Ferkans — Interactive Telecom Tutor

The Tunnel Is the Whole Story

Here is the punchline of BICM-ID convergence analysis, stated before the theorem: the iteration walks up a staircase between the demapper curve and the inverted decoder curve. If the staircase can get all the way to $(1, 1)$ , BER is zero at the converged fixed point. If the two curves TOUCH or CROSS somewhere below $(1, 1)$ , the staircase gets trapped at that crossing and BER is bounded away from zero. The gap between the curves — the TUNNEL — is therefore the SNR margin of the system. Close the tunnel by lowering SNR, the iteration stalls; open it by raising SNR, the iteration completes. The convergence threshold is the unique SNR at which the tunnel JUST closes.

What I want you to internalise is that this picture is not an approximation of the right picture — it IS the right picture, up to the Gaussian-LLR assumption. The convergence theorem below is almost a triviality once you have the EXIT chart in hand; what the theorem does is formalise the "almost" and pin down exactly which properties of the curves matter for stability.

Definition:
Tunnel Width $\Delta(I_A)$

For a given SNR, labelling $\mu$ , and code rate $R$ , the tunnel width at a-priori level $I_A \in [0, 1)$ is $\Delta(I_A) \triangleq T_{\mathrm{dem}}(I_A, \mathrm{SNR}) - T_{\mathrm{dec}}^{-1}(I_A, R).$ Equivalently, it is the vertical gap between the demapper curve (plotted $I_E$ vs. $I_A$ ) and the inverted decoder curve on the EXIT chart. The minimum tunnel width is $\Delta_{\min} \triangleq \min_{I_A \in [0, 1)} \Delta(I_A),$ and the convergence threshold $\mathrm{SNR}_{\rm conv}$ is the smallest SNR such that $\Delta_{\min} > 0$ .

Strictly, $\Delta_{\min}$ is taken over an open interval $[0, 1 - \varepsilon]$ and the limit $\varepsilon \to 0$ is needed to cope with curves that both end at $(1, 1)$ . The practical rule is: if the two curves do not cross anywhere except possibly at $(1, 1)$ , the tunnel is open; otherwise closed. Modern numerical tools sample $\Delta(\cdot)$ at $\sim 100$ points and report $\Delta_{\min}$ as the tunnel margin.

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Theorem: BICM-ID Converges iff the Tunnel Is Open

Consider BICM-ID with code rate $R$ , labelling $\mu$ , and SNR. Assume the Gaussian-LLR consistency, long interleaver, and symmetric labelling hypotheses of s02. Starting from $I_A^{(0)} = 0$ , the iteration $I_A^{(t+1)} = T_{\mathrm{dec}}\!\left(T_{\mathrm{dem}}(I_A^{(t)}, \mathrm{SNR}), R\right)$ converges monotonically to some limit $I_A^\infty \in [0, 1]$ . Then $I_A^\infty = 1$ if and only if the tunnel is open: $T_{\mathrm{dem}}(I_A, \mathrm{SNR}) > T_{\mathrm{dec}}^{-1}(I_A, R) \quad \text{for all } I_A \in [0, 1).$ Equivalently, $I_A^\infty = 1$ iff $\Delta(I_A) > 0$ on $[0, 1)$ . The convergence threshold $\mathrm{SNR}_{\rm conv}$ is the SNR at which the tunnel is marginally open — i.e., $\min_{I_A \in [0, 1)} \Delta(I_A) \to 0^+$ .

Two curves on the unit square, one steep (demapper under SP), one gentle (a short-LDPC decoder curve at moderate $R$ ). Start at $(0, 0)$ . The iteration first jumps UP to the demapper curve, then RIGHT to the decoder curve, then UP again, then RIGHT — a staircase. As long as the demapper curve is above the inverted decoder curve, each "up" step increases $I_E$ , and the monotone convergence theorem drives the trajectory toward $(1, 1)$ . If the demapper curve DIPS BELOW the decoder curve at some intermediate $I_A^*$ , the staircase gets clipped there — it cannot pass through a crossing of the two curves.

