UNIT 5: SHORT NOTES (Based on RGPV EC-503(A) Past Papers)

PART A: TRANSMISSION LINES & NETWORK ANALYSIS

1.0 Transmission Line Fundamentals & Parameters

Primary Constants (per unit length):

• R: Resistance (Ω/m) – conductor loss.

• L: Inductance (H/m) – magnetic flux linkage.

• G: Conductance (mho/m) – dielectric loss.

• C: Capacitance (F/m) – electric field storage.

Secondary Constants:

• Propagation Constant: $$\displaystyle \gamma = \alpha + j\beta $$

  • $\alpha$: Attenuation constant (Np/m or dB/m)

  • $\beta$: Phase constant (rad/m)

• Characteristic Impedance: $$\displaystyle Z_0 = \sqrt{\frac{R + j\omega L}{G + j\omega C}} $$ (Ω)

• Phase Velocity: $$\displaystyle v_p = \frac{\omega}{\beta} $$ (m/s)

• Wavelength: $$\displaystyle \lambda = \frac{2\pi}{\beta} = \frac{v_p}{f} $$

Telegrapher's Equations (Derivation Expected):

$$\frac{\partial V(x,t)}{\partial x} = -L\frac{\partial I(x,t)}{\partial t} - RI(x,t)$$

$$\frac{\partial I(x,t)}{\partial x} = -C\frac{\partial V(x,t)}{\partial t} - GV(x,t)$$

Voltage & Current at Any Point (for lossy line):

$$V(x) = V^+ e^{-\gamma x} + V^- e^{\gamma x}$$

$$I(x) = \frac{V^+}{Z_0} e^{-\gamma x} - \frac{V^-}{Z_0} e^{\gamma x}$$

Where $$\displaystyle V^+ $$, $$\displaystyle V^- $$ are forward/backward traveling waves.

Infinite Line & Matched Line:

• Infinite Line: $$\displaystyle Z_L = Z_0 $$. No reflection ($$\displaystyle \Gamma=0 $$), $$\displaystyle VSWR=1 $$, all power absorbed.

• Matched Line: Load impedance equals $$\displaystyle Z_0 $$. Same properties as infinite line.

Distortion in Transmission Lines:

1. Frequency Distortion: Different attenuation for different frequencies ($\alpha$ not constant with $\omega$).

2. Phase Distortion: Different phase velocities for different frequencies ($\beta$ not linear with $\omega$).

3. Delay Distortion: Group delay $$\displaystyle \tau_g = -\frac{d\beta}{d\omega} $$ not constant.

Condition for Distortionless Line:

$$\frac{R}{L} = \frac{G}{C} \quad \text{or} \quad \alpha = R\sqrt{\frac{C}{L}}, \quad \beta = \omega\sqrt{LC}$$

[!TIP] Exam Focus: Derive distortionless condition from $$\displaystyle \gamma = \alpha + j\beta = \sqrt{(R+j\omega L)(G+j\omega C)} $$ by setting $\alpha \propto \sqrt{f}$ and $\beta \propto f$.

Special Transmission Lines:

• Quarter-Wave Line ($$\displaystyle \ell = \lambda/4 $$):

$$Z_{in} = \frac{Z_0^2}{Z_L}$$

Used for impedance inversion/matching.

• Half-Wave Line ($$\displaystyle \ell = \lambda/2 $$):

$$Z_{in} = Z_L$$

Used as impedance repeater.

• Short-Circuited Line ($$\displaystyle Z_L=0 $$):

$$Z_{in} = jZ_0\tan(\beta\ell)$$

Acts as inductor (if $$\displaystyle \ell < \lambda/4 $$) or capacitor.

• Open-Circuited Line ($$\displaystyle Z_L=\infty $$):

$$Z_{in} = -jZ_0\cot(\beta\ell)$$

Acts as capacitor (if $$\displaystyle \ell < \lambda/4 $$) or inductor.

