AI & Signal Processing (EC-802 (A)) - Unit 3 Short Notes

UNIT 3: AI & Signal Processing for Wireless Communications

1.0 Artificial Intelligence Fundamentals

1.1 Introduction to AI, Machine Learning, and Deep Learning

• Artificial Intelligence (AI): Broad field creating systems capable of tasks requiring human intelligence (reasoning, learning, perception).

• Machine Learning (ML): Subset of AI where systems learn patterns from data without explicit programming.

• Deep Learning (DL): Subset of ML using multi-layered neural networks to learn complex representations.

• Relationship: AI ⊃ ML ⊃ DL. DL's success is driven by big data and computational power.

1.2 Types of Machine Learning

1.3 Real-world Applications of AI

• Healthcare: Medical diagnosis (imaging), drug discovery.

• Transport: Autonomous vehicles (perception, path planning).

• Finance: Fraud detection, algorithmic trading.

• Customer Service: Chatbots, recommendation systems (Netflix, Amazon).

• Smart Assistants: Siri, Alexa (NLP, speech recognition).

1.4 Human Intelligence vs Artificial Intelligence

1.5 Knowledge Representation: Declarative vs Procedural

• Declarative Knowledge: What is true. Facts, statements about the world.

  • Example: "The sky is blue." "Paris is the capital of France."

  • Represented as: Logic (predicates), Semantic Networks, Frames.

• Procedural Knowledge: How to do something. Rules, processes, skills.

  • Example: "To start a car, insert key and turn."

  • Represented as: Production Rules (IF-THEN), Scripts, Algorithms.

1.6 Utility Theory in AI

• Core Idea: Quantify an agent's preferences over outcomes using a utility function U(s), where s is a state.

• Principle of Maximum Expected Utility: An agent should choose the action that maximizes the expected utility.

$$ \text{Choose action } a^* = \arg\max_a \mathbb{E}[U(\text{result}(a))] $$

• Importance: Provides a rational framework for decision-making under uncertainty. Used in game theory, planning, and economics.

• Example: Choosing between a guaranteed ₹100 (utility U(100)) and a 50% chance to win ₹200 (expected utility 0.5U(200) + 0.5U(0)). If U is linear, both have same expected utility.

1.7 Reasoning Methods: Forward and Backward Chaining

[!TIP]

Common Pitfall: Confusing the starting point. Forward starts with data, backward starts with a hypothesis/goal.

1.8 Probabilistic Graphical Models

1.8.1 Markov Models

• Definition: Stochastic model where the future state depends only on the present state, not on the sequence of events that preceded it (Markov Property).

• Components:

  • States: S = {S1, S2, ..., SN}

  • Transition Probabilities: P(S_{t+1} | S_t)

  • Initial State Probabilities: P(S_1)

• Types: Markov Chain (discrete time, discrete states), Hidden Markov Model (HMM - states are hidden, observations are visible).

• Applications: Speech recognition (HMM), bioinformatics (gene prediction), queueing theory.

1.8.2 Bayesian Networks (Belief Networks)

• Definition: Directed acyclic graph (DAG) representing conditional dependencies among random variables.

• Nodes: Random variables.

• Edges: Direct conditional dependence (→ means "is a parent of" or "directly influences").

• Key Property: Conditional Independence. A node is conditionally independent of its non-descendants given its parents.

• Joint Probability:

$$ P(X_1, X_2, ..., X_n) = \prod_{i=1}^{n} P(X_i | \text{Parents}(X_i)) $$

• Example: Rain → WetGrass, Sprinkler → WetGrass. Rain and Sprinkler are independent.

• Importance: Efficient representation of joint distributions, probabilistic inference (computing posterior probabilities given evidence).

1.9 Machine Learning Algorithms

1.9.1 Learning Models and Factors Affecting Learning

• Learning Models:

  • Batch Learning: Train on entire dataset at once.

  • Online/Incremental Learning: Update model sequentially with each data point.

  • Passive Learning: Learner only observes data.

  • Active Learning: Learner queries for labels on strategic data points.

• Factors Affecting Learning:

  1. Data Quality: Noise, missing values, outliers.

  2. Feature Representation: Choice and engineering of input features.

  3. Model Complexity: Underfitting (too simple) vs Overfitting (too complex).

  4. Training Data Size: More data generally improves performance (up to a point).

  5. Optimization Algorithm: Learning rate, convergence criteria.

1.9.2 Decision Trees

• Idea: Hierarchical model using if-then rules. Splits data recursively based on feature values.