Show Hint

Monotonicity: both $T_{\mathrm{dem}}(\cdot, \mathrm{SNR})$ and $T_{\mathrm{dec}}(\cdot, R)$ are non-decreasing in $I_A$ (more a priori always means $\geq$ extrinsic), so the composition is non-decreasing.

Bounded: the iterates $I_A^{(t)}$ are bounded above by 1 (an MI), so the sequence converges.

At the limit $I_A^\infty$ , the iteration is stationary: $I_A^\infty = T_{\mathrm{dec}}(T_{\mathrm{dem}}(I_A^\infty, \mathrm{SNR}), R)$ . Equivalently, $(I_A^\infty, T_{\mathrm{dem}}(I_A^\infty, \mathrm{SNR}))$ is a crossing of the two EXIT curves.

If the tunnel is open, no crossing exists on $[0, 1)$ , so the limit must be $I_A^\infty = 1$ . If closed, there is a crossing at some $I_A^* < 1$ , and the iteration is trapped there.

Proof

Step 1: Monotone non-decreasing dynamics

Both EXIT curves are non-decreasing: supplying more a priori to the demapper cannot decrease its extrinsic MI (a standard data- processing argument), and the same holds for the decoder. The composition $T_{\mathrm{dec}} \circ T_{\mathrm{dem}}$ is therefore non-decreasing in $I_A$ . The claim $I_A^{(t+1)} \geq I_A^{(t)}$ follows if $I_A^{(1)} \geq I_A^{(0)} = 0$ , which holds because $T_{\mathrm{dec}}(T_{\mathrm{dem}}(0, \mathrm{SNR}), R) \geq 0$ . The sequence $\{I_A^{(t)}\}$ is monotone non-decreasing and bounded above by 1, hence convergent by the monotone convergence theorem.

Step 2: Limit is a fixed point

Continuity of the two EXIT maps (a Gaussian-LLR consequence, provable by dominated convergence on the defining integral) implies that the limit $I_A^\infty$ satisfies $I_A^\infty = T_{\mathrm{dec}}(T_{\mathrm{dem}}(I_A^\infty, \mathrm{SNR}), R)$ . Equivalently, the point $(I_A^\infty, T_{\mathrm{dem}}(I_A^\infty, \mathrm{SNR}))$ lies on both EXIT curves (demapper directly, decoder via the inverse convention). This is a CROSSING of the two curves on the EXIT chart.

Step 3: Tunnel open ⇒ no crossing ⇒ $I_A^\infty = 1$

If $\Delta(I_A) > 0$ for all $I_A \in [0, 1)$ , there is no crossing on this interval. The only available crossing is at $I_A = 1$ — where both curves end (at MI = 1, under mild assumptions on the labelling, s04). Hence $I_A^\infty = 1$ .

Step 4: Tunnel closed ⇒ crossing ⇒ $I_A^\infty < 1$

If $\Delta(I_A^*) = 0$ for some $I_A^* < 1$ (or the two curves cross each other), the iteration is stationary AT OR BEFORE $I_A^*$ : the monotone sequence cannot exceed the smallest crossing it encounters starting from 0. Hence $I_A^\infty \leq I_A^* < 1$ , and the BER is bounded away from zero by Thm. TFixed-Point Rate of a Converged BICM-ID Receiver.

