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Making the Theory Practical

The Heisenberg transform of Section 2 assumes an idealized bi-orthogonal pulse. In practice we must choose a concrete pulse and add a cyclic prefix to absorb the channel's delay spread. These two choices — pulse shape and CP — determine how closely the deployed OTFS system tracks the clean theoretical input-output relation of Chapter 4. Both choices directly affect pilot design (Chapter 7) and detection (Chapter 8).

The point is that OTFS's CP and pulse-shape trade-offs mirror OFDM's — same hardware, same engineering levers — with one crucial twist: OTFS uses a whole-frame CP (or a per-symbol CP with block structure), and the pulse's DD-domain properties (via the Zak transform of $g_{tx}$ ) determine the residual off-diagonal errors.

Definition:
Cyclic Prefix for OTFS

The cyclic prefix (CP) prepends the last $L_{CP}$ samples of each OFDM symbol before transmission, exactly as in OFDM. Two CP strategies exist for OTFS:

Per-symbol CP: each of the $N$ OFDM symbols has its own CP of length $L_{CP} \geq l_{\max}$ . Overhead: $N L_{CP}$ samples per frame. Compatible with 5G NR OFDM resource grids.
Frame-level CP (reduced CP): a single CP at the start of the frame of length $L_{CP,\text{frame}} \geq l_{\max}$ . Reduces overhead to $L_{CP}$ samples per frame. Requires a different detector (Chapter 20) but saves $\sim (N-1)L_{CP}$ samples per frame.

Per-symbol CP is the baseline. Frame-level CP is a more advanced design covered in Chapter 20.

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Theorem: Minimum Cyclic-Prefix Length

For the discrete DD input-output relation of Chapter 4 to hold exactly (doubly-circular, no inter-block interference), the CP length must satisfy $L_{CP} \;\geq\; l_{\max} \;=\; \lceil \tau_{\max} W \rceil.$ When $L_{CP} < l_{\max}$ , residual inter-block interference contaminates the first $l_{\max} - L_{CP}$ delay bins.

Same requirement as OFDM: the CP must absorb the channel memory so that the linear convolution within each OFDM symbol becomes circular. For OTFS the requirement is per OFDM symbol, and it is identical to OFDM's — OTFS inherits CP dimensioning from OFDM with no change.

Proof

Channel memory

The discrete channel has taps up to delay $l_{\max}$ . Within one OFDM symbol, the channel output is the circular convolution only if the previous $l_{\max}$ samples from the neighboring symbol are included — that is, the CP provides this context.

Insufficient CP

If $L_{CP} < l_{\max}$ , the first $l_{\max} - L_{CP}$ delay bins receive input from the previous symbol's last samples (inter-block interference). The circular convolution identity fails for these cells.

Consequence

In the DD channel matrix view (Chapter 4), insufficient CP produces off-block-diagonal terms in $\mathbf{H}_{DD}$ , breaking the clean Kronecker-product structure. The ISFFT no longer diagonalizes exactly; detection loses $\sim 1$ dB per unit of missing CP at typical parameters. $\blacksquare$

⚠️Engineering Note

CP Overhead in 5G NR

5G NR specifies two CP configurations:

Normal CP (numerology-dependent): $L_{CP}$ is chosen to give roughly $7\%$ overhead for each numerology. Supports $\tau_{\max} \leq 4.7\,\mu$ s at numerology 0.
Extended CP: longer CP at the cost of shorter symbol duration. Supports $\tau_{\max} \leq 16.7\,\mu$ s, with $25\%$ overhead.

OTFS inherits these configurations directly. For deployment scenarios with $\tau_{\max} > 5\,\mu$ s (rural macro, hills, hilly terrain in HST), extended CP is needed. For indoor / urban micro with $\tau_{\max} < 1\,\mu$ s, normal CP suffices.

The CP overhead is independent of $N$ — it is fixed by the physical delay spread and the symbol duration. Making the OTFS frame longer (larger $N$ ) does not increase per-symbol CP overhead. This is an advantage over per-frame pilot schemes.

