Ferkans — Interactive Telecom Tutor

Where the "Fully Interleaved" Assumption Breaks

The diversity theorem of s03 relied on the fully-interleaved Rayleigh assumption: every coded bit rides an INDEPENDENT fading realisation. Real propagation channels do not oblige. They have a coherence time $T_c$ — measured in symbol periods — during which the fading amplitude $|h|$ changes negligibly. Within a coherence block of $T_c$ symbols, all coded bits share the same $|h|$ . If the interleaver spans $N$ symbols with $N \gg T_c$ , the fully-interleaved picture is a good approximation. If $N \lesssim T_c$ , multiple coded bits see the same fade, and diversity is lost.

This section quantifies that loss: the effective diversity is $\min(d_H, \lceil N / T_c \rceil)$ . The formula has immediate engineering consequences — it tells the standards designer how long the interleaver must be to extract the code's full $d_H$ -order diversity on a specific channel.

The tradeoff is between diversity and latency: a longer interleaver gives more diversity but more buffering delay. Every wireless standard has had to make this tradeoff, and the choices made — 3G HSPA's 20 ms turbo interleaver, LTE's $\sim 1$ ms sub-block interleaver, 5G NR's $\leq 0.5$ ms URLLC interleaver — reveal how the application tolerances drive the design.

Definition:
Coherence Time $T_c$ of a Fading Channel

For a time-varying Rayleigh fading channel with Doppler spread $f_d$ , the coherence time is approximately $T_c \approx \frac{c}{f_d} \cdot \frac{1}{\text{symbol period}} \quad\text{(in symbols)},$ where $c$ is a proportionality constant of order $0.1$ – $0.4$ . More precisely, $T_c$ is the duration over which the normalised autocorrelation $|\mathbb{E}[h_i^* h_{i + T_c}]| \approx 1/2$ . A block-fading model idealises this: the channel is constant for $T_c$ symbols, then independently redrawn.

At 3 GHz carrier and pedestrian speed (1 m/s), $f_d \approx 10$ Hz, so at symbol rate 1 MHz we have $T_c \sim 30$ – $100 \times 10^3$ symbols. At vehicular speed (30 m/s) the same carrier gives $T_c \sim 10^3$ symbols. A minimum-latency interleaver of 1 ms at 1 Msymb/s spans $10^3$ symbols — marginal for vehicular UEs, fine for stationary ones.

Definition:
Effective Number of Independent Fading Samples $N_{\rm eff}$

For a block-fading channel with coherence $T_c$ and interleaver of length $N$ symbols, the effective number of independent fading samples seen by a single codeword is $N_{\rm eff} = \left\lceil \frac{N}{T_c} \right\rceil.$ This is the number of distinct fading blocks that the coded symbols span. In the limit $T_c \to 1$ (fast fading), $N_{\rm eff} = N$ , and we recover the fully-interleaved model. In the limit $T_c \to N$ (quasi- static fading), $N_{\rm eff} = 1$ .

Theorem: BICM Diversity with Finite Interleaver Depth

For BICM with binary code of minimum Hamming distance $d_H$ , labelling $\mu$ , and interleaver of length $N$ symbols, transmitted over a block- fading Rayleigh channel with coherence time $T_c$ symbols, the effective diversity order is $\boxed{\; d_{\rm eff} = \min\!\left(d_H \cdot L_{\min}(\mu), \; N_{\rm eff} \cdot L_{\min}(\mu)\right) = L_{\min}(\mu) \cdot \min(d_H, N_{\rm eff}), \;}$ where $N_{\rm eff} = \lceil N / T_c \rceil$ . In particular, for Gray labelling ( $L_{\min} = 1$ ), $d_{\rm eff}(\mu_G) = \min(d_H, N_{\rm eff}).$ When $N_{\rm eff} \ge d_H$ the interleaver extracts the full code diversity; when $N_{\rm eff} < d_H$ the interleaver length caps the diversity at $N_{\rm eff}$ .

Intuitively, the channel provides $N_{\rm eff}$ independent fading samples over the codeword span. The code can harvest at most $d_H$ of them (through its minimum Hamming distance), but it cannot harvest more than the channel provides. Hence the min. The message for the standards designer is: scale the interleaver so $N_{\rm eff} \ge d_H$ , or the code's full power is wasted. If latency caps $N$ , one should then cap $d_H$ accordingly — no point in paying for a rate- $1/6$ , $d_H = 30$ code if $N_{\rm eff} = 8$ .

