„Signal Theory”  Zdzisław
Papir

•Dirichlet’s convergence
conditions

•Fourier series behaviour
at discontinuity points

•Peter G. L. Dirichlet

•Mean square convergence

•Parseval theorem

•Fractional power

•Gibbs effect

•Fejer, Lanczos windows...

•Summary

Fourier Series
Convergence

„Signal Theory”  Zdzisław Papir

Fourier series of signal
x(t)
Exponential form

  



 

 



,

2

,

1

,

0

1

o

o

o





























n

dt

e

t

x

T

X

e

X

t

x

T

t

x

t

x

T

t

jn

n

n

t

jn

n







































m

n

T

m

n

dt

e

e

e

e

n

e

T

t

m

j

t

jn

t

jm

t

jn

t

jn

,

,

0

,

2

,

1

,

0

,

o

o

o

o

o

o













If the signal x(t) in the interval [0, T]:

Class A signals:
A1)

has a finite number of 1st order discontinuities,

A2)

has a finite number of extrema,

A3)

is bounded

Class B signals:
B1)

has a finite number of 2nd order discontinuities,

B2)

except to

B1

points fits

A1

,

A2

, and

A3

conditions,

B3)

is absolutely integrable

then the exponential Fourier series is
uniformly convergent to the signal x(t) in
its each continuity point.

„Signal Theory”  Zdzisław Papir

Dirichlet’s convergence
conditions (I)

 







T

dt

t

x

Dirichlet’s conditions are sufficient conditions.

 





t

x

Dirichlet’s conditions
(I)

„Signal Theory”  Zdzisław Papir

time

x(t)

0

T

class A signal

I

I

„Signal Theory”  Zdzisław Papir

time

x(t)

0

T

class B signal

I

II

Dirichlet’s conditions
(I)

„Signal Theory”  Zdzisław Papir









   































t

x

t

x

t

x

t

x

2

1

lim

lim

2

1

o

o









The value of Fourier series in

1st order discontinuity points

equals:

The theorem suggests that the value of the signal in its
discontinuity points should be equal by definition to
an

arithmetic mean of right and left limit values

(at the discontinuity point).

This definition guarantees the Fourier series convergence
to the signal at each time point (a uniform convergence in
continuity points only)

Dirichlet’s conditions
(I)

„Signal Theory”  Zdzisław Papir

Fourier series
behaviour
at discontinuity points

time

x(t)

0

T

t

x(t-)

x(t+)

 

   













t

x

t

x

t

x

2

1

„Signal Theory”  Zdzisław Papir

Discontinuity point

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0

2

4

6

8

10

12

Fourier series behaviour at discontinuity points

czas

(10 components)

time

„Signal Theory”  Zdzisław Papir

Discontinuity point

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0

2

4

6

8

10

12

Fourier series behaviour at discontinuity points

czas

(20 components)

time

If the signal x(t) in the interval [0, T]:

class A signals:
A1)

is of bounded variation

class B signals:
B1)

has a finite number of 2nd order discontinuities,

B2)

except for

B1

points fits the condition

A1

,

B3)

is absolutely integrable

then the exponential Fourier series is
uniformly convergent to the signal x(t) in
its each continuity point.

„Signal Theory”  Zdzisław Papir

 







T

dt

t

x

   

































1

0

1

1

2

1

o

0

n

i

i

i

n

i

i

t

x

t

x

T

t

t

t

t

t

t





VARIATION

BOUNDED

conditions II

conditions I

Dirichlet’s convergence
conditions (II)

„Signal Theory” 

Zdzisław Papir

Dirichlet’s conditions
(II)

 

 







2

0

,

2

sin

2

ln









t

t

t

x

Absolutely integrable signal according to G. M. Fichtenholz
„Rachunek różniczkowy i całkowy”, vol. II, p. 507

 









2

o

dt

t

x

time

„Signal Theory”  Zdzisław Papir

 

 







