Ferkans — Interactive Telecom Tutor

Definition:
Time Diversity via Repetition Coding

Time diversity transmits the same symbol (or coded version) at multiple time instants separated by more than the coherence time $T_c$ of the channel:

$s(t), \quad s(t + \Delta t), \quad s(t + 2\Delta t), \quad \ldots$

with $\Delta t > T_c$ . Each transmission experiences an approximately independent fade, providing diversity order equal to the number of repetitions $L_t$ .

The cost is a rate reduction by factor $1/L_t$ : to achieve diversity order $L_t$ , the effective data rate is reduced to $R/L_t$ . This makes pure repetition coding inefficient; in practice, coding with interleaving is preferred.

Time diversity requires the channel to vary (i.e., fading must be present). In a static channel, repeating the symbol provides no diversity — only an SNR (power) gain.

Definition:
Interleaving to Break Fade Correlation

An interleaver permutes coded symbols before transmission so that symbols that are adjacent in the code are separated by at least $T_c$ in time. After the channel, a deinterleaver restores the original order before decoding.

The interleaving depth $D$ must satisfy

$D \geq T_c / T_s$

where $T_s$ is the symbol period. This ensures that a single fade event (lasting approximately $T_c$ ) corrupts symbols that are well-separated in the codeword, allowing the code to correct them.

Interleaving converts a bursty error channel (where errors come in clusters during deep fades) into an approximately memoryless channel with independent errors — the channel model for which most error-correcting codes are designed.

Interleaving does not add redundancy and does not reduce the code rate. It is a "free" diversity technique in terms of rate, but it introduces latency equal to the interleaving depth.

Theorem: Rank and Determinant Criterion for Space-Time Code Design

Consider a space-time code that transmits codeword matrices $\mathbf{C}_i$ ( $T \times N_t$ ) over a quasi-static fading channel with $N_r$ receive antennas. Define the codeword difference matrix

$\mathbf{D}_{ij} = \mathbf{C}_i - \mathbf{C}_j$

and the product matrix $\mathbf{A}_{ij} = \mathbf{D}_{ij}^H \mathbf{D}_{ij}$ .

Rank criterion (diversity): The minimum rank of $\mathbf{A}_{ij}$ over all distinct codeword pairs determines the diversity order:

$d = N_r \cdot \min_{i \neq j} \operatorname{rank}(\mathbf{A}_{ij})$

For full diversity, $\mathbf{A}_{ij}$ must be full rank ( $= N_t$ ) for every pair.
Determinant criterion (coding gain): Subject to the rank criterion being satisfied, the coding gain is determined by

$\min_{i \neq j} \left(\prod_{k=1}^{r_{ij}} \lambda_k(\mathbf{A}_{ij})\right)^{1/r_{ij}}$

where $\lambda_1, \ldots, \lambda_{r_{ij}}$ are the nonzero eigenvalues of $\mathbf{A}_{ij}$ . This product should be maximised for the best coding gain.

The rank criterion ensures that the codeword difference matrix "spans" all transmit dimensions, so no direction in the spatial channel is wasted. The determinant criterion then measures how well-separated the codewords are in the space-time domain — the space-time analogue of minimum distance for scalar codes.

Proof

Pairwise error probability

The pairwise error probability (PEP) of confusing $\mathbf{C}_i$ with $\mathbf{C}_j$ is bounded by

$P(\mathbf{C}_i \to \mathbf{C}_j) \leq \prod_{k=1}^{r} \frac{1}{1 + \lambda_k \text{SNR}/(4N_t)}$

where $r = \operatorname{rank}(\mathbf{A}_{ij})$ and $\lambda_k$ are the nonzero eigenvalues.

High-SNR approximation

At high SNR:

$P(\mathbf{C}_i \to \mathbf{C}_j) \leq \left(\prod_{k=1}^{r} \lambda_k\right)^{-N_r} \left(\frac{\text{SNR}}{4N_t}\right)^{-r N_r}$

The diversity order is $r \cdot N_r$ , maximised when $r = N_t$ .

Design criteria

Maximise the minimum rank $\to$ rank criterion
Maximise the minimum eigenvalue product $\to$ determinant criterion $\blacksquare$

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Definition:
Frequency Diversity

Frequency diversity is obtained when the signal bandwidth $W$ exceeds the coherence bandwidth $B_c$ of the channel:

$W > B_c$

In this case, different frequency components of the signal experience approximately independent fading, providing frequency diversity of order approximately $L_f \approx \lceil W / B_c \rceil$ .

Frequency diversity arises naturally in:

Wideband/spread-spectrum systems: the signal bandwidth is intentionally made much larger than $B_c$
OFDM: subcarriers separated by more than $B_c$ experience independent fading; coding across subcarriers extracts diversity
Frequency hopping: the carrier frequency is changed every symbol or slot, sampling different parts of the spectrum

Example: Interleaver Depth for Mobile Channel

A mobile user moves at $v = 60$ km/h at carrier frequency $f_c = 2$ GHz. The symbol rate is $R_s = 15$ ksymbols/s (a voice channel).

(a) Compute the coherence time $T_c$ .

(b) Determine the minimum interleaving depth (in symbols).

(c) Compute the resulting latency.

Solution

Coherence time

Maximum Doppler shift: $f_d = v f_c / c = (60/3.6) \times 2 \times 10^9 / (3 \times 10^8) = 111$ Hz.

Coherence time: $T_c \approx 1/(4 f_d) = 1/444 = 2.25$ ms.