• Key Concepts:

  • Root Node: Top of tree.

  • Internal Node: Test on a feature.

  • Leaf Node: Class label or regression value.

  • Splitting Criteria: Gini Impurity, Information Gain (Entropy), Gain Ratio.

• Advantages: Easy to interpret, handles non-linear relationships, requires little data preprocessing.

• Disadvantages: Prone to overfitting (needs pruning), unstable (small data changes → different tree), biased towards features with more levels.

1.9.3 Naive Bayes

• Idea: Probabilistic classifier based on Bayes' Theorem with a strong (naive) independence assumption among features.

• Formula:

$$ P(C_k | \mathbf{x}) = \frac{P(C_k) \cdot P(\mathbf{x} | C_k)}{P(\mathbf{x})} \propto P(C_k) \prod_{i=1}^{n} P(x_i | C_k) $$

where `C_k` is class, `x` is feature vector.

• Training: Estimate P(C_k) (class prior) and P(x_i | C_k) (likelihood, often Gaussian for continuous, multinomial for discrete).

• Advantages: Very fast, works well with high-dimensional data, surprisingly effective despite naive assumption.

• Disadvantages: Independence assumption rarely holds; zero probability problem (needs smoothing).

1.9.4 Comparison of Decision Trees and Naive Bayes

2.0 Signal Processing Fundamentals

2.1 Discrete Fourier Transform (DFT)

• Definition: For a finite-length sequence x(n) of length N, the DFT is:

$$ X(k) = \sum_{n=0}^{N-1} x(n) \cdot e^{-j 2\pi kn/N} = \sum_{n=0}^{N-1} x(n) \cdot W_N^{kn}, \quad k=0,1,...,N-1 $$

where `W_N = e^{-j 2π/N}` (twiddle factor).

• Inverse DFT (IDFT):

$$ x(n) = \frac{1}{N} \sum_{k=0}^{N-1} X(k) \cdot W_N^{-kn} $$

2.1.1 Properties: Circular Convolution

• Property: Circular convolution of two length-N sequences in time domain corresponds to multiplication of their DFTs in frequency domain.

$$ x_1(n) \circledast x_2(n) \longleftrightarrow X_1(k) \cdot X_2(k) $$

• Computation: To compute linear convolution via DFT, sequences must be zero-padded to length L ≥ N1+N2-1 to avoid aliasing (circular convolution effect).

2.1.2 DFT of Real and Even Sequences

• Real Sequence: x(n) is real → X(N-k) = X^*(k) (conjugate symmetry). Magnitude is even, phase is odd.

• Even Sequence (x(n) = x(N-n)): X(k) is real and even.

• Odd Sequence (x(n) = -x(N-n)): X(k) is imaginary and odd.

2.1.3 Modulation Property and DFT Computation

• Modulation Property: Multiplication by a complex exponential in time shifts the DFT.

$$ x(n) \cdot e^{j 2\pi m n / N} \xrightarrow{\text{DFT}} X((k-m)_N) $$

where `(k-m)_N` denotes modulo-N.

• DFT Computation: Efficiently computed using Fast Fourier Transform (FFT) algorithms (Radix-2, mixed-radix), reducing complexity from O(N^2) to O(N log N).

2.1.4 Practical Computation Problems

• Spectral Leakage: Caused by finite observation window (windowing effect). Non-periodic signals within the window spread energy.

• Picket Fence Effect: DFT samples the continuous DTFT; if signal frequency falls between bins, energy spreads (similar to leakage).

• ** scalloping loss**: Variation in DFT magnitude for frequencies between bins.

• Mitigation: Use appropriate window functions (Hamming, Hanning), zero-padding for finer frequency grid.

2.2 Multirate Signal Processing

• Core Idea: Process signals at multiple sampling rates. Used for efficient filter banks, compression, transmultiplexers.

• Building Blocks: Decimator (downsample by M), Interpolator (upsample by L), Filter (anti-aliasing/imaging).

2.2.1 Quadrature Mirror Filter Banks (QMF)

• Goal: Split a signal into two subbands (lowpass/highpass) with perfect reconstruction (PR) after synthesis.