Show Hint

Re-do the proof of Thm. 3 of s03 but with the fading realisations of the $d_H$ differing coded bits no longer i.i.d. — they now share fading within each coherence block of $T_c$ .

Within a single coherence block, all symbols experience the same $|h|$ , so the PEP contribution from bits within that block is $P(d_{\text{within}}, |h|) =$ conditional AWGN PEP — but then averaging over the single $|h|$ gives only ONE power of $1/(1 + \alpha \text{SNR})$ , not $d_{\text{within}}$ powers.

Across coherence blocks, the $|h|$ 's are independent, so each contributes an independent power. The total diversity is the NUMBER OF DISTINCT BLOCKS the codeword hits, not the number of bits.

For an interleaver that uniformly spreads the $d_H$ differing bits over $N_{\rm eff}$ blocks, the diversity is $\min(d_H, N_{\rm eff})$ .

Proof

Step 1: Decompose the bit errors by coherence block

Partition the interleaver of length $N$ into $N_{\rm eff}$ blocks of size $T_c$ each (final block possibly truncated). Let $d_j$ be the number of differing coded bits that fall in block $j$ ; then $\sum_j d_j = d$ (the Hamming distance).

Step 2: Conditional PEP given fading

Given fadings $|h_1|, \ldots, |h_{N_{\rm eff}}|$ (one per block), the conditional PEP bound from Thm. 1 of s01 specialised to fading gives $P(\mathbf{c} \to \hat{\mathbf{c}} \mid \{|h_j|\}) \le \exp\!\left(-\sum_j \frac{|h_j|^2 d_j d^2_{\rm avg}(\mu)}{4 N_0}\right).$ Note that within block $j$ , all $d_j$ differing bits share $|h_j|^2$ — they sum to $d_j |h_j|^2$ rather than $d_j$ independent $|h|^2$ terms.

Step 3: Average over independent fading blocks

The $|h_j|^2$ are independent exponentials. Taking expectation: $P(\mathbf{c} \to \hat{\mathbf{c}}) \le \prod_{j=1}^{N_{\rm eff}} \frac{1}{1 + \frac{d_j d^2_{\rm avg}(\mu)}{4 N_0} \text{SNR}}.$ At high SNR this is approximately $\prod_j (d_j d^2_{\rm avg}(\mu) \text{SNR}/(4N_0))^{-1} \cdot \mathbb{1}\{d_j > 0\}$ .

Step 4: Count the nonzero $d_j$

The PEP at high SNR decays as $P \sim \text{SNR}^{-k}$ where $k = |\{j : d_j > 0\}|$ . This $k$ is the number of DISTINCT coherence blocks that the differing coded bits visit. Under a uniform interleaver over $N$ symbols and $d$ differing bits, the expected $k$ is $\min(d, N_{\rm eff}) \cdot (1 - (1 - 1/N_{\rm eff})^d)$ for large $N_{\rm eff}$ ; the worst-case (all bits land in the same block) is $k = 1$ .

Step 5: Worst-case and best-case diversity

The worst-case (sub-optimal interleaver) gives diversity $1$ ; the best-case (optimally spread interleaver) gives $k = \min(d, N_{\rm eff})$ . For a well-designed random interleaver with $N \gg d T_c$ , the probability of "bad spreading" ( $k < d$ ) is small, and with high probability $k = d$ . Combining with the $L_{\min}(\mu)$ factor from s03 gives the stated formula. $\blacksquare$

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Key Takeaway

A finite interleaver caps the diversity at $N_{\rm eff} = N / T_c$ . The code's minimum distance $d_H$ gives an UPPER BOUND on diversity, but the interleaver's ability to spread coded bits across independent coherence blocks gives another upper bound. The actual diversity is the smaller of the two: $d_{\rm eff} = L_{\min}(\mu) \cdot \min(d_H, N_{\rm eff})$ . Design rule: size $N$ so that $N \ge d_H \cdot T_c$ . Making $N$ larger than this does not help; making it smaller throws away code diversity.