2

0

,

cos

2

sin

2

ln

1

















t

n

nt

t

t

x

n

Dirichlet’s conditions
(II)

time

number of components -10

signal x(t)

Fourier series

„Signal Theory”  Zdzisław Papir

Peter Gustav Lejeune
Dirichlet

•

German mathematician, I half XIX century

• Most important achievements:

• number theory

– dzeta function

• set theory

– pigeon hole principle

• series theory

– series convergence

„Signal Theory”  Zdzisław Papir

Peter Gustav Lejeune
Dirichlet

Riemann’s dzeta function:

(specific case of the Dirichlet’s function)

 

 

1

Re

,

1

1











s

n

s

n

s



Euler’s identity:

 

 

set

number

prime

,

1

Re

,

1

1

1











p

s

p

s

p

s



„Signal Theory”  Zdzisław Papir

Peter Gustav Lejeune
Dirichlet

Riemann hypothesis:

(unproven till today)

 

 

1

Re

0

,

0

1

1















s

n

s

n

s



jb

s





2

1

The dzeta function has infinite number of roots given by
the formula:

Proof of the Riemann hypothesis would
change the number theory; computer
calculations verify that more than 1,5 x 10

9

figures satisfy the Riemann hypothesis.

„Signal Theory”  Zdzisław Papir

Peter Gustav Lejeune
Dirichlet

Number of prime numbers:

(proof based on dzeta function)

 

 

 



















x

x

x

x

x

x

x

x

x

x

x

for

ln

1

ln

lim

2

,

numbers

prime

of

number

-







Estimation error:

x = 10

10

4,5%
x = 10

14

3,0%
x = 10

18

2,5%

„Signal Theory”  Zdzisław Papir

Peter Gustav Lejeune
Dirichlet

Pigeon hole principle:

If N objects is placed in K < N containers,
then at least one container contains at least 2 objects.

N = 4

K = 3

Application:
Two Cracow inhabitants have the same number
of hairs on their heads (

N  800.000

).

Maximum number of hairs on a head -

K = 500 000

.

„Signal Theory”  Zdzisław Papir

Mean square
convergence

Truncated Fourier series
approximation

Fourier series

Mean square approximation error

 

 





















k

k

n

t

jn

n

n

t

jn

n

e

X

t

x

e

X

t

x

o

o

a





 

 





0

1

o

2

a

2









T

dt

t

x

t

x

T

e

„Signal Theory”  Zdzisław Papir

Truncated Fourier series is a mean square approximation
of a signal. Mean square convergence requests a square-
integrable signals:

 







T

dt

t

x

T

o

2

1

so it is valid for signals of limited energy (power).

Mean square
convergence

 

0

1

2

o

2

2



 





















k

k

k

n

n

T

X

dt

t

x

T

e

„Signal Theory”  Zdzisław Papir

Parseval theorem

 













n

n

T

X

dt

t

x

T

2

o

2

1

Using Parseval theorem we can determine a signal power:

 





T

dt

t

x

T

P

o

2

1

in the frequency domain:











n

n

X

P

2

„Signal Theory”  Zdzisław Papir

Fractional Power

 

1

2

2

o

f



 

























k

n

n

k

k

n

n

X

X

kf

P

0

5

10

15

20

25

30

35

0

0.05

0.1

0.15

0.2

kf

o

|X

k

|

Sawtooth signal – Fourier coefficients

T = 1

t

1

„Signal Theory”  Zdzisław Papir

Fractional power

 





t

n

n

e

n

j

t

x

n

t

jn

n

n

o

1

2

o

sin

1

1

2

1

1

2

2

1

































0

5

10

15

20

25

30

35

75

80

85

90

95

100

kfo

P

f

(k

fo

)

[%

]

Sawtooth signal – fractional power

 

 





%

99

16

%

95

4

%

90

2

o

f

o

f

o

f







f

P

f

P

f

P

T = 1

t

1

„Signal theory”  Zdzisław Papir

Fractional power
(sawtooth signal - 90%)

