Exercises

A 10 dB increase (factor of 10 in linear SNR) produces a $10^4$ improvement in BER: from $10^{-2}$ to $10^{-6}$.

Since $P_b \propto \bar{\gamma}^{-d}$: $(10)^d = 10^4 \implies d = 4$.

At $\bar{\gamma} = 30$ dB (factor 10 higher than 20 dB):

$P_b = 10^{-6} \times 10^{-4} = 10^{-10}$.

Each additional 10 dB reduces BER by four decades. $\blacksquare$

$P(\gamma < 1) = 1 - e^{-1/\bar{\gamma}} = 1 - e^{-1/100} \approx 1/100 = 10^{-2}$.

About 1% of the time, the link is in a deep fade.

For $\gamma_{\text{MRC}} \sim \text{Erlang}(4, 100)$:

$P(\gamma_{\text{MRC}} < 1) = \frac{\gamma(4, 1/100)}{3!} \approx \frac{(1/100)^4}{4!} = \frac{10^{-8}}{24} \approx 4.2 \times 10^{-10}$

(using the incomplete gamma function expansion for small arguments).

MRC reduces the deep fade probability from $10^{-2}$ to about $4 \times 10^{-10}$ — more than seven orders of magnitude improvement. $\blacksquare$

$\rho = 0$: full diversity, $d_{\text{eff}} = 2$.

$\rho = 1$: both branches fade together, so the system behaves as a single branch. $d_{\text{eff}} = 1$.

For $0 < \rho < 1$, the diversity order is still $d_{\text{eff}} = 2$ at sufficiently high SNR, because any $\rho < 1$ ensures that the probability of joint deep fading decays as $\bar{\gamma}^{-2}$.

However, the coding gain (horizontal shift of the BER curve) degrades with increasing $\rho$. The BER curve at finite SNR appears closer to $d = 1$, and the asymptotic $d = 2$ slope may only be visible at very high SNR.

Rule of thumb: $\rho < 0.5$ gives near-full diversity gain; $\rho > 0.7$ significantly reduces the practical diversity benefit. $\blacksquare$

Each $\gamma_l \sim \text{Exp}(\bar{\gamma})$ with $F(\gamma) = 1 - e^{-\gamma/\bar{\gamma}}$.

$F_{\gamma_{\text{SC}}}(\gamma) = P(\max_l \gamma_l \leq \gamma) = \prod_{l=1}^L F(\gamma) = (1 - e^{-\gamma/\bar{\gamma}})^L$.

$E[\gamma_{\text{SC}}] = \int_0^\infty [1 - (1 - e^{-\gamma/\bar{\gamma}})^L]\, d\gamma$

Using the binomial expansion:

$= \bar{\gamma} \sum_{k=1}^{L} \frac{(-1)^{k+1}}{k} \binom{L}{k} = \bar{\gamma} \sum_{l=1}^{L} \frac{1}{l}$

SC gain $= \sum_{l=1}^{L} 1/l$ (harmonic sum):

• $L = 1$: gain $= 1.0$ (0 dB)
• $L = 2$: gain $= 1.5$ (1.76 dB)
• $L = 4$: gain $= 2.083$ (3.19 dB)
• $L = 8$: gain $= 2.717$ (4.34 dB)

For comparison, MRC gain is always $L$ (6.02 dB for $L = 4$), so SC leaves $6.02 - 3.19 = 2.83$ dB on the table at $L = 4$. $\blacksquare$

For $L = 2$:

$$P_b = \left(\frac{1-\mu}{2}\right)^2 \sum_{k=0}^{1} \binom{1+k}{k} \left(\frac{1+\mu}{2}\right)^k$$

$= \left(\frac{1-\mu}{2}\right)^2 \left[1 + 2 \cdot \frac{1+\mu}{2}\right] = \left(\frac{1-\mu}{2}\right)^2 (2 + \mu)$

$\mu = \sqrt{10/11} = 0.9535$.

$(1-\mu)/2 = 0.02326$.

$P_b = (0.02326)^2 \times (2 + 0.9535) = 5.41 \times 10^{-4} \times 2.954 = 1.60 \times 10^{-3}$.

$\mu = \sqrt{100/101} = 0.99503$.

$(1-\mu)/2 = 0.002485$.

$P_b = (0.002485)^2 \times (2 + 0.99503) = 6.175 \times 10^{-6} \times 2.995 = 1.85 \times 10^{-5}$.

