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The Throughput of Hybrid-ARQ in Block Fading under Modulation Constraints

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Title: The Throughput of Hybrid-ARQ in Block Fading under Modulation Constraints


1
The Throughput of Hybrid-ARQ in Block Fading
under Modulation Constraints
  • March 22, 2006
  • Tarik Ghanim
  • Matthew Valenti
  • West Virginia University
  • Morgantown, WV 26506-6109
  • mvalenti_at_wvu.edu

2
Overview
  • Hybrid-ARQ
  • Combines FEC with ARQ.
  • Breaks the codeword into B distinct blocks
  • Incremental Redundancy Code combining
  • Repetition Coding Diversity combining
  • Block fading
  • Each block multiplied by the same fading
    coefficient.
  • On Coding for Block Fading Channels (Knopp and
    Humblet, 2000)
  • Extended to Hybrid-ARQ by Caire and Tuninetti
    2001.
  • Both of these references consider unconstrained
    inputs.
  • Modulation constraints
  • Block fading Coded Modulation in the Block
    Fading Channels (Fabregas Caire, 2006)
  • Hybrid-ARQ This paper.

3
System Model
4
Noisy Channel Coding Theorem
  • Claude Shannon, A mathematical theory of
    communication, Bell Systems Technical Journal,
    1948.
  • Every channel has associated with it a capacity
    C.
  • Measured in bits per channel use (modulated
    symbol).
  • The channel capacity is an upper bound on
    information rate r.
  • There exists a code of rate r lt C that achieves
    reliable communications.
  • Reliable means an arbitrarily small error
    probability.
  • The capacity is the mutual information between
    the channels input X and output Y maximized over
    all possible input distributions

5
Coded Modulation (CM)
  • ? log2 M bits are mapped to the symbol xk,
    which is chosen from the set S x1, x2, , xM
  • Examples QPSK, M-PSK, QAM
  • The signal y xk n is received
  • where n is Gaussian with variance No/2
  • x is a signal with average energy (variance) Es
  • For each signal in S, the receiver computes
    p(yxk)
  • This function depends on the modulation, channel,
    and receiver.
  • The modulation-constrained (CM) capacity is
  • E. is over all possible symbols and noise
    realizations

6
BICM
  • Most off-the-shelf capacity approaching codes are
    binary.
  • A pragmatic system would use a binary code
    followed by a bitwise interleaver and an M-ary
    modulator.
  • Bit Interleaved Coded Modulation (BICM) Caire
    1998.

Binary to M-ary mapping
Binary Encoder
Bitwise Interleaver
7
BICM Receiver
  • Like the CM receiver, the BICM receiver
    calculates p(yxk) for each signal in S.
  • Furthermore, the BICM receiver needs to calculate
    the log-likelihood ratio of each code bit
  • where represents the set of symbols whose
    nth bit is a 1.
  • and is the set of symbols whose nth bit is a
    0.

8
BICM Capacity
  • The BICM capacity is then Caire 1998
  • As with CM, this can be computed using a Monte
    Carlo integration.

For each bit, calculate
Modulator Pick xk at random from S
Receiver Compute p(yxk) for every xk ? S
xk
nk
For the symbol, calculate
Noise Generator
Unlike CM, the capacity of BICM depends on how
bits are mapped to symbols
After running many trials, calculate
9
5
4.5
Unconstrained
4
3.5
16QAM, CM (solid line)
3
Capacity
2.5
QPSK
2
1.5
16QAM, BICM w/ SP
1
16QAM, BICM w/ gray labeling
0.5
0
-10
-5
0
5
10
15
20
Es/No in dB
10
Block-Fading Channels
  • In a block-fading channel, the transmitter
    produces a codeword of length n-bits, which is
    broken up into B blocks of n/B bits each.
  • Mimics performance of slow fading wireless
    channels.
  • All bits within the same block are multiplied by
    the same fading coefficient.
  • is a complex scalar channel gain
    independent from block-to-block.
  • In Rayleigh fading, instantaneous
    SNR is exponentially distributed.
  • is a vector of complex Gaussian noise
  • Because now the fading is so slow, the channel is
    no longer ergodic

11
Instantaneous Capacity
  • Let ?b denote the instantaneous SNR of the bth
    block
  • Let C(?b) denote the instantaneous capacity of
    the block with SNR ?b
  • For a Gaussian input, C(?b) log2 (1 ?b)
  • With constrained modulation (e.g. QPSK, QAM),
    then the instantaneous capacity is equal to the
    mutual information between input and output.
  • Let (?1, ?2, ?B) describe the inst. SNR of all
    B blocks for one codeword.
  • Let C(?1,?B) denote the instantaneous capacity
    for the entire codeword.
  • This is equivalent to adding B parallel Gaussian
    channels.
  • Thus
  • Code-Combining
  • Diversity-Combining

12
Information Outage Probability
  • An information outage occurs whenever the
    instantaneous capacity is smaller than the code
    rate, e.g. when C(?) C(?1,?B) lt r
  • When an information outage occurs, no rate r code
    can reliably convey information over the channel.
  • The information outage probability is computed by
    integrating the joint pdf of the vector ? over
    the range defined by C(?) lt r
  • Where in the above, it is assumed that the ?i are
    i.i.d. exponential each with average SNR ?.
  • Monte Carlo integration is used for Bgt3.

