How does the BCJR algorithm work?

BCJR operates on the trellis representation of a convolutional code in three passes: 1) Forward recursion (α): computes α_k(s) = probability of being in state s at time k, processing the received sequence left to right. α_k(s) = Σ α_{k-1}(s') × γ_k(s',s). 2) Backward recursion (β): computes β_k(s) = probability of the future observations given state s at time k, processing right to left. β_k(s) = Σ β_{k+1}(s') × γ_{k+1}(s,s'). 3) Completion: the posterior LLR for bit d_k is L(d_k) = ln[Σ α_k(s)·γ_k·β_k(s') for d_k=1] - ln[Σ α_k(s)·γ_k·β_k(s') for d_k=0]. The branch metric γ_k(s',s) incorporates channel observations, code structure, and prior (extrinsic) information from the other decoder in turbo iterations.

What is the difference between BCJR and Viterbi?

Both operate on the code trellis but optimize different criteria: Viterbi: finds the most likely sequence (ML sequence), minimizing the probability of sequence error. Uses only a forward pass (no backward). Produces hard decisions (or approximate soft outputs via SOVA). Complexity: O(N × 2^v) per pass. BCJR: finds the most likely individual bit (MAP bit), minimizing the probability of bit error. Requires both forward and backward passes. Produces exact soft-output LLRs. Complexity: O(N × 2^v) per pass × 2 passes = 2x Viterbi. For standalone decoding, the performance difference is negligible (<0.1 dB). BCJR's advantage is critical in iterative decoding: its exact soft outputs enable turbo/LDPC decoders to converge within 0.5-1.0 dB of Shannon capacity.

Where is the BCJR algorithm used in modern RF?

Primary applications: 1) 3G UMTS/WCDMA turbo decoder: two BCJR decoders exchange extrinsic LLRs iteratively (6-8 iterations typical). 2) 4G LTE turbo decoder: same architecture, higher throughput (up to 150 Mbps per decoder instance). 3) DVB-RCS2 and DVB-SH satellite: turbo codes with BCJR component decoders. 4) LDPC decoding: the sum-product (belief propagation) algorithm is a generalization of BCJR to sparse graph codes, used in 5G NR, Wi-Fi 6, and DVB-S2. 5) Equalization: BCJR applied to the channel trellis performs optimal MLSE equalization with soft outputs for coded systems (turbo equalization). 6) Joint detection-decoding in MIMO systems.

Channel Coding

BCJR Algorithm

Q: Where is the BCJR algorithm used in modern RF?

Primary applications: 1) 3G UMTS/WCDMA turbo decoder: two BCJR decoders exchange extrinsic LLRs iteratively (6-8 iterations typical). 2) 4G LTE turbo decoder: same architecture, higher throughput (up to 150 Mbps per decoder instance). 3) DVB-RCS2 and DVB-SH satellite: turbo codes with BCJR component decoders. 4) LDPC decoding: the sum-product (belief propagation) algorithm is a generalization of BCJR to sparse graph codes, used in 5G NR, Wi-Fi 6, and DVB-S2. 5) Equalization: BCJR applied to the channel trellis performs optimal MLSE equalization with soft outputs for coded systems (turbo equalization). 6) Joint detection-decoding in MIMO systems.

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A maximum a posteriori (MAP) decoding algorithm (Bahl, Cocke, Jelinek, Raviv, 1974) that computes the posterior probability of each transmitted bit given the entire received sequence. Uses forward (α) and backward (β) trellis recursions with branch metrics (γ). Produces exact soft-output LLRs essential for turbo code iterative decoding. Practical variants: Log-MAP (exact, log domain) and Max-Log-MAP (~0.2 dB loss, simpler). Foundation of 3G/4G turbo decoding and 5G LDPC belief propagation.

Type: MAP bit-level

Complexity: O(N×2^v)×2

Output: Soft LLR

Understanding the BCJR Algorithm

The BCJR algorithm was published in 1974 but remained largely academic until the invention of turbo codes in 1993 by Berrou, Glavieux, and Thitimajshima. Turbo codes use two BCJR decoders that exchange soft information iteratively, approaching the Shannon capacity limit within 0.5 to 1.0 dB. The key insight is that BCJR produces exact posterior probabilities for each bit, not just the most likely sequence (which Viterbi provides).

The difference between "most likely sequence" (Viterbi) and "most likely bit" (BCJR) becomes critical in iterative decoding. Soft LLR outputs from BCJR serve as prior information for the next decoder iteration. Each iteration refines the probability estimates, progressively eliminating uncertainty until the decoder converges to the correct codeword.

BCJR Recursions

Forward Recursion:
α_k(s) = ∑_s' α_k−1(s') × γ_k(s',s)

Backward Recursion:
β_k(s) = ∑_s' β_k+1(s') × γ_k+1(s,s')

LLR Output:
L(d_k) = ln[∑_d=1 αγβ] − ln[∑_d=0 αγβ]

Log-MAP (practical):
ln(e^a+e^b) = max(a,b) + ln(1+e^−|a−b|)
Max-Log-MAP: drop correction term (−0.2 dB)

Decoding Algorithm Comparison

Algorithm	Criterion	Output	Passes	Used In
Viterbi	ML sequence	Hard / SOVA	1 (forward)	3G conv., GSM
BCJR (MAP)	MAP bit	Exact soft LLR	2 (fwd + bwd)	Turbo (3G/4G)
Max-Log-MAP	Approx MAP	Approx soft LLR	2	HW turbo decoders
Belief Prop.	MAP (graph)	Soft LLR	Iterative	LDPC (5G, Wi-Fi)

Common Questions

Frequently Asked Questions

How does BCJR work?

Three passes on trellis: forward (α), backward (β), combine with branch metrics (γ). Computes posterior probability per bit. LLR = ln(P₁/P₀). Log-MAP uses log domain. Max-Log-MAP approximates (~0.2 dB loss).

BCJR vs. Viterbi?

Viterbi: most likely sequence, 1 pass, hard/SOVA output. BCJR: most likely bit, 2 passes, exact soft LLR. BCJR = 2x complexity. Standalone: <0.1 dB difference. Iterative: BCJR converges within 0.5 to 1.0 dB of Shannon.

Modern RF applications?

3G/4G turbo decoder (6 to 8 iterations). DVB-RCS2 satellite. LDPC belief propagation (generalized BCJR). Turbo equalization. MIMO joint detection-decoding. Every modern iterative receiver derives from BCJR.

Related Terms

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Channel Coding

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