1

Esantionarea semnalelorDiscretizarea variatiei in timp a semnalului.

Teorema esantionarii

Esantionarea ideala

( )

( ) ( ) ( ) ( )( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( )

( ) ( ) ( )

$ ( ) ( ) ( ) ( ) ( )

0

0

12 2

0

;

e

e

e e e

e e ek k

e e Tk k

T e ek

u t t t

x t u t x u t

x t u t kT x kT u t kT

x t u t kT x kT u t kT

lim u t t

lim u t kT t kT t

x t x t t x kT t kT

Δ

Δ Δ

Δ Δ

∞ ∞

Δ Δ=−∞ =−∞

ΔΔ→

∞ ∞

ΔΔ→ =−∞ =−∞

∞

=−∞

⎡ ⎤Δ Δ⎛ ⎞ ⎛ ⎞= σ + −σ −⎜ ⎟ ⎜ ⎟⎢ ⎥Δ ⎝ ⎠ ⎝ ⎠⎣ ⎦≅

− ≅ −

− ≅ −

= δ

− = δ − = δ

= δ = δ −

∑ ∑

∑ ∑

∑

2

$ ( ) ( ) ( ) ( ) ( )eT e e

kx t x t t x kT t kT

∞

=−∞= δ = δ −∑

x(t) x(t)= x(t)δTe(t)

δTe(t)

Spectrul semnalului esantionatideal

( )

( ) ( ) ( ){ } ( )

( ) ∑∑

∑

∑

∞

−∞=

∞

−∞=

∞

−∞=

∞

−∞=

⎟⎟⎠

⎞⎜⎜⎝

⎛ π−ω=⎟⎟

⎠

⎞⎜⎜⎝

⎛ π−ωδ∗ω=

=⎟⎟⎠

⎞⎜⎜⎝

⎛ π−ωδ

π∗ω

π=δ=ω

ω=π

⎟⎟⎠

⎞⎜⎜⎝

⎛ π−ωδ

π↔δ

keee

ke

kee

T

^

ee

kee

T

TkX

TTkX

T

Tk

TXttxX

TTk

Tt

e

e

2121

2221

222

F

;

3

( ) ∑∞

−∞= ⎟⎟⎠

⎞⎜⎜⎝

⎛ π−ω=ω

kee

^

TkX

TX 21

Eroarea de aliere.

Teorema esantionarii semnalelorde banda limitata

( ) er

MecM

Me

TH =

ω−ω≤ω≤ωω>ω

0aliere. apareNu

2

4

( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) a.p.t ,

1

0

txtx

,XpTkXT

HX̂Xthtx̂tx

,,T

pTH

r

kee

e

rrrr

MecMc

ceer

c

c

=

ω=ωω−ω=

=ω⋅ω=ω↔∗=

ω−ω≤ω≤ω⎩⎨⎧

ω>ωω≤ω

=ω=ω

∑∞

−∞=ω

ω

alierea. Apare

MMe ω<ω−ω

5

( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )( )

( ) ( )( )

( ) ( ) ( )( )eM

eM

ker

Me

ec

ece

ke

c

e

ece

kee

ce

ke

ek

ec

err

cerer

kTtkTtsinkTxtx

kTtkTtsinkTx

kTtkTtsinTkTxkTt

ttsinTkTx

kTtkTxt

tsinTtx̂thtx

ttsinTthpTH

c

−ω−ω

=

ω=ω−ω−ω

ωω

=

=−π−ω

=−δ∗πω

=

=−δ∗πω

=∗=

πω

=↔ω=ω

∑

∑

∑∑

∑

∞

−∞=

∞

−∞=

∞

−∞=

∞

−∞=

∞

−∞=

ω

:devine

tiereconstruc de formulaNyquist frecventa la iiesantionar cazulIn Nyquist. eesantionar de frecventa de denumirea poarta si 2 este minima eesantionar de Frecventa

