Nano Electronics (EC-702 (C)) Unit 5 Notes

UNIT 5: EC-702(C) - Nano Electronics

I. Transmission Lines

Microstrip Lines

A microstrip line consists of a conducting strip on a dielectric substrate with a ground plane. It supports a quasi-TEM mode at microwave frequencies.

Effective Dielectric Constant ($$\displaystyle \epsilon_{eff} $$)

\[ > \epsilon_{eff} = \frac{\epsilon_r + 1}{2} + \frac{\epsilon_r - 1}{2} \cdot \frac{1}{\sqrt{1 + 12d/W}} > \]

where $d$ = substrate thickness, $W$ = strip width, $$\displaystyle \epsilon_r $$ = relative permittivity.

\boxed{\epsilon_{eff} = \frac{\epsilon_r + 1}{2} + \frac{\epsilon_r - 1}{2} \cdot F(d/W)}

Characteristic Impedance ($$\displaystyle Z_0 $$)

For $W/d \leq 1$:

\[ > Z_0 = \frac{60}{\sqrt{\epsilon_{eff}}} \ln \left( \frac{8d}{W} + \frac{W}{4d} \right) > \]

For $W/d \geq 1$:

\[ > Z_0 = \frac{120\pi}{\sqrt{\epsilon_{eff}} \left( \frac{W}{d} + 1.393 + 0.667 \ln \left( \frac{W}{d} + 1.444 \right) \right)} > \]

\boxed{Z_0 \propto \frac{1}{\sqrt{\epsilon_{eff}}} \cdot f(W/d)}

[!TIP] Derivation of $$\displaystyle \epsilon_{eff} $$ uses the concept of filling factor. $$\displaystyle Z_0 $$ derivation comes from conformal mapping. Remember: $$\displaystyle \epsilon_{eff} $$ lies between 1 and $$\displaystyle \epsilon_r $$.

Slot Lines

Construction: A slot (gap) in a ground plane on a dielectric substrate.
Comparison with Microstrip:

| Feature | Microstrip | Slot Line | |---------|------------|-----------| | Conductor | Strip on top | Slot in ground | | Mode | Quasi-TEM | Quasi-TEM (inverted field) | | $$\displaystyle Z_0 $$ | Lower for wide strip | Higher for narrow slot | | Integration | Easy with active devices | Difficult, less common |

Strip Lines

Construction: Conducting strips sandwiched between two ground planes (dielectric layers).
Modes:
- Dominant mode: TEM (true TEM, as fields are completely confined).
- Higher order modes: TE, TM modes appear at higher frequencies (cutoff determined by dielectric thickness and width).

Slotted Line

Construction: A coaxial line with a longitudinal slot in the outer conductor. A probe moves along the slot to sample the electric field.
Working: The slot allows measurement of the standing wave pattern inside the line.
Measurements:
1. VSWR: $$\displaystyle VSWR = \frac{V_{max}}{V_{min}} $$ measured by probe positions.
2. Wavelength ($$\displaystyle \lambda_g $$): Distance between two successive $$\displaystyle V_{max} $$ or $$\displaystyle V_{min} $$.
3. Impedance: Using $$\displaystyle Z = Z_0 \frac{1 + \Gamma e^{-j2\beta l}}{1 - \Gamma e^{-j2\beta l}} $$, where $\Gamma$ is found from VSWR and position of $$\displaystyle V_{min} $$.

[!TIP] Slotted line is a direct measurement tool for VSWR and $$\displaystyle \lambda_g $$. For impedance, you need $$\displaystyle Z_0 $$ and $\Gamma$.

TEM Mode

Definition: Transverse Electromagnetic mode. Both E and H fields are entirely transverse to the direction of propagation (no $$\displaystyle E_z $$ or $$\displaystyle H_z $$).
Condition: Exists only in transmission lines with two or more conductors (e.g., coaxial, two-wire, stripline). Cannot exist in single-conductor waveguides (hollow pipes).
Properties: Propagation constant $$\displaystyle \gamma = \alpha + j\beta = \sqrt{(R+j\omega L)(G+j\omega C)} $$. For lossless TEM: $$\displaystyle \gamma = j\omega\sqrt{LC} $$, $$\displaystyle v_p = 1/\sqrt{LC} $$.

II. Waveguides

Rectangular Waveguides (dimensions: $a \times b$, $$\displaystyle a > b $$)

TE/TM Modes: Denoted as TE$$\displaystyle _{mn} $$/TM$$\displaystyle _{mn} $$, where $m,n$ are integers (number of half-wave variations in $x$ and $y$).
Cutoff Wavelength ($$\displaystyle \lambda_c $$):

\[ \lambda_c = \frac{2}{\sqrt{(m/a)^2 + (n/b)^2}} \]
Cutoff Frequency ($$\displaystyle f_c $$):

[

f_c = \frac{c}{2a\sqrt{\epsilon_r}} \quad \text{for dominant TE$$\displaystyle _{10} $$ mode (n=0, m≥1)}

]
Characteristic Wave Impedance:

For TE$$\displaystyle _{mn} $$: $$\displaystyle Z_{TE} = \frac{\eta}{\sqrt{1 - (f_c/f)^2}} $$

For TM$$\displaystyle _{mn} $$: $$\displaystyle Z_{TM} = \eta \sqrt{1 - (f_c/f)^2} $$

where $$\displaystyle \eta = \sqrt{\mu/\epsilon} $$ is intrinsic impedance.