Step 5: Convergence threshold

As SNR increases, $T_{\mathrm{dem}}(I_A, \mathrm{SNR})$ increases pointwise (more channel information can only help). The tunnel widens. The threshold $\mathrm{SNR}_{\rm conv}$ is the unique SNR at which $\min_{I_A} \Delta(I_A) = 0$ — the critical SNR separating "stalls" from "converges". In the $E_b/N_0$ convention, $E_b/N_0|_{\rm conv}$ is the analogous threshold after normalising by rate. $\blacksquare$

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EXIT Chart: Convergence Trajectory

Animated demonstration of the BICM-ID iteration staircase under two SNR conditions for 16-QAM with set-partition labelling and a rate

R = 1/2

outer code. At

\mathrm{SNR} = \mathrm{SNR}_{\rm conv} + 0.5

dB (tunnel open), the trajectory climbs smoothly from

(0, 0)

to

(1, 1)

over a dozen iterations, each step shrinking as the tunnel narrows near the top. At

\mathrm{SNR} = \mathrm{SNR}_{\rm conv} - 0.5

dB (tunnel closed), the staircase runs straight into the touching point of the two curves at

I_A^* \approx 0.6

and stalls there — BER never decays below the error floor. Watch the two regimes side by side.

Left panel: tunnel open, trajectory escapes to

(1, 1)

; right panel: tunnel closed at

I_A^* \approx 0.6

, trajectory stalls. The difference is 1 dB of SNR.

BICM-ID BER at Successive Iterations

BER versus $E_b/N_0$ for BICM-ID over AWGN with 16-QAM, rate- $1/2$ outer convolutional code, and the selected labelling. Each curve corresponds to one outer iteration $t = 1, 2, \ldots$ . The one-iteration curve ( $t = 1$ ) is the ordinary non-iterative BICM performance of Ch. 6. The multi-iteration curves show the "waterfall cliff" that emerges as the tunnel opens: below $\mathrm{SNR}_{\rm conv}$ no amount of iteration lowers the BER; at and above, each extra iteration lowers BER by roughly an order of magnitude until the BER floor is hit. Set partitioning shows a steeper cliff than Gray under iteration.

Parameters

Iteration count5

Bit labelling

Minimum Tunnel Width vs. SNR

Numerical tunnel-width computation: for a fixed labelling, compute $\Delta_{\min}(\mathrm{SNR}) = \min_{I_A} (T_{\mathrm{dem}}(I_A, \mathrm{SNR}) - T_{\mathrm{dec}}^{-1}(I_A, R))$ across a range of SNRs, for several code rates $R \in \{1/3, 1/2, 2/3, 3/4\}$ . The zero-crossing of $\Delta_{\min}(\mathrm{SNR})$ is the convergence threshold. Higher-rate codes have steeper inverted decoder curves and demand more SNR; lower-rate codes have flatter curves and converge earlier. This plot is the "design dashboard" for picking a code rate at a target SNR.

Parameters

Bit labelling

Example: Convergence Threshold for 16-QAM BICM-ID at Rate $1/2$

For 16-QAM BICM-ID over AWGN with a rate- $1/2$ memory-2 convolutional outer code (generators $[7, 5]_{\rm octal}$ ), and set-partition labelling, estimate the convergence threshold $E_b/N_0|_{\rm conv}$ using the EXIT chart. Compare with the BICM-ID ML (one-shot) threshold and with the CM capacity limit at the same spectral efficiency.

Solution

Setup: EXIT curves at candidate SNRs

Spectral efficiency $= R \cdot \log_2 16 = 2$ bits/channel use, so the CM capacity limit is at $E_b/N_0$ satisfying $C_{16 \text{-QAM}}(\mathrm{SNR}) = 2$ bits, numerically $E_b/N_0|_{\rm CM} \approx 4.3$ dB. At each candidate $E_b/N_0 \in \{3.0, 3.5, 4.0, 4.5, 5.0\}$ dB, compute $T_{\mathrm{dem}}(I_A, \mathrm{SNR})$ by Monte Carlo over the 16-QAM demapper with $10^4$ symbols and consistent-Gaussian a-priori.