Example: Interleaver Depth for Diversity 4 at $T_c = 10$ Symbols

A system uses rate- $1/2$ convolutional code with $d_{H, {\rm free}} = 4$ and Gray-QPSK on a block-Rayleigh channel with coherence time $T_c = 10$ symbol periods. What is the minimum interleaver length $N$ needed to achieve full diversity $d_{\rm eff} = 4$ ? What happens to the BER if $N = 20$ (i.e., half the needed length)?

Solution

Minimum $N$ for full diversity

For full diversity $d_{\rm eff} = d_H \cdot L_{\min}(\mu_G) = 4$ , we need $N_{\rm eff} \ge 4$ , i.e., $N \ge d_H \cdot T_c = 4 \cdot 10 = 40 \text{ symbols.}$ At Gray QPSK (2 bits/symbol), this is $N = 80$ coded bits minimum.

Effect of $N = 20$

With $N = 20$ , $N_{\rm eff} = 20/10 = 2$ . Therefore $d_{\rm eff} = \min(4, 2) = 2$ . The BER slope at high SNR drops from $-4$ to $-2$ , i.e., each $3$ -dB SNR increase only halves the BER instead of doubling-each-4th. At BER $10^{-6}$ , the required SNR jumps from about $10 \text{ dB}$ (full diversity 4) to about $22 \text{ dB}$ (effective diversity 2) — a $12 \text{ dB}$ penalty, drastic by any wireless standard.

Design choice

Either use the full $N = 40$ interleaver (latency cost: 40 symbol periods $\approx 40 \mu s$ at 1 Msymb/s — tolerable for most applications except URLLC), or accept diversity-2 and choose the waveform parameters accordingly. A common middle ground in practice is to use $N \ge 2 d_H T_c$ to give margin against interleaver non-uniformity and variation in $T_c$ .

BER vs Interleaver Length at Fixed Coherence Time

Average BER versus $E_s/N_0$ for BICM with Gray-16-QAM and a binary code of $d_H = 8$ , over a block-Rayleigh channel with coherence time $T_c$ symbols, plotted for a range of interleaver lengths $N$ . Observe that: (a) curves for $N_{\rm eff} \ge d_H$ overlap (full diversity achieved); (b) curves for smaller $N$ exhibit progressively smaller slopes, converging onto the flat uncoded $\text{SNR}^{-1}$ line as $N \to T_c$ . The transition cliff at $N_{\rm eff} = d_H$ is the "latency-diversity knee" that every wireless standards designer must navigate.

Parameters

Coherence time [symbols]4

d_H

8

Common Mistake: Longer Interleavers Do NOT Always Help

Mistake:

A naive design rule "make the interleaver as long as possible" is both unhelpful and occasionally damaging. The rule is unhelpful because the diversity saturates at $d_H \cdot L_{\min}(\mu)$ once $N_{\rm eff} \ge d_H$ : making $N$ longer than $d_H \cdot T_c$ gives no diversity gain, only extra latency. It is damaging because very long interleavers couple with channel coherence VARIATION — the assumption of independent $|h|$ across blocks breaks if the statistics shift across $N$ symbols (e.g., due to changing Doppler in a moving terminal).

Correction:

Size the interleaver to just exceed $d_H \cdot T_c$ , with a modest margin (say, $2\times$ ) for interleaver non-uniformity and channel coherence variability. Further increases do not help and may expose the system to non-stationary fading. This is why real standards specify interleaver lengths matched to the expected fading rate, not maximum possible lengths.

⚠️Engineering Note

LTE Sub-Block Interleaver and HARQ

LTE's turbo code is rate $1/3$ with $d_{H,{\rm free}} = 10$ (approximate). The sub-block interleaver (TS 36.212 §5.1.4) is $32 \times \lceil N/32 \rceil$ rectangular, spanning up to $\sim 6000$ information bits. At LTE symbol rate (15 kHz subcarrier, OFDM), this corresponds to about 1 ms of time-domain span — the LTE TTI. Against vehicular-channel coherence time of $\sim 1$ ms (30 m/s, 2 GHz), this gives $N_{\rm eff} \approx 1$ within a single transmission. Full diversity is only recovered through HARQ retransmissions, which spread the codeword across multiple TTIs. This is why LTE with mobile UEs relies so heavily on HARQ: without it, the single-transmission diversity is insufficient for the turbo code's $d_H = 10$ .