2

Harry Nyquist , (February 7, 1889 – April 4, 1976) 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's Department of Development and Research from 1917 to 1934, and continued when it became Bell Telephone Laboratories in that year, until his retirement in 1954. As an engineer at Bell Laboratories, he did important work on thermal noise ("Johnson–Nyquist noise"), the stability of feedback amplifiers, telegraphy, facsimile, television, and other communications problems. In 1932, he published a classical paper on stability of feedback amplifiers (H. Nyquist, "Regeneration theory", Bell System Technical Journal, vol. 11, pp. 126-147, 1932). Nyquist stability criterion can now be found in all textbooks on feedback control theory. 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 Claude Shannon, which led to the development of information theory.

6

Teorema WKS (Whittaker, Kotelnikov, Shannon)

( ) ( )( )

( ){ }( )

( ) ( ) ( )( )

.kTt

kTtsinkTxtx

tx,ZnnTx

tx,Xtx

MecMc

ec

ec

e

c

ke

Me

M

M

ω−ω≤ω≤ωω−ω−ω

ωω

=

ω≥ω∈

ω>ω

≡ωω

∑∞

−∞=

: relatia satisfaca saincat ales astfel fie sa ca conditiacu

2:relatiaprin a.p.t sale, leesantioanedin ireconstitu poate se

initial semnalul sus mai de conditiileIn maxime. frecventei dublulputin cel este eesantionar de frecventa adica ,2 daca sale

lor esantioane multimea de determinat unic este atuncipentru 0 ca sensulin , la limitata banda de este semnalul Daca

e

Edmund Taylor Whittaker Vladimir Kotelnikov Claude Shannon

Wikipedia

7

Edmund Whittaker was educated to Trinity College, Cambridge starting 1892. After Whittaker became a Fellow of Trinity College he began to teach and give lecture courses and, among his first pupils were G H Hardy and J H Jeans. Whittaker made revolutionary changes to the topics taught at Cambridge. He taught a course based on his famous book A Course of Modern Analysis(1902). This work is important in the study of functions of a complex variable. It also develops the theory of special functions and their related differential equations. Other courses Whittaker taught at Cambridge included astronomy, geometrical optics, and electricity and magnetism. Hardy and Jeans were not the only famous mathematicians which Whitttaker taught at Cambridge. His pupils included Bateman, Eddington, Littlewood, Turnbull, and Watson. An application which interested him came through his association with actuaries in Edinburgh who were dealing with life assurance. This motivated him to study the mathematics lying behind somewhat ad hoc methods that the actuaries were using and Whittaker proved some important results on interpolation as a consequence.

Vladimir Aleksandrovich Kotelnikov (Russian, September 6, 1908 in Kazan –February 11, 2005 in Moscow) was an information theory pioneer from the Soviet Union. He was elected a member of the Russian Academy of Science, in the Department of Technical Science (radio technology) in 1953.• 1926-31 study of radio telecommunications at the Moscow Power Engineering Institute, dissertation in engineering science. • 1931-41 worked at the MEI as engineer, scientific assistant, laboratory director and lecturer. • 1941-44 worked as developer in the telecommunication industry. • 1944-80 full professor at the MEI. • 1953-87 deputy director and since 1954 director of the institute for radio technology and electronics at the Russian Academy of Science. • 1964 Lenin Prize• 1970-88 vice-president of the RAS; since 1988 adviser of the presidium. He is mostly known for having discovered, independently of others (e.g. Edmund Whittaker, Harry Nyquist, Claude Shannon), the sampling theorem in 1933. This result of Fourier Analysis was known in harmonic analysis since the end of the 19th century and circulated in the 1920s and 1930s in the engineering community. He was the first to write down a precise statement of this theorem in relation to signal transmission.

