The Ultimate GATE EC Cheatsheet

# Fundamental Laws
  KCL: Σ I_entering = Σ I_leaving  (current conservation at node)
  KVL: Σ V_rise = Σ V_drop       (voltage conservation in loop)

  V = IR,  I = V/R,  R = V/I

# Source Transformations
  V_source with series R → I_source with parallel R
    I_s = V_s / R,  R same value
  V_source R1 in series with R2 → combine: R_eq = R1 + R2
  I_source R1 parallel R2 → combine: 1/R_eq = 1/R1 + 1/R2

# Key Theorems
  Thevenin: V_th = V_oc, R_th = V_oc / I_sc (deactivate sources)
  Norton:   I_n = I_sc,  R_n = V_oc / I_sc
  Superposition: Consider one source at a time, add results
  Maximum Power Transfer: R_L = R_th → P_max = V_th² / (4 × R_th)
  Reciprocity: Response same when source and measurement swapped

# Star-Delta Transformation
  R_AB = (R_a R_b + R_b R_c + R_c R_a) / R_c  (Y → Δ)
  R_a = (R_AB × R_AC) / (R_AB + R_AC + R_BC)   (Δ → Y)

# AC Fundamentals
  v(t) = V_m sin(ωt + φ)
  V_rms = V_m / √2,  V_avg = 2V_m / π (full wave)

# Impedances
  Resistor:    Z_R = R
  Inductor:    Z_L = jωL     (V leads I by 90°)
  Capacitor:   Z_C = 1/(jωC)  (I leads V by 90°)

  Series RLC:  Z = R + j(ωL − 1/ωC)
  Parallel RLC: 1/Z = 1/R + 1/(jωL) + jωC

# Resonance
  Series Resonance (ω₀):
    ω₀ = 1/√(LC),  f₀ = 1/(2π√(LC))
    At resonance: Z = R (minimum), I = maximum
    Quality Factor: Q = ω₀L/R = 1/(ω₀CR) = (1/R)√(L/C)
    Bandwidth: BW = ω₀/Q = R/L

  Parallel Resonance (ω₀):
    ω₀ = 1/√(LC),  at resonance: Z = R (maximum)
    Quality Factor: Q = R/(ω₀L) = ω₀CR = R√(C/L)
    Bandwidth: BW = ω₀/Q = 1/(RC)

# Power in AC
  P = VI cos(φ)    [Real power, Watts]
  Q = VI sin(φ)    [Reactive power, VAR]
  S = VI            [Apparent power, VA]
  Power Factor: pf = cos(φ) = P/S = R/|Z|
  S² = P² + Q²

# RC Circuit
  Charging: v_C(t) = V₀(1 − e^(−t/RC))     i(t) = (V₀/R)e^(−t/RC)
  Discharging: v_C(t) = V₀ × e^(−t/RC)      i(t) = −(V₀/R)e^(−t/RC)
  Time constant τ = RC (63.2% charged in 1τ, 99.3% in 5τ)

# RL Circuit
  Charging: i(t) = (V₀/R)(1 − e^(−tR/L))
  Discharging: i(t) = (V₀/R)e^(−tR/L)
  Time constant τ = L/R

# RLC Circuit
  Characteristic equation: s² + 2αs + ω₀² = 0
    α = R/(2L),  ω₀ = 1/√(LC)
    ω_d = √(ω₀² − α²)   (damped frequency)

  Cases:
    α > ω₀  → Overdamped (two real negative roots)
    α = ω₀  → Critically damped (repeated real root)
    α < ω₀  → Underdamped (complex conjugate roots)
    α = 0    → Undamped (purely imaginary roots)

# Initial Conditions
  Inductor current CANNOT change instantaneously: i_L(0⁺) = i_L(0⁻)
  Capacitor voltage CANNOT change instantaneously: v_C(0⁺) = v_C(0⁻)

# Key Conditions
  Reciprocal Network: z₁₂ = z₂₁ (or y₁₂ = y₂₁, or AD−BC = 1)
  Symmetric Network:  z₁₁ = z₂₂ AND z₁₂ = z₂₁
  Open Circuit Z:     Z₁₁ = V₁/I₁ | I₂=0,   Z₂₂ = V₂/I₂ | I₁=0
  Short Circuit Y:    Y₁₁ = I₁/V₁ | V₂=0,   Y₂₂ = I₂/V₂ | V₁=0