• QMF Condition: Analysis lowpass H0(z) and highpass H1(z) filters are mirrored around π/2:

$$ H_1(z) = H_0(-z) \quad \text{or} \quad H_1(z) = H_0(z^{-1}) $$

• Perfect Reconstruction: Requires H0(z)H0(-z) + H1(z)H1(-z) = 2 and phase conditions. Often achieved with alias cancellation using polyphase components.

• Application: Subband coding (e.g., JPEG2000, speech coding).

2.2.2 Types of Filter Banks

2.2.3 Computationally Efficient Sampling Rate Converters

• Polyphase Implementation: Key to efficiency. Reshape filter coefficients into M (for decimation) or L (for interpolation) phases.

• Structure: For rate change by L/M:

  1. Upsampler (insert L-1 zeros).

  2. Polyphase filter (filter with L phases, operate at input rate).

  3. Downsampler (keep every M-th sample).

• Advantage: Reduces number of multiplications by factor of min(L, M). Avoids computing values that are later discarded.

2.3 Digital Filter Design and Structures

2.3.1 Basic FIR and IIR Filter Structures

• FIR (Finite Impulse Response):

  • Direct Form I: y(n) = \sum_{k=0}^{M} b_k x(n-k) (transversal).

  • Direct Form II: Lattice structure, uses fewer delays.

  • Advantages: Always stable, linear phase possible, no feedback.

• IIR (Infinite Impulse Response):

  • Direct Form I: y(n) = \sum_{k=0}^{M} b_k x(n-k) - \sum_{k=1}^{N} a_k y(n-k).

  • Direct Form II (Canonical): Minimal number of delays.

  • Cascade Form: Series of biquad sections (stable if each section stable).

  • Parallel Form: Sum of all-pass and feedforward terms.

  • Disadvantages: Can be unstable, non-linear phase.

2.3.2 Parallel All-Pass Realization of IIR Filters

• Concept: Any stable, minimum-phase IIR transfer function H(z) can be decomposed into a sum of all-pass sections and a constant.

$$ H(z) = \sum_{i=1}^{K} \frac{A_i(z) + B_i(z)}{2} + C $$

where `A_i(z)` and `B_i(z)` are all-pass functions (`|A_i(e^{jω})|=1`).

• Purpose: Used in perfect reconstruction filter banks and multirate systems for phase compensation. Ensures linear phase overall.

2.4 Interpolation Techniques

2.4.1 Spline Interpolation

• Idea: Fit low-degree polynomials (cubic most common) between data points, ensuring smoothness at joints (continuous first and second derivatives).

• Cubic Spline: For N points, N-1 cubic polynomials S_i(x) on intervals [x_i, x_{i+1}].

• Conditions:

  1. S_i(x_i) = y_i, S_i(x_{i+1}) = y_{i+1} (interpolation).

  2. S_i'(x_{i+1}) = S_{i+1}'(x_{i+1}) (first derivative continuous).

  3. S_i''(x_{i+1}) = S_{i+1}''(x_{i+1}) (second derivative continuous).

  4. Boundary Conditions: Natural (S''(x_0)=S''(x_N)=0), clamped (S'(x_0)=f'(x_0), etc.).

• Advantage over Polynomial: Avoids Runge's phenomenon (oscillations at ends). Globally smooth.

• Disadvantage: Computationally more intensive than linear interpolation.

3.0 Wireless Communication Channel Modeling

3.1 Introduction to Wireless Systems

3.1.1 Services, Requirements, and Evolution (1G to 5G)

3.1.2 Economic and Social Impact

• Economic: Trillion-dollar industry, job creation, enables digital economy (e-commerce, fintech), productivity boost.

• Social: Connectivity anytime/anywhere, social media, remote education/healthcare, emergency services, but also digital divide, privacy concerns.

3.1.3 Technical Challenges

• Multipath Propagation: Signals arrive via multiple paths → constructive/destructive interference (fading), delay spread (ISI).

• User Mobility: Causes Doppler shift and time-varying channel. Requires fast tracking and handoff.

• Other challenges: Limited spectrum, noise, interference, hardware limitations.

3.1.4 Spectrum Limitations and Regulatory Aspects

• Limitations: Spectrum is finite and expensive. Licensed bands are scarce; unlicensed bands are crowded.

• Regulatory: Governed by bodies (ITU, national regulators like TRAI in India). Allocates bands for services (e.g., 700 MHz for 5G). Enforces spectrum licensing (auction, administrative) and spectrum sharing (CBRS, LSA).

