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

2. Hybrid-ARQ Under Modulation Constraints
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.
What is hybrid-ARQ
Code (IR) vs diversity (RC)
Block fading (previously determined by Caire)
Modulation constraints
Performance of block fading (fabregas)
Hybrid-ARQ
Overview of talk:
Capacity under modulation constraints.
Bit interleaved coded modulation (BICM).
Block Fading Channels  Information Outage Probability
Hybrid-ARQ : Throughput Analysis
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Hybrid-ARQ Under Modulation Constraints

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 < C that achieves reliable communications.
Reliable means an arbitrarily small error probability.
The capacity is the mutual information between the channel’s input X and output Y maximized over all possible input distributions:
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Hybrid-ARQ Under Modulation Constraints

5. Hybrid-ARQ Under Modulation Constraints
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(y|xk)
This function depends on the modulation, channel, and receiver.
The modulation-constrained (CM) capacity is:
E[.] is over all possible symbols and noise realizations
Capacity can be expressed in this form…(good for us)
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Hybrid-ARQ Under Modulation Constraints

6. Hybrid-ARQ Under Modulation Constraints
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
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Hybrid-ARQ Under Modulation Constraints

7. Hybrid-ARQ Under Modulation Constraints
BICM Receiver
Like the CM receiver, the BICM receiver calculates p(y|xk) 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.
Summation whose nth symbol is a 1
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Hybrid-ARQ Under Modulation Constraints

8. Hybrid-ARQ Under Modulation Constraints
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 xk
Receiver:
Compute p(y|xk)
for every xk  S
Mu channels ||  one for each bit
Can obtain capacity Through numerical or MC integration 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:
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Hybrid-ARQ Under Modulation Constraints

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
-10 -5 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
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Hybrid-ARQ Under Modulation Constraints

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
Equality for Gaussian
distributed inputs
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Hybrid-ARQ Under Modulation Constraints

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)] < 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() < 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 B>3.
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Hybrid-ARQ Under Modulation Constraints

13. Modulation Constrained Input Unconstrained Gaussian Input10
Modulation Constrained Input
Unconstrained Gaussian Input -1 10
16-QAM
R=2
Rayleigh Block Fading -2 10 -3 10
Information Outage Probability
B=1 -4 10
Emphasize diversity  slope -5 10
B=10
B=4
B=3
B=2 -6 10 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 d=B
Under modulation constraints the diversity is upper-bounded by the Singleton bound
In this case d=1,2,2,3,6 for B=1,2,3,4,10, respectively.
e.g.: for B=3 it asymptotically has the same slope as the B=2 unconstrained case.
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Hybrid-ARQ Under Modulation Constraints

15. Hybrid-ARQ Under Modulation Constraints
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 = B*RB
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.
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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
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Hybrid-ARQ Under 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.
A two stage rate matching algorithm is used to puncture
the codeword, which is then modulated (after bitwise interleaving)
using either QPSK or 16-QAM. For each modulation
type, there are eight ways to perform rate matching, which
is specified by a three bit variable called the redundancy
version. In the case of 16-QAM, gray-labelling is used and
rate matching can be used to essentially rearrange the signal
constellation mapping. When a retransmission is requested,
the rate matching algorithm can either be run with the same
redundancy version, resulting in a repetition code which is
diversity-combined at the receiver, or a different redundancy
version can be used for each transmission, in which case code-combining
is used.
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Hybrid-ARQ Under Modulation Constraints

18. B=1 QPSK R = 3202/2400 B=2 B=4 B=3 Actual Coded HSDPA10
Actual Coded HSDPA
Modulation Constrained Input
Unconstrained Gaussian Input 10 -1
B=1
FER 10 -2
QPSK
R = 3202/2400 10 -3
B=2
B=4
B=3 10 -4
-10 -5 5 10 15 20 25 30
Es/No in dB

19. Hybrid-ARQ Under Modulation Constraints
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:
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Hybrid-ARQ Under Modulation Constraints

20. Normalized throughput
-10 -5 5 10 15 20 25 30
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9 1
Es/No in dB
Normalized throughput
Unconstrained Gaussian Input
Modulation Constrained Input
Simulated HSDPA Performance
16-QAM
QPSK
Bmax = 4
R = 3202/2400 for QPSK
R = 4664/1920 for QAM
QPSK Losses:
- Modulation Constraints = 0.35dB
- Code = 0.93dB
16QAM Losses:
- Modulation Constraints = 0.56dB
- Code = 1.04dB
0.93dB
0.56dB
1.04dB
0.35dB
The resukts show how much of the loss id due

21. Hybrid-ARQ Under Modulation Constraints
Discussion Cont’d
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.
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Hybrid-ARQ Under Modulation Constraints

22. Hybrid-ARQ Under Modulation Constraints
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
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Hybrid-ARQ Under Modulation Constraints

23. Hybrid-ARQ Under Modulation Constraints
About the Software
The software used to generate the results in this paper is available for free at the Iterative Solutions website:
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/
LDPC codes: DVB-S2.
Tarik:
Just mention that the software can be downloaded for free and runs in matlab.
Don’t need to discuss each feature … just say “and the supported features and standards are as listed here…”
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Hybrid-ARQ Under Modulation Constraints