For comparison, without diversity ($L = 1$): $P_b(10\text{ dB}) \approx 0.025$, $P_b(20\text{ dB}) \approx 2.5 \times 10^{-3}$. MRC with $L = 2$ provides approximately 15x and 135x improvement, respectively. $\blacksquare$

EGC co-phases and adds with equal weights: $r = \sum_l e^{-j\angle h_l} r_l = (\sum_l |h_l|) s + \tilde{w}$.

Signal power: $(\sum_l |h_l|)^2 E_s$. Noise power: $L N_0$ (sum of $L$ independent noise terms).

$\gamma_{\text{EGC}} = (\sum_l |h_l|)^2 E_s / (L N_0)$.

Using the Craig representation of $Q(\sqrt{2\gamma})$ and the MGF of $\gamma_{\text{EGC}}$:

For $L = 2$ Rayleigh branches, the MGF of $X = |h_1| + |h_2|$ involves the distribution of the sum of two Rayleigh random variables, which does not have a simple closed form.

However, using the alternative Q-function representation and numerical integration over $\theta$:

$P_b = \frac{1}{\pi} \int_0^{\pi/2} M_{\gamma_{\text{EGC}}}(-1/\sin^2\theta)\, d\theta$

This integral can be evaluated numerically.

At $\bar{\gamma} = 15$ dB ($= 31.6$):

• MRC ($L = 2$): $P_b \approx 6.3 \times 10^{-4}$ (from exact formula)
• EGC ($L = 2$): $P_b \approx 8.1 \times 10^{-4}$ (numerical integration)
• SC ($L = 2$): $P_b \approx 2.1 \times 10^{-3}$

EGC is about 0.5 dB worse than MRC — a very small penalty for not needing amplitude estimates. $\blacksquare$

In AWGN, each branch has the same deterministic channel gain $|h|$. MRC output SNR: $\gamma_{\text{MRC}} = L \cdot |h|^2 E_s / N_0 = L \gamma$.

This is a factor of $L$ ($10\log_{10} L$ dB) improvement. This is array gain — the coherent combining of signal energy from $L$ branches. There is no diversity gain because there is no fading to combat.

In fading, MRC provides both:

• Array gain: $E[\gamma_{\text{MRC}}] = L \bar{\gamma}$ ($10\log_{10} L$ dB increase in mean SNR)

• Diversity gain: the BER slope changes from $\bar{\gamma}^{-1}$ to $\bar{\gamma}^{-L}$, providing a much larger improvement at high SNR than array gain alone.

At $\bar{\gamma} = 20$ dB with $L = 4$: array gain alone would give $\sim 6$ dB improvement, but diversity changes the BER from $\sim 10^{-3}$ to $\sim 10^{-9}$ — a gain far exceeding 6 dB. $\blacksquare$

$r_1 = (0.5+0.5j)(+1) + (1.0-0.5j)(-1) + (0.1-0.05j)$ $= 0.5 + 0.5j - 1.0 + 0.5j + 0.1 - 0.05j = -0.4 + 0.95j$

$r_2 = -(0.5+0.5j)(-1)^* + (1.0-0.5j)(+1)^* + (-0.08+0.02j)$ $= 0.5 + 0.5j + 1.0 - 0.5j - 0.08 + 0.02j = 1.42 + 0.02j$

$\tilde{s}_1 = h_1^* r_1 + h_2 r_2^*$ $= (0.5-0.5j)(-0.4+0.95j) + (1.0-0.5j)(1.42-0.02j)$ $= (-0.2+0.475j+0.2j+0.475) + (1.42-0.02j-0.71j-0.01)$ $= 0.275+0.675j + 1.41-0.73j = 1.685 - 0.055j$

$|h_1|^2 + |h_2|^2 = 0.5 + 1.25 = 1.75$

$\tilde{s}_1 / 1.75 \approx 0.963 - 0.031j$

The real part is positive $\to \hat{s}_1 = +1$. Correct.

$\tilde{s}_2 = h_2^* r_1 - h_1 r_2^*$

After computation: $\tilde{s}_2 / 1.75 \approx -0.97 + \text{small imaginary}$

Real part negative $\to \hat{s}_2 = -1$. Correct.

Both symbols are correctly detected despite the noise. $\blacksquare$

$\gamma_{\text{MRC}} = (|h_1|^2 + |h_2|^2) P / N_0$

$E[\gamma_{\text{MRC}}] = 2P/N_0 = 2\bar{\gamma}$

Diversity order: $d = 2$.

$\gamma_{\text{Alam}} = (|h_1|^2 + |h_2|^2) (P/2) / N_0$

$E[\gamma_{\text{Alam}}] = 2 \cdot P/(2N_0) = P/N_0 = \bar{\gamma}$

Diversity order: $d = 2$.

The 3 dB loss relative to MRC comes from the power split.

