BME Signals : Background

The concepts reviewed in this section come from calculus, differential equations, and as high school mathematics. Most are not thoroughly explained in the Oppenheim and Willsky textbook, and will not be discussed in detail in the lectures. However, these concepts are extremely important for the materials taught in the Signals course. We highly recommended that you read through these notes in the first couple weeks of the course. If you encounter topics with which you are not comfortable, please meet with a TA.

You do not need to memorize the derivations, but it is good to have a general idea of where the formulas come from. As you work through examples in lecture and homework problems on your own, the formulas should become second nature to you in a few weeks. The highlighted equations are essential for understanding the topics developed in this course.

Complex numbers are written in the form

, where

and

. When B=0, the number is said to be (purely) real; when A=0 and B<>0, the number is said to be (purely) imaginary. Thus, a complex number is the sum of a real part and an imaginary part.

One may graphically represent a complex number as a two-dimensional vector. The x-axis represents the real component. The y-axis represents the imaginary component. (Note that both axes have real indices)

For time-dependent signals, Euler’s formula is often written as (just replace "x" with "w t":

This is an extremely important relation in the understanding of signals. (See the end of the section for a derivation of Euler’s formula)

Given the following specific values (k is an integer):

By plugging into Euler’s formula values for x, we can immediately see that (again, k is an integer):

(You should know the two relations above by heart.)

Like all vectors, a complex number can be represented as the product of its magnitude (a scalar) multiplied by the unit vector in its direction, therefore:

Plugging in the expression for θ, we arrive at the magnitude-angle representation of complex numbers:

The derivation of Euler’s formula (which you do not need to know for this course) is simply a comparison of the Taylor ’s series expansion for the exponential, cosine, and sine functions:

Derivation of Euler’s Formula:

Recall the formula for obtaining the Taylor series of a continuous function f(x):

Now, apply the Taylor series expansion to cos, sin, and exponential functions about x_0=0:

For more information:

Wikipedia:

Complex Numbers: http://en.wikipedia.org/wiki/Complex_number
Euler’s Formula: http://en.wikipedia.org/wiki/Euler's_formula
Taylor Series: http://en.wikipedia.org/wiki/Taylor_series

Mathworld:

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(1)	Equality
(2)	Sum
(3)	Product