Practical Constraints

•
Sub-block interleaver: $32 \times \lceil N/32 \rceil$ rectangular.
•
TTI: 1 ms (14 OFDM symbols, Type-1 FDD).
•
Maximum block: 6144 info bits (before segmentation).
•
HARQ: up to 4 retransmissions, adaptive asynchronous.

📋 Ref: 3GPP TS 36.212 §5.1.4

🔧Engineering Note

DVB-S2 Block LDPC Interleaver and the 64 800-Bit Frame

The DVB-S2 satellite-TV standard uses LDPC codes of length 64 800 bits (or short frames of 16 200 bits) with effective $d_H$ in the range $30$ – $50$ for the high-rate MODCODs. Combined with 16-APSK or 32-APSK modulation and a block-level bit interleaver, a single FEC frame spans $N = 64800/L$ symbols (e.g., 16 200 symbols for 16-APSK). Against atmospheric scintillation coherence times of 1–10 seconds at Ku-band (1 kHz to 10 kHz symbol rates), this is $N_{\rm eff} \sim 10^3$ — far exceeding $d_H$ , so the code achieves its full asymptotic diversity. This is why DVB-S2 performance under rain-fade scintillation is close to the theoretical BICM bound.

Practical Constraints

•
Normal frame: 64 800 coded bits (long LDPC).
•
Short frame: 16 200 coded bits (1/4 the span).
•
Modulations: QPSK, 8-PSK, 16-APSK, 32-APSK.
•
Per-stream interleaver: block-level, symbol-aligned.

📋 Ref: ETSI EN 302 307-1 §5.1–5.3

🚨Critical Engineering Note

5G NR URLLC: Diversity-Latency Tradeoff Cranked to 11

5G NR Ultra-Reliable Low-Latency Communications (URLLC) aims for 1 ms user-plane latency and $10^{-5}$ BER. The 1 ms budget includes transmission, decoding, and HARQ turnaround — leaving only $\sim 0.2$ ms for the interleaver itself. At 30 kHz subcarrier (120 μs OFDM symbol duration) this is only $\sim 2$ OFDM symbols, or about 400 REs (resource elements). With 30 m/s mobility at 3.5 GHz, $T_c \approx 1$ ms $\gg$ interleaver span — so URLLC must harvest frequency diversity instead of time diversity. The design explicitly spreads coded bits across multiple PRBs (12-subcarrier resource blocks) to get $N_{\rm eff}$ of order $d_H = 10$ – $15$ from frequency coherence of tens of subcarriers. The principle is the same as this section's theorem — the "fading samples" are just frequency-indexed instead of time-indexed.

Practical Constraints

•
Latency budget: 1 ms end-to-end.
•
BLER target: $10^{-5}$ for URLLC services.
•
Subcarrier spacing: 30 or 60 kHz (shorter symbols).
•
Interleaver: cross-PRB frequency spreading mandatory.

📋 Ref: 3GPP TS 38.212 §5.4, 3GPP TR 38.824

Why This Matters: Backward: Ch. 5 Capacity; Forward: Ch. 7 Error Exponents and Ch. 8 BICM-ID

This chapter's error-probability analysis mirrors Ch. 5's capacity analysis. Chapter 5 asked: "What is the achievable rate under BICM?" and answered $C_{\rm BICM}(\mu) = \sum_\ell I(Y; B_\ell)$ . Chapter 6 asked: "What is the achievable error probability?" and answered $P_b \propto \text{SNR}^{-d_H L_{\min}(\mu)}$ (on fading) or $P_b \le Q(\sqrt{d_H d^2_{\rm avg}(\mu)/2 \cdot E_s/N_0})$ (on AWGN). Together, these two results give the coded-modulation designer both a rate target and a BER slope.

Chapter 7 extends the error-probability analysis to ERROR EXPONENTS: the relationship between the rate at which BER can decay and how close one operates to capacity. The central quantity there is the cutoff rate $R_0$ , a tighter bound than the union bound at waterfall SNRs.

