Ferkans — Interactive Telecom Tutor

Definition:
Interleaving Depth for Fading Channels

The interleaving depth $D$ is the time span (in symbol periods or channel coherence times) over which coded bits are spread by the interleaver. For a channel with coherence time $T_c$ and symbol period $T_s$ , the interleaving depth should satisfy

$D \geq \frac{T_c}{T_s} \cdot d_{\min}$

so that the $d_{\min}$ coded bits that distinguish any two codewords experience approximately independent fading realisations.

In frequency-selective channels, interleaving can also be applied across frequency (subcarriers in OFDM) to exploit frequency diversity.

The interleaver does not add redundancy — it rearranges bits so that burst errors caused by deep fades are spread across multiple codewords, enabling the code to correct them.

Theorem: Coding Gain and Diversity from Channel Coding

On a Rayleigh fading channel with perfect interleaving (so that coded bits see independent fades), a code with minimum distance $d_{\min}$ and code rate $R_c$ provides:

Diversity order equal to $d_{\min}$ : the pairwise error probability decays as

$P(\mathbf{c} \to \hat{\mathbf{c}}) \leq \left(\frac{1}{4 R_c E_b/N_0}\right)^{d_{\min}}$

at high SNR.
Coding gain of $R_c \cdot d_{\min}$ in the exponent, changing the slope of the BER vs. SNR curve.

Without coding, uncoded BPSK on Rayleigh fading has diversity order 1 ( $\text{BER} \propto 1/\text{SNR}$ ). With a code of $d_{\min} = 10$ , the BER decays as $1/\text{SNR}^{10}$ — a dramatic improvement.

Each of the $d_{\min}$ differing bit positions sees an independent fade. The probability that all $d_{\min}$ positions simultaneously experience deep fades is the product of individual probabilities, giving diversity order $d_{\min}$ .

Proof

Pairwise error probability

The PEP for two codewords differing in $d$ positions, with independent Rayleigh fading on each, is

$P(\mathbf{c} \to \hat{\mathbf{c}}) = \prod_{l=1}^{d} \frac{1}{1 + \gamma_l}$

where $\gamma_l = |h_l|^2 \cdot 2R_c E_b/N_0$ and $|h_l|^2$ is Rayleigh distributed.

High-SNR approximation

At high SNR, $1/(1 + \gamma_l) \approx 1/\gamma_l$ , and averaging over $|h_l|^2$ :

$P(\mathbf{c} \to \hat{\mathbf{c}}) \leq \left(\frac{1}{4 R_c E_b/N_0}\right)^d$

The minimum over all codeword pairs gives $d = d_{\min}$ . $\blacksquare$

,

Outage-Limited vs. Ergodic Regime

The benefit of coding depends on the relationship between the codeword length and the channel coherence time:

Ergodic regime ( $n T_s \gg T_c$ ): The codeword spans many coherence intervals. With sufficient interleaving, each coded bit sees an independent fade. The relevant performance metric is the ergodic capacity, and good codes achieve the ergodic capacity of the fading channel.

Outage-limited regime ( $n T_s \ll T_c$ ): The entire codeword experiences a single fading realisation. No amount of coding or interleaving can overcome a deep fade. The relevant metric is the outage probability: $P_{\text{out}}(R) = P(\log_2(1 + |h|^2 \text{SNR}) < R)$ .

Intermediate regime: Partial interleaving provides some diversity. The effective diversity order is $\min(d_{\min}, L)$ , where $L$ is the number of independent fading blocks spanned by the codeword.

Example: Interleaving Depth Calculation

A system uses a rate-1/2 convolutional code with $d_{\text{free}} = 10$ over a Rayleigh fading channel with Doppler spread $f_d = 100$ Hz and symbol rate $R_s = 200$ ksymbols/s.

(a) Compute the coherence time $T_c$ .

(b) Determine the minimum interleaving depth (in symbols) to ensure independent fading across the $d_{\text{free}}$ coded bits.

(c) What is the resulting interleaving delay?

Solution

Coherence time

$T_c \approx \frac{0.423}{f_d} = \frac{0.423}{100} = 4.23$ ms.

Interleaving depth

Number of symbols per coherence time: $N_c = T_c \cdot R_s = 4.23 \times 10^{-3} \times 2 \times 10^5 = 846$ symbols.

Minimum interleaving depth: $D \geq N_c \times d_{\text{free}} = 846 \times 10 = 8460$ symbols.

Interleaving delay

Delay $= D / R_s = 8460 / 200000 = 42.3$ ms.

This is acceptable for data applications but may be too large for voice (typically requiring $< 40$ ms one-way delay). $\blacksquare$

Quick Check

Why is interleaving essential when using channel codes over a slowly fading channel?

Interleaving increases the code rate

Interleaving converts burst errors into random errors that the code can correct

Interleaving reduces the noise power

Interleaving adds additional parity bits

Correction:

Interleaving converts burst errors into random errors that the code can correct

A deep fade causes a burst of errors in consecutive bits. Interleaving spreads these errors across different codewords, turning them into isolated random errors within each codeword. The code is designed to correct random errors up to weight $t$ .

Common Mistake: Insufficient Interleaving Depth

Mistake:

Using a powerful code (e.g., turbo or LDPC with large $d_{\min}$ ) on a fading channel without ensuring that the interleaving depth is sufficient for the coherence time.

Correction:

If the interleaving depth is too small, multiple coded bits within the same codeword experience correlated fading. The effective diversity order is then limited to the number of independent fading blocks spanned by the codeword, not $d_{\min}$ .

Rule of thumb: the interleaver should span at least $10 \times T_c$ to provide approximately independent fading for all coded bits. For delay-sensitive applications, frequency-domain interleaving (across OFDM subcarriers) can provide diversity without time delay.

Key Takeaway

The combination of channel coding and interleaving provides two distinct benefits on fading channels: (1) coding gain (same as in AWGN) reduces the required SNR by the code's inherent redundancy, and (2) diversity gain (unique to fading) changes the BER slope from $1/\text{SNR}$ (uncoded) to $1/\text{SNR}^{d_{\min}}$ , providing enormous improvement at high SNR. In modern systems (LTE, 5G NR, Wi-Fi), LDPC or polar codes with time-frequency interleaving across OFDM subcarriers and symbols provide both gains simultaneously.

Interleaving Depth

The time or frequency span over which an interleaver distributes coded bits. Must exceed the product of the channel coherence time and the code's minimum distance to ensure independent fading across coded bits.

Coded Diversity

The diversity order achieved by channel coding on a fading channel with sufficient interleaving, equal to the minimum distance $d_{\min}$ of the code. Improves the BER slope from $1/\text{SNR}$ to $1/\text{SNR}^{d_{\min}}$ .

Coding for Fading Channels

Definition: Interleaving Depth for Fading Channels