Practical Constraints

•
$L_{CP} \geq l_{\max} = \lceil \tau_{\max} W\rceil$ for valid doubly-circular structure
•
Normal CP: $7\%$ overhead; Extended CP: $25\%$ overhead
•
Frame-level CP (Chapter 20): up to $N\times$ savings in overhead but requires advanced detector

📋 Ref: 3GPP TS 38.211 §4.3

Definition:
Common OTFS Prototype Pulses

Four prototype pulses are standard in OTFS:

Rectangular: $g_{tx}(t) = (1/\sqrt{T_s})\mathbf{1}_{[0, T_s)}(t)$ . Simple, bi-orthogonal at critical density. Poor spectral containment (high side lobes). The default in most literature.
Root-raised-cosine (RRC): $g_{tx}(t)$ is the impulse response of an RRC filter with roll-off $\alpha$ . Better spectral containment; not strictly bi-orthogonal.
Gaussian: $g_{tx}(t) \propto e^{-t^2/(2\sigma^2)}$ . Good simultaneous time-frequency localization (optimal for $\sigma = \sqrt{T_s/(2\pi\Delta f)}$ ). Not compactly supported.
Designed (optimized) pulses: numerically optimized for specific channel statistics (Chapter 20).

The choice affects: (i) spectral containment at band edges, (ii) PAPR, (iii) off-diagonal residuals in the DD channel matrix, (iv) pilot region leakage (Chapter 7).

OTFS Prototype Pulses: Trade-offs

Pulse	Time support	Freq. containment	DD grid cleanness	Default use
Rectangular	Compact	Poor (sinc side lobes)	Exact (bi-ortho)	Theoretical analysis
Raised-cosine	Extended (truncated)	Good	Near-exact	Practical OFDM
RRC	Extended (truncated)	Very good	Good	Advanced OTFS
Gaussian	Infinite tails	Good	Good w/ large $\sigma$	Information-theoretic proofs
Designed	Optimized	Optimized	Optimized	6G research

Theorem: Residual DD Cross-Talk From Non-Bi-Orthogonal Pulses

For a prototype pulse $g$ that does not satisfy the strict bi-orthogonality condition, the DD channel matrix $\mathbf{H}_{DD}$ acquires off-diagonal "cross-talk" entries whose magnitudes are bounded by $|\text{cross-talk}| \;\lesssim\; \max_{(t, \nu) \in \mathbb{T}^2}\left|Z_g(t, \nu) \overline{Z_{g^\perp}(t, \nu)} - 1\right|,$ where $g^\perp$ is the ideal bi-orthogonal dual window. The bound is zero iff $g = g^\perp$ (perfect bi-orthogonality).

The DD domain is where bi-orthogonality "lives" — the Zak transform's bounded, bounded-below condition from Chapter 2 exactly specifies it. A non-bi-orthogonal pulse has non-flat $|Z_g|$ on the fundamental domain, and the deviation from flatness quantifies the cross-talk.

Proof

Gabor-dual pair

The bi-orthogonal dual window $g^\perp$ satisfies $\langle g, g^\perp\rangle \cdot \text{identity} = Z_g \cdot \bar{Z_{g^\perp}}$ .

Cross-talk estimate

The deviation from the identity multiplier on the Gabor system bounds the off-diagonal error. Under the sup-norm bound above, the DD-cell coupling is of order $\epsilon = \text{sup-deviation}$ .

For rectangular $g$

$g = \text{rect}/\sqrt{T_s}$ , $g^\perp = g$ (self-dual), $|Z_g| = 1$ on $\mathbb{T}^2$ — perfect bi-orthogonality, zero cross-talk.

For Gaussian $g$

$|Z_g|$ is periodized Gaussian; $|Z_g - 1|$ is at most $\sim 0.1$ for $\sigma \sim T_s$ — moderate cross-talk. $\blacksquare$

How Pulse Choice Affects the DD Cross-Talk

For each prototype pulse (rectangular, RRC, Gaussian), plot (i) the time-domain pulse, (ii) its Zak transform magnitude on the fundamental domain, and (iii) the resulting DD cross-talk magnitude on a sample OTFS frame. Observe that rectangular pulses give exact bi-orthogonality (flat Zak, zero cross-talk), while RRC and Gaussian trade some cross-talk for spectral containment.

Parameters

Prototype pulse

Roll-off

\alpha

(RC / RRC)0.25

Gaussian

\sigma/T_s

0.5

Subcarriers

M

16

Common Mistake: Don't Assume CP Size Equals Channel Memory

Mistake:

Allocating $L_{CP} = \tau_{\max} W$ as if the channel's maximum delay is known exactly. In deployment, $\tau_{\max}$ is an upper bound; the actual channel fluctuates.

Correction:

Allocate $L_{CP} = \lceil 1.5 \tau_{\max}^{\text{max}} W\rceil$ as a safety factor, where $\tau_{\max}^{\text{max}}$ is the 99th percentile over the deployment environment. This is the OFDM practice adopted in 5G NR. For OTFS the same applies — do not under-allocate CP based on typical delay, over-allocate based on worst-case delay.

Cyclic Prefix and Pulse Shaping