Self-Study Syllabus

Phase 1 — Linear Algebra May – mid Jun 2026 · ~70 hrs

3Blue1Brown — Essence of Linear Algebra · Singh — Linear Algebra: Step by Step

Completed

Dot Product Vectors, angles, and projection 3B1B · Singh viz →

Cross Product Perpendicular vectors and parallelogram area 3B1B · Singh viz →

Determinant Area scaling, orientation, and collapse — ad − bc 3B1B · Singh viz →

Inverse Transformation A⁻¹ undoes A — round-trip identity, singular failure 3B1B · Singh viz →

Eigenvectors Av = λv — directions a matrix only stretches 3B1B · Singh viz →

Eigenspaces E_λ = Null(A − λI) — multiplicity & defective matrices 3B1B · Singh viz →

To do

Linear Transformations Rotations, reflections, shears, projections as a unified concept 3B1B v3

Matrix Multiplication as Composition Applying A then B = BA; why order matters geometrically 3B1B v4

Span, Basis & Linear Independence What a basis is; when vectors are redundant; dimension 3B1B v2 · Singh

Column Space & Null Space What A can reach; what it maps to zero; rank-nullity theorem 3B1B v7 · Singh viz → viz →

Change of Basis Same transformation, different coordinate frame; P⁻¹AP 3B1B v13 · Singh

Orthogonal Projection Projecting onto a subspace; least squares Singh

Eigendecomposition & Diagonalisation A = PDP⁻¹; matrix powers; why it matters 3B1B v14 · Singh

Singular Value Decomposition The most general decomposition; every matrix Singh

Phase 2 — Thomson: Introduction to Space Dynamics mid Jun – mid Jul 2026 · ~50 hrs

Thomson — Introduction to Space Dynamics (Dover)

Completed — Kinematics

Circular Motion v = rω, centripetal and tangential acceleration Thomson §1 viz →

Angular Velocity Why ω is orthogonal, and v = ω × r Thomson §1 viz →

Plane Motion Radial and transverse components in polar coordinates Thomson §1 viz →

Instantaneous Center The point about which the body rotates instantaneously Thomson §1 viz →

Euler Angles ψ, θ, φ and transformation of displacements Thomson §1 viz →

Areal Rate r × v, equal areas, and Kepler's second law Thomson §2 viz →

Completed — Orbital Mechanics

Kepler's First Law Orbits are conic sections — Newton's inverse-square law via Binet's substitution Thomson §4 viz →

Kepler's Second Law Equal areas in equal times — conservation of h = r²θ̇ Thomson §4 viz →

Kepler's Third Law T² = 4π²a³/K — period depends only on semi-major axis Thomson §4 viz →

Satellite Orbits Radial & transverse equations, Binet's substitution, orbit as a conic Thomson §4 viz →

Two-Body Problem CoM reduction, relative coordinate, and reduced mass Thomson §4 viz →

Orbit from Initial Conditions χ, eccentricity from v₀ and β₀, Fig 4.9-2 chart Thomson §4.9 viz →

Hohmann Transfer Cotangential transfer between coplanar circular orbits, Δv₁, Δv₂ Thomson §4 viz →

Repulsive Inverse-Square Force F = +K/r² produces repulsive hyperbola — Rutherford scattering Thomson §4.13 viz →

Impulsive Orbit Change Δv at θ* transfers between orbits sharing the apse line Thomson §4.13 viz →

Orbit from Burnout Given χ, β₀, r₀/R at engine cutoff — derive e, a/R, θ₀ Thomson §4.13 viz →

Completed — Gyrodynamics

Polhode & Herpolhode Motion of the instantaneous rotation axis in body and space frames Thomson §5 viz →

To do — Rigid Body Dynamics & Gyrodynamics

Euler's Equations for Rigid Body Rotation Body-frame equations of motion; torque-free case Thomson §3

Moment of Inertia Tensor Principal axes, eigenvalues of I, Steiner's theorem Thomson §3

Precession & Nutation Spinning top under gravity; steady precession conditions Thomson §5

Gyroscope Dynamics Gyroscopic couple, stabilisation, spacecraft attitude control Thomson §5