3.2 Propagation Mechanisms

• Reflection & Transmission: Occurs at boundaries between media with different permittivity/permeability. Governed by Fresnel equations (for plane waves). Reflection coefficient Γ depends on angle of incidence, polarization, and material properties.

• Diffraction: Bending of waves around obstacles (e.g., buildings). Modeled by knife-edge diffraction (Fresnel zones). Loss increases with obstacle sharpness and frequency.

• Scattering: Interaction with rough surfaces (walls, foliage) or small objects. Energy spreads in many directions.

  • Rough Surface Scattering: If surface height variation σ_h > λ/8, scattering is diffuse. Rayleigh criterion for rough surface.

  • Kirchhoff Theory: Treats surface as collection of point scatterers with random phases. Valid for large, gently undulating surfaces.

  • Perturbation Theory: For small surface roughness (σ_h << λ). Scattered field is first-order approximation.

3.3 Channel Characterization

3.3.1 Large-Scale Fading: Path Loss and Shadowing

• Path Loss: Average signal power decay with distance d. Log-distance model:

$$ PL(d) = PL(d_0) + 10n \log_{10}\left(\frac{d}{d_0}\right) + X_\sigma $$

where `n` is path loss exponent (2 for free space, 2-6 for urban), `X_σ` is shadowing (log-normal RV).

• Shadowing: Slow variations (λ-scale) due to obstructions (buildings, hills). Modeled as log-normal distribution: PL(dB) ~ N(μ(d), σ^2).

3.3.2 Small-Scale Fading: Multipath and Delay Spread

• Cause: Multiple delayed copies of signal due to reflectors within coherence area (λ-scale).

• Impulse Response: h(t, τ) (time-variant, delay-spread).

3.3.2.1 Delay Spread and Coherence Bandwidth (Derivation and Relationship)

• Mean Excess Delay (\bar{τ}): First moment of power delay profile P(τ).

$$ \bar{τ} = \frac{\sum_k P(τ_k) τ_k}{\sum_k P(τ_k)} $$

• RMS Delay Spread (τ_{rms}): Square root of second central moment. Key parameter.

$$ τ_{rms} = \sqrt{\bar{τ^2} - (\bar{τ})^2}, \quad \text{where} \quad \bar{τ^2} = \frac{\sum_k P(τ_k) τ_k^2}{\sum_k P(τ_k)} $$

• Coherence Bandwidth (B_c): Frequency range over which channel is flat (correlated fading). Approximate relationship:

$$ B_c \approx \frac{1}{k \cdot τ_{rms}} $$

where `k` is constant (typically 5-50, depending on correlation threshold, e.g., 0.9 or 0.5).

*   **Derivation**: From Fourier transform of autocorrelation function `R_h(Δf)`. `B_c` is width where `|R_h(Δf)|` remains high. For exponential PDP, `B_c ≈ 1/(5τ_{rms})` for 0.5 correlation.

3.3.2.2 Causes of Delay Dispersion and Mitigation

• Causes: Rich multipath environment (urban, indoor), high symbol rate (narrow pulses).

• Impact: Intersymbol Interference (ISI) → Error floor.

• Mitigation Techniques:

  1. Equalization (time-domain compensation).

  2. Orthogonal Frequency Division Multiplexing (OFDM) (frequency-domain, uses cyclic prefix to convert ISI to scalar multiplication).

  3. Spread Spectrum (RAKE receiver for CDMA).

3.3.3 Doppler Effect

3.3.3.1 Doppler Shift Derivation and Velocity Impact

• Derivation: Mobile moving with velocity v relative to fixed scatterer. Incident wave angle θ (with respect to direction of motion).

$$ f_d = f_c \cdot \frac{v}{c} \cos θ $$

where `f_c` is carrier frequency, `c` is speed of light.

• Maximum Doppler Shift: f_m = f_c \cdot v / c (when cosθ = ±1).

• Velocity Impact: f_m ∝ v. Higher speed → faster channel variation.

3.3.3.2 Doppler Spectra

• Doppler Power Spectral Density (PSD): S_h(f) describes power distribution vs Doppler shift.

• Classical Spectrum (Jakes Model): For isotropic scattering (uniform θ in [0, 2π]).

$$ S(f) = \frac{1}{\pi f_m \sqrt{1 - (f/f_m)^2}}, \quad |f| ≤ f_m $$

(U-shaped, singular at `±f_m`).

