Signal Processing - March 2017 - 106

Encoder

2) Sequential recovery: Upon recovering v [0], f, v [B - 1]
at time T - 1, the decoder can compute p v [T ], subtract it
from p [T ] = p v [T ] + u [0], and in turn recover u [0] at
time T. Similarly, the decoder can use p [T + 1], f,
p [T + B - 1] to sequentially recover u [1], f, u [B - 1]
with a delay of T packets.
Hence, s [i] = (u [i], v [i]) for i ! {0, f, B - 1} are recovered
at time i + T.
We summarize the error-correction property of the MS
code in the following section [24], [25].

1) Source splitting: Split each source packet into two subpackets u [i] and v [i] of sizes K u and K v, respectively,
where K u + K v = K, i.e.,
s [i] = (u [i], v [i]) .
2) RLC code: Apply a rate K v / (K v + K u) RLC code to the
v [·] stream of subpackets to generate parity-check packets
p v [·] of size K u .
3) Repetition code: Apply a shifted-repetition code to the u [·]
subpackets.
4) Parity combination: Combine the p v [·] parity-check packets with the repeated u [·] subpackets after shifting the latter
by T time slots to generate the overall parity-check packets,

Theorem 2 (error-correction properties of MS codes at a
given maximum delay)
Given a rate R and delay T, the MS code can recover from a
burst-erasure channel of maximum length B or an isolatederasure channel with N erasures provided that

p [i] = p v [i] + u [i - T ].
5) Channel packet: Generate the channel packets by appending the overall parity checks to the source packets, i.e.,
x [i] = (s [i], p [i]) is the packet transmitted at time i and is
of size N = K + K u .

(9)

As shown in Figure 10, the MS code splits the source packet
into two groups, i.e., s [i] = (u [i], v [i]) . It applies a shiftedrepetition code to u [i] and an RLC code to v [i] to generate
the parity-check packet p v [i] + u [i - T ]. The main weakness
of this construction is the shifted-repetition code applied to
the u [·] packets. When there are two isolated losses, at time
t = 0 and t = T, the MS code fails to recover the subpacket
u [0]. We discuss two ways in which these codes can be made
robust to correct from isolated losses.

Decoder
Consider a channel that introduces an erasure burst of length
B in the interval [0, B - 1] , as shown in Figure 10. The
decoder proceeds in two steps.
1) Simultaneous recovery: The decoder subtracts the unerased
u [B - T ], f, u [- 1] subpackets from the corresponding
parities p [B], f, p [T - 1] to recover the parity-check
packets p v [B], f, p v [T - 1] . These T - B parities, each
consisting of K u symbols, suffice to recover the B erased
v [·] symbols, since B · K v = (T - B) · K u holds.

(B-1)ts

N # 1.

Robust extensions of MS codes

In the above construction, we may select any value of K u and
K v such that their ratio is K u /K v = B/ (T - B) . The overall rate
is given by R = ^K u + K vh / ^2K u + K vh = T/ ^T + B h . We next
explain how the code can recover from a burst of length B
with a delay of T.

MIDAS codes
The main idea in the MIDAS construction is to apply
an additional RLC code of rate K u / (K u + K r) to the u [i]
subpackets. This generates a new set of parity-check
packets p u [i] consisting of K r symbols. These are then

Tts

(T+1)ts

Ku

(T-1)ts

u[0]

u[1]

...

u[B -1]

u[B ]

...

u[T-1]

u[T ]

u[T+1]

Kv

Bts

v[0]

v[1]

...

v[B -1]

v[B ]

...

v[T-1]

v[T ]

v[T+1]

Ku

1ts

(8)

Furthermore, the upper bound on B in (8) is the maximum
value that can be attained by any code of rate R and delay T.

Rate analysis

0ts

B # min c 1, 1 - R m T,
R

pv [0]+

pv [1]+

pv [B -1]+

pv [T ]+

pv [T+1]+

u[-T +1]

...

pv [T-1]+

u[-T ]

...

u[-1]

u[0]

u[1]

Simultaneously
Recover
v[0], ..., v[B-1]

Recover
u[0]

Recover
u[1]

u[B -T-1]

pv [B ]+
u[B -T ]

FIGURE 10. An illustration of the decoding steps in an MS code. Each column denotes a channel packet transmitted at the time index shown above it.
106

IEEE Signal Processing Magazine

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March 2017

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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
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Signal Processing - March 2017 - Cover3
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