Tri-Stream Decoder

• Tri-Stream Decoding is a novel architecture that coordinates three symbol streams to enable low-latency decoding in relay and MIMO systems.
• It leverages staggered diagonal embedding of MDS codes for erasure correction under strict delay constraints with reduced computational complexity.
• This approach is applicable in ultra-reliable low-latency communication scenarios like telesurgery, autonomous driving, and real-time control.

A Tri-Stream Decoder is a decoding architecture for streaming and space-time coded systems in relay networks and distributed MIMO, characterized by the coordinated processing of three packet or symbol streams—typically corresponding to source, relay, and destination or, in spatial multiplexing, to separate codeword groupings. This architecture is notably relevant in the decode-and-forward (DF) relay network under stringent delay constraints as well as distributed MIMO scenarios where structural partitioning enables complexity reduction and low-latency decoding. The Tri-Stream Decoder leverages code and channel properties such as orthogonality, symbol grouping, and diagonal embedding to enable parallel, independent, or conditional decoding of constituent streams for robust performance under burst and random erasures, with computational complexity matching that of optimal MDS-code-based decoders.

1. Network and Channel Model in Tri-Stream Decoding

The Tri-Stream Decoder is rooted in the three-node decode-and-forward relay network consisting of source ($s$), relay ($r$), and destination ($d$). The source transmits an infinite stream of packets $\{m_n\}$, which are relayed and re-encoded at $r$ before reaching $d$. Each link (source-to-relay and relay-to-destination) is modeled by a delay-constrained sliding-window (DCSW) erasure channel. The DCSW $(a, b, T)$ model assumes that, within any window of size $T+1$, up to $a$ arbitrary (random) erasures or a burst erasure of up to $b$ consecutive packets may occur, with $b \geq a$. This model is practical for streaming over networks with bursty and random loss patterns, closely approximating real-world wireless or congested network behavior (Singhvi et al., 2022).

Delay Constraints

A key operational requirement is that each packet transmitted at time $i$ by $s$ must be decoded at $d$ by time $i + T$, enforcing a per-packet maximum decoding latency. The relay employs a symbol-wise decode-and-forward method where individual symbols from source packets may be delayed or reordered to optimize delay profiles and reliability. Each symbol $l$ in a $k$-symbol packet is thus associated with a delay profile $(t_\ell, \tau_\ell)$, corresponding to hops $s \rightarrow r$ and $r \rightarrow d$ respectively, with $t_\ell + \tau_\ell \leq T$ for all $\ell$.

2. Theoretical Capacity Bound

For a point-to-point $(a, b, T)$ DCSW channel, the code rate is upper bounded by

$\mathcal{R} \leq \frac{T - a + 1}{T - a + 1 + b}$

This expression, denoted as $C_{(a, b, T)}$, governs the fundamental tradeoff between redundancy and delay required for resilience to both random and burst erasures.

In the three-node relay scenario, the effective rate is

$$\mathcal{R}_{(s, r, d)} = \frac{k}{\max\{n_1, n_2\}}$$

where $n_1, n_2$ are the codeword lengths on $s \rightarrow r$ and $r \rightarrow d$. The overall network capacity is given by

$$C_{(a_1, b_1, a_2, b_2, T)} \leq \min \left\{ C_{(a_1, b_1, T-b_2)},\; C_{(a_2, b_2, T-b_1)} \right\}$$

ensuring neither hop can become a bottleneck subject to DCSW constraints and the end-to-end delay $T$ (Singhvi et al., 2022).

3. Code Construction: Staggered Diagonal Embedding (SDE)

Tri-Stream decoding systems achieve rate-optimality by employing staggered diagonal embedding (SDE) of Maximum Distance Separable (MDS) codes. In SDE, the $n$ symbols of a base $[n, k]$ MDS codeword are dispersed diagonally across $N \leq T+1$ consecutive transmitted packets such that any pattern of up to $a$ random or $b$ burst erasures within a window can be corrected.

The embedding is specified by a subset $S \subseteq [0, N-1]$ and for each transmission index $i$,

$$(x_{i+i_0}(0),\; x_{i+i_1}(1),\; ...,\; x_{i+i_{n-1}}(n-1))$$

constitute a codeword in the base code. The embedding parameter $N$ is chosen via

$$N = n + (b - a) \left\lfloor \frac{n-1}{a} \right\rfloor$$

and decoding delay for symbol $j$ is $$t_j = N-1 - j - \left\lfloor \frac{b - a}{a} j \right\rfloor$$.

