Exercises

$\sigma_t \cdot \sigma_f \geq 1/(4\pi)$. No pulse can be arbitrarily compact in both domains.

Any transmit pulse has bounded time and frequency concentration. Time-compact pulses leak in frequency → ICI. Frequency-compact pulses leak in time → ISI. Design picks the trade-off.

Gaussian achieves equality: optimally localized in both domains. Good default for high-mobility + sensing.

$A_{g_{\mathrm{rx}}, g_{\mathrm{tx}}}(\ell T_s, m W_s) = \delta[\ell, m]$

$A_{g_{\mathrm{rx}}, g_{\mathrm{tx}}}(\tau, \nu) = \int g_{\mathrm{rx}}(t)^* g_{\mathrm{tx}}(t-\tau) e^{j2\pi\nu t} dt$

Symbol at DD cell $(\ell_0, m_0)$ reaches receiver at the same cell — no leakage to other cells.

CP-OFDM: 10-11 dB. DFT-s-OFDM: 5-6 dB. OTFS: 7-8 dB.

DFT-s $<$ OTFS $<$ CP-OFDM. OTFS is between.

OTFS PAPR 2 dB higher than DFT-s. Matters for UE mmWave uplink. OFDM-downlink: PAPR not critical.

For RRC: $T_s W_s = 1 + \gamma = 1.35$. Slightly over-critical. Bi-orthogonality holds.

Excess bandwidth: $\gamma \cdot 100\% = 35\%$. For every 1 MHz of useful bandwidth: 1.35 MHz allocated.

Rate: $1/(1 + \gamma) = 0.74$ of brickwall (hypothetical). Significant but acceptable for mobility benefits.

$\gamma = 0.2$: 20% penalty, adequate for vehicular. $\gamma = 0.5$: 50% penalty, over-engineered for most use cases.

$\mathrm{OOBE} = \mathrm{SLL} + 10 \log_{10} M$

$M = 1024$: $10 \log 1024 = 30$ dB.

3GPP mask: -30 dBc at 1 MHz. Required SLL: $-30 - 30 = -60$ dB.

Blackman (-58 dB): insufficient by 2 dB. Nuttall (-98 dB): passes with 38 dB margin. Nuttall required. Or: combine Blackman with steeper RF filter.

CP length $\geq$ delay spread. If CP = 1.5 $\mu$s (50% margin): ISI power from incomplete CP coverage $\to 0$ (all delay spread absorbed).

Tapered pulse. ISI from delay $\tau$ within CP-like region: $\mathrm{ISI} \leq -30$ dB (RRC suppression).

Rectangular + CP: ISI = 0 but ICI at Dirichlet level (-13 dB). RRC $\gamma = 0.25$: ISI -30 dB, ICI -30 dB. RRC is cleaner overall.

RRC total interference: -27 dB. Rectangular + CP: -13 dB (dominated by ICI). RRC better for high-mobility.

$g(t) = (2\sigma^2)^{-1/4} e^{-t^2/(4\sigma^2)}$. Gaussian.

$\sigma_t \sigma_f = 1/(4\pi)$. Achieves uncertainty lower bound. Any other pulse has strict inequality.

Ideal time-frequency concentration. Good for extreme mobility (LEO). Approximate orthogonality via Hermite basis extension.

Pure Gaussian: non-compact (finite but infinite-support). Truncated Gaussian: approximately compact with small leakage. Common in sensing-heavy OTFS.

$g(t) = 1$ for $|t| \leq T_s/2$. $G(f) = T_s \mathrm{sinc}(f T_s)$.

Peak at $f = 0$: $|G(0)| = T_s$. Width (3 dB): $\sim 0.89/T_s$.

At $f = 1.5/T_s$: $|G| = T_s/\pi \cdot (1/1.5) \approx T_s \cdot 0.21$. Ratio to main lobe: $0.21$. In dB: $-13.3$ dB.

-13 dB sidelobes are high; window smooths edges to reduce sidelobes at cost of main-lobe widening.

$\mathrm{RC}(t) = \mathrm{sinc}(t/T_s) \cdot \frac{\cos(\gamma \pi t/T_s)}{1 - (2\gamma t/T_s)^2}$

$\mathrm{RC}(\ell T_s) = \mathrm{sinc}(\ell) \cdot (\cdots) = 0$ for $\ell \neq 0$ (by $\mathrm{sinc}(\ell) = 0$). $\mathrm{RC}(0) = 1$.