(Using the common approximation $T_c \approx 0.423/f_d = 3.8$ ms gives a similar order of magnitude.)

Minimum interleaving depth

Symbol period: $T_s = 1/R_s = 1/15000 = 66.7\;\mu$ s.

Depth: $D \geq T_c / T_s = 2.25 \text{ ms} / 66.7\;\mu\text{s} \approx 34$ symbols.

In practice, the interleaver should span several coherence times for robust diversity. A typical choice is $D = 4 T_c / T_s \approx 135$ symbols.

Latency

Minimum latency $= D \times T_s = 34 \times 66.7\;\mu\text{s} = 2.3$ ms.

With $D = 135$ : latency $= 9$ ms. This is acceptable for voice (which tolerates up to $\sim$ 100 ms end-to-end delay) but may be problematic for ultra-low-latency applications. $\blacksquare$

Example: Frequency Diversity from Wideband Transmission

A channel has delay spread $\sigma_\tau = 1\;\mu$ s.

(a) Estimate the coherence bandwidth.

(b) For a signal bandwidth of $W = 5$ MHz, what is the approximate frequency diversity order?

(c) How does this compare with a narrowband signal ( $W = 200$ kHz)?

Solution

Coherence bandwidth

$B_c \approx \frac{1}{5\sigma_\tau} = \frac{1}{5 \times 10^{-6}} = 200 \text{ kHz}$ $

Wideband diversity order

$L_f \approx \lceil W / B_c \rceil = \lceil 5000/200 \rceil = 25$ $

The wideband signal resolves approximately 25 independent frequency bins, each experiencing independent fading. With appropriate coding across frequency, the system can exploit this diversity.

Narrowband comparison

For $W = 200$ kHz $\leq B_c$ : $L_f = 1$ (flat fading, no frequency diversity).

The wideband signal gains 25-fold frequency diversity — a dramatic advantage that explains why spread-spectrum and OFDM systems are inherently more robust in multipath channels. $\blacksquare$

Quick Check

A channel has coherence time $T_c = 5$ ms and the symbol period is $T_s = 100\;\mu$ s. What is the minimum interleaving depth?

5 symbols

50 symbols

500 symbols

10 symbols

Correction:

50 symbols

$D \geq T_c / T_s = 5 \text{ ms} / 100\;\mu\text{s} = 50$ symbols. This ensures that adjacent codeword symbols are separated by at least one coherence time, experiencing approximately independent fading.

Common Mistake: Interleaving Adds Latency

Mistake:

Designing deep interleavers without considering the end-to-end latency constraint of the application.

Correction:

The latency introduced by a block interleaver is at least $D \times T_s$ at both the transmitter and receiver, giving a minimum round-trip contribution of $2D \times T_s$ .

For voice ( $< 100$ ms), interleaving depths of a few hundred symbols are feasible. For 5G URLLC ( $< 1$ ms), deep interleaving is impossible — the system must rely on spatial diversity (multiple antennas) or frequency diversity (wideband OFDM) instead of time diversity.

This latency-diversity trade-off is a fundamental constraint in system design. Low-latency systems must use space or frequency diversity; time diversity is available only to delay-tolerant traffic.

OFDM Naturally Provides Frequency Diversity

In OFDM systems, the data is spread across $K$ subcarriers, each spaced by $\Delta f$ . If $K \Delta f > B_c$ , different subcarriers experience independent fading. Coding across subcarriers (e.g., turbo/LDPC codes spanning the full bandwidth) automatically extracts frequency diversity without any explicit repetition.

In LTE/5G NR, the resource grid spans up to 100 MHz (5G) across frequency, providing substantial frequency diversity even without interleaving in time. This is one reason why OFDM-based systems are inherently more robust than narrowband single-carrier systems.

Comparison of Diversity Dimensions

Property	Space Diversity	Time Diversity	Frequency Diversity
Mechanism	Multiple antennas	Repeat/interleave over time	Wideband/code across frequency
Independence condition	Spacing $> \lambda/2$	Separation $> T_c$	Separation $> B_c$
Typical order	$N_t \times N_r$	$L_t$ (repetitions)	$\lceil W/B_c \rceil$
Rate cost	None (MRC) or small (STBC)	$1/L_t$ (repetition)	None (inherent in wideband)
Latency cost	None	$L_t \times T_c$	None
Hardware cost	Extra antennas + RF chains	None	Wider bandwidth allocation

Key Takeaway

Interleaving is the cheapest form of diversity in terms of hardware: it requires no extra antennas, no extra bandwidth, and no extra transmit power — only memory and latency. Combined with a good error-correcting code, interleaving converts a bursty fading channel into an approximately memoryless one, enabling the code to operate near its AWGN performance. The price is latency, which limits time diversity to delay-tolerant applications.

Interleaving

A permutation applied to coded symbols before transmission that separates adjacent codeword symbols in time (or frequency) to break fading correlation. Enables time or frequency diversity without additional redundancy.

Coherence Time

The time duration over which the channel impulse response remains approximately constant. Two transmissions separated by more than $T_c$ experience approximately independent fading. $T_c \approx 1/(4 f_d)$ where $f_d$ is the maximum Doppler spread.

Frequency Diversity

Diversity obtained by transmitting over a bandwidth exceeding the channel coherence bandwidth $B_c$ . Different frequency components experience independent fading, providing diversity order approximately $W/B_c$ .

Time and Frequency Diversity

Definition: Time Diversity via Repetition Coding

Definition: Interleaving to Break Fade Correlation