Fourier Series
Umar Ansari
[email protected]
Properties of Sine and Cosine Functions
The graphs of y = sin x and y = cos x have similar properties:
1. The domain is the set of real numbers.
2. The range is the set of y values such that  1  y  1.
3. The maximum value is 1 and the minimum value is –1.
4. The graph is a smooth curve.
5. Each function cycles through all the values of the range
over an x-interval of 2 .
6. The cycle repeats itself indefinitely in both directions of the
x-axis.
Copyright © by Houghton Mifflin
Company, Inc. All rights reserved.
2
Graph of the Sine Function
To sketch the graph of y = sin x first locate the key points.
These are the maximum points, the minimum points, and the
intercepts.

3
x
0

2
2
sin x
0
2
1
0
-1
0
Then, connect the points on the graph with a smooth curve
that extends in both directions beyond the five points. A
single cycle is called a period.
y = sin x
y
3

2




1
2
2
1
Copyright © by Houghton Mifflin
Company, Inc. All rights reserved.
3

3
2
2
5
2
x
Graph of Tangent Function: Periodic
Vertical asymptotes
where cos θ = 0
tan θ
tan 
θ
tan θ
−π/2
−∞
−π/4
−1
0
0
π/4
1
π/2
∞
−3π/2
−π/2
0
π/2
One period: π
3π/2
θ
sin 
cos
Graph of Cotangent Function: Periodic
Vertical asymptotes
where sin θ = 0
cos
cot 
sin 
cot θ
θ
tan θ
0
∞
π/4
1
π/2
0
3π/4
−1
π
−∞
−3π/2
-π
−π/2
π/2
π
3π/2
Cosecant is the reciprocal of sine
Vertical asymptotes
where sin θ = 0
csc θ
−3π
θ
0
−2π
−π
π
One period: 2π
2π
3π
sin θ
Secant is the reciprocal of cosine
Vertical asymptotes
where cos θ = 0
sec θ
θ
−3π
−2π
−π
0
π
One period: 2π
2π
3π
cos θ
Example: Sketch the graph of y = 3 cos x on the interval [–, 4].
Partition the interval [-π,4] on your x-axis
x
y = 3 cos x
y

(0, 3)
2
1

0
3
0

-3
x-int
min
2
max
0
2
3
x-int
max
(2, 3)

1 
( , 0)
2
2
3
( , –3)
Copyright © by Houghton Mifflin
Company, Inc. All rights reserved.
3
2
2
( 3 , 0)
2
8
3
4 x
The amplitude of y = a sin x (or y = a cos x) is half the distance
between the maximum and minimum values of the function.
amplitude = |a|
If |a| > 1, the amplitude stretches the graph vertically.
If 0 < |a| < 1, the amplitude shrinks the graph vertically.
If a < 0, the graph is reflected in the x-axis.
y
4
y = 2sin x

2
y=
1
2
2
x
sin x
y = sin x
y = 4 sin x
y = – 4 sin x
reflection of y = 4 sin x
4
Copyright © by Houghton Mifflin
Company, Inc. All rights reserved.

3
2
9
The period of a function is the x interval needed for the
function to complete one cycle.
For k  0, the period of y = a sin kx is 2 .
k
For k  0, the period of y = a cos kx is also 2 .
k

For k  0, the period of y = a tan kx is
.
k

If k > 1, the graph of they function
is shrunk horizontally.
y  sin 2x
period: 2

period: 
y  sin x x




2
If 0 < k < 1, the graph of the function is stretched horizontally.
y
y  cos x
1
y  cos x
period: 2
2 
2
3
4

x
period: 4
Copyright © by Houghton Mifflin
Company, Inc. All rights reserved.
Use basic trigonometric identities to graph y = f (–x)
Example 1: Sketch the graph of y = sin (–x).
The graph of y = sin (–x) is the graph of y = sin x reflected in
the x-axis.
y = sin (–x)
y
Use the identity
sin (–x) = – sin x
x
y = sin x

2
Example 2: Sketch the graph of y = cos (–x).
The graph of y = cos (–x) is identical to the graph of y = cos x.
y
Use the identity
x
cos (–x) = – cos x

2
y = cos (–x)
Copyright © by Houghton Mifflin
Company, Inc. All rights reserved.
11
Example: Sketch the graph of y = 2 sin (–3x).
Rewrite the function in the form y = a sin kx with k > 0
y = 2 sin (–3x) = –2 sin 3x
Use the identity sin (– x) = – sin x:
2  2
period:
amplitude: |a| = |–2| = 2
=
3
k
Calculate the five key points.
x
0
y = –2 sin 3x
6


6
3
(0, 0)
Copyright © by Houghton Mifflin
Company, Inc. All rights reserved.

