Ferkans — Interactive Telecom Tutor

Diversity and Multiplexing, Now in 3D

Classical MIMO offers a choice: use the $N_t \times N_r$ antennas to multiplex multiple streams (spatial multiplexing) or to diversify a single stream (MRC, Alamouti). The Zheng-Tse tradeoff quantifies the tradeoff: at high SNR, diversity $d^*(r) = (N_t - r)(N_r - r)$ for multiplexing gain $r$ . MIMO-OTFS extends this to the delay-Doppler dimensions: you get additional diversity from the $P$ paths. This section derives the combined DD-angle diversity-multiplexing tradeoff and shows when MIMO-OTFS gains the most over MIMO-OFDM.

,

Definition:
Diversity-Multiplexing Tradeoff

The diversity-multiplexing tradeoff (DMT) for a channel is the function $d^*(r)$ defined by $d^*(r) \;=\; -\lim_{\text{SNR} \to \infty} \frac{\log P_{\text{err}}(\text{SNR})}{\log \text{SNR}},$ where $r = R/\log \text{SNR}$ is the multiplexing gain and $P_{\text{err}}$ is the error probability. Intuitively, $d^*(r)$ is the exponent at which error probability decays for a given rate scaling.

For a flat $N_t \times N_r$ MIMO channel: $d^*(r) = (N_t - r)(N_r - r)$ , the Zheng-Tse bound.

Theorem: MIMO-OTFS DMT

For a MIMO-OTFS channel with $N_t \times N_r$ antennas and $P$ resolvable DD-paths, the achievable DMT is $d^*(r) \;=\; (N_t - r)(N_r - r) \cdot P \qquad \text{for integer } r \leq \min(N_t, N_r).$ Interpretation: MIMO-OTFS multiplies the classical MIMO diversity by the path count $P$ .

For a $4 \times 4$ system with $P = 8$ paths: $d^*(0) = 16 \cdot 8 = 128$ (full diversity). This is the DD + spatial diversity stacking that MIMO-OFDM cannot match at high Doppler.

Classical MIMO provides diversity up to $N_t N_r$ : you can lose nearly all antenna pairs and still decode. MIMO-OTFS multiplies this by the number of paths: you can lose all antennas' signals on all but one DD-path tap and still decode. The total diversity $N_t N_r P$ is the product — an enormous reliability at high SNR.

Proof

Pair-wise error probability

PEP of a codeword error at multiplexing $r$ : $\mathrm{PEP}(r) \leq \binom{2 N_t N_r P - 1}{N_t N_r P} (4 \text{SNR})^{-(N_t N_r P - r(N_t N_r) )}$ by the standard pair-wise analysis for channels with $N_t N_r P$ diversity.

Diversity order

At $r = 0$ : $d^*(0) = N_t N_r P$ . At $r = \min(N_t, N_r)$ : $d^*(r) = 0$ (all diversity consumed by multiplexing).

Linear interpolation

$d^*(r) = (N_t - r)(N_r - r) \cdot P$ for integer $r$ . This is the MIMO-OTFS extension of the Zheng-Tse tradeoff. $\blacksquare$

,

Key Takeaway

MIMO-OTFS diversity multiplies by $P$ . A $4 \times 4$ MIMO-OTFS system with $P = 8$ paths achieves $d^*(0) = 128$ diversity — far beyond any fixed-topology MIMO-OFDM system. This is the quantitative reliability advantage for safety-critical high-mobility applications.

Definition:
Optimal Rate and Diversity Operating Points

For a MIMO-OTFS system at SNR $\text{SNR}$ with target BER $\beta$ , the operating point is:

Multiplexing gain $r^* = \min(N_t, N_r)$ — full multiplexing; maximum rate. Diversity $d^*(r^*) = \max(N_t, N_r) - \min(N_t, N_r) = |N_t - N_r|$ paths (close to $0$ if $N_t \approx N_r$ ).

Diversity-optimal $r^* = 0$ : single-stream transmission; full diversity $d^*(0) = N_t N_r P$ . Maximum reliability.

Balanced $r^* = \min(N_t, N_r)/2$ : half-multiplexing, partial diversity. Typical real-world operating point.

Example: DMT for Urban MIMO-OTFS

An urban MIMO-OTFS deployment: $N_t = 16$ , $N_r = 4$ , $P = 8$ paths. Compute DMT at three operating points: $r = 0$ (diversity- optimal), $r = 2$ (balanced), $r = 4$ (multiplexing-optimal).

Solution

$r = 0$

$d^*(0) = (16)(4)(8) = 512$ . Nearly all paths contribute to every codeword error. Extreme reliability.

$r = 2$

$d^*(2) = (16-2)(4-2)(8) = 14 \cdot 2 \cdot 8 = 224$ . Serving 2 streams with high diversity. Good balance.

$r = 4$

$d^*(4) = (12)(0)(8) = 0$ . Full multiplexing (4 streams); no diversity left. Basic-rate operation.

Engineering implication

Safety-critical: $r = 0$ (single stream, diversity 512). Typical data: $r = 2$ (balanced). High-throughput: $r = 4$ (no diversity but maximum rate). Modern BSs adapt based on link quality.

