UNIT 3: OPTICAL FIBRE COMMUNICATION - COMPREHENSIVE NOTES

Based on RGPV past papers (2022-2025), these notes cover all tested topics for Unit 3.

1. FIBER OPTICS FUNDAMENTALS & WAVE GUIDANCE

Normalized Frequency (V-number)

• Definition: A dimensionless parameter that determines the number of modes supported by a step-index fiber.

• Expression:

$$V = \frac{2\pi a}{\lambda} \cdot NA$$

where \(a\) = core radius, \(\lambda\) = wavelength, \(NA\) = Numerical Aperture.

• Significance:

  • \(V < 2.405\): Single-mode operation (only fundamental mode propagates).

  • \(V > 2.405\): Multimode operation.

  • For large \(V\), total number of guided modes \(M \approx \frac{V^2}{2}\).

• Cut-off Condition: The fundamental mode has no cut-off. The first higher-order mode (LP₁₁) cuts off at \(V = 2.405\).

[!TIP] Common Pitfall: Remember \(V\) is proportional to \(a/\lambda\). Smaller core or longer wavelength favors single-mode operation.

Numerical Aperture (NA) & Acceptance Angle

• Definition: NA measures the light-gathering ability of a fiber. It is the sine of the maximum acceptance angle \(\theta_{0,\text{max}}\) from air into the fiber core.

• Derivation (Snell's Law):

  1. At air-core interface: \(n_0 \sin\theta_0 = n_1 \sin\theta_1\)

  2. At core-cladding interface for critical angle \(\theta_c\): \(n_1 \sin\theta_c = n_2\)

  3. For guided ray, \(\theta_1 + \theta_c = 90^\circ\) → \(\sin\theta_1 = \cos\theta_c = \sqrt{1 - \sin^2\theta_c} = \sqrt{1 - (n_2/n_1)^2}\)

  4. Therefore, \(NA = n_0 \sin\theta_{0,\text{max}} = n_1 \sqrt{1 - (n_2/n_1)^2} = \sqrt{n_1^2 - n_2^2}\)

• Small Δ Approximation: For \(\Delta = (n_1 - n_2)/n_1 \ll 1\), \(NA \approx n_1 \sqrt{2\Delta}\).

• Key Relations:

  • \(NA = n_0 \sin\theta_{0,\text{max}}\)

  • Critical angle at core-cladding: \(\theta_c = \sin^{-1}(n_2/n_1)\)

\boxed{NA = \sqrt{n_1^2 - n_2^2} \approx n_1\sqrt{2\Delta}}

[!TIP] Exam Focus: This is a recurring 7m/14m derivation. Be prepared to derive NA from first principles using Snell's law at both interfaces. Numerical problems often combine NA, acceptance angle, and critical angle.

Modes of Propagation

• Planar Waveguide (Slab Guide):

  • Guiding occurs via total internal reflection between two parallel interfaces.

  • Guided Modes: Discrete set of angles \(\theta_m\) satisfying \(2k_1 d \cos\theta_m = 2m\pi\) (m = 0,1,2...), where \(d\) = slab thickness, \(k_1\) = wavevector in core.

  • Cut-off: Each mode has a minimum \(V\)-value (or \(d/\lambda\)) below which it becomes leaky.

  • Mode Field: Electric field distribution varies sinusoidally across the core and decays exponentially in the cladding.

• Step-Index Fiber:

  • Ray Theory:

    • Meridional Rays: Pass through fiber axis. Define maximum number of reflections.

    • Skew Rays: Do not pass through axis; follow helical path. More modes than meridional rays for same \(V\).

  • Wave Theory (LP Modes): Solutions to wave equation in cylindrical coordinates. Labeled as LP\(_{lm}\) (linearly polarized). Each LP mode has a specific cut-off \(V\)-value (e.g., LP₀₁: no cut-off, LP₁₁: 2.405).

