The requirements for the Ph.D. and Eng.Sc.D. degrees are identical.
Both require a dissertation based on the candidate’s original research,
conducted under the supervision of a faculty member. The work may be
theoretical or experimental or both. Students who wish to become
candidates for the doctoral degree in electrical engineering have the
option of applying for admission to the Eng.Sc.D. program or the Ph.D.
program. Students who elect the Eng.Sc.D. degree register in the School
of Engineering and Applied Science; those who elect the Ph.D. degree
register in the Graduate School of Arts and Sciences. Doctoral
candidates must obtain a minimum of 60 points of formal course credit
beyond the bachelor’s degree. A master’s degree from an accredited
institution may be accepted as equivalent to 30 points. A minimum of 30
points beyond the master’s degree must be earned while in residence in
the doctoral program. These requirements are summarized on the Ph.D. Requirements Checklist.
For application information click here.
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1. General Remarks
Admission as a graduate student generally requires a bachelor of
science degree. Most graduate students in the Department of Electrical
Engineering have, as undergraduates, majored in electrical engineering,
physics, or mathematics. Outstanding students with a bachelor's degree
in other fields of science and engineering may also be admitted as
graduate students. Usually, however, they will be required to remove
certain undergraduate deficiencies without gaining credit toward the
advanced degree. All applicants for the M.S., Ph.D., and Eng.Sc.D.
degrees, including Columbia undergraduates, are required to take the
graduate record examination (GRE) general test. The application
deadline is December 1 for students who wish to be considered for
financial aid. Only completed applications will be considered. All
students must have a satisfactory command of the English language.
Students whose native language is not English are required to submit
results of the Test of English as a Foreign Language (TOEFL) with their
application. Students who score below 550 on the TOEFL will not be
admitted. Students whose native language is not English must pass the
English Language Test given by the American Language Program (ALP) at
Columbia before they graduate. An arrangement to take the test must be
made with the ALP in 505 Lewisohn Hall (tel: 212-854-3584) to make an
appointment. Usually results are available three days following the
test. Candidates for the M.S. degree must reach level 8 and candidates
for the doctoral degree must reach level 10 before they defend their
thesis. Students failing to reach those levels on the placement
examination will usually be required to register for an English course.
2. Classification of Graduate Students
Students are admitted under the following classifications:
- M.S. Candidate
- M.S. Doctoral-Track Candidate
- Ph.D. Candidate
- Eng.Sc.D. Candidate
- Professional-Degree Candidate
- Matriculated Special Student
- One-Time Special Student
3. M.S. Doctoral-Track Candidates
Exceptionally qualified students may be admitted to M.S.
doctoral-track candidacy directly after receiving the B.S. degree.
These students may gain admission to the doctoral program upon
satisfactory completion of the requirements for the M.S. degree,
although a formal application (without new reference letters) must be
approved by their adviser. The usual practice for potential doctoral
degree candidates is, however, to apply for doctoral candidacy after
completing the requirements for the M.S. degree.
4. Doctoral Degree (Ph.D. and Eng.Sc.D.)
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The basic requirement is a master's degree in engineering, science,
or mathematics with a superb record. It is very difficult to meet all
doctoral-degree requirements by evening attendance only.
Rules Applicable to all Graduate Students1. Academic Regulations and Requirements
Rules and requirements, applicable to all graduate students, are
contained in the current bulletin of the School of Engineering and
Applied Science (SEAS) and the bulletin of the Graduate School of Arts
and Sciences (GSAS).
2. Scholastic Standing
All degree candidates must maintain a minimum scholastic standing,
measured by a grade-point average, as indicated under the various
A special student must maintain the following schedule of averages from
the time of first registration in order to be allowed to continue:
- 1-4 points. No minimum requirement on average
- 5 points 1.00
- 6 points 1.25
- 7 points 1.50
- 8 points 1.75
- 9 points + 2.00
Averages are calculated on the basis of A=4, B=3, C=2, D=1, and F=0.
Courses in which the candidate receives the grade of R (registration
only) or F (failure) are not credited toward a degree. However, F
grades are included in calculating the point average. A grade of P
(pass) gives course credit but does not affect the grade-point average.
