Hybrid ARQ

Definition:
Type I HARQ

In Type I HARQ, the transmitter sends a codeword that provides both error detection (via CRC) and error correction (via FEC).

If the receiver decodes successfully (CRC passes), it sends an ACK.
If decoding fails (CRC fails), it sends a NACK, discards the received packet, and the transmitter retransmits the identical codeword.

Type I HARQ does not combine retransmissions — each attempt is decoded independently. The coding redundancy is fixed, regardless of channel conditions.

Definition:
Type II HARQ — Chase Combining

In Type II HARQ (Chase combining), the transmitter sends the same codeword on each retransmission, but the receiver soft-combines all received copies before decoding:

$L_{\text{combined}}(j) = \sum_{r=1}^{R} L_{\text{ch}}^{(r)}(j)$

where $R$ is the number of transmission attempts and $L_{\text{ch}}^{(r)}(j)$ is the channel LLR for bit $j$ in the $r$ -th attempt.

Chase combining provides a diversity gain: after $R$ attempts, the effective SNR is approximately $R$ times the single-attempt SNR (assuming independent fading across attempts).

Chase combining is also called soft combining or maximal ratio combining at the packet level.

Definition:
Type III HARQ — Incremental Redundancy (IR)

In Type III HARQ (incremental redundancy), the transmitter sends different coded bits in each retransmission:

First transmission: Send a high-rate punctured codeword (e.g., rate 3/4).
First retransmission: Send additional parity bits that, combined with the first transmission, lower the effective rate (e.g., to 1/2).
Subsequent retransmissions: Send further parity bits, progressively lowering the rate (e.g., to 1/3, 1/4, ...).

The receiver combines all received bits and decodes using the accumulated lower-rate code. IR-HARQ provides both diversity gain (from multiple transmissions) and coding gain (from the progressively lower code rate).

IR-HARQ achieves higher throughput than Chase combining because each retransmission carries new information (additional parity).

IR-HARQ requires rate-compatible codes: a family of codes with nested codeword sets, so that lower-rate codes contain the higher-rate codes as subsets.

HARQ Throughput Comparison

Compare the throughput of Type I, Chase combining, and incremental redundancy HARQ as a function of SNR. The throughput is measured in information bits per channel use, accounting for retransmissions.

Parameters

SNR range (dB)

Initial code rate

Example: HARQ Throughput at SNR = 5 dB

Compare the throughput of the three HARQ types at $\text{SNR} = 5$ dB using a rate-1/2 base code over an AWGN channel.

Assume: initial packet error rate $P_1 = 0.3$ , after Chase combining (2 attempts) $P_2^{\text{CC}} = 0.02$ , after IR (2 attempts at effective rate 1/3) $P_2^{\text{IR}} = 0.005$ .

Solution

Type I throughput

Expected transmissions: $\bar{N} = 1/(1-P_1) = 1/0.7 = 1.43$ .

Throughput: $\eta = R_c / \bar{N} = 0.5 / 1.43 = 0.35$ bits/channel use.

Chase combining throughput

If 1st attempt succeeds (prob $0.7$ ): 1 transmission. If 2nd attempt succeeds (prob $0.3 \times 0.98$ ): 2 transmissions. If both fail (prob $0.3 \times 0.02$ ): more transmissions.

$\bar{N} \approx 0.7 \times 1 + 0.294 \times 2 + 0.006 \times 3 = 1.306$ .

Throughput: $\eta = 0.5 / 1.306 = 0.383$ bits/channel use.

Incremental redundancy throughput

1st attempt: rate 1/2, succeeds with prob 0.7. 2nd attempt: rate 1/3 (combined), succeeds with prob $0.3 \times 0.995 = 0.299$ .

Average resources: $0.7 \times 1 + 0.299 \times 1.5 + 0.001 \times 2 = 1.15$ (in units of the first transmission).

Throughput: $\eta = 0.5 / 1.15 = 0.435$ bits/channel use.

IR-HARQ provides the highest throughput. $\blacksquare$

Comparison of HARQ Types

Property	Type I	Type II (Chase)	Type III (IR)
Retransmission content	Same codeword	Same codeword	New parity bits
Receiver combining	No combining	Soft combining (LLR addition)	Code combining (lower rate)
SNR gain per retx	None (independent decoding)	~3 dB (doubling effective SNR)	>3 dB (coding + diversity gain)
Throughput efficiency	Low	Medium	Highest
Buffer requirement	None	Store soft values	Store soft values
Code design	Any FEC + CRC	Any FEC + CRC	Rate-compatible code family

Quick Check

In incremental redundancy HARQ, what does the transmitter send in a retransmission?

The exact same codeword as the first transmission

Additional parity bits not sent in previous transmissions

Only the information bits without parity

A different interleaved version of the same codeword

Correction:

Additional parity bits not sent in previous transmissions

IR-HARQ sends new redundancy bits, effectively lowering the code rate with each retransmission. The receiver combines all received bits for a stronger code.

Why This Matters: HARQ in 5G NR

5G NR uses Type III HARQ (incremental redundancy) as its retransmission strategy:

The LDPC code produces a large "mother code" at low rate (e.g., 1/5), and rate matching selects a subset of bits for each transmission.
Each retransmission sends bits from a different redundancy version (RV), with 4 RVs defined (RV0, RV1, RV2, RV3).
The receiver soft-combines all received bits and decodes with the accumulated lower-rate code.
HARQ operates per transport block with a round-trip time of about 4-8 OFDM symbols (~1 ms at 15 kHz SCS).
Up to 16 parallel HARQ processes allow continuous data flow despite retransmission delays.

HARQ

Hybrid automatic repeat request: a retransmission scheme that combines forward error correction (FEC) with ARQ, using CRC for error detection and soft combining of retransmissions.

Chase Combining

A HARQ strategy where identical codewords are retransmitted and the receiver soft-combines (adds LLRs of) all received copies before decoding, providing diversity gain.

Related: Type I HARQ, Soft Combining

Incremental Redundancy

A HARQ strategy where each retransmission carries new parity bits, progressively lowering the effective code rate. Provides both diversity and coding gain, achieving the highest throughput among HARQ types.

Bit-Interleaved Coded Modulation (BICM)Coding for Fading Channels