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Why Time-Invariance Matters

The physical channel is manifestly time-varying — reflectors move, paths come and go within a single OTFS frame. Yet the DD channel matrix $\mathbf{H}_{DD}$ has no time argument. The channel, viewed through the DD lens, is effectively time-invariant over the frame duration.

This is not a coincidence of notation. It is a structural property with direct engineering consequences: one channel estimate suffices for the whole frame (Chapter 7); one detector configuration works for all DD cells (Chapter 8); diversity order is determined by the path count, not by the fading realization over time (Chapter 9). In this section we make the time-invariance claim precise and compare it rigorously to the OFDM case.

Definition:
DD Time-Invariance

A channel is said to be DD time-invariant over a frame of duration $T$ if the input-output relation in the DD domain has the form $y_{DD}(\tau, \nu) \;=\; h(\tau, \nu) \star\star x_{DD}(\tau, \nu) \;+\; \mathbf{w}_{DD}(\tau, \nu),$ where $h(\tau, \nu)$ is a fixed (not time-dependent) 2D kernel that can be identified from any portion of the frame.

By contrast, the TF channel $H(f, t)$ of Bello's framework (Chapter 1) has an explicit $t$ argument and is therefore TF time-varying. Time-invariance is a property of a representation, not of the physical channel.

Theorem: The Physical Multipath Channel Is DD Time-Invariant

A physical multipath channel $y(t) = \sum_{i=1}^{P} h_i\,x(t - \tau_i)\,e^{j 2\pi \nu_i t} + w(t)$ with $\{h_i, \tau_i, \nu_i\}_{i=1}^P$ constant over an observation window of duration $T$ is DD time-invariant with kernel $h(\tau, \nu) = \sum_i h_i\,\delta(\tau - \tau_i)\,\delta(\nu - \nu_i)$ .

In particular, for any two disjoint subframes $[t_0, t_0 + T_1]$ and $[t_0 + T_1, t_0 + T]$ within the frame, the DD channel kernels estimated from each subframe are identical (in the absence of noise).

The Zak transform converts the time-domain delay-and-Doppler-shift operator into a pure translation on the DD plane. Multiple paths sum to a linear combination of translations, which is a 2D convolution with a fixed kernel. Since the kernel does not carry a $t$ argument, the channel is DD time-invariant.

This is the DD analog of what the LTI assumption buys for FIR filters: a fixed impulse response characterizes the entire system's action on any input. Here the "impulse response" is 2D.

Proof

Channel parameters constant over the frame

By hypothesis, $h_i, \tau_i, \nu_i$ do not vary over $[t_0, t_0 + T]$ . This is the block-fading assumption — reasonable when the path geometry changes slowly compared to the frame duration. For realistic OTFS frames ( $T \leq 10$ ms) and physical reflector velocities ( $\leq 300$ km/h), it is an excellent approximation.

Zak transform of the output

From Theorem TContinuous DD Input-Output Relation, $y_{DD}(\tau, \nu) = \sum_i h_i\,x_{DD}(\tau - \tau_i, \nu - \nu_i) + \mathbf{w}_{DD}$ . The coefficients $h_i$ , shifts $\tau_i, \nu_i$ are constants of the subframe — the expression has no explicit $t$ .

Sub-frame comparison

If the frame $[t_0, t_0 + T]$ is split into two subframes and each is analyzed by computing its Zak transform, both analyses produce the same kernel $h(\tau, \nu)$ (plus independent noise). This is what "DD time-invariant" means operationally — the channel estimate is the same regardless of which subframe yields it.

Contrast with TF

In the TF domain, $H(f, t)$ at time $t_1$ and $H(f, t_2)$ are related by $H(f, t_2) = H(f, t_1)\,e^{j 2\pi \nu_i (t_2 - t_1)}$ per path — different snapshots of a time-varying transfer function. No fixed 1D kernel suffices. Only the full 2D kernel $h(\tau, \nu)$ is invariant. $\blacksquare$

,

Key Takeaway

One DD channel for the whole frame. Under the block-fading assumption (stable path geometry over $T \leq 10$ ms), the DD kernel $h(\tau, \nu)$ is constant. A pilot that identifies $h$ at the start of the frame yields a valid channel estimate for the entire frame — no tracking, no per-symbol updates. This is the structural reason OTFS is robust to mobility: mobility makes the channel's TF representation change rapidly, but does not change the DD kernel itself within the frame. Only path geometry evolution (over tens of ms at typical velocities) changes $h(\tau, \nu)$ .

A Moving Reflector: TF Changes, DD Stays Put

Three reflectors in a 2D scene. As the ego vehicle moves over one OTFS frame (top-left animation), the time-frequency transfer function

|H(f, t)|

visibly evolves (top-right heatmap). Meanwhile, the delay-Doppler spreading function

|h(\tau, \nu)|

(bottom) is three static spikes — no change over the frame. Two different pictures of the same physical channel; one moves, the other does not.