[!NOTE] Both short/open lines are used as stubs for impedance matching.

Reflection Coefficient & VSWR:

• Voltage Reflection Coefficient: $$\displaystyle \Gamma = \frac{Z_L - Z_0}{Z_L + Z_0} $$

• Current Reflection Coefficient: $$\displaystyle \Gamma_I = -\Gamma $$

• VSWR: $$\displaystyle VSWR = \frac{1 + |\Gamma|}{1 - |\Gamma|} $$

• Reflection Loss (RL): $$\displaystyle RL(dB) = -20\log_{10}|\Gamma| $$

• Insertion Loss (IL): Loss due to insertion of a network; for matched line, $IL \approx 2\alpha\ell$ (dB).

Power Flow:

• Power at any point: $$\displaystyle P(x) = \frac{|V^+|^2}{2Z_0}\left(1 - |\Gamma|^2\right)e^{-2\alpha x} $$ (for matched load).

• For lossless line ($$\displaystyle \alpha=0 $$), power constant along line.

2.0 Two-Port Network Analysis & Parameters

Symmetrical Two-Port Network:

• $$\displaystyle Z_{11} = Z_{22} $$, $$\displaystyle Y_{11} = Y_{22} $$, $$\displaystyle A = D $$, $$\displaystyle B = C $$ (for ABCD).

• Image Impedance ($$\displaystyle Z_{i1} = Z_{i2} = Z_i $$ for symmetrical network):

$$Z_i = \sqrt{Z_{11}Z_{12} - Z_{12}^2} \quad \text{(from Z-params)}$$

$$Z_i = \sqrt{\frac{Z_{11}}{Y_{11}}} \quad \text{(general)}$$

• Characteristic Impedance ($$\displaystyle Z_0 $$): For infinite cascade of identical symmetrical networks, $$\displaystyle Z_0 = Z_i $$.

Asymmetrical Two-Port Network:

• $$\displaystyle Z_{11} \neq Z_{22} $$ etc.

• Image Impedances ($$\displaystyle Z_{i1} \neq Z_{i2} $$):

$$Z_{i1} = \sqrt{Z_{11}Z_{12} - Z_{12}^2} \quad \text{(with port 2 terminated in $$\displaystyle Z_{i2} $$)}$$

$$Z_{i2} = \sqrt{Z_{22}Z_{12} - Z_{12}^2} \quad \text{(with port 1 terminated in $$\displaystyle Z_{i1} $$)}$$

Derivation involves cascading two networks and solving for impedance seen at input when output terminated in its image impedance.

[!TIP] Common Mistake: Using symmetrical formula for asymmetrical network. Always check $$\displaystyle Z_{11} \neq Z_{22} $$.

T & π (Pi) Equivalent Circuits:

• T-Network: $$\displaystyle Z_1 $$, $$\displaystyle Z_2 $$ (series), $$\displaystyle Z_3 $$ (shunt).

$$Z_{11} = Z_1 + Z_3, \quad Z_{22} = Z_2 + Z_3, \quad Z_{12} = Z_3$$

• π-Network: $$\displaystyle Y_1 $$, $$\displaystyle Y_2 $$ (shunt), $$\displaystyle Y_3 $$ (series).

$$Y_{11} = Y_1 + Y_3, \quad Y_{22} = Y_2 + Y_3, \quad Y_{12} = -Y_3$$

• Relation: Convert T to π via:

$$Y_1 = \frac{1}{Z_1} + \frac{1}{Z_3}, \quad Y_2 = \frac{1}{Z_2} + \frac{1}{Z_3}, \quad Y_3 = -\frac{1}{Z_3}$$

Lattice Network:

• Bridge network: $$\displaystyle Z_a $$ and $$\displaystyle Z_b $$ cross-connected.

• Symmetrical by nature ($$\displaystyle Z_{11}=Z_{22} $$).

• Image Impedance:

$$Z_i = \sqrt{Z_a Z_b}$$

• Used in filters and equalizers.