Example (TM$$\displaystyle _{11} $$): $$\displaystyle m=1, n=1 $$, $$\displaystyle \lambda_c = 2/\sqrt{(1/a)^2+(1/b)^2} $$.

[!TIP] TE$$\displaystyle _{10} $$ is dominant (lowest $$\displaystyle f_c $$). TM modes cannot have $$\displaystyle m=0 $$ or $$\displaystyle n=0 $$ (would be TEM, not possible). For a given $f$, check $$\displaystyle f > f_c $$ for propagation.

Circular Waveguides (radius $r$)

Dominant mode: TE$$\displaystyle _{11} $$ (lowest cutoff).
Cutoff Wavelength:

\[ \lambda_c = \frac{2\pi r}{x'_{11}} \approx 2.61r \quad (x'_{11} \approx 1.841) \]
Cutoff Frequency:

\[ f_c = \frac{x'_{11} c}{2\pi r \sqrt{\epsilon_r}} \approx \frac{1.841 c}{2\pi r \sqrt{\epsilon_r}} \]
Guided Wavelength:

\[ \lambda_g = \frac{\lambda_0}{\sqrt{1 - (\lambda_0/\lambda_c)^2}} \]
Mode Identification: For given $r$ and $f$, compute $$\displaystyle f_c $$ for TE$$\displaystyle _{mn} $$/TM$$\displaystyle _{mn} $$ using Bessel function roots ($$\displaystyle x'_{mn} $$ for TE, $$\displaystyle x_{mn} $$ for TM). Propagating modes satisfy $$\displaystyle f > f_c $$.

[!TIP] TE$$\displaystyle _{11} $$ is always dominant. TM$$\displaystyle _{01} $$ has higher cutoff than TE$$\displaystyle _{11} $$. Use $$\displaystyle \lambda_c = 2\pi r / p'_{mn} $$ where $$\displaystyle p'_{mn} $$ are roots of derivative of Bessel function for TE.

III. Microwave Active Devices

Klystrons

Two-Cavity Klystron Amplifier:
- Velocity Modulation: Electrons entering the first cavity (buncher) are velocity-modulated by the RF field. Fast electrons catch up with slow ones, forming density bunches. These bunches induce voltage in the second cavity (catcher) at the same frequency, providing amplification.
Reflex Klystron:
- Construction: Single cavity, electron beam reflected back by negative repeller.
- Working: Electrons are velocity-modulated in the cavity. Some electrons are reflected back after giving energy to the cavity (if they are in proper phase), causing oscillations.
- Mode Curve: Output power vs. repeller voltage shows peaks (modes). Mode number $N$ corresponds to $N$ electrons returning per RF cycle.

[!TIP] Reflex klystron is a low-power oscillator. Mode spacing $$\displaystyle \Delta V \approx V_0 / N $$ (where $$\displaystyle V_0 $$ is beam voltage).

Traveling Wave Tube (TWT)

Interaction Mechanism: Electron beam travels in a helix (or coupled-cavity) slow-wave structure. RF signal propagates along the helix at nearly same speed as beam. Continuous energy transfer from beam to RF wave via velocity modulation and bunching along the entire length, providing broadband amplification.
Helix: Supports forward and backward waves; attenuator suppresses backward wave oscillations.

Magnetron

Types: Cylindrical, coaxial, rising-sun.
Oscillation Mechanism: Electrons from cathode under crossed $E$ (radial) and $B$ (axial) fields follow cycloidal paths. They interact with resonant cavities (anode block) via velocity modulation and bunching. Oscillations occur when the anode voltage satisfies the Hull condition: $$\displaystyle V_a = \frac{B^2 r_a^2}{8m/e} $$ (for π-mode oscillation in cylindrical magnetron).

Gunn Diode

Principle: Based on Gunn effect (transferred electron effect) in III-V materials (GaAs, InP). Has negative differential resistance (NDR) in $I$-$V$ characteristic due to transfer of electrons from high-mobility $\Gamma$-valley to low-mobility $L$-valley.
Domains of Operation:
1. Gunn Domain (Delay Domain): High field domain forms and travels, generating RF pulses (used in oscillators).
2. Quenched Domain: Domain absorbed at anode, used in amplifiers.
3. Limited Space Charge Accumulation (LSA): High frequency domain oscillation.

[!TIP] Gunn diode is transferred-electron device, not a PN junction. Requires $n$-type material with specific doping ($$\displaystyle 10^{14}-10^{16} $$ cm$$\displaystyle ^{-3} $$).

Tunnel Diode

Modes of Operation:
- Resonant Tunneling: In quantum well structures (RTD), peaks in $I$-$V$ due to resonant tunneling through discrete energy levels.
- Normal Tunnel Mode: NDR due to tunneling in heavily doped PN junction.
Uses: High-speed oscillators, amplifiers, digital circuits.

