These notes are synthesized from the ECE 414 / ECE 614 course offerings at the University of Waterloo (Spring 2025, instructor Prof. Guang Gong) and draw on the standard graduate-level treatments of Goldsmith, Tse–Viswanath, and Proakis–Salehi. They are intended to function as a self-contained mathematical reference, not as a substitute for working through the primary texts.

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Sources and References

The following publicly available texts and resources form the scholarly basis for these notes.

• D. Tse and P. Viswanath, Fundamentals of Wireless Communication, Cambridge University Press, 2005. Available open-access at https://web.stanford.edu/~dntse/wireless_book.html

• A. Goldsmith, Wireless Communications, Cambridge University Press, 2005 (reprinted 2007, 2009). Cambridge Core e-book available.

• J. G. Proakis and M. Salehi, Communication Systems Engineering, 2nd ed., Prentice Hall, 2002.

• A. F. Molisch, Wireless Communications, 2nd ed., Wiley–IEEE Press, 2011.

• T. L. Marzetta, E. G. Larsson, H. Yang, and H. Q. Ngo, Fundamentals of Massive MIMO, Cambridge University Press, 2016.

• L. Zheng and D. N. C. Tse, “Diversity and multiplexing: A fundamental tradeoff in multiple-antenna channels,” IEEE Trans. Inf. Theory, vol. 49, no. 5, pp. 1073–1096, May 2003.

• S. M. Alamouti, “A simple transmit diversity technique for wireless communications,” IEEE J. Sel. Areas Commun., vol. 16, no. 8, pp. 1451–1458, Oct. 1998.

• M. Costa, “Writing on dirty paper,” IEEE Trans. Inf. Theory, vol. 29, no. 3, pp. 439–441, May 1983.

• MIT OpenCourseWare 6.450: Principles of Digital Communications. https://ocw.mit.edu/courses/6-450-principles-of-digital-communications-i-fall-2006/

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Chapter 1: Foundations of Wireless Channel Modeling

1.1 The Wireless Propagation Environment

Radio propagation in a mobile environment is radically more complex than in a wired or free-space optical channel. The signal reaching a receiver is not a single clean copy of the transmitted waveform; it is the superposition of many copies, each having traveled a different path, each arriving with a different delay, amplitude, and phase. The physical mechanisms driving this are threefold.

The separation of scales is justified because the correlation distance of shadowing (tens of meters) far exceeds the correlation distance of multipath fading (on the order of half a wavelength, \(\lambda/2\), which at 2 GHz is approximately 7.5 cm).

1.2 Free-Space Path Loss

In free space — no reflections, no obstructions — the received power from an isotropic radiator falls off as the square of the distance according to the Friis transmission equation. Let \(P_t\) denote the transmitted power, \(G_t\) and \(G_r\) the transmit and receive antenna gains, \(d\) the separation, and \(\lambda = c/f_c\) the carrier wavelength. Then:

\[ P_r = P_t G_t G_r \left( \frac{\lambda}{4\pi d} \right)^2 \]

The factor \((\lambda / 4\pi d)^2\) is the free-space path loss (reciprocal of the link gain). Expressing this in decibels, with \(d\) in meters and \(f_c\) in MHz:

\[ PL_{\text{dB}}(d) = 20\log_{10}(d) + 20\log_{10}(f_c) - 27.55 \]

This \(d^{-2}\) decay is a consequence of energy spreading uniformly over the surface of a sphere. In realistic terrestrial environments, obstructions and reflectors cause the path loss exponent to deviate significantly from 2.

1.3 The Log-Distance Path Loss Model

Empirical measurements consistently show that the median path loss (averaged over shadowing) obeys a power law with exponent \(n\) in the range 2 to 6, depending on the propagation environment:

\[ PL(d) = PL(d_0) + 10n \log_{10}\!\left(\frac{d}{d_0}\right) \quad \text{[dB]} \]

Here \(d_0\) is a close-in reference distance (typically 1 m for indoor, 100 m or 1 km for outdoor systems) at which the path loss \(PL(d_0)\) is calculated or measured. The exponent \(n\) takes characteristic values: \(n \approx 2\) in free space, \(n \approx 2\)–\(3\) in open urban areas, \(n \approx 3\)–\(5\) in dense urban, and \(n \approx 4\)–\(6\) in buildings with obstructions.

1.4 Log-Normal Shadowing

Even for a fixed transmitter–receiver distance, the received power varies from location to location because of the shadowing effect of large obstacles (buildings, hills, foliage). Measurements show that this variation is well modeled by a log-normal random variable, i.e., the path loss in dB is normally distributed:

\[ PL(d) = \overline{PL}(d) + X_\sigma, \quad X_\sigma \sim \mathcal{N}(0, \sigma^2_{\rm dB}) \]

where \(\overline{PL}(d)\) is the mean (log-distance) path loss from Section 1.3 and \(\sigma_{\rm dB}\) is the shadowing standard deviation, typically in the range 4–12 dB. The received power in dBm is therefore:

\[ P_r[\text{dBm}] = P_t[\text{dBm}] - PL(d)[\text{dB}] \]

The log-normal model has an important consequence for coverage planning. The probability that the received power exceeds a sensitivity threshold \(\gamma_{\rm min}[\text{dBm}]\) at distance \(d\) is:

\[ \Pr\!\bigl[P_r \geq \gamma_{\rm min}\bigr] = Q\!\left(\frac{\overline{PL}(d) + \gamma_{\rm min} - P_t}{\sigma_{\rm dB}}\right) \]

where \(Q(x) = \frac{1}{\sqrt{2\pi}}\int_x^\infty e^{-t^2/2}\,dt\) is the Q-function.

1.5 Multipath Propagation and the Baseband Channel Model

Consider a narrowband transmitted signal \(s(t) = \mathrm{Re}\{u(t)e^{j2\pi f_c t}\}\) where \(u(t)\) is the complex baseband envelope. The received passband signal is the superposition of \(N\) multipath components, each with amplitude \(\alpha_i(t)\), delay \(\tau_i(t)\), and Doppler shift \(\nu_i(t)\). After downconversion and baseband filtering, the received complex envelope is:

\[ r(t) = \sum_{i=1}^{N} \alpha_i(t)\, u(t - \tau_i(t))\, e^{-j2\pi f_c \tau_i(t)} e^{j2\pi \nu_i(t) t} \]

This can be written as the convolution of \(u(t)\) with the time-varying channel impulse response:

\[ h(\tau, t) = \sum_{i=1}^{N} \alpha_i(t)\, e^{-j\phi_i(t)}\, \delta(\tau - \tau_i(t)) \]

where \(\phi_i(t) = 2\pi f_c \tau_i(t) - 2\pi \nu_i(t) t\) is the carrier-phase of the \(i\)-th path. The received signal is then \(r(t) = \int h(\tau,t) u(t-\tau)\,d\tau\).

1.6 Delay Spread and Coherence Bandwidth

The delay spread quantifies how “spread out” in time the channel is. A large \(\sigma_\tau\) means that energy from one symbol arrives spread over many symbol periods, causing inter-symbol interference (ISI). The reciprocal relationship between time and frequency gives the coherence bandwidth:

\[ B_c \approx \frac{1}{5\sigma_\tau} \]

(A factor between \(1/(2\pi\sigma_\tau)\) and \(1/\sigma_\tau\) is used in different texts depending on the correlation threshold adopted.) If the signal bandwidth \(B_s \ll B_c\), all frequency components of the signal fade identically — the channel is frequency-nonselective (flat fading). If \(B_s \gg B_c\), different frequency components fade independently — the channel is frequency-selective.

