Written Examination

Administration of the Written Exam

The written examination is given two times per year, usually a few days before classes begin each semester. Students must pass the written examination within one year of entry into the graduate program. They are given two chances to pass and three opportunities to take the exam. For instance, students entering in the fall semester may attempt the exam in August just before the start of the student’s first fall semester, in the end of January just before the start of spring semester, and the following August. Students must inform the ECE Graduate Studies Office of their intention to take the written examination at least one week before the exam is scheduled to be given unless an earlier registration deadline has been stated otherwise.

Format of Exam

The Ph.D. written qualifying exam is comprised of 10 sections, each covering a different topic related to Electrical and Computer Engineering. Students must answer five out of the 10 sections. Each section is graded out of 20 points, such that the maximum attainable cumulative score on the exam is 100 points. Calculators, cell phones and all other hand-held electronic devices are not permitted in the exam. The exam is closed-book and closed-notes. The questions shall be composed and graded with these constraints in mind. The total time allowed for the exam is 3 hours and 45 minutes (i.e., 45 minutes on average per section.)

Exam Sections and Topics

The questions are chosen to test understanding of standard undergraduate material at or below the junior level. The list provided below describes the 10 sections of the exam and the topics covered under each section.

Previous semesters’ exam questions are posted here.

Basic Mathematics

  1. Calculus
    1. Derivatives and integrals
    2. Taylor series
    3. Limits and convergence
  2. Elementary Linear Algebra
    1. Matrices and vectors – linear independence and orthogonality
    2. Eigenvalues and eigenvectors
  3. Differential Equations
    1. First order linear and nonlinear differential equations
    2. Second-order linear ordinary differential equations
    3. Systems of first order linear ordinary differential equations
  4. Vector Calculus
    1. Partial derivatives, curl, gradient, vector fields
    2. Line, surface, and volume integrals

Probability

  1. Basic probability
    1. Random experiments, axioms of probability
    2. Conditional probability, Bayes' theorem, independence
    3. Permutations and combinations, counting methods
    4. Discrete, independent, identically distributed trials: binomial, multinomial, and geometric distributions
    5. Poisson distribution and its applications
  2. Random Variables
    1. Probability density (PDF), probability mass (PMF) and cumulative distribution (CDF) functions
    2. Common continuous distributions: uniform, exponential, Gaussian, Laplace
    3. Expectation and variance; fundamental theorem of expectation
    4. Simple transformations of random variables
  3. Multiple Random Variables
    1. Joint and marginal distributions
    2. Conditional PDFs and PMFs; iterated (total) expectation
    3. Basic properties of independent random variables
    4. Correlation and covariance

Electromagnetism

  1. Electrostatics and Magnetostatics
    1. Gauss’s, Ampere’s, Biot-Savart’s, and Coulomb’s laws
    2. Boundary conditions at dielectric and conducting interfaces
    3. Scalar and vector potential
    4. Calculating capacitance and inductance
  2. Electrodynamics and Waves
    1. Maxwell’s equations
    2. Wave equation and Helmholtz equation
    3. Plane wave solutions in lossy and lossless materials
    4. Poynting vector and power density
  3. Reflection and Transmission of Plane Waves
    1. Normal incidence at dielectric and conducting interfaces
    2. Reflection from multiple layers (normal incidence)
    3. Oblique incidence at dielectric and conducting interfaces
    4. Snell’s law, Brewster’s angle and total internal reflection
  4. Transmission Lines
    1. Transmission line equations for V and I (Telegraphist’s equations)
    2. Characteristic inductance, capacitance, and impedance of transmission lines
    3. Input impedance of terminated transmission lines
  5. Waveguides
    1. TE and TM modes of metallic waveguides
    2. Dispersion relations and cutoff frequencies
    3. Parallel plate and rectangular waveguide