8

Shannon was born in Petoskey, Michigan. His childhood hero was Thomas Edison, whom he later learned was a distant cousin. In 1932 he entered the University of Michigan, where he took a course that introduced him to the works of George Boole. He graduated in 1936 with two bachelor's degrees, one in electrical engineering and one in mathematics, then began graduate study at the Massachusetts Institute of Technology (MIT), where he worked on Vannevar Bush's differential analyzer, an analog computer. A paper drawn from his 1937 master's thesis, A Symbolic Analysis of Relay and Switching Circuits, was published in the 1938 issue of the Transactions of the American Institute of Electrical Engineers. Next, Shannon worked on his dissertation at Cold Spring Harbor Laboratory, funded by the Carnegie Institution, to develop similar mathematical relationships for Mendelian genetics, which resulted in Shannon's 1940 PhD thesis at MIT, An Algebra for Theoretical Genetics. Shannon then joined Bell Labs to work on fire-control systems and cryptography during World War II, under a contract with section D-2 of the National Defense Research Committee. In 1948 Shannon published A Mathematical Theory of Communication, an article in two parts in the Bell System Technical Journal. He is also credited with the introduction of Sampling Theory. He returned to MIT to hold an endowed chair in 1956.

Hobbies and inventionsOutside of his academic pursuits, Shannon was interested in juggling, unicycling, and chess. He also invented many devices, including rocket-powered flying discs, a motorized pogo stick, and a flame-throwing trumpet for a science exhibition. One of his more humorous devices was a box kept on his desk called the "Ultimate Machine“. Otherwise featureless, the box possessed a single switch on its side. When the switch was flipped, the lid of the box opened and a mechanical hand reached out, flipped off the switch, then retracted back inside the box.

Shannon and his famous electromechanicalmouse Theseus, named after the Greek mythology hero of Minotaur and Labyrinthfame, and which he tried to teach to come out of the maze in one of the first experiments in artificial intelligence.

9

( ) ( ) ( )( )

( ) ( ) ( )( )

( ) ( ) ( )( )

( ) ( )

2

2

2

10

c ece

e c ek

c e ece e

e c e ek

eM M e

e ek

e n ,k ek

n ,k

sin t kTx t x kT

t kT

sin nT kTx nT x kT

nT kT

T

sin n kx nT x kT

n k

x kT x nT

, n k, n k

∞

=−∞∞

=−∞

∞

=−∞∞

=−∞

ω −ω=

ω ω −

ω −ω=

ω ω −

ωω = ⇒ ω = π

π −= =

π −

= δ =

=⎧δ = ⎨ ≠⎩

∑

∑

∑

∑

Reconstructia prin filtrare trece-jos ideala

Tema de curs: Demonstrati ca relatia de reconstructie reprezinta o descompunere a semnalului initial intr-o baza ortonormata a spatiului semnalelor de energie finita si banda limitata.

Reconstructia prin interpolare( )

2

2

2

e

r ee

TsinH T T

ω⎛ ⎞⎜ ⎟

ω = ⎜ ⎟ω⎜ ⎟⎜ ⎟⎝ ⎠

10

Reconstructia prin extrapolarede ordinul zero

( )

( )

2

2

2

2

2

22

2

2

2

2

e

e

e

e

e

er T

eTj

eTje

e

eTjr e

e

je

e

Th t p t

Tsine

Tsine T T

TsinH e T T

sine

ω−

ω−

ω−

ω− π

ω

⎛ ⎞= −⎜ ⎟⎝ ⎠

ω

↔ =ωω

=ω

ω

ω = =ω

ωπω

=ω

πω

11

Esantionarea ideala a semnalelor periodice

( )( )

( )

0 0 00

0 0 0

0 0

0 0 0

0

2 ; ;

Pentru ca sa nu apara suprapunerea lobilor centrali este necesar ca:N

Diferenta dintre si trebuie sa fie de forma:

, R=1,2,...sau

M e

e

e

N MT

N M N

M N N

M N N R

M

πω = ω ω = ω = ω

ω < ω − ω = ω −

ω − ω

ω − − ω = ω

ω = ω = ( )

( )