1.7 Doppler Spread and Coherence Time

When the mobile moves with velocity \(v\), the \(i\)-th multipath component arriving at angle \(\theta_i\) relative to the direction of motion acquires a Doppler shift:

\[ \nu_i = \frac{v}{\lambda} \cos\theta_i = f_D \cos\theta_i \]

where \(f_D = v/\lambda\) is the maximum Doppler frequency. At a carrier of 2 GHz with \(v = 100\) km/h, \(f_D \approx 185\) Hz.

The Doppler power spectrum \(S(\nu)\) (the Fourier transform of the time autocorrelation of \(h(\tau, t)\) over \(t\)) gives the distribution of power over Doppler shifts. For a 2D isotropic scattering model (Clarke’s model), all arrival angles \(\theta\) are uniformly distributed, yielding the Jakes spectrum:

\[ S(\nu) = \frac{1}{\pi f_D\sqrt{1-({\nu}/{f_D})^2}}, \quad |\nu| < f_D \]

This is U-shaped: most power arrives near \(\pm f_D\). The Doppler spread \(B_D\) is the width of \(S(\nu)\), approximately \(2f_D\) in the Clarke model. The coherence time is:

\[ T_c \approx \frac{1}{f_D} = \frac{\lambda}{v} \]

More precisely, adopting the convention that \(T_c\) is the time lag at which the time autocorrelation of \(h\) drops to 0.5 of its peak, \(T_c \approx 0.423/f_D\). If the symbol period \(T_s \ll T_c\), the channel is approximately constant over a symbol — slow fading. If \(T_s \gg T_c\), the channel changes significantly within a symbol — fast fading.

1.8 Statistical Models for Small-Scale Fading

When many independent scatterers contribute to the received signal and no single path dominates, the central limit theorem applies to the in-phase and quadrature components. With \(N\) equal-amplitude paths at random phases, the received complex envelope \(h = h_I + jh_Q\) has:

\[ h_I, h_Q \overset{\text{i.i.d.}}{\sim} \mathcal{N}(0, \Omega/2), \quad \Omega = \mathbb{E}[|h|^2] \]

The envelope \(|h|\) is therefore Rayleigh distributed:

\[ f_{|h|}(r) = \frac{2r}{\Omega} e^{-r^2/\Omega}, \quad r \geq 0 \]

and the received power \(|h|^2 \sim \text{Exponential}(\Omega)\).

When a strong line-of-sight (LOS) component is present alongside the scattered components, the in-phase component has a nonzero mean. Let the LOS component have amplitude \(s_0\) and the scattered spread be \(\sigma^2\) per dimension. Then \(|h|\) follows a Rician distribution parameterized by the K-factor:

\[ K = \frac{s_0^2}{2\sigma^2} \]\[ f_{|h|}(r) = \frac{2r(K+1)}{\Omega} e^{-K - (K+1)r^2/\Omega} I_0\!\left(2r\sqrt{\frac{K(K+1)}{\Omega}}\right) \]

where \(I_0\) is the modified Bessel function of the first kind, order zero. The Rayleigh model is recovered at \(K=0\) (no LOS), and as \(K \to \infty\) the envelope becomes deterministic (pure LOS, no fading).

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Chapter 2: Digital Modulation and Detection

2.1 Signal Space and Bandpass Representation

A digital modulation scheme transmits one of \(M\) symbols \(\{s_m(t)\}_{m=1}^M\), each of duration \(T_s\) seconds. The signal space is spanned by an orthonormal basis \(\{\phi_k(t)\}_{k=1}^K\) with \(K \leq M\), and each symbol maps to a point \(\mathbf{s}_m = (s_{m1}, \ldots, s_{mK})\) in \(\mathbb{R}^K\).

For AWGN reception, the sufficient statistic is the projection of the received signal onto each basis function, forming the matched-filter output vector. The minimum-distance detector chooses:

\[ \hat m = \arg\min_{m} \|\mathbf{r} - \mathbf{s}_m\|^2 \]

For M-QAM with square constellation and Gray coding, the symbol error probability in AWGN is:

\[ P_e \approx 4\left(1 - \frac{1}{\sqrt{M}}\right) Q\!\left(\sqrt{\frac{3 E_s}{(M-1) N_0}}\right) \]

where \(E_s\) is the energy per symbol and \(N_0/2\) is the noise power spectral density per dimension.

2.2 Probability of Error in Flat Fading Channels

In a flat-fading channel with complex gain \(h \sim \mathcal{CN}(0, 1)\) (Rayleigh, unit average power), the instantaneous received SNR for binary signaling is \(\gamma = |h|^2 E_b/N_0\). Since \(|h|^2\) is exponentially distributed with mean 1, the average bit error probability for BPSK is obtained by averaging the conditional BER \(Q(\sqrt{2\gamma})\) over the fading distribution:

\[ \bar P_b = \int_0^\infty Q\!\left(\sqrt{2\gamma}\right) e^{-\gamma/\bar\gamma}\,\frac{d\gamma}{\bar\gamma} = \frac{1}{2}\left(1 - \sqrt{\frac{\bar\gamma}{1+\bar\gamma}}\right) \]

where \(\bar\gamma = \mathbb{E}[|h|^2] E_b/N_0\) is the average SNR. At high SNR:

\[ \bar P_b \approx \frac{1}{4\bar\gamma} \]

This is the crucial contrast with AWGN: whereas BER decays exponentially in \(\bar\gamma\) in AWGN, it decays only as \(1/\bar\gamma\) in Rayleigh fading. Wireless communication design is fundamentally about converting this power-law decay into something steeper.

2.3 Coherent vs. Non-Coherent Detection

Coherent detection requires knowledge of the channel phase at the receiver. The matched filter is phase-aligned to the channel. For BPSK the coherent detector is:

\[ \hat b = \text{sgn}\!\left(\mathrm{Re}\{h^* r\}\right) \]

Non-coherent detection is used when the channel phase is unknown or varies rapidly. For binary FSK, the receiver computes the envelope of the matched filter outputs for each hypothesis and chooses the larger. The BER for non-coherent orthogonal FSK in AWGN is:

\[ P_b = \frac{1}{2} e^{-E_b/(2N_0)} \]

In Rayleigh fading, the average BER for non-coherent FSK becomes:

\[ \bar P_b = \frac{1}{2+\bar\gamma} \]

which decays as \(1/\bar\gamma\) — the same order as coherent detection, but with a 3 dB penalty.

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Chapter 3: Diversity Techniques

3.1 The Diversity Principle

The fundamental remedy for fading is diversity: the receiver is given multiple independently faded copies of the transmitted signal. If \(L\) independent copies are available, all must fade simultaneously for an error to occur. Since deep fades are rare events, the probability of \(L\) simultaneous deep fades falls dramatically with \(L\).

The proof follows from the fact that the distribution of the minimum of \(L\) i.i.d. exponential random variables has a PDF that behaves as \(\gamma^{L-1}\) near \(\gamma = 0\), so the outage probability at low instantaneous SNR scales as \(\gamma^L\).

3.2 Sources of Diversity

Diversity can be obtained from four distinct dimensions of the wireless channel.

Spatial (antenna) diversity uses multiple antennas separated by at least \(\lambda/2\) at the receiver (or transmitter), ensuring that the fading is approximately independent across antennas. This is the most common form because it requires no additional bandwidth or time resources.