Students in the Electrical Engineering Department may drop a course
after the change-of-program period, but before the first examination or
the midterm date (whichever occurs earlier), provided they receive the
permission of their adviser and the dean. Withdrawal from a course at a
later time requires the permission of the chair of the Department of
Electrical Engineering and the dean. Such permission is not generally
granted to students with poor performance in the course.
privilege of auditing courses is granted to two kinds of regular degree
candidates: Those who in any term are registered for 15 points or more
or those who have completed their course requirements for the doctorate
and are registered for full-time research.
is made at the registrar's office during the change of program period
(Monday through Friday following registration).
Teaching Assistants may not register for more than three courses
without the written permission of the chairman of the Financial Aid
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Requirements for the Doctoral Degree
Students who wish to become candidates for the doctoral degree in
electrical engineering have the option of applying for admission to
either the Eng.Sc.D. or the Ph.D. program. Students who elect the
Eng.Sc.D. degree register in the School of Engineering and Applied
Science, and those who choose the Ph.D., register in the Graduate
School of Arts and Sciences. Candidates for either degree must earn a
minimum of 60 points of credit beyond the bachelor's degree. A master's
degree from an accredited institution may be accepted as the equivalent
of 30 points. A minimum of 30 points beyond the master's degree must be
earned while in residence at Columbia , and at least 15 of these points
must be obtained while registered in the doctoral program. A student
for the Eng.Sc.D. must fulfill the additional point requirements stated
in Part 2 below. For Ph.D. candidates, the 60-point minimum will
represent a partial fulfillment of the residence-unit requirements
stated in part 3 below. One residence unit is essentially equivalent to
15 points, but the student should read carefully the detailed
specifications as given in the appropriate university bulletin and in
Part 3 below.
3. Residence-Unit Requirements for the Ph.D. Degree
with the semester immediately following the one in which the minimum
point requirements of Part 1 are completed, the candidate must register
continuously for EE E9800 (3, 6, 9, or 12 points) until 12 points have
- Registration for EE E9800 at times other than as prescribed in (a) is not allowed except by written permission of the dean.
part of the credit for EE E9800 may be counted toward the minimum
requirements stated in Part 1. These credits are in addition to the
required 30 credits of course and research work.
registration is required to maintain enrollment in any degree program.
Continuous registration refers to the autumn and spring semesters, but
not the summer session. Failure to register will result in withdrawal
from the program. If a student completes the 9800 requirement and has
not yet completed the thesis requirements he or she may opt to take
ELEN E9900 for 0 points until the dissertation is complete. ELEN E9800
and E9900 courses are for DES students only. Ph.D. students should not
register for these courses at any time.
A residence unit is awarded for one term of full-time study, as
determined by the student's adviser. In some cases a student may be
allowed to register for either one-quarter or one-half residence unit
for one semester of part-time study. Six residence units beyond the
bachelor's degree are required for the Ph.D. Continuous registration
until completion of all requirements is obligatory. Students whose
degree programs are not yet complete at the end of the minimum
residence requirements must register for extended residence. A maximum
of two residence units may be credited for academic work completed at
another university or in another division of Columbia University.
Students are directed to the Graduate School of Arts and Sciences
(GSAS) bulletin for further information
A candidate is required to maintain a grade-point average of at least
3.5 at any time in the program. Students with an unsatisfactory average
may be asked to withdraw.
A prospective doctoral candidate follows a program of study formulated
in consultation with his/her faculty adviser. The program must include
6. Doctoral Qualifying Examination (D.Q.E.)
- Courses used to satisfy the 30-point minimum beyond the M.S. degree will usually be at the 6000 level or higher.
- Not more than 6 points of seminar.
most 12 points of doctoral research (ELEN E9001, E9002, E9011, E9012,
E6001, and E6002) may be credited toward the degree. However, in
exceptional cases, in recognition of a student's advanced standing and
professional proficiency, this 12-point limitation may be waived at the
discretion of the chair of the doctoral committee.