With 2 RX antennas, the effective channel gain per symbol is $\sum_{i=1}^{2}\sum_{j=1}^{2} |h_{ij}|^2$, and each TX antenna uses power $P/2$:

$E[\gamma] = 4 \cdot P/(2N_0) = 2P/N_0 = 2\bar{\gamma}$

Diversity order: $d = 2 \times 2 = 4$.

The $2 \times 2$ Alamouti has the same average SNR as $1 \times 2$ MRC but twice the diversity order. $\blacksquare$

$R = K/T = 3/4 = 0.75$ symbols per channel use.

With full-rank OSTBC for $N_t = 4$ and $N_r = 1$: $d = N_t \times N_r = 4 \times 1 = 4$.

Rate-3/4 STBC with QPSK: $\eta = 0.75 \times 2 = 1.5$ bits/s/Hz.

Alamouti ($N_t = 2$) with QPSK: $\eta = 1.0 \times 2 = 2.0$ bits/s/Hz.

The rate-3/4 code loses 25% spectral efficiency.

The $4 \times 1$ STBC provides diversity order 4 (vs 2 for Alamouti) at the cost of 25% rate loss. Whether this trade-off is worthwhile depends on the operating SNR: at high SNR, the steeper BER slope from $d = 4$ eventually compensates for the rate loss, but at low-to-moderate SNR, the 25% rate reduction may outweigh the diversity benefit.

In practice, 5G NR does not use rate-3/4 STBC; instead, it uses precoding with CSIT for $N_t > 2$. $\blacksquare$

$f_d = \frac{120/3.6 \times 3.5 \times 10^9}{3 \times 10^8} = \frac{33.33 \times 3.5 \times 10^9}{3 \times 10^8} = 389$ Hz.

$T_c \approx 1/(4 f_d) = 1/1556 = 0.643$ ms.

$T_s = 1/30000 = 33.3\;\mu$s.

$D \geq T_c/T_s = 0.643/0.0333 = 19.3 \to D = 20$ symbols (minimum).

Latency $= D \times T_s = 20 \times 33.3\;\mu\text{s} = 0.67$ ms.

This is well within the limits for most applications, including VoIP ($< 100$ ms) and even many 5G services ($< 10$ ms). $\blacksquare$

$B_c \approx 1/(5 \times 2 \times 10^{-6}) = 100$ kHz.

(i) GSM ($W = 200$ kHz): $L_f \approx 200/100 = 2$.

(ii) LTE-5 ($W = 5$ MHz): $L_f \approx 5000/100 = 50$.

(iii) 5G NR ($W = 100$ MHz): $L_f \approx 100000/100 = 1000$.

5G NR's 100 MHz bandwidth provides approximately 1000 independent frequency diversity branches, compared to only 2 for GSM. With proper coding across the full bandwidth, 5G NR can exploit this massive frequency diversity, making deep fades virtually impossible.

This is why wideband OFDM systems rarely need spatial diversity for reliability — the frequency diversity alone is often sufficient. $\blacksquare$

$\mathbf{D} = \begin{bmatrix} 0 & 0 \\ 0 & 2 \end{bmatrix}$$

$\mathbf{A} = \mathbf{D}^H \mathbf{D} = \begin{bmatrix} 0 & 0 \\ 0 & 4 \end{bmatrix}$$$\operatorname{rank}(\mathbf{A}) = 1$.

$d = N_r \times \operatorname{rank}(\mathbf{A}) = 1 \times 1 = 1$.

No. Full diversity would require $\operatorname{rank}(\mathbf{A}) = N_t = 2$, but $\operatorname{rank}(\mathbf{A}) = 1$. The code fails to use the first antenna for distinguishing between codewords.

A full-diversity alternative would be the Alamouti code, which ensures $\operatorname{rank}(\mathbf{D}) = 2$ for all codeword pairs. $\blacksquare$

$P_{\text{out}} = Q(12/6) = Q(2) = 0.0228$.

$P_{\text{out,macro}} = (0.0228)^3 = 1.19 \times 10^{-5}$.

To achieve $P_{\text{out}} = 1.19 \times 10^{-5}$ with a single link:

$Q^{-1}(1.19 \times 10^{-5}) = 4.22$

Margin $= 4.22 \times 6 = 25.3$ dB.

Macrodiversity saves $25.3 - 12 = 13.3$ dB in fade margin. $\blacksquare$

$\text{SNR}_{1} = 20 - 120 - (-100) = 0$ dB.

$\text{SNR}_{2} = 20 - 123 - (-100) = -3$ dB.