Bridged-T Network:

• T-network with a bridge (series element $$\displaystyle Z_b $$) across series arms.

• Used for impedance matching (e.g., bridged-T attenuator).

• Can be converted to lattice.

3.0 Filter Design & Approximation

Filter Fundamentals:

• Prototype Filter: Low-pass filter with $$\displaystyle R_0=1\Omega $$, $$\displaystyle \omega_c=1 $$ rad/s.

• Frequency Transformations:

  | Filter Type | Transformation | Element Swap | |-------------|----------------|--------------| | LP→HP | $$\displaystyle s \to \frac{\omega_c^2}{s} $$ | L↔C | | LP→BP | $$\displaystyle s \to \frac{s^2 + \omega_0^2}{s\Delta\omega} $$ | L,C → series/parallel LC | | LP→BS | $$\displaystyle s \to \frac{s\Delta\omega}{s^2 + \omega_0^2} $$ | Series/parallel LC → L,C |

Constant-K Filters:

• Design Equations (LPF prototype, $$\displaystyle R_0=1\Omega $$, $$\displaystyle \omega_c=1 $$):

  • T-section: $$\displaystyle K = \sqrt{\frac{L}{C}} $$, $$\displaystyle Z_0 = \sqrt{L/C} $$ at $$\displaystyle \omega_c $$.

$$L = \frac{R_0}{\omega_c}, \quad C = \frac{1}{R_0\omega_c}$$

- **π-section**: $$\displaystyle K = \sqrt{\frac{L}{C}} $$ same.

$$L = \frac{R_0}{\omega_c}, \quad C = \frac{1}{R_0\omega_c}$$

• Limitations: Poor stopband attenuation (roll-off ~20 dB/dec), no attenuation at $$\displaystyle \omega=\infty $$ for LPF.

m-Derived Filters:

• Purpose: Sharper cutoff than constant-K by introducing attenuation pole at finite frequency.

• LPF T-section Design:

$$m = \text{damping factor} \ (0 < m < 1)$$

$$L' = m L_K, \quad C' = \frac{C_K}{m}$$

where $$\displaystyle L_K, C_K $$ from constant-K.

- **Attenuation pole**: $$\displaystyle \omega_\infty = \frac{\omega_c}{m} $$.

• Composite Filters: Combine constant-K and m-derived sections to get good stopband and impedance stability.

Chebyshev Approximation:

• Equiripple in Passband, monotonic in stopband.

• Chebyshev Polynomial: $$\displaystyle T_n(x) = \cos(n\cos^{-1}x) $$.

• LPF Design:

$$\epsilon = \sqrt{10^{A_p/10} - 1} \quad (A_p = \text{passband ripple in dB})$$

$$|H(j\Omega)|^2 = \frac{1}{1 + \epsilon^2 T_n^2(\Omega)}$$

• Element values from tables or synthesis.

• Advantage: Faster roll-off than Butterworth for same order, at cost of passband ripple.

4.0 Network Synthesis

Fundamental Concepts:

• Positive Real Function (PRF): Necessary for passive network realization.

  1. $F(s)$ real for real $s$.

  2. $\text{Re}[F(s)] \geq 0$ for $\text{Re}(s) \geq 0$.

  3. Poles in LHP or on $j\omega$ axis (simple).

  4. No poles at $$\displaystyle s=\infty $$ unless removable.

• Hurwitz Polynomial:

  1. Coefficients positive.

  2. All roots in LHP (no RHP roots).

  3. Continued fraction expansion all positive.

• Minimum PRF: No poles at $$\displaystyle s=\infty $$ or $$\displaystyle s=0 $$ (except simple poles on $j\omega$).

Foster Synthesis:

• Foster I: Partial fractions of $F(s)$ → parallel LC branches.

$$F(s) = K_\infty + \sum \frac{K_i}{s^2 + \omega_i^2} \quad \text{(even)} \quad \text{or} \quad \sum \frac{sK_i}{s^2 + \omega_i^2} \quad \text{(odd)}$$

• Foster II: Partial fractions of $1/F(s)$ → series LC branches.