0

0 0

2adica

2 2 ; R=1,2,...e M

N R

N R R

+ ω

ω = + ω = ω + ω

( ) ( ) 0 0

0 0 0

; N0

Pentru a evita aparitia erorilor de aliere este necesar ca: sau 2 2

Spre deosebire de semnalele aperiodice unde 2 pentru semnalele p

c

e cr e c e

c

e e M

e M

T ,H T p N

,

N N N,

ω⎧ ω ≤ ω⎪ω = ω = ω < ω < ω − ω⎨ ω > ω⎪⎩

ω − ω > ω ω > ω = ω

ω ≥ ω

eriodice trebuie sa esantionam astfel incat 2 Pe perioada celei mai rapide componente spectrale

trebuie sa prelevam mai mult de doua esantioane (adica cel putin3).

e M .ω > ω

12

( ) ( )0

00

0

Daca este perioada fundamentalei si daca esantionarea se2 2face conform relatiei 2 atunci 2 ; T

R=1,2,...sau 2

Doar 2 + esantioane pot fi distincte ca urmare a periodicitatii

ee

e

T

N R N RT

TT

N RN R

π πω = + ω = +

=+

0

semnalului supus esantionarii. Toate pot fi prelevate intr-o singura perioada a fundamentalei T .

( ) ( ) ( )0 0

00 0

Acelasi rezultat se poate obtine si preluand esantioane succesive din perioade succesive.

2Aceasta posibilitate este valorificata in constructia osciloscoapelo

e e e

e e

x kT x T kT x kT kTTT ' kT T kT

N R

= + = +

= + = ++

r cu esantionare.

13

http://www.jhu.edu/~signals/sampling/index.html

Tema de curs: Folositi acestapplet pentru ca sa studiatiesantionarea unui semnalsinusoidal.

Relatii energetice

( ) ( )

( ) ( )0

22

22

0

Pentru semnale aperiodice esantionate este adevaratarelatia de tip Rayleigh:

Pentru semnale periodice esantionate este valabila relatiade tip Parseval:

1 1 ;

e ek

eT

W x t dt T x kT

P x t dt x kT MT M

∞ ∞

=−∞−∞

= =

= =

∑∫

∫1

0=2 + , =1,2,...

Energia sau puterea pot fi calculate fie din forma de variatie in timpfie in domeniul frecventa.

M

kN R R

−

=∑

14

Esantionarea cu retinere

% ( ) ( ) ( ) ( ) $ ( ) ( )

( ) 2 2

2

22 2

22

eT

t tj jt

x t x t t h t x t h t

t tsin sinth t p t e e t t

ωΔ ωΔ− −

Δ

⎡ ⎤= δ ∗ = ∗⎣ ⎦ωΔ ωΔ

Δ⎛ ⎞= − ↔ = Δ⎜ ⎟ ωΔω⎝ ⎠

Spectrul semnalului esantionatcu retinere

15

Acest caz se numesteesantionare cu memorare.

Esantionarea naturala

%% ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) 2

2

22unde

2

e eT T e ek k

j t

t

x t x t q t x t h t t x t h t kT x t h t kT

tsinth t p t H e

∞

=−∞ =−∞

ωΔ−

Δ

⎡ ⎤= = ∗δ = − = −⎣ ⎦

ωΔΔ⎛ ⎞= − ↔ ω =⎜ ⎟ ω⎝ ⎠

∑ ∑

16

Spectrul semnalului esantionatnatural

17

Relatia dintre spectrul unuisemnal discret si spectrul

semnalului analogic din care provine

18

Intre cele doua axe de frecventa corespunzatoare spectrului semnalului analogic esantionat respectivspectrului semnalului discret exista relatia: Se explica acum si natura periodica a spectrulueT .Ω = ω

( )

i

semnalului discret Intre si exista relatia: ; d M M M M e eM

X . T T .πΩ Ω ω Ω = ω ≤

ω

19

Esantionarea semnalelordiscrete

In prelucrarea numerica a semnalelor apar situatii in care,ulterior achizitionarii esantioanelor, se constata ca frecventade esantionare a fost prea mare. In astfel de situatii, cand nuse mai poate es

[ ] [ ]

$ [ ]$ [ ] [ ] [ ] [ ] [ ]

antiona semnalul analogic, este posibila esantionarea semnalului numeric, retinandu-se tot a -a valoare. Fie:

Semnalul discret esantionat, se obtine prin produsul:

Nk

Nk

N

n n - kN

x n ,

x n x n n x n n kN

∞

=−∞

=

δ = δ

= δ = δ −

∑

[ ] [ ]k

x kN n kN .∞ ∞

−∞ =−∞= δ −∑ ∑

N=3.

