Wireless Communication (EC-802 (B)) - Unit 5 Short Notes

UNIT 5: WIRELESS CHANNEL CHARACTERISTICS AND RECEIVER TECHNIQUES

I. Foundations and Overview

Wireless Services and Applications

• Types:

  • Voice-centric: Traditional cellular telephony (circuit-switched).

  • Data-centric: Mobile internet, IoT, M2M (packet-switched).

  • Broadcast: Digital audio/video broadcasting (DAB/DVB).

  • Emergency & Public Safety: Dedicated networks for first responders.

• Economic & Social Impact: Enables ubiquitous connectivity, drives GDP growth, transforms industries (healthcare, transport), but raises concerns on privacy, digital divide, and health.

Evolution: 1G to 5G

Key Technical Challenges

1. Multipath Propagation: Causes intersymbol interference (ISI) and fading.

2. User Mobility: Induces Doppler shift, time-varying channel.

3. Spectrum Limitations: Scarcity, need for efficient sharing (cognitive radio).

Spectrum Limitations & Regulatory Aspects

• Limitation: Finite resource, high cost of licensed bands, interference.

• Regulation: Governed by bodies like ITU-R, national agencies (FCC, TRAI). Allocation in bands: licensed, unlicensed, shared.

[!TIP] Exam Focus: Be prepared to contrast 1G-5G services/requirements and explain how multipath/mobility cause fundamental system impairments (ISI, fading).

II. Radio Wave Propagation Mechanisms

Reflection & Transmission at Material Boundaries

• Governed by Fresnel's equations.

• Reflection Coefficient (Γ): $$\displaystyle \Gamma = \frac{\eta_2 - \eta_1}{\eta_2 + \eta_1} $$ (for normal incidence), where $\eta$ is wave impedance.

• Transmission Coefficient (τ): $$\displaystyle \tau = 1 + \Gamma $$.

• Critical Angle: For total internal reflection when $$\displaystyle \eta_1 > \eta_2 $$.

Scattering from Rough Surfaces

• Kirchhoff Theory (Physical Optics): Assumes surface irregularities small compared to wavelength. Scattering is coherent.

• Perturbation Theory: For slightly rough surfaces. Scattering power proportional to $$\displaystyle (\Delta h / \lambda)^2 $$, where $\Delta h$ is height variation.

Time-Variant Two-Path Propagation Model

• Model: Direct path + one reflected path.

• Received Signal: $$\displaystyle r(t) = \text{Re}\left\{ \left[ a_1 e^{j\theta_1(t)} + a_2 e^{j\theta_2(t)} \right] s(t) \right\} $$.

• Phase: $$\displaystyle \theta_i(t) = 2\pi f_c \tau_i(t) $$, where $$\displaystyle \tau_i(t) $$ is time-variant delay.

• Result: Small-scale fading due to constructive/destructive interference as mobile moves.

[!DIAGRAM: CANVAS] Two-Path Model: Show transmitter, mobile, reflector. Two paths with lengths $$\displaystyle d_1 $$, $$\displaystyle d_2 $$. Path difference $$\displaystyle \Delta d = d_2 - d_1 $$. As mobile moves, $\Delta d$ changes, causing phase difference $$\displaystyle \Delta \phi = 2\pi \Delta d / \lambda $$ to vary.

III. Fading Mechanisms

Large-Scale Fading (Path Loss & Shadowing)

• Path Loss: Average signal power decay with distance $d$. Log-distance path loss model: $$\displaystyle PL(d) = PL(d_0) + 10n \log_{10}(d/d_0) + X_\sigma $$.

  • $n$: Path loss exponent (2-6).

  • $$\displaystyle X_\sigma $$: Shadowing (log-normal random variable, std dev $\sigma$ dB).

• Represents: Signal variation over large distances (hundreds of wavelengths).

Small-Scale Fading

• Definition: Rapid fluctuations over short distances (order of $\lambda$) or time due to multipath.