6
3
2
2
3
–2
0
2
0
(  , 2)
2
2


0
y


(  , -2)
6
2
2
3

2
(  , 0) 2
3
( , 0)
3
12
5
6

x
The Graph of y = Asin(Kx - C)
The graph of y = A sin (Kx – C) is obtained by horizontally shifting the graph
of y = A sin Kx so that the starting point of the cycle is shifted from x = 0 to
x = -C/K. The number – C/K is called the phase shift.
y
amplitude = | A|
period = 2 /K.
y = A sin Kx
Amplitude: | A|
x
Starting point: x = -C/K
Period: 2/B
Copyright © by Houghton Mifflin
Company, Inc. All rights reserved.
13
Example
Determine the amplitude, period, and
phase shift of y = 2sin(3x-)
Solution:
asin(kx  c)
Amplitude = |A| = 2
period = 2/K = 2/3
phase shift = -C/K = /3 to the right

Copyright © by Houghton Mifflin
Company, Inc. All rights reserved.
14
Example cont.
• y = 2sin(3x- )
Copyright © by Houghton Mifflin
Company, Inc. All rights reserved.
15

asin(kx  c)  d
Amplitude
Period: 2π/k
Phase Shift:
-c/k
Vertical
Shift
Copyright © by Houghton Mifflin
Company, Inc. All rights reserved.
16
How to Represent Signals?
• Option 1: Taylor series represents any function using
polynomials.
• Polynomials are not the best - unstable and not very
physically meaningful.
• Easier to talk about “signals” in terms of its “frequencies”
(how fast/often signals change, etc).
Expansion of a Function
Example (Taylor Series)
constant
first-order
term
second-order
term
…
Fourier Series
Fourier series make use of the orthogonality relationships of
the sine and cosine functions
Examples
A Sum of Sinusoids
• Our building block:
Asin(x   
• Add enough of them to
get any signal f(x) you
want!
• How many degrees of
freedom?
• What does each control?
• Which one encodes the
coarse vs. fine structure of
the signal?
Fourier Transform
• We want to understand the frequency of our signal. So, let’s
reparametrize the signal by  instead of x:

Fourier
Transform
f(x)
F()
• For every  from 0 to inf, F() holds the amplitude A and phase
of the corresponding sine
Asin(x   
– How can F hold both? Complex number trick!
F ( )  R( )  iI ( )
A   R( )  I ( )
2
F()
2
Inverse Fourier
Transform
I ( )
  tan
R( )
1
f(x)
Time and Frequency
• example : g(t) = sin(2pi f t) + (1/3)sin(2pi (3f) t)
Time and Frequency
• example : g(t) = sin(2pi f t) + (1/3)sin(2pi (3f) t)
=
+
Frequency Spectra
• example : g(t) = sin(2pi f t) + (1/3)sin(2pi (3f) t)
=
+
Frequency Spectra
• Usually, frequency is more interesting than the phase
Frequency Spectra
=
=
+
Frequency Spectra
=
=
+
Frequency Spectra
=
=
+
Frequency Spectra
=
=
+
Frequency Spectra
=
=
+
Frequency Spectra

1
= A sin(2 kt )
k 1 k
Frequency Spectra
Fourier Transform – more formally
Represent the signal as an infinite weighted sum
of an infinite number of sinusoids

F u    f x e
i 2ux

dx
Note: e  cos k  i sin k
ik
i  1
Arbitrary function
Single Analytic Expression
Spatial Domain (x)
Frequency Domain (u)
(Frequency Spectrum F(u))
Inverse Fourier Transform (IFT)

f x    F u e

i 2ux
dx
Fourier Transform
• Also, defined as:

F u    f x e

iux
dx
Note: e  cos k  i sin k
ik
• Inverse Fourier Transform (IFT)
1
f x  
2



F u eiuxdx
i  1