Exercises

$\nu_i T = 0.6345$. Integer part: $0$. Fractional part: $0.6345$.

Since $0.6345 > 0.5$, use $k_i^{\text{int}} = 1, \epsilon_i = -0.3655$. Equivalent: $0.6345 - 1 = -0.3655$.

$|\epsilon_i| = 0.37 < 0.5$. Consistent with fractional-offset definition.

$|K_{\text{IDI}}(0, 0.25)|^2 \approx \text{sinc}^2(0.25) = [\sin(0.25\pi)/(0.25\pi)]^2 = 0.811$.

With $N = 32$: $|K_{\text{IDI}}(0, 0.25)|^2 = (1/32)^2 \cdot [\sin(0.25\pi)/\sin(0.25\pi/32)]^2 \approx 0.810$. Close to sinc.

81% of path energy lands in the main-lobe bin. 19% leaks to neighbors. Moderate IDI.

$\text{sinc}(\epsilon) = \frac{\sin(\pi\epsilon)}{\pi\epsilon} = 1 - (\pi\epsilon)^2/6 + O(\epsilon^4)$.

$|\text{sinc}(\epsilon)|^2 = (1 - (\pi\epsilon)^2/6)^2 \approx 1 - (\pi\epsilon)^2/3$.

$\rho_{\text{IDI}}(\epsilon) \approx (\pi\epsilon)^2/3$.

$\epsilon = 0.1$: $\rho = 0.033$ (3.3%). $\epsilon = 0.25$: $\rho = 0.205$ (20%). $\epsilon = 0.5$: Taylor breaks down; exact $\rho = 1 - |\text{sinc}(0.5)|^2 = 0.36$.

$c_0 = \text{sinc}(0.3) = 0.89$. $c_1 = \text{sinc}(0.3 - 1) = \text{sinc}(-0.7) = 0.37$. $c_{-1} = \text{sinc}(0.3 + 1) = \text{sinc}(1.3) = -0.22$. $c_2 = \text{sinc}(-1.7) = 0.21$. $c_{-2} = \text{sinc}(2.3) = -0.09$.

$|c_0|^2 + |c_1|^2 + |c_{-1}|^2 + |c_2|^2 + |c_{-2}|^2 = 0.79 + 0.14 + 0.048 + 0.044 + 0.008 = 1.03$. Roughly unity (rounding error in sinc approximation).

$Q = 5$ captures essentially all of the IDI energy for $\epsilon = 0.3$. $Q = 3$ would capture $0.79 + 0.14 + 0.05 = 0.98$, also essentially all. Both are adequate.

Main-lobe at $k = 0$: $|\text{sinc}(0.3)|^2 = 0.79$. Peak sidelobe at $k = 1$: $|\text{sinc}(-0.7)|^2 = 0.14$. Total IDI: $\sim 0.21$ (21%).

Main-lobe widened: ENBW 1.36, so effective main-lobe captures slightly more energy for moderate $\epsilon$. Peak sidelobe: $-43$ dB below main, so $\sim 10^{-4.3} = 5 \times 10^{-5}$ times smaller. Total IDI: dominated by sidelobes, now $\sim 0.5\%$.

Hamming reduces IDI from 21% to 0.5% — roughly 16 dB improvement. At cost of modest main-lobe widening. Almost always worth it.

$\beta \nu_i T = 2.4$ in the oversampled grid. Integer part: $2$. Fractional part: $0.4$. Since $0.4 > -0.5$, keep: $\epsilon^{(\beta)} = 0.4$. Integer bin: $k_i^{(\beta)} = 2$.

Original: $\epsilon = 0.6 - 1 = -0.4$, bin $= 1$. Oversampled: $\epsilon = 0.4$, bin $= 2$. Same underlying path, but in the finer grid.

$|\text{sinc}(0.4)|^2 = 0.76$ in oversampled (vs original 0.76). IDI power: same as original. So oversampling does NOT reduce IDI per se — it provides finer resolution for super-resolution algorithms.

Oversampling reduces IDI on average over fractional offsets, because a random $\nu_i$ in $(-0.5/T, 0.5/T)$ maps to smaller $|\epsilon^{(\beta)}|$ with probability $1/\beta$ (uniform distribution). The worst-case $\epsilon = 0.5/\beta$ has much less IDI than worst-case $\epsilon = 0.5$ in the original grid.

