Dirac Distribution

„Signal Theory” Zdzisław Papir

•Non-Dirichlet signals

•Distributions

•Dirac delta (Dirac pulse)

•Sampling property of a Dirac delta

•Other properties of a Dirac delta

•Comb distributions

•Special functions – Fourier transform
pairs

•Signal sampling
•Summary

Non-Dirichlet signals

 

 

























dt

t

x

dt

t

x

2

• Signals x(t) are often used in signal processing modeling,
therefore, it would be recommended to find
their Fourier transform.

• Distribution concept makes possible to extend
a class of Fourier transformable functions.

 

   

 

 

t

t

x

t

t

x

const

t

x

sgn







1

 

 

t

t

x

t

t

x

0

0

sin

cos









„Signal Theory” Zdzisław Papir

Distributions

Distribution D(·) assignes a
number V

D

{



(t)}

to some function



(t):

 

 

 

 

t

V

t

D

D







 





Distribution examples:

 

 

 

 

 

 

 

 

 

 

 

 

 





 





length

arc

1

area"

"

signal

sample

signal

2

0









 









 









 













dx

dx

x

d

x

V

x

dt

t

t

V

t

t

t

V

t

b

a

D

D

b

a

D

D

D

D



















„Signal Theory” Zdzisław Papir

Integral form of a distribution

 

 

 

 

t

V

t

D

D







 





We use an integral form of the
distribution:

 

 

 

 

   

dt

t

t

D

t

V

t

df

D

D





















 



in order to retain a linearity property:

 

 





 





 





t

V

t

V

t

t

V

D

D

D

2

1

2

1















„Signal Theory” Zdzisław Papir

Distribution D(·) assignes a
number V

D

{



(t)}

to some function



(t):

Dirac delta (Dirac pulse)

Dirac delta



(·) assignes to some signal



(t)

a number



(0):

 

 

 

 

   

 

0

































dt

t

t

t

V

t

Definition of the Dirac
delta
is identified with its
sampling property.

 

t



t

0



t

 

0



„Signal Theory” Zdzisław Papir

Paul Adrien Maurice DIRAC ( 1902 -

† 1984)

„Signal Theory” Zdzisław Papir

Paul Adrien Maurice Dirac (1902-1984), outstanding English
theoretician physicst, coorginator of quantum mechanics, predicted
positon existence and contributed considerably into a quantum
electrodynamics development. He was a professor at Cambridge and
Oxford universities and a member of a Royal Society. He was granted
in 1933 with the Nobel prize in recognition of quantum mechanics
development (together with E. Schrödinger).

   

 









 

 

0

0

2

2

lim

1

lim

π

lim

0

2

2

0

0





































































dt

t

dt

t

t

   

 





 

 

   

 

0

π

lim

π

lim

0

0























































dt

t

t

dt

t

t

dt

t

t

t



 



0



t

 

t



t

 

0



2





2







1

 

 

t

t







Dirac delta (Dirac pulse)

„Signal Theory” Zdzisław
Papir

Sampling property of a Dirac
delta

„Signal Theory” Zdzisław Papir

 

 

 

 

   

 

0

































dt

t

t

t

V

t



  

 

0

0

t

dt

t

t

t



















 

t



t

0

t

t 

 

0

t







0

t

t



 

 

 

 

   

 

0

































dt

t

t

t

V

t

  



  



0

0

0

t

t

t

t

t

t















0

t

t 

 

t



t

 

0

t







0

t

t



Sampling property of a Dirac
delta

„Signal Theory” Zdzisław Papir

 

 

 

 

   

 

0

































dt

t

t

t

V

t



  

 

 

 

















b

a

t

b

a

t

t

dt

t

t

t

b

a

,

,

0

,

,

0

0

0

0







 

t



t

 

0

t







0

t

t



a

b

0

t

t 

Sampling property of a Dirac
delta

„Signal Theory” Zdzisław Papir

   

  



 

t

d

t

t

t

































Dirac delta convolved

„Area” under Dirac delta

 

1











dt

t



 

