Ferkans — Interactive Telecom Tutor

When CSI Is Not Available

The point is that all the space-time coding results in Chapters 10-17 assumed the receiver knows the channel. In practice, CSI is never free: pilots cost bandwidth, and in very short bursts or very high mobility the channel changes before CSI estimation stabilises. Non-coherent space-time coding addresses this regime, where neither Tx nor Rx knows the instantaneous channel realisation. The subject has been studied for 25 years, has elegant Grassmannian geometry, and yet has NOT produced standards-ready explicit codes for arbitrary $(n_t, n_r)$ . This is an open problem.

Definition:
Non-Coherent MIMO Block Fading

In the non-coherent block-fading MIMO model, the channel $\mathbf{H} \in \mathbb{C}^{n_r \times n_t}$ remains constant for $T$ channel uses ("coherence block") and then takes a new independent realisation. Crucially, NEITHER Tx NOR Rx knows $\mathbf{H}$ . The input is an $n_t \times T$ signal matrix $\mathbf{X}$ and the output is $\mathbf{Y} = \mathbf{H}\mathbf{X} + \mathbf{w}, \quad \mathbf{w}_{ij} \sim \mathcal{CN}(0, \sigma^2)\ \text{i.i.d.}$ The receiver sees only $\mathbf{Y}$ ; the code must be designed so that $\mathbf{X}$ can be recovered without any channel estimate.

Theorem: Non-Coherent MIMO Capacity Pre-Log

For the non-coherent block-fading MIMO model with coherence time $T \ge 2 n_t$ , the capacity grows with SNR as $C(\text{SNR}) = n_t^* \left(1 - \frac{n_t^*}{T}\right)\log_2 \text{SNR} + O(1)$ where $n_t^* = \min\left(n_t, n_r, \lfloor T/2 \rfloor\right)$ is the effective number of transmit streams. The capacity-achieving input distribution lies on the Grassmannian manifold of $n_t^*$ -dimensional subspaces of $\mathbb{C}^T$ .

Proof

Input on the Grassmannian

Marzetta and Hochwald showed via a calculus-of-variations argument that the capacity-achieving input has the form $\mathbf{X} = \mathbf{\Phi} \mathbf{D}$ where $\mathbf{\Phi}$ is an isotropically-distributed $n_t \times T$ matrix with orthonormal columns (a point on the Stiefel manifold) and $\mathbf{D}$ is a diagonal energy-allocation matrix. Only the SUBSPACE of $\mathbf{X}$ carries information.

Pre-log factor

The degrees of freedom for subspace encoding are $n_t^*(T - n_t^*)$ real dimensions, divided among $T$ channel uses — giving $(n_t^*(T - n_t^*))/T = n_t^* - (n_t^*)^2/T = n_t^*(1 - n_t^*/T)$ effective DoF per channel use at high SNR.

When is $n_t^* = n_t$?

If $T \ge 2 n_t$ and $n_r \ge n_t$ , we get $n_t^* = n_t$ and the pre-log is $n_t(1 - n_t/T)$ . As $T \to \infty$ , this approaches $n_t$ — the coherent pre-log. Short coherence (small $T$ ) costs pre-log.

Short coherence degradation

For $T < 2 n_t$ the system is SYMMETRICALLY under-parameterised: $n_t^* = \lfloor T/2 \rfloor$ , halving the achievable DoF. $\blacksquare$

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Non-Coherent MIMO Capacity vs Coherence Time

Coherent capacity (top, dotted) and non-coherent capacities for several coherence times $T$ . As $T \to \infty$ , non-coherent approaches coherent; as $T$ shrinks toward $2 n_t$ , the pre-log halves.

Parameters

Transmit antennas

n_t

2

Receive antennas

n_r

4

Example: Non-Coherent 2×2 with $T = 8$

Compute the high-SNR pre-log for a $2 \times 2$ MIMO block-fading channel with coherence time $T = 8$ .

Solution

Effective streams

$n_t^* = \min(n_t, n_r, \lfloor T/2 \rfloor) = \min(2, 2, 4) = 2$ .

Pre-log

$n_t^* (1 - n_t^*/T) = 2(1 - 2/8) = 2 \cdot 0.75 = 1.5$ .

Comparison

Coherent pre-log: $\min(n_t, n_r) = 2$ . Non-coherent loses $0.5 \log_2 \text{SNR}$ at high SNR — a 25% penalty for not knowing the channel.

Historical Note: Non-Coherent STC: A 25-Year Research Effort

The non-coherent MIMO line of research traces:

Hochwald-Marzetta 1999: the pre-log result above.
Hochwald-Marzetta 2000: unitary space-time codes for non-coherent block fading — explicit constructions for $n_t \in \{2, 3, 4\}$ based on group codes.
Zheng-Tse 2002: non-coherent DMT for the same model. Showed that for $T \ge n_t + n_r$ , the non-coherent DMT equals the coherent DMT (Ch 12) MINUS a "CSI penalty" of $n_t$ per stream.
Yang-Belfiore 2007: explicit diagonal-plus-Grassmannian codes for $n_t = 2, 4$ . What is STILL OPEN in 2026: a general constructive framework for $(n_t, n_r, T)$ that achieves the DMT simultaneously for every multiplexing gain $r$ . Partial results exist (USTM, diagonal, unitary space-time), but no analogue to Ch 13's CDA universality.

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Common Mistake: Differential STC Is Not Always the Answer

Mistake:

"Use differential modulation: we don't need CSI."

Correction:

Differential modulation (e.g., DQPSK) achieves zero pre-log loss for $T \gg n_t$ but incurs a 3 dB SNR penalty relative to coherent, even at high SNR. For small $T$ , differential is far from capacity- achieving. Unitary space-time codes (Grassmannian packings) close the gap, at the cost of non-trivial decoder complexity.

🔧Engineering Note

Non-Coherent Detection in Narrow-Band IoT (NB-IoT)

Non-coherent detection appears in niche deployed systems: NB-IoT (3GPP Rel-13, 2016) uses differential QPSK on the NPUSCH to enable battery-powered sensors without channel estimation. The pre-log loss is negligible because the coherence time greatly exceeds the packet duration. For high-capacity MIMO deployments, non-coherent techniques remain research-stage. This is a gap the theory can close but standards have not pursued.

Key Takeaway

Non-coherent MIMO achieves pre-log $n_t^*(1 - n_t^*/T) \log \text{SNR}$ — a factor of $(1 - n_t/T)$ below coherent. Capacity-achieving inputs live on Grassmannian manifolds. Explicit DMT-optimal non-coherent codes for general $(n_t, n_r, T)$ are an OPEN problem 25 years after the Marzetta-Hochwald pre-log result.

Non-Coherent Space-Time Codes