Temporal diversity exploits the time variation of the channel by transmitting the same information at time separations greater than the coherence time \(T_c\). This is practical in fast-fading environments but wastes spectral efficiency in slow-fading environments. Channel coding with interleaving is the standard implementation.

Frequency diversity transmits redundant copies on carriers separated by more than the coherence bandwidth \(B_c\). Multipath channels where \(B_s > B_c\) offer inherent frequency diversity that can be exploited by appropriate receivers (RAKE, equalization, OFDM with coding across subcarriers).

Code diversity (spread spectrum) spreads a narrow-band signal over a wide bandwidth, effectively seeing many independent frequency-selective fades. The RAKE receiver collects these as distinct resolvable paths.

3.3 Receive Combining: MRC, EGC, SC

Consider a receive antenna array with \(L\) branches. Let \(h_\ell\) be the complex channel gain on branch \(\ell\) and \(n_\ell \sim \mathcal{CN}(0, N_0)\) the noise. The received signal on branch \(\ell\) is:

\[ r_\ell = h_\ell s + n_\ell, \quad \ell = 1, \ldots, L \]

Maximum Ratio Combining (MRC). The combiner weights each branch by the complex conjugate of its channel gain \(w_\ell = h_\ell^*\), then sums:

\[ \tilde r = \sum_{\ell=1}^L h_\ell^* r_\ell = \left(\sum_{\ell=1}^L |h_\ell|^2\right) s + \sum_{\ell=1}^L h_\ell^* n_\ell \]

The output SNR is:

\[ \gamma_{\rm MRC} = \frac{E_s}{N_0} \sum_{\ell=1}^L |h_\ell|^2 \]

MRC maximizes the output SNR and is therefore the optimal linear combiner in the minimum-probability-of-error sense. For i.i.d. Rayleigh branches, \(\gamma_{\rm MRC}\) follows a chi-squared distribution with \(2L\) degrees of freedom:

\[ f(\gamma) = \frac{\gamma^{L-1}}{(L-1)!\,\bar\gamma^L} e^{-\gamma/\bar\gamma} \]

and the average BER for BPSK is:

\[ \bar P_b = \left[\frac{1-\mu}{2}\right]^L \sum_{k=0}^{L-1}\binom{L-1+k}{k}\left[\frac{1+\mu}{2}\right]^k, \quad \mu = \sqrt{\frac{\bar\gamma}{1+\bar\gamma}} \]

At high SNR this yields the \(1/\bar\gamma^L\) decay characteristic of order-\(L\) diversity.

Equal Gain Combining (EGC). All branches are co-phased (multiplied by \(e^{-j\angle h_\ell}\)) but given equal weight. The output is \(\tilde r = \sum_\ell |h_\ell| s + \sum_\ell e^{-j\angle h_\ell} n_\ell\). EGC achieves order-\(L\) diversity but with an SNR loss relative to MRC, because suboptimal weighting is used.

Selection Combining (SC). The receiver selects only the branch with the highest instantaneous SNR:

\[ \gamma_{\rm SC} = \max_{1 \leq \ell \leq L} \gamma_\ell \]

For i.i.d. Rayleigh branches, the CDF of \(\gamma_{\rm SC}\) is:

\[ F_{\gamma_{\rm SC}}(\gamma) = \left(1 - e^{-\gamma/\bar\gamma}\right)^L \]

SC is the simplest to implement (only one RF chain needed), but it achieves order-\(L\) diversity with a further SNR penalty compared to MRC. In practice, SC saves hardware cost at the expense of about 1–3 dB in performance.

3.4 Transmit Diversity

When the transmitter has multiple antennas but the receiver has only one, transmit diversity requires either feedback (so the transmitter knows the channel) or space-time coding (open-loop).

Alamouti’s space-time block code (STBC). For two transmit antennas, over two consecutive symbol periods the transmitter sends:

Time \(t\)	Antenna 1	Antenna 2
\(t\)	\(s_1\)	\(s_2\)
\(t+T_s\)	\(-s_2^*\)	\(s_1^*\)

The received signals (assuming channel is constant over two periods: \(h_1\) from Tx 1, \(h_2\) from Tx 2) are:

\[ r_1 = h_1 s_1 + h_2 s_2 + n_1 \]\[ r_2 = -h_1 s_2^* + h_2 s_1^* + n_2 \]

After forming the vector \((r_1, r_2^*)\) and applying the matrix:

\[ \begin{pmatrix} h_1^* & h_2 \\ h_2^* & -h_1 \end{pmatrix} \begin{pmatrix} r_1 \\ r_2^* \end{pmatrix} = (|h_1|^2 + |h_2|^2)\begin{pmatrix} s_1 \\ s_2 \end{pmatrix} + \text{noise} \]

Both symbols are decoded from a single scalar decision variable with SNR \(\gamma = (|h_1|^2 + |h_2|^2)E_s/N_0\) — identical to two-branch MRC at the receiver. The Alamouti code achieves full diversity order 2 with no feedback and no reduction in data rate. Its elegance lies in the orthogonality of the code matrix \(\mathbf{G} = \begin{pmatrix} s_1 & s_2 \\ -s_2^* & s_1^* \end{pmatrix}\), for which \(\mathbf{G}^\dagger \mathbf{G} = (|s_1|^2 + |s_2|^2)\mathbf{I}\).

3.5 RAKE Receivers for Frequency-Selective Channels

In a wideband channel where the delay spread \(\sigma_\tau > 1/B_s\), individual multipath components become resolvable. A matched receiver for the multipath channel consists of a bank of correlators, one per resolvable path, each matched to the pulse shape at the corresponding delay. The outputs are combined in a RAKE structure analogous to MRC across the \(L_p\) resolvable paths:

\[ \tilde r = \sum_{\ell=1}^{L_p} \alpha_\ell^* r_\ell(t - \tau_\ell) \Big|_{\text{sampled}} \]

The RAKE effectively transforms frequency diversity (available because \(B_s > B_c\)) into a diversity gain \(L_p \leq \lfloor \sigma_\tau B_s \rfloor + 1\). This is the core receiver structure in CDMA systems (IS-95, UMTS/WCDMA).

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Chapter 4: Equalization for Dispersive Channels

4.1 The ISI Channel Model

When the channel is frequency-selective and a single-carrier waveform is transmitted, each received sample is a linear combination of multiple transmitted symbols. In discrete time, after matched filtering and sampling at symbol rate:

\[ r[n] = \sum_{k=0}^{L-1} h[k] s[n-k] + w[n] \]

where \(\{h[k]\}_{k=0}^{L-1}\) is the sampled channel impulse response and \(w[n] \sim \mathcal{CN}(0, N_0)\). In matrix form for a block of \(N\) received samples:

\[ \mathbf{r} = \mathbf{H}\mathbf{s} + \mathbf{w} \]

where \(\mathbf{H}\) is an \(N \times (N+L-1)\) Toeplitz convolution matrix. The goal of equalization is to recover \(\mathbf{s}\) from \(\mathbf{r}\).

