Asta Ziel Uni

8/16/2019 Asta Ziel Uni

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LUCRAREA DE LABORATOR NR. 3CONVOLUŢII ŞI CORELAŢII......................................................................23

OBIECTIVELE LUCRĂRII...................................................................................23DESFĂŞURAREA LUCRĂRII...............................................................................23

-nterpretarea polinomial% a convoluţiei liniare şi ciclice...........................2'#eorema lui 3lanc&erel...............................................................................24

alculul convoluţiilor cu a5utorul transformatei ourier..........................2

6ecanismul de calcul al convoluţiei...........................................................2

onvoluţia corelaţia şi filtra5ul adaptat....................................................27

Studiul operatorilor de convoluţie şi corelaţie utiliz,nd mediul

D-D+#--E/............................................................................................27

#em%............................................................................................................27

LUCRAREA DE LABORATOR NR. 4

STUDIUL TRANSFORMATELOR APLICATE SECVENŢELORNUMERICE.......................................................................................................28


#ransformata ourier discret% a unei secvenţe numerice 1D....................28

Densitatea spectral% de putere şi transformata ourier............................2

3ropriet%ţi ale transformatei ourier.........................................................2

-nfluenţa lunimii semnalului asupra spectrului estimat cu #D..............'0

-nfluenţa tipului ferestrei asupra spectrului estimat cu #D......................'1

#ransformata cosinus discret% 1D..............................................................''

ompresia de imaini cu transformata cosinus discret% 2D......................''

ompresia de imaini utiliz,nd #/..........................................................'*

Studiul transformatei ourier discrete utiliz,nd mediul D-D+#--E/. .'4

#em%............................................................................................................'4

LUCRAREA DE LABORATOR NR. 5FILTRE NUMERICE FIR...............................................................................36

OBIECTIVELE LUCRĂRII...................................................................................36

DESFĂŞURAREA LUCRĂRII...............................................................................36omenzi 6+#/+9 pentru proiectarea :;-R.........................................'

3roiectarea :;-R prin metoda ferestrelor..............................................'

3roiectarea :;-R prin metoda eşantion%rii !n frecvenţ%......................'8

3roiectarea :;-R tip cosinus ridicat......................................................'

3roiectarea :;-R prin optimizare folosind aloritmul Remez ..............*0

3roiectarea :;-R prin optimizare folosind minimizarea erorilor

p%tratice......................................................................................................*0

3roiectarea :;-R cu faz% liniar% prin interpolare.................................*0

Studiul filtrelor numerice de tip -R utiliz,nd mediul D-D+#--E/.......*1#em%............................................................................................................*1

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LUCRAREA DE LABORATOR NR. 6FILTRE NUMERICE IIR................................................................................42


3roiectarea -/#RE/


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LUCRAREA DE LABORATOR NR. !PRELUCRAREA MULTIRATĂ A SECVENŢELOR SAU SC"IMBAREARATEI DE EŞANTIONARE............................................................................61

OBIECTIVELE LUCRĂRII...................................................................................61

DESFĂŞURAREA LUCRĂRII...............................................................................61omenzi 6+#/+9 pentru prelucrarea multirat%.......................................1

Decimarea si interpolarea secventelor.......................................................1

-nterpolatorul de ordinul zero.....................................................................2

#ransmultiplexorul numeric........................................................................2

3roiectarea filtrelor de )and% !nust%........................................................*

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SECVENŢE NUMERICE 1D ŞI 2D

Obiectivele lucrrii1) Asimil!" #$%&'iil(! ATLAB s*"&i#i&" l$&!$l$i &$ s"&+"%'" ,is&!"-"

1D i 2D/2) D"*!i%,"!" -"0%i&il(! ," %li s"&+"%'"l(! ,is&!"-" % ,(m"%iil"

-"m*(!l i s*"&-!l/3) S-$,i$l i%-"!&-i+ l *!(*!i"-'il(! s"&+"%'"l(! ,is&!"-" i l (*"!-(!il(! *li&'i &"s-(! $-ili%, m",i$l DIDACTICIEL.

De!"#ur$re$ lucrrii

Eşantionarea semnalelor continue

S s" !"*!"i%-" #(!m"l" ," $%, *"%-!$ $% s"m%l si%$s(i,l &(%-i%$$( ) x t i +!i%- s "%-i(%- % -im* ,is&!"-) [ ] x n -ii%, &

( ) ) si% x t + t = Ω + ϕS"m%l$l "%-i(%- s" s&!i"

( ) ( ) ( ) ( )

si% si% 2

si% 2 si% 219

t n#

e

x t x n# + n# + n#

+ n + n

#

= = = Ω + ϕ = π + ϕ =

= π + ϕ = π + ϕ

S" (:'i%" ,"&i

[ ] ( ) ( ) si% si% 2 x n + n + f n= ω + ϕ = π + ϕ

$%," 2 2 9 &$ 19e e f # ω = π = π = .D"(!"&" t n# = i # ω = Ω !"$l-

[ ]

[ ]

[ ]

!, s!,9"%-i(%

s "%-i(%

t

nω = Ω = =

O im;i%" ;!#i& s$;"s-i+ s$*! &(!"s*(%,"%'"l(! % #!"&+"%' ,i%-!",(m"%iil" -im* &(%-i%$$ i -im* ,is&!"- "s-" !",- % #i;$! 1.1.

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Figura 1.1

C(,$l ATLAB $!m-(! !"*!"i%- #(!m"l" ," $%, *"%-!$ $% s"m%lsi%$s(i,l &(%-i%$ i ,is&!"- *"%-!$ $!m-(!ii *!m"-!i

12 9 4n⋅ ⋅ + ") s"m%l si%$s &!,i%l [ ]si%& n #) s"m%l "?*(%"%'il " n− ;) s"m%l *$-"!" 9 22 n−

#) s"m%l l(;!i-m %-$!l [ ]l% n 0) s"m%l l"-(! &$ !"*!-i'i" %(!ml &$ m",i 1.5 i ,is*"!si.25.

8


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=igue("nits","o#","osition",.01 .01 . .5);set(,"enu!","none"); !1=!es("osition",.05 . .25 .25);ste#(eos(1,14) 1 eos(1,5));set(!1,"7i#",0 1.5,"89i$",0:5:20,... "89i$7!bel",-10:5:10,"Font'ie",);

legen&("#puls


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Secvenţe complexe

S s" ;"%"!"" s"m%l$l ) "?* x n c n= ⋅ ⋅

$%," B2 1912 > 9 6c = − + i 4n n∈ ≤ ≤N .

$ = -(1/12)+(pi/6)*i;? = 2; n = 0:40; = ?*ep($*n);subplot(2,1,1);ste#(n,e!l());l!bel(" n&e te#po!l n");%l!bel("@#plitu&ine");title("!te e!l!");subplot(2,1,2);ste#(n,i#!g());l!bel(" n&e te#po!l n");%l!bel("@#plitu&ine");title("!te i#!gin!!");

C!" "s-" s"m%i#i&'i *!'ii !"l" i *!'ii im;i%!" l$i &

Secvenţe modulate

S s" ;"%"!"" s"m%l$l m(,$l- % m*li-$,i%" ) 1 si%2> )) si%2> )) & C n m f n f n= + ⋅ ⋅

$%," mB .4 .1 .1) & f f = = i 1n n∈ ≤ ≤N .

n = 0:100 ; # = 0.4 ; h = 0.1 ; b = 0.01 ;h = sin(2*pi*h*n) ; b = sin(2*pi*b*n) ;% = (1+#*b).*h ; ste#(n,%) ; gi& ;l!bel(" n&e te#po!l n") ; %l!bel("@#plitu&ine");

Reprezentarea secvenţelor numerice 1D şi 2D folosind seria

Fourier în timp discret SF!D"O s"&+"%' 1D *"!i(,i& [ ] x n *(-" #i !"*!""%-- !"*!""%-- *!i% s"!iF($!i"! % -im* ,is&!"- 1D 1DSFTD )

[ ] [ ] ( )1

2>

" :

: n

x n x n : c−

=

= + = ∑$%,"

[ ] ( )1

2>

1" 1

: : n

n

c x n :

:

−−

=

= = −

∑1


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@% m(, simil! ( s"&+"%' 2D *"!i(,i& [ ]1 2 x n n *(-" #i !"*!""%-- *!i% s"!i F($!i"! % -im* ,is&!"- 2D 2DSFTD )

[ ] [ ]

[ ] 1 21 2

1 2

1 2 1 1 2

1 11 2

1 2 2 1 21 2

"?* 2> 2> : :

x n n x n : n

x n n : c n n : :

− −

= =

= + =

= + = + ∑ ∑

$%,"

[ ]1 2

1 2

1 2

1 11 2

1 2 1 21 2 1 2

1 "?* 2> 2>

: :

n n

c x n n n n

: : : :

− −

= =

= − −

∑ ∑

{ }

1

12i i :

i

= −

∈

) S s" !"li"" ,"s&(m*$%"!" % 1DSFTD $%"i s"!ii ," im*$ls$!i *"!i(,i&" &$ #&-(!$l ," $m*l"!" .5. (-i+$l ," : + #i ,"#i%i- #(l(si%, 64 ,"

"%-i(%". S s" +"!i#i&" *(i *!(*!i"-'il" ;"%"!l" l" &("#i&i"%'il(! ,"s&(m*$%"!ii % 1DSFTD $%"i s"&+"%'" *"!i(,i&" &$ +l(!i !"l" *"%-!$&!"