• Flat Spectrum: For one dominant path (e.g., Rician with strong LOS).

3.3.3.3 Impact on System Performance

• Time Variation: Channel impulse response h(t,τ) changes over time. Coherence time T_c ≈ 1/(2f_m) is time duration over which channel is stationary.

• Fading Rate: Fast fading if symbol duration T_s < T_c; slow fading if T_s >> T_c.

• System Impact: Requires channel tracking (pilot symbols, decision-directed). Fast fading causes time-selective fading, degrading coherent detection.

3.4 Channel Models

3.4.1 Narrowband, Wideband, and Directional Models

3.4.2 Deterministic Models: Ray Tracing and Efficiency Considerations

• Idea: Use electromagnetic theory (ray optics) to compute propagation from transmitter to receiver via reflection, diffraction, scattering.

• Process:

  1. Environment Database: 3D map with material properties (permittivity, conductivity).

  2. Ray Launching/Shooting: Launch rays from TX in all directions.

  3. Intersection Testing: Find interactions with objects (walls, buildings).

  4. Field Calculation: Compute electric field for each ray (Fresnel, diffraction coefficients).

  5. Summation: Coherent sum at receiver location.

• Efficiency Considerations:

  • Computational Cost: Very high (millions of rays). Requires acceleration structures (BSP trees, KD-trees).

  • Accuracy vs Speed Trade-off: Use image method for simple geometries, shooting and bouncing rays (SBR) for complex.

  • Frequency Dependence: Ray optics valid when object dimensions >> λ. For lower frequencies, need diffraction models.

3.4.3 Stochastic Models

3.4.3.1 WSSUS Model (Wide-Sense Stationary Uncorrelated Scattering)

• Assumptions:

  1. Wide-Sense Stationary (WSS): Statistics (mean, autocorrelation) invariant to time shift.

  2. Uncorrelated Scattering (US): Scattering components with different delays are uncorrelated.

• Implication: Scattering function S(τ, f_D) (delay-Doppler) separable:

$$ S(τ, f_D) = P(τ) \cdot S(f_D) $$

where `P(τ)` is power delay profile (PDP), `S(f_D)` is Doppler spectrum.

• Condensed Parameters:

  • Delay Spread: τ_{rms} from P(τ).

  • Doppler Spread: f_{rms} from S(f_D).

  • Coherence Time: T_c ≈ 1/(2f_{rms}).

  • Coherence Bandwidth: B_c ≈ 1/(k τ_{rms}).

3.4.3.2 Time-Variant Two-Path Model

• Simplest model: One direct LOS path + one reflected path.

$$ h(t) = a_1 e^{jφ_1} + a_2(t) e^{jφ_2(t)} $$

• a_1, φ_1 constant (LOS), a_2(t), φ_2(t) vary due to mobility.

• Useful for: Understanding basic fading mechanisms, Doppler shift derivation.

3.4.3.3 Models with Dominant Component (Rician Fading)

• Scenario: Strong LOS component plus many scattered rays.

• Amplitude Distribution: Rician Distribution.

$$ p(r) = \frac{r}{σ^2} e^{-(r^2 + A^2)/(2σ^2)} I_0\left(\frac{rA}{σ^2}\right), \quad r ≥ 0 $$

where `A` is LOS amplitude, `σ^2` is power of scattered components, `I_0` is modified Bessel function.

• Rician K-factor: K = A^2/(2σ^2) (ratio of deterministic to scattered power).

  • K → ∞ → AWGN (pure LOS).

  • K = 0 → Rayleigh fading (no LOS).

• Phase Distribution: Uniform over [0, 2π) if LOS and scattered are independent.

3.4.4 AWGN Channel Model and Error Probability

• AWGN (Additive White Gaussian Noise): Baseline model. Channel: y(t) = x(t) + n(t), where n(t) is white Gaussian noise with PSD N_0/2.

• Error Probability (for coherent BPSK):

$$ P_b = Q\left(\sqrt{\frac{2E_b}{N_0}}\right) $$

where `Q(x) = (1/√(2π)) ∫_x^∞ e^{-t^2/2} dt`.

• For QPSK (same E_b as BPSK):

$$ P_b = Q\left(\sqrt{\frac{E_b}{N_0}}\right) $$

(QPSK has same BER as BPSK but double spectral efficiency).