$\mathrm{RC}(\ell T_s) = \delta[\ell]$: ISI = 0 at integer sample times.

OTFS 2D: $\mathrm{RC}_{2D}(\ell T_s, m W_s) = \delta[\ell, m]$. Bi-orthogonality of RRC pair. $\blacksquare$

99%-tile PAPR: ~11 dB for QPSK + rectangular.

Smoother envelope: Gaussian-like. PAPR reduces to ~8-9 dB.

Reduction: ~2 dB from rectangular. Matches empirical measurements.

RRC suppresses time-domain discontinuities. Signal is smoother. Peaks less pronounced.

Hermite $\psi_0$ (Gaussian), $\sigma = 0.3 T_s$. Wide bandwidth, short duration. Maximizes data rate.

Hermite $\psi_1$, $\sigma = 0.8 T_s$. Narrow bandwidth, long duration. Doppler resolution: $1/T_{\text{frame}}$.

Hermite $\psi_2$, $\sigma = 0.5 T_s$. Moderate bandwidth, moderate duration. Coarse-velocity detection.

Bandwidth: Data 60%, Coarse 25%, Fine 15%. Orthogonality: Hermite family → zero cross-coupling.

Data: 0.6 × rate of single-pulse. Sensing: 1.5× better fine-Doppler resolution. Trade-off: 40% data rate for 50% better sensing.

-43 dB per single-tone.

$M = 1024$: add 30 dB (multiple-tone sum).

$-43 + 30 = -13$ dB. Far below 3GPP -30 dB requirement.

Need window with SLL ≤ -60 dB: Nuttall (-98 dB). Or combined Blackman + steep RF filter to gain additional 10-15 dB.

$g(t) = (2\sigma^2)^{-1/4} e^{-t^2/(4\sigma^2)}$.

At $t = T_s$ (next symbol): $|g(T_s)|^2 = (2\sigma^2)^{-1/2} e^{-T_s^2/(2\sigma^2)}$. $\sigma = 0.3 T_s$: exponent $= -T_s^2/(2 \cdot 0.09 T_s^2) = -5.56$. $|g(T_s)|^2 \approx 3.9 \times 10^{-3}$. ISI power: $\sim -24$ dB.

Similar analysis in frequency: $|G(W_s)|^2 \approx 3 \times 10^{-4}$ (Gaussian in frequency is narrow). ICI: $\sim -35$ dB.

Combined interference: -21 dB. Acceptable for LEO where Doppler is extreme but paths are sparse (P = 1-3).

The collection $\{g(t - \ell T_s) e^{j 2\pi m W_s t}\}_{\ell, m \in \mathbb{Z}}$ is a Weyl-Heisenberg system with density $1/(T_s W_s)$.

System is a Riesz basis of $L^2(\mathbb{R})$ iff $T_s W_s = 1$. At $T_s W_s < 1$: overcomplete (frame). At $> 1$: incomplete.

OTFS symbols live in this system. At critical sampling: unique representation (bi-orthogonal dual exists). Below: symbols overlap non-trivially (under-sampled). Above: symbols are redundant.

$T_s W_s = 1$. Gabor pulse + bi-orthogonal partner gives unique clean signal decomposition.

FB-OTFS complexity $3-5\times$ single-pulse OTFS. 3GPP prefers single-pulse for standardization simplicity.

FB-OTFS shines in multi-service ISAC (sensing + data + URLLC). This is a niche, not mainstream 6G.

Rel. 21 (2028-2030): single-pulse OTFS only. Rel. 22 (2030+): potential study item for FB-OTFS. Rel. 23+ (2032+): possible FB-OTFS spec.

Specialized ISAC deployments (automotive, healthcare) may adopt FB-OTFS independently of 3GPP. Mass 6G stays single- pulse.

Adaptive RRC with $\gamma$ selected by scheduler based on expected mobility. Stored $\gamma$ values: 0, 0.15, 0.25, 0.35.

Static: $\gamma = 0$ (rectangular + CP). Pedestrian: $\gamma = 0.15$. Vehicular: $\gamma = 0.25$. HST / LEO: $\gamma = 0.35$ + Gaussian component.

FIR filter coefficients programmable. ROM stores 4-5 pulse families. Per-slot selection. Modest compute.

Scheduler estimates UE mobility from recent channel estimates. Updates $\gamma$ every few frames. No hard switching of hardware path.