  • Number of Modes: \(M \approx \frac{V^2}{2}\) for large \(V\).

• Graded-Index Fiber:

  • Ray Transmission: Core refractive index decreases gradually from center (\(n_1\)) to cladding (\(n_2\)). Rays follow curved paths (continuous refraction) → reduces intermodal dispersion.

  • Advantages: Higher bandwidth than step-index multimode fiber.

Dispersion

• Intermodal Dispersion (Modal Dispersion):

  • Cause in Step-Index: Different modes travel different optical path lengths → pulse broadening.

  • RMS Pulse Broadening:

$$\Delta t_{\text{modal}} \approx \frac{n_1 \Delta}{c} \cdot L$$

where \(L\) = fiber length, \(\Delta = (n_1 - n_2)/n_1\).

*   **Reduction:** Use graded-index profile (ideal \(\alpha=2\) profile minimizes dispersion).

• Intramodal Dispersion (Chromatic Dispersion):

  • Material Dispersion: \(n(\lambda)\) varies with wavelength → different colors travel at different group velocities. Zero-dispersion wavelength \(\lambda_0 \approx 1.3\ \mu m\) for silica.

  • Waveguide Dispersion: Due to wavelength dependence of mode confinement. Shifts zero-dispersion wavelength.

  • Impact: Limits bandwidth in single-mode fibers, especially at 1.55 μm where attenuation is lowest but material dispersion is high.

• System Bandwidth: Overall rms pulse broadening: \(\Delta t_{\text{total}} = \sqrt{\Delta t_{\text{modal}}^2 + \Delta t_{\text{material}}^2 + \Delta t_{\text{waveguide}}^2 + \Delta t_{\text{receiver}}^2 + ...}\)

  • Maximum bit rate (NRZ): \(B_{\text{max}} \approx \frac{0.44}{\Delta t_{\text{total}}}\)

[!TIP] Common Pitfall: Intermodal dispersion is only in multimode fibers. Intramodal (chromatic) dispersion affects all fibers but is dominant in single-mode.

2. ATTENUATION & LOSS MECHANISMS

Attenuation (Loss)

• Definition: Power reduction per unit length, expressed in dB/km: \(\alpha = \frac{10}{L} \log_{10} \frac{P_{\text{in}}}{P_{\text{out}}}\).

• Impact: Directly determines maximum link distance without amplification/repeaters.

Absorption Losses

[!TIP] Exam Focus: OH⁻ absorption peak at 1383 nm is a classic exam question. Modern fibers have <0.1 ppm OH⁻.

Scattering Losses

• Rayleigh Scattering:

  • Cause: Microscopic density/composition fluctuations frozen into the glass during manufacture (<< λ).

  • Dependence: \(\alpha_R \propto \lambda^{-4}\). Fundamental loss limit in silica (~0.15 dB/km at 1550 nm).

• Mie Scattering:

  • Cause: Large inhomogeneities (e.g., imperfections, bubbles) with size ~λ.

  • Mitigation: Improved fabrication quality; not significant in modern fibers.

Bending Losses

• Macrobending:

  • Cause: Fiber curvature causes some guided rays to exceed critical angle at outer bend → radiation loss.

  • Critical Radius of Curvature \(R_c\): Approximate condition for significant loss: \(R_c \approx \frac{3\lambda}{4\pi n_1 \sqrt{2\Delta}}\) for step-index fiber.

  • Reduction: Larger \(R_c\), smaller \(\Delta\), longer \(\lambda\).

• Microbending:

  • Cause: Small-scale (<1 cm) distortions due to lateral pressure, cabling stresses.

  • Mitigation: Proper cable design, protective coatings.

[!TIP] Recurring Problem: Given \(n_1, \Delta, \lambda\), estimate \(R_c\). Use the formula above. Remember \(R_c \propto 1/\Delta\) and \(1/\lambda\).

