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

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@wvu.edu. Overview. Hybrid-ARQ Combines FEC with ARQ. Breaks the codeword into B distinct blocks

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

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  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@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. 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: Hybrid-ARQ Under Modulation Constraints

  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(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 Hybrid-ARQ Under Modulation Constraints

  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 Hybrid-ARQ Under Modulation Constraints

  7. 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. Hybrid-ARQ Under Modulation Constraints

  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(y|xk) 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: 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 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 Hybrid-ARQ Under Modulation Constraints

  11. Equality for Gaussian distributed inputs 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 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. Hybrid-ARQ Under Modulation Constraints

  13. 0 10 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 -5 10 B=10 B=4 B=3 B=2 -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 d=B • Under modulation constraints the diversity is upper-bounded by the Singletonbound • 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. Hybrid-ARQ Under Modulation Constraints

  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 = 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. Hybrid-ARQ Under Modulation Constraints

  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 • bis 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 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. Hybrid-ARQ Under Modulation Constraints

  18. 0 10 Actual Coded HSDPA Modulation Constrained Input Unconstrained Gaussian Input -1 10 B=1 -2 10 FER QPSK R = 3202/2400 -3 10 B=2 B=4 B=3 -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: Hybrid-ARQ Under Modulation Constraints

  20. 1 0.9 0.8 0.7 0.93dB 0.56dB 1.04dB Bmax = 4 R = 3202/2400 for QPSK R = 4664/1920 for QAM 0.6 0.35dB 0.5 Normalized throughput 0.4 QPSK 16-QAM 0.3 Unconstrained Gaussian Input 0.2 Modulation Constrained Input 0.1 Simulated HSDPA Performance 0 -10 -5 0 5 10 15 20 25 30 Es/No in dB QPSK Losses: - Modulation Constraints = 0.35dB - Code = 0.93dB 16QAM Losses: - Modulation Constraints = 0.56dB - Code = 1.04dB

  21. 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. Hybrid-ARQ Under Modulation Constraints

  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 Singletonbound • 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 Hybrid-ARQ Under Modulation Constraints

  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. Hybrid-ARQ Under Modulation Constraints

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