Nano Electronics (EC-702 (C)) Unit 4 Notes

UNIT 4: Microwave Engineering and Information Theory (Nano Electronics Context)

A. Microwave Engineering Topics

1. Transmission Lines and Waveguide Systems

TEM Mode and Transmission Line Fundamentals

Primary Constants:
- $R$: Resistance per unit length (Ω/m)
- $L$: Inductance per unit length (H/m)
- $C$: Capacitance per unit length (F/m)
- $G$: Conductance per unit length (S/m)
Propagation Constant: $$\displaystyle \gamma = \alpha + j\beta = \sqrt{(R + j\omega L)(G + j\omega C)} $$
- $\alpha$: Attenuation constant (Np/m)
- $\beta$: Phase constant (rad/m)
Phase Velocity: $$\displaystyle v_p = \omega / \beta $$
Characteristic Impedance (lossy line): $$\displaystyle Z_0 = \sqrt{\frac{R + j\omega L}{G + j\omega C}} $$

[!TIP] For low-loss lines ($R \ll \omega L$, $G \ll \omega C$), $$\displaystyle Z_0 \approx \sqrt{L/C} $$ and $\gamma \approx j\omega\sqrt{LC}$.

Microstrip Lines

Structure: Conductor strip on dielectric substrate with ground plane.
Field Distribution: Quasi-TEM; partly in dielectric, partly in air.
Effective Dielectric Constant:

$$\epsilon_{\text{eff}} = \frac{\epsilon_r + 1}{2} + \frac{\epsilon_r - 1}{2} \frac{1}{\sqrt{1 + 12d/W}}$$

where $d$ = substrate height, $W$ = strip width.

Characteristic Impedance (for $W/d \leq 1$):

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

For $W/d \geq 1$: $$\displaystyle Z_0 = \frac{120\pi}{\sqrt{\epsilon_{\text{eff}}}(W/d + 1.393 + 0.667\ln(W/d + 1.444))} $$

[!TIP] $$\displaystyle \epsilon_{\text{eff}} $$ lies between 1 (air) and $$\displaystyle \epsilon_r $$ (dielectric). $$\displaystyle Z_0 $$ decreases as $$\displaystyle \epsilon_r $$ or $W/d$ increases.

Slot Lines and Strip Lines

Slot Line: Inverted microstrip; slot in ground plane. Supports quasi-TEM. Higher loss than microstrip.
Strip Line: Sandwiched conductor between two ground planes. Pure TEM mode possible. $$\displaystyle Z_0 \propto 1/(\text{width}) $$.
Strip Line Modes: Dominant TEM; higher-order modes appear at higher frequencies.

Rectangular Waveguides

Dimensions: $a$ (broad), $b$ (narrow), $$\displaystyle a > b $$.
Cutoff Wavelength (TE$$\displaystyle _{mn} $$/TM$$\displaystyle _{mn} $$):

$$\lambda_c = \frac{2}{\sqrt{(m/a)^2 + (n/b)^2}}$$

Cutoff Frequency: $$\displaystyle f_c = c/(2a\sqrt{\epsilon_r}) $$ for dominant TE$$\displaystyle _{10} $$.
Wave Impedance:
- TE: $$\displaystyle \eta_{TE} = \frac{\eta}{\sqrt{1 - (f_c/f)^2}} $$
- TM: $$\displaystyle \eta_{TM} = \eta \sqrt{1 - (f_c/f)^2} $$
where $$\displaystyle \eta = \sqrt{\mu_0/\epsilon_0} $$ (free space impedance).
Example: TM$$\displaystyle _{11} $$ impedance: $$\displaystyle \eta_{TM_{11}} = \eta \sqrt{1 - (\lambda/\lambda_{c_{11}})^2} $$.

Circular Waveguides

Radius: $r$.
Dominant Mode: TE$$\displaystyle _{11} $$.
Cutoff Wavelength: $$\displaystyle \lambda_c = \frac{2\pi r}{x'_{11}} \approx 3.41r $$ ($$\displaystyle x'_{11} \approx 1.841 $$).
Guided Wavelength: $$\displaystyle \lambda_g = \lambda / \sqrt{1 - (\lambda/\lambda_c)^2} $$.
Mode Determination: For given $f$ and $r$, compute $$\displaystyle \lambda_c $$ for all TE$$\displaystyle _{mn} $$/TM$$\displaystyle _{mn} $$ using Bessel function roots. Modes with $$\displaystyle \lambda > \lambda_c $$ propagate.