20

N=3.

[ ]Cum , unde este frecventa maxima din spectrul semnalului analogic din care provine ,iar pasul cu care acest semnal analogic a fostesantionat, rezulta:

; ;

S-ar fi

M e M M

e

e e e eM M

Tx n

T

NT T ' T ' NT

Ω = ω ω

π π≤ ≤ =ω ω

( ) e

respectat teorema WKS chiar daca semnalul ar fi fost esantionat cu pasul Daca

apare suprapunerea lobilor spectrali vecini, adicaerori de tip "alias".

e M Mx t T ' . Ω −Ω <Ω

21

Reconstruirea semnaluluidiscret din esantioanele sale

( ) 2 .

0 in restc

r M c e MN , k

H,

⎧ Ω − π ≤ ΩΩ = Ω ≤Ω ≤Ω −Ω⎨

⎩

[ ]

[ ] $ [ ] [ ] [ ]

[ ] $ [ ] [ ]

$ [ ] $ [ ] [ ]

[ ] $ [ ] [ ] [ ]

c

Raspunsul la impuls al filtrului de reconstructie este:

; .2

Dar 0 pentru si si deci

c er

c

r r

rk

rm m

sin nh n

n N

x n x n h n x n

x n x k h n k

x k k Nm x Nm x Nm

sin n mNx n x Nm h n Nm x Nmn m

N

∞

=−∞

∞ ∞

=−∞ =−∞

Ω Ω π= Ω = =

Ω

= ∗ = ⇔

= −

= ≠ =

π⎛ ⎞− π⎜ ⎟⎝ ⎠= − =π

− π

∑

∑ ∑

22

Esantionarea si decimarea unuisemnal discret

23

N=2.

24

Esantionarea spectrului unuisemnal discret de durata finita

25

[ ]% [ ]

Fie cu suportul 0 1 In urma esantionarii spectrului acestui semnal se obtine 2semnalul periodic de perioada Daca nu se produce suprapunerea

grupurilor temporale corespunzatoare die

x n n M .

x n N . N M

≤ ≤ −

π= ≥Ω

verselor valori k.

% [ ] [ ]

[ ] [ ] [ ] [ ] % [ ] [ ]

2 0 1Prin multiplicarea semnalului cu fereastra temporala rectangulara

0 in rest

se obtine semnalul reconstruit , identic cu semnalul

r

r r r

, n Nx n w n N

,

x n x n : x n x n x n w n .

π⎧ ≤ ≤ −⎪= ⎨⎪⎩

= =

26

( )[ ]

Daca spectrul se esantioneaza prea rar, , apare suprapunerea

grupurilor temporale, adica erori de tip "alias". Semnalul nu mai poate fi reconstruit din spectrul esantionat.

X M N

x n

Ω <

Masuri practice la esantionareasemnalelor analogice

De obicei nu se cunoaste largimea de banda a semnalului ce urmeaza a fi esantionat. Acesta poate avea componente spectralede frecventa mare, neinteresante in aplicatia considerata.Acestea pot fi de exemplu cauzate de zgomotul ce insotestesemnalul util. Exista deci riscul aparitiei erorilor de tip "alias".Pentru evitarea lor se prevede in structura lantului de prelucrarea semnalului, inaintea circuitului de esantionare, un filtru trece josnumit filtru "anti-alias" sau filtru de garda.