[ ] [ ]1 1

1 11) 1 2) 1 1)

: : n

:

n n

c x n c : x n c c : :

− −∗

−= =

= + = + − =∑ ∑ :) S s" !"li"" ,"s&(m*$%"!" % 2DSFTD $%"i s"&+"%'" 2D

*"!i(,i&" l &!"i m(-i+ ," : "s-" ,"#i%i- ," ( s"&+"%' !"&-%;$l! $%i-!," 8×8 +%, $% s$*(!- 2D ," 16×16.

)=ones(1,32),eos(1,32);=64;o =0:-1 $(+1)=0; o n=0:-1

$(+1)=$(+1)+(n+1)*ep(-A*pi*2*/*n); en& $(+1)=$(+1)/en&ste#(!bs($(1:/2)))

:)=eos(16);(1:,1:)=ones();$=eos(16);=16;o 1=0:-1o 2=0:-1

o n1=0:-1o n2=0:-1

$(1+1,2+1)=$(1+1,2+1)+(n1+1,n2+1)*ep(-A*2*pi*1/*n1-A*2*pi*2/*n2);

en&en&

en&

en&$=!bs($)

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!eorema eşantion#rii

1. S s" ;"%"!"" $% s"m%l ," .5 s &(m*$s ,i% s$m ,($ si%$s(i,"," #!"&+"%'" 1


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!ransformata Fourier discret# în timp discret !F!D"

S s" "l:(!"" ( #$%&'i" ATLAB *"%-!$ &l&$l$l TFTD $%"i s"&+"%'"#i%i-" ," l$%;im" : *"%-!$ : #!"&+"%'" "&0i,is-%-" *" &"!&$l $%i--". S s"

$-ili"" #$%&'i &!"- *"%-!$ &l&$l$l TFTD s"&+"%'"i [ ] .88n x n = %128 : = *$%&-".

un$tion D,E=tt&(h,)E=(2*pi/)*0:-1"; #i&=$eil(/2)+1;E(#i&:)=E(#i&:)-2*pi; E=tshit(E);D=tshit(t(h,));

nn=0:40; n=0..nn;8,E=tt&(n,12);subplot(211),plot(E/2/pi,!bs(8))

l!bel("e$Cent! no#!li!t!")%l!bel("!#plitu&ine B")subplot(212),plot(E/2/pi,10/pi*!ngle(8))l!bel("e$Cent! no#!li!t!")%l!bel("!! g!&e")

!ransformata %il&ert

Fi" s"&+"%' !"l [ ]2>

&(s x n n :

=

. T!%s#(!m- s


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title("Fun$ti! pon&ee"); l!bel("n")subplot(312); se#ilog%(,!#p);l!bel("uls!tie no#!li!t!");%l!bel("@#plitu&ine (&)"); gi&subplot(313); plot(,ph!se);

l!bel("uls!tie no#!li!t!");%l!bel("F!! (g!&e)"); gi&

F$%&'i *(%,"!" $%$i -!%s#(!m-(!


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lo!& $lo>n ; = 8;lo!& g!tlin2 ; % = 8;l = #in(sie(,1),sie(%,1));$ = #in(sie(,2),sie(%,2)); = (1:l,1:$);% = %(1:l,1:$);

8=t2() ; =t2(%);G1 = !bs(8).*ep(i*!ngle());G2 = !bs().*ep(i*!ngle(8));1 = e!l(it2(G1));2 = e!l(it2(G2));igue(1)h = (1:l-l/2)/l;C = (1:$-$/2)/$;subplot(3,2,1); i#!ge(); !is("i#!ge");title("i#!gine "); !is osubplot(3,2,2); i#!ge(%); !is("i#!ge");title("i#!gine %"); !is o

subplot(3,2,3);i#!ges$(h,C,log(!bs(tshit(8.2))));!is("i#!ge"); title("o&ul 8");l!bel("e$Cente oiont!le");%l!bel("e$Cente Ceti$!le")subplot(3,2,4);i#!ges$(h,C,log(!bs(tshit(.2))));!is("i#!ge"); title("o&ul ");l!bel("e$Cente oiont!le");%l!bel("e$Cente Ceti$!le")subplot(3,2,5);

i#!ges$(h,C,!ngle(tshit(8))); !is("i#!ge");title("F!! 8");l!bel("e$Cente oiont!le");%l!bel("e$Cente Ceti$!le")subplot(3,2,6);i#!ges$(h,C,!ngle(tshit()));!is("i#!ge"); title("F!! ")l!bel("e$Cente oiont!le");%l!bel("e$Cente Ceti$!le")igue(2) ;subplot(2,2,1); i#!ge(); !is("i#!ge");title("i#!gine "); !is o

subplot(2,2,2); i#!ge(%); !is("i#!ge");title("i#!gine %"); !is osubplot(2,2,3); i#!ge(1);!is("i#!ge"); !is otitle("i#!gine: #o&ul + !! %")subplot(2,2,4); i#!ge(2);!is("i#!ge"); !is otitle("i#!gine: #o&ul % + !! ");$olo#!p("g!%")

C(%&l$i(%'i % *!i+i%' im*(!-%'"i s*"&-!$l$i ," # *"%-!$

!"&(%s-!$&'i im;i%il(!.

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Studiul secvenţelor numerice utiliz(nd mediul D)D'*!)*)E+ , "

1) S" l%s" DIDACTICIEL$l *!i% i%-!(,$&"!" &(m"%iididact

2) S" s-$,i i%-"!&-i+ *!(*!i"-'il" ,i+"!s"l(! s"&+"%'" %$m"!i&" &$

$-(!$l m"%i$!il(! ,"#i%i-" %Signal generator Numerical signalsRandom signals

!em#

V"!i#i&'i *!(*!i"--" ," li%i!i--" 1DTFTD

[ ] [ ]{ } [ ]{ } ( ){ }1D 1 1 2 2 1 1D 1 2 1D 2TFTD TFTD TFTDa x n a x n a x n a x n+ = +

#(l(si%, $!m-(!"l" s"m%l"

[ ]

[ ]

1

2

4 ,& 1234

% !"s-

2 ,& 1234

% !"s-

n n x n

n n x n

− ==

=

=

i &(%si,"!%, 1 21i .5a a= = .

$% !(;!m$l DIDACTICIEL #(s- i%i'i- ," *!(#. A%,!M N$i%$is i s-$,"%'ii si ," l

EPSIETA B!"s- F!%' % 18. Ul-"!i(! % &,!$l &((*"!!ii -"0%i&(-ii%'i#i&" ,i%-!"EPSIETA B!"s- i AT B$&$!"-i *!(;!m$l DIDACTICIEL s$#"!i- %$m"!(s"m:$%-'i!i.

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SISTEME NUMERICE 1D ŞI 2D

Obiectivele lucrrii1) Asimil!" #$%&'iil(! ATLAB s*"&i#i&" l$&!$l$i &$ sis-"m" ,is&!"-"

1D i 2D/2) S-$,i$l *!(*!i"-'il(! ," li%i!i--" i%+!i%' % -im* i s-:ili--" %

&$l sis-"m"l(! %$m"!i&"/3) D"*!i%,"!" -"0%i&il(! ," ,"-"!mi%!" #$%&'i"i *(%,"!" #$%&'i"ii%,i&il" *!"&$m i !s*$%s$l$i sis-"m"l(! %$m"!i&" l ,i+"!s" s"&+"%'",is&!"-"/

4) S-$,i$l i%-"!&-i+ l *!(*!i"-'il(! s"&+"%'"l(! ,is&!"-" i l (*"!-(!il(! *li&'i &"s-(! $-ili%, m",i$l DIDACTICIEL.

De!"#ur$re$ lucrrii

Sisteme numerice liniare 1DS s" +"!i#i&" & sis-"m$l %$m"!i& !"*!""%-- *!i% #$%&'i ," -!%s#"!1 2

1 2

2.243 2.48 2.243 )

1 .4 .75

z z ( z

z z

− −

− −+ +

=− +

"s-" $% sis-"m li%i! i i%+!i%- % -im*.

1)$le!; n = 0:40; ! = 2;b = -3;1 = $os(2*pi*0.1*n); 2 = $os(2*pi*0.4*n); = !*1 + b*2;nu# = 2.2403 2.40 2.2403; &en = 1 -0.4 0.5; i$ = 0 0;%1 = ilte(nu#,&en,1,i$); %2 = ilte(nu#,&en,2,i$);% = ilte(nu#,&en,,i$); %t = !*%1 + b*%2; & = % - %t;iguesubplot(3,1,1); ste#(n,%); %l!bel("@#plitu&ine");title("esie! pt int!e! pon&e!t!: ! H$&ot IJ1Kn + b H$&otIJ2Kn");subplot(3,1,2); ste#(n,%t); %l!bel("@#plitu&ine");title("esie: ! H$&ot %IJ1Kn + b H$&ot %IJ2Kn");subplot(3,1,3); ste#(n,&);l!bel("n&e te#po!l n"); %l!bel("@#plitu&ine");title(" 'e#n!l &ieent!");

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2)$le!; n = 0:40; < = 10;! = 3.0;b = -2; = !*$os(2*pi*0.1*n) + b*$os(2*pi*0.4*n);& = eos(1,


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2)$le!;b,!=butte(5,0.5);h=i#p(b,!,50);igue;subplot(211);ste#(h); title("butte>oth ")b,!=ellip(5,1,20,0.5);h=i#p(b,!,50);subplot(212);ste#(h); title("$!ue ")

Sta&ilitatea sistemelor numerice

S s" !-" & sis-"m$l %$m"!i& &!&-"!i- *!i% #$%&'i ," -!%s#"!1

1 2

1 .8 )

1 1.5 .

z ( z

z z

−

− −−

=+ +

"s-" s-:il.

