The official prerequisites for this course are:

Principles of EE II (14:332:222 and 14:332:224)
Differential Equations (01:640:244 or 01:640:252 or 01:640:292)

To take differential equations, you also should have taken Calculus I - III.

Important Note: this course is going to involve building on what you have seen before. Students come into this class having taken their prereqs at different times with different instructors. We want to make sure everyone has a chance to brush up their knowledge of mathematical tools we'll be using in this class.

This is a list of things we expect you to be able to do before starting this class. Some of the prereqs were a while ago or you may have taken them at another school, so it's understandable that you may have forgotten some things. We will try to provide some resources for you to refresh your knowledge, but we want to focus time in this class teaching new ideas and material and not re-teaching material from previous classes.

You can try to self-rate your familiarity skill with each task:

0: I have no idea what this means
1: I have heard of this but don't know how to do it.
2: I remember learning this once, maybe.
3: I used to be able to do this, but it was a while ago.
4: I can figure this out when I need to.
5: I can do this in my sleep.

The goal is to be at level 4-5 on these basic skills. The important thing is that you need to know how and when to use a formula. For example, you can write the quadratic formula in your notes but the more important part is to (quickly) identify where you should use it. If you feel you're operating at levels 0-3 on any item, here are a couple of things to try:

Look through your previous textbooks or notes to see if things start to look familiar.
Use the online resources below to do a little bit of practice and work through some problems.
Talk to your classmates! Maybe the way it's described here is unfamiliar.

If a lot things are completely unfamiliar then you should contact the instructors ASAP: it might be that you need to re-take a prereq before taking this class. You will learn a lot more from this class if you are comfortable with the mathematical basics.

Basic functions and algebra

Knowing what basic functions look like and how to draw them is crucial for building intuition. It's not enough to just say "I can plot this": you should be able to picture what the function looks like, where important points on the function are, the asymptotic behavior, etc.

Rules for plots:

Label your axes.
Label important points on the axes like intercepts/crossings.
Label asymptotes and discontinuities.
Label maxima and minima.
Plotting and visualizing functions.
- Plot linear, affine, and quadratic functions.
- Plot exponential functions and identify where they cross the axis.
- Plot trigonometric functions: sine, cosine, tangent, etc. For sines and cosines, show the effect of phase shifts and changing the frequency.
- Plot functions that are defined piece-wise (e.g. x(t) = t^2 for 0 < t < 1 and x(t) = t for t > 1)
Basic algebraic manipulations.
- Multiply polynomials and simplify the result.
- Factor quadratic equations and find roots (real and imaginary).
- Partial fraction expansion of rational functions.
Trigonometric operations and formulae
- Use sum and difference formulas for sines and cosines.
- Use double-angle formulas for sines and cosines.
- Rewrite sines and cosines as a cosine with a phase shift.
Scaling of functions.
- Find the asymptotic scaling behavior of functions.
- Compare the scaling of different functions to identify which is larger.
Logarithms in different bases (natural, base-10, base-2).
- Perform basic computations like addition and subtraction of logarithms.
- Transform logarithms between bases.
- Use logarithmic scaling of units such as decibels, nepers, and octaves.

Complex numbers

Complex numbers are as “realistic” as real numbers in engineering: the algebra/geometry of complex numbers matches up with the physics of electromagnetic waves. Just because we call components "real" and "imaginary" doesn't mean the latter is somehow fictious: it's just an unfortunate choice of words that can be confusing at first.

Calculations on complex numbers.
- Express a complex number in Cartesian (rectangular) form.
- Express a complex number in magnitude-phase (polar) form.
- Convert between the two forms.
- Compute the complex conjugate of a complex number.
- Add, subtract, multiply, and divide complex numbers.
- Use Euler's formula to express sines and cosines in terms of complex exponentials.
Graphical view of complex numbers.
- Interpret the two forms, Cartesian and magnitude-phase, in terms of vectors on the plane made up of the real and imaginary axes.
- Interpret conjugation graphically.
- Interpret arithmetic operations graphically.
- Interpret Euler's formula in terms of sines and cosines.

Geometry and Linear Algebra

A lot of the mathematics in engineering is actually geometry. Especially in signals and systems, geometric intuition is a very useful way to understand what different systems are doing and how signals relate to each other. For example, in communication systems we model a radio signal as a vector. The squared length of the vector is the energy used to transmit the signal (much like power is proportional to the squared voltage). We will encounter some of these concepts as we progress through the class. Note, however, that while linear algebra is not a prerequisite to this course, it is very useful in understanding linear systems.

Distances and angles
- Calculate the distance between two vectors
- Measure the angle between two vectors in radians and degrees
- Express a vector in polar coordinates
Dot products
- Compute the dot product of two vectors
- Calculate the length of a vector using the dot product
- Determine if two vectors are orthogonal using the dot product
- Use the law of cosines to calculate the length of a vector or the dot product of two vectors
- Use the dot product to calculate the component of a vector in the direction of another vector
Matrices and vectors
- Identify if a set of vectors is linearly independent.
- Multiply a vector by a matrix.
- Explain what an eigenvector/eigenvalue is.
- Verify that a given vector is an eigenvector of a matrix and find its eigenvalue.

Calculus

Calculus is used all over engineering, which is why there are 4 semesters of calculus as prerequisites for this course. However, the parts of calculus that you actually need to use varies a lot from field to field. Here are some of the topics which are most useful in this course.

Integral and differential calculus
- Take derivatives of all basic functions.
- Compute definite and indefinite integrals of polynomial functions, rational functions, and trigonometric functions.
- Use integration by parts to compute a definite integral.
- Use L'Hospital's rule to find the limit of a function.
Optimization
- Calculate the maxima and minima of functions using calculus.
- Use the method of Lagrange multipliers to solve for maxima and minima under a constraint.
Convergent and divergent series.
- Use the series expansions of sines, cosines, and exponentials.
- Compute the limit of the geometric series as well as truncations and segments of the series.
- Use simple tests to determine if a series converges.
- Use the fact that the harmonic series diverges to show other series diverge as well.
Differential equations.
- Solve a linear constant coefficient differential equation with and without initial conditions.
- Use Laplace transforms to solve linear constant-coefficient differential equations.