IMPATT Diode

Principle: Impact Ionization Avalanche Transit Time. Uses avalanche breakdown and carrier transit time to produce NDR. Structure: $$\displaystyle p^+ $$-$n$-$$\displaystyle n^+ $$ or $p$-$i$-$n$.
Comparison with TRAPATT:

| Feature | IMPATT | TRAPATT | |---------|--------|---------| | Mechanism | Avalanche + drift | Trapped plasma avalanche triggered transit | | Frequency | Lower (X-band) | Higher (above X-band) | | Efficiency | Lower (5-10%) | Higher (15-30%) | | Noise | High | Lower |

BARITT Diode

Principle: Barrier Injection and Transit Time. Uses thermionic emission over a barrier (not avalanche). Lower noise than IMPATT but lower power. Structure: $p$-$i$-$n$ with intrinsic layer.

Schottky Barrier Diode

Structure: Metal-semiconductor junction (e.g., Pt on n-GaAs).
Working: Majority carriers (electrons in n-type) flow over barrier. Fast response due to minority carrier storage absence.
Use as Mixer/Detector:
- Mixer: Non-linear $I$-$V$ characteristic produces sum/difference frequencies when RF and LO are applied. Low conversion loss.
- Detector: Rectifies RF signal to DC (video detection).

Microwave BJT

Working: Similar to low-frequency BJT but with reduced base width to minimize transit time. Uses heterojunction (e.g., AlGaAs/GaAs) for high-frequency performance.
Frequency Limitations:
- Base transit time $$\displaystyle \tau_b = W_b^2/(2D_n) $$
- Collector depletion layer transit time $$\displaystyle \tau_c = W_{depl}/v_{sat} $$
- RC time constants from parasitic capacitances ($$\displaystyle C_{cb}, C_{be} $$).
Applications: Low-noise amplifiers, oscillators up to ~100 GHz.
Basic Relations:

\[ f_T = \frac{g_m}{2\pi(C_{be} + C_{bc})} \approx \frac{1}{2\pi \tau_{total}} \]

where $$\displaystyle f_T $$ is transition frequency.

Microwave FET (MESFET, HEMT)

Working: Voltage-controlled device. Gate voltage modulates channel conductivity. Short channel ($$\displaystyle <1\ \mu m $$) to reduce transit time.
Basic Relations:

\[ g_m = \frac{\partial I_D}{\partial V_{GS}}, \quad f_{max} = \frac{f_T}{2\sqrt{R_g(R_s + R_d)C_{gd}}} \]

where $$\displaystyle R_g, R_s, R_d $$ are gate, source, drain resistances; $$\displaystyle C_{gd} $$ is gate-drain capacitance.

Frequency Multipliers

Derivation: For a nonlinear device $$\displaystyle i(t) = a_1 v(t) + a_2 v^2(t) + ... $$, if $$\displaystyle v(t) = V_0 \cos \omega t $$, then $i(t)$ contains $2\omega, 3\omega,...$ terms. Output at $n\omega$ is proportional to $$\displaystyle a_n V_0^n $$.
Operating Principle: Use diode nonlinearity (varactor, step-recovery) to generate harmonics. Input at $f$, output at $nf$ (e.g., 2nd, 4th harmonic for efficient multiplication).

[!TIP] Multipliers use nonlinear capacitance (varactor) or current switching (step-recovery diode) for high efficiency.

IV. Network Analysis: Scattering Parameters (S-parameters)

Definition and Necessity

Definition: S-parameters describe the reflected and transmitted waves at each port when other ports are terminated in matched loads ($$\displaystyle Z_0 $$).
Necessity at Microwave Frequencies: At high $f$, $V$ and $I$ are not measurable due to transmission line effects, parasitic reactances, and standing waves. S-parameters use incident and reflected voltage waves ($$\displaystyle a_i, b_i $$) which are measurable.

Properties

Reciprocity: $$\displaystyle S_{ij} = S_{ji} $$ for reciprocal networks (e.g., passive, linear, isotropic).
Losslessness: For lossless network, $$\displaystyle \mathbf{S}^\dagger \mathbf{S} = \mathbf{I} $$ (unitary matrix). In 2-port: $$\displaystyle |S_{11}|^2 + |S_{21}|^2 = 1 $$, $$\displaystyle |S_{22}|^2 + |S_{12}|^2 = 1 $$, $$\displaystyle S_{11}S_{11}^* + S_{12}S_{12}^* = 1 $$.
Symmetry: For reciprocal networks, S-matrix is symmetric.

S-parameters for Two-Port Networks

\[ \begin{bmatrix} b_1 \\ b_2 \end{bmatrix} = \begin{bmatrix} S_{11} & S_{12} \\ S_{21} & S_{22} \end{bmatrix} \begin{bmatrix} a_1 \\ a_2 \end{bmatrix} \]

where $$\displaystyle a_i = \frac{V_i^+}{\sqrt{Z_0}} $$, $$\displaystyle b_i = \frac{V_i^-}{\sqrt{Z_0}} $$.

General Form for Reciprocal and Lossless Two-Port:

For a lossless reciprocal network, S-matrix can be expressed as:

\[ > \mathbf{S} = > \begin{bmatrix} > \Gamma & T \\ > T & \Gamma > \end{bmatrix} > \]

with $$\displaystyle |\Gamma|^2 + |T|^2 = 1 $$. For example, a matched transmission line: $$\displaystyle \Gamma=0 $$, $$\displaystyle |T|=1 $$.

