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

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 12/6/2018 Hybrid-ARQ Under Modulation Constraints

System Model

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: 12/6/2018 Hybrid-ARQ Under Modulation Constraints

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) 12/6/2018 Hybrid-ARQ Under Modulation Constraints

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 12/6/2018 Hybrid-ARQ Under Modulation Constraints

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 12/6/2018 Hybrid-ARQ Under Modulation Constraints

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: 12/6/2018 Hybrid-ARQ Under Modulation Constraints

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

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 12/6/2018 Hybrid-ARQ Under Modulation Constraints

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 12/6/2018 Hybrid-ARQ Under Modulation Constraints

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. 12/6/2018 Hybrid-ARQ Under Modulation Constraints

Modulation Constrained Input Unconstrained Gaussian Input 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 Emphasize diversity  slope -5 10 B=10 B=4 B=3 B=2 -6 10 10 20 30 40 50 Es/No in dB

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. 12/6/2018 Hybrid-ARQ Under Modulation Constraints

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. 12/6/2018 Hybrid-ARQ Under Modulation Constraints

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 12/6/2018 Hybrid-ARQ Under Modulation Constraints

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. 12/6/2018 Hybrid-ARQ Under Modulation Constraints

B=1 QPSK R = 3202/2400 B=2 B=4 B=3 Actual Coded HSDPA 10 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

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: 12/6/2018 Hybrid-ARQ Under Modulation Constraints

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

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. 12/6/2018 Hybrid-ARQ Under Modulation Constraints

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 12/6/2018 Hybrid-ARQ Under Modulation Constraints

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: 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. 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…” 12/6/2018 Hybrid-ARQ Under Modulation Constraints