$le!; nu# = 1 -0.; &en = 1 1.5 0.;

= 200; h = i#p(nu#,&en,+1); p!su# = 0;o = 1:+1;

p!su# = p!su# + !bs(h());i !bs(h()) L 10(-6),

be!,en&

en&iguen = 0:;ste#(n,h)l!bel("n&e te#po!l n"); %l!bel("@#plitu&ine");igue

pl!ne(nu#,&en);&isp("'u# =");&isp(!bs(h()));

*onectarea în cascad# a sistemelor numerice

Fi" ,($ sis-"m" %$m"!i&" ,"#i%i-" *!i% #$%&'iil" ," -!%s#"! $!m-(!"

1 2

1 1 2

1 2

2 1 2

.3 .2 .4 )

1 . .8

.2 .5 .3 )

1 .7 .85

z z ( z

z z

z z ( z

z z

− −

− −

− −

− −

− +=

+ +− +

=+ +

A!-'i & #$%&'i ," -!%s#"! sis-"m$l$i %$m"!i& (:'i%$- *!i% &(%"&-!"% &s&, &"l(! ,($ sis-"m" "s-" ";l &$ *!(,$s$l #$%&'iil(! ," -!%s#"! l"&"s-(!.

$le!; = 1 eos(1,40);n = 0:40;&en = 1 1.6 2.2 1.325 0.6;nu# = 0.06 -0.1 0.2 -0.26 0.12;% = ilte(nu#,&en,);nu#1 = 0.3 -0.2 0.4;&en1 = 1 0. 0.;

nu#2 = 0.2 -0.5 0.3;&en2 = 1 0. 0.5;%1=ilte(nu#1,&en1,);%2=ilte(nu#2,&en2,%1);&=%-%2;

1


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igue;subplot(3,1,1); ste#(n,%);%l!bel("@#plitu&ine");title("esie! siste#ului eult!t"); gi&;subplot(3,1,2); ste#(n,%2);%l!bel("@#plitu&ine");title("esie! $elo &ou! siste#e $one$t!te in $!s$!&!"); gi&;subplot(3,1,3); ste#(n,&)

l!bel("n&e te#po!l n");%l!bel("@#plitu&ine");title(" 'e#n!l &ieent!"); gi&;

Funcţia indicial# a unui sistem numeric

S s" &(m*!" #$%&'iil" i%,i&il" *"%-!$ ,($ FTB ," -i* B$--"!(!-0 iC$"! ," (!,i%$l 5 +%, #!"&+"%' ," -i"!" %(!m- ," .5.

$le!;b,!=butte(5,0.5);unit=ones(50,1); h=ilte(b,!,unit);subplot(211);ste#(h); title("butte>oth")b,!=ellip(5,1,20,0.5); h=ilte(b,!,unit);subplot(212);ste#(h); title("$!ue")

Efectul zerourilor şi polilor asupra funcţiei de transfer a unuisistem numeric

1) Fi" sis-"m$l !"*!""%-- *!i% #$%&'i ," -!%s#"! ( ) 11 .5 ( z z −= −

*!""%-%, $% si%;$! "!($ l .5 z = . S s" s-$,i"" +!i'i l$i ( ) " ( ω l*!(*i"!" "!($l$i ," &"!&$l $%i--".

2) !(&",'i % &"li m(, *"%-!$ sis-"m$l !"*!""%-- *!i% #$%&'i ,"

-!%s#"! ( )1

1

1 .5 ( z

z −=

− &!" !" $% si%;$! *(l l .5 z = .

b(1,1:2)=1 -0.4;b(2,1:2)=1 -0.6;b(3,1:2)=1 -0.;b(4,1:2)=1 -1;igue;hol& on;tip$ul="b";"";"g";"";o =1:4 h=e(b(,:),1); plot(!bs(h),tip$ul(,:))

en&!is(0 512 0 2.2);gi& onlegen&("eou=.4","eou=.6","eou=.","eou=1",2)title("nluent! eouilo !sup! un$tiei &e t!nse")!(1,1:2)=1 -0.5;!(2,1:2)=1 -0.6;!(3,1:2)=1 -0.;!(4,1:2)=1 -0.;igue;hol& ono =1:4 h=e(1,!(,:));plot(!bs(h),tip$ul(,:))en&!is(0 512 0 5.5);gi& on

legen&("pol=.5","pol=.6","pol=.","pol=.")title("nluent! polilo !sup! un$tiei &e t!nse")

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Sisteme numerice liniare 2D

F$%&'i ," -!%s#"! $%$i 2DSPLI ," -i* %"!"&$!si+ "s-" ," #(!m

( ) ( ) 1 21 2

1 2

1 1

1 2 1 2 1 2

m mm mm m

( z z 9 z z ) z z − −

= =

= =

∑ ∑S s" ,"-"!mi%" #$%&'i *(%,"!" #$%&'i i%,i&il i !s*$%s$l l (

"?*(%"%'il &(m*l"? *"%-!$ $% 2DSPLI +%, m-!i&" &("#i&i"%'il(!

11 12

21 22

1 1

1

) ))

) )

= =

$le!; b=1,1;1,0;&i!$2


20/63

subplot(323);i#sho>(?1);title("Filt!e &e o&ine (st!tisti$! 5 - #e&i!n)")subplot(324);i#sho>(?2);title("Filt!e &e o&ine (st!tisti$! 1 - #in)")subplot(325);i#sho>(?3);

title("Filt!e &e o&ine (st!tisti$! - #!)")subplot(326);i#sho>(?4);title("Filt!e &e o&ine (st!tisti$! 4)")? = #e&ilt2(M);iguesubplot(211);i#sho>(M);title("#!gine initi!l!")subplot(212);i#sho>(?);title("#!gine ilt!t!")

Studiul sistemelor numerice utiliz(nd mediul D)D'*!)*)E+


2) S" s-$,i i%-"!&-i+ s-:ili--" i i%#l$"%' "!($!il(! i *(lil(! s$*! &!&-"!is-i&ii ," -!%s#"! $%$i SP &$ $-(!$l m"%i$!il(! ,"#i%i-" %

Filtering

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S" ,$ #$%&'iil" ," -!%s#"! $!m-(!" &!" &(!"s*$%, $%(! 1DSPLI

( )

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z ( z

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− −

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− +

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l" &"s-(! #$%&'ii ," -!%s#"!. !"&i'i &" -i* ," sis-"m" !"*!"i%- #i"&!".&) D"-"!mi%'i !s*$%s$l sis-"m$l$i ( )5 ( z l s"m%l$l im*$ls $%i--" i

-!"*- $%i--".

22


21/63

LUCRAREA DE LABORATOR NR. %

CONVOLUŢII ŞI CORELAŢII

Obiectivele lucrrii1) Asimil!" #$%&'iil(! ATLAB *"%-!$ !"li!" &(%+(l$'iil(!

&(!"l'iil(! 1D i 2D/2) S-$,i$l l;(!i-mil(! *"%-!$ &l&$l$l &(%+(l$'iil(! i &(!"l'iil(!/

3) S-$,i$l i%-"!&-i+ l &(%+(l$'iil(! i -!%s#(!m-"l(! $-ili%, m",i$lDIDACTICIEL.

De!"#ur$re$ lucrrii

)nterpretarea polinomial# a convoluţiei liniare şi ciclice

O*"!'iil" ," &(%+(l$'i" li%i! i &i&li& *!"&$m i ," ,"&(%+(l$'i" s" *(- i%-"!*!"- *!i% *!ism (*"!'iil(! &$ *(li%(m". S &(%si,"!m s"&+"%'"l"1D i *(li%(m"l" s(&i-" ," mi (s

{ } ( ){ } ( )

2 1 2 1 2

1 1

x x x x F z x x z x z

& & & ( z & & z

= ⇒ = + += ⇒ = +

C(%#(!m ,"#i%i'i"i1

:

n m n m

m

C x &−

−=

= ∑ . S" (:'i%" *!i% $!m!"{ } { } 1 2 3 1 1 1 1 2 2 1 C C C C C x & x & x & x & x & x &= = + +

A&"ii &("#i&i"%'i *(- #i (:'i%$'i & $!m!" *!(,$s$l$i *(li%(m"l(! ( ) ( )si F z ( z

( ) ( ) ( ) ( ) ( )

( ) ( )

2 1 2 1

2 3 1 1 1 1 2 2 1

z F z ( z x x z x z & & z

x & x & x & z x & x & z x & z

= = + + + =

= + + + + +

F$%&'i ATLAB $onC *"%-!$ &l&$l$l &(%+(l$'iil(! li%i!" 1D&l&$l" ," #*- &"s- *!(,$s. D" "?"m*l$ *"%-!$ !"li &(%+(l$'is"&+"%'"l(! { } { }1 2 3i 11 x & = = s" $-ili"

=1,2,3 ; h=1,1 ; %=$onC(,h)

23


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D"&(%+(l$'i *(-" #i !"li- &$ #$%&'i ATLAB &e$onC i *(-" #i+$- & !"$l--$l m*!'i!ii &$ !"s- *(li%(m"l(! i .