Applications

Oscillator Design:
- Condition: $$\displaystyle |\Gamma_{in}| \geq 1 $$ (input reflection coefficient magnitude ≥1).
- With feedback: $$\displaystyle \Gamma_{in} = S_{11} + \frac{S_{12}S_{21}\Gamma_L}{1 - S_{22}\Gamma_L} \geq 1 $$.
Impedance Matching: Use $$\displaystyle S_{11} $$ to design matching network to minimize reflection ($$\displaystyle S_{11}=0 $$).

[!TIP] For a two-port, $$\displaystyle S_{21} $$ is forward transmission, $$\displaystyle S_{12} $$ reverse transmission. For reciprocal: $$\displaystyle S_{12}=S_{21} $$.

V. Microwave Components and Circuits

Matching Networks

Construction: Lumped elements (L, C) or distributed elements (stubs, transmission lines) to transform impedance.
Matching Techniques:
1. Lumped Element Matching: Use series/ shunt L, C to achieve conjugate match.
2. Transmission Line Transformers: Quarter-wave transformer ($$\displaystyle Z_{in} = Z_0^2/Z_L $$), single-stub matching (shunt or series stub at specific distance).

Impedance Transformers

Single-Section: Quarter-wave transformer. Bandwidth limited by $\lambda/4$ condition.
Multi-Section: Cascaded $\lambda/4$ sections with intermediate impedances. Bandwidth increases with more sections (e.g., binomial, Chebyshev designs).

Multi-section Bandwidth: $$\displaystyle \Delta f/f_0 \propto 2^{n-1} $$ for $n$ sections (approx).

Hybrid Tee (Magic Tee)

Structure: Combination of E-plane tee (series) and H-plane tee (shunt) with matched ports.
Working:
- Ports 1,2,3,4: 1→(2+3) in-phase (E-plane), 1→4 in-phase (H-plane).
- Isolations: 1↔4, 2↔3.
Scattering Matrix:

\[ \mathbf{S} = \frac{1}{\sqrt{2}} \begin{bmatrix} 0 & 1 & 1 & 0 \\ 1 & 0 & 0 & 1 \\ 1 & 0 & 0 & -1 \\ 0 & 1 & -1 & 0 \end{bmatrix} \]

(Assuming ports 1=H-plane input, 4=E-plane input, 2,3=series/shunt outputs).

[!TIP] Magic tee combines E-plane (sum) and H-plane (difference) properties. S-matrix shows 3 dB coupling and perfect isolation.

Directional Couplers

Coupling Factor ($C$): $$\displaystyle C = -20 \log |S_{14}| $$ (for coupled port 4 when input at 1).
Directivity ($D$): $$\displaystyle D = C - 20 \log |S_{24}| $$ (isolation between coupled and isolated ports).
S-matrix for Matched 4-port Coupler:

\[ \mathbf{S} = \begin{bmatrix} 0 & j & 1 & 0 \\ j & 0 & 0 & 1 \\ 1 & 0 & 0 & j \\ 0 & 1 & j & 0 \end{bmatrix} \]

(for 3 dB directional coupler).

Circulators and Isolators

Symbol: 3-port (circulator), 2-port (isolator).
Working using two magic tees and phase shifter:
- Two magic tees connected via phase shifter (90° or 180°). Signal from port 1 goes to port 2, port 2 to port 3, port 3 to port 1 (circulation). Reverse signal is canceled.
Simplified S-matrix (3-port circulator):

\[ \mathbf{S} = \begin{bmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{bmatrix} \]

(clockwise circulation).

Diode Phase Shifters

Use varactor diodes (voltage-controlled capacitance) in transmission line to change electrical length.
Types: Reflection type (using hybrid tee), transmission type (diode in line).
Phase shift: $$\displaystyle \Delta \phi = \beta \Delta l_{eff} $$, where $$\displaystyle \Delta l_{eff} $$ changes with varactor bias.

Microwave Mixers

Working Principle: Nonlinear mixing of RF ($$\displaystyle f_{RF} $$) and LO ($$\displaystyle f_{LO} $$) signals in diode or transistor produces sum ($$\displaystyle f_{RF}+f_{LO} $$) and difference ($$\displaystyle |f_{RF}-f_{LO}| $$) frequencies. IF filter selects desired output.
Roles:
- LO: High-power local oscillator.
- RF: Input signal.
- IF: Intermediate frequency output.
Conversion Loss: $$\displaystyle L_c = P_{RF}/P_{IF} $$ (typically 6-9 dB for diode mixer).

Schottky Diode Mixer Block Diagram:


RF → [Bandpass Filter] → [Shunt Diode] → [IF Filter] → IF

              ↑

             LO (via coupler)

YIG Resonator

Structure: Yttrium Iron Garnet (YIG) sphere placed in a magnetic field (from electromagnet) and coupled to microwave transmission line (e.g., microstrip loop).
Working: Precession of magnetic moments in YIG at Larmor frequency $$\displaystyle f = \gamma B_0/(2\pi) $$ (γ = gyromagnetic ratio). Magnetic field $$\displaystyle B_0 $$ tunes resonance frequency.
Tuning: $$\displaystyle f \propto B_0 $$. Wide tuning range (2-40 GHz).
Applications: Tunable filters, oscillators, frequency synthesizers.