• Classification:

  • Flat Fading: Channel bandwidth $$\displaystyle B_c \ll B_{channel} $$ (coherence bandwidth $$\displaystyle B_c $$ large). All frequencies fade equally.

  • Frequency-Selective Fading: $$\displaystyle B_c < B_{channel} $$. Different frequency components fade differently → ISI.

  • Fast Fading: Channel changes during symbol duration ($$\displaystyle T_s < T_c $$, coherence time).

  • Slow Fading: Channel constant over $$\displaystyle T_s $$ ($$\displaystyle T_s \ll T_c $$).

Rayleigh Fading Model

• Assumption: No dominant line-of-sight (LOS) component; many scattered paths.

• Amplitude Distribution: Rayleigh distribution.

$$p_R(r) = \frac{r}{\sigma^2} e^{-r^2/(2\sigma^2)}, \quad r \geq 0$$

where $$\displaystyle \sigma^2 $$ is average power of in-phase/quadrature components.

• Power Distribution: Exponential distribution.

$$p_P(P) = \frac{1}{\bar{P}} e^{-P/\bar{P}}, \quad P \geq 0$$

$$\displaystyle \bar{P} = 2\sigma^2 $$ = average power.

• Phase Distribution: Uniform over $[0, 2\pi)$.

Rician Fading Model

• Assumption: One dominant LOS component plus many scattered components.

• Complex Envelope: $$\displaystyle z = x + jy = z_0 e^{j\theta_0} + x_n + jy_n $$.

  • $$\displaystyle (x_n, y_n) $$: Zero-mean Gaussian (scattered).

  • $$\displaystyle z_0 $$: Deterministic LOS magnitude.

• Amplitude Distribution: Rician distribution.

$$p_R(r) = \frac{r}{\sigma^2} e^{-(r^2+z_0^2)/(2\sigma^2)} I_0\left(\frac{r z_0}{\sigma^2}\right)$$

$$\displaystyle I_0(\cdot) $$: Modified Bessel function of first kind, order zero.

• Rician K-factor: $$\displaystyle K = \frac{z_0^2}{2\sigma^2} $$. $$\displaystyle K=0 $$ → Rayleigh; $K \to \infty$ → AWGN.

• Phase Distribution: Not uniform; depends on $K$.

Doppler Effect

• Doppler Shift Formula: $$\displaystyle f_d = \frac{v}{\lambda} \cos \theta = f_c \frac{v}{c} \cos \theta $$.

  • $v$: Mobile speed, $\theta$: angle between mobile velocity and incident wave direction.

  • $$\displaystyle f_c $$: carrier frequency, $c$: speed of light.

• Maximum Doppler Shift: $$\displaystyle f_{d,\max} = \frac{v}{\lambda} = \frac{v f_c}{c} $$.

• Impact of Velocity: Higher $v$ → larger $$\displaystyle f_{d,\max} $$ → faster channel variation (smaller coherence time $$\displaystyle T_c \approx 1/(2f_{d,\max}) $$).

Doppler Power Spectral Density (PSD)

• Clarke's Model (Isotropic scattering): $$\displaystyle S(f) \propto \frac{1}{\sqrt{f_{d,\max}^2 - f^2}} $$ for $$\displaystyle |f| \leq f_{d,\max} $$. "U-shaped" spectrum.

• Jakes' Model: Practical simulation method for Clarke's spectrum.

[!TIP] Exam Focus: Derive Doppler shift from geometry. Distinguish Rayleigh vs. Rician (K-factor). Remember: $$\displaystyle f_{d,\max} = v/\lambda $$. Coherence time $$\displaystyle T_c \approx 0.423/f_{d,\max} $$ for Jakes' model.

IV. Wireless Channel Modeling Approaches

Deterministic Channel Modeling: Ray Tracing

• Principle: Solve Maxwell's equations (or use geometric optics) for a specific environment map (buildings, terrain).

