Chapter 5 - The Fourier Transform

Transcript

1 M. J. Robrs - /7/ Chapr 5 - Th ourir Trasorm Soluios (I his soluio maual, h symbol,, is usd or priodic covoluio bcaus h prrrd symbol which appars i h x is o i h o slcio o h word procssor usd o cra his maual.). Th rasiio rom h CTS o h CTT is illusrad by h sigal, or x()= rc comb w T T ()= x T rc w =. Th complx CTS or his sigal is giv by X[ k]= Aw kw sic T T. Plo h modiid CTS, [ ]= ( ), T X k Awsic w k or w = ad = 5.,. ad. vrsus k or h rag < k <. 5-

2 M. J. Robrs - /7/ T X[k] - k T X[k] - k - = 5. T X[k] - k T X[k] - k - =. T X[k] - k T X[k] - k - =. 5-

3 M. J. Robrs - /7/. Suppos a ucio, m( x), has uis o kg 3 ad is a ucio o spaial posiio, x, i m mrs. Wri h mahmaical xprssio or is CTT, M( y). Wha ar h uis o M ad y? M = yx y m x dx Th uis o M ar kg m ad h uis o y ar m. 3. Usig h igral diiio o h ourir rasorm, id h CTT o hs ucios. (a) x()= ri() = () = ( + ) X ri d cos si d X( )= ( cos ) d = cos d cos d X si cos d = si X( )= + si cos = ( ) ( ) cos X( )= si d d si d = ( ) cos x Th, usig si( x) si( y)= [ cos( x y) cos( x + y) ] si ( x )= ( ) si X( )= = sic ( ) 5-3

4 M. J. Robrs - /7/ (b) x()= δ+ δ X= + d si δ δ = =. I igur E hr is o xampl ach o a lowpass, highpass, badpass ad badsop sigal. Idiy hm. x() (a) x() (b) x() (c) x() (d) (a) (b) (c) (d) badsop badpass lowpass highpass igur E Sigals wih dir rqucy co 5. Sarig wih h diiio o h CTT id h radia-rqucy orm o h gralizd CTT o a cosa. Th vriy ha a chag o variabl, ω, yilds h corrc rsul i cyclic-rqucy orm. Chck your aswr agais h ourir rasorm abl i Appdix E. L g()= A. Th = = G A ω σ ω d lim A cos ω si ω d + σ σ G lim A σ ( ω)= { cos( ω) d si( ω) d σ + v 3 { v 3 v odd 5-

5 M. J. Robrs - /7/ = σ G ω lim A cos ω d σ + = σ G ω lim A cos ω d σ + az az Usig cos( bz) dz= ( acos( bz)+ bsi( bz) ), a + b σ or ω, lim A =. σ σ ω + + σ G( ω)= lim A si cos σ ω ω ω σ ω + σ + ( ) σ G( ω)= lim A σ + σ + ω σ Th ara udr A σ + ω is Ara = σ A d σ + ω ω. Usig dx bx = a a + b x ab a ω Ara = Aσ a A =. σ σ Thror h CTT pair is A Aδ ω. or h orm h pair is Lig = ω i h scod pair, A Aδ. A Aδ ω A = δ ( ω ). QED. 5-5

6 M. J. Robrs - /7/ 6. Sarig wih h diiio o h CTT, id h gralizd CTT o a si o h orm, Asi( ω ) ad chck your aswr agais h rsuls giv abov. Chck your aswr agais h ourir rasorm abl i Appdix E. ()= L g Asi. Th ω = ( ) = ( ) + G Asi ω σ ω ω d lim A si ω cos ω si ω d σ σ G A lim σ ( ω)= { si( ω ) cos( ω) d si( ω ) si( ω) d σ + v 3 3 { v 3 3 odd v odd odd = σ G ω A lim si ω si ω d σ + σ + σ G( ω)=a lim cos( ( ω ω) ) cos( ( ω + ω) ) d az az Usig cos( bz) dz= acos bz bsi bz a + b [ ] ( + ) σ si cos σ ω ω ω (( ω ω ) ) σ (( ω ω ) ) + ( ) G( ω)=a lim+ σ σ si cos σ + ( ω + ω) ω (( ω + ω ) ) σ ( ω + ω ) ( ) σ σ G( ω)=alim lim + + σ σ + ( ω ω) σ σ + ( ω + ω) By h sam rasoig as i Exrcis 5, wh ω ω, G( ω)= lim σ + ad, whω ω G( ω)= lim σ + σ σ + ω ω = σ =. σ + ω + ω 5-6