3. FIBER FABRICATION & CHARACTERIZATION

Fiber Fabrication Techniques

• Modified Chemical Vapour Deposition (MCVD):

  1. Process: Rotating silica tube (substrate) heated externally. Gases (SiCl₄, GeCl₄, etc.) flow inside. Oxidation forms soot deposit on inner wall.

  2. Soot layer sintered to form dense glass.

  3. Tube collapsed to form solid preform rod.

  4. Preform drawn into fiber in drawing tower.

  • Advantages: Excellent control of refractive index profile, low loss, high purity.

• Vapour Axial Deposition (VAD):

  1. Process: Porous soot preform built axially by depositing reacting gases onto a rotating seed rod from below.

  2. Sintered as it grows.

  3. No collapse step needed.

  • Advantages: Higher production rate, larger preform size, no tube distortion.

  • Comparison vs. MCVD: VAD is faster but MCVD offers finer index profile control.

• Outside Vapour Deposition (OVD): Soot deposited externally on a rod, then sintered. Similar advantages to VAD.

Fiber Measurement & Testing

• Optical Time-Domain Reflectometer (OTDR):

  • Principle: Injects short optical pulse, measures backscattered (Rayleigh) and reflected light intensity vs. time/distance.

  • Trace Analysis:

    • Slope: Fiber attenuation (dB/km).

    • Events: Sharp drops (splices/connectors - loss), spikes (reflections - e.g., open connector), breaks (large drop + reflection).

    • Distance: Calculated from pulse travel time: \(L = \frac{c \cdot \Delta t}{2n_1}\).

  • Limitations: Dead zones after high-reflection events, limited dynamic range.

• Cut-back Method:

  • Principle: Measure output power \(P_{\text{out,long}}\) for long fiber length \(L\). Then cut fiber to short length (~2 m) and measure \(P_{\text{out,short}}\). Loss \(\alpha = \frac{10}{L} \log_{10} \frac{P_{\text{out,short}}}{P_{\text{out,long}}}\).

  • Use: Laboratory standard for accurate attenuation measurement.

4. OPTICAL SOURCES & DETECTORS

Light Emitting Diode (LED)

• Structure: P-n junction (homojunction or double heterostructure - DH). Surface-emitting (SLED) or edge-emitting (ELED).

• Principle: Spontaneous emission. Electron-hole recombination emits incoherent, wide-spectrum light.

• Characteristics:

  • Output Power: Moderate (~mW).

  • Spectral Width: Wide (30-100 nm) → high chromatic dispersion.

  • Modulation Bandwidth: Limited (~100 MHz) by carrier lifetime.

  • Efficiency: Low (~10-20%).

• Internal Optical Power Generation (Derivation):

  • Radiative recombination rate \(R_{\text{rad}} = B n p\) (B = coefficient).

  • Under forward bias, \(n \approx N_D\), \(p \approx N_A\) (heavily doped).

  • Internal power: \(P_{\text{int}} = \eta_i \cdot \frac{I}{q} \cdot E_{\text{photon}} = \eta_i \cdot \frac{I}{q} \cdot \frac{hc}{\lambda}\)

  • where \(\eta_i\) = internal quantum efficiency, \(I\) = current.

LASER Diodes (LD)

• Basic Principle: Stimulated emission. Requires population inversion and optical feedback (resonant cavity).

• Laser Diode Structures:

  • Homojunction: Simple, high threshold current.

  • Heterojunction (DH): Confinement of carriers and optical field → lower threshold, higher efficiency.

  • Quantum Well (QW): Further confinement → discrete energy states, lower threshold, temperature stability.

• Characteristics vs. LED:

  • Coherent: Narrow spectral width (<0.1 nm), high directionality.

  • Higher Power: ~10-100 mW.

  • Higher Modulation Speed: >10 GHz.

• Resonant Frequency & Mode Spacing:

  • Cavity length \(L\), effective index \(n\). Allowed frequencies: \(\nu_q = \frac{q c}{2nL}\) (q = integer).