• Example: $$\displaystyle F(s) = \frac{(s^2+1)(s^2+6)}{s^2(s+3)} $$

  • Decompose → Foster I form (parallel combination of series LC and capacitor).

Cauer Synthesis:

• Cauer I: Continued fraction expansion of $F(s)$ → ladder (series-parallel) network.

• Cauer II: Continued fraction of $1/F(s)$ → ladder starting with shunt element.

• Procedure: Polynomial division, remove poles at infinity or finite poles.

• Example: $$\displaystyle F(s) = \frac{s^3 + 2s}{s^2 + 3s + 2} $$ → Cauer I gives series L, shunt C, series L.

Brune's Method:

1. Remove pole at infinity (series inductor or shunt capacitor).

2. Remove finite pole (series or shunt branch).

3. Remainder is PRF → repeat.

4. Brune's Synthesis Coefficient: For removing a pole at $$\displaystyle s=p $$, add impedance $$\displaystyle Z = \frac{K}{s-p} $$ (series) or admittance $$\displaystyle Y = \frac{K}{s-p} $$ (shunt), where $K$ chosen so remainder is PRF.

Bott-Duffin Method:

• For non-realizable PRFs (with zeros on $j\omega$ axis).

• Uses extra element (ideal transformer or gyrator) to realize.

• Principle: $$\displaystyle F(s) = \frac{a s^2 + b}{c s^2 + d} $$ can be realized with transformer.

5.0 Impedance Matching & Attenuators

Impedance Matching Fundamentals:

• Need: Maximum power transfer ($$\displaystyle Z_L = Z_G^* $$), minimize reflections ($$\displaystyle Z_L = Z_0 $$).

• Matching Network: Lossless (L, C) or resistive (attenuator) network between source and load.

Stub Matching:

• Single-Stub:

  • Shunt Stub: Place at distance $\ell$ from load where $$\displaystyle Y_{in} $$ is $1 \pm jB$, then add shunt susceptance $jB$ (open/short stub).

  • Series Stub: Place at distance $\ell$ where $$\displaystyle Z_{in} $$ is $R \pm jX$, then add series reactance $jX$.

  • Smith Chart Procedure: Normalize $$\displaystyle z_L $$, move towards generator on constant $|z|$ circle to $$\displaystyle r=1 $$ (or $$\displaystyle g=1 $$), read $\ell$, compute stub length.

• Double-Stub Matching:

  • Two stubs fixed at distances $d$ (usually $\lambda/8$ or $\lambda/4$) apart.

  • First stub at load, second at distance $d$.

  • Advantage: No need to move stubs; wider bandwidth; fixed locations convenient.

  • Limitation: Some loads cannot be matched (forbidden region on Smith Chart).

Quarter-Wave Transformer:

• Single $\lambda/4$ line with $$\displaystyle Z_{T} = \sqrt{Z_0 Z_L} $$.

• Bandwidth limitation: Good only near $\lambda/4$ frequency.

• Multi-section: Improves bandwidth.

Attenuators:

• Purpose: Reduce power level, provide matching.

• Symmetrical T-Attenuator ($$\displaystyle R_1 $$, $$\displaystyle R_2 $$, $$\displaystyle R_1 $$):

$$R_1 = R_0 \frac{K-1}{K+1}, \quad R_2 = R_0 \frac{2K}{K^2-1}$$

where $$\displaystyle K = 10^{IL/20} $$ (voltage attenuation factor), $$\displaystyle R_0 $$ = image impedance.

• Symmetrical π-Attenuator ($$\displaystyle R_1 $$, $$\displaystyle R_2 $$, $$\displaystyle R_1 $$):

$$R_1 = R_0 \frac{K^2-1}{2K}, \quad R_2 = R_0 \frac{K+1}{K-1}$$

• Asymmetrical T-Attenuator (for $$\displaystyle R_{01} \neq R_{02} $$):

  Solve using T-to-π conversion or direct equations from Z-params.