# Interconnections
  Series:     Z_total = Z_a + Z_b
  Parallel:   Y_total = Y_a + Y_b
  Cascade:    [ABCD]_total = [ABCD]_a × [ABCD]_b

# Diode Equation (Shockley)
  I = I_s (e^(V/(nV_T)) − 1)
  I_s = reverse saturation current (~nA for Si)
  V_T = kT/q = 26mV at 300K (room temp)
  n = ideality factor (1 for ideal, 2 for real)

  Forward voltage: Si ≈ 0.7V, Ge ≈ 0.3V

# Rectifier Formulas
  Half Wave:    V_dc = V_m/π,   Ripple factor γ = 1.21,   PIV = V_m
  Full Wave:    V_dc = 2V_m/π,  Ripple factor γ = 0.48,   PIV = 2V_m
  Bridge:       V_dc = 2V_m/π,  PIV = V_m
  With C filter: V_dc ≈ V_m − I_dc/(4fC)

# BJT (Bipolar Junction Transistor)
  Modes:
    Active:        BE forward, BC reverse   (Amplifier)
    Saturation:    BE forward, BC forward   (Switch ON)
    Cut-off:       BE reverse, BC reverse   (Switch OFF)
    Reverse Active: BE reverse, BC forward  (rarely used)

  Key Relations (npn):
    I_E = I_B + I_C
    I_C = β × I_B       (β = h_FE = common emitter current gain)
    I_E = (1+β) × I_B
    α = I_C/I_E = β/(1+β),  β = α/(1−α)

  Ebers-Moll Model:
    I_C = α_F × I_ES(e^(V_BE/V_T) − 1) − I_CS(e^(V_BC/V_T) − 1)

# Voltage Divider Bias (Self Bias) — Most Common
  R₂ → base (to GND),  R₁ → V_CC → base
  V_B = V_CC × R₂ / (R₁ + R₂)
  V_E = V_B − V_BE  (≈ 0.7V for Si)
  I_E ≈ I_C = V_E / R_E
  V_CE = V_CC − I_C(R_C + R_E)

  Stability Factor S = (1+β)(1 + R_B/R_E + β×R_C/R_E)
  For good stability: R_E >> R_B/(1+β)

# Small Signal Model (Hybrid-π)
  g_m = I_C / V_T = β / r_π
  r_π = β / g_m = βV_T / I_C
  r_o = V_A / I_C  (Early voltage V_A, typically 50–100V)

  A_v = −g_m × (r_o || R_C || R_L)  for CE

# Ideal Op-Amp Properties
  A_OL = ∞ (open loop gain),  Z_in = ∞,  Z_out = 0
  Virtual Ground: V+ = V− (when negative feedback)

# Key Configurations
  Inverting:    V_out = −(R_f/R_in) × V_in
  Non-Inverting: V_out = (1 + R_f/R_in) × V_in
  Buffer:       V_out = V_in  (unity gain, R_in = ∞, R_out = 0)
  Summing:      V_out = −R_f(V₁/R₁ + V₂/R₂ + ... + Vₙ/Rₙ)
  Difference:   V_out = (R_f/R)(V₂ − V₁)  (when R_f/R ratios match)
  Integrator:   V_out = −(1/RC)∫V_in dt
  Differentiator: V_out = −RC × dV_in/dt

# Important Op-Amp Parameters
  CMRR = |A_d / A_c| in dB = 20 log₁₀(|A_d/A_c|)
  Slew Rate SR = max dV_out/dt (V/μs)
    Full power bandwidth: f_max = SR / (2πV_m)
  Input offset voltage: V_os (typically mV range)
  PSRR = ΔV_os / ΔV_supply

# Oscillators (Barkhausen Criterion)
  |Aβ| ≥ 1  AND  ∠Aβ = 0° (360°)
  RC Phase Shift: f = 1/(2πRC√6),  |A| = 29
  Wien Bridge: f = 1/(2πRC),  |A| = 3