2. S-Parameter Theory and Network Analysis

Scattering Matrix (S-Matrix)

Definition: Relates incident ($$\displaystyle a_n $$) and reflected ($$\displaystyle b_n $$) voltage waves at ports: $$\displaystyle \mathbf{b} = \mathbf{S}\mathbf{a} $$.
Significance: At microwave frequencies, S-parameters are measurable with matched terminations; avoid open/short circuit issues of Z/Y parameters.
Properties:
- Reciprocity: $$\displaystyle S_{mn} = S_{nm} $$ (for passive, linear networks).
- Losslessness: $$\displaystyle \mathbf{S}^\dagger\mathbf{S} = \mathbf{I} $$ (unitary matrix).
- Symmetry (for reciprocal networks): $$\displaystyle \mathbf{S} = \mathbf{S}^T $$.

Two-Port S-Matrix (Reciprocal & Lossless)

General form:

$$\mathbf{S} = \begin{bmatrix} S_{11} & S_{12} \\ S_{21} & S_{22} \end{bmatrix}$$

For reciprocal: $$\displaystyle S_{12} = S_{21} $$.

For lossless: $$\displaystyle |S_{11}|^2 + |S_{12}|^2 = 1 $$, $$\displaystyle |S_{22}|^2 + |S_{12}|^2 = 1 $$, $$\displaystyle S_{11}S_{11}^* + S_{21}S_{21}^* = 1 $$.

Applications

Oscillator Design: Oscillation condition: $$\displaystyle \Gamma_{in} \Gamma_{out} \geq 1 $$, where $$\displaystyle \Gamma_{in} = S_{11} + \frac{S_{12}S_{21}\Gamma_L}{1 - S_{22}\Gamma_L} $$. Often simplified to $$\displaystyle |S_{11}S_{22}| \geq |S_{12}S_{21}| $$ and $$\displaystyle \angle(S_{11}S_{22}) = \angle(S_{12}S_{21}) $$.
Return Loss (RL): $$\displaystyle RL = -20\log_{10}|S_{11}| $$ (for matched output).

[!TIP] S-parameters are defined for matched ports. For other terminations, use $$\displaystyle \Gamma = (Z_L - Z_0)/(Z_L + Z_0) $$ and $$\displaystyle b_2 = S_{21}a_1 + S_{22}a_2 $$, $$\displaystyle a_2 = \Gamma_L b_2 $$.

3. Microwave Active Devices

Vacuum Tubes

Klystron Amplifier (Two-cavity):
- Velocity Modulation: Electron beam accelerated through first cavity (buncher); velocity modulated → density modulated (bunching) → induces RF in second cavity (catcher).
- Construction: Electron gun, buncher cavity, drift space, catcher cavity, collector.
Reflex Klystron:
- Working: Single cavity; electron beam reflected by negative repeller. Bunching occurs in cavity; energy extracted on return.
- Mode Curve: $$\displaystyle V_{repeller} $$ vs. output power; modes correspond to electron transit time = $n$ RF cycles.
Traveling Wave Tube (TWT):
- Interaction: Electron beam and RF wave on slow-wave structure (helix) maintain synchronism; continuous energy transfer → amplification.
- Applications: High-power, broadband amplifiers.
Magnetron:
- Oscillation Mechanism: Crossed E-field (anode) and B-field (magnet). Electrons follow curved paths; π-mode oscillation ($$\displaystyle m = \pi $$) gives maximum output. Straps (alternating vane connections) ensure π-mode operation.