6.0 Smith Chart & Special Transmission Lines

Smith Chart Construction:

• Plot of normalized impedance $$\displaystyle z = Z/Z_0 $$ or normalized admittance $$\displaystyle y = Y/Y_0 $$.

• Circles of constant $|z|$ (resistance), constant $r$ (conductance), constant $x$ (reactance), constant $b$ (susceptance).

• Outer circle: $$\displaystyle |Γ|=1 $$, center: $$\displaystyle Γ=0 $$.

Applications:

1. Finding $$\displaystyle Z_L $$, $$\displaystyle Y_L $$, Γ, VSWR from given point.

2. Transmission line problems: Move along line (clockwise for lossless) by electrical length $$\displaystyle \theta = \beta\ell $$.

3. Stub matching: As described above.

4. Impedance matching design.

Microstrip Lines:

• Structure: Conductor strip on dielectric substrate with ground plane.

• Effective Dielectric Constant:

$$\epsilon_{eff} = \frac{\epsilon_r + 1}{2} + \frac{\epsilon_r - 1}{2}\frac{1}{\sqrt{1 + 12\frac{h}{w}}}$$

for $w/h \leq 1$.

• Characteristic Impedance:

  For $w/h \leq 1$: $$\displaystyle Z_0 = \frac{60}{\sqrt{\epsilon_{eff}}} \ln\left(\frac{8h}{w} + \frac{w}{4h}\right) $$

  For $w/h \geq 1$: $$\displaystyle Z_0 = \frac{120\pi}{\sqrt{\epsilon_{eff}} \left[ \frac{w}{h} + 1.393 + 0.667\ln\left(\frac{w}{h} + 1.444\right) \right]} $$

• Advantages: Easy fabrication, integrate with active devices, used in RF/microwave circuits (up to ~100 GHz).

PART B: MOBILE COMMUNICATION SYSTEMS

1.0 Cellular System Fundamentals

Basic Cellular Concept:

• Cell: Coverage area with base station.

• Cluster: $N$ cells using all available channels once.

• Frequency Reuse: Same channels reused in non-adjacent cells to increase capacity.

• Base Station (BS): Fixed transceiver in cell.

• Mobile Switching Center (MSC): Connects calls, manages handoffs.

Frequency Reuse:

• Reuse Factor $$\displaystyle N = i^2 + ij + j^2 $$ (for hexagonal cells, $i,j$ integers).

• Reuse Distance $D$:

$$D = R \sqrt{3N}$$

where $R$ = cell radius.

• Co-Channel Cells: Cells using same frequency set. Located at $(m\Delta i + n\Delta j)$ offsets.

[!TIP] Diagram: Draw 19-cell cluster ($$\displaystyle N=19 $$, $$\displaystyle i=3,j=1 $$ or $$\displaystyle i=3,j=2 $$ etc.), show co-channel cells at distance $D$.

Capacity:

• Total channels $$\displaystyle S = \text{total spectrum} / \text{channel bandwidth} $$.

• Channels per cell $$\displaystyle S_c = S/N $$.

• Capacity = $$\displaystyle S_c \times \text{number of cells} $$.

• Trunking & Grade of Service (GoS):

  • Erlang B Formula (blocked calls cleared):

$$B(E, M) = \frac{\frac{E^M}{M!}}{\sum_{k=0}^{M} \frac{E^k}{k!}}$$

    where $E$ = offered traffic (Erlangs), $M$ = number of channels.

- GoS = probability of call blocking.

Capacity Expansion:

1. Cell Splitting: Divide congested cells into smaller cells (reduce $R$). Requires new BS, frequency plan adjustment.

2. Cell Sectoring: Replace omnidirectional antenna with directional (120° or 60°). Reduces $D/R$ ratio, increases $N$ effectively.

