Exercises

$6.02 \cdot 4 + 1.76 = 25.84$ dB.

$1/\mathrm{SNR}_{\rm eff} = 1/\mathrm{SNR}_{\rm in} + 1/\mathrm{SNR}_q = 1/100 + 1/384 \approx 0.0126$. $\mathrm{SNR}_{\rm eff} \approx 79.5 \to 19$ dB.

1 dB loss from 4-bit quantisation at 20 dB input. Acceptable for most communication links.

1-bit / Shannon = 0.735 / 1 = 73.5%.

At 0 dB, 1-bit ADC recovers 73.5% of Shannon capacity. At very low SNR this ratio approaches $2/\pi \cdot \log_2 e \approx 0.92$ (low-SNR asymptote in nats). The 0 dB point is intermediate.

$R_{\rm SM} = 3 + 6 = 9$ bits/ch.use.

$\binom{6}{3} = 20$; $\lfloor \log_2 20 \rfloor = 4$ bits.

$3 \cdot 4 = 12$ bits.

$R_{\rm GSM} = 4 + 12 = 16$ bits/ch.use.

$\mathrm{SNR}_q \approx \mathrm{SNR}_{\rm in}$ at about 50 dB input SNR.

Below 50 dB, thermal noise dominates and 8-bit ADC is sufficient. Above 50 dB (optical coherent systems!), quantisation matters and higher-resolution ADCs are required.

$\sqrt{1/n_t} = 1/\sqrt{64} = 0.125 = 12.5\%$.

At $n_t = 64$, the SNR fluctuation is 12.5% around the mean — roughly 1 dB peak-to-peak. From the user's perspective, the channel is nearly deterministic.

BICM design at $n_t = 64$ is essentially AWGN design. The ~1 dB BICM-Shannon gap on flat channels (Ch 5) is nearly the full gap; no additional fading penalty.

$\frac{dR}{dn_a} = -\log(n_a/(n_t-n_a)) + \log_2 M$ (using Stirling for the binomial term). Setting to zero: $n_a/(n_t - n_a) = M$, i.e. $n_a^*/(n_t - n_a^*) = M$.

$M = 2$ (BPSK): $n_a^* = 2n_t/3$; $M = 16$: $n_a^* = 16n_t/17$, very close to $n_t$; $M = 256$: $n_a^* \approx n_t$.

Larger constellation → more marginal value per antenna → $n_a^* \to n_t$. At $M = 2$ the optimum is near $n_t/2$; at large $M$ it saturates near $n_t$ (use all antennas).

$C_{\rm 1bit} \approx 2 \cdot (2/\pi) \cdot 1 \cdot 1.44 = 1.83$ bits/use.

Actually, the BS combines across antennas BEFORE quantisation consequence matters. Effective combined SNR is $128 \cdot 1 = 128$ ($\sim 21$ dB). At 21 dB, the 1-bit effect saturates both dimensions: $C_{\rm 1bit} \approx 2 \cdot (1 - h_2(Q(\sqrt{128}))) \approx 2 \cdot 1 = 2$ bits.

The MASSIVE combining PUSH the effective SNR high enough that 1-bit saturates at its hard upper bound (QPSK-equivalent). Hence 1-bit massive MIMO recovers near-full performance at moderate input SNR — provided joint BS processing is used.

$C_{\rm Shannon} = \log_2(1 + \rho) \approx \rho / \ln 2 = \rho \log_2 e$ (in bits per channel use, nats-based).

$Q(\sqrt{\rho}) = 1/2 - \sqrt{2\rho/\pi}/2 + O(\rho)$. So $1 - 2Q(\sqrt{\rho}) \approx \sqrt{2\rho/\pi}$.

$h_2(1/2 - \epsilon) \approx 1 - \epsilon^2 \log_2 e \cdot 2$. With $\epsilon = \sqrt{2\rho/\pi}/2$: $h_2 \approx 1 - 2 \rho/(\pi \ln 2)$.