3.5 Channel Measurement and Sounding

3.5.1 Channel Sounding Process

1. Probe Signal: Known signal transmitted (pulse, chirp, PN sequence).

2. Propagation: Signal traverses channel, gets distorted.

3. Reception: Receiver captures distorted signal.

4. Correlation/Processing: Cross-correlate received signal with known probe (or its matched filter) to estimate channel impulse response (CIR) h(τ).

5. Parameter Extraction: From CIR, compute PDP, delay spread, etc.

3.5.2 Time Domain Measurement Methods

• Direct Pulse: Transmit short pulse (wideband). Measure received pulse shape. Limited by pulse width and noise.

• Pseudo-Noise (PN) Sequence: Transmit long binary sequence (e.g., m-sequence). At receiver, correlate with delayed replica. High processing gain, good noise immunity. Most common for wideband sounding.

• Swept Time-Delay Cross-Correlation (STDCC): Use two antennas, sweep local oscillator frequency to create virtual array.

3.5.3 Frequency Domain Measurement Methods

• Network Analyzer (S-parameter measurement): Measure S21 (transmission) over frequency band. Inverse FFT gives CIR.

• Multitone (Frequency-Domain Sounding): Transmit multiple narrowband carriers (tones) simultaneously. Measure amplitude/phase on each tone. Directly gives frequency response H(f). Efficient for MIMO.

• Advantage over Time Domain: Better SNR per tone, easier to avoid PAPR issues.

4.0 Signal Processing for Wireless Transceivers

4.1 Transceiver Architecture

4.1.1 Block Diagram of a Wireless Transceiver

[TX]                                      [RX]

Source → Encoder → Interleaver → Mapper → ↑ → RF Up → Antenna → Channel → Antenna → RF Down → ↓ → Equalizer → Demapper → Deinterleaver → Decoder → Sink

                                    (Modulator)                                    (Demodulator)

• Key Blocks:

  • Modulator: Maps bits to complex symbols (QPSK, QAM).

  • Up/Down Conversion: Mix to/from RF carrier.

  • Equalizer: Compensates for channel distortion (ISI).

  • Channel Estimation: Uses pilot symbols to estimate h(t).

4.1.2 Modulation and Demodulation Techniques

• Linear Modulation: QPSK, QAM. Spectrally efficient, but sensitive to nonlinearities (PA saturation).

• Constant Envelope Modulation: MSK, GMSK. Robust to nonlinearities, used in GSM (GMSK).

• Impact on Performance:

  • Spectral Efficiency: Bits/s/Hz. QPSK (2), 16-QAM (4), 64-QAM (6).

  • Error Performance: Higher-order QAM needs higher E_b/N_0 for same BER.

  • Robustness: Constant envelope better for low-cost PAs.

4.1.3 Impact on System Performance

• Modulation Choice: Trade-off between data rate, power efficiency, and bandwidth.

• Demodulation: Coherent (needs channel estimate) vs Non-coherent (DPSK, no estimate but worse BER).

• Overall: Determines link budget, required SNR, hardware complexity.

4.2 Fading Mitigation Techniques

4.2.1 Equalization

• Need: Frequency-selective fading causes ISI. Equalizer inverts channel effect: G(z) ≈ 1/H(z).

• Linear Equalizers:

  • Zero-Forcing (ZF): Forces h * g = δ(n). Inverts channel exactly → noise enhancement for deep fades.

  • Minimum Mean Square Error (MMSE): Balances ISI and noise. G(z) = H^*(z) / (|H(z)|^2 + N_0/E_s).

• Decision-Directed Equalization: Uses detected symbols (after slicer) as training data after initial convergence. Adapts to slow channel changes.

• Blind Equalization: No training sequence. Uses higher-order statistics (e.g., Constant Modulus Algorithm - CMA). Preferred when: Spectrum efficiency is critical (no pilot overhead), or channel varies rapidly during data burst.

• Fractional Spaced Equalizer (FSE): Sampler runs at T_s/M (M>1) before slicer. Advantages: Avoids timing sensitivity, can correct for carrier phase offset, better performance in multipath.