# MOSFET (Enhancement Type, n-channel)
  Cutoff:         V_GS < V_th  →  I_D = 0
  Triode (Linear): V_GS > V_th, V_DS < V_GS − V_th
    I_D = μₙCₒₓ(W/L)[(V_GS−V_th)V_DS − V_DS²/2]
  Saturation:     V_GS > V_th, V_DS ≥ V_GS − V_th
    I_D = (μₙCₒₓ/2)(W/L)(V_GS − V_th)²
    Drain current independent of V_DS

  V_th (threshold) ≈ 0.4–4V depending on technology
  g_m = 2I_D/(V_GS−V_th) = √(2μₙCₒₓ(W/L)I_D)

# JFET (n-channel)
  I_D = I_DSS(1 − V_GS/V_P)²  (saturation region)
  V_P = pinch-off voltage
  g_m = g_m0(1 − V_GS/V_P) = −2I_DSS(1−V_GS/V_P)/V_P

# MOSFET Amplifier Configurations
  CS (Common Source): Similar to BJT CE, A_v = −g_m(r_o||R_D)
  CD (Source Follower): Similar to CC, A_v ≈ 1
  CG (Common Gate): Similar to CB

# Number Representations (n bits)
  Unsigned:             0 to 2^n − 1
  Signed Magnitude:    −(2^(n−1)−1) to +(2^(n−1)−1), two zeros
  1's Complement:      −(2^(n−1)−1) to +(2^(n−1)−1), two zeros
  2's Complement:      −2^(n-1) to +(2^(n-1)−1), one zero

  2's complement shortcut: flip bits, add 1
  Range of 8-bit 2's comp: −128 to +127
  Range of 16-bit 2's comp: −32768 to +32767

# Gray Code
  Binary to Gray: MSB same, then G_i = B_i ⊕ B_(i+1)
  Gray to Binary: B_MSB = G_MSB, B_i = G_i ⊕ B_(i+1)

# BCD (Binary Coded Decimal)
  Each decimal digit → 4-bit binary
  Invalid codes: 1010–1111 (A–F)
  BCD Addition: if sum > 1001, add 0110 (6) to correct

# Multiplexer (MUX)
  2^n:1 MUX → n select lines
  4:1 MUX: Y = S₁'S₀'I₀ + S₁'S₀I₁ + S₁S₀'I₂ + S₁S₀I₃
  Any Boolean function of n variables → 2^n:1 MUX
  Any Boolean function of n variables → 2^(n-1):1 MUX (using 1 variable on select)

# Demultiplexer (DEMUX)
  1:2^n DEMUX → n select lines
  Decoder with enable = DEMUX

# Encoder
  Priority Encoder: highest priority input gets encoded
  8:3 priority encoder: 3 output bits, 1 valid bit

# Comparators
  n-bit comparator: outputs A>B, A=B, A<B
  Cascadable: use (A>B)_i, (A=B)_i, (A<B)_i from lower bits

# Adders
  Half Adder:  S = A⊕B,  C = AB
  Full Adder:  S = A⊕B⊕C_in, C_out = AB + C_in(A⊕B)
  Ripple Carry Adder: n FAs in series, delay = O(n)
  Carry Lookahead Adder:
    C_{i+1} = G_i + P_i × C_i
    G_i = A_i B_i (Generate),  P_i = A_i ⊕ B_i (Propagate)

# Flip-Flops
  SR Latch:  S=1,R=1 is FORBIDDEN (invalid state)
  D FF:      Q_next = D
  JK FF:     Q_next = JQ' + K'Q  (toggle when J=K=1)
  T FF:      Q_next = T⊕Q  (toggle when T=1)

  Excitation Table:
  Q→Qn  SR    JK    D    T
  0→0   0x    0x    0    0
  0→1   10    1x    1    1
  1→0   01    x1    0    1
  1→1   x0    x0    1    0

# Counters
  Mod-N: counts 0 to N-1, uses ⌈log₂N⌉ FFs
  Ring Counter: N states with N FFs
  Johnson Counter: 2N states with N FFs (twisted ring)
  BCD Counter: Mod-10, counts 0-9

  Synchronous vs Asynchronous:
    Synchronous: common clock, faster, no ripple delay
    Asynchronous: clock from previous FF output, ripple delay

# State Machines
  Moore Machine: output depends only on current state
    # states may be more, output logic is simpler
  Mealy Machine: output depends on state AND input
    # states fewer, output depends on input → glitches possible

  Conversion: Mealy → Moore (add states for each output combination)