27

Esantionarea trebuie facuta cu o frecventa de cel putin 2 ori mai mare decat frecventa deoprire 2De asemenea trebuie sa avem

Deci:

2Cu cat banda de tranzitie

este mai m

s e s

M p

eM p s

s p

.

.

.

ω ω ≥ ω

ω ≤ ω

ωω ≤ ω < ω ≤

ω −ω

are, cu atat frecventade esantionare trebuie sa fie mai mare decat frecventa Nyquist2 M .ω

2

Banda de tranzitie mai mare ordin de filtru mai redus, mai putine elemente constructive, mai ieftin.Cu scaderea lui scaderorile de tip "alias" darcresc si deci si s e .

⇒

ε

ω ω

Sisteme de telefonie numerica - 3 4 KHz, 8 KHz.Sisteme de televiziune - 5 MHz, 18 MHz.

M e

M e

f , ff f

= =

≅ =

28

Semnal de vorbire fara aliasing.

Semnal de vorbire cu aliasing.

Semnal muzical fara aliasing.

Semnal muzical cu aliasing.

Esantionarea semnalelor trecebanda

[ ] [ ]Semnale de tip "trece jos" - spectrul concentrat in benzi care includ frecventa nula.Semnale de tip "trece banda" - au suportul spectrului de forma M m m M, , .−ω −ω ∪ ω ω

Reconstructia perfecta a unui semnal trece banda esantionat ideal se poate realiza pe baza teoremei WKS, 2Uneori semnalele trece banda pot fi reconstruite din esantioanele lor chiar daca s-a fol

e M .ω ≥ ω

osit o frecventa de esantionare mai mica decat frecventa Nyquist.

29

Cazul semnalelor trece bandade banda ingusta

( ){ } [ ] [ ]{ }

1

Suportul spectrului unui semnal trece banda de banda ingustaesantionat ideal este de forma:

supp

M m

m

e M e m e m e M en Z

.

X n , n n , n .∈

ω −ω<

ω

ω = −ω + ω −ω + ω ∪ ω + ω ω + ωU

Semnalul trece banda de banda ingusta poate fi reconstruitperfect din esantioanele sale chiar daca a fost folosita o frecventa de esantionare mai mica decat frecventa Nyquist.

[ ] [ ][ ] [ ]

( )M

Conditia de reconstructie perfecta este: ,

Pentru 0 conditia devine adica:

- 22sau 1 1

Daca exista

M e m e m e M e

M e m e m M

e m mMe

M e M

k , k l , l , k l Z .

l , k , k , k Z .

k.

k k k

−ω + ω −ω + ω ω + ω ω + ω =∅ ∀ ∈

= −ω + ω −ω + ω ω ω =∅ ∀ ∈

ω + ω ≤ ω⎧ ωω≤ ω ≤⎨−ω + + ω ≥ ω +⎩

I

I

valori intregi , pentru care aceasta conditie este satisfacuta, atunci exista valori ale frecventei de esantionare inferioare frecventei Nyquist pentru care semnalele trece banda de banda ingusta po

k

t fi reconstruite in urma esantionarii ideale.

30

( )

0

Solutia din multimea numerelor intregi a dublei inecuatii

obtinute este: 0 Notand cu partea intreaga

a fractiei , rezulta ca frecventa de esantionare

va apartine unor intervale de f

m

M m

m M m

k . n

/

ω< ≤

ω −ω

ω ω −ω

{ }0

m 0

22orma cu 11

Exemplu

8 si 10 Valoarea factorului este 4

Valorile admisibile pentru sunt 1, 2, 3 si 4. Acestor valori lecorespund urmatoarele domenii pentru

mM

mM

m M

, k ,...,n .k k

. n .

k

ωω⎡ ⎤ ∈⎢ ⎥+⎣ ⎦

ωω = π ω = π =

ω −ω

{ } [ ] [ ] [ ] [ ]frecventa de esantionare:

4 5 , 5,33 6 66 , 8 10 , 16 20 , , .π π π π π π π π ∞U U U U