• Equalizer Structures and Comparison:

  | Structure | Description | Pros | Cons | | :--- | :--- | :--- | :--- | | Linear (ZF/MMSE) | FIR filter, no feedback. | Simple, stable. | Noise enhancement (ZF), residual ISI. | | DFE (Decision Feedback) | Feedforward + feedback (uses past decisions). | Eliminates post-cursor ISI, no noise enhancement. | Error propagation if decisions wrong. | | MLSE (Viterbi) | Maximum Likelihood Sequence Estimation. | Optimal (minimizes sequence error). | Complexity grows exponentially with channel memory. |

4.2.2 Diversity Techniques

• Goal: Combat fading by providing multiple independent (or nearly) copies of signal.

• Microdiversity vs Macrodiversity:

  | Feature | Microdiversity | Macrodiversity | | :--- | :--- | :--- | | Scale | Small (within a cell, λ-scale). | Large (between cell sites, km-scale). | | Correlation | Low (antennas spaced > λ/2). | Very low (independent fading). | | Purpose | Mitigate fast fading (small-scale). | Mitigate shadowing, extend coverage. | | Example | Antenna array at BS, MIMO. | Soft handoff in CDMA, cooperative relaying. |

• Types of Diversity:

  1. Time Diversity: Same information sent at different times (interleaving, coding). Requires T_s > T_c.

  2. Frequency Diversity: Same information on different frequencies (CDMA spreading, OFDM subcarriers). Requires Δf > B_c.

  3. Space Diversity: Multiple antennas (SIMO, MIMO). Requires antenna spacing > coherence distance.

  4. Polarization Diversity: Orthogonal polarizations (vertical/horizontal). Compact, but some correlation.

• How Diversity Improves Reception: Combines multiple faded signals (selection, MRC, EGC) → higher average SNR, lower error probability. Diversity order d → BER ∝ (SNR)^{-d}.

4.2.3 Error Probability in Fading Channels

• AWGN Channel: P_b = Q(√(2γ)) for BPSK (γ = E_b/N_0).

• Fading Channel: Average BER by integrating over fading distribution p(γ).

  • Rayleigh Fading (no LOS): P_b ≈ (1/(2γ)) for BPSK (high SNR). 3 dB worse than AWGN at same average SNR.

  • Rician Fading (K-factor): P_b decreases as K increases.

• Impact of Delay Spread (Frequency-Selective Fading): Causes ISI → error floor even at high SNR if not equalized. Equalizer reduces but doesn't eliminate.

4.3 Advanced Detection and Coding

4.3.1 Viterbi Detector (Maximum Likelihood Sequence Estimation - MLSE)

• Purpose: Optimal detection in presence of ISI (memory channel). Finds most likely transmitted sequence given received signal.

• Algorithm: Dynamic programming on trellis diagram (states = channel memory). Computes path metrics (Euclidean distance) and retains survivor paths.

• Complexity: O(2^ν) states, where ν is channel memory (number of taps - 1). Used in GSM (GMSK), digital subscriber lines (DSL).

• Advantage: Near-optimal performance (minimizes sequence error probability).

• Disadvantage: Complexity high for long channels.

4.3.2 Comparison of Modulation Schemes: MSK vs QPSK

4.4 Multiple Access and Data Services

4.4.1 TDMA and CDMA Systems

4.4.2 Data Services in Cellular Communication

• Circuit-Switched Data (CSD): Dedicated channel for duration of call (e.g., 9.6 kbps in GSM). Inefficient.

• Packet-Switched Data:

  • GPRS (2.5G): "Always-on", shared channels, best-effort. Speeds up to ~100 kbps.

  • EDGE (2.75G): Enhanced GPRS with 8PSK, up to ~200 kbps.

  • UMTS/HSPA (3G): High-Speed Packet Access. HSPA+ up to 42 Mbps downlink.

  • LTE (4G): All-IP, OFDMA/SC-FDMA. Peak rates >100 Mbps.

  • 5G NR: eMBB (enhanced Mobile Broadband), URLLC (Ultra-Reliable Low-Latency), mMTC (massive Machine-Type Communications).

5.0 Antennas and System Design Considerations

5.1 Antennas for Mobile Stations

• Constraints: Small size, low cost, omnidirectional pattern (usually), robustness.

• Common Types:

  • Monopole/Whip: Quarter-wave, external, omnidirectional.

  • Patch (Microstrip): Low-profile, conformal, directional (hemispherical). Used in smartphones, laptops.

  • PIFA (Planar Inverted-F Antenna): Modified patch with shorting pin. Compact, good impedance bandwidth. Dominant in modern phones.