Students enrolled in the Ph.D., Eng.Sc.D., or doctoral-track M.S.
programs must pass the Ph.D. qualifying examination, as part of the
requirements for the doctoral degree. The exam is given each January,
during the week before classes start. Students in the above programs
must take this exam in their first year at Columbia. It is emphasized
that research and academic advisers, or the student coordinator, are
not empowered to allow a student to postpone taking this examination. A
student who does not take the examination during his/her first year
following admission, without an extraordinary legitimate excuse
approved by the doctoral committee, will be assumed to have failed it.
Students studying for their M.S. degree, not currently in the
M.S.-Ph.D. track, but who plan to apply for a position in the Ph.D.
program, can also take the exam with the permission of the chair of the
doctoral committee. Passing the exam is not connected to Ph.D.
admissions (see Admission Requirements). However, if a student is
admitted to the Ph.D. program and has already passed the exam, he/she
will not have to take it again.
The qualifying exam consists of two parts: a written part and an oral part.
The written part is a breadth exam, and is the same for all students. Its duration is four hours. Its purpose is to test:
- Breadth of knowledge
- Understanding of fundamentals
- Ability to do research
The exam is based only on topics typically covered in undergraduate
classes, but the types of questions asked in it assume graduate level
maturity in terms of thinking ability and initiative. The examination
consists of 18 problems, three from each of the following six areas:
The exam is not tied to particular
courses. A syllabus for the material is given in Section D below.
Students must work out six and only six out of the 18 problems. If more
than six problems are handed in, six of those problems will be picked
at random for grading.
- Circuits and Electronics
- Signals, Systems, and Communications
- Solid-State Devices and Electromagnetics
- Digital Computing Systems
- Systems Biology and Neuroengineering
If a student comes from a department other than electrical engineering
and computer science, and brings significant expertise in an area
distinct from those covered in the EE written D.Q.E. examination, and
that knowledge is expected to be usable in a fundamental manner in the
student's Ph.D. thesis, then the student may request from the
department to only be responsible for four problems in the written
D.Q.E. exam. If the student research adviser concurs with that request,
then the student will have to make an oral presentation prior to the
D.Q.E., to a committee of three faculty members explaining the topic of
his/her proposed research, and demonstrating the use of that knowledge
in combination with a sufficiently broad area within electrical
engineering. The student's performance in that oral examination will
then be considered by the department in lieu of the additional problems
in the written D.Q.E. exam.
No calculators will be allowed into the written examination.
The oral part is a depth exam in one of the above areas, selected by
the student. This examination is not limited to the material of the
written examination. The oral exam is given within three working days
of the written exam. Each student is examined by several designated
faculty members on a one-to-one basis. These exams take place in
faculty offices, and each typically lasts no more than 15 minutes. The
scheduling of the slots for the oral exam is done by the graduate
The evaluation of the results of the oral and written exams is made by
the entire faculty, and students are judged on a department-wide basis.
Faculty members are not empowered to discuss any aspect of the exam or
its evaluation with the students. Questions concerning procedural
matters are to be addressed to the graduate student coordinator in the
Electrial Engineering Department office.
The department notifies the students of the results of the exam in
writing, within two weeks of the date of the written part. Students who
fail the exam the first time may take it once more in the following
year. Under no circumstances will a student be allowed to take the exam
more than twice.
Students who must take the qualifying exam in January are expected to
have taken by that time at least 6 credits of graduate (6000-level) or
senior/graduate (4000-level) courses at Columbia, and are expected to
obtain an average score of A- or better in those courses. However,
failure to meet this expectation is not a valid reason for postponing
to take the examination.
Students who want to take the qualifying exam must file the required
form by November 30. If they already have identified an adviser who is
tentatively willing to supervise their research, they should indicate
his/her name on the form. However, having a research adviser is not a
requirement for taking the D.Q.E.; in many instances a research adviser
will be identified after a student has passed the examination.
Students who are admitted to the Ph.D. program and pass the D.Q.E. must
still find an adviser willing to supervise their work in order to
continue in the Ph.D. program. Those who pass the D.Q.E. exam and do
not yet have a adviser should speak to potential research advisers and
identify one willing to supervise them as soon as possible. Once the
name of the adviser is known, it should be reported to the Graduate
Student Coordinator in the EE office. Students are expected to find and
declare a research adviser within three semesters (not including
summer) after passing their qualifying exam.