Phase 3 — Differential Equations mid Jul – Oct 2026 · ~130 hrs

Arnold — Ordinary Differential Equations · 3Blue1Brown — Differential Equations

Completed

ODE Overview Slope fields, solution curves, initial conditions, order and linearity Arnold · 3B1B viz →

The Pendulum SHM, phase portrait, separatrix, nonlinear period corrections Arnold · 3B1B viz →

Heat Equation ∂u/∂t = α ∂²u/∂x² — Fourier modes, diffusion, decay rates 3B1B viz →

To do — 3Blue1Brown

Fourier Series Decomposing functions into sine/cosine modes; why they solve the heat equation 3B1B

Laplace Transforms Solving ODEs via algebraic manipulation in s-domain 3B1B

To do — Arnold

Phase Portraits & Vector Fields Qualitative analysis of autonomous systems; equilibria and orbits Arnold §1–2

Stability of Equilibria Lyapunov stability; linearisation; stable vs unstable fixed points Arnold §3

Linear ODE Systems x' = Ax; eigenvalue classification of phase portraits Arnold §4

Limit Cycles Isolated closed orbits; Poincaré–Bendixson theorem Arnold §5

Structural Stability & Bifurcations How phase portraits change as parameters vary Arnold §6

Differential Equations on Manifolds Flows on surfaces; torus; global qualitative behaviour Arnold §7–8

Phase 4 — Oppenheim & Willsky: Signals and Systems Oct – Dec 2026 · ~80 hrs

Oppenheim & Willsky — Signals and Systems (2nd ed.)

Continuous-time signals & systems

Signals & Systems — Classification Memory, causality, stability, invertibility; continuous vs discrete; energy and power signals O&W §1

CT LTI Systems & Convolution Convolution integral, impulse response, BIBO stability, causality from h(t) O&W §2

Fourier Series (CT) Orthogonality, synthesis and analysis equations, Gibbs phenomenon, Parseval's theorem O&W §3

Continuous-Time Fourier Transform CTFT pair, convolution theorem, Parseval, duality; spectrum of standard signals O&W §4

Frequency Response of CT LTI Systems H(jω), ideal filters, first- and second-order system frequency response O&W §3–4

Sampling Theorem Nyquist rate, aliasing, reconstruction; the bridge from CT to DT O&W §7

Laplace & Z-transforms (system perspective)

Laplace Transform & Transfer Functions Region of convergence, poles and zeros, partial fractions, system stability from pole locations O&W §9

Block Diagrams & System Interconnections Series, parallel, feedback; unilateral Laplace for IVPs; signal flow graphs O&W §9–10

Z-Transform ROC, poles and zeros, inverse Z-transform; DT system stability O&W §10

Phase 5 — Control Systems Dec 2026 – Feb 2027 · ~80 hrs

Ogata — Modern Control Engineering (5th ed.) · Franklin — Feedback Control of Dynamic Systems

Classical control

Mathematical Models of Systems Transfer functions from ODEs; block diagram algebra; linearisation around operating points Ogata §2

Time-Domain Response First- and second-order step response; damping ratio, natural frequency, settling time, overshoot Ogata §4

Root Locus Closed-loop poles as gain varies; construction rules; designing for desired transient specs Ogata §6

Frequency Response & Bode Plots Magnitude and phase vs frequency; asymptotic Bode construction; gain and phase margins Ogata §7

Nyquist Stability Criterion Encirclement condition; stability margins in the frequency domain; robustness Ogata §8

PID Controllers Proportional, integral, derivative action; Ziegler–Nichols tuning; practical limitations Ogata §8

Modern (state-space) control

State-Space Representation ẋ = Ax + Bu, y = Cx + Du; eigenvalues as poles; solution via matrix exponential Ogata §9

Controllability & Observability Controllability matrix rank; observability matrix rank; duality theorem Ogata §10

State Feedback & Pole Placement Full-state feedback u = −Kx; placing closed-loop poles; integral action for zero steady-state error Ogata §10

State Observers (Luenberger) Estimating states from outputs; observer gain; separation principle — direct precursor to Kalman filter Ogata §10 · Franklin §7

LQR & Optimal Control Quadratic cost minimisation; Riccati equation; relationship between LQR and Kalman filtering (LQG) Franklin §9

Phase 6 — Abbott: Understanding Analysis Feb – Apr 2027 · ~90 hrs

Abbott — Understanding Analysis (Springer UTM, 2nd ed.)

