Exercises

By Theorem TWhen Hybrid Matches Fully Digital, $N_{\text{RF}} \geq 2K$ suffices to realize any fully-digital precoder exactly.

${N_{\text{RF}}}_{\min} = 2 \cdot 4 = 8$ RF chains.

Only 8 RF chains are needed to match a 128-antenna digital array's spectral efficiency, yielding a 16x reduction in data-converter power. $\blacksquare$

$N_{\text{PS}}^{\text{FC}} = 8 \cdot 256 = 2048$ phase shifters.

$N_{\text{PS}}^{\text{SA}} = 256$ phase shifters, a factor 8 reduction.

At $\sim 40$ mW per phase shifter, FC dissipates $\sim 82$ W in shifters alone; SA dissipates $\sim 10$ W. $\blacksquare$

$\eta(1) = \text{sinc}^2(0.5) = (2/\pi)^2 \approx 0.405$. Loss $= -10\log_{10}(0.405) \approx 3.92$ dB.

$\eta(2) = \text{sinc}^2(0.25) \approx 0.811$. Loss $\approx 0.91$ dB.

$\eta(3) = \text{sinc}^2(0.125) \approx 0.950$. Loss $\approx 0.22$ dB.

$\eta(4) = \text{sinc}^2(0.0625) \approx 0.987$. Loss $\approx 0.056$ dB.

Three bits give sub-0.25 dB loss - the practical sweet spot. $\blacksquare$

$\Delta\theta \approx (45 \cdot \pi/180) \cdot (2/28) = 0.785 \cdot 0.0714 \approx 0.056$ rad $\approx 3.2^\circ$.

$\theta_{3\text{dB}} \approx 2/N_t = 2/64 = 0.031$ rad $\approx 1.8^\circ$.

$\Delta\theta = 3.2^\circ > \theta_{3\text{dB}} = 1.8^\circ$: squint is larger than one beamwidth, so the edge-of-band beam misses its target. For this array, either reduce bandwidth, add sub-band precoding, or switch to true-time-delay (Rotman lens). $\blacksquare$

Depth $K = \log_2 256 = 8$, symbols $= 2K = 16$.

$M = 256$ symbols.

$256 / 16 = 16$x reduction. For a $N_t = 1024$ array, the ratio grows to $1024 / 20 = 51$x. The savings scale as $N_t/\log_2 N_t$. $\blacksquare$

$\|\mathbf{F}_{\text{RF}}\|_F^2 = \sum_{m=1}^{N_t}\sum_{n=1}^{N_{\text{RF}}} |[\mathbf{F}_{\text{RF}}]_{m,n}|^2$.

Each entry has magnitude $1/\sqrt{N_t}$, so each contributes $1/N_t$ to the squared norm.

Total $= N_t \cdot N_{\text{RF}} \cdot (1/N_t) = N_{\text{RF}}$. Independent of phases, as claimed. $\blacksquare$

$\sin 15^\circ \approx 0.259$. DFT beam positions: $\sin\theta_m = 2m/32 - 1 \in \{-1, -0.9375, \ldots, 0.9375\}$ for $m = 0, \ldots, 31$ (unambiguous region scaled for $\lambda/2$).

The nearest DFT beam is at $\sin\theta_m = 0.25$ (or $0.3125$); midpoint is $0.28125$. Mismatch to beam at $0.25$: $\Delta(\sin\theta) = 0.009$.

Relative to DFT grid spacing $2/32 = 0.0625$, this is $0.009/0.0625 \approx 0.14$ of a beam width.

Using $\text{sinc}^2(0.14) \approx 0.968$, loss $\approx 0.14$ dB. If the target landed exactly at the midpoint $0.28125$, the mismatch would be $0.5$ beam and the loss would be $\text{sinc}^2(0.5) \approx 0.405 \Rightarrow 3.92$ dB. $\blacksquare$

$\mathbf{A}_t^H \mathbf{W}_{\text{res}}$ is $(G \times N_t) \cdot (N_t \times K)$, $G \cdot N_t \cdot K = 4N_t^{2} \cdot K$ complex muls.

At iteration $i$, $\mathbf{F}_{\text{RF}}^H \mathbf{W}_{\text{opt}}$ is $i \cdot N_t \cdot K$ complex muls; the $i \times i$ inverse is $O(i^3)$. For $i \leq N_{\text{RF}} = 4$, the inversion is negligible.

$\approx 4N_t^{2} \cdot K$ complex muls, dominated by correlation. For $N_t = 64$, $K = 4$: about $65\,000$ complex muls per iteration - very cheap by DSP standards. $\blacksquare$

For any matrices $\mathbf{A} \in \mathbb{C}^{m \times p}$ and $\mathbf{B} \in \mathbb{C}^{p \times n}$, $\text{rank}(\mathbf{A}\mathbf{B}) \leq \min(\text{rank}(\mathbf{A}), \text{rank}(\mathbf{B})) \leq p$.