# DAC (Digital to Analog)
  R-2R Ladder: only 2 resistor values needed
    V_out = −V_ref × (D_n−1/2 + D_n−2/4 + ... + D₀/2^n)
  Resolution = V_ref / (2^n − 1)

# ADC (Analog to Digital)
  SAR (Successive Approximation): n clock cycles for n-bit conversion
  Flash ADC: 2^n − 1 comparators, fastest but most expensive
  Dual Slope: integrator + counter, slow but accurate, rejects noise
  Resolution = V_ref / 2^n

# Memory Types
  SRAM:  Fast, expensive, 6T per cell, used for cache
  DRAM:  Slower, cheap, 1T+1C per cell, needs refresh, main memory
  ROM:   Non-volatile, read-only
  PROM:  Programmable once
  EPROM: Erasable by UV light
  EEPROM: Electrically erasable
  Flash: Block-erasable EEPROM

# Standard Form
  G(s) = K / (s(Ts + 1))  →  Type-1 system
  G(s) = K / (s²)         →  Type-2 system

  Type = number of pure integrators (poles at s=0)

# Block Diagram Algebra
  Series:      G(s) = G₁(s) × G₂(s)
  Parallel:    G(s) = G₁(s) + G₂(s)
  Feedback:    G(s) = G(s) / (1 + G(s)H(s))   [negative feedback]
               G(s) = G(s) / (1 − G(s)H(s))   [positive feedback]

# Signal Flow Graph (Mason's Gain Formula)
  T = Σ (P_k × Δ_k) / Δ
  Δ = 1 − Σ L₁ + Σ L₂ − Σ L₃ + ...
  P_k = forward path gain of k-th path
  Lᵢ = loop gain
  Δ_k = Δ with paths touching k-th forward path removed

# Standard Second-Order System
  G(s) = ωₙ² / (s² + 2ζωₙs + ωₙ²)

  Parameters:
  ωₙ = natural frequency (undamped)
  ζ  = damping ratio

  Cases:
  ζ < 1  → Underdamped (oscillatory)
  ζ = 1  → Critically damped (fastest no overshoot)
  ζ > 1  → Overdamped (slow, no overshoot)
  ζ = 0  → Undamped (sustained oscillations)

# Time Domain Specifications (Underdamped)
  Rise Time:      t_r = (π − φ) / (ωₙ√(1−ζ²))
  Peak Time:      t_p = π / (ωₙ√(1−ζ²))  = π / ω_d
  Peak Overshoot: M_p = e^(−πζ/√(1−ζ²))
  Settling Time:  t_s = 4/(ζωₙ) for 2% band,  = 3/(ζωₙ) for 5% band

# Steady State Error
  System Type →  Step    Ramp    Parabola
  Type 0     →  1/(1+Kp)  ∞      ∞
  Type 1     →  0        1/Kv    ∞
  Type 2     →  0        0       1/Ka

  Kp = lim G(s)H(s) at s→0
  Kv = lim s·G(s)H(s) at s→0
  Ka = lim s²·G(s)H(s) at s→0

# Bode Plot
  Corner frequency (break frequency): ω = 1/T
  At corner: magnitude drops by −20 dB/decade for each pole
             magnitude rises by +20 dB/decade for each zero

  Transfer function: G(s) = K(s+z₁)(s+z₂)... / [s^N(s+p₁)(s+p₂)...]
  Magnitude (dB): 20 log₁₀|G(jω)|
  Phase: ∠G(jω)

  Phase Margin (PM): PM = 180° + ∠G(jω_gc)  at gain crossover (|G|=1)
  Gain Margin (GM): GM = 1/|G(jω_pc)| in dB = −20log₁₀|G(jω_pc)|
                    at phase crossover (∠G = −180°)

  For stability: PM > 0° AND GM > 0 dB (minimum phase system)

# Routh-Hurwitz Criterion
  For characteristic equation: aₙsⁿ + aₙ₋₁sⁿ⁻¹ + ... + a₁s + a₀ = 0

  Necessary condition: ALL aᵢ > 0
  Routh array: construct rows, check sign of first column

  Sign changes in first column = number of RHP poles

  Special cases:
  1. First element of row = 0 (rest non-zero): use ε → 0⁺
  2. Entire row = 0: auxiliary polynomial, take derivative

