Signal Processing - March 2017 - 97
source packets, while also maintaining the
The communication channel considered is a
The dynamics of
same overhead as in sequence 1. However,
packet erasure channel. Each transmitted
packet-loss patterns,
the delay incurred with this code is considpacket is either erased or perfectly received
and not just the average
erably higher than the previous case. Thus,
at the destination. This is motivated by the
fraction of losses, must
the dynamics of packet-loss patterns, and
fact that erroneous packets are discarded at
be considered in
not just the average fraction of losses, must
lower layers in the communication protocol
be considered in streaming applications.
stack. In particular, the channel output at
streaming applications.
We discuss coding techniques that can
(discrete) time t is given by y [t] =* if the
repair burst losses with a much shorter delay than RS codes.
channel introduces an erasure at time t, and by y [t] = x [t] if it
We also show that codes that are optimal for burst losses in
does not. Throughout this article, we will use the term channel
terms of minimizing the delay are rather sensitive to other
to denote the packet-loss sequence, as is the convention in the
loss patterns. In practice, communication links introduce both
coding theory literature. To develop insights into the perfortypes of erasure patterns illustrated in Figure 1. Thus, we dismance of different coding schemes, we focus on a simple class
cuss coding schemes that enable fast recovery from burst lossof channels defined below.
es and are also robust to isolated losses [13]-[15].
Definition 1 (burst-erasure channel)
Case study: Why traditional FEC is not enough
A burst-erasure channel with parameter B is a channel that
introduces a single contiguous sequence of erasures of maximum length B. That is, starting from some arbitrary time j $ 0
and 0 # Bl # B , we have that y [t] = * for t ! [ j, j + Bl - 1]
and that y [t] = x [t] otherwise.
In this section, we study the performance of various errorcorrecting codes in a streaming setup via an example. To provide a common point of comparison, we focus on the streaming
setup shown in Figure 2. In this model, a source packet s [t]
for t = 0, 1, 2, f arrives at the FEC encoder every t s seconds-i.e., s [t] arrives at time t · t s seconds. For simplicity, we
will assume that each source packet is of the same size and
consists of k symbols. The encoder generates a channel packet
x [t] of size n symbols and transmits it in the interval
[t · t s, (t + 1)· t s) . The encoding function is causal
Sequence 1:
Sequence 2:
FIGURE 1. Two examples of erasure sequences that have the same
x [t] = ft ^s [t - m], f, s [t]h, t $ 0,
number of erasures but different erasure patterns. The shaded boxes
denote the erasures, while the white boxes denote packet reception. In
sequence 1, the erasures are mostly isolated, while in sequence 2, they
occur in a single burst. One can use a short (3, 2) RS code to recover for
sequence 1, but a longer (15, 10) RS code is required over sequence 2,
resulting in a higher delay.
(1)
where ft (·) is the encoding function at time t and m denotes
the memory of the encoder. Furthermore, the rate of the code
is given by R = k/n , and its redundancy is 100 (n - k) /k% .
ts
T · ts
(T+1)ts
(T+2)ts
(T+3)ts
(T+4)ts
s[1]
s[2]
...
...
s[T ]
s[T +1]
s[T +2]
s[T +3]
s[T +4]
s[0]
s[1]
s[2]
...
...
s[T ]
s[T +1]
s[T +2]
s[T +3]
s[T +4]
p[0]
p[1]
p[2]
...
...
p[T ]
p[T +1]
p[T +2]
p[T +3]
p[T +4]
Propagation
Delay (tp)
s[0]
s[1]
s[2]
...
...
s[T ]
s[T +1]
s[T +2]
p[0]
p[1]
p[2]
...
...
p[T ]
p[T +1]
p[T +2]
"
2ts
"
0
Time
(seconds)
Symbols
s[0]
k
Encoder
n
Channel
n
Decoder
Decoding Delay
"
k
s[0]
s[1]
s[2]
FIGURE 2. The source stream s 6 t @ for t $ 0 is encoded to a channel stream x 6t @, which is transmitted over an erasure channel. The decoder tolerates a
maximum decoding delay of T packets.