0

0.5

1

1.5

0

0.2

0.4

0.6

0.8

1

Sawtooth signal approximation

2 harmonics

90% of total signal power

T = 1

t

1

„Signal Theory”  Zdzisław Papir

0

0.5

1

1.5

0

0.2

0.4

0.6

0.8

1

Sawtooth signal approximation

4 harmonics

95% of total signal power

T = 1

t

1

Fractional power
(sawtooth signal - 95%)

„Signal Theory”  Zdzisław Papir

0

0.5

1

1.5

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

Sawtooth signal approximation

16 harmonics

99% of total signal power

T = 1

t

1

Fractional power
(sawtooth signal - 99%)

„Signal Theory”  Zdzisław Papir

Gibbs effect

-0.5

0

0.5

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Gibbs effect

Rectangular pulse
11 harmonics

„Signal Theory”  Zdzisław Papir

Gibbs effect

-0.5

0

0.5

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Gibbs effect

Rectangular pulse
39 harmonics

„Signal Theory”  Zdzisław Papir

Gibbs effect

-0.5

0

0.5

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Gibbs effect

Rectangular pulse
79 harmonics

„Signal Theory”  Zdzisław Papir

Gibbs effect

Gibbs effect appears at signal discontinuity

points when

approximating the signal by a truncated

Fourier series.

Gibbs effect is visible as excessive

osscilations close to

discontinuity points.

Gibbs effect does not depend on the how

the truncated

Fourier series is long.

„Signal Theory”  Zdzisław Papir

Fejer, Lanczos...
windows

 

 





















k

k

n

t

jn

n

n

n

t

jn

n

e

X

w

t

s

e

X

t

x

o

o





Window function (a set of different weights)

k

n

w

n



 ,

0

is tuned to minimize the Gibbs effect.

In a classical case a rectangular window is used:

k

n

w

n



 ,

1

„Signal Theory”  Zdzisław Papir

Rectangular windowFejer window

k

n

w

n



 ,

1

k

n

k

n

w

n







,

/

1

Lanczos window

von Hann, Hamming, Kaiser... windows





 

x

x

x

Sa

k

n

k

n

Sa

w

n

/

sin

,

/









Fejer, Lanczos...
windows

„Signal Theory”  Zdzisław Papir

-15

-10

-5

0

5

10

15

0

0.2

0.4

0.6

0.8

1

Fourier coefficients

w

e

ig

h

t

w

n

Rectangular window

Fejer window

Lanczos window

von Hann window

Fejer, Lanczos...
windows

„Signal Theory”  Zdzisław Papir

-0.5

0

0.5

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Rectangular window

Rectangular pulse

Fejer window

Lanczos window

Impuls prostokątny
11 harmonicznych

Rectangular pulse
7 harmonics

Fejer, Lanczos...
windows

„Signal Theory”  Zdzisław Papir

-0.5

0

0.5

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Rectangular window

Rectangular pulse

Fejer window

Lanczos window

Rectangular pulse

15 harmonics

Fejer, Lanczos...
windows

„Signal Theory”  Zdzisław Papir

Windows, Gibbs effect
approximation error...

Fejer, Lanczos etc. windows

decrease the Gibbs effect

, however,

at the cost of

increasing of the approximation error

.

The Fourier series (

rectangular window

) is the best signal approximation

in a mean-square sense.

„Signal Theory”  Zdzisław Papir

Summary

•

Dirichlet conditions are sufficient conditions for Fourier series

being convergent to a signal; at signal discontinuities they generate
an arithmetic mean of left and right signal limits.

•

The practical condition for Fourier series convergence is limited power

of a signal (mean-square convergence).

•

Mean-square convergence results in the Parseval theorem that splits

total signal power over its harmonics.

•

Fractional power is a tool for evaluating a signal bandwidth.

•

Truncated Fourier series approximation reveals the Gibbs effect being

excessive oscillations close to signal discontinuity points.

•

Fejer, Lanczos etc. windows decrease the Gibbs effect, however,

at the cost of increasing of the approximation error.