VI. Microwave Measurements

Voltage Standing Wave Ratio (VSWR)

Definition: Ratio of maximum to minimum voltage in standing wave pattern: $$\displaystyle VSWR = V_{max}/V_{min} $$.
Derivation from Reflection Coefficient ($\Gamma$):

\[ V_{max} = |1 + \Gamma| V^+, \quad V_{min} = |1 - \Gamma| V^+ \]

\[ \therefore VSWR = \frac{|1 + \Gamma|}{|1 - \Gamma|} \]

For real $\Gamma$ (lossless line): $$\displaystyle VSWR = \frac{1 + |\Gamma|}{1 - |\Gamma|} $$.

\boxed{VSWR = \frac{1 + |\Gamma|}{1 - |\Gamma|}}

Slotted Line for Parameter Measurement

Determination:
1. VSWR: Measure $$\displaystyle V_{max} $$, $$\displaystyle V_{min} $$.
2. Wavelength ($$\displaystyle \lambda_g $$): Distance between two $$\displaystyle V_{min} $$ (or $$\displaystyle V_{max} $$).
3. Impedance:
  - Find $$\displaystyle \Gamma = |\Gamma| e^{j\theta} $$, where $$\displaystyle |\Gamma| = (VSWR - 1)/(VSWR + 1) $$, $$\displaystyle \theta = 2\beta d $$ ($d$ = distance from load to first $$\displaystyle V_{min} $$).
  - Then $$\displaystyle Z_L = Z_0 \frac{1 + \Gamma}{1 - \Gamma} $$.

Power Measurement

Bridges: Bolometer bridge (measuring RF power via heating), microwave bridge circuits.
Tuned Detectors: Crystal detector + wavemeter (narrowband, high sensitivity).
Broadband Detectors: Thermocouples, thermistors (calorimetric), square-law detectors.

Challenges in Measuring Z, Y, h, ABCD at Microwave Frequencies

Distributed Effects: Transmission line behavior dominates; lumped parameter models invalid.
Parasitics: Stray inductances/capacitances affect measurements.
Standing Waves: VSWR ≠ 1 even for matched loads due to connectors.
Calibration: Need precise calibration standards (SOLT, TRL).
Direct Measurement: $V$ and $I$ not measurable directly; use S-parameters instead.

[!TIP] S-parameters are preferred at microwave frequencies because they are measured with terminated ports (matched conditions).

VII. Information Theory and Coding

Fundamentals of Information Theory

Uncertainty: Measure of unpredictability of a message.
Information: $$\displaystyle I(x) = \log_2 \frac{1}{P(x)} $$ bits (for binary log).
Entropy ($H(X)$): Average information/uncertainty.

\[ H(X) = -\sum_{i=1}^{M} P(x_i) \log_2 P(x_i) \ \text{bits/symbol} \]
Proof: Entropy Maximum for Equiprobable Messages (M=3)

Let $$\displaystyle P(x_1)=p $$, $$\displaystyle P(x_2)=q $$, $$\displaystyle P(x_3)=1-p-q $$.

\[ H = -[p \log p + q \log q + (1-p-q) \log(1-p-q)] \]

Maximize $H$ w.r.t $p,q$ using $$\displaystyle \partial H/\partial p=0 $$, $$\displaystyle \partial H/\partial q=0 $$:

\[ \log p + 1 = 0 \Rightarrow p = 1/2? \ \text{Wait, check:} \]

Actually, $$\displaystyle \frac{\partial H}{\partial p} = -\log_2 p - \frac{1}{\ln 2} = 0 \Rightarrow p = 1/e $$? Correction: Using Lagrange multiplier for constraint $$\displaystyle p+q+r=1 $$:

\[ \mathcal{L} = -\sum p_i \log p_i + \lambda (\sum p_i - 1) \]

\[ \frac{\partial \mathcal{L}}{\partial p_i} = -\log_2 p_i - \frac{1}{\ln 2} + \lambda = 0 \Rightarrow p_i = 2^{\lambda - 1/\ln 2} = \text{constant} \]

So $$\displaystyle p_1 = p_2 = p_3 = 1/3 $$. Thus $$\displaystyle H_{max} = \log_2 3 \approx 1.585 $$ bits.

Morse Code Example:

Dot duration = 1 unit, dash = 3 units.

$$\displaystyle P(\text{dash}) = 3 P(\text{dot}) $$? Actually: "probability of dash is 1 of the probability of a dot" → likely means $$\displaystyle P(\text{dash}) = 1 - P(\text{dot}) $$? Re-read: "The probability of occurrence of a dash is 1 of the probability of a dot." → Probably typo: "1/3 of the probability of a dot"? Standard problem: $$\displaystyle P(\text{dash}) = 3 P(\text{dot}) $$? Let's assume: $$\displaystyle P(\text{dot}) = p $$, $$\displaystyle P(\text{dash}) = 3p $$, and $$\displaystyle p + 3p = 1 \Rightarrow p=0.25 $$, $$\displaystyle P(\text{dash})=0.75 $$.