7 M. J. Robrs - /7/ Th aras udr hs ucios ar Ara σω d =A σlim + σ + ω ω σω d ad Ara =A σlim + σ + ω + ω Makig h chag o variabl, ω ω = λ dω = d λ, i h irs igral, Ara σλ d =A σlim + σ + λ =A Similarly, Ara = σλ d A σlim + σ + λ = A. Ths aras ar idpd o h valu o σ. Thror [ ] G( ω)= A δ( ω + ω)δ( ωω ). QED. 7. id h CTS ad CTT o ach o hs priodic sigals ad compar h rsuls. Ar idig h rasorms, ormula a gral mhod o covrig bw h wo orms or priodic sigals. ()= (a) x Acos A Th CTS is simply wo impulss, X[ k]= ( δ[ k]+ δ [ k + ] ). A Th CTT is X( )= ( δ( )+ δ( + ) )= X[] δ( )+ X[ ] δ ( + ). ()= () (b) x comb Th CTS is oud rom X = [ ] ( ) X k δ k k = 5-7

8 M. J. Robrs - /7/ T ( k ) ( k ) X[ k]= comb() d= () d= T δ. T Th CTT is = = ( )= [ ] ( ) X comb δ k X k δ k k= k=. L a sigal b did by ()= + x cos 5cos 5. id h CTT s o x ad x + ad idiy h rsula phas shi o ach siusoid i ach cas. Plo h phas o h CTT ad draw a sraigh li hrough h phas pois which rsul i ach cas. Wha is h gral rlaioship bw h slop o ha li ad h im dlay? x = cos cos x = cos + 5cos5 { { phas phas shi shi 5 5 x ( )+ ( + ) [ ] δ δ δ δ x ( ) + ( + ) δ δ δ δ x+ cos cos =

9 M. J. Robrs - /7/ 3 x+ = cos + + 5cos5+ { 5 { phas phas shi shi 5 5 x+ ( )+ ( + ) [ ] δ δ δ δ x+ ( ) + ( + ) δ δ δ δ X( ) Slop = Slop = Th slop o h li is ims h dlay. 9. Usig h rqucy-shiig propry, id ad plo vrsus im h ivrs CTT o + X( )= rc + rc. ()= + = x sic sic sic cos x()

10 M. J. Robrs - /7/. id h CTT o x()= sic(). Th mak h rasormaio,, i x() ad id h CTT o h rasormd sigal. sic rc () sic rc rc( ) rc rc( ) - rc( ) -. Usig h muliplicaio-covoluio dualiy o h CTT, id a xprssio or y which dos o us h covoluio opraor,, ad plo y(). ()= () (a) y rc cos y()= sic δ + δ + y()= sic sic + + δ δ y()= + + = + + δ δ δ δ y()= cos y() ()

11 M. J. Robrs - /7/ ()= () (b) y rc cos y()= sic [ δ( )+ δ( + ) ] y()= δ sic()+ δ + sic = y() - - (c) y()= sic() sic y()= { rc rc }= { rc }= sic y() ()= () (d) y sic sic y()= { rc ri }= { rc ri }== { ri }= sic y() - ()= () () y u si ()= y δ( + )δ + [ ] = δ + + δ + 5-

12 M. J. Robrs - /7/ ()= y δ( + ) δ( ) ( + )( + ) ( )( ) δ δ = + ( ) ( + ) ()= y [ ] δ( + )δ( )+ δ( + )+ δ + ( ) ()= y [ ] δ( + )δ( ) ( + )+ δ δ + ( ) + ( ) Th, usig w g ()= y + si cos Acos( x)+ Bsi( x)= A + B cos xa ()= y + ( ) cos +. 5 B A. y() Usig h CTT o h rcagl ucio ad h diriaio propry o h CTT id h ourir rasorm o x()= δ δ( + ). Chck your aswr agais h CTT oud usig h abl ad h im-shiig propry. L y()= rc. Th x d ()= ( d y () ). rc Usig h diriaio propry o h CTT, sic( ) 5-

13 M. J. Robrs - /7/ d d rc sic( ) sic [ ]= d d rc si( ) = si ( ) Usig h dirc approach, + δ δ( + ) si =. Chck. 3. id h CTS ad CTT o hs priodic ucios ad compar aswrs. (a) x()= rc() comb T k X[ k]= rc() comb d rc comb T = () T k X[ k]= d = [ cos( k) si( k) ] d si k X[ k]= cos( k) d= k k si k sic = = k X ( )= sic ( ) k comb ( )= sic ( ) δ k = k k X( )= sic X k k δ = [ ] δ k= ()= (b) x ri comb T k= k ( k ) k X[ k]= ri comb d= ri d T T d 5-3

