Ferkans — Interactive Telecom Tutor

From DD to DD-Angle

Chapter 12 treated each target as a point in the delay-Doppler plane: a pair $(\tau_i, \nu_i)$ with complex amplitude $a_i$ . That description is complete for a single-antenna transceiver, which can tell how far a scatterer is and how fast it is moving, but cannot tell in which direction it lies. With an antenna array, the third question becomes answerable. The channel becomes a tensor in three dimensions — delay, Doppler, and angle — and the ISAC optimization problem grows a spatial dimension. This section sets up that tensor and states the MIMO DD input-output relation.

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Definition:
MIMO Delay-Doppler Channel

Consider a transceiver with $N_t$ transmit antennas, $N_r$ receive antennas, and a scatter environment of $P$ resolvable paths. Path $i$ has delay $\tau_i$ , Doppler $\nu_i$ , angle of departure $\phi_i$ (azimuth at the transmitter), angle of arrival $\theta_i$ (azimuth at the receiver), and complex gain $a_i$ .

The MIMO delay-Doppler spreading function is the tensor-valued map $\mathbf{H}(\tau, \nu) \;=\; \sum_{i=1}^{P} a_i \,\mathbf{a}_r(\theta_i)\,\mathbf{a}_t(\phi_i)^H \,\delta(\tau - \tau_i)\, \delta(\nu - \nu_i),$ taking values in $\mathbb{C}^{N_r \times N_t}$ . Here $\mathbf{a}_t(\phi) \in \mathbb{C}^{N_t}$ and $\mathbf{a}_r(\theta) \in \mathbb{C}^{N_r}$ are the transmit and receive array response vectors — for a uniform linear array with half-wavelength spacing, $[\mathbf{a}(\phi)]_n = e^{j\pi n\sin\phi}$ .

The MIMO DD input-output relation is $\mathbf{y}(\tau, \nu) \;=\; \iint \mathbf{H}(\tau', \nu')\, \mathbf{x}(\tau-\tau', \nu-\nu')\, d\tau'\, d\nu' \,+\, \mathbf{w}(\tau, \nu),$ with $\mathbf{x}(\tau, \nu) \in \mathbb{C}^{N_t}$ the DD transmit signal and $\mathbf{y}(\tau, \nu) \in \mathbb{C}^{N_r}$ the DD receive signal.

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The Three Signal Dimensions

The MIMO DD channel lives in three dimensions: delay $\tau$ (discretized to $M$ bins by bandwidth $W$ ), Doppler $\nu$ (discretized to $N$ bins by frame duration $T$ ), and angle $\theta, \phi$ (sampled coarsely by the array aperture — angular resolution $\sim 2/N_t$ at the transmitter, $2/N_r$ at the receiver). The channel is still sparse: $P$ paths place $P$ points in the $M \times N \times \min(N_t, N_r)$ cube. Sparsity is more pronounced than in SISO OTFS because a path that separates in angle need no longer be resolved in delay or Doppler for the receiver to disambiguate it.

Theorem: MIMO-OTFS Channel Parameter Count

The discrete MIMO-OTFS channel over a frame of $MN$ DD cells and $N_r \times N_t$ antennas is described by $P \cdot (1 + 1 + 1 + 2 + 2) \;=\; 7P$ real parameters: $P$ complex gains ( $2P$ real), $P$ delays, $P$ Dopplers, $P$ AoDs, $P$ AoAs. The full time-varying MIMO channel matrix has $N_r N_t MN \cdot 2$ real degrees of freedom.

For a representative automotive scenario $(P = 6, M = N = 64, N_t = N_r = 8)$ , the DD-angle parameter count is $42$ versus $524{,}288$ — four orders of magnitude fewer parameters. This is the sparsity the DD-angle representation exposes.

The point is that MIMO does not add complexity proportional to $N_t \times N_r$ ; it adds exactly $4P$ new real numbers — two angles per path. This is why the MIMO-OTFS estimator has tractable sample complexity even when the nominal MIMO channel matrix is enormous. The DD-angle representation collapses the combinatorial explosion of time-varying MIMO channels into a handful of geometric parameters.

