Fourier Series for Periodic Functions
Lecture #8
5CT3,4,6,7
BME 333 Biomedical Signals and Systems
- J.Schesser
2
Fourier Series for Periodic Functions
• Up to now we have solved the problem of approximating a
function f(t) by fa(t) within an interval T.
• However, if f(t) is periodic with period T, i.e., f(t)=f(t+T), then the
approximation is true for all t.
• And if we represent a periodic function in terms of an infinite
Fourier series, such that the frequencies are all integral multiples
of the frequency 1/T, where k=1 corresponds to the fundamental
frequency of the function and the remainder are its harmonics.
1
ak 
T
T
2
T

2

f (t )e

 j 2 kt
T
f (t )  a0   [a k e
dt
j 2 kt
T
k 1

f a (t )  a0   Ck cos(
k 1
 ak * e
 j 2 kt
T
]

ae
k 
k
j 2 kt
T

 a0   2 Re[a k e
j 2 kt
T
]
k 1
j 2 kt
 k ), where 2a k  Ck e j k and a0  C0
T
BME 333 Biomedical Signals and Systems
- J.Schesser
3
Another Form for the Fourier Series
2 kt
f (t )  C0   Ck cos(
 k )
T
k 1

2 kt
2 kt
)  Bk sin(
)
 A0   Ak cos(
T
T
k 1
where Ak  Ck cos k and Bk  -Ck sin k and A0  C0

BME 333 Biomedical Signals and Systems
- J.Schesser
4
Fourier Series Theorem
• Any periodic function f (t) with period T
which is integrable ( f (t)dt   ) can be
represented by an infinite Fourier Series
• If [f (t)]2 is also integrable, then the series
converges to the value of f (t) at every point
where f(t) is continuous and to the average
value at any discontinuity.

BME 333 Biomedical Signals and Systems
- J.Schesser
5
Properties of Fourier Series
• Symmetries
– If f (t) is even, f (t)=f (-t), then the Fourier Series
contains only cosine terms
– If f (t) is odd, f (t)=-f (-t), then the Fourier Series
contains only sine terms
– If f (t) has half-wave symmetry, f (t) = -f (t+T/2), then
the Fourier Series will only have odd harmonics
– If f (t) has half-wave symmetry and is even, even
quarter-wave, then the Fourier Series will only have
odd harmonics and cosine terms
– If f (t) has half-wave symmetry and is odd, odd quarterwave,then the Fourier Series will only have odd
harmonics and sine terms
BME 333 Biomedical Signals and Systems
- J.Schesser
6
Properties of FS Continued
• Superposition holds, if f(t) and g(t) have coefficients fk and
gk, respectively, then Af(t)+Bg(t) => Afk+Bgk
j 2 kt

• Time Shifting: f (t )   ak e T
k 
f (t  t1 ) 

ae
k 
k
j 2 kt
T
(ak )delayed  (ak )original e
e
 j 2 kt1
T


ae
k 
k
 j 2 kt1
T
e
j 2 kt
T
 j 2 kt1
T
• Differentiation and Integration
j 2 k
(ak )original
T
T

(ak )original
j 2 k
(ak )derivative 
(ak )integral
BME 333 Biomedical Signals and Systems
- J.Schesser
7
FS Coefficients Calculation Example
f (t )  0,  2  t  0
f (t )  E , 0  t  1
E
-4 -3 -2 -1
1
2
3
f (t )  0, 1  t  2
f (t )  f (t  4)
4
 j 2 kt
 j 2 kt
 j 2 kt
1
4
4
ak  [  0e
dt   Ee
dt   0e 4 dt ]
4 2
0
1
0

1
1 E
e
4  j 2 k
4
2
 j 2 kt
1
4
0
 j k
2
|
Since 2ak  Ck  , then
f (t ) 
 j k
4
j k
4
E E
 
4 2 k 1
2
 j k
4
E
Ee
e e
1.5
 1] 
[e
[
]
j2
 j 2 k
k
1
k
sin( )  j k
E
4 e 4 , k 0
0.5

4 k
0
4
0 1 2 3 4 5
1
E
E
-0.5
a 0   dt 
40
4
BME 333 Biomedical Signals and Systems