 

t

a

at





1



„Symmetry” of a Dirac delta

Other properties of a Dirac delta

„Signal Theory” Zdzisław Papir

Comb distribution

„Signal Theory” Zdzisław Papir

t





nT

t



 

















nT

t

t

T





 





T

e

T

nT

t

t

n

t

jn

T











2

,

1

0

0



























Exponential Fourier series of the comb distribution

Signal sampling

„Signal Theory” Zdzisław Papir

t

nT

  



nT

t

nT

x





 

  



 



    

t

t

x

nT

t

t

x

nT

t

nT

x

t

x

T

































s

The comb distribution supports
signal sampling description:

Special functions – Fourier
transform pairs

Dirac delta

 

 

  



 

1

1

exp

















t

dt

t

j

t

t









F

Constant signal (dc component)

 





2

1

Unit step

 

 

 

 

 









j

t

j

t

t

t

1

2

sgn

,

sgn

2

1

2

1











1

1

„Signal Theory” Zdzisław Papir

Harmonic signals

 







































0

0

0

0

2

exp

2

exp

2

1





















t

j

t

j



 









 







0

0

0

0

0

0

sin

cos



















































j

t

t



0









0









Special functions – Fourier
transform pairs

„Signal Theory” Zdzisław Papir

Comb distribution

 





T

e

T

nT

t

t

n

t

jn

T











2

,

1

0

0



























 



























n

n

t

jn

T

n

T

e

T

t

0

2

1

0













 

 











0

0



t

T

Special functions – Fourier
transform pairs

„Signal Theory” Zdzisław Papir

Signal sampling

„Signal Theory” Zdzisław Papir

t

nT

  



nT

t

nT

x





 

  



 



    

t

t

x

nT

t

t

x

nT

t

nT

x

t

x

T

































s

 

 





   





 

 





 

 































0

0

s

1

2

1

2

1

0



























n

X

T

X

t

F

X

t

t

x

t

x

X

T

T

s

F

F

 

















0

s

1





n

X

T

t

x

Oversampling

„Signal Theory” Zdzisław Papir

 

















0

s

1





n

X

T

t

x

 



s

X

g



0



g

0

2









oversampling

„Signal Theory” Zdzisław Papir

 

















0

s

1





n

X

T

t

x

g



0



g

N

0

2















N

– Nyquist frequency

Critical sampling

Critical sampling

 



s

X

„Signal Theory” Zdzisław Papir

 

















0

s

1





n

X

T

t

x

g



Sampling the lowpass signal at a frequency
equal at least the Nyquist frequency makes
possible to recover signal from its samples.

undersampling

Undersampling

 



s

X

0



g

N

0

2













aliasing

„Signal Theory” Zdzisław Papir

His early theoretical work on determining the bandwidth
requirements for transmitting information, as published in "Certain
factors affecting telegraph speed„
(Bell System Technical Journal, 3, 324-346, 1924), laid the
foundations for later
advances by Shannon, which led to the development of information
theory.

In 1927 Nyquist determined that an analog signal should be
sampled at regular
intervals over time and at twice the frequency of the signal's
bandwidth in order to be
converted into an adequate representation of the signal in digital
form. Nyquist
published his results in the paper Certain topics in Telegraph
Transmission Theory
(1928). This rule is now known as the Nyquist-Shannon sampling
theorem.

Harry NYQUIST (1889 - †1976)

Harry Nyquist was an important contributor to
information theory. He was born in Nilsby, Sweden.
He emigrated to the USA in 1907 and entered the
University of North Dakota in 1912. He received a
Ph.D. in physics at Yale University in 1917. He
worked at AT&T from 1917 to 1934, then moved to
Bell Telephone Laboratories. As an engineer at Bell
Laboratories, he did important work on thermal
noise (Johnson-Nyquist noise) and the stability of
feedback amplifiers.