3. Overlay/Underlay: Use different cell sizes with same frequencies (microcells under macrocells).

4. Microcells/Picocells: Very small cells for high density (indoor, urban).

2.0 Channel Assignment & Handoff

Channel Assignment:

• Fixed Channel Assignment (FCA): Channels permanently allocated to cells. Simple, but inefficient under varying traffic.

• Dynamic Channel Assignment (DCA): Channels allocated on demand from pool. Complex, but lower blocking, better utilization.

  • Algorithms: Borrow from neighbor, locate free channel in cluster.

  • Disadvantage: High signaling overhead.

Handoff (Handover):

• Necessity: Maintain call when mobile moves between cells.

• Performance: Seamless, no dropped calls.

• Types:

  1. Hard Handoff (Break-before-make): GSM, CDMA. Release old channel before acquiring new.

  2. Soft Handoff (Make-before-break): CDMA only. Mobile communicates with multiple BSs simultaneously.

  3. Intra-cell: Within same cell (rare, due to interference).

  4. Inter-cell: Between cells (common).

  5. Inter-MSC: Between MSCs (more complex).

• Handoff Decision Parameters:

  • RSSI (Received Signal Strength Indicator)

  • Signal quality (BER)

  • Distance from BS

  • Timing advance (GSM)

• Mobile-Assisted Handoff (MAHO): Mobile measures neighbor BS signals, reports to MSC. Reduces network processing.

• Queuing in Handoff: Prioritize handoff requests over new call requests to reduce dropped calls.

3.0 Radio Wave Propagation Models

Large-Scale Path Loss:

• Free Space:

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

or $$\displaystyle PL(d) = 32.45 + 20\log_{10}(f_{MHz}) + 20\log_{10}(d_{km}) $$ (dB).

• Two-Ray (Ground Reflection):

$$P_r \propto \frac{1}{d^4} \quad \text{(far field)}$$

**Critical Distance** $$\displaystyle d_c = \frac{4\pi h_t h_r}{\lambda} $$ (where $$\displaystyle h_t $$, $$\displaystyle h_r $$ are antenna heights). Before $$\displaystyle d_c $$, fading due to constructive/destructive interference.

• Log-Distance:

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

$n$ = path loss exponent (2=free space, 4=urban), $$\displaystyle X_\sigma $$ = log-normal shadowing (dB).

Small-Scale Multipath:

• Causes: Reflection, diffraction, scattering.

• Parameters:

  • Multipath Delay Spread $$\displaystyle \tau_m $$: Max excess delay.

  • Coherence Bandwidth $$\displaystyle B_c \approx \frac{1}{5\tau_m} $$ (for correlated fading). Flat fading if $$\displaystyle B_s << B_c $$.

  • Coherence Time $$\displaystyle T_c \approx \frac{1}{f_m} $$ ($$\displaystyle f_m $$ = max Doppler shift). Slow fading if $$\displaystyle T_s >> T_c $$.

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

  • Doppler Spread $$\displaystyle B_D \approx 2f_m $$ (for isotropic scattering).

• Fading Types:

  • Flat Fading: $$\displaystyle B_s << B_c $$ (narrowband). All frequencies fade equally.

  • Frequency-Selective Fading: $$\displaystyle B_s > B_c $$ (wideband). Different fades at different frequencies.

  • Fast Fading: $$\displaystyle T_s < T_c $$ (rapid channel change).

  • Slow Fading: $$\displaystyle T_s > T_c $$ (channel constant over symbol).

Specific Scenarios:

• Foliage Losses: Additional attenuation through trees (up to 20 dB at 1 GHz). Empirical: $$\displaystyle L_{foliage} \propto f^{0.5} $$ or $$\displaystyle f^{0.6} $$.

• Near-In-Distance: Close-in reference distance model:

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

where $$\displaystyle d_0 $$ is close-in distance (1-10 m).

• Mobile-to-Mobile: No fixed BS; both ends move. Path loss exponent similar but with different shadowing statistics.