14 M. J. Robrs - /7/ = [ ]= X k ri cos k si k d ri cos k d X[ k]= ( ) cos( k) d = cos( k) d cos( k) d k si( k) si( k) X[ k]= cos k d cos d k ( ) = k ( k ) λ λ λ si k k 5 5 k X[ k]= si si d k ( k ) λ λ 5 λ λ si k 5 k X[ k]= ksi k cos k ( k ) [ λ ] si k 5 X[ k]= ksi k cos k + k ( k ) cos k 5 5 X[ k]= k X( )= X( )= sic comb = sic 5 k= X( )= 5 k = k si 5 5 δ( k)= k 5 k sic δ( k) 5 k = k= δ( k) k cos 5 δ( k) k 5-

15 M. J. Robrs - /7/ X( )= k cos 5 5 δ ( k). Chcks wih CTS. k k =. Usig Parsval s horm, id h sigal rgy o hs sigals. (a) x()= sic 5 = () = = = x E x d X d rc 5 d rc 5 d ()= (b) x sic 3 Ex = d = E = () d = ( x ) d = x X ri d = d ri E x = 3 3 d = d = + d ri E = x + = = Wha is h oal ara udr h ucio, g()= sic 3 g d G () = ()? G = rc G()= 3 = g() d 6. Usig h igraio propry, id h CTT o hs ucios ad compar wih h CTT oud usig ohr propris. 5-5

16 M. J. Robrs - /7/ (a) ()= g, <, < <, lswhr g() g'() g 3 3 g = rc+ rc 3 3 g = rc+ rc sic sic () sic 3 3 sic ( )= 3 3 () g 3sic 3 sic si 3 si( 3 )= sic( ) Alra Mhod: g()= rc rc sic () sic Chck. (b) g()= rc g = δ + δ 6si 3 () g = si( 3 ) si = ( 3 ) 6 = sic ( 3 ) 3 Alra Mhod: () g sic( 3 ) Chck. 5-6

17 M. J. Robrs - /7/ 7. Skch h magiuds ad phass o h CTT s o hs sigals i h orm. (a) x()= δ( ) X= X( ) - Phas o X( ) - - (b) x()= u() u X = + δ x()= u() u X= ( ), (impulss cacl) + x()= u() u X= = sic X( ) -5 5 Phas o X( )

18 M. J. Robrs - /7/ + (c) x()= rc X sic 5 = ( ) X( ) - Phas o X( ) - - (d) x()= 5sic( ) X =. 5rc X( ) 3 - Phas o X( ) - - [ ] () x()= 6si( ) X= 3 δ( + )δ 5-

19 M. J. Robrs - /7/ X( ) 3 - Phas o X( ) - - () u() + 3 x()= u( 3) X= = X( ) - Phas o X( ) - (g) x()= = X= =

20 M. J. Robrs - /7/ X( ) Phas o X( ) Skch h magiuds ad phass o h CTT s o hs sigals i h ω orm. (a) x()= comb X comb k ( )= ω ω = δ ω k = X(ω) - Phas o X(ω) ω - ω - ()= (b) x sg X( ω)= ω 5-

21 M. J. Robrs - /7/ X(ω) 6 - Phas o X(ω) ω - ω - (c) x()= ri X( ω)= sic ω ω X(ω) - ω Phas o X(ω) - ω - (d) ()= x + sic X( ω)= ri ω ω 5-

22 M. J. Robrs - /7/ X(ω).3 - ω Phas o X(ω) - ω - () ()= x cos X( )= + ω δ ω δ ω + ω ()= x ()= x cos X( )= [ ( )+ ( + )] ω ω δω δω cos X( ω)= δω ( ) + δω ( + ) + X(ω) ω -7 7 Phas o X(ω) ω () x()= u()= u( 3) X( ω)= 3 ω

23 M. J. Robrs - /7/ X(ω) - ω Phas o X(ω).57 - ω -.57 (g) x()= 7 X( )= ω 5 + ω 7 = 5 + ω X(ω) 3 - ω Phas o X(ω) - ω - 9. Skch h ivrs CTT s o hs ucios. (a) x()= 6sic X=5rc x()

24 M. J. Robrs - /7/ (b) ()= x rc sic X( )= x() - (c) a a a +,, x() x 3 x()= 6 X= 9 + x() () (d) a u, a > a +,, x() x x()= u() X= + x() () cos δ( )+ δ + [ ] 5-

25 M. J. Robrs - /7/ δ( 3)+ δ + 3 x()= cos( 6) X= 3 6 x() () x()= X δ 5 δ 5 5 = = x() - (g) sg () x()= 3sg() X= 3 x() Skch h ivrs CTT s o hs ucios. ω (a), ω ω, x() x x()= X( 6 ω 6 ω ω )= = ω = 5-5

26 M. J. Robrs - /7/ x(). - ω (b) ri() sic, ω ω (), x x 7 x()= ri X sic = ω ω 7 x() - [ ] (c) si( ω ) δ( ω + ω )δ ωω [ ] x()= si( ) X( ω)= δ( ω + )δ ω x() ω (d) comb() comb, ω ω (), x x ω comb x()= comb X ( )= ω 5 x()