  • Frequency Spacing: \(\Delta \nu = \nu_{q+1} - \nu_q = \frac{c}{2nL}\)

  • Wavelength Spacing: \(\Delta \lambda = \frac{\lambda^2}{2nL}\)

• Modulation: Direct modulation (vary injection current). Fast, simple.

[!TIP] Key Difference: LED = incoherent, wide spectrum, slow, cheap. LD = coherent, narrow spectrum, fast, expensive, requires temperature control.

Photodetectors

• PIN Photodiode:

  • Structure: P⁺-I-N⁻. Intrinsic (I) layer is wide, undoped.

  • Working (Photoconductive Mode): Reverse biased. Absorbed photons generate electron-hole pairs in I-layer. High electric field sweeps carriers → photocurrent.

  • I-V Characteristic: Dark current (small, reverse), photocurrent (proportional to light) under reverse bias.

  • Responsivity (R): \(R = \frac{I_{\text{ph}}}{P_{\text{opt}}} = \frac{\eta q}{h\nu} = \frac{\eta \lambda}{1.24}\) (A/W, with \(\lambda\) in μm).

  • Quantum Efficiency (η): \(\eta = \frac{\text{electrons collected}}{\text{photons incident}} = \frac{I_{\text{ph}} q}{P_{\text{opt}} h \nu}\).

  • Bandwidth: Limited by carrier transit time and RC time constant.

• Avalanche Photodiode (APD):

  • Structure: P⁺-I-N⁺-P⁺ (or similar). High reverse bias near breakdown.

  • Principle: Impact ionization. Primary carriers (from photon absorption) gain enough energy to create secondary electron-hole pairs → internal multiplication.

  • Multiplication Factor (M): \(I_{\text{ph,APD}} = M \cdot I_{\text{ph,PIN}}\). \(M\) depends on bias voltage and material (ionization coefficients).

  • Excess Noise Factor (F): Statistical variation in multiplication process. \(F(M) \geq 1\), increases with \(M\).

  • Comparison with PIN:

    | Parameter | PIN | APD | | :--- | :--- | :--- | | Gain | 1 (no gain) | \(M > 1\) (10-1000) | | Noise | Lower (shot noise only) | Higher (excess noise) | | Speed | Faster (no multiplication delay) | Slower (avalanche buildup) | | Bias | Moderate (~50V) | High (~100-400V) | | Cost | Lower | Higher | | Use | Short-reach, low-cost | Long-reach, low-light |

[!TIP] Exam Derivation: Quantum efficiency in terms of carrier lifetime \(\tau\) for a PIN with I-layer width \(W\): \(\eta = (1 - e^{-\alpha W}) \cdot (1 - \frac{\tau_{\text{tr}}}{\tau_{\text{rec}}})\) where \(\alpha\) = absorption coefficient, \(\tau_{\text{tr}}\) = transit time, \(\tau_{\text{rec}}\) = recombination lifetime.

5. OPTICAL LINK DESIGN & ANALYSIS

Link Power Budget (Very High Priority - 14m)

Design Steps:

1. Transmitter Power (\(P_T\)): Specify output power (dBm).

2. Receiver Sensitivity (\(P_{R,\text{min}}\)): Minimum average power for required BER (e.g., -28 dBm for 10⁻⁹).

3. Total Loss (\(L_{\text{total}}\)):

   • Fiber attenuation: \(\alpha \cdot L\) (dB)

   • Connector loss: \(N_c \cdot \alpha_c\) (dB, \(\alpha_c \approx 0.3-0.5\) dB/connector)

   • Splice loss: \(N_s \cdot \alpha_s\) (dB, \(\alpha_s \approx 0.1\) dB/splice)

   • Safety Margin: 3-6 dB (for aging, unforeseen losses).

