-repetition codes. However, unlike the shiftR
=
1
/
2
,
A repetition code is a simple construction with rate
ed-repetition codes, they can correct from more than one
where each source packet is repeated with a unit delay-i.e.,
isolated loss. As an example, consider the rate-1/2 code
x [i] = (s [i], s [i - 1]) for all i $ 1. While simple in implementation, such a construction cannot recover from burst lossx [i] = (s [i], p [i]), where we select
es of length B $ 2. Interestingly, a simple variation of this
construction achieves optimal recovery over the burst-erasure
p [i] = s [i - 5] + s [i - 4] .
channel. Some generalizations of repetition codes, where lowbit-rate redundant audio packets are used as parities, are
This code is similar to the (n = 2, k = 1, m = 2) RLC code,
studied in [23].
but the parity-check packets are further delayed by T = 3
102
IEEE Signal Processing Magazine
|
March 2017
|
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