Chapter 8 revisits the labelling question once iterative decoding is added: BICM-ID feeds soft information from the decoder back to the demapper, and with enough iterations, set-partition labelling (!) can win over Gray in a regime where the EXIT-chart analysis shows convergence. The $L_{\min}(\mu)$ quantity of this chapter will be augmented by an iterative "tunneling" condition in Ch. 8.

Historical Note: The Long Road of Interleaver Design: from GSM to 5G

1991–present

The first cellular standard — GSM, deployed in 1991 — used a diagonal convolutional interleaver of length 8 time slots (about 40 ms). This spanned roughly $N_{\rm eff} \approx 10$ coherence intervals at pedestrian speed and 2 GHz carrier — enough for the GSM (rate 1/2, $d_H = 5$ ) convolutional code to achieve close to full diversity on typical urban channels. At vehicular speeds the 40 ms interleaver helped less, and GSM's BER floor at high mobility was a known problem.

The IS-95 CDMA standard (1993) used a long block interleaver (20 ms span, 576 bits) designed to extract full diversity from its rate- $1/3$ convolutional code ( $d_H = 15$ ) with the Rake receiver contributing additional path diversity. The combination was famously robust — IS-95 worked better than GSM at high mobility — at the cost of added end-to-end latency.

3G UMTS/HSPA introduced the turbo code (rate 1/3, $d_H \approx 20$ ) with a sophisticated contention-free permutation interleaver of up to 5114 bits, spanning one TTI of 20 ms initially (later 2 ms in HSPA+). This gave effective-diversity budgets matched to the turbo code's distance profile.

4G LTE shortened the TTI to 1 ms and kept the turbo code, trading some diversity for latency — compensated by aggressive HARQ combining. 5G NR replaced the turbo code with LDPC and shortened the TTI further still, pushing interleaver depth to $< 0.5$ ms for URLLC.

The thread through these five generations is the same theorem of this section: effective diversity $\le N / T_c$ . Each generation re-tuned $N$ , $T_c$ (by moving to higher frequency or wider subcarrier spacing), and the code's $d_H$ to match its latency requirements. What looks like a parade of specialised engineering choices is, underneath, a single tradeoff curve sampled at different operating points.

Quick Check

A BICM system uses a rate- $1/2$ convolutional code with $d_{H, {\rm free}} = 6$ and Gray-QPSK on a fading channel with coherence time $T_c = 5$ symbols. The interleaver length is $N = 40$ symbols. What is the effective diversity order?

$3$

$5$

$6$

$8$

Correction:

6

By Thm. 5, $d_{\rm eff} = L_{\min}(\mu_G) \cdot \min(d_H, N_{\rm eff}) = 1 \cdot \min(6, 40/5) = \min(6, 8) = 6$ . The interleaver is long enough to extract the code's full diversity.

Interleaver Depth and Finite-Coherence Fading

Where the "Fully Interleaved" Assumption Breaks

Definition: Coherence Time TcT_cTc​ of a Fading Channel

Definition: Effective Number of Independent Fading Samples NeffN_{\rm eff}Neff​

Theorem: BICM Diversity with Finite Interleaver Depth

Step 1: Decompose the bit errors by coherence block

Step 2: Conditional PEP given fading

Step 3: Average over independent fading blocks

Step 4: Count the nonzero $d_j$

Step 5: Worst-case and best-case diversity

Key Takeaway

Example: Interleaver Depth for Diversity 4 at Tc=10T_c = 10Tc​=10 Symbols

Minimum $N$ for full diversity

Effect of $N = 20$

Design choice

BER vs Interleaver Length at Fixed Coherence Time

Parameters

Common Mistake: Longer Interleavers Do NOT Always Help

LTE Sub-Block Interleaver and HARQ

DVB-S2 Block LDPC Interleaver and the 64 800-Bit Frame

5G NR URLLC: Diversity-Latency Tradeoff Cranked to 11

Why This Matters: Backward: Ch. 5 Capacity; Forward: Ch. 7 Error Exponents and Ch. 8 BICM-ID

Historical Note: The Long Road of Interleaver Design: from GSM to 5G

Quick Check

Definition:
Coherence Time $T_c$ of a Fading Channel

Definition:
Effective Number of Independent Fading Samples $N_{\rm eff}$

Example: Interleaver Depth for Diversity 4 at $T_c = 10$ Symbols