4.2 Zero-Forcing Equalizer

The zero-forcing (ZF) equalizer inverts the channel completely, eliminating ISI at the cost of noise enhancement. In the frequency domain, the ZF equalizer tap at frequency \(f_k\) is:

\[ W_{\rm ZF}(f_k) = \frac{1}{H(f_k)} \]

The output SNR at frequency \(f_k\) is \(\gamma_k = |H(f_k)|^2 \cdot (E_s/N_0)\), but after ZF equalization the effective noise power at \(f_k\) is \(N_0/|H(f_k)|^2\). Thus:

\[ \text{SNR}_{\rm ZF}(f_k) = \frac{|H(f_k)|^2 E_s}{N_0} \]

At frequencies where \(|H(f_k)|^2 \ll 1\) (deep nulls), the ZF equalizer amplifies noise severely. The ZF equalizer is unbiased but can give extremely poor SNR in frequency-selective channels with spectral nulls.

4.3 Minimum Mean-Square-Error Equalizer

The MMSE equalizer minimizes \(\mathbb{E}[|\hat s - s|^2]\) jointly with noise. In the frequency domain:

\[ W_{\rm MMSE}(f_k) = \frac{H^*(f_k)}{|H(f_k)|^2 + N_0/E_s} \]

At high SNR the MMSE approaches the ZF; at low SNR the MMSE reduces to a matched filter. The practical advantage of MMSE is robustness at channel nulls.

4.4 Decision Feedback Equalization

The decision feedback equalizer (DFE) uses past detected symbols to cancel already-decoded ISI. It consists of a feed-forward filter \(F(z)\) applied to the received signal and a feedback filter \(B(z)\) applied to the detected sequence:

\[ \hat s[n] = \text{Decide}\!\left\{\sum_{k=0}^{K_f} f_k r[n-k] - \sum_{k=1}^{K_b} b_k \hat s[n-k]\right\} \]

When past decisions are correct, the DFE achieves performance between MMSE and MFB (matched filter bound — the maximum achievable SNR if ISI were completely absent). Error propagation — where incorrect past decisions corrupt future decisions — is the primary practical limitation of the DFE.

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Chapter 5: Multiple-Input Multiple-Output (MIMO) Systems

5.1 The MIMO Channel Model

A system with \(N_t\) transmit antennas and \(N_r\) receive antennas is characterized by an \(N_r \times N_t\) complex channel matrix \(\mathbf{H}\). Assuming flat fading (or per-subcarrier in OFDM):

\[ \mathbf{y} = \mathbf{H}\mathbf{x} + \mathbf{n} \]

where \(\mathbf{x} \in \mathbb{C}^{N_t}\) is the transmitted vector with power constraint \(\mathbb{E}[\|\mathbf{x}\|^2] \leq P\), \(\mathbf{n} \sim \mathcal{CN}(\mathbf{0}, N_0 \mathbf{I}_{N_r})\), and \(\mathbf{H}\) has entries \(h_{ij} \sim \mathcal{CN}(0,1)\) for the i.i.d. Rayleigh model.

5.2 Singular Value Decomposition and MIMO Capacity

The key to understanding MIMO is the singular value decomposition (SVD):

\[ \mathbf{H} = \mathbf{U} \mathbf{\Sigma} \mathbf{V}^\dagger \]

where \(\mathbf{U} \in \mathbb{C}^{N_r \times N_r}\) and \(\mathbf{V} \in \mathbb{C}^{N_t \times N_t}\) are unitary matrices, and \(\mathbf{\Sigma}\) is an \(N_r \times N_t\) matrix with non-negative real entries \(\sigma_1 \geq \sigma_2 \geq \cdots \geq \sigma_{\min(N_t,N_r)} \geq 0\) on the diagonal. These are the singular values of \(\mathbf{H}\).

By precoding the transmit vector as \(\mathbf{x} = \mathbf{V}\tilde{\mathbf{x}}\) and receive processing as \(\tilde{\mathbf{y}} = \mathbf{U}^\dagger \mathbf{y}\), the MIMO channel decomposes into \(r = \text{rank}(\mathbf{H}) \leq \min(N_t, N_r)\) independent parallel channels:

\[ \tilde y_k = \sigma_k \tilde x_k + \tilde n_k, \quad k = 1, \ldots, r \]

Each sub-channel has SNR \(\lambda_k E_k / N_0\), where \(\lambda_k = \sigma_k^2\) is the \(k\)-th eigenvalue of \(\mathbf{H}\mathbf{H}^\dagger\) and \(E_k\) is the power allocated to that mode.

Proof sketch. The capacity of the parallel Gaussian channel is maximized by treating each mode independently. For Gaussian inputs, the mutual information of sub-channel \(k\) is \(\log_2(1 + \lambda_k p_k/N_0)\). Maximizing the sum subject to \(\sum p_k = P\) via Lagrange multipliers yields the water-filling solution \(p_k^* = (\mu - N_0/\lambda_k)^+\).

5.3 Spatial Multiplexing

When the transmitter has no channel knowledge (or chooses equal power), it transmits independent streams on the \(\min(N_t, N_r)\) eigenmodes with equal power \(P/N_t\). The resulting capacity is:

\[ C = \sum_{k=1}^r \log_2\!\left(1 + \frac{\lambda_k P}{N_t N_0}\right) \]

This grows approximately linearly in \(\min(N_t, N_r)\) at high SNR — the spatial multiplexing gain \(r = \min(N_t, N_r)\). Each additional antenna pair adds approximately \(\log_2(1 + \text{SNR})\) bits/s/Hz.

More precisely, for i.i.d. Rayleigh fading with \(N_t = N_r = N\), the ergodic capacity scales as:

\[ \bar C \approx N \log_2\!\left(1 + \frac{P}{N_0}\right), \quad P/N_0 \gg 1 \]

reflecting the \(N\)-fold multiplexing advantage. At low SNR the growth is more modest.

5.4 Ergodic and Outage Capacity

For fading channels where the channel \(\mathbf{H}\) is random, two notions of capacity are relevant.

The outage capacity is more relevant for delay-sensitive applications (voice, video) where the codeword length is constrained. A system operating at rate \(R\) experiences outage whenever the instantaneous channel capacity falls below \(R\).

5.5 Diversity–Multiplexing Tradeoff

The fundamental tension in MIMO is between using the extra degrees of freedom for diversity (combating fading) and multiplexing (increasing data rate). Zheng and Tse (2003) characterized this precisely for i.i.d. Rayleigh fading with \(N_t\) transmit and \(N_r\) receive antennas in the high-SNR regime.

5.6 SIMO and MISO Beamforming

SIMO (Single-Input Multiple-Output). One transmit antenna, \(N_r\) receive antennas. The channel is a vector \(\mathbf{h} \in \mathbb{C}^{N_r}\). The received vector is \(\mathbf{y} = \mathbf{h} s + \mathbf{n}\). Optimal processing is MRC: multiply by \(\mathbf{h}^\dagger\). The output SNR is \(\|\mathbf{h}\|^2 P/N_0\), which is the sum of \(N_r\) independent exponentials — \(N_r\)-fold diversity, no multiplexing gain.

MISO (Multiple-Input Single-Output). \(N_t\) transmit antennas, one receive antenna. Channel is a vector \(\mathbf{h}^T\). If the transmitter knows \(\mathbf{h}\), it performs transmit beamforming: \(\mathbf{x} = \mathbf{w} s\) where \(\mathbf{w} = \mathbf{h}^*/\|\mathbf{h}\|\). The received signal is:

\[ y = \mathbf{h}^T \mathbf{w} s + n = \|\mathbf{h}\| s + n \]

SNR is \(\|\mathbf{h}\|^2 P/N_0\) — identical to SIMO-MRC by the duality of \(\mathbf{h}\mathbf{h}^\dagger\) and \(\mathbf{h}^T \mathbf{h}^*\). Transmit beamforming requires full CSIT; without CSIT the transmitter spreads power uniformly across antennas, achieving diversity but not the beamforming gain.