With coherent JT, the received power adds coherently (amplitude addition):

Signal amplitude from BS1: $\propto 10^{0/20} = 1$. Signal amplitude from BS2: $\propto 10^{-3/20} = 0.708$.

Combined amplitude: $1 + 0.708 = 1.708$. Combined power: $1.708^2 = 2.917$ (4.65 dB).

$\text{SNR}_{\text{JT}} = 4.65$ dB.

Best single link: $\text{SNR}_{1} = 0$ dB.

CoMP JT gain: $4.65 - 0 = 4.65$ dB.

This includes both array gain (coherent combining of two signals) and the contribution of the second, weaker link. $\blacksquare$

With 2 BSs $\times$ 2 antennas = 4 independent branches combined optimally (MRC across all):

$d = 4$.

(i) 4-antenna MRC at single BS: $d = 4$ (same diversity order).

(ii) 2-antenna MRC at each BS, then select best BS: Each BS provides $d_{\text{micro}} = 2$. Selection between 2 BSs provides an additional diversity gain. Total $d = 4$ (since selection of the max of two order-2 branches gives order 4 asymptotically, though with worse coding gain than full MRC).

• 4-antenna MRC at one BS: Simplest backhaul (no inter-BS coordination), but no shadowing protection.

• 2 BS $\times$ 2 ant with full combining: Best performance (macrodiversity + microdiversity), but requires high-capacity, low-latency backhaul for joint processing.

• 2 BS $\times$ 2 ant with selection: Simpler backhaul (only need to compare aggregate SNR), still protects against shadowing, slight coding gain loss vs full combining.

In practice, configuration (ii) with selection is the most common because it balances diversity gain against backhaul requirements. $\blacksquare$

$\bar{\gamma} = 12$ dB $= 15.85$. $P_b \approx 1/(4 \times 15.85) = 0.0158$.

Far above the $10^{-5}$ target. Not achievable.

Try $L = 2$: $P_b \approx 3/(4 \times 15.85)^2 = 3/4016 = 7.5 \times 10^{-4}$. Too high.

Try $L = 3$: $P_b \approx 10/(4 \times 15.85)^3 = 10/253600 = 3.9 \times 10^{-5}$. Close but too high.

Try $L = 4$: $P_b \approx 35/(4 \times 15.85)^4 = 35/16 \times 10^6 = 2.2 \times 10^{-6}$. Meets target.

Minimum MRC: $L = 4$ branches.

SC has the same diversity order as MRC but worse coding gain (approximately 2.8 dB loss at $L = 4$). At $\bar{\gamma} = 12$ dB, SC effectively operates at $\bar{\gamma}_{\text{eff}} \approx 9.2$ dB.

With this effective SNR, even $L = 4$ SC gives approximately $P_b \approx 2 \times 10^{-4}$. Need $L = 5$ or $L = 6$.

SC requires approximately $L = 6$ branches (numerical computation confirms $P_b < 10^{-5}$ at $L = 6$).

Alamouti $2 \times 1$: diversity order 2, but with 3 dB penalty (effective $\bar{\gamma} = 9$ dB per branch).

$P_b \approx 3/(4 \times 7.94)^2 = 3/1008 = 3.0 \times 10^{-3}$.

Not sufficient. Alamouti $2 \times 2$ (4 receive antennas) would be needed, which gives $d = 4$ and meets the target. $\blacksquare$

$d(r) = (2-r)(2-r) = (2-r)^2$ for $0 \leq r \leq 2$.

• $r = 0$: $d = 4$ (full diversity)
• $r = 1$: $d = 1$
• $r = 2$: $d = 0$ (full multiplexing, no diversity)

Maximum diversity order: 4. Maximum multiplexing gain: 2.

Alamouti transmits at rate $R = 1$ symbol per channel use with full diversity $d = 2 N_r$. For $N_r = 2$: $d = 4$.

In DMT terms, Alamouti does not increase the multiplexing rate beyond a single-antenna system ($r = 0$), but extracts all available diversity. It operates at $(r, d) = (0, 4)$.

$d(2) = (4-2)(4-2) = 4$.

Even at half the maximum multiplexing gain, a $4 \times 4$ system retains diversity order 4 — substantial fading protection.

• Low SNR / cell edge / reliability-critical: Prioritise diversity ($r \to 0$). Use STBC or diversity combining.

• High SNR / cell center / throughput-driven: Prioritise multiplexing ($r \to \min(N_t, N_r)$). Use spatial multiplexing.

• Adaptive: Modern systems (LTE, 5G NR) switch dynamically: transmit diversity at cell edge, spatial multiplexing at cell center, based on channel quality feedback.

The DMT is revisited in detail in Chapter 13 (MIMO). $\blacksquare$