# Nyquist Criterion
  N = Z − P
  N = number of clockwise encirclements of (−1, 0)
  P = number of open-loop RHP poles
  Z = number of closed-loop RHP poles

  For stability: Z = 0, i.e., N = −P

# PID Controller
  G_c(s) = K_p + K_i/s + K_d·s
  Proportional (P): reduces error, may cause oscillations
  Integral (I):      eliminates steady-state error, may slow response
  Derivative (D):    improves stability, reduces overshoot, noise-sensitive

# Compensators
  Lead Compensator:   G_c(s) = (1+sT)/(1+sαT),  α < 1
    Adds phase lead, increases bandwidth, improves transient response

  Lag Compensator:    G_c(s) = (1+sT)/(1+sβT),  β > 1
    Reduces steady-state error, decreases bandwidth

  Lag-Lead: combines both advantages

# Signal Energy & Power
  Energy signal: E < ∞, P = 0    (e.g., e^(−at)u(t), a>0)
  Power signal:  E = ∞, P < ∞    (e.g., sin(ωt), u(t))
  Neither:       E = ∞, P = ∞    (e.g., e^(at), a>0)

  Any periodic signal with period T:
    P = (1/T) ∫₀ᵀ |x(t)|² dt

# System Properties
  Linear:     T{ax₁+bx₂} = aT{x₁} + bT{x₂}
  Time-Inv:   x(t)→y(t) implies x(t−t₀)→y(t−t₀)
  Causal:     output depends only on present and past inputs
  Stable (BIBO): bounded input → bounded output
    Necessary & sufficient: ∫|h(t)|dt < ∞ (continuous)
                            Σ|h[n]| < ∞  (discrete)

# Fourier Series (Periodic Signals)
  x(t) = a₀ + Σ[aₙcos(nω₀t) + bₙsin(nω₀t)]
  ω₀ = 2π/T (fundamental frequency)

  a₀ = (1/T)∫x(t)dt
  aₙ = (2/T)∫x(t)cos(nω₀t)dt
  bₙ = (2/T)∫x(t)sin(nω₀t)dt

# Key Transform Pairs
  δ(t)          ↔  1
  1             ↔  2πδ(ω)
  e^(jω₀t)     ↔  2πδ(ω − ω₀)
  cos(ω₀t)      ↔  π[δ(ω−ω₀) + δ(ω+ω₀)]
  sin(ω₀t)      ↔  jπ[δ(ω+ω₀) − δ(ω−ω₀)]
  u(t)          ↔  πδ(ω) + 1/(jω)
  rect(t/T)     ↔  T·sinc(ωT/2π)
  e^(−at)u(t)   ↔  1/(a+jω),     a>0
  te^(−at)u(t)  ↔  1/(a+jω)²,    a>0
  e^(−a|t|)     ↔  2a/(a²+ω²),   a>0

# Key Properties
  Time shift:   x(t−t₀) ↔ X(ω)·e^(−jωt₀)
  Freq shift:   x(t)·e^(jω₀t) ↔ X(ω−ω₀)
  Scaling:      x(at) ↔ (1/|a|)X(ω/a)
  Convolution:  x₁*x₂ ↔ X₁(ω)·X₂(ω)
  Multiplication: x₁·x₂ ↔ (1/2π)X₁*X₂
  Differentiation: d/dt x(t) ↔ jωX(ω)
  Time reversal: x(−t) ↔ X(−ω)
  Parseval's:    ∫|x(t)|²dt = (1/2π)∫|X(ω)|²dω

# Laplace Transform
  e^(−at)u(t)  ↔  1/(s+a),        ROC: Re(s)>−a
  t·e^(−at)u(t) ↔ 1/(s+a)²
  u(t)          ↔  1/s,            ROC: Re(s)>0
  δ(t)          ↔  1,              ROC: all s

  ROC properties:
    - ROC doesn't contain any poles
    - Right-sided signal: ROC is right of rightmost pole
    - Left-sided signal: ROC is left of leftmost pole
    - Two-sided signal: ROC is a strip/band

# Sampling Theorem (Nyquist)
  f_s ≥ 2f_max   (sampling rate ≥ twice max frequency)
  f_s/2 = Nyquist rate = minimum sampling rate

  Aliasing: if f_s < 2f_max, frequencies fold (alias)