27 M. J. Robrs - /7/ () sg () ω, δ( ω) 5 x()= sg()+ 5 X ω 5 ω = + δ( ω) x() - - () () a u, a > a + ω 3 6 x()= 6 u() X( ω)= 3 + ω x() (g) sic () ri ω, ω 6ω, x() x x()= sic X( ω)= ri( ω) 6 x().5 -. id h CTT s o hs sigals i ihr h or ω orm, whichvr is mor covi. 5-7

28 M. J. Robrs - /7/ ()= + (a) x 3cos si 3 5 X= δ δ δ δ 3 X( )= δ δ.. X= δ δ X = ( ) ω 5 δ ω 5 δ ω (b) x()= comb comb = = X comb comb comb = ( )= X comb comb si X ω ω ω ( ω)= comb si x() 6 (a) X(ω) -6 (b) x() - Phas o X(ω) - - X( ) ω ω Phas o X( ) - - (c) x()= sic sic sic + 5-

29 M. J. Robrs - /7/ X( )= rc rc rc X( )= rc rc + rc rc cos = ()= + ω ω ω X( ω)= rc rc cos [ ( ) ] () + (d) x u Usig a u (), a > a + ω X( ω)= + = + ω + + ω Alra Soluio: X( ω)= X( )= ()= + ω + ω ω + ω ω ( + + ) [ ( ) ] ()= + x u cos u a Usig cos ω u a+ ω ω + a ω ( ) () + X( ω)= ω + ω

30 M. J. Robrs - /7/ (c) x() X( ) - - x() (d) - Phas o X( ) - - X( ) ()= () x Phas o X( ) Usig a a a ( a)> + ω, R, 6 + ω ω 6 () x() = + 56ω X(ω) ω Phas o X(ω) ω Skch h magiuds ad phass o hs ucios. Skch h ivrs CTT s o h ucios also. 3 5 (a) x()= ( ) u() X( ω)= 3 + ω 5 + ω 5-3

31 M. J. Robrs - /7/ x() 6 X(ω) - - Phas o X(ω) - - ω ω (b) x()= 6rc cos X= sic + sic + x() X( ) Phas o X( ) - + (c) x ()= 6. sic ( ) si X= ri ri x() X( ) Phas o X( ) (d) δ( + 5)+ δ( + 95) x()= [ cos( )+ cos( 9) ] X= + δ( 95)+ δ( 5) or δ( + 5)+ δ( + 95) x()= cos( ) cos( ) X= + δ( 95)+ δ( 5) 5-3

32 M. J. Robrs - /7/ x() X( ) - Phas o X( ) - - () or ()= x ()= x cos + cos( 9) cos + cos cos + cos( ) δ( + 5)+ δ( + ) X( )= + δ( + 95)+ δ( 95) + δ( )+ δ( 5) δ( + 5)+ δ( + ) X( )= + δ( + 95)+ δ( 95) + ( )+ ( δ δ 5) x() X( ) - Phas o X( ) Skch hs sigals vrsus im. Skch h magiuds ad phas o hir CTT s i ihr h or ω orm, whichvr is mor covi. (a) x rc comb rc comb ()= () X( )= sic comb ( ) X( )= sic comb( ) si k k k X= sic si k ( ) δ ( ) k = 5-3

33 M. J. Robrs - /7/ No-zro oly or odd valus o k. A hos odd valus, always valuas o +. Thror ( k) k si, k X( )= sic k δ ( ) k = k x() (b) x rc comb ()= + () X( )= sic comb δ X( ) - Phas o X( ) - - k k X= + sic k sic k ( δ )= δ δ( ) k= Sam as aswr i par (a). k= k x() X( ) Phas o X( ) - - ()= () (c) x u si X( ω)= X ω δω + = δω ( + )δω + ω [ ] δω

34 M. J. Robrs - /7/ Raioalizig h domiaor, X( ω)= X( ω)= ( ) + δω ( + ) δω + ( ) 6 [ δω ( + )δω ] x()= si cos + ( ) + ( ) ( ) [ δω ( + )+ δω ] ()= x si 3cos + 6 x(). X(ω) Phas o X(ω) - - ω ω [ ] ()= () (d) x rc comb X= sic comb k k X= sic ( k)= sic δ k δ ( ) k = k = k k k x sic k k ()= sic cos k = + k = k = 5-3

35 M. J. Robrs - /7/ x() X( ) - Phas o X( ) [ ] ()= () () () x rc ri comb X= sic sic comb All h comb impulss hav zro wigh xcp h o a zro. X= δ x()= x() X( ) - - Phas o X( ) - - ()= () () x sic. comb X( )= rc comb= δ( + )+ δ+ δ... x cos. ()= + [ ] 5-35