„Signal Theory” Zdzisław Papir

g



0



g

N

0

2













Recovering signal from its samples

 



X

Ideal low-

pass filter

  



nT

t

nT

x





Ideal low-

pass filter

 

t

x

 

 

 





T

T

t

t

h

T

H











N

2

2

,

Sa

g













 

















0

s

1







n

X

T

X

  



nT

t

nT

x





Ideal low-

pass filter

 

t

x

 

 

 





T

t

t

h

T

H









Sa

g

2









T

T

















g

g

0

2

2

     

  











 

 





 















































nT

t

nT

x

T

nT

t

nT

x

t

x

T

t

nT

t

nT

x

t

x

t

h

t

x

g

s

Sa

Sa

Sa









„Signal Theory” Zdzisław Papir

Recovering signal from its samples

„Signal Theory” Zdzisław Papir

Orthogonal set of Sampling functions

The set of sampling functions:

is orthogonal in an interval (-, +).

The proof is based on the Rayleigh theorem:









T

n

nT

t

















g

g

;

,

2

,

1

,

0

:

Sa











































0

Sa

Sa

Sa

Sa

g

g

g

g

dt

nT

t

kT

t

nT

t

kT

t













 





 

























jnT

jkT

e

nT

t

e

kT

t

















g

g

2

g

g

2

g

g

Sa

Sa









 

 

































































n

k

n

k

T

k

n

T

d

e

d

e

e

dt

nT

t

kT

t

T

k

n

j

jnT

jkT

,

0

,

Sa

2

2

Sa

Sa

g

g

g

g

2

g

2

2

2

g

g

g





































 

 





 





























nT

t

nT

x

T

nT

t

nT

x

t

x

g

Sa

Sa





The Kotielnikov-Shannon series:

is a Fourier series over a set of orthogonal sampling
functions; Fourier coefficients are equal to signal samples.

„Signal Theory” Zdzisław Papir

Orthogonal set of Sampling functions

Władymyr A. KOTELNIKOV
(1908 -

Prof. Vladimir A. Kotelnikov has been making
fundamental contributions to his field for over
70 years, despite working for many years in relative
isolation from the global engineering community.
V. Kotelnikov led the formulation and proof of the
sampling theorem, spearheaded the development
of the theory of optimum noise immunity, and then
applied his findings to both radar and communi-
cations.

As a leader of several institutions, including the
Moscow Power Engineering Institute, the Research
Institute of the Ministry of Communications, and
the Institute of Radioengineering and Electronics
of the Russian Academy of Sciences, he created
innovative communications equipment, jet technology,
and devices for the control of rocket trajectories.
He also improved radiotelegraphic lines, perfected
code systems, and played a leading role in radar
astronomy, designing planet radar equipment
that led to close observations of planets.

„Signal Theory” Zdzisław Papir

Claude E. SHANNON (1916 -

†2001)

Claude Elwood Shannon, prof. at the MIT, has,
in a long and celebrated career, developed the
mathematical theories and techniques that make
possible the analysis of switching circuits,
computers and communications. His most significant
piece of work is "A Mathematical Theory of
Communication," published in two parts in 1947-48.
With this paper, Shannon laid down the theoretical
foundation for communications engineering opening
a new mathematical field for engineering applications.
Shannon's work compares only to that of
Norbert Wiener in the theory of time series and to that
of Von Neumann and Morgenstern in the theory of games.

„Signal Theory” Zdzisław Papir

• Several signals of practical interest does not fit
Dirichlet conditions to be Fourier transformable.

• The concept of the Dirac delta function (a generalized
function) makes possible to determine Fourier transforms
of some non-Dirichlet signals.

• The Dirac delta assignes to a signal its sample.

• The comb distribution (a sequence of periodically repeated
Dirac deltas) is useful when modelling a signal sampling
process and deriving a sampled signal spectrum.

• The Nyquist frequency is equal to a doubled signal cutoff
frequency; signal sampling has to be performed at frequency
exceeding the Nyquist frequency in order to avoid
an aliasing effect.

Summary

„Signal Theory” Zdzisław Papir

• Signal sampling at the Nyquist frequency does not result
in a loss of intersample signal values; an ideal lowpass filtering
is sufficient for a signal reconstruction.

•

A continuous signal obtained out of lowpass filtering

of its samples is a Fourier series over a set of orthogonal
sampling functions; Fourier coefficients are equal to
signal samples .

Summary

„Signal Theory” Zdzisław Papir