  Reconstruction: x_r(t) = Σ x(nT_s) × sinc((t−nT_s)/T_s)

# Z-Transform Pairs
  δ[n]        ↔  1
  u[n]        ↔  z/(z−1),      |z|>1
  aⁿu[n]      ↔  z/(z−a),      |z|>|a|
  naⁿu[n]     ↔  az/(z−a)²,    |z|>|a|
  cos(ω₀n)u[n] ↔ z(z−cosω₀)/(z²−2zcosω₀+1)
  sin(ω₀n)u[n] ↔ zsinω₀/(z²−2zcosω₀+1)

# Properties
  Time shift:   x[n−k] ↔ z^(−k)X(z)
  Scaling:      aⁿx[n] ↔ X(z/a)
  Convolution:  x₁*x₂ ↔ X₁(z)·X₂(z)
  Differentiation: nx[n] ↔ −z·dX(z)/dz

# Inverse Z-Transform
  Long division, Partial fractions, Residue method

# Maxwell's Equations (Differential Form)
  ∇ · D = ρ_v              (Gauss's Law)
  ∇ · B = 0                 (No magnetic monopoles)
  ∇ × E = −∂B/∂t           (Faraday's Law)
  ∇ × H = J + ∂D/∂t        (Ampere's Law + Maxwell)

# Maxwell's Equations (Integral Form)
  ∮ D · dS = Q_enc           (Gauss)
  ∮ B · dS = 0                (Gauss, magnetic)
  ∮ E · dl = −d/dt ∮ B · dS  (Faraday)
  ∮ H · dl = I_enc + d/dt ∮ D · dS  (Ampere)

# Constitutive Relations
  D = εE,   B = μH,   J = σE
  ε = ε₀εᵣ (ε₀ = 8.854 × 10⁻¹² F/m)
  μ = μ₀μᵣ (μ₀ = 4π × 10⁻⁷ H/m)
  σ = conductivity (S/m)

# Wave Equation
  ∇²E − με ∂²E/∂t² = 0
  ∇²H − με ∂²H/∂t² = 0
  Wave velocity: v = 1/√(με)
  Intrinsic impedance: η = √(μ/ε)
  Free space: v = c = 3×10⁸ m/s, η = 377Ω ≈ 120πΩ

# Telegrapher's Equations
  γ = √(ZY) = α + jβ
  Z₀ = √(Z/Y) = characteristic impedance

  Z = R + jωL (series impedance per unit length)
  Y = G + jωC (shunt admittance per unit length)

  For lossless: α = 0, β = ω√(LC)
    Z₀ = √(L/C),  v = 1/√(LC)

# Key Parameters
  VSWR = (1 + |Γ|) / (1 − |Γ|)
  Γ = (Z_L − Z₀) / (Z_L + Z₀)  (reflection coefficient)
  Return loss = −20 log₁₀|Γ| dB
  Insertion loss = 10 log₁₀(P_in/P_out) dB

  Standing wave: V_max/V_min = VSWR
  Power: P_avg = |V⁺|²(1−|Γ|²) / (2Z₀)

# Quarter Wave Transformer
  Z_in = Z₀²/Z_L  (at λ/4, acts as impedance inverter)
  Used for impedance matching

# Smith Chart
  Normalized impedance: z = Z/Z₀
  Constant resistance circles, constant reactance arcs
  Open circuit: rightmost point
  Short circuit: leftmost point

# Antenna Parameters
  Directivity: D = U_max / U_avg = 4π × U_max / P_rad
  Gain: G = η × D  (η = efficiency)
  Effective aperture: A_e = (λ²/4π) × G
  Radiation resistance: R_rad = P_rad / I²_rms

# Hertzian Dipole (Short Dipole, l << λ)
  E_θ = (jηωI₀l sinθ) e^(−jβr) / (4πr)
  D = 1.5 (or 10log₁₀(1.5) = 1.76 dBi)
  R_rad = 80π²(l/λ)²

# Half-Wave Dipole (l = λ/2)
  D = 1.64 (or 2.15 dBi)
  R_rad = 73Ω
  HPBW ≈ 78°

# Antenna Arrays
  Array factor for N elements with spacing d:
    AF = Σ e^(jn(kd cosθ + α)), n = 0 to N−1

  Broadside: α = 0, maximum perpendicular to array
  End-fire: α = −kd, maximum along array axis