36 M. J. Robrs - /7/ x() X( ) Phas o X( ) - - ()= () (g) x sic 99. comb X( )= rc comb x 99. ()= = δ = x() X( ) - - Phas o X( ) - - ()= (h) x X= = x()= = x() X( ) - - Phas o X( ) - -. Skch h magiuds ad phass o hs ucios. Skch h ivrs CTT s o h ucios also. 5-36

37 M. J. Robrs - /7/ (a) X( )= sic δ( )+ δ + [ ] ()= ( + )= x rc rc cos X( ) - Phas o X( ) - - x() = (b) X sic comb x()= rc comb()= rc δ = X( ) x() - Phas o X( ) Skch hs sigals vrsus im. Skch h magiuds ad phass o h CTT s o hs sigals i ihr h or ω orm, whichvr is mor covi. I som cass h im skch may b covily do irs. I ohr cass i may b mor covi o do h im skch ar h CTT has b oud, by idig h ivrs CTT. ()= (a) x si ( ) ( ) X + = [ δ( + )δ( ) ]= 5-37

38 M. J. Robrs - /7/ x() X( ) Phas o X( ) - - (b) x cos comb cos δ ()= = X= [ δ( )+ δ( + ) ] comb + X( )= comb + comb 3 3 = comb = comb X( )= comb x(). -.. X( ). - Phas o X( ) - - [ ] ()= + (c) x cos cos X= + [ ( )+ ( + )] δ δ δ δ ( )+ δ + δ [ δ( )+ δ( + ) ] X( )= + δ( ) δ( )+ δ + + ( + ) ( )+ + δ δ δ [ ] [ ] [ ] 5-3

39 M. J. Robrs - /7/ δ( )+ δ( + ) X( )= + δ( )+ δ( + )+ δ( )+ δ + [ ] x() X( ) Phas o X( ) [ ] ()= + (d) x rc 5comb 5 cos 5 X= + sic comb δ δ ( 5 )+ δ [ ] k X= + sic k ( ) k δ δ 5 δ ( 5 )+ δ ( + 5 ) = [ ] [ δ( 5)+ δ( + 5) ] X( )= k + sic k k ( δ 5 5 )+ δ k = [ ]. x() X( ) -5 5 Phas o X( ) () x()= rc comb 7 () = = ( ) X 7sic 7 comb 7sic 7 δ k X= 7 sic 7 k k = ( ) k = 5-39

40 M. J. Robrs - /7/ x() X( ) 7 - Phas o X( ) Skch h magiuds ad phass o hs ucios. Skch h ivrs CTT s o h ucios also. (a) X( )= sic comb( ) ()= ()= ( ) x rc comb rc δ ()= ( ) x rc = = X( ) x() -6 6 Phas o X( )

41 M. J. Robrs - /7/ + (b) X( )= sic + sic comb ()= ( + ) ()= ( ) x rc comb rc cos δ ()= ( ) x rc cos = ( ( )) = X( ) x() - Phas o X( ) = (c) X sic sic x()= rc() rc rc rc = () Th rsul o h graphical covoluio ca b xprssd i h orm, x()= ri ri 3 3 X( ).5 x() - Phas o X( ) Skch hs sigals vrsus im ad h magiuds ad phass o hir CTT s. d (a) x()= [ sic() ]= d d d X= rc = si cos si 5-

42 M. J. Robrs - /7/ (a) x() X( ) Phas o X( ) - - d (b) x()= rc d 6 si( 6 X= ) sic( 6 )= = si ( 6 ) 6 Alra Soluio: d d rc = [ ( + ) ( ) δ δ ] [ δ( + 3)δ( 3) ] si x() X( ) Phas o X( ) (c) d x()= [ ri comb() ]= rc+ rc comb d () k X= sic comb= ksic k δ ( ) k = 5-

43 M. J. Robrs - /7/ x() X( ) Phas o X( ) - -. Skch hs sigals vrsus im ad h magiuds ad phass o hir CTT s. (a) x()= si( λ) dλ X( )= [ δ( + )δ( ) ]+ ( si ) δ = [ ] = δ( + ) δ X( )= = δ ( + )+ δ [ ] δ( + )δ x(). X( ) Phas o X( ) - -, < (b) x()= rc( λ) dλ = +, <, > sic( ) sic X( )= + rc [ ( ())] δ= = + δ x() X( ) Phas o X( ) - 5-3