5.7 Massive MIMO

In massive MIMO, the number of base station antennas \(M\) is much larger than the number of served users \(K\) (\(M \gg K\)). Key properties in the large-antenna limit:

1. Channel hardening. The effective channel gain \(\|\mathbf{h}_k\|^2/M \to 1\) almost surely as \(M \to \infty\) (law of large numbers), so small-scale fading averages out.

2. Favorable propagation. The inner products \(\mathbf{h}_k^\dagger \mathbf{h}_j / M \to 0\) for \(k \neq j\) as \(M \to \infty\), making users asymptotically orthogonal. Simple linear processing (MRC at receiver, matched filter precoding at transmitter) approaches the optimal performance.

3. Energy efficiency. For fixed total SNR, the per-antenna transmit power can be reduced by \(M\), since the array gain scales with \(M\). This enables green communication.

4. Pilot contamination. In time-division duplex (TDD) massive MIMO, uplink pilots are used for channel estimation. With many cells, pilots from neighboring cells contaminate the channel estimates — this is pilot contamination, a fundamental limit of massive MIMO at finite \(M\).

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Chapter 6: Orthogonal Frequency Division Multiplexing (OFDM)

6.1 Motivation and Basic Principle

OFDM addresses the ISI problem in frequency-selective channels by converting a single wideband channel into many narrowband parallel sub-channels, each of width \(\Delta f\) chosen so that \(\Delta f \ll B_c\) (flat fading per subcarrier). The symbol duration on each subcarrier \(T_{\rm OFDM} = 1/\Delta f\) is much longer than the delay spread \(\sigma_\tau\), so ISI across OFDM symbols is eliminated by the cyclic prefix.

6.2 Discrete Fourier Transform Implementation

An OFDM system with \(N\) subcarriers transmits the complex data symbols \(\{X_k\}_{k=0}^{N-1}\) simultaneously. The time-domain OFDM symbol is formed by the inverse DFT:

\[ x[n] = \frac{1}{\sqrt{N}} \sum_{k=0}^{N-1} X_k e^{j2\pi kn/N}, \quad n = 0, 1, \ldots, N-1 \]

At the receiver, the DFT recovers the data:

\[ Y_k = \frac{1}{\sqrt{N}} \sum_{n=0}^{N-1} y[n] e^{-j2\pi kn/N} \]

The computational efficiency of the IFFT/FFT (O(\(N \log N\)) versus O(\(N^2\)) for brute force) makes large \(N\) (e.g., \(N = 2048\) in LTE) practical.

6.3 Cyclic Prefix and ISI Elimination

A cyclic prefix (CP) of length \(L_{\rm CP} \geq L-1\) (where \(L\) is the channel length in samples) is prepended to each OFDM symbol by copying the last \(L_{\rm CP}\) samples of \(x[n]\) to the front:

\[ \tilde x[n] = x[(n \bmod N)], \quad n = -L_{\rm CP}, \ldots, N-1 \]

At the receiver, after discarding the CP, the remaining \(N\) samples represent a circular convolution of the transmitted symbol with the channel:

\[ y[n] = h[n] \circledast x[n] + w[n] \]

The DFT of a circular convolution is a pointwise product: \(Y_k = H_k X_k + W_k\), where \(H_k = \sum_{\ell=0}^{L-1} h[\ell] e^{-j2\pi k\ell/N}\) is the channel frequency response at subcarrier \(k\). This reduces equalization to a per-subcarrier scalar division:

\[ \hat X_k = \frac{Y_k}{H_k} = X_k + \frac{W_k}{H_k} \quad \text{(ZF equalization)} \]

6.4 Subcarrier Orthogonality

The subcarriers \(e^{j2\pi k n/N}\) are orthogonal over the DFT window:

\[ \frac{1}{N}\sum_{n=0}^{N-1} e^{j2\pi kn/N} e^{-j2\pi mn/N} = \delta[k-m] \]

This orthogonality holds exactly when the DFT window begins at a sample free of inter-symbol interference — which the CP guarantees. In a continuous-time formulation, the subcarrier spacing \(\Delta f = 1/T_{\rm OFDM}\) is chosen so that the sinusoids \(e^{j2\pi k\Delta f t}\) complete an integer number of cycles in the interval \([0, T_{\rm OFDM}]\), ensuring zero inter-carrier interference (ICI) in ideal conditions.

6.5 PAPR and Mitigation

A major practical disadvantage of OFDM is its high peak-to-average power ratio (PAPR). When \(N\) subcarriers add constructively, the peak amplitude is \(\sqrt{N}\) times the RMS amplitude, giving PAPR up to \(10\log_{10}N\) dB. For \(N = 2048\), this is about 33 dB — far beyond what a practical power amplifier can handle linearly.

The PAPR is formally defined as:

\[ \text{PAPR} = \frac{\max_{0\leq nClipping and filtering is the simplest mitigation: hard-limit the amplitude to \(A_{\rm clip}\) and then filter to remove out-of-band spectral regrowth. This introduces in-band distortion proportional to the clipping ratio \(\text{CR} = A_{\rm clip}/\sqrt{2\sigma_x^2}\).

Selected mapping (SLM) generates \(U\) candidate OFDM symbols by multiplying the data vector with different phase sequences \(\{b_k^{(u)}\}\), transmits the one with lowest PAPR, and conveys the choice \(u\) as side information.

Partial transmit sequences (PTS) partition the subcarriers into disjoint subsets, multiplies each subset by a phase factor from \(\{+1,-1,+j,-j\}\), and exhaustively searches for the combination minimizing PAPR.

6.6 Pilot-Based Channel Estimation

The channel frequency response \(\{H_k\}_{k=0}^{N-1}\) must be estimated from received pilots. Pilot symbols \(X_p\) are inserted at known subcarrier positions \(\mathcal{P} \subset \{0,\ldots,N-1\}\). The received pilot observations are:

\[ Y_p = H_p X_p + W_p, \quad p \in \mathcal{P} \]

The LS estimate at pilot positions is \(\hat H_p^{\rm LS} = Y_p/X_p\). To estimate \(H_k\) at data subcarriers, one of the following interpolations is used:

• Linear interpolation between adjacent pilot estimates (simple, works for slowly varying channels).
• MMSE interpolation using the channel power delay profile as prior: \(\hat{\mathbf{H}} = \mathbf{R}_{HD}(\mathbf{R}_{DD} + \sigma_n^2 \mathbf{I})^{-1}\hat{\mathbf{H}}_D^{\rm LS}\), where \(\mathbf{R}\) matrices are the correlation matrices derived from the delay profile.
• DFT-based channel estimation: Transform LS estimates to the delay domain, retain only the \(L\) significant taps (below the noise floor), zero-pad, and transform back. This gives an SNR improvement roughly equal to \(N/L\) relative to LS.

The minimum pilot density required for alias-free channel estimation is one pilot per \(\lfloor N/L \rfloor\) subcarriers (Nyquist sampling in the delay domain).