4. Power Margin: \(P_M = P_T - (P_{R,\text{min}} + L_{\text{total}})\)

   • Requirement: \(P_M > 0\) (typically 3-6 dB).

5. Maximum Link Length: \(L_{\text{max}} = \frac{P_T - P_{R,\text{min}} - (N_c\alpha_c + N_s\alpha_s + \text{margin})}{\alpha}\)

Solved Numerical Example (Template):

Given: \(P_T = -3\) dBm, \(P_{R,\text{min}} = -28\) dBm, \(\alpha = 0.2\) dB/km, 4 connectors (0.5 dB each), 2 splices (0.1 dB each), margin = 5 dB.

\[ > L_{\text{max}} = \frac{(-3) - (-28) - (4\times0.5 + 2\times0.1 + 5)}{0.2} = \frac{25 - (2 + 0.2 + 5)}{0.2} = \frac{17.8}{0.2} = 89\ \text{km} > \]

System Considerations

• Point-to-Point Links: Simple transmitter-fiber-receiver configuration.

• Power Penalties: Additional loss factors reducing link margin:

  • Modal Noise: From coherent sources in multimode fiber.

  • Dispersion Penalty: Due to pulse broadening.

  • Reflection Penalty: From connector reflections (especially with coherent sources).

  • Intensity Noise: Source power fluctuations.

• Eye Pattern:

  • Use: Visual tool for digital transmission quality.

  • Interpretation:

    • Eye Opening: Vertical height = noise margin; horizontal width = timing jitter margin.

    • Jitter: Horizontal eye closure → timing uncertainty.

    • Noise: Vertical eye closure → signal distortion.

    • Intersymbol Interference (ISI): Eye partially closed due to dispersion.

6. OPTICAL AMPLIFICATION

Erbium-Doped Fiber Amplifier (EDFA)

• Principle: Er³⁺ ions doped in silica fiber core. Pumped at 980 nm or 1480 nm to create population inversion. Signal at 1550 nm stimulates emission → amplification.

• Configuration:

  • Co-propagating: Pump and signal same direction.

  • Counter-propagating: Pump opposite to signal (reduces ASE noise at receiver).

  • Bidirectional: Combination.

• Gain & Noise:

  • Gain Spectrum: ~30 nm around 1550 nm (C-band). Flattened using gain-flattening filters.

  • Amplified Spontaneous Emission (ASE): Major noise source. Power spectral density: \(S_{\text{ASE}} = n_{\text{sp}} h \nu (G - 1)\) where \(n_{\text{sp}}\) = spontaneous emission factor.

• Derivation - Gain & Conversion Efficiency:

  • Small-signal gain: \(G = \exp[(\sigma_e N_2 - \sigma_a N_1) \cdot L_{\text{eff}}]\)

    where \(\sigma_e, \sigma_a\) = emission/absorption cross-sections, \(N_1,N_2\) = population densities, \(L_{\text{eff}}\) = effective length.

  • Conversion Efficiency (\(\eta\)): \(\eta = \frac{P_{\text{signal,out}} - P_{\text{signal,in}}}{P_{\text{pump,in}}} \approx \frac{\lambda_s}{\lambda_p} \cdot \frac{G - 1}{G}\) for high pump.

Raman Amplifier

• Principle: Based on Stimulated Raman Scattering (SRS). High-power pump photon (\(\lambda_p\)) scatters off silica molecule, creating signal photon (\(\lambda_s > \lambda_p\)) and optical phonon.

• Types:

  • Discrete: Separate Raman pump laser module.

  • Distributed Raman Amplification (DRA): Pump light transmitted with signal in same fiber → distributed gain along span.

• Advantages:

  • Broad gain bandwidth (can amplify any band).

  • Low noise (quantum-limited).

  • Can extend reach, reduce nonlinear effects.

• Challenges: Requires high pump power (>100 mW), pump-signal isolation.

7. OPTICAL NETWORKING & SYSTEMS

SONET / SDH

• Architecture: Hierarchical, synchronous multiplexing.