$\mathbf{F}_{\text{RF}}$ is $N_t \times N_{\text{RF}}$ and $\mathbf{F}_{\text{BB}}$ is $N_{\text{RF}} \times K$, so the inner dimension is $N_{\text{RF}}$. Hence $\text{rank}(\mathbf{F}_{\text{RF}} \mathbf{F}_{\text{BB}}) \leq N_{\text{RF}}$.

If $N_{\text{RF}} < K$, the hybrid precoder cannot independently serve $K$ streams - spatial multiplexing caps at $N_{\text{RF}}$. The bound $N_{\text{RF}} \geq K$ is thus a fundamental constraint, not an engineering convenience. $\blacksquare$

For $N = 16$, centered: $\sin\theta_m = 2(m - N/2)/N = (m - 8)/8$ for $m = 0, \ldots, 15$, giving values $\{-1, -0.875, -0.75, \ldots, 0, 0.125, \ldots, 0.875\}$.

$\theta_m = \arcsin(\sin\theta_m)$: beam 8 points to $0^\circ$ (broadside), beam 9 to $\arcsin(0.125) \approx 7.2^\circ$, beam 15 to $\arcsin(0.875) \approx 61.0^\circ$, beam 0 to $-90^\circ$ (endfire), etc.

The 16 beams uniformly tile $\sin\theta \in [-1, 1]$ in steps of $0.125$, corresponding to angles clustered near broadside and widening toward endfire. Beam endfire beams have reduced gain due to the ULA element pattern. $\blacksquare$

Write $|a|/A = \cos\beta$ for a unique $\beta \in [0, \pi/2]$ (possible since $|a| \leq A$). Set $\alpha = \arg(a)$, so $a = A e^{j\alpha}\cos\beta$.

$\cos\beta = \frac{1}{2}(e^{j\beta} + e^{-j\beta})$, so $a = \frac{A}{2}(e^{j(\alpha + \beta)} + e^{j(\alpha - \beta)})$.

Set $\phi_1 = \alpha + \beta$, $\phi_2 = \alpha - \beta$. Both are real angles; the representation $a = (A/2)(e^{j\phi_1} + e^{j\phi_2})$ holds. $\blacksquare$

Two unit-modulus vectors sum to any complex number of modulus $\leq A$ where $A$ is the sum of their magnitudes. This is the elementary identity behind the hybrid-matches-digital result.

Let $\phi_\ell$ be the true direction, $\hat\phi_\ell$ the nearest grid direction. Expand $\mathbf{a}(\phi_\ell) = \mathbf{a}(\hat\phi_\ell) + (\phi_\ell - \hat\phi_\ell) \cdot \mathbf{a}'(\hat\phi_\ell) + O((\Delta\theta)^2)$. The derivative norm is $\|\mathbf{a}'\|^2 \sim N_t^{2}/3$ for a ULA of size $N_t$.

$\|\mathbf{a}(\phi_\ell) - \mathbf{a}(\hat\phi_\ell)\|^2 \leq (\Delta\theta/2)^2 \cdot N_t^{2}/3 = N_t^{2} (\Delta\theta)^2/12$ (worst case, midpoint mismatch).

OMP picks the nearest dictionary atom for each path and absorbs its amplitude. The residual contributed by path $\ell$ is at most $|\alpha_\ell|^2 \cdot N_t^{2} (\Delta\theta)^2/12$. Summing: $$\|\mathbf{W}_{\text{opt}} - \mathbf{F}_{\text{RF}}\mathbf{F}_{\text{BB}}\|_F^2 \leq \frac{L \cdot N_t^{2} (\Delta\theta)^2}{12} \cdot \mathbb{E}[|\alpha|^2].$$

For $\Delta\theta = 2\pi / G$ with $G = 4N_t$, the relative residual is $O(1/N_t^{2})$, which is below the numerical precision floor for $N_t \geq 64$. Coarser dictionaries ($G = N_t$) yield $O(1)$ residual, destroying OMP optimality. $\blacksquare$

$P_{\text{tot}}(N_{\text{RF}}) = P_t + N_{\text{RF}}(P_{\text{RF}} + N_t P_{\text{PS}}) = 0.5 + N_{\text{RF}}(1 + 64 \cdot 0.04) = 0.5 + N_{\text{RF}} \cdot 3.56$ W.

Assume multi-stream gain: $R(N_{\text{RF}}) = N_{\text{RF}} \log_2(1 + \eta N_t P_t/(N_{\text{RF}}\sigma^2))$, where we split the transmit power equally across streams. With $\eta = 0.95$, $N_t = 64$, $P_t = 0.5$ W, $\sigma^2 = 10^{-12}$ W: per-stream SNR $= 0.95 \cdot 64 \cdot 0.5/(N_{\text{RF}} \cdot 10^{-12}) = 3.04 \times 10^{13}/N_{\text{RF}}$.