• Process: Launch rays from Tx, trace reflections/diffractions, sum contributions at Rx.

• Efficiency Considerations: Computationally intensive for complex environments. Requires accurate 3D database. Often uses image theory for reflections.

Stochastic Channel Models

• Narrowband Model: Flat fading. Channel as a single complex gain $h(t)$ (e.g., Rayleigh/Rician process). Used when $$\displaystyle B_{signal} \ll B_c $$.

• Wideband Model: Frequency-selective fading. Characterized by delay spread. Uses tapped-delay-line model: $$\displaystyle h(t,\tau) = \sum_{i=0}^{L-1} a_i(t) \delta(\tau - \tau_i(t)) $$.

• Directional Model: Includes angle-of-arrival (AoA) and angle-of-departure (AoD). Uses clusters of paths.

WSSUS Model (Wide-Sense Stationary Uncorrelated Scattering)

• Assumptions:

  1. Wide-Sense Stationary (WSS): Statistical properties (mean, autocorrelation) independent of absolute time $t$.

  2. Uncorrelated Scattering (US): Scattering components with different delays $\tau$ are uncorrelated. $$\displaystyle R_h(\tau_1, \tau_2; \Delta t) = P(\tau) \delta(\tau_1 - \tau_2) R_h(0, \Delta t) $$.

• Significance: Simplifies channel characterization to two functions:

  • Delay Power Spectrum (DPS): $P(\tau)$ (average power vs. excess delay).

  • Time-Autocorrelation Function: $$\displaystyle R_h(0, \Delta t) $$ (related to Doppler spread).

Time-Variant Linear System Characterization

• Wireless channel is a linear time-variant (LTV) system.

• Input: Transmitted signal $s(t)$.

• Output: Received signal $$\displaystyle r(t) = h(t,\tau) \otimes s(t) = \int_{-\infty}^{\infty} h(t,\tau) s(t-\tau) d\tau $$.

• Impulse Response: $h(t,\tau)$ is the response at time $t$ to an impulse at time $t-\tau$. It is a random process in $t$ and $\tau$.

V. Channel Characterization Parameters

Delay Spread & Coherence Bandwidth

• Mean Excess Delay: $$\displaystyle \bar{\tau} = \frac{\sum_i P_i \tau_i}{\sum_i P_i} $$.

• RMS Delay Spread: $$\displaystyle \sigma_\tau = \sqrt{\bar{\tau^2} - (\bar{\tau})^2} $$, where $$\displaystyle \bar{\tau^2} = \frac{\sum_i P_i \tau_i^2}{\sum_i P_i} $$.

• Coherence Bandwidth ($$\displaystyle B_c $$): Frequency separation over which channel is highly correlated.

  • Approximation 1: $$\displaystyle B_c \approx \frac{1}{5\sigma_\tau} $$ (for correlation > 0.5).

  • Approximation 2: $$\displaystyle B_c \approx \frac{1}{50\sigma_\tau} $$ (for correlation > 0.9).

• Relationship: $$\displaystyle \boxed{B_c \propto \frac{1}{\sigma_\tau}} $$. Large $$\displaystyle \sigma_\tau $$ → small $$\displaystyle B_c $$ → frequency-selective fading.

Doppler Spread & Coherence Time

• Doppler Spread ($$\displaystyle B_D $$): Range of frequencies over which received Doppler spectrum is non-zero. $$\displaystyle B_D \approx 2f_{d,\max} $$.

• Coherence Time ($$\displaystyle T_c $$): Time duration over which channel impulse response is essentially invariant.

  • $$\displaystyle T_c \approx \frac{0.423}{f_{d,\max}} $$ (for Jakes' model, 50% correlation).

  • $$\displaystyle T_c \approx \frac{0.187}{f_{d,\max}} $$ (for 90% correlation).

• Relationship: $$\displaystyle \boxed{T_c \propto \frac{1}{f_{d,\max}}} $$. High mobility (large $$\displaystyle f_{d,\max} $$) → small $$\displaystyle T_c $$ → fast fading.