Solid-State Devices

Gunn Diode:
- Gunn Effect: In GaAs/InP, transferred electron mechanism: high-field domain (charge accumulation) forms and travels, causing oscillations.
- Domains: Stable (continuous), transit-time (pulsed), delayed-domain (for amplifiers).
IMPATT Diode:
- Principle: Impact ionization + avalanche transit time. High-field avalanche region + drift region → negative resistance at RF.
TRAPATT Diode:
- Principle: Trapped plasma avalanche transit time; higher efficiency than IMPATT, lower noise.
BARITT Diode:
- Principle: Barrier injection and transit time; lower efficiency, lower noise than IMPATT.
Schottky Barrier Diode:
- Structure: Metal-semiconductor junction. Low capacitance, fast response.
- I-V: $$\displaystyle I = I_s(e^{qV/nkT} - 1) $$.
- Applications: Microwave mixer (heterodyne), detector (envelope detection).
Microwave Transistors:
- BJT: Current-controlled. Frequency limits: $$\displaystyle f_T = g_m/(2\pi C_\pi) $$, $$\displaystyle f_{max} \approx f_T/\sqrt{2R_g/r_\pi} $$.
- FET: Voltage-controlled. $$\displaystyle f_T = g_m/(2\pi C_{gs}) $$. Higher $$\displaystyle f_{max} $$ than BJT.
Tunnel Diode: Negative resistance due to tunneling; modes: oscillator, amplifier, mixer.
MASER: Microwave amplification by stimulated emission of radiation; uses paramagnetic medium (e.g., ruby) in resonant cavity; very low noise.

4. Passive Components and Circuit Design

Matching Networks

Purpose: Match load impedance to line $$\displaystyle Z_0 $$ to minimize reflection.
Techniques:
- Single-Stub: Shorted/open stub placed at specific distance from load.
- Quarter-Wave Transformer: $$\displaystyle Z_{T} = \sqrt{Z_0 Z_L} $$; narrowband.
Working: Use lumped (L, C) or distributed (transmission line) elements to transform impedance.

Impedance Transformers

Single-Section: $$\displaystyle \Gamma(\theta) = \frac{Z_0 - Z_L}{Z_0 + Z_L} $$; bandwidth limited.
Multi-Section: Cascaded sections with intermediate impedances (e.g., binomial, Chebyshev). Bandwidth increases with number of sections $N$.
Bandwidth Enhancement: Use multi-section with optimized impedance profile (e.g., tapered).

Directional Couplers

Four-Port: Coupling factor $$\displaystyle C = -20\log_{10}|S_{14}| $$ (or $$\displaystyle S_{12} $$), Directivity $$\displaystyle D = C - 20\log_{10}|S_{13}| $$.
S-Matrix (completely matched, symmetric, reciprocal):

$$\mathbf{S} = \begin{bmatrix} 0 & \tau & j\kappa & 0 \\ \tau & 0 & 0 & j\kappa \\ j\kappa & 0 & 0 & \tau \\ 0 & j\kappa & \tau & 0 \end{bmatrix}$$

where $$\displaystyle \tau^2 + \kappa^2 = 1 $$, $$\displaystyle \tau = \sqrt{1 - |\kappa|^2} $$.

Hybrid Tee (Magic Tee)

Construction: Combination of E-plane and H-plane tees.
Working:
- Port 1→2,3: In-phase equal split; 1→4: 180° phase shift.
- Ports 2 & 3 isolated; 4 matched.
S-Matrix:

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

Circulators and Isolators

Circulator: Non-reciprocal 3/4-port; signal flows clockwise: 1→2, 2→3, 3→1.
Isolator: 2-port circulator with port 2 terminated; allows 1→2, blocks 2→1.
Working (using two magic tees + phase shifter): Magic tee 1 combines input and coupled signals; phase shifter introduces 90°; magic tee 2 separates outputs → circulator action.
Simplified S-Matrix (3-port):

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

Phase Shifters

Diode Phase Shifter: Uses varactor diodes (voltage-controlled capacitance) or switching diodes (binary phase states).
- Broadband: Switched-line (digital phase states).
- Tuned: Varactor-loaded line (continuous phase).

Microwave Resonators

YIG Resonator:
- Structure: Yttrium Iron Garnet sphere on dielectric rod, in magnetic field.
- Working: Precession frequency $$\displaystyle f = \gamma B_0/(2\pi) $$; tunable by $$\displaystyle B_0 $$.
- Applications: Tunable filters, oscillators, frequency meters.
General Types: Cavity (high Q), dielectric (low loss).