6.7 OFDM in LTE and 5G NR

LTE employs a sub-carrier spacing of \(\Delta f = 15\) kHz and normal CP of 4.7 μs (or extended CP of 16.7 μs). The resource grid is structured as OFDM symbols in time × subcarriers in frequency, grouped into resource blocks (RBs) of 12 subcarriers × 7 symbols = 84 resource elements. Channel coding (turbo codes in LTE, LDPC in 5G NR) is applied before OFDM modulation.

5G NR generalizes the subcarrier spacing to multiples of 15 kHz: \(\Delta f = 2^\mu \cdot 15\) kHz for numerology \(\mu = 0, 1, 2, 3, 4\). Larger \(\mu\) reduces OFDM symbol duration (from 66.7 μs at \(\mu=0\) to 4.17 μs at \(\mu=4\)), accommodating higher carrier frequencies (mmWave) where coherence time is shorter and higher Doppler is expected.

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Chapter 7: Spread Spectrum and Multiple Access Techniques

7.1 Direct Sequence Spread Spectrum

In direct sequence spread spectrum (DSSS), the data signal \(d(t)\) (bandwidth \(W_d\)) is multiplied by a pseudo-noise (PN) spreading code \(c(t)\) with chip rate \(W_c \gg W_d\), spreading the signal over bandwidth \(W_c\):

\[ s(t) = \sqrt{2P}\, d(t)\, c(t) \cos(2\pi f_c t) \]

The processing gain (or spreading gain) is:

\[ G_p = \frac{W_c}{W_d} = \frac{T_b}{T_c} \]

where \(T_b\) is the bit period and \(T_c\) is the chip period. \(G_p\) represents the improvement in effective SNR against a narrowband interferer: a jammer of power \(J\) spread over bandwidth \(W_d\) is seen by the despreader as an effective noise density \(J/G_p\) relative to the signal. The jamming margin is \(10\log_{10}G_p\) dB.

Pseudo-noise (PN) sequences are deterministic binary sequences \(\{c_k\} \in \{+1,-1\}\) of period \(N_c = 2^m - 1\) (for an \(m\)-stage linear feedback shift register) with the properties: (1) balanced (equal numbers of \(\pm 1\) per period), (2) run property (runs of length \(\ell\) occur \(2^{m-\ell}\) times), and (3) ideal autocorrelation: \(R_c(\tau) = N_c \delta[\tau \bmod N_c] - 1\). The near-ideal autocorrelation is what makes DSSS robust to delay: a despreader using a locally generated copy synchronized to the wanted signal sees \(G_p\) gain, while a delayed copy (by more than one chip) sees near-zero correlation (interference).

7.2 Code Division Multiple Access (CDMA)

In CDMA, \(K\) users simultaneously occupy the same time-frequency resource, distinguished only by their spreading codes \(\{c_k(t)\}\). The received signal at the base station is:

\[ r(t) = \sum_{k=1}^K \sqrt{P_k}\, d_k(t)\, c_k(t)\cos(2\pi f_c t) + n(t) \]

The despreader for user \(k\) multiplies by \(c_k(t)\) and integrates. The output for user 1 is:

\[ z_1 = \sqrt{P_1} d_1 + \sum_{k=2}^K \sqrt{P_k} d_k \underbrace{\int_0^{T_b} c_1(t)c_k(t)\,dt}_{\rho_{1k}} + \tilde n \]

If codes are orthogonal (\(\rho_{1k} = 0\)), multiple-access interference (MAI) vanishes. In practice (asynchronous uplink, multipath), codes lose orthogonality. The MAI is modeled as Gaussian noise for large \(K\), giving an effective \(\text{Eb/No}\):

\[ \left(\frac{E_b}{N_0}\right)_{\rm eff} = \frac{G_p}{K - 1 + G_p N_0/P} \]

This shows that as \(K\) increases, effective SNR decreases — CDMA is self-interference limited.

7.3 The Near-Far Problem and Power Control

A fundamental challenge in CDMA uplink is the near-far problem: if user 1 is close to the base station and user 2 is far away, user 1’s signal may be 30–40 dB stronger at the receiver, completely overwhelming user 2.

The remedy is power control: each user adjusts its transmit power so that all users arrive at the base station with equal received power. Closed-loop power control measures the path loss and commands the mobile to increase or decrease power in steps of 1 dB. In CDMA systems (WCDMA, cdma2000) power control operates at 1500 Hz to track fast fading, making it one of the most rapid feedback mechanisms in commercial wireless.

With perfect power control, the near-far problem is eliminated and all users contribute equal MAI. The capacity of a power-controlled CDMA cell (in the absence of intercell interference) is:

\[ K_{\max} = 1 + G_p / (E_b/N_0)_{\rm target} \]

7.4 FDMA, TDMA, and Spectral Efficiency

FDMA (Frequency Division Multiple Access) assigns each user a unique sub-band of bandwidth \(B_k\). Guard bands between users prevent adjacent-channel interference. FDMA is inherently orthogonal on a per-user basis; capacity is limited by the total available bandwidth. It is the basis of first-generation (1G) cellular (AMPS).

TDMA (Time Division Multiple Access) assigns each user a unique time slot in a repeating frame. Within a slot, the user occupies the full bandwidth. TDMA requires timing synchronization across users. GSM uses TDMA with 8 users per 200 kHz carrier; IS-136 uses 3 users per 30 kHz carrier.

Spectral efficiency comparison. For a single-cell system with \(K\) equal-rate users in bandwidth \(B\):

Scheme	Interference	Spectral eff.
FDMA	None (orthogonal)	Same as TDMA in AWGN
TDMA	None (orthogonal)	Same as FDMA in AWGN
CDMA	MAI (non-orthogonal)	Soft capacity — users can be added at cost of higher BER

The spectral efficiency advantage of CDMA over FDMA/TDMA in cellular systems arises not from the per-cell capacity (which is the same in theory) but from the universal frequency reuse: all CDMA cells reuse the same frequency, while FDMA/TDMA systems require frequency planning (reuse factor \(q > 1\)).

7.5 OFDMA and SC-FDMA

OFDMA (Orthogonal Frequency Division Multiple Access) assigns different sets of OFDM subcarriers to different users. Because each subcarrier is narrowband (flat fading), frequency-selective multiuser diversity can be exploited: user \(k\) is assigned the subcarriers where its channel is strong. This multiuser diversity gain increases with the number of users because the probability that at least one user has a strong channel on every subcarrier increases.

SC-FDMA (Single Carrier FDMA) is used in LTE uplink. Each user’s data is first transformed by a DFT of size \(M\) (the number of assigned subcarriers), and the output is mapped onto \(M\) contiguous subcarriers before the OFDM IFFT. The result has the frequency orthogonality of OFDMA but the lower PAPR of single-carrier signaling — critical for battery-powered handsets.

7.6 Random Access: ALOHA and CSMA

In random access protocols, users transmit without coordination, leading to collisions. Slotted ALOHA achieves maximum throughput \(S^* = 1/e \approx 0.37\) packets per slot, attained at offered load \(G = 1\) packet per slot. For CDMA systems, random access is handled by the RACH (Random Access Channel), where users transmit random preambles; the base station detects them using matched filtering against a Zadoff-Chu sequence bank.