Proof

Parameter counting for one path

A single scattering path contributes a rank-one matrix $\mathbf{a}_r(\theta_i)\mathbf{a}_t(\phi_i)^H$ at a single DD location. Its full description requires: one complex gain (2 real), one delay (1 real), one Doppler (1 real), one AoD (1 real), one AoA (1 real) — total 7 reals.

Sum over paths

$P$ paths contribute independently, so the channel is fully specified by $7P$ real parameters.

Comparison with dense MIMO channel

The full time-varying MIMO channel matrix $\mathbf{H}[\ell, k] \in \mathbb{C}^{N_r \times N_t}$ has $MN \cdot N_r N_t$ complex entries, or $2 MN N_r N_t$ real degrees of freedom. For the automotive numbers above, this is $524{,}288$ .

Ratio

$7P / (2MN N_r N_t) = 42/524288 \approx 8 \times 10^{-5}$ . This is the compression factor the DD-angle representation achieves. $\blacksquare$

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Key Takeaway

MIMO-OTFS is extremely sparse. The channel has $7P$ degrees of freedom, independent of $MN N_t N_r$ . This is the single most important fact driving low-overhead estimation, compressed-sensing channel acquisition, and tractable joint beamforming in the chapters that follow.

Definition:
Discrete MIMO DD Input-Output

With integer delay/Doppler offsets $(\ell_i, k_i)$ , the discrete MIMO DD input-output relation at the $(\ell, k)$ cell is $\mathbf{y}[\ell, k] \;=\; \sum_{i=1}^{P} a_i \,\mathbf{a}_r(\theta_i)\,\mathbf{a}_t(\phi_i)^H \,\mathbf{x}[\ell - \ell_i, k - k_i]\, e^{-j2\pi\nu_i \ell_i / (MN)} \,+\, \mathbf{w}[\ell, k].$ Stacking into vectors $\mathbf{y}, \mathbf{x} \in \mathbb{C}^{MN N_r}$ (or $MN N_t$ ), this becomes $\mathbf{y} \;=\; \mathbf{H}_{DD}\, \mathbf{x} \,+\, \mathbf{w},$ with $\mathbf{H}_{DD} \in \mathbb{C}^{MN N_r \times MN N_t}$ a structured block-sparse matrix with $P$ nonzero blocks.

Example: Automotive MIMO-OTFS Channel Count

A 77 GHz automotive ISAC transceiver has $N_t = N_r = 16$ antennas, a frame with $M = 256$ , $N = 32$ , and a scene with $P = 8$ scatterers (vehicles, pedestrians, road infrastructure).

(a) Count the DD-angle parameters. (b) Count the full time-varying MIMO channel degrees of freedom. (c) Compute the compression ratio.

Solution

DD-angle parameters

$7 P = 7 \cdot 8 = 56$ real parameters. The scene is completely described by $8$ complex gains, $8$ delays, $8$ Dopplers, $8$ AoDs, and $8$ AoAs.

Full MIMO channel parameters

$2 MN N_r N_t = 2 \cdot 256 \cdot 32 \cdot 16 \cdot 16 = 4{,}194{,}304$ real degrees of freedom.

Compression ratio

$56 / 4.2 \times 10^6 \approx 1.3 \times 10^{-5}$ — the DD-angle representation is $75{,}000$ times more compact than the dense channel description.

Implication for estimation

Pilot overhead scales with the number of parameters, not the nominal channel size. A MIMO-OTFS estimator needs $\mathcal{O}(P)$ observations per frame, not $\mathcal{O}(MN N_t)$ . This is what makes sensing-aware pilot design in Chapter 14 practical.

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MIMO-OTFS Channel Tensor: Delay × Doppler × Angle

Visualize the $P$ resolvable paths as points in the 3D $(\tau, \nu, \theta)$ cube. Move sliders for $P$ (number of paths), AoA spread, and Doppler spread to see how the sparse scene fills the cube. Compare with the dense TF-space representation's parameter count.

Parameters

P

6

AoA spread (°)45

Max Doppler (Hz)800

N_t = N_r

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Theorem: Beam-Doppler Coupling

For an ISAC transceiver tracking a target moving with radial velocity $v_r$ at angle $\theta$ , the Doppler $\nu = (2 v_r/\lambda)$ and the angle $\theta$ are physically coupled: the same target generates joint $(\nu, \theta)$ observations.

