fo vr,t,- ~ rt~ <
::::f.;- \'.)er-, c:o, c..~ ,9 r-c\s
. rePca~@ ~-,"'j..~cl ,·rv-\---c-rvo )S
.f ( +) = J l + ,-\(. T) w'0~re '-<... ~ - 2 , - , , o , 1 , 2. , . - •
·use 01 il""'lfinr+ese,,es ol-sil"'lvSC)oS L-fcvr 1er s en e s )
Cit::,
XC.+)-: q 0
+- L.. (_ Om co~ ~ru t
l"I"'= I
0 -t bl"T'\<;;lr, m G\..l 0 t) . - '2...--rT
/ VJo---;:-
c oeff-i C.\et)ts are d~te"rn' ned "'o. ,n ~ro-\-'lo<1 ·.
0 :: _ I j X( t) cit
T
'o"" ~ ~ ~ x ( +)
0 T
'& i n ( m oo O -1:. ) cH::
T
0,0
e )(.•• 2-5
Z Co l"rlcos rowO t ~ b <s. ,r> fYIW,,-\:-)
~
rr,
• c:-o s;n (rt\TT /2)
[· { i X(t)-=-... 2.,S-ttD· . ~ - - - - CO'S (.mtftJ
- . m-., . rnT\" .
• +~Hs vs per·1od,c CSi<:3na.l J<.lt) ·, s prec..\sel'I 't-e..?li ca-\ed 'o"'f Surnmirg
o..n_ ·inf-i ni \-e, n....n--oe-r .::~ <.:f,nvRol ds.~•, ·· -
• ~vtadamen:tol f-reque"(")cy S>-t x (--t') : w 0 ~ '1...)1/T
• nar:rnanic·s : f--r-:e-'1Y".'i--te9er. _--rn~\~p\e-s. cir- e,u @ V'-)Y"~c"' ~;1nvsoid
0
.f-voc+icns 9'-~VS .. oa.c.,\.!.Jr '-
• rnony pe-r:·,oa,c. ~ Qvos, -per\odi~ 'b1a\~ic;:o\ sigoa\s c..anoe- acc.uro-\-e\y
a:,ppr-cx\ma+ed w/ ohh; Q +e'N 'no\-~""b1"1\c ·,Qq{Y')~t1e,'f"\.+S
1
c:..omPOC+-Four,,ey- ser,e s
0,0
X (4):: A
O
-+ L
m:1
Arn co~ (rnw 0 t 1-q:>m)
.
i.
~X. c. on+d: ·h·· Dr,, above T{
. 1Q. '°" ~ ,,, ,5
lo \ m'TY ) \ -
f·;
rn1r
Ani~ loml: tn'fr s1nl_~ - o rv,~2.,+,~
(J) --: t·on-1(-bm ·J ~·t-on1_./ ·l _, o < j ,-, • .•) . ~ m-:~o, t·, 4,'5 ' ,
. _rri • _<?arm . \'5sin(mTT/2)/rr,1T . ·. . _-tn _
~.2.,?l,~_,l
1 1
Per-i od·,·c
eqi_gn::::i.\ •expressed
. +, rn'e ' 'dDmt'.),'r-t'' '·,· -. l / ( ~ ' ' , .,. • \'t"r-~q+.,~~C''f ,<:l'\orr.)(l\'f"\
4s,gnors time funcHon ,- ,. . , /' · r ·4 : A rt'' ec- q)-m · QS a '-fvnch(:)n
4 X (+) ot- t"" w r ::j-:-
0
. I
r .' I
\
, fo u r\ey- -+rons f c:,,r m f or o~,m,c. s\gno1.S
'b\J,-.·1 cr Dn\y opp\\cs -to f:e'f',Q:\\c signa\s
• b\O\~ica\ s,gno\s
~ -fu"c-t, ens o\-- +,me
con-t-i nuovs .
4
r 00
- Jc.o{-
don '-\- <"e~- in +\rre 'j(w) '.: -~ x_tt )e dt
.. four\e-r- -t-roD'31'0,\f"l '.
-, d e c c:Yn PD'S~ a c. of')+\ "1\J ovs ape -r\ cx:h c s; g n a \ \ n ..L 'J
f,eqvc'f'C'y comP:)nents
c<,ns~,+'--JCf")+
X Cw):. cornp,ex vo\ fxn. ~ con..-t\nu~ freo o-\- w
½ con be -+-r-a'f"Sfo,rred bac\< -t-o t,rre dc:>'<~10·1 n Yv / :
\ (o ~wt
x C-+) :: 21'" J X ( w) e d uJ
- 00
d iscre\-e -fovner -t-ron~f- o n-,, ( C>'FT)
N-1 .:., 2:n-r111<..