IEEE Signal Processing Magazine
|
March 2017
|
97
Table of Contents for the Digital Edition of Signal Processing - March 2017
Signal Processing - March 2017 - Cover1
Signal Processing - March 2017 - Cover2
Signal Processing - March 2017 - 1
Signal Processing - March 2017 - 2
Signal Processing - March 2017 - 3
Signal Processing - March 2017 - 4
Signal Processing - March 2017 - 5
Signal Processing - March 2017 - 6
Signal Processing - March 2017 - 7
Signal Processing - March 2017 - 8
Signal Processing - March 2017 - 9
Signal Processing - March 2017 - 10
Signal Processing - March 2017 - 11
Signal Processing - March 2017 - 12
Signal Processing - March 2017 - 13
Signal Processing - March 2017 - 14
Signal Processing - March 2017 - 15
Signal Processing - March 2017 - 16
Signal Processing - March 2017 - 17
Signal Processing - March 2017 - 18
Signal Processing - March 2017 - 19
Signal Processing - March 2017 - 20
Signal Processing - March 2017 - 21
Signal Processing - March 2017 - 22
Signal Processing - March 2017 - 23
Signal Processing - March 2017 - 24
Signal Processing - March 2017 - 25
Signal Processing - March 2017 - 26
Signal Processing - March 2017 - 27
Signal Processing - March 2017 - 28
Signal Processing - March 2017 - 29
Signal Processing - March 2017 - 30
Signal Processing - March 2017 - 31
Signal Processing - March 2017 - 32
Signal Processing - March 2017 - 33
Signal Processing - March 2017 - 34
Signal Processing - March 2017 - 35
Signal Processing - March 2017 - 36
Signal Processing - March 2017 - 37
Signal Processing - March 2017 - 38
Signal Processing - March 2017 - 39
Signal Processing - March 2017 - 40
Signal Processing - March 2017 - 41
Signal Processing - March 2017 - 42
Signal Processing - March 2017 - 43
Signal Processing - March 2017 - 44
Signal Processing - March 2017 - 45
Signal Processing - March 2017 - 46
Signal Processing - March 2017 - 47
Signal Processing - March 2017 - 48
Signal Processing - March 2017 - 49
Signal Processing - March 2017 - 50
Signal Processing - March 2017 - 51
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Signal Processing - March 2017 - 53
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Signal Processing - March 2017 - 55
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Signal Processing - March 2017 - 57
Signal Processing - March 2017 - 58
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Signal Processing - March 2017 - 60
Signal Processing - March 2017 - 61
Signal Processing - March 2017 - 62
Signal Processing - March 2017 - 63
Signal Processing - March 2017 - 64
Signal Processing - March 2017 - 65
Signal Processing - March 2017 - 66
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Signal Processing - March 2017 - 68
Signal Processing - March 2017 - 69
Signal Processing - March 2017 - 70
Signal Processing - March 2017 - 71
Signal Processing - March 2017 - 72
Signal Processing - March 2017 - 73
Signal Processing - March 2017 - 74
Signal Processing - March 2017 - 75
Signal Processing - March 2017 - 76
Signal Processing - March 2017 - 77
Signal Processing - March 2017 - 78
Signal Processing - March 2017 - 79
Signal Processing - March 2017 - 80
Signal Processing - March 2017 - 81
Signal Processing - March 2017 - 82
Signal Processing - March 2017 - 83
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Signal Processing - March 2017 - 85
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Signal Processing - March 2017 - 89
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Signal Processing - March 2017 - 92
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Signal Processing - March 2017 - 96
Signal Processing - March 2017 - 97
Signal Processing - March 2017 - 98
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Signal Processing - March 2017 - 100
Signal Processing - March 2017 - 101
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Signal Processing - March 2017 - 103
Signal Processing - March 2017 - 104
Signal Processing - March 2017 - 105
Signal Processing - March 2017 - 106
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Signal Processing - March 2017 - 118
Signal Processing - March 2017 - 119
Signal Processing - March 2017 - 120
Signal Processing - March 2017 - 121
Signal Processing - March 2017 - 122
Signal Processing - March 2017 - 123
Signal Processing - March 2017 - 124
Signal Processing - March 2017 - Cover3
Signal Processing - March 2017 - Cover4
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