For the relay network, SDE codes are constructed on both $s \rightarrow r$ and $r \rightarrow d$ links, with the relay decoding, reordering, and re-encoding before final transmission, tightly controlling per-symbol delays to respect the global constraint $t_\ell + \tau_\ell \leq T$.

4. Decoding Architecture and Complexity

The Tri-Stream Decoder operates by interlacing the three constituent streams—source symbols, relay re-encoded symbols, and destination-recovered symbols. Decoding is performed in a streaming (symbol-wise) fashion as packets are received, reconstructing the original stream from any (a, b, T) erasure process.

The decoding procedure inherits the complexity of MDS decoding, such as Reed–Solomon decoding, which is polynomial in $n$. Notably, the required finite field size grows only linearly with the delay constraint $T$, facilitating low-latency operation even for large $T$. This property ensures the approach is computationally feasible for real-time and embedded implementations, supporting strict delay and reliability requirements (Singhvi et al., 2022).

Another dimension is seen in space-time coded MIMO, where related tri-stream (or multi-stream) decoders reduce complexity by partitioning the detection task. For instance, embedded block orthogonality allows division of the maximum-likelihood search space into independent metric components, reducing the complexity from $O(M^8)$ to $O(M^{4.5})$ for square QAM by parallelizing real and imaginary parts, as shown for the 3D MIMO code—a technique explained in detail in (1401.1381). This highlights a general paradigm: when code structure enables group-orthogonality or partial decoupling, tri-stream (and multi-stream) architectures exploit this to enable independent or conditional decoding tasks.

5. Performance and Delay Analysis

The Tri-Stream Decoder achieves the minimum possible delay consistent with the global constraint $T$. Each message symbol is associated with a delay profile, and the SDE construction ensures $t_\ell + \tau_\ell \leq T$ for all $\ell$. The final rate is optimal for the DCSW network under the divisibility condition

$$\max\{b_1, b_2\} \mid (T - b_1 - b_2 - \max\{a_1, a_2\} + 1)$$

(with $a_1 = a_2$ or $b_1 = b_2$). Under these conditions, the SDE-MDS construction attains the theoretical cut-set bound. Practical decoding time is kept minimal by early termination in group searches and by the lack of substantial overhead due to field size or window tracking (Singhvi et al., 2022, 1401.1381).

In space-time-coded contexts, simulation results indicate that the partitioned, parallelized maximum-likelihood decoder architecture (sharing principles with Tri-Stream Decoders) can reduce the average number of nodes visited—and hence processing time—by up to 80% at low SNR and 23% at higher SNRs, compared to classical serial sphere decoding (1401.1381).

6. Application Scenarios

Tri-Stream Decoders underpin streaming code solutions for ultra-reliable low-latency communication (URLLC) in diverse domains. Applications include telesurgery, industrial control, virtual/augmented reality, autonomous driving, and mission-critical wireless where end-to-end latency, burst resilience, and per-packet reliability are essential (Singhvi et al., 2022).

The architecture's modular design, based on SDE-MDS construction and real-time waveform processing, means it is agnostic to whether the streams are spatial (as in MIMO codewords) or temporal/networked (as in relay channels). With its computational efficiency and strong delay guarantees, Tri-Stream Decoding is particularly suited to deployments in real-time systems with severe timing and reliability demands.

7. Further Considerations and Generalization

The Tri-Stream Decoder concept is generalizable to systems where code or channel structure enables partitioning of the joint decoding metric into smaller, independent or parallelizable tasks. For any system—whether networked streaming, space-time coding, or multi-relay communication—where the encoding structure, channel model, or orthogonality property permits group-wise decoupling, the methodologies of group metric decomposition, sorted candidate enumeration, and adaptive pruning can provide substantial reductions in computational and practical latency (1401.1381).

A plausible implication is that ongoing advances in code design, exploiting greater structure and group-wise orthogonality, may further expand the range of systems where tri- or multi-stream decoder architectures achieve optimal performance under computational and delay constraints.