Information content:

\[ > I(\text{dot}) = \log_2 \frac{1}{0.25} = 2 \ \text{bits} > \]

\[ > I(\text{dash}) = \log_2 \frac{1}{0.75} \approx 0.415 \ \text{bits} > \]

Average information (entropy):

\[ > H = 0.25 \times 2 + 0.75 \times 0.415 \approx 0.811 \ \text{bits/symbol} > \]

Transmission rate: Dot lasts 1 ms, pause between symbols = 1 ms → symbol duration = 2 ms → rate = $$\displaystyle 1/(0.002) = 500 $$ symbols/sec → $$\displaystyle R = 500 \times 0.811 = 405.5 $$ bits/sec.

Mutual Information

Proofs:
1. $$\displaystyle I(X;Y) = H(X) + H(Y) - H(X,Y) $$:
  
  \[ I(X;Y) = H(X) - H(X|Y) = -\sum p(x) \log p(x) + \sum \sum p(x,y) \log p(x|y) \]
  
  \[ = -\sum p(x) \log p(x) + \sum \sum p(x,y) \log \frac{p(x,y)}{p(y)} \]
  
  \[ = -\sum p(x) \log p(x) + \sum \sum p(x,y) \log p(x,y) - \sum p(y) \log p(y) \]
  
  \[ = H(X) + H(Y) - H(X,Y) \]
2. $$\displaystyle I(X;Y) = H(X) - H(X|Y) = H(Y) - H(Y|X) $$ by definition.

Source Coding

Huffman Coding (with Minimum Variance)

Procedure:
1. List probabilities in descending order.
2. Combine two smallest probabilities repeatedly.
3. Assign 0/1 to branches (shorter code for higher probability).
4. To minimize variance, combine smallest probabilities first (standard Huffman already minimizes weighted path length; variance is secondary but can be optimized by choosing which branch gets 0/1 at each step).
Code Efficiency: $$\displaystyle \eta = \frac{H}{\bar{l}} $$ (average code length $$\displaystyle \bar{l} = \sum p_i l_i $$).
Code Variance: $$\displaystyle \sigma^2 = \sum p_i (l_i - \bar{l})^2 $$.

Example: Probabilities $\{0.25, 0.25, 0.125, 0.125, 0.125, 0.0625, 0.0625\}$.

Standard Huffman:

0.25 → 00, 0.25 → 01, 0.125 → 100, 0.125 → 101, 0.125 → 110, 0.0625 → 1110, 0.0625 → 1111

$$\displaystyle \bar{l} = 0.25\times2 + 0.25\times2 + 0.125\times3 \times 3 + 0.0625\times4 \times 2 = 2.5 $$ bits.

$$\displaystyle H = -[0.25\log0.25\times2 + 0.125\log0.125\times3 + 0.0625\log0.0625\times2] \approx 2.219 $$ bits.

Efficiency $$\displaystyle \eta = 2.219/2.5 = 88.76\% $$.

Variance: compute $$\displaystyle (l_i - 2.5)^2 $$ weighted.

Arithmetic Coding

Example: Encode sequence "ABCA" with $$\displaystyle P(A)=0.5, P(B)=0.25, P(C)=0.25 $$.
- Interval [0,1):
  - A: [0, 0.5)
  - B: [0.5, 0.75)
  - C: [0.75, 1)
- "A": [0, 0.5)
- "AB": [0.25, 0.375)
- "ABC": [0.3125, 0.328125)
- "ABCA": [0.3125, 0.3203125) → choose binary fraction 0.0101... (0.3125=0.0101₂).

Lempel-Ziv Coding (Brief)

Dictionary-based: Build dictionary of strings as they appear. Encode new string by referencing previous dictionary entry + new character.
Adaptive: No prior probability needed.

Extended Huffman Coding

Groups $k$ source symbols together, treats each group as a single super-symbol. Probabilities are joint probabilities. Code efficiency improves for non-stationary sources.

Channel Models and Capacity

Binary Symmetric Channel (BSC)

Error probability $p$: $$\displaystyle P(Y=0|X=1)=P(Y=1|X=0)=p $$.
Input equiprobable: $$\displaystyle P(X=0)=P(X=1)=0.5 $$.
Output probabilities: $$\displaystyle P(Y=0)=0.5(1-p)+0.5p = 0.5 $$, same for $$\displaystyle Y=1 $$.
Efficiency (for a code): $$\displaystyle \eta = \frac{H(X)}{n \cdot H(X|Y)} $$? Actually channel efficiency often means rate/capacity.
Capacity:

\[ C = 1 - H_b(p) \ \text{bits/channel use} \]

where $$\displaystyle H_b(p) = -p \log_2 p - (1-p) \log_2 (1-p) $$.

Example: $$\displaystyle p=0.2 $$, $$\displaystyle H_b(0.2) \approx 0.7219 $$, $C \approx 0.2781$ bits/use.

Binary Erasure Channel (BEC)

Input $X \in \{0,1\}$, output $Y \in \{0,1,e\}$ (erasure).
$$\displaystyle P(Y=e|X=0)=P(Y=e|X=1)=\epsilon $$ (erasure probability).
Capacity Derivation:

\[ I(X;Y) = H(Y) - H(Y|X) \]

$$\displaystyle H(Y|X) = H_b(\epsilon) $$ (since given $X$, $Y$ is erasure with $\epsilon$).