Mixers and Frequency Converters

Mixer Working: Nonlinear device combines RF ($$\displaystyle f_{RF} $$) and LO ($$\displaystyle f_{LO} $$) → sum/difference frequencies ($$\displaystyle f_{IF} = |f_{RF} - f_{LO}| $$).
Conversion Loss: $$\displaystyle L_c = P_{RF}/P_{IF} $$ (typically 6–9 dB).
Schottky Mixer: Fast switching, low noise; often used in sub-harmonic mixers.

Frequency Multipliers

Basic Relation: For $n$-th harmonic, $$\displaystyle P_n \propto (V_{in})^n $$.
Principle: Nonlinear device (diode) generates harmonics; output tuned to desired harmonic via filter.

5. Detectors and Measurement Techniques

Detectors

Broadband Detector: Diode detector with wideband load; measures average power over wide frequency range.
Tuned Detector: Diode + resonant circuit; sensitive to specific frequency.

Slotted Line

Construction: Coaxial/rectangular waveguide with movable probe along slot.
Working: Probe samples $|V|$; min/max indicate VSWR; distance between min gives $$\displaystyle \lambda_g $$.
Impedance Measurement: Measure VSWR $S$ and distance $d$ from reference plane to voltage minimum; $$\displaystyle Z_L = Z_0 \frac{S - j\tan(\beta d)}{1 + jS\tan(\beta d)} $$.

Power Measurement

Bridges: Bolometer (resistive element temperature change), thermocouple, diode detector calibrated.

Standing Waves & VSWR

Standing Wave: Result of incident and reflected waves interference.
VSWR in terms of reflection coefficient $\Gamma$:

$$VSWR = S = \frac{1 + |\Gamma|}{1 - |\Gamma|}$$

$$\boxed{S = \frac{1 + |\Gamma|}{1 - |\Gamma|}}$$

$$\displaystyle |\Gamma| = (S-1)/(S+1) $$.

6. Oscillators and Amplifiers (Detailed)

Oscillator Design using S-Parameters

Kurokami Conditions: For two-port with feedback:
1. $$\displaystyle |S_{11}S_{22} - S_{12}S_{21}| \geq 1 $$
2. $$\displaystyle \angle(S_{11}S_{22}) = \angle(S_{12}S_{21}) $$
Reflection Coefficient Criteria: $$\displaystyle \Gamma_{in}\Gamma_{out} \geq 1 $$, where $$\displaystyle \Gamma_{in} = S_{11} + \frac{S_{12}S_{21}\Gamma_L}{1 - S_{22}\Gamma_L} $$, $$\displaystyle \Gamma_{out} = \Gamma_L $$.

TWT Amplifier

Interaction Mechanism: Electron beam velocity $$\displaystyle v_e $$ ≈ phase velocity $$\displaystyle v_p $$ of RF wave on helix. Space-charge waves (beam density variations) interact with RF electric field → energy transfer from beam to wave (amplification). Helix provides slow-wave structure ($$\displaystyle v_p \ll c $$).

Reflex Klystron Mode Curve

Plot: $$\displaystyle P_{out} $$ vs. $$\displaystyle V_{repeller} $$ for fixed $$\displaystyle V_{anode} $$.
Modes: Peaks correspond to electron round-trip time = $n$ RF cycles ($$\displaystyle n=1,2,... $$). Mode number increases with $$\displaystyle |V_{repeller}| $$.
Frequency Pulling: $f$ varies with $$\displaystyle V_{repeller} $$ due to transit time change.

B. Information Theory and Coding Topics

1. Information Theory Fundamentals

Uncertainty: Measure of unpredictability of a message.
Information Content of message $$\displaystyle x_i $$: $$\displaystyle I(x_i) = \log_2(1/p(x_i)) $$ bits.
Entropy $H(X)$: Average uncertainty/information.

$$H(X) = -\sum_{i=1}^M p(x_i) \log_2 p(x_i)$$

Entropy Maximum for Equiprobable Messages ($$\displaystyle M=3 $$):

Let $$\displaystyle p_1 = p_2 = p_3 = 1/3 $$.

$$\displaystyle H = -3 \cdot (1/3)\log_2(1/3) = \log_2 3 \approx 1.585 $$ bits.