-J- N -l
X(rn') :: Lx(t)e N j m-:o,,, .. . /
\(:6 .
· u-sed t-o ooo\yz.~ b ,epr-e~c1J+ d iSC\"e-\-e s ·1gnq 1s ·, n -the
-f reqve{lc'{ dt>rna,n
1
4 rn = dig',\-a\ frcq~r'Cy i()d-e')Z . 4 N ::.e:ven Nvmter- -tru-\- rep-e:e,n+-;
w x C\t..>:: sa rnp\ed da+Cl. cl- x C+1 -+re nurn of 9:lrnt=b ~ r x CK)
-t-os+ fovr\er +TOnSTon-n (FFTJ
· DFT needs O(Ni.) ooe~a~c11S ·
~ bf c i+ ms N/2 outputs' xk ~ eac'n requires O sum d- ~j
·, s a p::>weir ~ 2 1 so~eused --1:-rlc\.<.s .can be
~ ge+- the tO.TT'e ·y-e:su\+ W/ l (N/2)Log-i.("f) Oper0t1or--s
_ _ _ _ _ ______,
~
F\=:T
• -t-n.\ 'o ·~ . s,gn,f, a::Jni-ly -f-as \--"Cr'
-es-t-'1 rn::::l-\-1on ·, ssves , 1 f
. h ighest- frequency \ ho+ con re 8S'1 rn:l:\ec:f. j rna~ 2 Ts : ;-
L" • .t. - ;tS .
• -f-requen°'{ r-e~\.Jlu,,an. A.J -N
,-..,.1
Ts== sarr:ip,,n;t
J.s.: som~\i"9 frea ~ tq- ,n+er o\
o+
-, I"' - \,e ng+n
• ·,t- ·\ ~ ~ -to p\c\£. a l-P...ngn+-+rc+- y ·, e\d'3 a - f ~ , nc:rer-nen+ ao-\-Q
• ntC> -tre. <;; iqro\ s -fund orrtn\-0 \ f ~
L, -for FTT
L-, nuf"'Y'l mus\- 'oe ?C'\I\Jer cl -+wo
of- Gomp\es
1.4 1rod-\-o estima,-es@ s+ra~ -f-r-e,C\;
::::f.;- \'.)er-, c:o, c..~ ,9 r-c\s
. rePca~@ ~-,"'j..~cl ,·rv-\---c-rvo )S
.f ( +) = J l + ,-\(. T) w'0~re '-<... ~ - 2 , - , , o , 1 , 2. , . - •
·use 01 il""'lfinr+ese,,es ol-sil"'lvSC)oS L-fcvr 1er s en e s )
Cit::,
XC.+)-: q 0
+- L.. (_ Om co~ ~ru t
l"I"'= I
0 -t bl"T'\<;;lr, m G\..l 0 t) . - '2...--rT
/ VJo---;:-
c oeff-i C.\et)ts are d~te"rn' ned "'o. ,n ~ro-\-'lo<1 ·.
0 :: _ I j X( t) cit
T
'o"" ~ ~ ~ x ( +)
0 T
'& i n ( m oo O -1:. ) cH::
T
0,0
e )(.•• 2-5
Z Co l"rlcos rowO t ~ b <s. ,r> fYIW,,-\:-)
~
rr,
• c:-o s;n (rt\TT /2)
[· { i X(t)-=-... 2.,S-ttD· . ~ - - - - CO'S (.mtftJ
- . m-., . rnT\" .
• +~Hs vs per·1od,c CSi<:3na.l J<.lt) ·, s prec..\sel'I 't-e..?li ca-\ed 'o"'f Surnmirg
o..n_ ·inf-i ni \-e, n....n--oe-r .::~ <.:f,nvRol ds.~•, ·· -
• ~vtadamen:tol f-reque"(")cy S>-t x (--t') : w 0 ~ '1...)1/T
• nar:rnanic·s : f--r-:e-'1Y".'i--te9er. _--rn~\~p\e-s. cir- e,u @ V'-)Y"~c"' ~;1nvsoid
0
.f-voc+icns 9'-~VS .. oa.c.,\.!.Jr '-
• rnony pe-r:·,oa,c. ~ Qvos, -per\odi~ 'b1a\~ic;:o\ sigoa\s c..anoe- acc.uro-\-e\y
a:,ppr-cx\ma+ed w/ ohh; Q +e'N 'no\-~""b1"1\c ·,Qq{Y')~t1e,'f"\.+S
1
c:..omPOC+-Four,,ey- ser,e s
0,0
X (4):: A
O
-+ L
m:1
Arn co~ (rnw 0 t 1-q:>m)
.
i.