44 M. J. Robrs - /7/ (c) x()= 3 sic( λ ) d λ L u = λ. Th x or : ()= x ()= x 3 or : ()= x ()= 3 u 3 sic du = si u du u si u si u 3 si u si u du du = du + du u u u u 3 si( λ) si( u) 3 d + du = + Si Si u = 3 ( ) λ λ 3 3 = Si = Si= 3 si u si u 3 si u si u du + du = du + du u u u u 3 x()= + Si 3 Thror, or ay, x()= + Si X( )= rc + [ 3sic ] δ= rc + δ( ) = x() X( ) Phas o X( ) rom h diiio, id h DTT o x[ ]= rc [ ]. 5-

45 M. J. Robrs - /7/ ad compar wih h ourir rasorm abl i Appdix E. rom h abl, X x rc = = = = [ ] = [ ] = m m X= ( ) = = m = m = si 9 X= = drcl, si = 9 9 ( + ) rc [ ] N N + w drcl, N w w rc [ ] 9drcl, 9 rc [ ] 9drcl, 9 Chck 3. rom h diiio, driv a gral xprssio or h ad orms o h DTT o ucios o h orm, x Asi Asi. [ ]= = ()= = (I should rmid you o h CTT o x Asi Asi ω.) Compar wih h ourir rasorm abl i Appdix E. X x Asi A = = = [ ] = ( ) = = Th, usig [ ( + ) ] = A X= x = comb( x) = 5-5

46 M. J. Robrs - /7/ w g A X= [ comb( )comb ( ) A comb comb ]= ( )+ X= A comb( + )comb( ) [ ] [ ] Th orm ca b oud by h rasormaio,. X A ( )= comb + comb [ ] X( )= A comb( + )comb 3. A DT sigal is did by x[ ]= sic. Skch h magiud ad phas o h DTT o x[ ]. rom h abl o rasorm pairs, sic wrc w comb w sic rc( ) comb sic [ comb ] rc sic k rc k = ( ) 5-6

47 M. J. Robrs - /7/ 3. A DT sigal is did by x[] -3 3 x[ ]= si. 6 X( ) - Phas o X( ) - - Skch h magiud ad phas o h DTT o x[ ] [ ] 3 ad x +. rom h abl, si( ) comb( + )comb [ ] Similarly, si comb + comb 6 3 si comb + comb 6 6 si 3 6 δ + k δ k 6 6 k = si 3 6 k k δ + k δ k k = = 3 = si 3 δ + k δ k + 6 k = 3 si comb + + comb 6 si + + k k δ + k δ k k = 5-7

48 M. J. Robrs - /7/ si + + ( k) k δ + k δ k + ( + ) 6 k = si + δ + k δ k 6 k = + si comb + comb 6 This is h sam as h rasorm o h ushid ucio bcaus a shi o is a shi ovr xacly o priod. x[] X( ) Phas o X( ) - - x[] X( ) Phas o X( ) Th DTT o a DT sigal is did by Skch x[ ]. X( )= rc rc comb + +. rom h abl, w sic wrc comb w sic rc comb 5-

49 M. J. Robrs - /7/ 5-9 sic comb rc + sic comb rc sic comb rc rc sic cos comb + + rc rc Thror x sic cos [ ]= -6 6 x[] - 3. Skch h magiud ad phas o h DTT o x rc cos [ ]= [ ] 6. Th skch x [ ]. rom h abl, rc drcl, N w w w N N [ ] + +

50 M. J. Robrs - /7/ ad rc N ( ) si Nw + [ ] w si cos( ) comb+ comb + [ ] Th ucio, si 9 si Th si 9 X= comb comb si ( ) is priodic wih priod, o. To prov ha, l k b ay igr. si 9 + k si 9 9k si 9 cos 9k si 9k cos 9 = si + k si + k si cos k si k cos = ( ) si 9 + k si 9 si 9 si + k si si = =, k odd ( ) si 9 + k si 9 si + k si =, k v X= Sic si 9 si Thror, or ay k, X= si 9 k k si + + δ δ k = 6 6 si9 k + si k δ k + δ + k k = 6 6 sik + si k 6 6 si9 k 6 si9 k + 6 is priodic wih priod, o, so ar ad. sik 6 sik + 6 si9 k 6 sik 6 3 si = si 6 = = ad si9 k + 6 sik si = = = si 6 5-5

51 M. J. Robrs - /7/ ad hror X= comb comb comb comb + = Th, usig ad, hror, cos( ) comb+ comb + [ ] + + cos comb comb x[ ]=cos 6 x[] X( ) Phas o X( ) Skch h ivrs DTT o = [ ] X rc comb comb. rom h abl, sic wrc( w) comb ad combn comb w [ ] N Thror usig muliplicaio-covoluio dualiy, x[ ]= sic comb [ ]. 5-5