  • Loop Antennas: Small, less sensitive to hand/human body effects.

• Challenges: Detuning by user's hand/head (body loading), SAR (Specific Absorption Rate) limits.

5.2 Antenna Parameters: Gain and Rayleigh Distance (Derivation and Computation)

• Gain (G): Ratio of radiation intensity in a direction to average radiation intensity. G = η * D, where η is efficiency, D is directivity.

  • dBi: Gain relative to isotropic radiator.

  • Relationship to Effective Aperture: G = (4π A_e)/λ^2.

• Rayleigh Distance (Far-Field Distance) (R_f): Distance beyond which angular field distribution is essentially independent of distance.

  • Derivation: From Fresnel zones. For an antenna of largest dimension D and wavelength λ:

$$ R_f = \frac{2D^2}{λ} $$

*   **Physical Meaning**: Region where wavefronts are approximately planar.

• Computation Example (from May 2024 paper):

  • Given: Square antenna, Gain G = 20 dB = 100 (linear).

  • For square antenna of side a, area A = a^2. Directivity D ≈ 4πA/λ^2 (assuming high efficiency, G≈D).

  • So, A = G λ^2 / (4π).

  • Largest dimension D_ant = a = √A = λ √(G/(4π)).

  • Rayleigh distance: R_f = 2 D_ant^2 / λ = 2 * (λ^2 G/(4π)) / λ = (λ G)/(2π).

  • Without λ, cannot compute numerical R_f. Need frequency/carrier. If frequency f given, λ = c/f. Example: if f=2 GHz, λ=0.15 m, then R_f ≈ (0.15 * 100)/(2π) ≈ 2.39 m.

5.3 Impact of Channel Parameters on System Design

5.3.1 Delay Spread and Coherence Bandwidth

• Impact:

  • Large τ_{rms} (small B_c): Channel is frequency-selective. Requires equalization or OFDM (with cyclic prefix ≥ τ_{rms}).

  • System Design: Choose modulation bandwidth B relative to B_c. If B > B_c, use wideband techniques.

  • Coding: Use interleaving depth > τ_{rms} to spread errors.

5.3.2 Doppler Shift and Mobility

• Impact:

  • Large f_m (small T_c): Channel changes within symbol duration → fast fading. Requires fast channel tracking (frequent pilots), robust modulation (DPSK), or diversity (time/frequency).

  • System Design: Coherence time T_c limits frame length for channel estimation. High mobility → higher handoff rate.

  • Doppler Spread determines maximum allowable subcarrier spacing in OFDM to avoid ICI (Inter-Carrier Interference).

5.3.3 Spectrum Efficiency and Limitations

• Spectrum Efficiency (η): η = (bits/s/Hz) = R / B, where R is data rate, B is bandwidth.

• Limitations:

  • Shannon Capacity: C = B log_2(1 + SNR). Sets fundamental limit.

  • Practical Limits: Modulation order (QAM), coding rate, overhead (pilots, guards), interference, regulatory masks.

• Design Trade-offs:

  • Higher η → higher-order modulation → needs higher SNR, more linear PA.

  • 5G Design: Uses massive MIMO (spatial multiplexing), mmWave (large bandwidth), advanced coding (LDPC, Polar) to push η towards Shannon limit.

[!NOTE]

Exam Strategy:

1. Definitions First: Always start answers with clear definitions (e.g., "Coherence bandwidth is...").

2. Derivations: Show step-by-step. For Doppler shift, start from f_d = (v/λ) cosθ. For τ_{rms}, define PDP and compute moments.

3. Comparisons: Use tables (like Decision Trees vs Naive Bayes, TDMA vs CDMA, Micro vs Macro diversity). Highlight key differences.

4. Diagrams: Sketch block diagrams (transceiver, WSSUS model, two-path model, Viterbi trellis). Label clearly.

5. Formulas: Box critical ones: τ_{rms}, B_c ≈ 1/(kτ_{rms}), f_d = f_c v cosθ / c, R_f = 2D^2/λ, Rician PDF, BER expressions.

6. Units: Remember τ_{rms} in seconds, B_c in Hz, f_m in Hz, R_f in meters.

7. Context: Link concepts to applications (e.g., "OFDM is used because B > B_c").

8. Short Notes: For 7-mark questions, cover 4-5 key points with one example/application. For 14-mark, include derivation + explanation + implications.