Students who pass the qualifying exam but cannot find an adviser in the
area of their choice willing to supervise them and/or support them, are
allowed to change to a different area without having to retake any
portion of the qualifying exam. Students who are receiving financial
support but fail the qualifying exam in January will generally continue
to be supported through the spring; however, support beyond that point
is not automatic.
Doctoral students must pass the Ph.D. thesis-proposal exam within two
years after they have passed the D.Q.E. Students who want to take this
exam will be expected to have a grade point average of at least 3.5 for
courses taken at Columbia. The thesis proposal exam is oral, and its
topic is chosen in coordination with the student's research adviser.
The exam is administered by a committee of at least three faculty
members, including the student's research adviser, and can last up to
two hours. The student makes an oral presentation of the state of the
art and his/her proposed research, possibly with new results. This oral
presentation typically lasts no more than twenty minutes, and is
followed by a period of questions by the committee. These can be any
questions the committee feels are necessary to judge the candidate's
knowledge and readiness for doctoral research in his/her chosen field,
and are not limited to the topic of the presentation. The student is
notified of the outcome shortly after the exam.
Doctoral students must complete a breadth requirement through course
work in a technical area clearly outside their main area of interest.
This course work should normally be outside the EE department, or even
outside the FFSEAS. In special cases, some of the courses can be inside
the EE department if it is clear that, together with the courses
outside the department, they form a cohesive whole in an area clearly
distinct from the main area of interest. In such cases, the student may
use up to two courses inside the EE department to fulfill the Breadth
Course Requirement, subject to the approval of the student’s adviser.
This course work should be planned in consultation with the student's
research or academic adviser in such a way as to expose the student to
concepts in a different discipline, and especially to different
approaches and ways of thinking than those prevalent in the student's
main area of research. This program should consist of at least three
graduate courses (4000- or 6000-level) in one area. If the area chosen
is truly far removed from the student's main area, the student may be
allowed to replace one of the graduate courses with an undergraduate
course although such a course will not count toward the 60-point
graduate course requirement. The courses should be selected so that
they form a cohesive whole.
Every candidate in the the Eng.Sc.D. or the Ph.D. program who did not
get his/her bachelor's degree in a country where English is the native
language must take the American Language Examination and reach Level 10.
The performance of each student in the doctoral program will be
reviewed regularly, at least once a semester, to ensure satisfactory
progress towards completion of study. When necessary, students failing
the requirements may be placed on probation or asked to withdraw from
the doctoral program.
The degree of master of philosophy is conferred upon a student who has
successfully fulfilled all of the following requirements of GSAS and
- A minimum of 30 points beyond the M.S. must have been taken
residence units (R.U.s) must have been completed; two of these are
awarded for the M.S. degree and four are accumulated while the student
is a Ph.D. degree candidate
- The research proposal examination must have been passed
the student did not get his/her machelor degree in a country where
English is the native language he/she must take the American Language
Examination and reach Level 10.
When the above requirements are met, students should see the Graduate Student Coordinator to apply for this degree.
The research should be conducted on the Columbia University campus. In
unusual cases the thesis work may be performed at the student's place
of employment subject to the following conditions:
- The research must be unclassified.
- The dissertation must represent the student's own efforts.
employer must have full cognizance that the student is using company
facilities and possibly company time to carry out doctoral research.
- The employer must have no objections to the publication of results.
committee members must be free to visit the location where the research
is being conducted whenever they wish to do so.
student must be prepared to consult with his/her thesis adviser on the
campus as often as two half-days per week, if the adviser so requires.
the time the research program is submitted for approval, the student
must obtain a letter from a responsive member of his/her firm,
addressed to the chair of the doctoral committee, stating that the
company will abide by the conditions set forth in this section.
Upon completion of a thesis acceptable to the candidate's thesis
adviser, the candidate should file application for the final oral
thesis defense examination in the office of the appropriate dean.