52 M. J. Robrs - /7/.5 x[] Usig h dircig propry o h DTT ad h rasorm pair, ri + cos( ), id h DTT o ( δ[ + ]+ δ[ ]δ[ ]δ( ) ). Compar i wih ourir rasorm oud usig h abl i Appdix E. Th irs backward dirc o ri is ( δ[ + ]+ δ[ ]δ[ ]δ( ) ). Applyig h dircig propry, ri ri cos ( + ) ( δ[ + ]+ δ[ ]δ[ ]δ[ ] ) cos ( δ[ + ]+ δ[ ]δ[ ]δ[ ] ) + ( + ) + + ( δ[ + ]+ δ[ ]δ[ ]δ[ ] ) + + ( δ[ + ]+ δ[ ]δ[ ]δ[ ] ) + Ohr rou o h DTT: ( δ[ + ]+ δ[ ]δ[ ]δ[ ] ) + Chck. 37. Usig Parsval s horm, id h sigal rgy o 5-5

53 M. J. Robrs - /7/ x[ ]= sic si. E x X d x = [ ] = = rom h abl, ad sic wrc w comb w si( ) comb( + )comb [ ] Usig h muliplicaio-covoluio dualiy o h DTT, sic si rc( ) comb comb + comb Priodic covoluio is h sam as apriodic covoluio wih o priod. sic si 5rc( ) comb + δ δ sic si 5rc( ) comb + comb sic si 5 ( ) comb + ( ) comb rc rc Ex = + d 5rc comb rc comb Sic w ar igraig oly ovr a rag o o, oly o impuls i ach comb is sigiica. E = 5 ( x ) + d rc δ rc δ 5-53

54 M. J. Robrs - /7/ Ex = 5 rc + d rc Th squar o h sum quals h sum o h squars bcaus hr is o cross produc; h wo rcagls do o ovrlap. E = x + d d + 5 rc rc + + Ex = 5 d + d = + 5 = 5 3. Skch h magiud ad phas o h CTT o ad o h CTS o ()= () x rc x()= rc() comb. or compariso purposs, skch X ( ) vrsus ad T k o axs. (T is h priod o x () ad T =.) X sic = k X[ k]= sx( sk)= sic X [ ] vrsus k o h sam s 5-5

55 M. J. Robrs - /7/ X ( ) T X [ k] - - k Phas o X ( ) Phas o T X [ k] - - k Skch h magiud ad phas o h CTT o ()= x cos ad o h DTT o x[ ]= x( T s ) whr T s = 6. or compariso purposs skch X sam s o axs. X = δ( )+ δ + [ ] ad T T s X ( x[ ]= cos( Ts)= cos X= comb comb + + X ( )= k k k + + δ δ = X ( T )= s k k k δ δ = X ( T s )= 3 δ( 6k)+ δ + 6k k = [ ] TsX ( Ts)= δ( 6k)+ δ + 6k k = [ ] s ) vrsus o h 5-55

56 M. J. Robrs - /7/ X ( ) T s X (T s ) Phas o X ( ) Phas o T s X (T s ) Skch h magiud ad phas o h DTT o ad o h DTS o sic 6 x[ ]= sic 6 x [ ]= comb [ ]. 3 or compariso purposs skch X o axs. vrsus ad N X [ k] vrsus k o h sam s rom h abl, sic wrc w comb w sic 6 rc( 6) comb 5-56

57 M. J. Robrs - /7/ hror ( ) X = rc 6 comb = rc 6 q. q = Th udamal rqucy o h priodic sigal is h rciprocal o is priod, = =. N 3 Usig h rsuls o h aalysis o priodic xsios o apriodic DT sigals, X X [ k]= X( k) N k k k k [ ]= X = 6 q q rc = rc q= X q= - Phas o X - - N X [k] - k Phas o X [k] -3 3 k

58 M. J. Robrs - /7/. A sysm is xcid by a sigal, x()= rc ad is rspos is ( + ) y()= ( ) u( + ) ( ) u + y()= ( ) u ( ) ( ) u Wha is is impuls rspos? Y ω [ + ( )] ω δ ω ω ω ω + δ( ω) + ω = + + ω ω Y( ω)= ( si ω ω + ω )= ω + ω. ω H( ω)= si( ω) ω + ω = ω sic 5 si( ω) ω + ω 5 = ω si( ω) ω + ω ω 5 5 H( ω )= ω ω + ω ω = + 5 h u ()= (). Skch h magiuds ad phass o h CTT s o h ollowig ucios. ()= (a) g 5δ 5 5δ + 3 (b) g()= comb comb + 3 comb comb comb comb 6 5-5

59 M. J. Robrs - /7/ + 3 comb comb comb + 3 comb comb si comb= + 3 Also, g()= comb comb = (a) G( ) (b) G( ) - - Phas o G( ) Phas o G( ) ()= + ( ) (c) g u u u+ u δ δ (d) g sg sg u u + ( ) ( + )+ δ ()= () ( ) sg () sg = 5-59