,=&e$onC(%,)

Şi &(%+(l$'i &i&li& ,mi-" ( i%-"!*!"-!" *(li%(mil. S &(%si,"!ms"&+"%'"l"

{ } ( )

{ } ( )

1 1

1 1

&$ 2 x x x F z x x z :

& & & F z & & z

= ⇒ = + =

= ⇒ = +

C(%#(!m ,"#i%i'i"i ( )( )

1

:

n m n mm

C x &−

−=

′ = ∑ s" (:'i%"{ } { } 1 1 1 1 1 C C C x & x & x & x &′ ′ ′= = + +

A&"li !"$l-- s" (:'i%" & !"s- l (*"!'i"i *(li%(mil"( ) ( ) ( ) ( )2m(, 1 z F z ( z z ′ = − ,i& & !"s- l m*!'i!ii *(li%(m$l$i

( )21 1 1 1 & x z & x & x z & x+ + + *!i% ( )2 1 z − .U% "?"m*l$ ," &(%+(l$'i" &i&li& 1D "s-" *!""%-- mi (s

=1,2; h=3,4; #=1,0,-1;,%=&e$onC(lipl($onC(,h)),#)

A&"l"i &(%si,"!'ii s$%- +l:il" i % &$l s"&+"%'"l(! 2D. As-#"l %&$l &(%+(l$'i"i li%i!"

{ } [ ]

{ } [ ]1 2

1 2

1 2

1 2

1/ 1 2 1 2 / 3 4 5

1/ 1 2 1 1/1 1

n n

n n

x x n n

& & n n

= = = =

= = = =

(,$l ," &l&$l l *!im"l(! "l"m"%-" l" s"&+"%'"i 2D C & x= ∗ ∗ "s-" *!""%-- mi (s

1 2 1 21 2

1 2 1 2

1 2

1 2 1 2

1 2

1 2

1 2

1 1 1 1

1 2

2 2 2 1 1 2

1 1

1 2 2

r r r r r r

r r r r

r r

r r r r

r r

C & x & x

C & x & x & x

C & x & x & x & x

− −= =

− −= =

− −= =

= = =

= = + = + =

= = + + = + + ⋅ =

∑ ∑∑ ∑

∑ ∑R"$l--$l &(m*l"- s" (:'i%" $-ili%, &(m"%il" ATLAB

=0,1,2;3,4,5 ; h=1,0,1;1,0,1 ;%=$onC2(,h)

24


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A&"li !"$l-- *(-" #i (:'i%$- !"*!""%-%, &(%+(l$'i & *!(,$s ," *(li%(m"

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

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1 1

1

1 1

1

1 1

1

12 3 4

12 3 4

1 1 1 1

12 3 4

2 2 1 1

2 2

3 5 1 5 7

3 4 8 4 5

r r

r

r r

r

r r

r

z ( z F z ( z F z z z z z

z ( z F z ( F ( F z z z z

z ( z F z ( z F z z z z z

−

=

−=

−=

= = = + + +

= = + = + + + +

= = = + + + +

∑

∑

∑R"*!""%-%, #i"&!" s"&+"%' 2D & $% *(li%(m ," ,($ +!i:il" s"

(:'i%"

( )

( )

1 2 1 1 1 1 21 2 1 2 1 2 1 2 1 2 1 2 1 2

2 22 2 1 1 2 1 2

1 2 1 1 1 1 21 2 1 2 1 2 1 2 1 2 1 2 1 2

2 22 1 1 2

1 2 3 4 52 3 4 5

1 1 1 1

1

F z z z z z z z z z z z z z z

z z z z z z z

( z z z z z z z z z z z z z z

z z z z

= + + + + += + + + +

= + + + + +

= + + +S" *(-" ,"&i s&!i" &(%+(l$'i 2D s$: #(!m *(li%(m$l$l$i ," ,($

+!i:il"( ) ( ) ( )

( ) ( )

1 2 1 2 1 2

2 2 2 22 2 1 1 2 1 2 2 1 1 2

2 3 4 5 1

z z ( z z F z z

z z z z z z z z z z z

=

= + + + + + + +

!eorema lui lanc$erel

"%-!$ +"!i#i& -"(!"m l$i l%&0"!"l s" *(-" &(%si,"! $!m-(!$l"?"m*l$

1

2

) 2 1 1

) 1 2 3 2 1 ...

x n n n

x n

= − ∈= − −

> = -pi:2*pi/255:pi;1 = 1 3 5 11 13 15 1;2 = 1 -2 3 -2 1;% = $onC(1,2);h1 = e(1, 1, >);h2 = e(2, 1, >);hp = h1.*h2;h3 = e(%,1,>);subplot(2,2,1) ;plot(>/(2*pi),!bs(hp));gi&title("o&usul spe$telo &e !#plitu&ine")subplot(2,2,2) ;plot(>/(2*pi),!bs(h3));gi&title("'pe$tul &e !#plitu&ine ! eult!tului $onColutiei")subplot(2,2,3) ;plot(>/(2*pi),!ngle(hp));gi&title("'u#! spe$telo &e !!")

subplot(2,2,4) ;plot(>/(2*pi),!ngle(h3));gi&title("'pe$tul &e !! ! eult!tului $onColutiei")

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24/63

*alculul convoluţiilor cu a/utorul transformatei Fourier

S s" &l&$l"" &(%+(l$'i &i&li& ,($ s"&+"%'" x i & ," &""i *"!i(, : $-ili%, 1DTFD .

=1,2; h=3,4;8=t();D=t(h);=8.*D;%=!bs(it())

S s" $-ili"" &""i m"-(, *"%-!$ !"li &(%+(l$'i &i&li& 2D,i%-!" s"&+"%'"l"

{ } [ ]

{ } [ ]1 2

1 2

1 2

1 2

1/ 1 2 1 2 / 3 4 5

1/ 1 2 1 1/1 1

n n

n n

x x n n

& & n n

= = = =

= = = =

=1,2,3;4,5,6; h=1,0,1;1,0,1;8=t2(); D=t2(h);=8.*D;%=!bs(it2())

0ecanismul de calcul al convoluţiei

S s" &l&$l"" *s &$ *s &(%+(l$'i ,($ s"&+"%'" ,is&!"-". O!i;i%"&"l(! ,($ #$%&'ii "s-" *!"s$*$s l n = .

=ones(3,1); h=ep(-1:10);

7%=length()+length(h)-1;lh=length(h);l=length(); %=eos(7%,1);=lipl();&isp("@p!s!ti pe oi$e t!st! pentu ! $!l$ul! $onColuti!");o i=1:1:7%

in&lo=#!(0,i-l); in&hi=#in(i-1,lh-1);o A=in&lo:in&hi

%(i)=h(A+1)*(i-A)+%(i);en&subplot(311);ste#(-l+i:1:i-1,);%l!bel(" in oglin&!");li#1= #!(lh,l) ;li#2= #in(#in(#in(),#in(h)),0) ;

li#3= #!(#!(),#!(h)) ;!is(-l,li#1,li#2,li#3);subplot(312);ste#(0:1:lh-1,h);%l!bel("h");!is(-l,li#1,li#2,li#3);subplot(313);ste#(0:1:7%-1,%);%l!bel("%");p!use

en&

26


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*onvoluţia- corelaţia şi filtra/ul adaptat

S s" s&!i" $% *!(;!m &!" s !-" & (*"!'i ," &(%+(l$'i" !"li- ,"$% #il-!$ ,*-- "s-" "&0i+l"%- &$ &(!"l'i ,i%-!" s"m%l$l &$ &!" &"s- "s-",*-- i s"m%l$l ," i%-!!".

S" + &(%si,"! s"m%l$l ," i%-!!" ) 1234321... x n =

i #il-!$l ,*-- &!&-"!i- *!i% #$%&'i *(%,"!" ) 3212143...& n =

h = 3 2 1 -2 1 0 -4 0 3; = 1 -2 3 -4 3 2 1;! = lipl(h);%1 = $onC(h,) ;%2 = ilte(h,1, eos(1,length()+1));

%3 = $o(,!);subplot(311);ste#(%1); %l!bel("@#plitu&ine");title("esie obtinut! pin $onColutie"); gi&;subplot(312);ste#(%2); %l!bel("@#plitu&ine");title("esie obtinut! pin ilt!A"); gi&;subplot(313);ste#(%1)l!bel("n&e te#po!l n"); %l!bel("@#plitu&ine");title("esie obtinut! pin $oel!tie"); gi&;

Studiul operatorilor de convoluţie şi corelaţie utiliz(nd mediul

D)D'*!)*)E+


2) S" s-$,i i%-"!&-i+ (*"!-(!ii ," &(%+(l$'i" i &(!"l'i" &$ $-(!$lm"%i$!il(! ,"#i%i-" %

Math Tools

!em#

S" ,$ s"&+"%'"l" 1D

[ ]

[ ]

1 ,& 12

% !"s-

5 ,& 1234

% !"s-

n x n

n n& n

==

− =

=

Cl&$l'i %li-i& &(%+(l$'i l(! &i&li& *"%-!$ 6 : = i +"!i#i&'i!"$l--$l % ATLAB.

27


26/63

LUCRAREA DE LABORATOR NR. &

STUDIUL TRANS'ORMATELOR APLICATESECVENŢELOR NUMERICE

Obiectivele lucrrii1) Asimil!" #$%&'iil(! ATLAB *"%-!$ !"li!" -!%s#(!m-"l(!

*li&-" s"&+"%'"l(! 1D i 2D/

2) S-$,i$l l;(!i-mil(! *"%-!$ &l&$l$l -!%s#(!m-"l(!/3) S-$,i$l *!(*!i"-'il(! -!%s#(!m-"i F($!i"! ,is&!"-" i &-(!+*li&'ii l" -!%s#(!m-"l(!/

4) S-$,i$l i%-"!&-i+ l -!%s#(!m-"l(! *li&-" s"&+"%'"l(! %$m"!i&"$-ili%, m",i$l DIDACTICIEL.