For any other distribution, say $$\displaystyle p_1=0.5, p_2=0.3, p_3=0.2 $$:

$$\displaystyle H = -(0.5\log_2 0.5 + 0.3\log_2 0.3 + 0.2\log_2 0.2) \approx 1.485 < 1.585 $$.

$$\displaystyle \boxed{H_{\max} = \log_2 M \text{ when } p_i = 1/M \ \forall i} $$

Mutual Information

Definition: Information about $X$ gained from $Y$.
Proofs:

$$\displaystyle I(X;Y) = H(X) - H(X|Y) $$

$$\displaystyle = \sum_{x,y} p(x,y) \log \frac{p(x|y)}{p(x)} = \sum_{x,y} p(x,y) \log \frac{p(x,y)}{p(x)p(y)} $$

$$\displaystyle = \sum_{x,y} p(x,y) \log p(x,y) - \sum_{x,y} p(x,y) \log p(x) - \sum_{x,y} p(x,y) \log p(y) $$

$$\displaystyle = -H(X,Y) + H(X) + H(Y) $$.

$$\boxed{I(X;Y) = H(X) + H(Y) - H(X,Y)}$$

Also $$\displaystyle I(X;Y) = H(Y) - H(Y|X) $$.

2. Source Coding

Huffman Coding

Construction:
1. List probabilities in descending order.
2. Combine two smallest probabilities → new node (sum).
3. Repeat until single node; assign 0/1 to branches.
Code Variance: $$\displaystyle \sigma^2 = \sum p_i l_i^2 - (\sum p_i l_i)^2 $$ (measure of length variation).
Code Efficiency: $$\displaystyle \eta = H(X) / \bar{L} $$, where $$\displaystyle \bar{L} = \sum p_i l_i $$.
Example (from past paper):

$$\displaystyle P = \{0.25, 0.25, 0.125, 0.125, 0.125, 0.0625, 0.0625\} $$

Huffman code (min variance): 00, 01, 100, 101, 110, 1110, 1111.

$$\displaystyle \bar{L} = 2.25 $$, $H \approx 2.25$, $\eta \approx 1.0$, $$\displaystyle \sigma^2 = 0.1875 $$.

Arithmetic Coding

Principle: Represent message as interval $[0,1)$; subdivide based on symbol probabilities.
Example: For $$\displaystyle P(A)=0.7, P(B)=0.3 $$, message "AB":

Start: $[0,1)$

A: $[0,0.7)$

B: $[0.49, 0.7)$ → code = binary of any number in $[0.49,0.7)$ (e.g., 0.5 = 0.1).

Lempel-Ziv Coding

Principle: Dictionary-based; parse input into unique substrings; assign fixed-length codes to new substrings.
Example: "ABABABA..." → code: A(1), B(2), AB(3), BA(4), ABA(5), ...

Morse Code Example (Past Paper)

Dot duration = 1 unit, dash = 3 units. $$\displaystyle p(\text{dash}) = 1/3 \cdot p(\text{dot}) $$? Actually: "probability of dash is 1 of probability of dot" → likely $$\displaystyle p_d = p_\text{dot}/3 $$? Clarify: If $$\displaystyle p_d = k p_\text{dot} $$, and $$\displaystyle p_\text{dot} + p_d = 1 $$ → $$\displaystyle p_\text{dot} + k p_\text{dot} = 1 $$ → $$\displaystyle p_\text{dot} = 1/(1+k) $$, $$\displaystyle p_d = k/(1+k) $$. Given "dash is 1 of dot" → ambiguous; assume $$\displaystyle p_d = p_\text{dot}/3 $$? Then $$\displaystyle p_\text{dot} = 3/4 $$, $$\displaystyle p_d = 1/4 $$.
Information Content:

$$\displaystyle I(\text{dot}) = \log_2(1/0.75) \approx 0.415 $$ bits,

$$\displaystyle I(\text{dash}) = \log_2(1/0.25) = 2 $$ bits.
Average Information: $$\displaystyle \bar{I} = 0.75 \times 0.415 + 0.25 \times 2 \approx 0.811 $$ bits/symbol.
Transmission Rate: Dot lasts 1 ms, pause 1 ms → 2 ms per symbol → rate = $$\displaystyle \bar{I} / 0.002 = 405.5 $$ bits/sec.