Per-stream SNR in dB: $N_{\text{RF}}=1$: 134.8 dB; $N_{\text{RF}}=16$: 122.8 dB. Rates: $R(1) \approx 44.8$, $R(2) \approx 89.5$, $R(4) \approx 178.6$, $R(8) \approx 355.8$, $R(16) \approx 707.2$ bits/s/Hz (enormous at these SNRs). EE (bits/J): EE(1) $\approx 44.8/4.06 = 11.0$; EE(4) $\approx 178.6/14.74 = 12.1$; EE(16) $\approx 707.2/57.46 = 12.3$.

The EE is remarkably flat in $N_{\text{RF}}$ because the log-rate gain from multiplexing approximately cancels the linear phase-shifter power increase. In realistic conditions with lower SNR and fewer channel paths, a pronounced EE peak appears near $N_{\text{RF}} = L$ (matching the channel rank). $\blacksquare$

Each subarray of size $N_t/N_{\text{RF}}$ has beamwidth $\theta_{3\text{dB}} \approx 2N_{\text{RF}}/N_t$ radians. This is the angular resolution each subarray can exploit.

If the $L$ channel paths are angularly separated by more than one sub-aperture beamwidth, one path per subarray can be aligned without inter-path leakage: $\min_{\ell \neq \ell'} |\phi_\ell - \phi_{\ell'}| > 2N_{\text{RF}}/N_t$.

Assign subarray $n$ to path $\ell_n$ such that $\mathbf{v}_{n} = \mathbf{a}(\phi_{\ell_n})$ restricted to subarray $n$'s antenna set. The digital precoder combines them. Effective aperture per path $= N_t/N_{\text{RF}}$, but coherent combining across subarrays - when the paths are resolved - recovers the full aperture gain.

Subarray architectures are near-optimal for mmWave channels with $L \leq N_{\text{RF}}$ well-separated clusters. They pay the $10\log_{10}(N_{\text{RF}})$ penalty only when the channel is rank-1 but transmitted streams spread across subarrays incoherently. $\blacksquare$

$\lambda = 1$ mm. FSPL at 100 m: $20\log(4\pi \cdot 100/10^{-3}) = 122$ dB plus $10\log(W/1\text{Hz})$ of noise. With $W = 10$ GHz and noise figure 10 dB, thermal noise is $-174 + 100 + 10 = -64$ dBm. For 20 dB SNR: $P_r = -44$ dBm. With $P_t = 20$ dBm and path loss 122 dB: required antenna gain $G_t + G_r = 58$ dB, split 29 dBi each side.

$G = 29$ dBi $= 794$ linear $\Rightarrow$ equivalent $N_t \approx 800$ antennas per side. At $\lambda/2$ spacing (0.5 mm), this is a $20 \times 20$ planar array of 2 cm $\times$ 2 cm - mechanically feasible but a phase-shifter network would cost $800 \cdot N_{\text{RF}}$ shifters.

At 300 GHz, phase-shifter loss is prohibitive ($> 10$ dB insertion) and bandwidth limitations (squint) are severe for 10 GHz signals. Choose an array-fed parabolic reflector (CommIT contribution): a 5 cm dish with $N_{\text{RF}} = 4$ active chains in the focal plane. The reflector provides the 29 dBi gain losslessly, and the 4 chains allow spatial multiplexing across up to 4 directions or users.

The reflector is achromatic (geometric optics), so zero squint. With $N_{\text{RF}} = 4$, the system can serve 4 users simultaneously or provide 4x spatial multiplexing to a single user with a well-conditioned channel. At 20 dB SNR and 10 GHz bandwidth, per-stream rate $\approx 10 \cdot \log_2(1 + 100) \approx 67$ Gbps, well above the 10 Gbps target.

Recommended architecture: 5 cm parabolic reflector + $N_{\text{RF}} = 4$ active RF chains at the focal plane, digital baseband precoding. No phase shifters. Achromatic operation across 10 GHz, spatial multiplexing, target spectral efficiency met with large margin. $\blacksquare$

A block-diagonal $\mathbf{F}_{\text{RF}}^{\text{SA}}$ has off-diagonal blocks that are zero, violating the constant-modulus constraint $|\cdot| = 1/\sqrt{N_t}$. SA is thus not literally a special case of FC at the analog level.

Construct an FC matrix $\mathbf{F}_{\text{RF}}^{\text{FC}}$ where off-diagonal blocks are random constant-modulus phases, and design $\mathbf{F}_{\text{BB}}$ such that the product $\mathbf{F}_{\text{RF}}^{\text{FC}} \mathbf{F}_{\text{BB}}$ equals the desired SA composite precoder $\mathbf{F}_{\text{RF}}^{\text{SA}} \mathbf{F}_{\text{BB}}^{\text{SA}}$. This requires the off-diagonal contributions to cancel through the $\mathbf{F}_{\text{BB}}$ pairing.

The set of composite precoders realizable by SA is a lower-dimensional manifold (block-structured) than the full FC set. Any SA precoder can be matched by an FC precoder (by construction), but not every FC precoder has an SA representation. Hence: $$\{\text{SA hybrid precoders}\} \subsetneq \{\text{FC hybrid precoders}\}.$$ The inclusion is strict. $\blacksquare$