De!"#ur$re$ lucrrii

!ransformata Fourier discret# a unei secvenţe numerice 1D

Cl&$l'i i +i$li'i s*"&-!$l s"&+"%'"i 1D1 *"%-!$ 1

% !"s-

n : x n

= −=

&$ 8 16 64 128 512 : = .

=;=ones(1,);8=t();subplot(311);

ste#();title("Oepeent!e! se$Centei &is$ete n");subplot(312);ste#(!bs(8));title("Oepeent!e! #o&ulului t!nso#!tei Fouie");subplot(313);ste#(!ngle(8));title("Oepeent!e! !ei t!nso#!tei Fouie");

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Densitatea spectral# de putere şi transformata Fourier

R"*!""%-'i ,"%si--" s*"&-!l ," *$-"!" $%$i s"m%l #(!m- ,i% ,($si%$s(i," ," 5


28/63

h1 = e(nu#1, 1, >);n = 0:7-1;nu#2 = ep(>o*i*n).*nu#1;h2 = e(nu#2, 1, >);subplot(2,2,1)

plot(>/(2*pi),!bs(h1));gi&title("'pe$tul &e !#plitu&ine !l se$Centei initi!le")subplot(2,2,2)plot(>/(2*pi),!bs(h2));gi&title("'pe$tul &e !#plitu&ine !l se$Centei #o&ul!te")subplot(2,2,3)plot(>/(2*pi),!ngle(h1));gi&title("'pe$tul &e !! !l se$Centei initi!le")subplot(2,2,4)plot(>/(2*pi),!ngle(h2));gi&title("'pe$tul &e !! !l se$Centei #o&ul!te")

)nfluenţa lungimii semnalului asupra spectrului estimatcu !FD

S s" s-$,i"" i%#l$"%' l$%;imii s"m%l$l$i s$*! s*"&-!$l$i s$ "s-im- *!i% i%-"!m",i$l TFD. S" + &(%si,"! $% s"m%l si%$s(i,l ," #!"&+"%' 1


29/63

)nfluenţa tipului ferestrei asupra spectrului estimat cu !FD

) S s" ;"%"!"" ( si%$s(i, ," m*li-$,i%" 1V ," #!"&+"%' 1


30/63

%Ih!##=!bs(tshit((t(sig,t))));sig=%1.*bl!$#!n(32)";%Ibl!$=!bs(tshit((t(sig,t))));=-Fe/2:Fe/t:(Fe/2-Fe/t);subplot(511)

plot((513:1024),%Ie$t(513:1024));gi&;!is(0 12 0 20)legen&("ee!st! e$t!ngul!!",-1)subplot(512);plot((513:1024),%Iti!(513:1024));gi&;!is(0 12 0 )legen&("ee!st! ti!ngul!ie ",-1)subplot(513);plot((513:1024),%Ih!nn(513:1024));gi&;!is(0 12 0 )

legen&("ee!st! D!nning ",-1)%l!bel("!#plitu&ine spe$t!l!")subplot(514);plot((513:1024),%Ih!##(513:1024));gi&;!is(0 12 0 )legen&("ee!st! D!##ing ",-1)subplot(515);plot((513:1024),%Ibl!$(513:1024));gi&;!is(0 12 0 6)

legen&("ee!st! l!$#!n",-1)l!bel("e$Cent! (D)")

&) S s" ;"%"!"" s"m%l$l $!m-(! % 32 ," *$%&-"

1 74 si% 2> .1si% 2>

256 256 x n n n

= +

R"*!""%-'i s*"&-!$l s$ $-ili%, ,i+"!s" *(%,"!!i i i%-"!*!"-'i !"$l--"l".

t=(1:32);1=4;2=100 ;Fe=256;t=1024 ;

%1= 0.1*sin(2*pi*1/Fe*t)+ sin(2*pi*2/Fe*t);sig=%1.*bo$!(32)" ; %Ie$t=!bs(tshit(t(sig,t)));sig=%1.*ti!ng(32)"; %Iti!=!bs(tshit(t(sig,t)));sig=%1.*h!nning(32)" ; %Ih!nn=!bs(tshit(t(sig,t)));sig=%1.*h!##ing(32)" ; %Ih!##=!bs(tshit(t(sig,t)));sig=%1.*bl!$#!n(32)" ; %Ibl!$=!bs(tshit(t(sig,t)));=-Fe/2:Fe/t:(Fe/2-Fe/t);subplot(511)plot((513:1024),%Ie$t(513:1024));gi&;!is(0 12 0 20)legen&("ee!st! e$t!ngul!!",-1)subplot(512);plot((513:1024),%Iti!(513:1024));gi&;

!is(0 12 0 10)legen&("ee!st! ti!ngul!ie ",-1)

32


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subplot(513);plot((513:1024),%Ih!nn(513:1024));gi&;!is(0 12 0 10)legen&("ee!st! D!nning ",-1)%l!bel("!#plitu&ine spe$t!l!")subplot(514);plot((513:1024),%Ih!##(513:1024));gi&;

!is(0 12 0 10)legen&("ee!st! D!##ing ",-1)subplot(515);plot((513:1024),%Ibl!$(513:1024));gi&;!is(0 12 0 )legen&("ee!st! l!$#!n",-1) ; l!bel("e$Cent! (D)")

!ransformata cosinus discret# 1D

K"%"!'i ( s"&+"%' si%$s(i,l ," 1


32/63

:)lo!& i#&e#os &ots=1024; 1,2=sie(&ots); sl=&ots; s=eos(1,2);s(sl)=ones(1,length(in&(sl)));%=&$t2(s); l%,$%=sie(%); %C=%(:);

%Cs,i&s=sot(!bs(%C)); %Cs=lipu&(%Cs);i&s=lipu&(i&s); $ont=0;%C(i&s(+1:1*2))=eos(1*2-,1);o =1:2

%(:,)=%C((-1)*1+1:*1,1);en&s=i&$t2(%);subplot(121); i#!ges$(s)$olo#!p(g!%); title("#!gine iniQi!lP")subplot(122); i#!ges$(s)$olo#!p(g!%); title("#!gine e$onstuitP")

*ompresia de imagini utiliz(nd !+

S s" !"li"" &(m*!"si &"l""i im;i%i $-ili%, -!%s#(!m!"=!0$%"%L(+" *li&- +"&-(!il(! (:'i%$'i *!i% m*!'i!" im;i%ii % :l(&$!i," 4×4 *i?"li.

) S s" !"*!"i%-" +!i'i "!(!ii m",ii *-!-i&" % #$%&'i" ," %$m!$l ,"&("#i&i"%'i !"'i%$'i.

:) S s" !"li"" &(m*!"si im;i%ii &(%s"!+%, %$mi &(m*(%"%- &"mi im*(!-%- #i"&!$i +"&-(! ,i% s*'i$l -!%s#(!m-.

)lo!& i#&e#os &ots;1,2=sie(&ots); sl=&ots; s=eos(1,2);s(sl)=ones(1,length(in&(sl)));=i#2$ol(s,4 4,"&istin$t"); ="; sig=$oC();Ce$tp,C!lp=eig(sig);Clp=&i!g(C!lp); l&,i&=sot(Clp);i&=lipu&(i&); l&=lipu&(l&);o =1:16

Ce$tps="=";o =1:

no=nu#2st();eC!l("Ce$tps=Ce$tps ""Ce$tp(:,i&(",no,")) "";")

en&Ce$tps=Ce$tps """;";eC!l("#!t",Ce$tps);#!t$=eos(16,16); #!t$(1:,:)=#!t; s$=#!t*;s$ol=#!t"*s$;s=$ol2i#(s$ol,4 4,1 2,"&istin$t");e#()=(su#(su#(s-s).2))/(1*2);

en&plot(e#); l!bel("u#P &e $oei$ienQi eQinuQi");%l!bel("Ro!e")title("B!i!Qi! eoii #e&ii pPt!ti$e")

34


33/63

:)lo!& i#&e#os &ots=1;1,2=sie(&ots); sl=&ots; s=eos(1,2);s(sl)=ones(1,length(in&(sl)));=i#2$ol(s,4 4,"&istin$t"); ="; sig=$oC();

Ce$tp,C!lp=eig(sig);Clp=&i!g(C!lp); l&,i&=sot(Clp);i&=lipu&(i&); l&=lipu&(l&);Ce$tps="=";o =1:

no=nu#2st();eC!l("Ce$tps=Ce$tps ""Ce$tp(:,i&(",no,")) "";")

en&Ce$tps=Ce$tps """;";eC!l("#!t",Ce$tps);#!t$=eos(16,16);#!t$(1:,:)=#!t;s$=#!t*;s$ol=#!t"*s$;

s=$ol2i#(s$ol,4 4,1 2,"&istin$t");subplot(121); i#!ges$(s);$olo#!p(g!%); title("#!gine iniQi!lP")subplot(122); i#!ges$(s);$olo#!p(g!%); title("#!gine e$onstuitP")

Studiul transformatei Fourier discrete utiliz(nd mediul D)D'*!)*)E+


2) S" s-$,i i%-"!&-i+ -!%s#(!m- F($!i"! ,is&!"- &$ $-(!$lm"%i$!il(! ,"#i%i-" %

Fourier analysis

!em#

S s" +"!i#i&" !"l'i l$i !&"+l

[ ] [ ]

1 12 2

1 : :

n x n F :

− −

= ==∑ ∑% &$l s"&+"%'"i %$m"!i&"

[ ] *"%-!$ 7

% !"s-

n n x n

==

35


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LUCRAREA DE LABORATOR NR. (

'ILTRE NUMERICE 'IR

Obiectivele lucrrii1) S-$,i$l (*"!-(!il(! ," #il-!!" %$m"!i&/2) Asimil!" #$%&'iil(! ATLAB *"%-!$ *!(i"&-!" #il-!"l(! &$ !s*$%s

#i%i- l im*$ls/

3) R"li!" #il-!$l$i %$m"!i& l s"&+"%'"l(! 1D i 2D #(l(si%, #il-!"FIR/4) S-$,i$l i%-"!&-i+ l #il-!"l(! %$m"!i&" FIR 1D i 2D $-ili%, m",i$l

DIDACTICIEL.

