Exercises

$\arg\max_m \Big(\log \pi_m - \tfrac{1}{2\sigma^2}\|\mathbf{y}-\mathbf{s}_m\|^2\Big)$.

$\arg\max_m \Big(\log \pi_m - \tfrac{1}{2\sigma^2}\|\mathbf{s}_m\|^2 + \tfrac{1}{\sigma^2}\mathbf{s}_m^{\mathsf T}\mathbf{y}\Big)$.

With equal energies the rule becomes ML on $\mathbf{s}_m^{\mathsf T}\mathbf{y}$ shifted by $\sigma^2\log \pi_m$. Symbols with larger prior have larger (tilted) decision regions.

$\|s_0\|^2 = 1$, so $\phi_1(t) = 1$.

$\langle s_1, \phi_1\rangle = 0$, so $\phi_2(t) = s_1(t)/\|s_1\|$, $\|s_1\|^2 = 1/3$, giving $\phi_2(t) = \sqrt{3}(2t-1)$.

$\langle s_2, \phi_1\rangle = 1/3$; $\langle s_2, \phi_2\rangle = 0$. Residual $\psi_3 = s_2 - 1/3$, $\|\psi_3\|^2 = 4/45$, giving $\phi_3(t) = (3\sqrt{5}/2)(4t^2-4t+2/3)$.

$N = 3 = M$: the three signals are linearly independent.

$d_{mn} = 2\sin(\pi|m-n|/M)$ for $M$-PSK.

$d_{\min} = 2\sin(\pi/8) \approx 0.765$.

Each point has exactly $K_{\min} = 2$ nearest neighbors.

$P(m\to m'\mid m) = Q(d_{mm'}/(2{\sigma^2}^{1/2}))$.

$P_e(m) \leq \sum_{m'\neq m} Q(d_{mm'}/(2{\sigma^2}^{1/2}))$; averaging over $m$ gives the SER bound.

At high SNR, pairs with $d_{mm'} > d_{\min}$ contribute exponentially less. Only nearest-neighbor terms survive, and the bound matches the true SER to leading order.

QPSK points at $(\pm 1,\pm 1)/\sqrt{2}$ and 4-PSK points at $\{1, j, -1, -j\}$ are related by a $\pi/4$ rotation.

At 10 dB, $P_e \approx 2Q(\sqrt{10}) \approx 1.6\times 10^{-3}$.

$P_e = 1 - (1 - 2\cdot(3/4)Q(\sqrt{3\cdot 10^{1.5}/15}))^2 \approx 2.5\times 10^{-3}$.

$P_e \approx 2Q(\sqrt{2\cdot 10^{1.5}}\sin(\pi/16)) \approx 1.4\times 10^{-2}$.

16-QAM is $\sim 4$ dB better than 16-PSK at this SNR --- QAM packs more efficiently in $\mathbb{R}^2$.

With unit spacing, $\bar{E}_s = (M-1)/6 = 63/6 = 10.5$ (including scaling by 2 for 2D).

Cross-constellation has lower peak-to-average power; the average energy is roughly $9.75$ for the standard cross-64.

$10\log_{10}(10.5/9.75) \approx 0.32$ dB --- the typical shaping gain of cross-QAM over square at moderate $M$.

$\|\mathbf{s}_m - \mathbf{s}_{m'}\|^2 = 2E_s$ (orthogonality).

$P(m\to m') = Q(\sqrt{2E_s/(2N_0)}) = Q(\sqrt{E_s/N_0})$.

Multiply by $M-1$ terms per transmitted signal.

$\bar{P}_e = E_\gamma[Q(\sqrt{2\gamma})] = \frac{1}{\pi}\int_0^{\pi/2} E_\gamma[\exp(-\gamma/\sin^2\phi)]\,d\phi$.

$E_\gamma[\exp(-\gamma/\sin^2\phi)] = 1/(1 + \bar{\gamma}/\sin^2\phi)$.

$\bar{P}_e = \tfrac{1}{2}\big(1 - \sqrt{\bar{\gamma}/(1+\bar{\gamma})}\big)$; decays as $1/(4\bar{\gamma})$ at high SNR.

$\bar{P}_e \approx 1/(4\bar{\gamma})$; at 20 dB this gives $\approx 2.5\times 10^{-3}$.

$P_e = Q(\sqrt{200}) \approx 7\times 10^{-24}$.

Rayleigh fading loses many orders of magnitude at high SNR --- diversity order $1$ is the slowest possible decay, motivating diversity and coding.

$\bar{P}_e = \frac{1}{\pi}\int_0^{\pi/2}(1 + \bar{\gamma}/(m\sin^2\phi))^{-m}\,d\phi$.

Diversity order equals $m$: $\bar{P}_e \sim C/\bar{\gamma}^m$.

For every $s\in[0,1]$: $P_e \leq e^{-n\mu(s)}$, where $\mu(s) = -\log\int f_0^{1-s}f_1^s$. Maximise over $s$: $P_e \leq e^{-n C(f_0,f_1)}$.

By Cramer's theorem, $-\tfrac{1}{n}\log P_e \to C(f_0,f_1)$.

The exponent is tight: neither bound can be improved to leading order in $n$.

The union bound is within 0.5 dB of exact at about 10 dB and above, for 8-PSK.

At high SNR only nearest-neighbor pairs dominate, so union bound becomes a constant multiple of the exact SER.

Unit-energy: $\|\mathbf{s}_m\| = 1$. Four points on $S^1$ ($\mathbb{R}^2$).

Equidistant on the circle: $d_{mm'} = 2\sin(\pi|m-m'|/4)$. $d_{\min} = 2\sin(\pi/4) = \sqrt{2}$ for 4-PSK (square).

No rearrangement can increase $d_{\min}$ without violating energy constraints (Fejes Tóth inequality for $S^1$).

Each symbol error causes approximately 1 bit error out of $\log_2 M = 4$: BER $\approx$ SER$/\log_2 M$.

BER can be as high as SER$\cdot$(average Hamming distance) $\approx$ SER$/2$ --- worse by $2$--$3$ dB at equivalent SER.

Gray labeling is a near-costless way to minimise BER at fixed SER.

$H_2(p) = -p\log_2 p - (1-p)\log_2(1-p)$.

At low SNR, $p \approx 1/2 - d/(2\sqrt{2\pi})$; Taylor expansion gives $C \approx d^2/(2\ln 2)$ nats.

The Chernoff exponent $d^2/8$ (detection) and the capacity $d^2/(2\ln 2)$ (coding) differ by a constant factor --- both proportional to $d^2$.