Examinations will not be scheduled during the summer except in unusual
cases. Seven copies of the dissertation are required for use by the
members of the final examination committee. The examination committee
is approved by the Dean and must consist of at least two faculty
members from departments other than electrical engineering as well as
at least three regular members of the electrical engineering
department. It may also include scientists or engineers from outside
the University who are interested in the research problem. The
candidate must distribute thesis copies to each member of the final
examination committee at least three weeks before the date of the
examination. Eng.Sc.D. candidates should consult the academic calendar
of the School of Engineering and Applied Science for pertinent
The candidate stands for examination before the final examination
committee on the date agreed upon by all members of the committee. It
is customary for the candidate to present initially a brief and concise
summary of his/her research with emphasis on original contributions to
the particular area. The oral presentation part is typically not more
than forty minutes. The examination is either passed, passed with
revision required, or failed. (NOTE: Candidates must be registered at
the time of the thesis defense.)
Two copies of the accepted dissertation must be submitted to the
dissertation secretary in the dean's office of the Graduate School of
Arts of Sciences after the defense. Rules governing the form and
deposit of the dissertation can be found in a number of booklets
available from the Graduate School of Arts and Sciences, as well as in
mimeographed information available from the dean's office of the School
of Engineering and Applied Science, or the dissertation secretary of
the Graduate School of Arts and Sciences. It is absolutely essential to
secure these rules and to prepare the dissertation in accordance with
them. Acknowledgement should be made to indicate that the work was
carried out in the Department of Electrical Engineering.
All degree requirements must be completed within seven
full-time-equivalent years from the beginning of the first course
credited toward the doctoral degree. Students who do not meet this time
limit will no longer be considered degree candidates.
A student who must interrupt studies for a compelling reason, for
example, sustained ill health, may be granted a leave of absence for a
stated period, usually not to exceed one year. If a leave is granted,
the fact is entered on the student's permanent academic record. The
period of leave of absence is not counted as part of the time allowed
for the completion of degree requirements. A student who leaves the
University without securing a leave of absence must apply for
readmission, which is not easily granted.
Doctoral candidates may transfer to professional degree candidacy at
any time and may apply all accumulated credits toward meeting the point
requirements specified in section C1.
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Syllabus for the Written Part of the Ph.D. Qualifying Examination
NOTE: In the topics listed below, the emphasis is meant to be on
FUNDAMENTAL PRINCIPLES, and not on dry facts. Proper understanding of
these principles, in the familiar context outlined below, should enable
one to apply them to less familiar situations.
- Resistors, capacitors, inductors, ideal transformers, independent sources, dependent sources and operational amplifiers
- Charge, current, voltage, and power
- Kirchhoff's voltage and current laws
- Node and mesh analysis
- Analysis of RL, RC and RLC circuits (up to second order) using differential equations
- Natural and forced response
- Linearity and superposition; Thevenin and Norton equivalents
- Phasor analysis
- Transfer functions, the complex plane, poles and zeros
- Frequency response and Bode plots
- Resonant circuits
- Diodes and diode circuits
- MOS and bipolar transistor characteristics
- Basic logic gate circuits (CMOS, TTL, ECL)
- Noise margins, logic delay, rise and fall times, fanin and fanout
- Circuits for latches and flip-flops
- Transient response of logic gate circuits
- Small-signal equivalent circuits for diodes and transistors
amplifiers and differential pairs, and their dc bias analysis,
large-signal analysis, small-signal analysis and frequency response
- Circuits using operational amplifiers
- Continuous-time and discrete-time systems
- Linear time-invariant systems
- Fourier series
- Continuous-time Fourier transform
- Discrete-time Fourier transform
- Filters and difference and differential equations
- One-sided and two-sided Laplace transforms (including the handling of initial conditions)
- One-sided and two-sided z-transforms (including the handling of initial conditions)