De!"#ur$re$ lucrrii

*omenzi 0'!+' pentru proiectarea F34F)R

fir1 m"-(, #"!"s-!"l(!

fir2 m"-(, #"!"s-!"l(! &$ &!&-"!is-i& ," #!"&+"%' *!"s&!iskaiserord "s-im" *!m"-!ii *"%-!$ #i!1 &$ #"!"s-! =is"! firls $-ili" m"-(, &"l(! mi mi&i *-!-"remez $-ili" l;(!i-m$l R"m" *"%-!$ *!(i"&-!" FIR remezord "s-im" (!,i%$l FIR *"%-!$ $-ili!" l;(!i-m$l$i R"m"fircls $-ili" m"-(, &"l(! mi mi&i *-!-" cu constr,nerifircls1 $-ili" m"-(, &"l(! mi mi&i *-!-" *"%-!$ FT i FTScremez *!(i"&-" FIR &$ # %"li%i! "&0i!i*l$firrcos *!(i"&-" $% FIR &$ &!&-"!is-i& -i* &(si%$s !i,i&-

roiectarea F34F)R prin metoda ferestrelor

A&'()*+,-

) S" , " )d ( ω

:) S" ,"-"!mi% !s*$%s$l 1 1D TFD W " )Xd & n ( ω

∞ =

1

) ,2>nd & n ( e e

+∞

ω ω∞−∞

= ω∫ 36


35/63

&) S" ,"-"!mi% !s*$%s$l & n l FPFIR i%,& n & n= ⋅

,) S" &l&$l" !s*$%s$l % #!"&+"%' l FPFIR 1D" ) TFD W " )Xd ( (

ω ω=

37

%

%


36/63

E?"m*l$ ," *!(i"&-!" $%$i FPFIR &$ 57n = i .4>t ω = !,9s"&

h=0.4*sin$ (0.4* (-25:25));ste#(h)

[D,>]=e (h,1,512,2);plot (>,!bs (D)) , gi&

s$ #(l(si%, #"!"s-! ]=e (h,1,512,2);plot (>,!bs (D)) , gi&

E#,/& 0 /)(*+) FNFIR /)* ,+(0 )+)&()#(l(si%, #$%&'i *)1 9:9;+]=e (&1,1,512,2);plot (>,!bs (F&1))

roiectarea F34F)R prin metoda eşantion#rii în frecvenţ#

1 1 1 ( =

38


37/63

[ ]1

2> 1

1 )

1 "

: :

:

( z ( z

: z

−−

= −

−=

−∑

=[1,1,0,1];=4;o =0:-1

!=+1;p(!)=ep (A**2*pi/);?(!)=0;en&

[b,!]=esi&ue (,p,);bb (1:)=b (1:);bbb=[1,0,0,0,-1];=$onC (bbb,bb);@=.*!;D=&e$onC (,@);e (,@)

!(i"&-!" $%$i FPFIR -i* -!"&" (s &$ PG3 &$ &!&-"!is-i& ,"#!"&+"%' *!"s&!is

= [0 0.6 0.6 1]; #= [1 1 0 0];b= i2 (30,,#);

[h,>]= e (b,1,12);plot (,#,>/pi,!bs (h));

!(i"&-!" $%$i FPFIR -i* -!"&" s$s *!i% m"-(, "%-i(%!ii %#!"&+"%' *l"&%, ," l $!m-(!"l" ,-" ," *!(i"&-!"

(!,i%$l 24 : =#!"&+"%' limi- ," -!"&"!" 6 =


38/63

plot((0:1023)/1024*2,!bs(D)),!is ([0 1 0 1.2]);gi&

roiectarea F34F)R prin optimizare folosind algoritmul Remez ela&orat de ar5s40c*lellan

),> 9 9 9 : 9 ;+]=e (b,1,512);plot (,!,>/pi,!bs(h));

roiectarea F34F)R prin optimizare folosind minimizareaerorilor p#tratice

*)& 9 9 9 : 9 ;+


39/63

Studiul filtrelor numerice de tip F)R utiliz(nd mediul D)D'*!)*)E+

1) S" l%s" DIDACTICIEL$l *!i% i%-!(,$&"!" &(m"%iididact2) S" s-$,i i%-"!&-i+ (*"!-(!ii ," #il-! %$m"!i& FIR 1D i 2D &$

$-(!$l m"%i$!il(! ,"#i%i-" %FilteringImage processing

!em#

S s" *!(i"&-"" $% FT FIR ," (!,i%$l 32 +%, #!"&+"%' ," -i"!" ,"

5 =


40/63

LUCRAREA DE LABORATOR NR. )

'ILTRE NUMERICE IIR

Obiectivele lucrrii1) S-$,i$l (*"!-(!il(! ," #il-!!" %$m"!i&/2) Asimil!" #$%&'iil(! ATLAB *"%-!$ *!(i"&-!" #il-!"l(! &$ !s*$%s

i%#i%i- l im*$ls/

3) R"li!" #il-!$l$i %$m"!i& IIR l s"&+"%'"l(! 1D i 2D/4) S-$,i$l i%-"!&-i+ l #il-!"l(! %$m"!i&" IIR 1D i 2D $-ili%, m",i$lDIDACTICIEL.

De!"#ur$re$ lucrrii

roiectarea F)+!RE+6R 370ER)*E tip R)) ))R"

M+(0 0 /)(*+) FN +*/ IIR 0*+)? *&+)? /)(+(+*/

P)(*+) 0*)+@ /&?& )**&* >

)

) a)utter n ?n options

z p =

C D1 )

C D

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z p D

=

2 )

) ac&e)C n Rp Rs ?n options

z p

=

C D

)C D

) a

ellip n Rp Rs ?n options z p D

=

42

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:$--* &0":1*"lli** &0":2*


41/63

i!

1 )

2

)utterd

c&e) ord n ?n ?p ?s Rp Rs

c&e) ord

ellipord

=

E#,/& 0 /)(*+) ?() FA /)(+(+*/

!(i"&-'i $% FAFTB C":"+ -i* I ," (!,i%$l 1n = &$ 3,B p R = i

1 1 1.1 2> .2>) = Ω = = i 2 2.5 >) = Ω = .

b,!=$heb%1(n,Op,>n,"t%pe","s")b,!=$heb%1(10,3,1,"s")es(b,!)

>1=0.2*pi ;>2=pi;>=>2->1;>0=st(>1*>2);bt,!t=lp2bp(b,!,>0,>);>=linsp!$e(0.01,1,500)*2*pi;D=es(bt,!t,>);se#ilog%(>/2/pi,!bs(D));gi&

roiectarea F3 prin aproximare în planul z !(i"&-'i $% FP FTB &$ :%, ," -!"&"!" 1 =


42/63

roiectarea F34))R prin metoda invarianţei la impuls

!(i"&-'i $% FP ,i%-!$% FA-i* B$--"!(!-0 &$ 4 .3 ,B pn R= = *!i%m"-(, i%+!i%'"i l im*$ls$l $%i--" l";%, 1


43/63

E?"m*l$ !(i"&-'i $% FN FTS &$s G ωs G .25π δs G .2 G ,s* G ω * G .35π δ * G .1 G ,*

,i%-!$% FA [ FTS ," !"#"!i%' *!i% m"-(, i%+!i%'"i l im*$ls$l $%i--" *"%-!$

1 =p*Fs;o#gs=>s*Fs;!=20*log10(1/(1-&p));!#=20*log10(1/&s);n o#gt=$heb%1o&(o#gp,o#gs,!,!#,SsS);!,@!=$heb%1(n,!,o#gp,ShighS,SsS);igue(1); es(!,@!)&,@&=i#pinC!(!,@!,Fs);igue(2); e(&,@&)

roiectarea F3 8 ))R prin transformarea &iliniar#

!(i"&-!" FP -i* IIR ,i%-!$% FA #(l(si%, -!%s#(!m!" :ili%i!