Decoder EXIT for the $[7,5]$ code at rate $1/2$

Use the BCJR algorithm on the 4-state trellis with consistent- Gaussian a-priori inputs at $I_A \in \{0, 0.05, \ldots, 0.95, 0.99\}$ ; for each $I_A$ , pass $10^4$ coded blocks, measure the extrinsic LLR MI, and record $T_{\mathrm{dec}}(I_A, R = 1/2)$ . Invert by plotting $I_A$ on the vertical axis: this is the curve to compare with $T_{\mathrm{dem}}$ .

Read off the threshold

For SP labelling, the tunnel first opens at $E_b/N_0 \approx 4.7$ dB: at this SNR the demapper curve just kisses the inverted decoder curve near $I_A \approx 0.55$ . Below, the two curves cross; above, they are separated by a monotonically widening tunnel. Thus $E_b/N_0|_{\rm conv}^{\rm SP} \approx 4.7$ dB, about 0.4 dB above the CM limit (4.3 dB). The corresponding Gray-labelled BICM-ID needs $E_b/N_0 \approx 5.9$ dB (the Gray demapper curve is flat near $I_A = 0$ , so the tunnel closes earlier).

Compare with one-shot BICM

One-shot BICM with the same code and Gray labelling has an ML threshold (where the union bound predicts BER = $10^{-5}$ ) at $E_b/N_0 \approx 6.7$ dB. BICM-ID with SP labelling saves $\sim 2.0$ dB over that, which is the operational value of iterative decoding for 16-QAM at rate $1/2$ . Most of the remaining 0.4 dB to CM can only be recovered by a stronger outer code (s05). $\blacksquare$

⚠️Engineering Note

How Many Iterations Does BICM-ID Actually Need?

The EXIT chart says the iteration converges "in the limit," but practical receivers must commit to a fixed iteration budget. The empirical rule is 5–10 outer iterations at 0.5 dB above the convergence threshold. Each iteration adds $\sim 1$ ms of receiver latency on a modest DSP; on a phone-class modem the iteration budget is typically 3–5 for throughput reasons. DVB-S2X's optional iterative-APSK mode specifies a maximum of 8 outer iterations with an early-stopping CRC check, which terminates the loop as soon as the decoder agrees with itself on two consecutive passes. For short-block applications (URLLC in 5G NR, 100-byte control channel messages), even 2–3 iterations already capture most of the gain — the law of diminishing returns sets in fast because the tunnel narrows near the top.

Practical Constraints

•
Iteration latency: 1–2 ms per pass on a commercial modem chipset
•
Memory per iteration: $\sim N \cdot L$ soft bits of extrinsic LLR storage
•
Typical budget: 5–10 passes near threshold, 3–5 passes at operating SNR
•
Early-stopping via CRC matching saves 20–40% of worst-case passes

📋 Ref: ETSI EN 302 307-2 (DVB-S2X); 3GPP TS 38.212 (5G NR)

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Local Stability vs. Global Convergence

The convergence theorem is a GLOBAL statement: it says that under tunnel-open conditions, the iteration reaches $(1, 1)$ from ANY starting point $I_A^{(0)} \leq 1$ . This is stronger than local stability of the $(1, 1)$ fixed point, which is equivalent to the product of local slopes at $(1, 1)$ being less than 1. For regular monotone EXIT curves the two notions coincide; for codes with non-monotone decoder curves (some irregular LDPC constructions) local stability can hold while global convergence fails — the iteration stalls at a lower-valued attracting fixed point. Density evolution catches these cases; the EXIT chart can miss them if the Gaussian- LLR assumption hides a bimodal output distribution. When designing irregular LDPC for BICM-ID (s05), VALIDATE local stability at $(1, 1)$ numerically — the "stability condition" $\lambda'(0) \rho'(1) < 1$ of Richardson–Urbanke is the fast check.

Common Mistake: Demapper Curve Ending Below $(1, 1)$

Mistake:

A BICM-ID design that "closes the tunnel everywhere except possibly at $(1, 1)$ " is often declared convergent, on the grounds that the only crossing is at $(1, 1)$ itself.