$K_{\text{IDI}}(k, 0) = \frac{1}{N}\sum_{n=0}^{N-1} e^{-j 2\pi k n/N}$.

The sum of $N$-th roots of unity: $\sum_{n=0}^{N-1} e^{-j 2\pi k n/N} = N \delta_{k \bmod N, 0}$. For $k \in [-N/2, N/2)$: this is $N \delta_{k, 0}$.

$K_{\text{IDI}}(k, 0) = (1/N) \cdot N \cdot \delta_{k, 0} = \delta_{k, 0}$. Integer Doppler gives a clean delta — no IDI. $\blacksquare$

$|K_{\text{IDI}}(0, 0.35)|^2 = |\text{sinc}(0.35)|^2 = 0.73$.

$\sum_i |h_i|^2 \cdot 0.73 = (0.5 + 0.25 + 0.125 + 0.0625 + 0.0625) \cdot 0.73 = 1.0 \cdot 0.73 = 0.73$. 27% of channel power is "invisible" (scattered to IDI bins).

If the detector treats invisible power as noise: $\text{SNR}_{\text{eff}} = \text{SNR}_{\text{true}} \cdot 0.73/(1 + 0.27 \cdot \text{SNR}_{\text{true}})$. At true SNR = 20 dB (100): effective SNR $= 100 \cdot 0.73/(1 + 27) = 2.6$ (4.2 dB). 16 dB penalty for ignoring fractional.

BEM captures 95%+ of each path's energy. Effective SNR penalty: $\sim 0.3$ dB. Full diversity retained.

Uniform $\nu_i T \in [-0.5, 0.5]$. Oversample: $\beta \nu_i T \in [-1, 1]$. Take mod 1: $\epsilon^{(\beta)} \in [-0.5, 0.5]$. BUT: integer bin shifts, so effective offset in the closest bin is $\epsilon^{(\beta)} \in [-0.25, 0.25]$.

Original: $\mathbb{E}[\rho_{\text{IDI}}] = \mathbb{E}[(\pi\epsilon)^2/3] = \pi^2/(3 \cdot 12) = 0.27$. Oversampled: $\mathbb{E}[(\pi\epsilon^{(\beta)})^2/3]$ with $\epsilon^{(\beta)}$ uniform in $[-0.25, 0.25]$: $\mathbb{E}[\epsilon^{(\beta)2}] = 1/48$. So $\mathbb{E}[\rho_{\text{IDI}}^{(\beta)}] = \pi^2/144 = 0.068$.

$0.068/0.27 = 1/4$. Oversampling by $\beta = 2$ reduces IDI by factor $\beta^2 = 4$ (in SNR terms).

Grid size $2\times$ larger: compute $\sim 2\log 2 = 2\times$ per stage. 6 dB IDI reduction for 2× compute — excellent trade-off.

$P \cdot Q_{\text{BEM}} = 6 \cdot 3 = 18$ effective paths.

Per effective path: $MN = 2048$ non-zeros. Total: $18 \cdot 2048 = 36{,}864$ non-zeros. Matrix size: $(MN)^2 = 4{,}194{,}304$. Density: $36864/4194304 = 0.88\%$. Still very sparse.

Integer-Doppler with $P = 6$: $6 \cdot 2048 = 12{,}288$ non-zeros. Density: 0.29%. BEM increases non-zeros by $3\times$. Still sparse enough for fast detection.

$\nu T = 246.94$. $k^{\text{int}} = 247, \epsilon = -0.06$. (Small fractional — well-behaved.)

Main lobe: $|\text{sinc}(-0.06)|^2 = 0.988$. $Q = 1$ captures 98.8%. Already above 99% threshold. $Q = 3$: $0.988 + 2\cdot 0.0001 \approx 0.989$. Negligible gain.

At such large $k_{\max}$ (247), $Q = 1$ suffices for 99% energy. The fractional-Doppler problem at LEO is not about BEM order; it's about estimating $k_{\max} \gg 1$. Need full super-resolution Doppler estimation (Chapter 18).