&,p&,&=biline!(,p,,Fs,Fp)

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b&,!&=biline!(b,!,Fs,Fp)

) )1

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s f z

=−

=+

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s

p

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f z ( z ( s s

f z

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π −= =+

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p

s

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f

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ω =

P)(*+) FN IIR 0*+)? FA (&(*0 ,+(0 *&**)@S s" *!(i"&-"" $% FP [ FT *l"&%, ," l $% FA [ FT ," -i*

B$--"!(!-0 C":"+ I C":"+ II i Eli*-i& *"%-!$* G ω * G .2π δ * G . 1 G ,*s G ωs G .3π δs G .1 G ,s

S ,"-"!mi%m *!m"-!ii FA2l;1 ) 2l;. .15 ,B 6 pa = − δ = − =

2l; 2l;.1 4 ,Bm sa = − δ = − =2 9 )-; 9 2)# Ω = ω

45


44/63

S *!(i"&-m FA*!(-(-i*

nb,>b=butteo&(>p,>s,!,!#,SsS)n$1,>$1=$heb%1o&(>p,>s,!,!#,SsS)n$2,>$2=$heb%2o&(>p,>s,!,!#,SsS)

nel,>el=ellipo&(>p,>s,!,!#,SsS)

S *!(i"&-m &-" $% FA *!(-(-i*

b,@b=butte(nb,>b,SsS)$1,@$1=$heb%1(n$1,!,>$1,SsS)$2,@$2=$heb%2(n$2,!,>$2,SsS)el,@el=ellip(nel,!,!#,>el,SsS)b&,@b&=biline!(b,@b,1)$1&,@$1&=biline!($1,@$1,1)$2&,@$2&=biline!($2,@$2,1)

el&,@el&=biline!(el,@el,1)e(I&,@I&)

E?"m*l$ !(i"&-'i $% FN FTS *"%-!$s G ωs G .25π δs G .2G ,s* G ω * G .35π δ * G . 1 G ,*

,i%-!$% FA [ FTS ," !"#"!i%' *!i% m"-(, -!%s#(!m!ii :ili%i!". S" +&(%si,"! 1# = .

9=1; o#gp=(2/9)*t!n(>p/2); o#gs=(2/9)*t!n(>s/2);!=20*log10(1/(1-&p)); !#=20*log10(1/&s);

n,ongt=$heb%1o&(o#gp,o#gs,!,!#,ShighS,"s");!,@!=$heb%1(n,!,o#gt,"high","s");>!=linsp!$e(0.1,1,500)*2*pi; igue(1),es(!,@!);&,@&=biline!(!,@!,1/9); igue(2),e(&,@&,);

roiectarea F34 ))R prin optimizarea în planul z

!(i"&-!" $%$i FP [ IIR &!" *!(?im" % s"%s$l "!(!il(! *-!-i&"mi%im" ( &!&-"!is-i& ," #!"&+"%' #(l(s"-" #$%&'i

b,!=%ule>!l(n,,#)

$%," m [ $% +"&-(! &$ +l(!il" m*li-$,i%il(! ,(!i-" f [ $% +"&-(! &$ +l(!il" #!"&+"%'"l(! s*"&i#i&-"

E?"m*l$ !(i"&-'i $% FP ," ;!,$l 8n = ," -i* -!"&" (s &!$i&!&-"!is-i& ," #!"&+"%' "s-" s*"&i#i&- *!i%

=0 0.6 0.6 1; #=1 1 0 0 ;b,!=%ule>!l(,,#);h,>=e(b,!,12);plot(,#,>/pi,!bs(h),S--S)

46


45/63

!(i"&-!" $%$i FP -i* B$--"!(!-0 ;"%"!li-

b,!=#!l!t(nb,n!,En)b,!,b1,b2=#!l!t(n!,nb,En)b,!,b1,b2=#!l!t(nb,n!,En,S&esignIl!tS)

Studiul filtrelor numerice ))R utiliz(nd mediul D)D'*!)*)E+


2) S" s-$,i i%-"!&-i+ (*"!-(!ii ," #il-! %$m"!i& IIR 1D i 2D &$$-(!$l m"%i$!il(! ,"#i%i-" %

FilteringImage processing

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S s" *!(i"&-"" $% FP [ FT IIR *l"&%, ," l $% FA [ FT ," -i* "li*-i& *!i% m"-(, i%+!i%'"i l im*$ls i m"-(, -!%s#(!m!ii :ili%i!" *"%-!$

* G ω * G .1π δ * G .1 G ,*s G ωs G .2π δs G .1 G ,s

S" + &(%si,"! #!"&+"%' ," "%-i(%!" 1 =


46/63

LUCRAREA DE LABORATOR NR. *

ESTIMARE SPECTRAL+ ŞI ANALI,+ TIMP-'RECVENŢ+

Obiectivele lucrrii1) S-$,i$l "s-im-(!il(! s*"&-!li &lsi&i *!m"-!i&i i %"*!m"-!i&i/2) S-$,i$l m"-(,"l(! ," %li -im*#!"&+"%' *!(&"s"l(! %"s-'i(%!"/3) U-ili!" #$%&'iil(! ATLAB *"%-!$ im*l"m"%-!" -"0%i&il(! ,"

"s-im!" s*"&-!l i ," %li -im*#!"&+"%'/4) S-$,i$l i%-"!&-i+ l m"-(,"l(! ," "s-im!" s*"&-!l i ," %li-im*#!"&+"%' $-ili%, m",i$l DIDACTICIEL.

De!"#ur$re$ lucrrii

Estimatorul spectral simplu

Fi" 3 $% *!(&"s l: ;$ssi% &"%-!- pm = ) i ," ,is*"!si" 2 25 pσ = ,"#i%i- *" 248 "%-i(%". S s" !"li"" %li s*"&-!l &"s-$i *!(&"s

$-ili%, "s-im-(!$l s*"&-!l sim*l$.) S s" ,"-"!mi%" ,"*ls!" i ,is*"!si "s-im-(!$l$i.

:) S s" !"#& &""i %li *"%-!$ 512 124 i 46 "%-i(%".&) S s" +"!i#i&" & &"s- "s-im-(! "s-" ,"*ls- i & ,is*"!si s %$

,"*i%," ," ,$!- (:s"!+'i"i "s-im-(! i%&(%sis-"%-).

longueuI=204;=!n&n(1,longueuI)*st(0.25);spe$teIsi#ple,Fe$Cent!=peio&oIsi#ple(,longueuI);se#ilog%(Fe$Cent!,spe$teIsi#ple,"b");l!bel("Fe$Cent! no#!li!t!");%l!bel("@#plitu&ine (&)");title("eio&og!#! si#pl!");bi!is=#e!n(spe$teIsi#ple)C!i!n$e=st&(spe$teIsi#ple)2

Estimatorul spectral mediat

@% &$l &"l$ii *!(&"s l"-(! s s" !"li"" %li s*"&-!l #(l(si%,"s-im-(!$l s*"&-!l m",i-. S s" s-$,i"" i%#l$"%' %$m!$l$i ," s"&+"%'"#(l(si-" *"%-!$ m",i"!". S s" ,"-"!mi%" ,"*ls!" i ,is*"!si "s-im-(!$l$i.

48


47/63

longueuIseuen$e=256;spe$teI#o%enne,Fe$Cent!=peio&oI#o%enne(,longueuIseuen$e);se#ilog%(Fe$Cent!,spe$teI#o%enne,"b");l!bel("e$Cent! no#!li!t!");%l!bel("@#plitu&ine (&)");

hol& on;longueuIseuen$e=12;spe$teI#o%enne,Fe$Cent!=peio&oI#o%enne(,longueuIseuen$e);se#ilog%(Fe$Cent!,spe$teI#o%enne,"--");title("eio&og!#! #e&i!t!");legen&("#e&iee pe eeste","#e&iee pe 16 eeste");

S s" +"!i#i&" & &"s- "s-im-(! "s-" -(- ,"*ls- ,! &(%sis-"%-. C" s" *(-" s*$%" ,"s*!" +!i'i !"(l$'i"i #!"&+"%'il" % !*(!- &$ %$m!$l ,"s"&'i$%i m",i-"

R"$l--"l" s" +(! -!"&" %-!$% -:"l ," -i*$l

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f

f

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Estimatorul spectral modificat

S s" !"li"" %li s*"&-!l &"l$ii *!(&"s l"-(! #(l(si%,"s-im-(!$l s*"&-!l m(,i#i&-. S s" s-$,i"" i%#l$"%' %$m!$l$i ," s"&+"%'"#(l(si-" *"%-!$ m",i"!" i ,i#"!i-"l(! #"!"s-!" $-ili-" *"%-!$ %"-"i!". S s","-"!mi%" ,"*ls!" i ,is*"!si "s-im-(!$l$i.

enete=bo$!(longueuIseuen$e);enete=enete*st(longueuIseuen$e/ su#(enete.*enete));enete=enete.";

spe$teI#o&iie,Fe$Cent!=peio&oI#o&iie(,longueuIseuen$e,enete);se#ilog%(Fe$Cent!,spe$teI#o&iie,"b");l!bel("e$Cent! no#!li!t!");%l!bel("@#plitu&ine (&)");hol& on;title("eio&og!#! #o&ii$!t!");

4


48/63

Rezoluţia dinamic#

S s" !"li"" %li s*"&-!l $%$i s"m%l &(m*$s ,i% ,($ si%$s(i,"#"&--" ," ;(m(-. !m"-!ii &(m*(%"%-"l(! s"m%l$l$i s$%- $!m-(!ii

*!im si%$s(i, ,($ si%$s(i, ;(m(- l: ;$ssi%

#!"&+"%' G 25


49/63

b=1;!=1 0.5 0.5;%=ilte(b,!,buit); %=%.";$oe,sig#!2=p!!#eteI@O(%,2);title("olii si eouile unui po$es @.O. &e o&inul 2");

o&e=2;&sp=0;o nu=0:200

&eno#in!teu=1;o in&=1:o&e

$oe(in&+1,1);$oe#=$oe(in&+1,1)*ep(-2*pi*A*in&*nu/200);&eno#in!teu=&eno#in!teu+$oe# ;

en&;&sp=&sp sig#!2/(!bs(&eno#in!teu).2);

en&;nu=0:1/200:0.5;

plot(nu,&sp(1,1:length(nu)));title("O!spunsul in e$Cent! !l iltului");l!bel("e$Cent! no#!li!t!");%l!bel("@#plitu&ine");

'naliza spectral# de înalt# rezoluţie parametric#

S s" ;"%"!"" ( si%$s(i, *$! ," #!"&+"%' 5


50/63

Spectrograma unui semnal c$irp

U% s"m%l LF !" $!m-(!" "?*!"si" %li-i&

1 ) si%2 2 ) ) s t f t t = π × + πβ + Φ$%," 2 1 ) 92 ) f f 3ulselent&β = − × 3ulselent& #ii%, ,$!- s"m%l$l$i.