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Chapter 8: Cellular Systems

8.1 Frequency Reuse and Cluster Size

In a cellular network, the total bandwidth is divided among cells to limit co-channel interference. The cluster size \(N_c\) (also called reuse factor \(q\)) determines how many cells can use a given frequency channel simultaneously. For a hexagonal cell grid:

\[ N_c = i^2 + ij + j^2, \quad i, j \in \mathbb{Z}_{\geq 0} \]

giving valid cluster sizes \(N_c \in \{1, 3, 4, 7, 9, 12, \ldots\}\). The distance to the nearest co-channel cell is:

\[ D = R\sqrt{3 N_c} \]

where \(R\) is the cell radius. The co-channel interference ratio (C/I) for a user at the cell edge, with one tier of \(K_0 = 6\) interferers at distance \(D\):

\[ \frac{C}{I} \approx \frac{R^{-n}}{6 D^{-n}} = \frac{(D/R)^n}{6} = \frac{(3N_c)^{n/2}}{6} \]

For \(n = 4\) and the commonly used \(N_c = 7\): \(C/I = (21)^2/6 \approx 73.5 = 18.7\) dB, which comfortably exceeds the GSM minimum of 9 dB.

8.2 Handoff (Handover)

When a mobile moves from one cell to another, the call must be transferred — a handoff. Hard handoff (used in FDMA/TDMA) briefly interrupts the connection during the switch. Soft handoff (used in CDMA) allows the mobile to be connected simultaneously to two or more base stations (macro diversity), with the best signal selected or combined.

The handoff decision is based on a received signal strength (RSS) hysteresis rule to avoid ping-pong (rapid back-and-forth handoff between two cells):

\[ \text{Hand off: } \text{RSS}(B_2) > \text{RSS}(B_1) + \Delta_H \]

The hysteresis margin \(\Delta_H\) (typically 3–6 dB) prevents excessive signaling at the cost of a slightly suboptimal serving cell.

8.3 Cell Sectorization

Replacing an omnidirectional antenna at each cell site with directional sector antennas (typically three 120° sectors or six 60° sectors) reduces the number of interferers seen per sector:

• 3-sector cell: co-channel interferers reduced from 6 to 2 per tier (in the worst case), improving \(C/I\) by approximately \(10\log_{10}(6/2) \approx 4.8\) dB, or equivalently allowing a smaller cluster size while maintaining the same \(C/I\).
• 6-sector cell: further reduction to 1 interferer per tier, but with higher antenna and installation cost.

Sectorization also provides antenna gain (\(G_a = 120°/360° = 3\) dB equivalent for 3-sector), compounding the interference reduction.

8.4 Interference Management

Fractional frequency reuse (FFR) partitions the bandwidth into an inner zone (reuse factor 1, serves cell-center users) and an outer zone (reuse factor \(N_c > 1\), protects cell-edge users from co-channel interference). LTE employs a soft version of FFR where power on the outer-zone frequencies is boosted at cell edges.

Intercell interference coordination (ICIC) in LTE uses X2 interface messages between eNBs to exchange resource usage and interference measurements, allowing neighboring cells to coordinate frequency assignment dynamically.

Successive interference cancellation (SIC) in CDMA downlink (or the uplink in theory) decodes users sequentially — the strongest user first, subtracts it, then decodes the next. SIC achieves the sum-rate capacity of the MAC (multiple access channel) but requires large receiver complexity and accurate power control.

8.5 Heterogeneous Networks (HetNets) and Small Cells

The exponential growth of mobile data traffic is largely addressed by network densification: deploying small cells (pico cells, femto cells, relay nodes) within the macro cell coverage. A heterogeneous network (HetNet) consists of macro base stations (typical power 40 W, radius 1–2 km) overlaid with small base stations (typical power 0.25 W, radius 50–200 m).

The main challenge is cross-tier interference: a femto cell operating on the same frequency as the macro cell may severely interfere with macro UEs near the femto. Mitigation strategies include:

• Closed subscriber group (CSG): femto only serves registered users.
• Almost blank subframes (ABS): macro cells mute some subframes, allowing small cells to serve their UEs without macro interference.
• Cell range expansion (CRE): small cells use a positive offset in the cell selection criterion, attracting more UEs and offloading the macro.

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Chapter 9: Wireless Channel Capacity

9.1 AWGN Channel Capacity

The capacity of a complex baseband AWGN channel with bandwidth \(B\) Hz, signal power \(P\) watts, and noise PSD \(N_0/2\) W/Hz is given by Shannon’s formula:

\[ C = B \log_2\!\left(1 + \frac{P}{N_0 B}\right) \quad \text{bits/s} \]

The quantity \(\text{SNR} = P/(N_0 B)\) is the SNR per complex dimension. At high SNR, \(C \approx B\log_2(\text{SNR})\) grows logarithmically with power; at low SNR (bandwidth-unlimited regime), \(C \approx P/(N_0 \ln 2)\) grows linearly with power, independent of bandwidth.

9.2 Fading Channel Capacity: CSI at Receiver Only

When the channel is a random fading process \(h \sim \mathcal{CN}(0,1)\) and the receiver knows \(h\) but the transmitter does not (CSIR only), the transmitter cannot adapt power. The ergodic capacity is:

\[ C = \mathbb{E}_{|h|^2}\!\left[\log_2\!\left(1 + \frac{|h|^2 P}{N_0}\right)\right] \]

For Rayleigh fading (\(|h|^2 \sim \text{Exp}(1)\)), this integral does not have a closed form but can be expressed as:

\[ C = e^{1/\bar\gamma} E_1(1/\bar\gamma) \log_2 e \quad \text{bits/s/Hz} \]

where \(E_1(x) = \int_x^\infty e^{-t}/t\,dt\) is the exponential integral and \(\bar\gamma = P/N_0\). At high SNR, fading costs approximately \(0.83\) bits/s/Hz compared to AWGN — a constant penalty, not growing with SNR.

9.3 Capacity with Full CSI: Water-Filling in Fading

When both transmitter and receiver know the instantaneous channel state \(h\), the transmitter performs water-filling in time/frequency: allocate more power when the channel is strong, less when weak.

The ergodic capacity with full CSI is:

\[ C_{\rm FCSI} = \mathbb{E}_{|h|^2}\!\left[\log_2\!\left(\frac{\mu |h|^2}{N_0}\right)^+\right] \]

where \(\mu\) is the water level satisfying \(\mathbb{E}[({\mu} - N_0/|h|^2)^+] = P\). For Rayleigh fading at high SNR, the gain from water-filling over fixed power is small (the Jensen-gap is small when the capacity function is nearly linear in SNR). At low SNR, water-filling provides substantial gain by concentrating power on good channel states.

9.4 Successive Interference Cancellation and MAC Capacity

The multiple access channel (MAC) with \(K\) users is the uplink channel where all \(K\) mobiles transmit to a single base station receiver. The capacity region is the set of rate tuples \((R_1, \ldots, R_K)\) achievable by all users simultaneously.

MAC Capacity Region. For a two-user Gaussian MAC with individual power constraints \(P_1, P_2\) and channel gains \(h_1, h_2\), the capacity region is \[ R_1 \leq \log_2\!\left(1 + \frac{|h_1|^2 P_1}{N_0}\right) \]\[ R_2 \leq \log_2\!\left(1 + \frac{|h_2|^2 P_2}{N_0}\right) \]\[ R_1 + R_2 \leq \log_2\!\left(1 + \frac{|h_1|^2 P_1 + |h_2|^2 P_2}{N_0}\right) \]

The corner points of this pentagon are achieved by successive interference cancellation (SIC): decode user 1 treating user 2 as noise, then subtract user 1 and decode user 2.

