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]
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