Condensed Parameters of WSSUS Model

• RMS Delay Spread ($$\displaystyle \sigma_\tau $$): Key parameter for frequency dispersion.

• Doppler Spread ($$\displaystyle B_D $$ or $$\displaystyle f_{d,\max} $$): Key parameter for time dispersion.

• These define the channel's scattering function $$\displaystyle S(\tau, f_D) = P(\tau) \cdot S(f_D) $$ under WSSUS.

[!TIP] Exam Focus: Derive RMS delay spread from power delay profile. Memorize approximate formulas: $$\displaystyle B_c \approx 1/(5\sigma_\tau) $$, $$\displaystyle T_c \approx 0.423/f_{d,\max} $$. Know which parameter causes flat vs. selective fading ($$\displaystyle \sigma_\tau $$ vs. symbol duration).

VI. Channel Measurement and Sounding

Channel Sounding Process & Objectives

• Objective: To measure the time-variant impulse response $h(t,\tau)$ or transfer function $H(t,f)$ of a real channel.

• Process: Transmit a known probe signal $p(t)$, measure received signal $r(t)$. Estimate channel via correlation: $$\displaystyle \hat{h}(t,\tau) \propto r(t) \star p^*(-t) $$.

Time-Domain Measurement Methods

• Pulse Sounding: Transmit short pulses. Simple but low power efficiency, high peak-to-average power.

• Pseudorandom Noise (PN) Sequence Sounding: Transmit long PN sequence. Use sliding correlator at receiver. Good average power, but high complexity.

Frequency-Domain Measurement Methods

• Multitone (OFDM-like) Sounding: Transmit signal on multiple narrowband subcarriers. Measure amplitude/phase on each → estimate frequency response $H(f)$.

• Swept Spectrum: Transmit a tone, sweep frequency, measure at each step.

VII. Transceiver Architecture and Modulation

Block Diagram of a Wireless Link

Source → Channel Encoder → Interleaver → Modulator → Upconverter → Tx Antenna → [Wireless Channel] → Rx Antenna → Downconverter → Equalizer → Demodulator → Deinterleaver → Channel Decoder → Destination

Modulation Techniques for Wireless

• Requirements: Bandwidth efficiency, power efficiency, robustness to fading/ISI.

• Comparison (Spectral Efficiency):

  • MSK (Minimum Shift Keying): $$\displaystyle R_s = 0.5 $$ b/s/Hz (binary). Continuous phase → compact spectrum.

  • QPSK (Quadrature PSK): $$\displaystyle R_s = 2 $$ b/s/Hz (since 2 bits/symbol). Same bandwidth as BPSK but double rate.

  • Higher-order QAM: e.g., 16-QAM → 4 b/s/Hz, but less power-efficient.

• Impact: Higher spectral efficiency often sacrifices power efficiency (higher $$\displaystyle E_b/N_0 $$ required for same BER) and robustness (smaller Euclidean distance).

Error Probability Analysis

• AWGN Channel: For coherent detection, $$\displaystyle P_b = Q\left(\sqrt{\frac{2E_b}{N_0}}\right) $$ for BPSK. For $M$-ary QAM, $$\displaystyle P_s \approx 4\left(1 - \frac{1}{\sqrt{M}}\right) Q\left(\sqrt{\frac{3E_b}{(M-1)N_0}}\right) $$.

• Impact of Delay Dispersion (Frequency-Selective Fading): Causes ISI. Even if average $$\displaystyle E_b/N_0 $$ is high, deep fades at specific frequencies cause burst errors. Equalization is required.

• Impact of Flat Fading: Channel gain $$\displaystyle |h|^2 $$ is random. Average BER over fading: $$\displaystyle P_b^{\text{fading}} = \int_0^\infty P_b(\gamma) p_\gamma(\gamma) d\gamma $$, where $$\displaystyle \gamma = |h|^2 E_b/N_0 $$. For Rayleigh fading, BPSK BER: $$\displaystyle P_b = \frac{1}{2}\left(1 - \sqrt{\frac{\bar{\gamma}}{1+\bar{\gamma}}}\right) $$.

