Title: The Throughput of Hybrid-ARQ in Block Fading under Modulation Constraints
1The 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
2Overview
- 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.
3System Model
4Noisy 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
5Coded 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
6BICM
- 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
7BICM 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.
8BICM 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
95
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
10Block-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
11Instantaneous 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
12Information 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.
130
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
14Information 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.
15Hybrid-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.
16Info 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
17High 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.
180
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
19Throughput 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 -
20QPSK Losses - Modulation Constraints 0.35dB
- Code 0.93dB
16QAM Losses - Modulation Constraints 0.56dB
- Code 1.04dB
21Discussion 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.
22Conclusions
- 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
23About 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.