The sum-rate capacity \(\log_2(1 + (|h_1|^2 P_1 + |h_2|^2 P_2)/N_0)\) is achieved by SIC in any order, since the sum-rate is symmetric. The order determines which user gets the higher individual rate.

9.5 Broadcast Channel (Downlink) Capacity

The broadcast channel (BC) is the downlink from one base station to \(K\) users. Unlike the MAC (where the noise processes are independent), the BC users each receive a superposition of signals intended for all other users.

Dirty Paper Coding (DPC) Capacity. For the Gaussian BC, the capacity region is achieved by dirty paper coding (Costa, 1983): if the transmitter knows the interference to user 2 caused by user 1's signal, it can pre-subtract this interference in the codeword for user 2 at no power cost. The resulting DPC capacity region is the same as the MAC capacity region via the MAC–BC duality.

In practice, DPC is complex to implement. Linear precoding approximations (zero-forcing beamforming, regularized ZF) are used in LTE Multi-User MIMO (MU-MIMO), achieving near-DPC performance at moderate SNR.

9.6 Multiuser Diversity

In a multiuser system with scheduling, the base station can assign the channel resource at each time slot to the user with the highest instantaneous SNR. The multiuser diversity gain comes from the fact that, with \(K\) users whose channels fade independently, the maximum instantaneous SNR among all users is much larger than any individual’s average SNR. For i.i.d. Rayleigh fading:

\[ \mathbb{E}\!\left[\max_{1\leq k\leq K} \gamma_k\right] \approx \bar\gamma \ln K + O(1) \]

This logarithmic growth in \(K\) — the opportunistic scheduling gain — is the basis for proportional-fair scheduling in LTE/5G, where users are served when their instantaneous channel is near its peak relative to their long-term average.

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Chapter 10: Advanced Topics

10.1 Reconfigurable Intelligent Surfaces

A reconfigurable intelligent surface (RIS) is a planar array of passive, software-controlled reflective elements. Unlike active relays, an RIS does not amplify signals — it introduces controllable phase shifts on the reflected signal, enabling passive beamforming toward a desired receiver.

Consider a single-antenna transmitter, an RIS with \(N\) elements, and a single-antenna receiver. Let \(h_d \in \mathbb{C}\) be the direct path and \(\mathbf{h}_r \in \mathbb{C}^N\), \(\mathbf{g} \in \mathbb{C}^N\) be the transmitter-to-RIS and RIS-to-receiver channels. The received signal is:

\[ y = \left(h_d + \mathbf{g}^T \mathbf{\Phi} \mathbf{h}_r\right) s + n \]

where \(\mathbf{\Phi} = \text{diag}(e^{j\theta_1}, \ldots, e^{j\theta_N})\) is the RIS phase shift matrix. The SNR is maximized by choosing:

\[ \theta_n = \angle h_d - \angle g_n - \angle h_{r,n} \]

so that all reflected paths align in phase with the direct path. The maximum SNR scales as \((|h_d| + N \cdot \mathbb{E}[|g_n h_{r,n}|])^2 E_s/N_0\), showing that the RIS provides an \(N^2\) beamforming gain (in power), not just \(N\) as for active arrays. This squared gain makes RIS attractive for range extension in millimeter-wave communications where the direct path may be blocked.

10.2 Space-Time Coding

The Alamouti code (Section 3.4) achieves full transmit diversity order 2 with two antennas and rate 1 (no bandwidth expansion). For more transmit antennas, orthogonal space-time block codes (OSTBCs) generalize the Alamouti design: the code matrix \(\mathbf{G}\) satisfies \(\mathbf{G}^\dagger \mathbf{G} = c(|s_1|^2 + \cdots + |s_K|^2)\mathbf{I}\), enabling linear decoupling of all \(K\) symbols.

However, for \(N_t > 2\) antennas and complex constellations, OSTBCs can achieve full diversity only at reduced rates (Tarokh 1999). Space-time trellis codes (STTCs) jointly design coding and modulation to achieve both diversity and coding gain, but require Viterbi decoding over a product trellis.

10.3 Link Budget Analysis

A link budget is the accounting of all gains and losses along the propagation path to determine whether the received SNR meets the required threshold. The link budget equation in dB is:

\[ P_r[\text{dBm}] = P_t[\text{dBm}] + G_t[\text{dBi}] + G_r[\text{dBi}] - PL(d)[\text{dB}] - L_{\rm misc}[\text{dB}] \]

where \(L_{\rm misc}\) captures cable losses, body loss, polarization mismatch, and implementation losses. The link margin is:

\[ \mathcal{M} = P_r - S_{\min} \]

where \(S_{\min}\) is the receiver sensitivity. A positive link margin means the link is viable; margin is allocated to counter shadowing, hardware degradation, and interference. Typically 10–20 dB of margin is budgeted for shadowing (based on the log-normal model of Section 1.4).

Uplink link budget for an LTE macro cell:

• Transmit power (UE): 23 dBm
• Antenna gain (UE): 0 dBi, (eNB): 15 dBi
• Required \(E_b/N_0\): 1 dB (QPSK, rate 1/3 turbo code)
• Noise figure (eNB): 5 dB
• Thermal noise density: −174 dBm/Hz
• Resource block bandwidth: 180 kHz → noise floor: −174 + 52.5 + 5 = −116.5 dBm
• Required received power: −116.5 + 1 = −115.5 dBm
• Maximum path loss: 23 + 0 + 15 − (−115.5) = 153.5 dB
• For \(n = 3.5\) and \(d_0 = 100\) m: \(d_{\max} = d_0 \cdot 10^{(153.5 - PL(d_0))/(10 \times 3.5)}\)

This gives a cell edge at approximately 1.8 km, consistent with LTE macro deployment targets.

10.4 Wireless Standards: LTE, 5G NR, and Wi-Fi 6

LTE (Long-Term Evolution / 4G). Downlink uses OFDMA; uplink uses SC-FDMA. Peak rates: 150 Mbps DL (MIMO 2×2, 20 MHz, 64-QAM). Key features: OFDM with CP, HARQ (hybrid automatic repeat request), AMC (adaptive modulation and coding), MU-MIMO on the downlink.

LTE-Advanced (4G+). Adds carrier aggregation (up to 100 MHz), enhanced MIMO (up to 8×8 DL), CoMP (coordinated multipoint transmission). Peak rates: 1 Gbps DL.

5G NR (New Radio). Two frequency ranges: FR1 (450 MHz–6 GHz, sub-6 GHz) and FR2 (24.25–52.6 GHz, mmWave). Flexible numerology (\(\mu = 0\) to 4). Downlink: OFDMA with up to 256-QAM, massive MIMO beamforming (up to 64 antenna elements). Uplink: CP-OFDM or DFT-spread-OFDM (SC-FDMA). Peak rates: 20 Gbps DL in mmWave with 400 MHz bandwidth and 256-QAM. Key new features: beam management, analog/digital beamforming, non-terrestrial networks (NTN), integrated access and backhaul (IAB).

Wi-Fi 6 (IEEE 802.11ax). Operates in 2.4 GHz, 5 GHz, and 6 GHz (Wi-Fi 6E). Key features: OFDMA (for multiple simultaneous downlink/uplink users), 1024-QAM, MU-MIMO (up to 8 streams), BSS coloring to reduce intra-system interference, target wake time (TWT) for IoT power saving. Peak rates: 9.6 Gbps. Unlike LTE/NR which use FDD (frequency division duplex) or TDD, Wi-Fi uses CSMA/CA random access — fundamentally different MAC layer.