- Time and frequency characteristics of linear time-invariant systems
- Frequency domain analysis
- Transfer function
- Sampling theorem
- Frequency response, Bode plots
- Filtering, allpass, minimum-phase, linear phase systems
- Linear feedback systems
- Gain and phase margin
- State-space analysis of continuous-time and discrete-time systems
- Analysis of basic communication systems
- Amplitude modulation
- Pulse amplitude modulations (PAM), pulse code modulation (PCM)
- Multiplexing signals: time division multiplexing (TDM), frequency division multiplexing (FDM)
- Elementary probability, sample spaces
- Discrete and continuous random variables
- Distribution functions (Bernoulli, Poisson, geometric, binomial, normal, exponential)
- Conditional probability and Bayes's formula
- Independent events and random variables
- Expectation, conditional expectation
- Covariance, variance, correlation
- Electronic classification of solids: metals, semiconductors, insulators
- Density-of-states, Fermi-Dirac distribution function, quasi-Fermi levels
- Donors and acceptors, electrons and holes, majority and minority carrier
- Drift and diffusion, fundamental transport equations
- Recombination and generation
- P-N junctions: I-V characteristics and small-signal models, breakdown
- Bipolar junction transistors: Basic structure and operation, I-V characteristics and small-signal models
devices: C-V characteristics of MOS capacitors, accumulation,
depletion, inversion, threshold voltage, I-V characteristics, and
- Fundamental fabrication processes: photolithography, oxidation, epitaxy, diffusion, ion implantation
- Electric and magnetic fields
- Gauss, Coulomb, Ampere, Biot-Savart laws
- Electromotive force; magnetomotive force
- Current, current density, conservation of charge
- Current element; dipole
- Ohm's law, conductivity, electric power density
- Faraday's law, Maxwell's equations
- Transmission lines; wave propagation
- Standing waves; impedance matching
- Poynting theorem: real, complex
- Plane waves; polarization: linear, circular
- Snell's laws
- Elementary logic design
- Digital coding, parity, error correction
- Switching functions, truth tables, boolean algebra.
- Logic gates, canonical forms, K-maps and 2-level logic minimization
- Digital building blocks
- Decoders, multiplexers, encoders, logic arrays, read-only memory
- Latches, flip-flops, setup and hold times
- Registers, shift registers, counters, register files, read-write memories
- State machine design
- Synchronous and asynchronous systems, sequential systems
- State transition diagrams, finite-state machines, state minimization
- Computer architecture
- Data paths and control, busses, bit slices, register transfer language
- Microcode, microprogrammed controllers, hardwired controllers
- Instruction set architecture, assembly language, CPU, main memory, cache, I/O, interrupts
- Fundamental data structures and algorithms
- Arrays, linked lists, stacks, hash tables, queues, trees
- Sorting, searching, storage management
- Software engineering and operating systems concepts
programming and use of compilers/interpreters; operator precedence,
naming, control structures, nesting and scope, definition of new data
structures; subroutines, iteration, recursion, exception handling;
virtual memory, processor scheduling, process management, interprocess
communication, device management, file systems
- Basic probability and combinatorics
- Bernoulli and Poisson processes
- Binomial, geometric, exponential distributions
- Error detecting and correcting codes
- 1D and 2D parity check codes
- CRC check codes
- Hamming codes
- Multiple access resolution protocols
- Slotted ALOHA / round-based methods
- Carrier sensing MAC protocols
- Routing and spanning tree algorithms
- Minimum spanning tree
- Dijkstra's algorithm
- Bellman-Ford / distance-vector algorithms
- Reliable transfer protocols
- Alternating-bit protocol
- Selective repeat
- Finite state methods
- Congestion and flow control
- Fluid models of congestion collapse
- Flow control (application of Little's law)
- AIMD adaptive congestion control
- Queueing and fairness
- FIFO queuing (M/M/k/n queueing systems)
- Virtual clock
- (Weighted) fair queueing
- Markov model representations of queueing systems
- Max-min fairness
- Network analysis
- Little's law
- Kleinrock's independence approximation
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- Biomolecular sequence alignment
- Building of phylogenetic trees
- Gene prediction
- Biomolecular structure prediction
- Analysis of microarray data
- Modeling of biomolecular networks
- Hodgkin-Huxley neuron
- Reduced Hodgkin-Huxley models
- Integrate-and-fire neuron
- Stimulus representation and the neural code
- Algorithms for stimulus recovery
- Synaptic plasticity and learning algorithms
- Computation by excitatory and inhibitory networks