6.003: Signal Processing

Continuous-Time Fourier Series

(Complex Exponential Form)

10 September 2020

Complex Numbers

Complex numbers consist of real and imaginary parts. You may

already be familiar with complex numbers written in their rectangular

form:

a0 + b0j

where j =√

−1. Here, a0 is called the real part and b0 is called theimaginary part.

We will often represent these numbers using a 2-d space we’ll call the

complex plane. For example, here is the number 3 + 4j representedas a point:

1 2 3 4

1

2

3

4

Re

Im

Complex Numbers: Polar Form

Importantly, we can also think of these numbers as vectors in the

complex plane. For example, here is 3 + 4j represented as a vector:

1 2 3 4

1

2

3

4real part

imaginary part

Re

Im

θ

r

We have labeled this number with a magnitude (r) and an angle (θ).

Let’s also note that we can define relationships between a0, b0, r,

and θ:

r =�

a20 + b20 θ = tan−1

�b0a0

�

a0 = r cos θ b0 = r sin θ

Complex Numbers: Polar Form

From there, we can rewrite a0 + b0j as: r (cos(θ) + j sin(θ)).

Then we can use Euler’s equation (ejx = cos(x) + j sin(x)) to expressour complex number as:

rejθ

This representation of complex numbers is known as the polar form.

Here, r is called the magnitude of the number, and θ is called the

phase (or angle, or argument) of the number.

Operations on Complex Numbers

Addition/Subtraction: real and imaginary parts sum independently

c1 + c2 = a1 + b1j + a2 + b2j = (a1 + a2) + (b1 + b2)j

rectangular form is nice for addition/subtraction.

Operations on Complex Numbers

Addition/Subtraction: real and imaginary parts sum independently

c1 + c2 = a1 + b1j + a2 + b2j = (a1 + a2) + (b1 + b2)j

rectangular form is nice for addition/subtraction.

Multiplication/Division: complicated in rectangular form!

c1c2 = (a1 + b1j)(a2 + b2j) = (a1a2 − b1b2) + (a1b2 + a2b1)jGROSS!

Operations on Complex Numbers

Addition/Subtraction: real and imaginary parts sum independently

c1 + c2 = a1 + b1j + a2 + b2j = (a1 + a2) + (b1 + b2)j

rectangular form is nice for addition/subtraction.

Multiplication/Division: complicated in rectangular form!

c1c2 = (a1 + b1j)(a2 + b2j) = (a1a2 − b1b2) + (a1b2 + a2b1)jGROSS!

Polar form is much nicer: magnitudes multiply, angles add

c1c2 = (r1ejθ1)(r2ejθ2) = (r1r2)ej(θ1+θ2)

Complex Exponentials

From Euler’s formula, we have several useful ways of converting

between trig functions and complex exponentials:

ejx = cos(x) + j sin(x)

cos(x) = ejx + e−jx

2

sin(x) = ejx − e−jx

2j = −j

2

�ejx − e−jx

�

Complex Numbers

How many of the following are true?

• 1cos θ + j sin θ = cos θ − j sin θ

• (cos θ + j sin θ)n = cos(nθ) + j sin(nθ)

• Im�

jj�

> Re�

jj�

• tan−1�1

2

�+ tan−1

�13

�= tan−1 1

Continuous-Time Fourier Series

Complex exponential form.

Synthesis Equation (making a signal from components):

x(t) = x(t + T ) =∞�

k=−∞X[k] e j

2πkT t

Analysis Equation (finding the components):

X[k] = 1T

�

Tx(t) e−j

2πkT t dt

Warm Up

Find the Fourier series components X[k] for

x(t) = cos(t)

Warm Up

Find the Fourier series components X[k] for

x(t) = sin(t)

Pulse Train

Find the Fourier series coefficients X[k] for x(t):x(t)

t−T −S TS

1

Pulse Train

What would happen to Fourier series if you delayed x(t) by T/2?

x(t − T/2)

t−T T

T/2

−S

T/2

+ST

2

1

In fact, what about an arbitrary shift t0?

Delay Property of Fourier Series

Complex exponential form simplifies expression of delay property.

delay complex exponential form trig form

T/2

T/4

t0

Fourier Series Matching

Match the signals (left column) to Fourier series coefficients (right).

t

x1(t)

0 11/4

1

k

Re XA[k]

k

Im XA[k]

t

x2(t)

0 11/4

1

k

Re XB [k]

k

Im XB [k]

t

x3(t)

0 11/4

1

k

Re XC [k]

k

Im XC [k]

t

x4(t)

1/4

1

k

Re XD[k]

k

Im XD[k]