Circuits

  1. Linear Circuit Analysis
    1. Series and parallel combinations, voltage and current dividers
    2. Node-voltage and mesh-current methods
    3. Norton and Thevenin equivalent sources
  2. AC Circuits Analysis
    1. Phasor representation of sinusoidally varying signals
    2. Complex impedance
    3. Time averaged power
    4. Transfer functions and frequency response
  3. Transient Analysis
    1. 1st order RC and RL circuits
    2. 2nd order RLC circuits
  4. Ideal Operational Amplifier Circuits
    1. Common op-amp circuits: buffer, inverting, non-inverting, summing, integrating, differentiating amplifier circuits
    2. Analyzing ideal op-amp circuits
  5. Diode Circuits
    1. Ideal and non-ideal idiode operation
    2. Rectifers
    3. Load lines and biasing
  6. BJT and CMOS Transistors
    1. DC analysis and biasing
    2. Small signal equivalent models
  7. Transistor circuits
    1. CMOS digital gate circuits and inverters
    2. Amplifiers and small signal equivalent circuits
    3. Current sources, mirrors, differential pairs

Linear Systems and Signals

  1. Linear Time-Invariant (LTI) Systems
    1. Basic concepts: linearity, time invariance, causality
    2. Convolution (discrete and continuous)
    3. Impulse and step response (discrete and continuous)
  2. Fourier Analysis in Continuous Time
    1. Fourier series of a periodic signal; determination of coefficients
    2. Fourier transform; basic properties and pairs
  3. Fourier Analysis in Discrete Time
    1. Discrete-time Fourier transform (DTFT) of a sequence; basic properties and pairs
  4. Continuous-Time LTI Systems in the s-Domain
    1. Laplace transform and its properties; regions of convergence
    2. Systems described by differential equations
    3. Transfer function: poles, zeros; causality and stability
    4. Determination of system output using the Laplace transform
    5. Relationship between Laplace and Fourier transforms; response of stable LTI systems to exponential and sinusoidal inputs; input-output relationship in the frequency domain

Devices

  1. Elementary Properties of Materials
    1. Semiconductors
    2. Conductors
    3. Insulators
  2. Basic Solid State Devices: PN Junctions, Bipolar Transistors, MOS Capacitors, MOSFETs
    1. Fundamental structure of these device
    2. Their internal operation in terms of electrons and holes, drift and diffusion currents, and electromagnetic principles
    3. Operation of these devices as circuit elements

Computer Architecture and Systems

  1. Pipelines and their analysis
  2. Caches and their analysis
  3. Assembly code and program analysis
  4. Virtual memory
  5. Multitasking and process management

Digital Logic

  1. Boolean Algebra and Boolean Simplification (K-Maps and Quinn McCluskey methods)
  2. Complex Logic Design With Simple Boolean Functions (Adders, Subtractors etc.)
  3. Flip Flops
  4. Synchronous Sequential Systems

Basic Physics

  1. Newtonian Mechanics
    1. Kinematics
    2. Newtons laws of motion
    3. Work, energy and power
    4. Momentum and center of mass
    5. Circular motion and rotation
    6. Harmonic oscillation
  2. Waves and Optics
    1. Wave equations
    2. Traveling waves and standing waves
    3. Interference, superposition and diffraction
  3. Thermodynamics and Heat
    1. Ideal gas law
    2. Pressure, work, and heat
    3. Laws of thermodynamics
  4. Modern Physics and Quantum Mechanics
    1. Schrodinger wave equation
    2. Operators, eigenfunctions and eigenvalues
    3. One dimensional potentials - bound and unbound solutions
    4. Angular momentum and spin

Software

  1. Data types, Variables and Operators
  2. Program Control and Structure
    1. Expressions, declarations, and statements
    2. Functions, arguments, return values, and recursion
    3. Storage classes and variable scope
    4. Loop structures (for, while, do)
    5. Conditional execution (if-then-else, switch)
  3. Input and Output
    1. Formatted input and output (printf, scanf)
    2. Basic file I/O (fopen, fclose, fprintf, fscanf)
  4. Arrays
    1. Strings
    2. Arrays
    3. Pointers
  5. Dynamic Memory Allocation (malloc, calloc, free, realloc)
  6. Structures
  7. Linked lists
  8. The C pre-processor
    1. #include, #define, #ifdef
    2. Standard library header files

The software section covers undergraduate programming at a level treated in most Electrical and Computer Engineering curricula. In this exam, ANSI C is used consistently as the specific language for programs. It is insufficient for students to convey the “general idea” of a solution through pseudocode or code in some other programming language; syntactically and logically correct ANSI C code must be provided to receive full credit.