Consequence. The rank-one model $\mathbf{H}_{i} \;=\; a_i \mathbf{a}_r(\theta_i) \mathbf{a}_t(\phi_i)^H \,\delta(\tau - \tau_i)\,\delta(\nu - \nu_i)$ exploits this coupling: the estimator of $\theta_i$ is correlated with the estimator of $\nu_i$ through the joint observation. This correlation is quantified by the off-diagonal Fisher information $J_{\theta\nu}$ , derived in §3.

A target at angle $\theta$ moving with velocity $v_r$ at that angle radiates the same complex waveform from the array into a specific DD bin. The receiver sees one scattered copy, not two. So the information about $\theta$ and $\nu$ is mixed in the observation. Exploiting this mixing — rather than estimating $\theta$ and $\nu$ separately — is the essence of MIMO-OTFS sensing.

Proof

Signal model

The received signal from path $i$ is $\mathbf{y}_i[\ell, k] \;=\; a_i \,\mathbf{a}_r(\theta_i)\, [\mathbf{a}_t(\phi_i)^H \mathbf{x}[\ell-\ell_i, k-k_i]]\, e^{-j2\pi k \nu_i T/N}.$ Both $\theta_i$ and $\nu_i$ modulate the same waveform energy.

Fisher information

Taking gradients of the log-likelihood with respect to $(\theta_i, \nu_i)$ and applying the Cauchy-Schwarz inequality yields off-diagonal Fisher matrix entries $[J]_{\theta\nu} \;=\; \Re\left\{\frac{\partial \mathbf{s}_i}{\partial \theta}^H \frac{\partial \mathbf{s}_i}{\partial \nu}\right\},$ which is generically nonzero.

Operational interpretation

The joint estimator achieves lower combined variance than two independent estimators. The rate-CRB tradeoff in §3 exploits this by designing the transmit beamformer to decorrelate the two dimensions when joint estimation is paramount. $\blacksquare$

Common Mistake: Do Not Force the Angle onto a Grid

Mistake:

Treating AoAs/AoDs as discretized like delays and Dopplers, setting $\theta_i \in \{2\pi k/N_r\}_{k=0}^{N_r-1}$ . This replaces a continuous parameter with a grid and incurs discretization bias that grows as $1/N_r$ — severe for small arrays ( $N_t = N_r = 8$ has $\sim 12°$ grid spacing).

Correction:

Treat angles as continuous parameters and estimate them via super-resolution (Newton iteration on the beamforming gain) or compressed-sensing algorithms (OMP, ANM). The DD delays/Dopplers are grid-aligned (enforced by the OTFS structure) but angles are not. Distinguishing the two is essential for correct algorithm design.

⚠️Engineering Note

Array Calibration for OTFS-ISAC

MIMO-OTFS-ISAC is sensitive to array calibration errors — unknown per-element gain and phase offsets. Practical implications:

Communication: The precoder $\mathbf{F}$ is set using the estimated channel and therefore compensates for calibration errors automatically. Comms performance is robust.
Sensing: The AoA/AoD estimates depend directly on the array manifold. A 1° calibration error in the array response translates to a 1° error in every estimated angle.

Engineering practice: run a calibration phase (known pilot from a reference direction) at boot and periodically during operation (e.g., every few seconds for automotive). Kalibration techniques from classical radar (MUSIC with calibration model) apply.

Practical Constraints

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Comms: robust to calibration (precoder absorbs the error)
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Sensing: angle errors propagate 1-for-1 from calibration
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Practical: calibration phase at boot + periodic refresh

Why This Matters: Connection: Telecom Ch. 18-19 MIMO, DD Extension

Telecom Chapter 18 introduced MIMO precoding (SVD, water-filling, BD) under the assumption that the channel matrix $\mathbf{H}$ is time-invariant within a coherence window. Chapter 19 introduced MIMO-OFDM: apply the SISO-OFDM trick per subcarrier. The MIMO-OTFS framework here neither assumes coherence nor works per subcarrier — it works on the DD grid, where the channel is genuinely sparse even under high Doppler. The precoding design in §2 is a DD-domain generalization of the TF-domain designs you know from Telecom 18-19.

MIMO-OTFS Channel: The Delay-Doppler-Angle Tensor