[!TIP] Exam Focus: Compare MSK vs. QPSK spectral efficiency. Explain why ISI increases error probability beyond AWGN prediction. Know average BER expressions for Rayleigh fading.

VIII. Equalization for Mitigating ISI

Need for Equalization

To compensate for channel distortion (frequency-selective fading) that causes ISI, by inverting or mitigating the channel's effect.

Classification of Equalizers

Comparison of Equalizer Structures

• Complexity: MLSE > DFE > Linear/FSE.

• Performance: MLSE (optimal) > DFE > MMSE > ZF.

• Stability: Linear/FSE always stable. DFE/MLSE stability depends on channel.

• Robustness to Channel Variation: Blind/Decision-Directed > Training-only.

[!TIP] Exam Focus: Explain DFE operation (FF vs. FB). Why FSE is better than symbol-spaced? When is blind equalization preferred (e.g., bursty data, no training overhead allowed). State Viterbi detector's role in MLSE.

IX. Diversity Techniques for Fading Mitigation

Concept of Diversity & Diversity Gain

• Diversity: Provide multiple independent (or partially correlated) signal replicas to receiver.

• Diversity Gain: Improvement in average SNR or reduction in fade probability. For $N$ branches, diversity gain ≈ $$\displaystyle 10 \log_{10} N $$ dB (for high SNR, selection combining).

Classification of Diversity

• Microdiversity (Intra-cell): Combats small-scale fading within a cell/sector.

  • Implementation: Space diversity (multiple antennas at BS/MS), polarization diversity, frequency diversity (spread spectrum).

• Macrodiversity (Inter-cell): Combats large-scale shadowing. Used in soft handoff (CDMA).

  • Implementation: Multiple base stations simultaneously communicate with mobile.

• Space, Time, Frequency Diversity:

  • Space: Multiple antennas (MIMO).

  • Time: Same channel at different times (interleaving, coding).

  • Frequency: Same channel at different frequencies (FHSS, multi-carrier).

How Diversity Improves Reception

• Probability that all branches fade deeply is much lower than for a single branch.

• Combining Techniques:

  • Selection Combining (SC): Choose branch with highest SNR. Simple, near-optimal for large N.

  • Maximal Ratio Combining (MRC): Weighted sum (weights ∝ SNR). Optimal, but needs channel knowledge.

  • Equal Gain Combining (EGC): Same weight, phase-aligned. Suboptimal but simpler than MRC.

• Result: Reduces fade duration and depth, improves outage probability and BER.

[!TIP] Exam Focus: Differentiate micro vs. macro diversity (scale, purpose, example). List implementation methods for space/time/frequency diversity. Explain why diversity reduces fade probability mathematically (product of independent probabilities).

X. System-Level Components and Multiple Access

Antennas for Mobile Stations (MS)

• Design Constraints: Small size, low cost, omnidirectional pattern (usually), impedance matching to 50Ω, low SAR (Specific Absorption Rate).

• Types:

  • Monopole/Whip: Common, λ/4 long.

  • PIFA (Planar Inverted-F Antenna): Compact, used in phones.

  • Patch Antenna: Low profile, directional.

• Impedance Matching: Use matching network (LC circuit) to transform antenna impedance to 50Ω for efficient power transfer.

Data Services in Cellular Communication

• Circuit-Switched: Dedicated channel for entire session (e.g., traditional voice call, early GSM data). Inefficient for bursty data.

• Packet-Switched: Shared channel, data sent in packets (e.g., GPRS, 3G/4G data). Efficient, supports always-on, QoS classes.

• Examples: GSM (CS), GPRS/EDGE (PS), UMTS (CS+PS), LTE (all-IP PS).