The Real Numbers Axiom of completeness, Archimedean property, uncountability of ℝ Abbott §1

Sequences & Series Convergence, Cauchy sequences, monotone convergence, infinite series Abbott §2

Basic Topology of ℝ Open and closed sets, compactness, connectedness — the real line version before Armstrong's general treatment Abbott §3

Functional Limits & Continuity ε–δ definition, uniform continuity, extreme value and intermediate value theorems Abbott §4

The Derivative Rigorous definition, mean value theorem, L'Hôpital's rule Abbott §5

Sequences & Series of Functions Pointwise vs uniform convergence; power series; Taylor series with rigorous remainder Abbott §6

The Riemann Integral Darboux definition, integrability criteria, fundamental theorem of calculus Abbott §7

Phase 7 — Armstrong: Basic Topology Apr – Sep 2027 · ~160 hrs

Armstrong — Basic Topology (Springer UTM) · Abbott §3 provides the ℝ warm-up

Topological Spaces & Continuity Open sets, neighbourhoods, continuous maps — without distance Armstrong §1–2

Homeomorphisms Topological equivalence; invariants that tell spaces apart Armstrong §2

Connectedness Path-connected vs connected; intermediate value theorem revisited Armstrong §3

Compactness Open covers; Heine–Borel; why compact spaces behave nicely Armstrong §3

Identification Spaces & Quotient Topology Gluing constructions; torus, Möbius band, Klein bottle Armstrong §4

The Fundamental Group π₁ — loops based at a point; homotopy; simply connected spaces Armstrong §5

Classification of Surfaces Genus, orientability; every compact surface is a sphere, connected sum of tori, or projective planes Armstrong §7

Simplicial Homology Triangulations; boundary operator; H₀, H₁, H₂ — holes in different dimensions Armstrong §8

Phase 8 — Blitzstein & Hwang: Introduction to Probability Sep – Nov 2027 · ~90 hrs

Blitzstein & Hwang — Introduction to Probability (CRC Press) · Stat 110 Harvard lectures (YouTube)

Counting & Naive Probability Multiplication rule, permutations, combinations; birthday problem B&H §1

Conditional Probability & Bayes P(A|B), law of total probability, Bayes' theorem B&H §2

Random Variables & Distributions PMF, CDF, Bernoulli, Binomial, Geometric, Poisson B&H §3–4

Expectation & Variance E[X], linearity of expectation, LOTUS, Var(X) B&H §4–5

Continuous Distributions PDF, Uniform, Normal, Exponential, Beta, Gamma B&H §5–8

Joint Distributions & Independence Joint PDF/PMF, marginals, covariance, correlation B&H §7

Law of Large Numbers & Central Limit Theorem Convergence in probability; normal approximation B&H §10

Markov Chains Transition matrices, stationary distributions, PageRank B&H §11

Phase 9 — RF & Communications Fundamentals Nov 2027 – Jan 2028 · ~70 hrs

Proakis & Salehi — Communication Systems Engineering (2nd ed.) · Haykin — Communication Systems

Foundations

Channel Models & Noise AWGN channel, SNR, noise power spectral density; thermal noise and N₀/2 Proakis §3

Analog Modulation AM, DSB-SC, SSB, FM, PM; bandwidth and power tradeoffs; coherent vs envelope detection Proakis §3

Digital communications

Digital Modulation — BPSK, QPSK, QAM Signal constellations, decision regions, BER vs Eb/N₀; matched filter receiver Proakis §5

Spread Spectrum — DSSS & FHSS PN sequences, processing gain, jamming margin; CDMA; GPS L1 C/A signal structure Proakis §9

Channel Coding & Shannon Capacity Shannon's theorem, capacity–bandwidth tradeoff; Hamming codes; convolutional codes and Viterbi Proakis §8

RF propagation & link budgets

Antenna Fundamentals Gain, directivity, effective aperture, beam pattern; dipole and patch antennas for GNSS receivers Proakis §2 · Balanis §1–2

Link Budget & Friis Equation EIRP, path loss, receiver sensitivity, C/N₀; computing GPS received power from 20,200 km Proakis §2