~X. c. on+d: ·h·· Dr,, above T{
. 1Q. '°" ~ ,,, ,5
lo \ m'TY ) \ -
f·;
rn1r
Ani~ loml: tn'fr s1nl_~ - o rv,~2.,+,~
(J) --: t·on-1(-bm ·J ~·t-on1_./ ·l _, o < j ,-, • .•) . ~ m-:~o, t·, 4,'5 ' ,
. _rri • _<?arm . \'5sin(mTT/2)/rr,1T . ·. . _-tn _
~.2.,?l,~_,l
1 1
Per-i od·,·c
eqi_gn::::i.\ •expressed
. +, rn'e ' 'dDmt'.),'r-t'' '·,· -. l / ( ~ ' ' , .,. • \'t"r-~q+.,~~C''f ,<:l'\orr.)(l\'f"\
4s,gnors time funcHon ,- ,. . , /' · r ·4 : A rt'' ec- q)-m · QS a '-fvnch(:)n
4 X (+) ot- t"" w r ::j-:-
0
. I
r .' I
\
, fo u r\ey- -+rons f c:,,r m f or o~,m,c. s\gno1.S
'b\J,-.·1 cr Dn\y opp\\cs -to f:e'f',Q:\\c signa\s
• b\O\~ica\ s,gno\s
~ -fu"c-t, ens o\-- +,me
con-t-i nuovs .
4
r 00
- Jc.o{-
don '-\- <"e~- in +\rre 'j(w) '.: -~ x_tt )e dt
.. four\e-r- -t-roD'31'0,\f"l '.
-, d e c c:Yn PD'S~ a c. of')+\ "1\J ovs ape -r\ cx:h c s; g n a \ \ n ..L 'J
f,eqvc'f'C'y comP:)nents
c<,ns~,+'--JCf")+
X Cw):. cornp,ex vo\ fxn. ~ con..-t\nu~ freo o-\- w
½ con be -+-r-a'f"Sfo,rred bac\< -t-o t,rre dc:>'<~10·1 n Yv / :
\ (o ~wt
x C-+) :: 21'" J X ( w) e d uJ
- 00
d iscre\-e -fovner -t-ron~f- o n-,, ( C>'FT)
N-1 .:., 2:n-r111<..
-J- N -l
X(rn') :: Lx(t)e N j m-:o,,, .. . /
\(:6 .
· u-sed t-o ooo\yz.~ b ,epr-e~c1J+ d iSC\"e-\-e s ·1gnq 1s ·, n -the
-f reqve{lc'{ dt>rna,n
1
4 rn = dig',\-a\ frcq~r'Cy i()d-e')Z . 4 N ::.e:ven Nvmter- -tru-\- rep-e:e,n+-;
w x C\t..>:: sa rnp\ed da+Cl. cl- x C+1 -+re nurn of 9:lrnt=b ~ r x CK)
-t-os+ fovr\er +TOnSTon-n (FFTJ
· DFT needs O(Ni.) ooe~a~c11S ·
~ bf c i+ ms N/2 outputs' xk ~ eac'n requires O sum d- ~j
·, s a p::>weir ~ 2 1 so~eused --1:-rlc\.<.s .can be
~ ge+- the tO.TT'e ·y-e:su\+ W/ l (N/2)Log-i.("f) Oper0t1or--s
_ _ _ _ _ ______,
~
F\=:T
• -t-n.\ 'o ·~ . s,gn,f, a::Jni-ly -f-as \--"Cr'
-es-t-'1 rn::::l-\-1on ·, ssves , 1 f
. h ighest- frequency \ ho+ con re 8S'1 rn:l:\ec:f. j rna~ 2 Ts : ;-
L" • .t. - ;tS .
• -f-requen°'{ r-e~\.Jlu,,an. A.J -N
,-..,.1
Ts== sarr:ip,,n;t
J.s.: som~\i"9 frea ~ tq- ,n+er o\
o+
-, I"' - \,e ng+n
• ·,t- ·\ ~ ~ -to p\c\£. a l-P...ngn+-+rc+- y ·, e\d'3 a - f ~ , nc:rer-nen+ ao-\-Q
• ntC> -tre. <;; iqro\ s -fund orrtn\-0 \ f ~
L, -for FTT
L-, nuf"'Y'l mus\- 'oe ?C'\I\Jer cl -+wo
of- Gomp\es
1.4 1rod-\-o estima,-es@ s+ra~ -f-r-e,C\;