  • Basic Rate: STS-1 (SONET, 51.84 Mbps) / STM-1 (SDH, 155.52 Mbps).

  • Higher Rates: STS-3c/STM-4 (3×STS-1), STS-12c/STM-16, etc. ("c" = concatenated).

• Frame Structure:

  • Overhead: Section (regenerator sections), Line (line between multiplexers), Path (end-to-end).

  • Payload: Actual user data (e.g., ATM cells, Ethernet frames).

• Working: Synchronous byte-interleaved multiplexing. Key elements:

  • ADM (Add/Drop Multiplexer): Insert/drop lower-rate signals without de-multiplexing entire signal.

  • Regenerator: Re-timers, re-shapers, re-generators (3R).

• Layers:

  • Path Layer: End-to-end transport, path overhead.

  • Line Layer: Line protection (e.g., UPSR, BLSR), line overhead.

  • Section Layer: Physical section (fiber span), section overhead.

Wavelength Division Multiplexing (WDM)

• Principle: Simultaneous transmission of multiple optical carrier wavelengths (channels) on same fiber.

• Components:

  • MUX/DEMUX: Combine/separate wavelengths (AWG, thin-film filters, diffraction gratings).

  • Optical Amplifiers: EDFAs to boost all channels simultaneously.

  • Transponders: O-E-O conversion for wavelength conversion and regeneration.

• Types:

  • CWDM (Coarse WDM): 18 channels, 20 nm spacing, 1270-1610 nm. Lower cost, no amplification.

  • DWDM (Dense WDM): 40-160 channels, 0.4-0.8 nm spacing (100/50 GHz grid). Uses EDFAs, high capacity.

• MEMS Technology: Micro-Electro-Mechanical Systems. Used for tunable filters and optical cross-connects (OXC). Micromirrors tilt to route specific wavelengths.

Passive Optical Networks (PON)

• Architecture: Point-to-multipoint. OLT (Optical Line Terminal) at central office → ODN (Optical Distribution Network, passive splitter) → multiple ONUs (Optical Network Units) at users.

• Types: GPON (2.5 Gbps down / 1.25 Gbps up), EPON (1 Gbps symmetric).

• Key Feature: No active elements in ODN → lower cost, higher reliability. Uses TDMA for uplink.

8. FIBER CONNECTIVITY & JOINING

Fiber Connectors

• Purpose: Temporary, demountable connection between fibers.

• Common Types:

  • FC/PC: Ferrule Connector with Physical Contact (angled polish 8° to reduce back reflection). Screw-on.

  • SC: Subscriber Connector. Push-pull, snap-in. Square ferrule.

  • ST: Straight Tip. Bayonet-lock, round ferrule.

  • LC: Lucent Connector. Small form factor (like RJ-45), latch.

• Performance Parameters:

  • Insertion Loss: Loss due to misalignment, gap, etc. (typical 0.2-0.5 dB).

  • Return Loss (Back Reflection): Ratio of reflected to incident power. FC/APC > 60 dB, others ~30-40 dB.

Fiber Splicing

• Splicing Steps:

  1. Preparation: Strip coating, cleave fiber end (perfect 90° end-face).

  2. Alignment: Align cores axially and laterally (using microscope or V-groove).

  3. Joining: Fusion (arc) or mechanical (gel/adhesive).

  4. Protection: Sleeve (fusion) or enclosure (mechanical).

• Techniques:

  • Fusion Splicing:

    • Principle: Electric arc melts fiber ends, they fuse together.

    • Process: Alignment → arc pre-heat → main arc → pull to smooth.

    • Advantages: Very low loss (<0.1 dB), high strength, permanent.

  • Mechanical Splicing:

    • Principle: Precision alignment sleeve (V-groove) holds fibers in contact. Index-matching gel reduces reflection.

    • Types: V-groove, elastic sleeve, biconic.