  HPBW ≈ 0.886λ/(Nd) for uniform linear array

# Friis Transmission Equation
  P_r/P_t = G_t G_r (λ/(4πR))²

# Radar Equation
  P_r = (P_t G²_t λ² σ) / ((4π)³ R⁴)

# Key Modulation Parameters
  Modulation Index:
    AM: m = Vₘ/V_c (m ≤ 1 for no over-modulation)
    FM: β = Δf/fₘ = frequency modulation index

  AM Power Distribution:
    P_total = P_c + P_USB + P_LSB
    P_c = carrier power = V_c²/(2R)
    P_USB = P_LSB = P_c × m²/4
    P_total = P_c(1 + m²/2)
    Max efficiency: η = m²/(2+m²), at m=1 → η = 33.3%

  FM Bandwidth (Carson's Rule):
    BW = 2(Δf + fₘ) = 2fₘ(1 + β)
    For narrowband FM: β << 1, BW ≈ 2fₘ
    For wideband FM:   β >> 1, BW ≈ 2Δf

  FM Power: always A_c²/2R (constant, independent of β)
    Power distributes in infinite sidebands

# Pre-emphasis & De-emphasis
  Pre-emphasis: boost high frequencies before modulation
  De-emphasis: attenuate high frequencies after demodulation
  Improves SNR by ~13 dB for FM

# Information Theory (Shannon)
  Entropy: H(X) = −Σ p(xᵢ) log₂ p(xᵢ)
  Max entropy (equiprobable): H_max = log₂ N
  Channel capacity: C = B × log₂(1 + S/N) bits/sec

  For S/N in dB: (S/N)_linear = 10^(SNR_dB/10)

# Sampling & Quantization
  PCM:
    Bit rate = n × f_s  (n bits per sample, f_s sampling rate)
    SQNR = 1.76 + 6.02n dB
    Minimum B/W (using sinc pulse): n×f_s/2

  Companding: μ-law (US) and A-law (Europe)
    Improves SQNR for small signals

  DM (Delta Modulation):
    Step size Δ = fixed
    Slope overload: when signal slope > Δ/T_s
    Granular noise: when signal nearly constant

# Line Coding & Baseband
  Polar NRZ: BW = f_b, DC present
  Manchester: BW = 2f_b, no DC, self-clocking
  AMI: BW = f_b/2, no DC

  Matched Filter: maximizes SNR at sampling instant
    h(t) = x(T−t) (time-reversed input)
    Peak SNR = 2E/N₀

# Error Probability (AWGN)
  BPSK: P_e = Q(√(2E_b/N₀))
  BFSK (coherent): P_e = Q(√(E_b/N₀))
  BFSK (non-coherent): P_e = (1/2)e^(−E_b/(2N₀))
  QPSK: P_e ≈ Q(√(2E_b/N₀)) (same as BPSK but double data rate)

  Q(x) = (1/2)erfc(x/√2)

# Noise Figure & Friis Formula
  Noise Figure: F = (SNR)_in / (SNR)_out = 1 + T_e/T₀

  Friis Noise Formula (cascaded stages):
    F_total = F₁ + (F₂−1)/G₁ + (F₃−1)/(G₁G₂) + ...

  Effective Noise Temperature:
    T_e = T₀(F − 1)    where T₀ = 290K

  Noise Power: N = kTB
  k = Boltzmann's constant = 1.38 × 10⁻²³ J/K

# Key SNR Comparisons
  AM envelope detector: SNR_out/SNR_in = m²/(2+m²)
  DSB-SC coherent:      SNR_out/SNR_in = 1
  SSB coherent:          SNR_out/SNR_in = 1
  FM (wideband):         SNR_out/SNR_in = 3β² (processing gain)

# Matrix Properties
  Rank of matrix: max number of linearly independent rows/cols
  For m×n matrix: rank ≤ min(m, n)
  Rank(A) = Rank(Aᵀ) = Rank(AAᵀ) = Rank(AᵀA)

  Invertible (non-singular): det(A) ≠ 0, rank(A) = n
  Singular: det(A) = 0, rank(A) < n

  Eigenvalues: det(A − λI) = 0 (characteristic equation)
  Eigenvectors: (A − λI)v = 0