4.0 Fading Models & Statistics

Clarke's Model (Rayleigh Fading):

• Assumptions: Dense multipath, no LOS component.

• Envelope Distribution:

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

$$\displaystyle \sigma^2 $$ = average power.

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

• Level Crossing Rate (LCR):

$$N_R = \sqrt{2\pi f_m} e^{-R^2/(2\sigma^2)}$$

• Average Fade Duration (AFD):

$$\tau_{avg} = \frac{1 - e^{-R^2/(2\sigma^2)}}{\sqrt{2\pi f_m} R/\sigma}$$

Rician Fading:

• Presence of LOS component with power $K$ times scattered power.

• Envelope PDF:

$$p(r) = \frac{r}{\sigma^2} e^{-(r^2 + A^2)/(2\sigma^2)} I_0\left(\frac{rA}{\sigma^2}\right)$$

$A$ = LOS amplitude, $$\displaystyle I_0 $$ = modified Bessel function.

• Rician K-factor: $$\displaystyle K = A^2/(2\sigma^2) $$ (dB). $$\displaystyle K=0 $$ → Rayleigh, $$\displaystyle K=\infty $$ → AWGN.

Nakagami-m Distribution:

• Generalized: $$\displaystyle p(r) = \frac{2m^m}{\Gamma(m)\Omega^m} r^{2m-1} e^{-m r^2/\Omega} $$

  $m$ = fading figure ($$\displaystyle m=1 $$ → Rayleigh, $$\displaystyle m>1 $$ → less fading).

• Fading Statistics: LCR, AFD expressions available.

5.0 Multiple Access Techniques

FDMA:

• Principle: Each user gets dedicated frequency band.

• Channel Bandwidth: $$\displaystyle B_c = B_T/N - B_{guard} $$.

  Given $$\displaystyle B_T $$, $$\displaystyle B_{guard} $$, $$\displaystyle B_c $$ → $$\displaystyle N = \frac{B_T}{B_c + B_{guard}} $$ (integer).

• Guard Bands: Prevent adjacent channel interference.

• Adv: Simple, low latency.

• Disadv: Fixed allocation, inefficient for bursty traffic.

• Example: Analog AMPS.

TDMA:

• Principle: Users share frequency in time slots.

• Frame Structure:

  • Time Slot: Basic unit.

  • Frame: Collection of slots (e.g., GSM: 8 slots/frame).

  • Multiframe: 26 or 51 frames for traffic/control.

• Adv: Higher capacity than FDMA, flexible.

• Disadv: Synchronization required, guard times.

• Example: GSM, IS-136.

CDMA:

• Principle: Spread spectrum; all users share same frequency/time, separated by unique PN codes.

• DSSS: Data multiplied by high-rate chip sequence ($C$).

$$Spread\ Signal = d(t) \cdot C(t)$$

• Channels:

  • Forward (BS→MS): Walsh codes (orthogonal) for channels, PN sequence for privacy.

  • Reverse (MS→BS): Unique PN sequences (not orthogonal).

• Processing Gain (PG):

$$PG = \frac{\text{Chip Rate}}{\text{Data Rate}}$$

Example: IS-95: Chip rate = 1.2288 Mcps, Data rate = 9.6/13 kbps → $PG \approx 128$.

• Near-Far Problem: Strong signal masks weak signal. Solved by power control.

  • Open Loop: MS adjusts based on received forward signal.

  • Closed Loop: BS commands MS to adjust power.

• Capacity: Soft capacity (quality degrades gracefully with more users).

• Adv: Security, interference limited, soft handoff.

• Disadv: Complex power control, self-interference.

Spread Spectrum Multiple Access:

• FHSS: Frequency hops according to PN sequence.

  • Slow Hopping: Several symbols per hop.

  • Fast Hopping: Multiple hops per symbol.

Advanced Access:

• OFDM:

  • Principle: High-rate data stream split into many parallel low-rate subcarriers (orthogonal).