    • Advantages: Quick, no power needed, good for emergencies/repairs. Loss higher (~0.3-0.5 dB).

• Comparison:

  | Aspect | Fusion Splicing | Mechanical Splicing | | :--- | :--- | :--- | | Loss | Very low (0.05-0.1 dB) | Higher (0.3-0.5 dB) | | Speed | Slower (minutes) | Fast (seconds) | | Equipment | Expensive splicer | Simple tool kit | | Strength | High (like fiber) | Lower | | Use | Permanent, long-haul | Temporary, repair, multimode |

Optical Coupler Parameters (May 2023)

• Splitting Ratio: Power division between output ports (e.g., 50:50, 90:10).

• Excess Loss: Total power lost in coupler relative to input: \(L_{\text{ex}} = -10 \log_{10}(P_{\text{out,total}}/P_{\text{in}})\).

• Insertion Loss: Loss for a specific path: \(IL = -10 \log_{10}(P_{\text{out,port}}/P_{\text{in}})\).

• Crosstalk ( Isolation): Power coupled from one port to unintended port: \(XT = -10 \log_{10}(P_{\text{crosstalk}}/P_{\text{in}})\).

9. SYSTEM PERFORMANCE & RECEIVER DESIGN

Digital Receiver Performance

• Model: Photocurrent \(i(t) = R \cdot P(t) + i_n(t)\) where \(R\) = responsivity, \(P(t)\) = received optical power, \(i_n\) = total noise current.

• Noise Sources:

  • Shot Noise: From photocurrent and dark current. Variance: \(\langle i_{sh}^2 \rangle = 2q(I_{\text{ph}} + I_d) B\).

  • Thermal (Johnson-Nyquist) Noise: From load resistor \(R_L\). \(\langle i_{th}^2 \rangle = \frac{4kTB}{R_L}\).

  • APD Excess Noise: \(\langle i_{APD}^2 \rangle = 2q I_{\text{ph}} M^2 F(M) B\).

• Total Noise Variance: \(\sigma_i^2 = \langle i_{sh}^2 \rangle + \langle i_{th}^2 \rangle + \langle i_{ex}^2 \rangle\).

• Signal-to-Noise Ratio (SNR): For "1" bit: \(SNR_1 = \frac{(R \cdot P_1)^2}{\sigma_i^2}\), for "0" bit: \(SNR_0 = \frac{(R \cdot P_0)^2}{\sigma_i^2}\) (usually \(P_0=0\)).

• Bit Error Rate (BER): For non-return-to-zero (NRZ), Gaussian noise:

  \[ BER = Q\left( \sqrt{SNR_1} \right) \quad \text{or} \quad BER = \frac{1}{2} \text{erfc}\left( \sqrt{\frac{SNR_1}{2}} \right) \]

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

• Receiver Sensitivity: Minimum \(P_{\text{avg}}\) for specified BER (e.g., 10⁻⁹). \(P_{\text{avg}} = \frac{P_1 + P_0}{2}\). For \(P_0=0\), \(P_{\text{avg}} = P_1/2\).

Eye Pattern Analysis

• Generation: Overlay of many bit periods on oscilloscope (triggered by clock).

• Interpretation:

  • Eye Opening (Vertical): Noise margin. Larger opening → lower BER.

  • Eye Opening (Horizontal): Timing jitter margin. Determines clock recovery window.

  • Eye Closure: Caused by:

    • Noise: Vertical closure.

    • Dispersion/ISI: Both vertical and horizontal closure (diagonal).

    • Insufficient Bandwidth: Rounded corners.

  • Best Sampling Point: Center of eye opening (maximum margin).

[!TIP] Final Formula Box: For Gaussian noise, BER is related to Q-factor: \(BER = Q(\sqrt{SNR}) = \frac{1}{2} \text{erfc}(\sqrt{SNR/2})\). Receiver sensitivity improves with lower noise (higher SNR).