3. Channel Models and Capacity

Binary Symmetric Channel (BSC)

Model: Crossover probability $p$; $$\displaystyle P(y \neq x | x) = p $$.
Input/Output Probabilities (given input distribution $$\displaystyle P(x=0)=q $$, $$\displaystyle P(x=1)=1-q $$):

$$\displaystyle P(y=0) = q(1-p) + (1-q)p $$,

$$\displaystyle P(y=1) = qp + (1-q)(1-p) $$.
Capacity: $$\displaystyle C = 1 - H(p) $$ bits/channel use, where $$\displaystyle H(p) = -p\log_2 p - (1-p)\log_2(1-p) $$.
Example (Past Paper): $$\displaystyle p=0.2 $$, equiprobable input messages (4 messages, each 3 bits). But BSC is binary → input bits? Likely: input bits 0/1 equiprobable? Then $$\displaystyle q=0.5 $$, $$\displaystyle P(y=0)=0.5(0.8)+0.5(0.2)=0.5 $$. Capacity $$\displaystyle C = 1 - H(0.2) \approx 0.722 $$ bits/use.
Code Efficiency: $$\displaystyle \eta = H(X)/C $$? For equiprobable input, $$\displaystyle H(X)=1 $$ bit → $$\displaystyle \eta = 1/0.722 \approx 1.385 > 1 $$? Actually efficiency $\leq 1$; perhaps they mean rate $R$? Clarify: If source entropy $H(X)$ and channel capacity $C$, max efficiency = $H(X)/C \leq 1$. With $$\displaystyle H(X)=1 $$, $$\displaystyle C=0.722 $$, efficiency $$\displaystyle = 1/0.722 > 1 $$ impossible. Maybe they ask for code efficiency of a specific code? Past question: "calculate the efficiency of the code" – likely for given code? Not clear. Assume they mean normalized rate $R/C$? For equiprobable input, $$\displaystyle R=1 $$, $$\displaystyle C=0.722 $$ → $R/C \approx 1.385$ >1 means code cannot achieve capacity. But efficiency usually $$\displaystyle \eta = H(X)/\bar{L} \leq 1 $$. Possibly misinterpretation. Skip numeric.

Binary Erasure Channel (BEC)

Model: Input $x \in \{0,1\}$; output $y \in \{0,1,e\}$ where $e$ = erasure with probability $p$.
Capacity: $$\displaystyle C = 1 - p $$ bits/channel use.
Derivation: Maximize $$\displaystyle I(X;Y) = H(Y) - H(Y|X) $$. $$\displaystyle H(Y|X)=H(p) $$ (binary erasure entropy). $$\displaystyle H(Y) \leq \log_2 3 $$ but constrained by input distribution; optimum $$\displaystyle P(X=0)=P(X=1)=0.5 $$ gives $$\displaystyle H(Y)=H(p) + (1-p) $$ → $$\displaystyle C = (1-p) $$.

Channel Capacity Theorem (Infinite Bandwidth)

Capacity: $$\displaystyle C = \frac{P}{N_0} \log_2 e \approx 1.44 \frac{P}{N_0} $$ bits/sec, where $P$ = signal power, $$\displaystyle N_0 $$ = noise spectral density.
Explanation: As bandwidth $B \to \infty$, noise power $$\displaystyle N = N_0 B \to \infty $$, but $C/B \to 0$; total capacity approaches finite limit.

4. Error Control Coding

Block Codes

Generator Matrix $\mathbf{G}$: $k \times n$, codeword $$\displaystyle \mathbf{c} = \mathbf{u}\mathbf{G} $$, $\mathbf{u}$ = message.
Parity-Check Matrix $\mathbf{H}$: $(n-k) \times n$, $$\displaystyle \mathbf{c}\mathbf{H}^T = \mathbf{0} $$.
Example (Past Paper): $(6,3)$ code with $$\displaystyle \mathbf{G} = \begin{bmatrix}1&0&0&1&1&0\\0&1&0&0&1&1\\0&0&1&1&1&1\end{bmatrix} $$.

All code vectors: $$\displaystyle \mathbf{u} = (0,0,0) \to (0,0,0,0,0,0) $$; $(1,0,0) \to (1,0,0,1,1,0)$; etc. (8 total).