When Block-Fading Fails

The DD time-invariance claim requires constant path geometry over the frame. This fails for:

Very long frames ( $T \gg T_c$ ): the reflectors move enough within the frame to change $\tau_i(t)$ or $\nu_i(t)$ appreciably.
Accelerating vehicles: $\nu_i(t) = (v_i(t)/c)f_0$ drifts if $v_i$ is not constant. A vehicle accelerating at $10\,\text{m/s}^2$ over a 10 ms frame changes velocity by 0.1 m/s — negligible. A hypersonic missile at 100 m/s² changes velocity by 1 m/s, producing a Doppler drift of $\sim 33$ Hz at 10 GHz — non-negligible.
LEO satellites with fast arc traversal: the angle to the UE can sweep several degrees per second, producing a $\nu_i(t)$ that is quasi-linearly varying over a 10 ms frame.

For these regimes, basis expansion models (BEMs) represent the paths as slow time-varying $h_i(t), \tau_i(t), \nu_i(t)$ with known trajectories. See Chapter 10 and Chapter 18. The block-fading assumption used in this chapter remains a useful first-order model for most terrestrial mobility.

TF Matrix Drifts; DD Matrix Stays Sparse

Parameters

Number of paths

P

3

Max Doppler

f_D

(Hz)500

Time in frame

t

(ms)0

Frame duration

T

(ms)10

Seed3

Example: Block-Fading Validity at Urban Vehicular Scales

A vehicle at $v = 100$ km/h experiences reflectors at delays $\tau_i \in [0, 5\,\mu\text{s}]$ . The OTFS frame duration is $T = 5$ ms at carrier $f_0 = 6$ GHz. Verify that (a) the delay change over the frame is negligible, (b) the Doppler change due to possible angle change is negligible.

Solution

Delay change

Position change over the frame: $\Delta d = v T = 27.8 \cdot 0.005 = 0.139$ m. Delay change: $\Delta\tau = \Delta d / c = 4.6 \times 10^{-10}$ s = 0.46 ns. Delay resolution at $W = 10$ MHz: $\Delta\tau_{\text{grid}} = 100$ ns. Ratio: $0.46/100 = 0.46\%$ . Delay is effectively constant.

Doppler change

Assume the angle to a reflector changes by at most 1° over the frame (a strict upper bound for a 140-m displacement relative to a 100-m-away reflector). Doppler change: $\Delta \nu_i \approx f_0\,(v/c)\sin(\Delta\theta) \approx 6 \times 10^9 \cdot 27.8/3 \times 10^8 \cdot 0.017 = 9.5$ Hz. Doppler resolution: $1/T = 200$ Hz. Ratio: $9.5/200 = 4.7\%$ — small but non-negligible; flat block-fading is a reasonable but not perfect approximation.

Conclusion

For urban vehicular mobility with $T \leq 5$ ms, block-fading is accurate to within a few percent of the DD grid resolution. Longer frames (or higher mobility / carrier) require the basis-expansion refinement of Chapter 10.

🎓CommIT Contribution(2022)

Crystal Symmetry of the DD-Domain Channel

S. K. Mohammed, R. Hadani, A. Chockalingam, G. Caire — IEEE BITS the Information Theory Magazine

Mohammed–Hadani–Chockalingam–Caire (2022) framed the DD time-invariance property in a language borrowed from solid-state physics: the DD channel admits a "crystalline" structure in which the spreading function $h(\tau, \nu)$ plays the role of a unit cell, tiled across the torus $\mathbb{T}^2$ by the quasi-periodic extension. The analogy is precise — the Zak transform, after all, originated as a tool for Bloch-type periodicity in crystals.

The CommIT contribution here is threefold: (i) formalizing the DD channel as a convolution on the fundamental domain of a discrete symmetry group (the Heisenberg–Weyl lattice), (ii) clarifying when block-fading holds at the level of the crystalline symmetry, and (iii) providing a single unifying framework for both the communication and radar-sensing aspects of OTFS — a conceptual bridge exploited in Chapter 12 of this book (OTFS-ISAC).

zak-otfstime-invariancecrystal-symmetry

Why This Matters: Link to Information Theory

The DD time-invariance we establish here is the structural reason the capacity analysis of Chapter 9 works. In the ITA book, Chapter 13 derived the ergodic capacity of a fading channel by averaging over random channel realizations. For OTFS the analysis is per-realization — the channel is fixed over the frame, so within the frame we have a deterministic linear channel with known (or estimated) kernel. The full diversity and capacity results of Chapter 9 follow from this fixed-kernel structure.

Common Mistake: DD Time-Invariance $\neq$ Physical Time-Invariance

Mistake:

Concluding that "OTFS eliminates the time-variation of the channel." It does not. The physical channel is still time-varying — paths still have Doppler shifts, reflectors still move. OTFS just represents the channel in a domain (DD) where the time-variation collapses into a fixed 2D kernel.

Correction:

The right statement is: the channel's effect on data symbols is, in the DD domain, a fixed 2D convolution. The Doppler shifts haven't disappeared — they are encoded in the kernel's Doppler axis. What OTFS buys is a representation in which the same equalization or detection works for all DD cells, rather than a TF representation where each cell experiences a different time-varying gain.

Time-Invariance of the DD Channel