- J.Schesser
6
k
4 cos( 2 kt  k )
k
4
4
4
sin
7 8
9 10 11 12 13 14 15 16 17 18 19 20
8
Example Continued
2
1.5
1 term
1
0.5
0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
-0.5
2
1.5
10 terms
1
0.5
0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
-0.5
BME 333 Biomedical Signals and Systems
- J.Schesser
9
Frequency Spectrum of the Pulse Function
• In the preceding example, the coefficients
for each of the cosine terms was
proportional to sin (k 4) (k 4)
• We call the function Sa( x)  sin x
x
the Sampling Function
• If we plot these coefficients along the
frequency axis we have
the frequency spectrum of
f(t)
1.2
1
0.8
0.6
0.4
0.2
0
-30
-20
-10
-0.2
0
10
20
-0.4
BME 333 Biomedical Signals and Systems
- J.Schesser
10
30
Frequency Spectrum
Note that the periodic signal in
the time domain exhibits a
discrete spectrum (i.e., in the
frequency domain)
1.5
1
0.5
0
-15
-10
-5
0
5
10
15
-0.5
Frequency (k/4)
BME 333 Biomedical Signals and Systems
- J.Schesser
11
Another Example
V
This signal is odd, H/W symmetrical, and
mean=0; this means no cosine terms, odd
harmonics and mean=0.
T

1 0
ak  [ T Ve
T 2
 j 2 kt
T
dt   Ve
 j 2 kt
0
T
T
2
V
1
e
[
|
j
k
2


T
T
V
[1  cos  k ] 

j k

T
2
0
 j 2 kt
T
dt ]
 j 2 kt T
T
2
0
1
e
|
j 2 k
T
2V
for k odd,
j k
0 for k even.

f (t )  2  2V cos( 2kt  90 )
k
T
k  odd


4V sin( 2kt )
k
T
k  odd

2
1.5
1
0.5
0
-0.5
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
-1
2V

  90 for k odd.
k
BME 333 Biomedical Signals and Systems
-1.5
- J.Schesser
12
Fourier Series of an Impulse Train or
Sampling Function

x(t ) 
  (t  nT )
n 
 j 2 k t
1 T2
ak  T x(t )e T dt
T
2
-3.5
-2.5
-1.5
-0.5
0.5
1.5
2.5
3.5
 j 2 k t
1 T2
 T  (t )e T dt
T
2
1
 for all k
T
1  j 2Tk t
x(t )   e
T k 
BME 333 Biomedical Signals and Systems
- J.Schesser
13
Fourier Series for Discrete Periodic
Functions

x[n]  a0   ak e
j 2kn
T
k 1

 j 2kn

T
  ak e
k 1


a e
k  
k
j 2kn
T
T
 j 2kn
1 2
ak 
x[n]e T

T n  T
2
Since x[n] is discrete, this
becomes a summation
Notice the similarities to the previous example.
BME 333 Biomedical Signals and Systems
- J.Schesser
14
•
Homework
Problem (1)
–
•
Compute the Fourier Series for the periodic functions
a) f(t) = 1 for 0<t<π, f(t)=0 for π<t<2π
b) f(t) = t for 0<t<3
Problem (2)
–
Compute the Fourier series of the following Periodic Functions:
•
f(t)= t, 2nπ < t < (2n+1)π for n ≥ 0
= 0, (2n+1)π < t < (2n+2)π for n ≥ 0
f(t)= e-t/π, 2nπ < t < (2n+2)π for n ≥ 0 Use Matlab to plot f(t) using ak for
maximum number of components, N=5,10, 100, and1000. Show your code.
•
•
Problem (3)
–
Problems: 4.1/3 Find the Fourier series of the following waveforms
(choose tc=To/4):
1.5
1
0.5
0
0
1
2
3
4
5
-To
-0.5
-1
-(To+ tc)
-To/2
-tc
- (To- tc)
tc
To/2
To
To- tc
To+ tc
-1.5
BME 333 Biomedical Signals and Systems
- J.Schesser
15
Homework
•
Problem (4)
–
Deduce the Fourier series for the functions shown (hint: deduce the second one
using superposition):



-π+a -a

-π
•
π+a 2π-a
a
π-a
π
4π/3 5π/3
-π -2π/3 -π/3
2π

π/3
2π/3
π
2π
5CT.7.1
BME 333 Biomedical Signals and Systems
- J.Schesser
16