Multipath & Ionospheric Effects Rayleigh fading, Doppler shift, ionospheric delay model; how these degrade GNSS pseudorange Proakis §13 · Kaplan §7

Phase 10 — Signal Processing & Estimation Theory Dec 2027 – Mar 2028 · ~140 hrs

Oppenheim & Schafer — Discrete-Time Signal Processing · Kay — Statistical Signal Processing · Kaplan & Hegarty — Understanding GPS/GNSS

Discrete-time signals & systems

Sampling & Reconstruction Nyquist theorem, aliasing, reconstruction from samples Oppenheim §1

Convolution & LTI Systems Impulse response, linearity, time-invariance, BIBO stability Oppenheim §2

Z-Transform Region of convergence, poles and zeros, inverse Z-transform Oppenheim §3

Discrete Fourier Transform & FFT DFT as sampled spectrum; FFT algorithm; spectral leakage and windowing Oppenheim §8

Statistical signal processing

Estimation Theory MVUE, Cramér–Rao bound, maximum likelihood estimation, bias vs variance Kay Vol.1 §1–3

Detection Theory Hypothesis testing, Neyman–Pearson, ROC curves, matched filter Kay Vol.2 §1–3

Wiener Filter Optimal linear filter for stationary processes; MMSE estimation Kay Vol.1 §12

Kalman filtering — the core target

Linear Kalman Filter State-space model, predict–update cycle, optimal gain, covariance propagation Kay · Welch & Bishop tutorial

Extended & Unscented Kalman Filter Linearisation for nonlinear systems; UKF sigma-point approach; orbit determination Kay · Crassidis & Junkins

Particle Filters Sequential Monte Carlo; non-Gaussian state estimation; tracking highly nonlinear dynamics Arulampalam et al. tutorial

Navigation & PNT

GNSS Architecture & Signal Structure GPS/Galileo/GLONASS signal design; pseudorange, carrier phase, satellite geometry (DOP) Kaplan & Hegarty §1–4

Spoofing, Jamming & Resilient PNT Threat taxonomy; detection algorithms; alternative and complementary positioning (IMU, eLoran, LEO PNT) Kaplan §9 · CISA advisories

Orbit Determination & SSA Batch least squares, sequential estimation, conjunction analysis — Thomson dynamics + Kalman estimation combined Vallado · Crassidis & Junkins

Phase 11 — Prince: Understanding Deep Learning Mar 2028+ · ~120 hrs

Prince — Understanding Deep Learning (MIT Press, 2024) · free PDF at udlbook.github.io

Foundations

Supervised Learning Input–output mappings, loss functions, empirical risk minimisation Prince §2

Shallow Neural Networks One hidden layer, activation functions, universal approximation Prince §3

Deep Neural Networks Composing layers, depth vs width, piecewise linear regions Prince §4

Loss Functions MSE, cross-entropy, maximum likelihood perspective Prince §5

Training

Gradient Descent & Backpropagation SGD, mini-batch, chain rule through computation graphs Prince §6

Gradients & Initialisation Vanishing/exploding gradients, He/Xavier init, batch norm Prince §7

Measuring Performance Bias–variance tradeoff, train/val/test split, double descent Prince §8

Regularisation L1/L2, dropout, data augmentation, early stopping Prince §9

Architectures

Convolutional Networks Convolution, pooling, receptive field, translation equivariance Prince §10

Residual Networks Skip connections, batch norm, modern CNN architectures Prince §11

Transformers Self-attention, multi-head attention, positional encoding, encoder–decoder Prince §12

Graph Neural Networks Message passing, node/edge/graph classification, relational inductive bias Prince §13

Generative models

Unsupervised Learning Clustering, dimensionality reduction, self-supervised pretraining Prince §14

Generative Adversarial Networks Minimax game, training instability, mode collapse, Wasserstein GAN Prince §15

Normalizing Flows Bijective mappings, change-of-variables formula, exact likelihood Prince §16

Variational Autoencoders Latent variable models, ELBO, reparameterisation trick Prince §17

Diffusion Models Forward noising process, denoising score matching, DDPM Prince §18

Reinforcement learning

Reinforcement Learning MDP, policy/value functions, Q-learning, policy gradient, RLHF Prince §19