MIMO-OTFS DMT vs MIMO-OFDM

Plot $d^*(r)$ for MIMO-OTFS and MIMO-OFDM, varying antenna counts and path count. Shows the $\times P$ diversity multiplier of MIMO-OTFS.

Parameters

N_t

4

N_r

4

P

8

Theorem: Ergodic Capacity of MIMO-OTFS

The ergodic capacity of MIMO-OTFS with $N_t \times N_r$ antennas and $P$ -path DD channel is $C \;=\; \mathbb{E}\!\left[\log \det\!\left(\mathbf{I}_{N_r} + \frac{\text{SNR}}{N_t}\, \mathcal{H}_{\text{eff}} \mathcal{H}_{\text{eff}}^H\right)\right],$ where $\mathcal{H}_{\text{eff}}$ is the per-stream effective channel after DD-domain equalization.

Asymptotic scaling: As $\text{SNR} \to \infty$ and $N_t = N_r = N \to \infty$ : $C \to N \log \text{SNR} + \mathcal{O}(1)$ — linear capacity scaling with antenna count. Same as MIMO-OFDM; MIMO-OTFS does not add capacity, it preserves it under high mobility.

Capacity is the ultimate information-theoretic limit. MIMO-OFDM achieves linear capacity scaling under block-fading, but loses this under high Doppler (ICI corrupts each subcarrier). MIMO-OTFS maintains linear scaling even under extreme Doppler — this is the capacity-theoretic case for MIMO-OTFS in mobile scenarios.

Proof

Channel normalization

Normalize the DD-channel matrix to unit average power per transmit antenna. The effective channel for MIMO capacity is $\mathcal{H}_{\text{eff}}$ where inter-cell interference is handled by the detector.

Telatar formula

Telatar's capacity formula applies to the effective channel: $C = \mathbb{E}[\log\det(\mathbf{I} + \frac{\text{SNR}}{N_t} \mathcal{H}_{\text{eff}} \mathcal{H}_{\text{eff}}^H)]$ .

High-SNR scaling

$\log\det \approx N \log\text{SNR}$ for $N = \min(N_t, N_r)$ and $\text{SNR} \gg 1$ . Linear scaling with antenna count. $\blacksquare$

,

Definition:
Spatial Modulation and Antenna Selection

An alternative to parallel spatial multiplexing is spatial modulation (SM): at each DD cell, only one antenna is active, and the antenna index itself carries $\log_2 N_t$ bits of information. SM provides $\log_2(N_t) + \log_2(M)$ bits/cell for an $M$ -QAM constellation — fewer than full multiplexing, but with the benefit of only one RF chain active at a time.

For MIMO-OTFS: SM extends by treating antenna selection per DD cell. The DD-spatial factor graph handles the joint detection naturally. Energy-efficient — important for UE-side operation.

🔧Engineering Note

Rate-Adaptive MIMO-OTFS in 5G/6G

Real systems adapt the rate-diversity operating point based on channel conditions:

Cell-edge (low SINR): Single stream, full diversity ( $r = 0$ , $d^* = N_t N_r P$ ). Safety-critical reliability.
Cell-center (high SINR): Full multiplexing ( $r = \min(N_t, N_r)$ ), basic reliability.
Medium SINR: Balanced ( $r = \min(N_t, N_r)/2$ ), moderate diversity.

Rate adaptation algorithm:

Estimate channel quality (SINR) per user per frame.
Consult lookup table: $(SINR, \text{target BER}) \mapsto r$ .
Configure precoder and detector for chosen $r$ .
Re-evaluate every 10-100 frames.

5G NR Rel. 17 supports link adaptation at $\sim 100$ Hz granularity; 6G expected to support $\sim 1$ kHz (per-frame). MIMO-OTFS fits naturally — the DD-domain sparsity gives stable channel estimates for fast rate adaptation.

Practical Constraints

•
Cell-edge: $r = 0$ , full diversity
•
Cell-center: $r = \min(N_t, N_r)$ , full mux
•
Adaptation rate: 100 Hz (5G), 1 kHz (6G target)

Common Mistake: DMT Saturates at Finite SNR

Mistake:

Assuming the DMT bound holds at operational SNR. The formula $d^*(r) = (N_t - r)(N_r - r) P$ is an asymptotic result — it describes the slope of the BER-vs-SNR curve at $\text{SNR} \to \infty$ . At 20 dB SNR, the curve is not yet in the asymptotic regime.

Correction:

Plot actual BER-vs-SNR curves for your target operating range (e.g., 15-25 dB for 5G). At finite SNR, diversity gain manifests as curve shape, not just slope. For practical design, compare:

Slope at BER = $10^{-4}$ → matches asymptotic $d^*(r)$ for $P \leq 5$ .
SNR gap at BER = $10^{-4}$ → coding gain (different from diversity). The MIMO-OTFS advantage at moderate SNR is often $\sim 10$ dB coding gain plus slope; the slope alone undersells the benefit.

Spatial Multiplexing and Diversity