  • Cyclic Prefix: Copy end of symbol to front to combat ISI (length > delay spread).

  • Adv: Robust to multipath, efficient FFT implementation.

  • Used in: 4G/5G, Wi-Fi.

• MIMO:

  • Spatial Multiplexing: Multiple data streams on same frequency via multiple antennas → capacity increase.

  • Diversity: Multiple antennas combat fading (selection, combining).

  • Beamforming: Directional transmission/reception.

6.0 GSM Architecture

Network Subsystems:

1. Mobile Station (MS): Mobile Equipment (ME) + SIM.

2. Base Station Subsystem (BSS):

   • BTS (Base Transceiver Station): Radio hardware.

   • BSC (Base Station Controller): Manages BTSs, handoffs, frequency hopping.

3. Network Switching Subsystem (NSS):

   • MSC (Mobile Switching Center): Call switching, mobility management.

   • HLR (Home Location Register): Permanent MS data.

   • VLR (Visitor Location Register): Temporary MS data in visited area.

   • AUC (Authentication Center): Security.

   • EIR (Equipment Identity Register): IMEI tracking.

4. OSS (Operation & Support Subsystem): Maintenance.

GSM Radio Subsystem:

• Traffic Channels (TCH):

  • TCH/F (Full Rate, 22.8 kbps)

  • TCH/H (Half Rate, 11.4 kbps)

• Control Channels (CCH):

  • BCCH (Broadcast Control): BS→MS, system info.

  • CCCH (Common Control): PCH (paging), AGCH (access grant), RACH (random access).

  • DCCH (Dedicated Control): SDCCH (standalone), SACCH (slow associated), FACCH (fast associated, steals TCH slot).

GSM Frame Structure:

• Timeslot: 156.25 bits, 577 µs.

• Frame: 8 timeslots, 4.615 ms.

• Multiframe:

  • Traffic: 26 frames (120 ms) → 26 TDMA frames.

  • Control: 51 frames (235 ms) → 51 TDMA frames.

• Burst Types:

  • Normal Burst: 148 bits (data) + 3 tail + 8.25 guard.

  • Frequency Correction Burst: FCCH.

  • Synchronization Burst: SCH.

  • Access Burst: RACH (short, for initial access).

Call Processing & Handoff:

• Mobile Originated: MS → BTS (RACH) → BSC (AGCH) → SDCCH → TCH.

• Handoff: MSC decides based on measurements (MAHO). New channel assigned, old released (hard handoff).

7.0 Antennas & Interference Management

Antennas:

• Cell Site:

  • Omnidirectional: 360° coverage, used in rural.

  • Sectoral: 120° or 60°, reduces interference, increases capacity.

  • Height: Trade-off between coverage and interference.

• Mobile: Monopole (λ/4), diversity (two antennas for space diversity).

Co-Channel Interference (CCI):

• Cause: Frequency reuse ($N$ small → $D/R$ small → same frequency in nearby cells).

• SIR Calculation:

$$SIR = \frac{S}{\sum_{i=0}^{i_0} I_i}$$

where $$\displaystyle I_i $$ = interference from $i$-th co-channel cell, $$\displaystyle i_0 $$ = number of co-channels.

Assuming path loss exponent $n$, and hexagonal grid:

$$SIR \approx \frac{1}{2\left(\frac{D}{R}\right)^n} \quad \text{(for first tier only)}$$

• Reduction:

  • Increase $D/R$ (increase $N$ → reduces capacity).

  • Power control.

  • Antenna tilting (down-tilt reduces interference to distant cells).

  • Diversity (reduces fade depth).

Adjacent Channel Interference (ACI):

• Caused by imperfect receiver filters; nearby channel leaks into desired channel.

• Reduced by guard bands and good filtering.

Frequency Management:

• Assign frequencies to minimize CCI/ACI.

• Use channel borrowing (DCA).

• Frequency hopping (GSM) to average interference.

END OF UNIT 5 NOTES