# Solving Systems: Ax = b
  Unique solution: rank(A) = rank(A|b) = n
  Infinite solutions: rank(A) = rank(A|b) < n
  No solution: rank(A) < rank(A|b)

  Cramer's Rule: xᵢ = det(Aᵢ)/det(A) where Aᵢ = A with col i replaced by b

# Cayley-Hamilton Theorem
  Every matrix satisfies its own characteristic equation
  A^n can be expressed as polynomial of degree (n−1) in A

# Differentiation
  d/dx (xⁿ) = nx^(n−1)
  d/dx (eˣ) = eˣ,  d/dx (ln x) = 1/x
  d/dx (sin x) = cos x,  d/dx (cos x) = −sin x
  Chain rule: d/dx f(g(x)) = f'(g(x)) × g'(x)
  Product rule: (uv)' = u'v + uv'
  Quotient rule: (u/v)' = (u'v − uv')/v²

  L'Hôpital's: lim f(x)/g(x) = lim f'(x)/g'(x)  (0/0 or ∞/∞)

# Integration
  ∫xⁿdx = x^(n+1)/(n+1),  ∫eˣdx = eˣ
  ∫sin x dx = −cos x,  ∫cos x dx = sin x
  ∫(1/x)dx = ln|x|
  ∫₀^∞ e^(−ax)dx = 1/a
  ∫₋∞^∞ e^(−ax²)dx = √(π/a)

# Multivariable (Partial Derivatives)
  Gradient: ∇f = (∂f/∂x, ∂f/∂y, ∂f/∂z)
  Divergence: ∇·F = ∂Fx/∂x + ∂Fy/∂y + ∂Fz/∂z
  Curl: ∇×F
  Laplacian: ∇²f = ∂²f/∂x² + ∂²f/∂y² + ∂²f/∂z²

# Laplace Transform (Common)
  L{1} = 1/s,    L{tⁿ} = n!/s^(n+1)
  L{e^(at)} = 1/(s−a)
  L{sin(at)} = a/(s²+a²)
  L{cos(at)} = s/(s²+a²)
  L{t·f(t)} = −dF(s)/ds

# Probability Rules
  P(A∪B) = P(A) + P(B) − P(A∩B)
  P(A|B) = P(A∩B) / P(B)          (Conditional)
  P(A∩B) = P(A) × P(B|A)          (Multiplication)
  Bayes': P(A|B) = P(B|A)P(A) / P(B)

  Independent: P(A∩B) = P(A)P(B)
  Mutually Exclusive: P(A∩B) = 0

# Random Variables
  E[X] = Σ xᵢp(xᵢ)  (discrete),  ∫ xf(x)dx (continuous)
  Var(X) = E[X²] − (E[X])²
  E[aX+b] = aE[X]+b,  Var(aX+b) = a²Var(X)

# Distributions
  Uniform [a,b]:  E = (a+b)/2,  Var = (b−a)²/12
  Binomial(n,p):  E = np,      Var = np(1−p)
  Poisson(λ):     E = λ,       Var = λ
  Exponential(λ): E = 1/λ,     Var = 1/λ²
  Normal(μ,σ²):   E = μ,       Var = σ²
    P(X ≤ x) = Φ((x−μ)/σ)

# Central Limit Theorem
  ΣXi (n iid) → Normal(μ, σ²/n) as n → ∞

# ODE Solution Methods
  First Order Linear: y' + P(x)y = Q(x)
    IF = e^(∫P dx),  y = (1/IF)∫(IF × Q)dx

  Second Order: ay'' + by' + cy = 0
    Characteristic: ar² + br + c = 0
    Distinct real r₁,r₂: y = C₁e^(r₁x) + C₂e^(r₂x)
    Repeated r: y = (C₁ + C₂x)e^(rx)
    Complex α±jβ: y = e^(αx)(C₁cosβx + C₂sinβx)

# Complex Variables
  Cauchy-Riemann: ∂u/∂x = ∂v/∂y,  ∂u/∂y = −∂v/∂x
  Analytic: f(z) is differentiable everywhere in domain
  Cauchy's Integral: f(a) = (1/2πj)∮f(z)/(z−a)dz
  Residue Theorem: ∮f(z)dz = 2πj × Σ Residues