1. W. Bergholz GenEE2 (Spring 12)
12_Fourier Series
Objective:
Understand how Fourier Series can be used to model arbitrary periodic functions
Make the connect to linear function spaces
Syllabus:
Periodic functions
Even and odd functions
Fourier series, function spaces and sinusoidal functions as base vectors
W. Bergholz GenEE2 (Spring 12)

2. W. Bergholz GenEE2 (Spring 12)
12_Fourier Series
Periodic functions are important in many areas of EE-applications:
Example 1:
Sawtooth in a TV cathode ray tube
Period T [s]
Frequency F [s-1] = [Hz]
W. Bergholz GenEE2 (Spring 12)

3. Signals picked up by electrodes placed on the skin at defined places
12_Fourier Series
Example 2:
Electrocardiogram
Signals picked up by electrodes placed on the skin at defined places
Signals generated by the system: control system around the heart + the beating heart muscle
EE task. Design amplifiers which are
Sensitive enough
Right filtering to reduce noise
High enough input resistance
From: Faller, der Körper des Menschen
Period T
W. Bergholz GenEE2 (Spring 12)

4. W. Bergholz GenEE2 (Spring 12)
12_Fourier Series
Example 2:
Electrocardiogram
Pathological
50mV
W. Bergholz GenEE2 (Spring 12)

5. W. Bergholz GenEE2 (Spring 12)
12_Fourier Series
Example 3:
A periodic function constructed from straight lines
W. Bergholz GenEE2 (Spring 12)

6. W. Bergholz GenEE2 (Spring 12)
12_Fourier Series
Example 4:
A periodic function constructed from pieces of parabolas
W. Bergholz GenEE2 (Spring 12)

7. W. Bergholz GenEE2 (Spring 12)
12_Fourier Series
Example 5:
A periodic function constructed from 2 sinosoidal functions
W. Bergholz GenEE2 (Spring 12)

8. W. Bergholz GenEE2 (Spring 12)
12_Fourier Series
Listen to a sound wave of different frequencies (musical scale)
W. Bergholz GenEE2 (Spring 12)

9. W. Bergholz GenEE2 (Spring 12)
12_Fourier Series
Our Engineering Tasks:
Analysis of circuits for arbitrary periodic functions
Strategy to solve this task:
Use our knowledge about harmonic circuit analysis = AC circuit analysis
Use the superposition principle
 Attempt to synthesize the periodic function by summing up harmonic (sinosoidal) functions
W. Bergholz GenEE2 (Spring 12)

10. Synthesize the periodic function form sinosoidal functions :
12_Fourier Series
Synthesize the periodic function form sinosoidal functions :
Step 1: feasibility check
saw tooth signal as a test case
with one sine function f(t)=2sin(t)
five
three
functions
f(t)=2[sin(t) - sin(2t)/2 + sin(3t)/3 – sin(4t)/4 + sin(5t)/5]
f(t)=2[sin(t) - sin(2t)/2 + sin(3t)/3]
W. Bergholz GenEE2 (Spring 12)

11. W. Bergholz GenEE2 (Spring 12)
12_Fourier Series
W. Bergholz GenEE2 (Spring 12)

12. W. Bergholz GenEE2 (Spring 12)
12_Fourier Series
Step 1: feasibility check results
This looks like a promising idea!
How do we go about
Finding the right coefficients for the sine functions?
The mathematical structure / foundation behind this?
W. Bergholz GenEE2 (Spring 12)

13. W. Bergholz GenEE2 (Spring 12)
12_Fourier Series
Step 2: Review a few mathematical relevant mathematical concepts
Concept 1: Odd and even functions
Sine: sin(t) = -sin(-t) Cosine: cos(t) = cos(-t)
A function f(t) is odd, if f(t) = - f(-t)
A function f(t) is even, if f(t) = f(-t)
This concept is a general one and will be useful for making life easier for finding the right coefficients
Examples:
f(t) = t2 → f(-t) = (-t)2 = t2 = f(t) → f(t) is even
f(t) = t3 → f(-t) = (-t)3 = -t3 = -f(t) → f(t) is odd
W. Bergholz GenEE2 (Spring 12)

14. W. Bergholz GenEE2 (Spring 12)
12_Fourier Series
Concept 2: linear algebra and orthogonal sets of base vectors
For vectors we are familiar with orthogonality:
the scalar product is zero if the vectors are
orthogonal:
Any vector in ℝ3 can be composed of the 3 base orthogonal base vectors
x1 = (1,0,0)
x2 = (0,1,0)
x3 = (0,0,1)
xixj = ij with ij = 1 for i=j
ij = 0 for ij
example:
r = (3, 4, 5) = 3x1 + 4x2 + 5x3
the coefficients can be calculated by taking the scalar product with the of r with the respective base vector, due to the construction of the base vectors: X2 x3 ℝ3 x2 x1 X1
W. Bergholz GenEE2 (Spring 12)

15. W. Bergholz GenEE2 (Spring 12)
12_Fourier Series
Concept 2: linear algebra and orthogonal sets of base vectors
Now transfer these ideas from vectors r within the total space of position vectors in ℝ3 to our original problem
The following analogies must exist:
vector r
vector space ℝ3
base vectors xi
periodic function f(t)
function space F of all periodic functions over a period T (with a finite number of discontinuities)
base vectors must be special functions
Guess which?
W. Bergholz GenEE2 (Spring 12)