$|\text{sinc}(0.4)|^2 = 0.62$. $\rho_{\text{eff}} = 10^{1.5} \cdot 0.62 = 19.6$. In dB: $\rho_{\text{eff,dB}} = 12.9$ dB.

$Q(\sqrt{2 \cdot 19.6}) = Q(6.3) \approx 1.5 \times 10^{-10}$. Very low — but this is only the main-lobe contribution.

IDI energy = $1 - 0.62 = 0.38$ (treated as noise). $\text{SINR} = 0.62/(0.38) = 1.63$. $Q(\sqrt{2 \cdot 1.63}) = Q(1.8) = 0.036$ (3.6%).

Ignore-fractional detector: BER ~ 3.6% (high floor from IDI treated as noise). Fractional-aware detector: BER ~ $10^{-10}$ (all IDI recovered as signal). The gap is 8+ orders of magnitude — detection matters.

Hamming sidelobes are $\sim -43$ dB, eliminating almost all cross-bin leakage. Main-lobe widens by 36%. Typical residual SNR loss: $\sim 0.5$ dB.

Captures $\geq 95\%$ of IDI energy. Residual: $\leq 5\%$ energy lost. SNR loss: $\sim 0.3$ dB.

Halves effective $\epsilon$. Reduces residual IDI by $4\times$. SNR loss: $\sim -1$ dB (i.e., 1 dB gain).

Sum: $0.5 + 0.3 - 1 = -0.2$ dB (slight improvement over ideal? No — these are upper bounds; actual gap $\sim 0.3$ dB to ideal integer-Doppler $P = 4$ diversity.)

$6\times$ compute ($3 \times 2 = 6$). $\sim 1.5 \times 10^7$ ops per frame for $MN = 10^4$. Well within realtime.

ENBW = 1.73 times rectangular main lobe. Rectangular: $\sim 1$ bin at $\epsilon = 0.5$. Blackman: $1.73$ bins.

Peak sidelobe $= -58$ dB below main = $10^{-5.8} = 1.6 \times 10^{-6}$ of main-lobe energy. Total IDI $\lesssim 10^{-5}$ — effectively zero.

With Blackman, the fractional-Doppler problem effectively vanishes — the kernel is essentially a discrete delta. The detector can use the integer-Doppler assumption with $<0.1$ dB penalty. Cost: 73% wider main lobe, i.e., each path covers ~2 bins instead of 1. Moderate overhead.

Each effective path contributes a shift-phase matrix $h_i K_{\text{IDI}}(q, \epsilon_i) \boldsymbol{\Psi}_{\ell_i, k_i + q}$. $\boldsymbol{\Psi}_{\ell, k}$ is block-circulant with circulant blocks.

Block-circulant matrices form an algebra (closed under addition). The sum of block-circulant matrices is block-circulant.

All block-circulant matrices share the 2D DFT as a common eigenbasis (Horn-Johnson, matrix analysis).

$\mathbf{H}_{DD}^{\text{ext}} = \mathbf{F}^H \boldsymbol{\Lambda} \mathbf{F}$ with $\boldsymbol{\Lambda}$ diagonal. Eigenvalues are the sum over paths and IDI shifts. MMSE, detection, etc. work as in the integer case. $\blacksquare$

For each estimated path, compute $Q_i = \lceil |\epsilon_i|/0.1\rceil + 1$. Small-offset paths ($|\epsilon| \leq 0.1$): $Q_i = 1$. Large-offset paths ($|\epsilon| = 0.5$): $Q_i = 6$.

For uniform $\epsilon \in [-0.5, 0.5]$: $\mathbb{E}[Q] \approx 3.5$. Typical $P = 8$: total effective paths $\sim 28$. Manageable.

Each path achieves $> 95\%$ energy capture with its own $Q_i$. Total diversity: $\min(P, 2k_{\max}+1)$, same as fixed $Q = 5$.

vs fixed $Q = 5$: $3.5/5 = 30\%$ compute savings. vs fixed $Q = 3$: more paths captured at similar cost. The adaptive scheme is an active research topic; it requires precise per-path $\epsilon_i$ estimation (Chapter 11 super- resolution techniques).