Correction:

Check the DEMAPPER endpoint: $T_{\mathrm{dem}}(1, \mathrm{SNR}) = ?$ . For labelings like the natural binary labelling of 8-PSK, or certain Gray labelings on multi-ring APSK, the demapper with FULLY KNOWN a priori on the other bits cannot distinguish the target bit — two constellation points share the target bit value and a shared set of "other" bit values, so the demapper returns zero LLR. In such cases $T_{\mathrm{dem}}(1, \mathrm{SNR}) < 1$ and the iteration CANNOT reach $(1, 1)$ regardless of SNR. The labelling is flawed: no amount of iteration can push BER to zero. See [?chindapol-ritcey-2001] for a catalogue of 16-QAM labelings and their $T_{\mathrm{dem}}(1, \cdot)$ values.

Quick Check

The tunnel in a BICM-ID EXIT chart is "open" when:

The demapper curve lies strictly above the inverted decoder curve on $[0, 1)$ .

The decoder curve has positive slope everywhere.

The convergence threshold is below the CM capacity.

The iteration reaches any fixed point.

Correction:

The demapper curve lies strictly above the inverted decoder curve on

[0, 1)

.

This is the defining condition of the convergence theorem. Equivalently, $\Delta(I_A) > 0$ on $[0, 1)$ , i.e., $\Delta_{\min} > 0$ .

Quick Check

When is the Gaussian-LLR assumption underlying EXIT analysis LEAST reliable?

At very high SNR, far above the convergence threshold.

For regular LDPC codes with moderate rate on BI-AWGN.

At very low SNR with short block lengths.

For turbo codes with two convolutional components.

Correction:

At very low SNR with short block lengths.

At low SNR the demapper LLRs are heavy-tailed, and short blocks mean the interleaver cannot fully decorrelate — two compounding violations of the Gaussian-LLR model. Density evolution is the correct tool in this regime.

Key Takeaway

Convergence = open tunnel. BICM-ID converges to BER = 0 iff the demapper curve strictly dominates the inverted decoder curve across $[0, 1)$ . The convergence threshold $\mathrm{SNR}_{\rm conv}$ is the SNR at which the two curves first touch. Below threshold, the iteration stalls at the touching point and an error floor persists no matter how many passes you run. Above threshold, roughly 5–10 iterations are enough to climb to $(1, 1)$ and the BER collapses. The tunnel geometry is the central quantitative tool of this chapter and of iterative-coding design more generally.

Convergence Analysis and the Tunnel

The Tunnel Is the Whole Story

Definition: Tunnel Width Δ(IA)\Delta(I_A)Δ(IA​)

Theorem: BICM-ID Converges iff the Tunnel Is Open

Step 1: Monotone non-decreasing dynamics

Step 2: Limit is a fixed point

Step 3: Tunnel open ⇒ no crossing ⇒ $I_A^\infty = 1$

Step 4: Tunnel closed ⇒ crossing ⇒ $I_A^\infty < 1$

Step 5: Convergence threshold

EXIT Chart: Convergence Trajectory

BICM-ID BER at Successive Iterations

Parameters

Minimum Tunnel Width vs. SNR

Parameters

Example: Convergence Threshold for 16-QAM BICM-ID at Rate 1/21/21/2

Setup: EXIT curves at candidate SNRs

Decoder EXIT for the $[7,5]$ code at rate $1/2$

Read off the threshold

Compare with one-shot BICM

How Many Iterations Does BICM-ID Actually Need?

Local Stability vs. Global Convergence

Common Mistake: Demapper Curve Ending Below (1,1)(1, 1)(1,1)

Quick Check

Quick Check

Key Takeaway

Definition:
Tunnel Width $\Delta(I_A)$

Example: Convergence Threshold for 16-QAM BICM-ID at Rate $1/2$

Common Mistake: Demapper Curve Ending Below $(1, 1)$