16. The function space of periodic functions over the interval [a,b]
12_Fourier Series
Concept 2: linear algebra and orthogonal sets of base vectors
Obvious candidates for the base vectors are sin(nt) and cos(nt)
The angular frequency  is related to the period T by  = 2f = 2/T where f is the frequency
Are the orthogonal relations valid for these?
Definition of the scalar product in a function space?
The function space of periodic functions over the interval [a,b]
W. Bergholz GenEE2 (Spring 12)

17. sin(mωt) and cos(nωt) are orthogonal provided m n
12_Fourier Series
Concept 2: linear algebra and orthogonal sets of base vectors
Apply this to our suspected base vectors
case 1: cosine for m n, m and n larger or equal to 1:
case 2: sine for m n, m and n larger or equal to 1, by integration we can show
case 3: sine and cosine mixed, by integration we can show
Conclusion: the harmonics sin(nt) and cos(nt) can serve as base vectors to compose all „well-behaved“ periodic functions with a period T (see third bullet point for definition of „well behaved“)
In fact it has been shown by Fourier (and you learn this in a more rigorous manner in ESM2B and FunEE):
Any periodic function with a period T can be synthesised by the sin and cos base vectors, if it meets the Dirichlet condition over [0,T]:
Most engineering functions meet the Dirichlet conditions
Example for a function which does not:
sin(mωt) and cos(nωt) are orthogonal provided m n
sin(mωt) and sin(nωt) are orthogonal provided m n
Therefore, cos(mωt) and cos(nωt) are orthogonal provided m n
W. Bergholz GenEE2 (Spring 12)

18. W. Bergholz GenEE2 (Spring 12)
12_Fourier Series
Concept 2: linear algebra and orthogonal sets of base vectors
f has infinitely many discontinuieties, therefore it does not fulfill the Dirichlet condition
(it is also intuitively clear, that there is no way how one can synthesize such a function only with sines and cosines)
Most engineering functions meet the Dirichlet conditions
Example for a function which does not:
W. Bergholz GenEE2 (Spring 12)

19. t in the argument forgotten
12_Fourier Series
Concept 2: linear algebra and orthogonal sets of base vectors
How to find the coefficients for the base vector function?
Answer: by scalar multiplication of the base vectors with the function, as for space vectors)
Note: a0 is the DC component and obviously the average of the f(t) over the interval [0,T]:
t in the argument forgotten
W. Bergholz GenEE2 (Spring 12)

20. Example: square wave function f(t)
12_Fourier Series
Example: square wave function f(t)
f(t) = Vm for t[0,T/4]
= 0 for t[T/4,3T/4]
= Vm for t[3T/4,T]
First thought: odd or even function?
ω = 2f
T = 1 / f = 2 / ω
f(t) is an even function, since f(t) = f(-t)
Therefore, it can be expanded in terms of the cosine functions only, so bn are zero
(this would follow from the integrations, but this way we can almost save half the work)
Calculate an:
Same idea: perform the integration over the 3 parts of the interval [0,T]
Calculate a0:
In fact, we can already state by inspection of the function, that its average is Vm/2.
Proof:
Note: for t=0, t=T/2 and t=T sin( ) is always 0
for t=T/4 and 3T/4: 1, 0 or -1, so that:
Result: for n even, (i.e. n=2k) an=0 because then sin ()=0 for T/4, 3T/4
for n odd, (i.e. n=2k+1) an= 2Vm/n , with alternating signs
W. Bergholz GenEE2 (Spring 12)

21. W. Bergholz GenEE2 (Spring 12)
12_Fourier Series
Example: square wave function f(t)
Result:
an = (-1)k x (2Vm)/n for n= 2k+1 (odd)
= 0 for n = 2k (even)
Hence we can write f(t) as:
Note:
The convergence is comparatively slow, since the are proportional to 1/n.
The root cause of this is the discontinuity. We will the next time that the triangle function converges much faster
W. Bergholz GenEE2 (Spring 12)

22. W. Bergholz GenEE2 (Spring 12)
12_Fourier Series
Example: square wave function f(t) with Vm = 1 and period = 2,
approximated by f9= [(1/) cosx – (1/3) cos3x + (1/5) cos5x - (1/7) cos7x + (1/9) cos9x ]
(i.e up to 9th harmonic)
approximated by f5= [(1/) cosx – (1/3) cos3x + (1/5) cos5x ] (i.e up to 5th harmonic)
approximated by f1= (2/) cosx (i.e up to 1st harmonic)
approximated by f7= [(1/) cosx – (1/3) cos3x + (1/5) cos5x - (1/7) cos7x ] (i.e up to 7th harmonic)
approximated by f3= [ (1/) cosx – (1/3) cos3x ] (i.e up to 3rd harmonic)
-0.2
0.2
0.4
0.6
0.8 1
1.2 2 4 6
f7(x)
f9(x)
f1(x)
f3(x)
f5(x)
Note: Gibb‘s
Phenomenon
W. Bergholz GenEE2 (Spring 12)