Multiple Access Techniques

Comparative Analysis: TDMA vs. CDMA

[!TIP] Exam Focus: Draw TDMA frame structure. Explain CDMA spreading/despreading. List 3-4 key differences in a table. Why is CDMA called "interference-limited"?

XI. Specialized Derivations and Parameters

Derivation of Rayleigh Distance for Antenna Arrays

• Definition: Distance at which far-field (Fraunhofer) region begins. Path difference between array extremities ≈ $\lambda/16$.

• For linear array of length $L$: $$\displaystyle d_R = \frac{2L^2}{\lambda} $$.

• For rectangular array ($$\displaystyle L_x \times L_y $$): $$\displaystyle d_R = \frac{2}{\lambda} \max(L_x^2, L_y^2) $$.

• Derivation: From far-field condition: maximum phase error $$\displaystyle < \pi/8 $$ (or path diff $$\displaystyle < \lambda/16 $$). For two elements at ends, path diff $$\displaystyle \approx \frac{L^2}{2d} $$ for $d \gg L$. Set $$\displaystyle \frac{L^2}{2d_R} = \frac{\lambda}{16} $$ → $$\displaystyle d_R = \frac{8L^2}{\lambda} $$. (Common factor 2 varies by definition; $$\displaystyle d_R \propto L^2/\lambda $$ is key).

Derivation of RMS Delay Spread & Relation to Coherence Bandwidth

• Given Power Delay Profile (PDP): $$\displaystyle P(\tau_i) $$ at delays $$\displaystyle \tau_i $$.

• Mean Excess Delay: $$\displaystyle \bar{\tau} = \frac{\sum_i P_i \tau_i}{\sum_i P_i} $$.

• Mean Square Delay: $$\displaystyle \overline{\tau^2} = \frac{\sum_i P_i \tau_i^2}{\sum_i P_i} $$.

• RMS Delay Spread: $$\displaystyle \boxed{\sigma_\tau = \sqrt{\overline{\tau^2} - (\bar{\tau})^2}} $$.

• Coherence Bandwidth ($$\displaystyle B_c $$): From channel autocorrelation $$\displaystyle R_h(\Delta f) \approx \int P(\tau) e^{-j2\pi \tau \Delta f} d\tau $$. Correlation bandwidth is inverse of delay spread. Hence, $$\displaystyle \boxed{B_c \approx \frac{1}{\alpha \sigma_\tau}} $$, where $\alpha$ is constant (5 or 50).

Statistical Derivation: Rician Amplitude & Phase

• Model: $$\displaystyle z = x + jy = (x_0 + x_n) + j(y_0 + y_n) $$.

  • $$\displaystyle x_n, y_n \sim \mathcal{N}(0, \sigma^2) $$ i.i.d.

  • $$\displaystyle x_0 = z_0 \cos \theta_0 $$, $$\displaystyle y_0 = z_0 \sin \theta_0 $$ (deterministic LOS).

• Amplitude: $$\displaystyle R = |z| = \sqrt{(x_0+x_n)^2 + (y_0+y_n)^2} $$.

  • PDF derived via transformation of Gaussian variables. Result is Rician distribution:

$$p_R(r) = \frac{r}{\sigma^2} e^{-(r^2+z_0^2)/(2\sigma^2)} I_0\left(\frac{r z_0}{\sigma^2}\right)$$

• Phase: $$\displaystyle \phi = \tan^{-1}(y/x) $$. Conditional on $R$, $\phi$ has a distribution. Marginal distribution is not uniform. For $K \to 0$, approaches uniform; for $K \to \infty$, approaches deterministic $$\displaystyle \theta_0 $$.

[!TIP] Exam Focus: Derive Rayleigh distance from far-field condition. Compute $$\displaystyle \sigma_\tau $$ from given PDP table. State Rician PDF and define K-factor. Practice numerical problems for Rayleigh distance (e.g., given antenna gain, find $L$, then $$\displaystyle d_R $$).