$H(Y)$ maximized when $$\displaystyle P(Y=0)=P(Y=1)=(1-\epsilon)/2 $$.

\[ \therefore C = \max_{P(X)} I(X;Y) = (1-\epsilon) - H_b(\epsilon) \ \text{bits/use}. \]

\boxed{C_{BEC} = 1 - \epsilon}

Channel Capacity Theorem (Infinite Bandwidth)

For AWGN channel with bandwidth $B$, SNR = $$\displaystyle P/N_0B $$:

\[ C = B \log_2 \left(1 + \frac{P}{N_0 B}\right) \]
As $B \to \infty$, $$\displaystyle C \to \frac{P}{N_0} \log_2 e $$ (finite limit).

\[ \boxed{C_{\infty} = \frac{P}{N_0} \log_2 e \ \text{bits/sec}} \]

Block Codes

Generator Matrix and Code Vectors

Given $G$ (k×n), code vectors $$\displaystyle \mathbf{c} = \mathbf{u} G $$, where $\mathbf{u}$ is message vector.

Example (6,3) code:

\[ > G = \begin{bmatrix} > 1 & 0 & 0 & 1 & 1 & 0 \\ > 0 & 1 & 0 & 0 & 1 & 1 \\ > 0 & 0 & 1 & 1 & 1 & 1 > \end{bmatrix} > \]

Messages (000) to (111) → code vectors:

000 → 000000

001 → 001111

010 → 010011

011 → 011100

100 → 100110

101 → 101101

110 → 110010

111 → 111001

Hamming Codes

Design: $$\displaystyle (2^m - 1, 2^m - m - 1) $$ with $$\displaystyle d_{min}=3 $$. Parity-check matrix $H$ has all non-zero $m$-bit columns.
For message length $$\displaystyle k=4 $$: Need $$\displaystyle n=2^m-1 \geq k+m+1 $$. Try $$\displaystyle m=3 $$: $$\displaystyle n=7 $$, $$\displaystyle k=7-3=4 $$. So (7,4) Hamming code.

$H$ (3×7) with columns all non-zero 3-bit combos:

\[ H = \begin{bmatrix} 1 & 0 & 1 & 0 & 1 & 0 & 1 \\ 0 & 1 & 1 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 & 1 & 1 & 1 \end{bmatrix} \]

$G$ systematic: $$\displaystyle G = [I_k | P] $$, $$\displaystyle P = H^T $$ (first $k$ columns of $H$ inverted? Actually $$\displaystyle GH^T=0 $$).

Cyclic Codes

Generator Polynomial $g(X)$: Degree $n-k$, divides $$\displaystyle X^n+1 $$.
Encoder: Shift register with feedback based on $g(X)$.
Syndrome Calculator: Same shift register with input zero; output is syndrome $$\displaystyle S(X) = R(X) \mod g(X) $$.
Systematic Form Matrices:

\[ G = [I_k | P], \quad H = [P^T | I_{n-k}] \]

Example (7,4) with $$\displaystyle g(X)=X^3+X+1 $$:

$$\displaystyle g(X)=X^3+X+1 $$ → $$\displaystyle n-k=3 $$, $$\displaystyle k=4 $$.

$g(X)$ divides $$\displaystyle X^7+1 $$? Check: $$\displaystyle X^7+1 = (X^3+X+1)(X^4+X^3+X^2+1) $$? Multiply: $$\displaystyle (X^3+X+1)(X^4+X^3+X^2+1) = X^7 + X^6 + X^5 + X^3 + X^5 + X^4 + X^3 + X + X^4 + X^3 + X^2 + 1 = X^7 + X^6 + (X^5+X^5) + (X^4+X^4) + (X^3+X^3+X^3) + X^2 + X + 1 = X^7 + X^6 + 3X^3 + X^2 + X + 1 $$ → not exactly. Actually $$\displaystyle X^7+1 = (X+1)(X^3+X+1)(X^3+X^2+1) $$? Standard (7,4) Hamming code has $$\displaystyle g(X)=X^3+X+1 $$? Wait, Hamming (7,4) is cyclic with $$\displaystyle g(X)=X^3+X+1 $$? Check: $$\displaystyle X^7+1 = (X+1)(X^3+X^2+1)(X^3+X+1) $$. Yes, so $$\displaystyle g(X)=X^3+X+1 $$ gives (7,4) code? Degree 3, n-k=3, k=4. But Hamming (7,4) has $$\displaystyle d_{min}=3 $$, cyclic. So $$\displaystyle g(X)=X^3+X+1 $$ is correct for (7,4) cyclic code (not Hamming? Actually Hamming (7,4) is cyclic with this $g$). Systematic $G$: $$\displaystyle g(X)=X^3+X+1 $$ → $P$ from $$\displaystyle X^3 \mod g(X) $$? Standard: $$\displaystyle g(X)=X^3+X+1 $$, then $$\displaystyle X^3 \equiv 1 \mod g(X) $$? Actually $$\displaystyle X^3 = -X-1 \mod g(X) $$. So $P$ matrix? Better: For systematic, $$\displaystyle c(X) = u(X)X^{n-k} + r(X) $$, $$\displaystyle r(X) = u(X)X^{n-k} \mod g(X) $$. With $$\displaystyle u(X)=u_0 + u_1 X + u_2 X^2 + u_3 X^3 $$, compute $$\displaystyle u(X)X^3 \mod g(X) $$. Since $$\displaystyle X^3 \equiv X+1 \mod g(X) $$, $$\displaystyle X^4 \equiv X^2+X $$, $$\displaystyle X^5 \equiv X^3+X^2 \equiv (X+1)+X^2 = X^2+X+1 $$, $$\displaystyle X^6 \equiv X^3+X^2+X \equiv (X+1)+X^2+X = X^2+1 $$. So:

$$\displaystyle u(X)X^3 = u_0 X^3 + u_1 X^4 + u_2 X^5 + u_3 X^6 \equiv u_0(X+1) + u_1(X^2+X) + u_2(X^2+X+1) + u_3(X^2+1) $$

Collect:

$$\displaystyle X^2: u_1 + u_2 + u_3 $$

$$\displaystyle X: u_0 + u_1 + u_2 $$

const: $$\displaystyle u_0 + u_2 + u_3 $$

So $P = \begin{bmatrix}

1 & 1 & 0 \

1 & 1 & 1 \

0 & 1 & 1 \

1 & 0 & 1

\end{bmatrix}$? Actually rows correspond to $u_0,u_1,u_2,u_3$? Wait, $r_0, r_1, r_2$ from above:

$$\displaystyle r_0 = u_0 + u_2 + u_3 $$

$$\displaystyle r_1 = u_0 + u_1 + u_2 $$

$$\displaystyle r_2 = u_1 + u_2 + u_3 $$

So $P = \begin{bmatrix}

1 & 0 & 1 \ % r0: u0, u1, u2, u3? Actually r0 depends on u0,u2,u3 → so row for r0: [1,0,1,1]? But P is 4×3? No, systematic G: [I4 | P] where P is 4×3. So rows of P correspond to each message bit.

For $$\displaystyle u_0 $$: coefficient in r0=1, r1=1, r2=0 → [1,1,0]

For $$\displaystyle u_1 $$: r0=0, r1=1, r2=1 → [0,1,1]

For $$\displaystyle u_2 $$: r0=1, r1=1, r2=1 → [1,1,1]

For $$\displaystyle u_3 $$: r0=1, r1=0, r2=1 → [1,0,1]

So $P = \begin{bmatrix}

1 & 1 & 0 \

0 & 1 & 1 \

1 & 1 & 1 \

1 & 0 & 1

\end{bmatrix}$.

Then $$\displaystyle G = [I_4 | P] $$, $$\displaystyle H = [P^T | I_3] $$.

Convolutional Codes

(2,1,3) Encoder using Transform Domain

Structure: 1 input, 2 outputs, constraint length $$\displaystyle K=3 $$ (3 shift registers).
Generators: $$\displaystyle g_1 = (1,1,1) $$, $$\displaystyle g_2 = (1,0,1) $$ (example).
Transform Domain: Use polynomial representation $$\displaystyle G_1(X)=1+X+X^2 $$, $$\displaystyle G_2(X)=1+X^2 $$.
Encoder operation: Input sequence $u(D)$ (delay operator), output sequences $$\displaystyle v_1(D)=u(D)G_1(D) $$, $$\displaystyle v_2(D)=u(D)G_2(D) $$ modulo-2.

Code Tree and Viterbi Algorithm

Code Tree: Shows all possible state transitions for each input bit. Branches labeled with output bits.
Viterbi Algorithm: Maximum likelihood sequence estimation. Uses add-compare-select on trellis (compressed tree). Path with maximum metric (e.g., Hamming distance) is survivor.

BCH Codes

Definition: Bose-Chaudhuri-Hocquenghem codes. Powerful cyclic codes correcting $t$ errors. Designed over GF($$\displaystyle 2^m $$).
Parameters: $(n, k, t)$ with $$\displaystyle n=2^m-1 $$, $k \geq n - mt$, $$\displaystyle d_{min} \geq 2t+1 $$.
Generator Polynomial: $$\displaystyle g(X) = \text{lcm}(m_1(X), m_2(X), ..., m_{2t}(X)) $$, where $$\displaystyle m_i(X) $$ are minimal polynomials of $$\displaystyle \alpha^i $$ ($\alpha$ primitive element in GF($$\displaystyle 2^m $$)).
Example: (15,7,2) BCH code over GF($$\displaystyle 2^4 $$), $$\displaystyle t=1 $$, $$\displaystyle g(X) = \text{lcm}(m_1(X), m_2(X)) $$ where $$\displaystyle m_1(X)=X^4+X+1 $$, $$\displaystyle m_2(X)=X^4+X^3+X^2+X+1 $$ → $$\displaystyle g(X)=X^8+X^7+X^6+X^4+1 $$ (degree 8, so k=15-8=7). Can correct 1 error.

[!TIP] BCH codes are multiple-error-correcting cyclic codes. Design involves finding minimal polynomials of consecutive powers of $\alpha$.

END OF UNIT 5 NOTES
Focus on derivations (microstrip, S-parameters, hybrid tee, VSWR, BEC capacity) and problem-solving (Huffman, BSC, code construction).