) S s" ;"%"!"" $% s-#"l ," s"m%l *"%-!$ Φ = . :) S s" !"*!"i%-" s"m%l$l % ,(m"%iil" -"m*(!l i #!"&+"%'il.&) S s" !"*!"i%-" &"li s"m%l % *l%$l -im*#!"&+"%' &$ $-(!$l

s*"&-!(;!m"i.

)1=2000;2=000;pulselength=0.025;

Fe=20000;t=(0:1/Fe:pulselength);bet!=(2-1)/(2*pulselength);$hip=sin(2*pi*(1+bet!*t).*t);$hip2=C$o(s!>tooth((2*pi/pulselength)*t,1), 1/Fe,2/Fe*Fe,Fe);igue(1);$l;subplot(211);plot(t,$hip);l!bel("9i#p");%l!bel("@#plitu&ine");title("@n!li! unei #o&ul!tii lini!e &e e$Cent!")

:)U=tshit(!bs(t($hip)).2);l$=length($hip);#$=l$/2;e=(-#$:1:#$-1)*Fe/l$;subplot(212);plot(e,U);l!bel("Fe$Cent! D");%l!bel("


51/63

*ompromisul rezoluţie spectral#9rezoluţie temoral#

S s" ;"%"!"" $% s"m%l &(m*$s ,i%-!( si%$s(i, i $% Di!&.S s" %li"" &"s- s"m%l &$ $-(!$l s*"&-!(;!m"i *"%-!$ ,i+"!s"

,im"%si$%i l" #"!"s-!"i ," %li.

S s" i%-"!*!"-"" !"$l--"l" (:'i%$-".

Fe=10000; 1=1000; 2=4000;9=0.015; t=0.005:1/Fe:9;&elt!=0.005*Fe;sig=eos(1,&elt!), sin(2*pi*1*t), eos(1,&elt!),5*ones(1,1), eos(1,2*&elt!);subplot(311);plot(0:1000*(1/Fe):1000*(length(sig)/Fe-1/Fe),sig);title("'e#n!l te#po!l");l!bel("9i#p #s");%l!bel("@#plitu&ine");',F,9 = spe$g!#(sig,12,Fe,64);subplot(312); i#!ges$(9*1000,F/1000,!bs('));l!bel("9i#p #s"); %l!bel("Fe$Cent! D");9itle("'pe$tog!#! $u o ee!st! &e 64 pun$te"),',F,9 = spe$g!#(sig,12,Fe,16);subplot(313); i#!ges$(9*1000,F/1000,!bs('));l!bel("9i#p #s"); %l!bel("Fe$Cent! D");title("'pe$tog!#! $u o ee!st! &e 16 pun$te")

ScalogramaS&l(;!m "s-" ( ,is-!i:$'i" ," "%"!;i" &!" s" (:'i%" & m(,$l$l l *-!- l -!%s#(!m-"i +"l"- &(%-i%$".

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− = − − A&"s- %$ +"!i#i& *!(*!i"--" ," m",i" %$l. "%-!$ *!(?im &"s-

&(%,i'i" &(%s"!+%, $% %$m! mi& ," (s&il'ii (!l"- *!(*$s &(m*!(mis$l

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= .S s" +i$li"" l$! &"s-"i $%,i(!" *"%-!$ ,i+"!i *!m"-!i. :) $%"'i % "+i,"%' +!i'iil" !"(l$'iil(! -"m*(!l i #!"&+"%'il %

#$%&'i" ," *(i'i % *l%$l -im*#!"&+"%'. S" + $-ili $% s"m%l &(m*$s,i%-!$% Di!& i ,($ si%$s(i," -!$%&0i-" ," #!"&+"%'" ,i#"!i-".

)on&e,ts= on&I#o(12, 10, 100);igue(1);$l;subplot(211);plot(ts,e!l(on&e));

hol& on;plot(ts,!bs(on&e),"");gi&;title("n&iso!! olet")

53


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:)Fe=64000;1=500; 2=000; 9=0.01;t=0:1/Fe:9;&elt!=0.005*Fe;

sig1=eos(1,&elt!), sin(2*pi*1*t), eos(1,2*&elt!);sig2=eos(1,&elt!),sin(2*pi*2*t), eos(1,2*&elt!);sig=sig1+sig2;sig(4*&elt!+1:4*&elt!+5)=10*ones(1,5);igue(1);subplot(212); plot(sig);title("'e#n!l stu&i!t: su#! ! &ou! sinusoi&e si un igne(,Fe);#!(#!(e!l(E)))#!(#!(i#!g(E)))su#(su#(E))su#(.2)

:)1=1000;&elt!=3000;2=1+&elt!;Fe=oun&(3*2);9=0.01;t=0:1/Fe:9;&elt!t=0.01*Fe;#!ge=20;sig1=eos(1,#!ge), sin(2*pi*1*t),eos(1,9*Fe+&elt!t+#!ge);sig2=eos(1,#!ge),sin(2*pi*2*t), eos(1,9*Fe+&elt!t+#!ge);sig3=eos(1,#!ge+9*Fe+&elt!t),sin(2*pi*1*t), eos(1,#!ge);

sig=sig1+sig2+sig3;E,E,>C,!bs$isse=>igne(sig,Fe); 1,2=sie(E);

54


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igue(2);$l; oo# on ;subplot(311); #ilieu=oun&(((1+2)/2)/(Fe/2)*1);plot(!bs$isse,E(#ilieu,:)); l!bel("9i#p (#s)");title("'tu&iul inteeentelo")subplot(312);#ilieut=oun&(#!ge+9*Fe+&elt!t);

plot(>C,E(:,#ilieut));l!bel("Fe$Cent! D")subplot(313);epee=oun&(#!ge+9/2*Fe);plot(>C,E(:,epee));l!bel("Fe$Cent! D")

'naliza multirezoluţie

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title("


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Studiul operatorilor de estimare spectral# şi analiz#timp4frecvenţ# utiliz(nd mediul D)D'*!)*)E+


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LUCRAREA DE LABORATOR NR.

DETECŢIA SECVENŢELOR NUMERICE

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57


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:)=>!CeIgen(1,"ti!ngle",1000);#=#!t$h("ti!ngle",);iguesubplot(3,1,1)

>!Ceplot()subplot(312)#!t$h("ti!ngle")subplot(313)>!Ceplot(#)

&)=>!CeIgen(1,"#!n$heste",100);#=#!t$h("#!n$heste",);iguesubplot(3,1,1)>!Ceplot()

subplot(312)#!t$h("#!n$heste")subplot(313)>!Ceplot(#)

,)5=>!CeIgen(1 0 0 1 0,"pol!In",1000);#=#!t$h("pol!In",5);iguesubplot(211)>!Ceplot(5)subplot(212)>!Ceplot(#)

")5=>!CeIgen(1 0 0 1 0,"unipol!In",1000);#=#!t$h("pol!In",5);iguesubplot(211)>!Ceplot(5)subplot(212)>!Ceplot(#)

Detecţia semnalelor

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>!Ceplot(10)%10=$h!nnel(10,1,2,400);subplot(312)>!Ceplot(%10)subplot(313)

10=#!t$h("pol!In",%10);>!Ceplot(10)

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Receptor cu filtru trece /os

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LUCRAREA DE LABORATOR NR. /

PRELUCRAREA MULTIRAT+ A SECVENŢELOR SAUSC0IMBAREA RATEI DE EŞANTIONARE

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A%li'i -!%s#(!m-"l" F($!i"! l" s"&+"%'"l(! i%i'il ,"&im-i%-"!*(l- i !"#&$-.

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)nterpolatorul de ordinul zero

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title("'pe$tul se#n!lului n")b0=i2(50,0 .5 .5 1,1 1 0 0);b1=i2(50,0 .5 .5 1,0 0 1 1);h0,>0=e(b0,1,512);h1,>1=e(b1,1,512);subplot(212),plot(>0/pi,!bs(h0),>1/pi,!bs(h1)),

title("'pe$tele D0 si D1")0=ilte(b0,1,);1=ilte(b1,1,);0&=0(1:2:length(0));%0=0&;1&=1(1:2:length(1));t&=(0:length(0&)-1)/Fe;80=!bs(tshit(t(0,512)));81=!bs(tshit(t(1,512)));igue(2);subplot(211),plot(C,80),title("'pe$tul se#n!lului 0 s!u %0")subplot(212),plot(C,81);title("'pe$tul se#n!lului 1")

C=((-255:256)/512)*(Fe/2);80&=!bs(tshit(t(0&,512)));81&=!bs(tshit(t(1&,512)));igue(3);subplot(211),plot(C,80&),title("'pe$tul se#n!lului 0&")subplot(212),plot(C,81&);title("'pe$tul se#n!lului 1&")2=length(1&);o =0:2-1%1(+1)=((-1))*1&(+1);

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