13
0
10
Modulation Constrained Input
Unconstrained Gaussian Input
-1
10
16-QAM R2 Rayleigh Block Fading
-2
10
-3
10
Information Outage Probability
B1
-4
10
-5
10
B2
B3
B4
B10
-6
10
0
10
20
30
40
50
Es/No in dB
14
Information Outage Probability Observations
  • Diversity is reduced under modulation
    constraints.
  • Fabregas and Caire, Jan. 2006, Trans. Info.
    Theory.
  • For an unconstrained Gaussian input channel, the
    Block Diversity dB
  • Under modulation constraints the diversity is
    upper-bounded by the Singleton bound
  • In this case d1,2,2,3,6 for B1,2,3,4,10,
    respectively.
  • e.g. for B3 it asymptotically has the same
    slope as the B2 unconstrained case.

15
Hybrid-ARQ
  • Combines FEC with ARQ
  • Encode data into a low-rate RB code
  • Implemented using rate-compatible puncturing.
  • Break the codeword into B distinct blocks
  • Each block has rate R BRB
  • Source begins by sending the first block.
  • If destination does not signal with an ACK, the
    next block is sent.
  • After bth transmission, effective rate is Rb
    R/b
  • This continues until either the destination
    decodes the message or all blocks have been
    transmitted.

16
Info Theory of Hybrid-ARQ
  • Throughput of hybrid-ARQ has been studied by
    Caire and Tuninetti (IT 2001).
  • Let ?b denote the received SNR during the bth
    transmission
  • ?b is a random variable.
  • Let C(?b ) be the capacity of the channel with
    SNR ?b
  • C(?b ) is also random.
  • The code-combining capacity after b blocks have
    been transmitted is
  • This is because the capacity of parallel Gaussian
    channels adds.
  • An outage occurs after the bth block if
  • When using Hybrid-ARQ, RB R/B, so the upper
    bound on diversity becomes
  • Hence, there is no loss in diversity due to
    modulation constraints

17
High Speed Downlink Packet Access
  • With HSDPA, the message is first encoded with by
    a rate 1/3 UMTS turbo code.
  • Rate matching used to produce a higher block rate
    R.
  • Uses two modulation types QPSK, gray-labeled
    16QAM
  • The encoder is binary and separated from the
    modulator by a bitwise interleaver, an example of
    BICM
  • Uses Hybrid ARQ First block encoded with a rate
    1/3 UMTS turbo encoder and then sent, if not
    decoded, another block encoded using different
    rate matching parameters then sent. Information
    combined at receiver.

18
0
10

Actual Coded HSDPA
Modulation Constrained Input
Unconstrained Gaussian Input
-1
10
B1
-2
10
FER
QPSK R 3202/2400
-3
10
B2
B3
B4
-4
10

-10
-5
0
5
10
15
20
25
30
Es/No in dB
19
Throughput Analysis
  • Throughput and delay depend on the average number
    of blocks required to get out of an outage.
  • Given the pmf of the random variable B indicating
    the number of hybrid-ARQ transmissions until
    successful decoding given an upper limit Bmax is
  • where
  • Then the Throughput Efficiency which is the ratio
    of correct bits to transmitted bits can be
    expressed as

20
QPSK Losses - Modulation Constraints 0.35dB
- Code 0.93dB
16QAM Losses - Modulation Constraints 0.56dB
- Code 1.04dB
21
Discussion Contd
  • Other key factors contributing to losses relative
    to the information theoretic
  • Some of the loss is due to finite block length
    effects,
  • The rate matching algorithm of HSDPA produces up
    to eight redundancy versions for each modulation
    type, these blocks are not mutually exclusive,
    i.e. some code bits will appear in more than one
    block. As a consequence, the processing at the
    receiver will actually be a combination of
    code-combining and diversity-combining.

22
Conclusions
  • Steps for determining the throughput of
    Hybrid-ARQ under modulation constraints
  • Determine the AWGN capacity under modulation
    constraints
  • Determine information outage probability
  • Determine throughput
  • In block fading, modulation constraints cause a
    loss relative to the unconstrained input bound
    (Caire and Tuninetti)
  • Under modulation constraints the diversity is
    upper-bounded by the Singleton bound
  • There is a loss of diversity when a fixed rate
    code is used.
  • However, when hybrid-ARQ is used, there is no
    loss in diversity.
  • Future work
  • Extension to Hybrid-ARQ based relay networks

23
About the Software
  • The software used to generate the results in this
    paper is available for free at the Iterative
    Solutions website
  • www.iterativesolutions.com
  • Runs in matlab, but uses c-mex for efficiency.
  • Supported features
  • Simulation of BICM
  • Turbo, LDPC, or convolutional codes.
  • PSK, QAM, FSK modulation.
  • BICM-ID Iterative demodulation and decoding.
  • Generation of ergodic capacity curves (BICM/CM
    constraints).
  • Information outage probability in block fading.
  • Calculation of throughput of hybrid-ARQ.
  • Implemented standards
  • Binary turbo codes UMTS/3GPP, cdma2000/3GPP2.
  • Duobinary turbo codes DVB-RCS, wimax/802.16.
  • LDPC codes DVB-S2.
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