$C_{\rm 1bit} = 2(1 - h_2) \approx 2 \cdot 2\rho/(\pi \ln 2) = (2/\pi) \rho \log_2 e$.

$C_{\rm 1bit} / C_{\rm Shannon} \to 2/\pi \approx 0.637$ as $\rho \to 0$. In dB: $10 \log_{10}(2/\pi) = -1.96$ dB. $\blacksquare$

At diversity $n_r = 2$, SNR 15 dB: ML BER $\approx 10^{-3}$ for 16-QAM.

3 dB worse effective SNR: 12 dB equivalent. BER at 12 dB, diversity 2: $\sim 3 \times 10^{-3}$.

For SM to approach its theoretical performance, ML detection or high-complexity sphere decoding is required. Two-stage decoding is cheap but costly in BER. This is the "hidden cost" of SM.

4 phase levels per element (0°, 90°, 180°, 270°).

Optimal unquantised beamforming aligns phases perfectly. With 2-bit quantisation, worst-case phase error is 45°; average error is ~22°.

Beamforming power gain with phase error $\theta$ is $\cos^2(\theta)$. For 22° average error: $\cos^2(22°) \approx 0.86$. Loss: ~0.65 dB — modest for 2-bit phase. 3-bit phase reduces the loss to ~0.2 dB.

For i.i.d. $|h_i|^2$ with mean 1 and variance 1: $\frac{1}{n_t}\sum_{i=1}^{n_t} |h_i|^2 \xrightarrow{n_t \to \infty} 1$ almost surely.

$\sqrt{n_t}(\frac{1}{n_t}\sum |h_i|^2 - 1) \xrightarrow{d} \mathcal{N}(0, 1)$.

$\mathrm{SNR}_{\rm MRC} / n_t = (\rho/n_t) \sum |h_i|^2 \to \rho$ as $n_t \to \infty$. The SNR distribution concentrates around $\rho$ (per-antenna SNR) times $n_t$ — a deterministic limit. $\blacksquare$

$1/\mathrm{SNR}_{\rm eff} = 1/10^{0.5} + 1/10^{1.38} = 0.316 + 0.042 = 0.358$. $\mathrm{SNR}_{\rm eff} \approx 2.79 \approx 4.5$ dB.

$256 \cdot 2.79 = 714 \approx 28.5$ dB.

$\log_2(1 + 714) \approx 9.5$ bits/use. This is about 80% of full-resolution ADC capacity (10.5 bits/use). 2-bit ADCs trade ~1 dB for 5-6× power savings.

Index bits: $\lfloor \log_2 \binom{16}{4} \rfloor = \lfloor \log_2 1820 \rfloor = 10$ bits. Modulation: $4 \cdot 2 = 8$ bits. Total: 18 bits per OFDM symbol. Per-subcarrier: $18 / 16 = 1.125$ bits/subcarrier.

Full 16-subcarrier QPSK: $16 \cdot 2 = 32$ bits per symbol. Per-subcarrier: $32/16 = 2$ bits/subcarrier.

OFDM-IM has 44% less rate but activates only 4 subcarriers — a 75% reduction in PAPR (peak-to-average power ratio) and simplified power amplifier operation. Attractive for low-power IoT devices.

(1) Joint BS detection — 1-bit signals must be combined across antennas BEFORE making hard decisions. (2) Transmit constellation matched to the 1-bit output alphabet — QPSK-centric with shaping. (3) LDPC or polar code matched to the joint BS detector output statistics.

(i) Model the full 1-bit MIMO link as a neural network. (ii) Jointly learn Tx constellation + BS combiner + decoder via cross-entropy loss over random channels. (iii) Regularise for robustness via channel distribution randomisation.

(a) Training convergence in high-dim latent space. (b) Generalisation across different channel distributions. (c) Complexity of receiver during deployment.

Early results (Mezghani-Swindlehurst 2019+) show ~5% capacity loss at 20 dB SNR with learned schemes vs analytical optimum. Full-scale deployment unlikely before 2030.