Hamming Codes

Design: $$\displaystyle d_{\min}=3 $$, single-error correcting. $$\displaystyle n = 2^m - 1 $$, $$\displaystyle k = n - m $$.
For 4-bit message: $$\displaystyle k=4 $$ → $$\displaystyle n=2^m-1 \geq 4 $$ → $$\displaystyle m=3 $$ gives $$\displaystyle n=7 $$. So $(7,4)$ Hamming code.
Parity-Check Matrix $\mathbf{H}$: All non-zero 3-bit columns → $3 \times 7$ matrix with columns 1 to 7 in binary.

Cyclic Codes

Generator Polynomial $g(X)$: Degree $n-k$, divides $$\displaystyle X^n+1 $$.
Example: $(7,4)$ code with $$\displaystyle g(X)=X^3+X+1 $$.
Encoder: Shift register with feedback based on $g(X)$.
Syndrome Calculator: Compute $$\displaystyle S(X) = R(X) \mod g(X) $$ via shift register.
Systematic Form:

$$\displaystyle \mathbf{G} = [I_k | P] $$, $$\displaystyle \mathbf{H} = [P^T | I_{n-k}] $$.

For $$\displaystyle g(X)=X^3+X+1 $$: $$\displaystyle g(X)=X^3 + 0\cdot X^2 + X + 1 $$. Systematic encoding: $$\displaystyle c(X) = u(X)X^{n-k} + r(X) $$, $$\displaystyle r(X) = u(X)X^{n-k} \mod g(X) $$. Compute $\mathbf{P}$.

BCH Codes

Primitive Polynomial: e.g., $$\displaystyle p(X)=X^3+X+1 $$ (primitive of degree 3 for $$\displaystyle m=3 $$).
Generator Polynomial: $$\displaystyle g(X) = \text{lcm}(m_1(X), m_2(X), ...) $$ where $$\displaystyle m_i(X) $$ are minimal polynomials of $$\displaystyle \alpha^i $$, $\alpha$ root of $p(X)$.
Example: $(7,4)$ BCH with $$\displaystyle t=1 $$ (correct 1 error) → $$\displaystyle g(X) = X^3+X+1 $$ (same as Hamming).

Convolutional Codes

$(2,1,3)$ Encoder:
- Input: 1 bit/clock.
- Output: 2 bits/clock.
- Constraint length $$\displaystyle K=3 $$ (memory 2).
- Generator polynomials: $$\displaystyle g_1 = 7 $$ (111), $$\displaystyle g_2 = 5 $$ (101) in octal.
Transform Domain: $$\displaystyle G(D) = [g_1(D), g_2(D)] = [1+D+D^2, 1+D^2] $$.
Code Tree/State Diagram: 4 states (00,01,10,11); each input produces two outputs and next state.

5. Decoding Algorithms

Viterbi Algorithm

Principle: Maximum likelihood decoding for convolutional codes.
Steps:
1. Compute branch metrics (e.g., Hamming distance) for each possible input.
2. Add to path metrics; select survivor path for each state.
3. After $N$ steps, trace back from state with best metric.
Path Metric: Accumulated distance; Traceback: Decode by following survivor path.

Cyclic Code Decoding

Syndrome Calculation: $$\displaystyle S = R \mod g(X) $$.
Error Correction: If $S \neq 0$, compare to precomputed syndrome table → error pattern → correct.

6. Performance Evaluation

Code Metrics

Code Efficiency: $$\displaystyle \eta = k/n $$ (rate) or $$\displaystyle \eta = H(X)/\bar{L} $$ for source coding.
Code Variance: $$\displaystyle \sigma^2 = \sum p_i l_i^2 - \bar{L}^2 $$ (for Huffman).

Example Calculations

BSC Efficiency: For given code rate $$\displaystyle R = k/n $$ and channel capacity $C$, efficiency $$\displaystyle \eta = R/C $$ (if $R \leq C$).
BEC Capacity: $$\displaystyle C = 1 - p $$ erasure probability.

END OF UNIT 4 NOTES
Focus on derivations and numerical problems from past papers. Practice S-parameter transformations, microstrip calculations, and coding constructions.