60 M. J. Robrs - /7/ (c) G( ) (d) G( ) - - Phas o G( ) Phas o G( ) () g()= rc + rc + rc + rc sic sic + + rc + rc sic cos () g()= rc rc sic () G( ) () G( ) - - Phas o G( ) Phas o G( )

61 M. J. Robrs - /7/ (g) g()= 5 ri ri 5 5ri ri 5sic 5 sic 5 3 (h) g()= rc rc 3 rc rc sic sic (g) G( ) 5 (h) G( ) Phas o G( ) Phas o G( ) Skch h magiuds ad phass o h CTT s o h ollowig ucios. (a) rc rc sic ω rc sic 5-6

62 M. J. Robrs - /7/ sic rc() - sic ( ) - - (b) rc δ () rc δ() sic = sic ω rc δ() sic sic - sic rc() δ() - - δ (c) rc rc δ sic = sic ω rc δ sic ω 5-6

63 M. J. Robrs - /7/ sic - rc() δ(-) - sic (d) rc δ rc δ sic = sic ω rc δ = sic sic - sic rc() δ() - - () rc comb() rc comb() sic comb( )= sic k rc comb() sic δ( k) k = k = δ( k) k k rc comb() sic δ ω k = sic k δω k= k= 5-63

64 M. J. Robrs - /7/ sic comb( ) - rc() comb() sic comb( ) () rc comb rc comb sic comb = sic k δ = k = k rc comb sic k k ( ) ( k )= sic ( k) 3 δ δ k= k= k rc comb() sic δωk k = Sam as par (). sic comb( ) - rc() comb(-) sic comb( ) (g) rc comb rc comb sic comb = sic δ k = k k rc comb sic ( k)= sic k δ δ ( ) k= k= 5-6

65 M. J. Robrs - /7/ k k rc comb sic k sic k δ ω = δω k= k= sic comb( )... - rc() comb() sic - comb( ) (h) rc() comb rc() comb sic comb sic δ () = k = rc comb sic δ k sic k δ k δ k= rc() comb δ ω k= k = ( )= = δ( ω) δ( ) rc() comb() δ( ) - -. Plo hs sigals ovr wo priods crd a =. (a) x()= cos( )+ si( )+ 3cos( )3si ()= + (b) x 5cos 7si 5-65

66 M. J. Robrs - /7/ x() (a) - x() (b) - Compar h rsuls o pars (a) ad (b). 5-66

67 M. J. Robrs - /7/ 5. A priodic sigal has a priod o our scods. (b) Hz Wha is h x-lows posiiv rqucy a which is CTT could b Wha is h lows posiiv rqucy a which is CTT could b o- (a) zro? ozro? Hz 6. Skch h magiud ad phas o h CTT o ach o h ollowig sigals (ω orm): (a) x ()=. rc X= sic( ) ω X( ω)= sic sic( ω ) ω sic( ) x() + 5 (b) x()= 3rc. - ω X = + sic

68 M. J. Robrs - /7/ ω X( ω)= 3sic 5 ω + 5 3sic( 5ω ) 5ω 3 ω 3 x() 3sic( 5ω ) 5ω ω 7 (c) x()= comb 5 5 X= 7comb( 5) 5ω X( ω)= 7comb = 7 k= 5ω δ k = 5 k δω 5 k= X(ω) x() X(ω) ω ω (d) x()= comb = comb = = X 7comb 5 7comb ω ω ω ω k X( ω)= comb δ k δω = 5 = k = k = k 5 5-6

69 M. J. Robrs - /7/ 5ω ω ω ω k X( ω)= comb δ k δω + 3 = = k = k = 6k + 5 X(ω) x() X(ω) ω ω 7. Skch h ivrs CTT s o h ollowig ucios: (a) X( )= rc x()= 6sic( ) X( ) - X( ) 6sic() (b) X( )= rc X( ) x()= 6sic+ 6 - X( ) 6sic[( + )]

70 M. J. Robrs - /7/ [ ] (c) X 5 = δ( + 5)+ δ( 5) X= [ δ( + 5) + δ( 5) ]= δ( + 5)+ δ 5 x()= cos( ) X( ) [ ] 5 5 X( ) (d) X= δ+ 5δ( 5)+ 5δ x()= + cos( ) 5 5 X( ) cos() X( ) +cos() id h ivrs CTT o his ral, rqucy-domai ucio (igur E) ad skch i. (L A =, = 95kHz ad = 5kHz.) + X= Arc + rc + [ ] ()= + x A sic sic 5-7

71 M. J. Robrs - /7/ ()= [ + ]= ( ) + x A sic A sic cos ()= ( ) x, sic, cos 5 - X( ) Α - igur E A ral rqucy-domai ucio x x() x id h CTT (ihr orm) o his sigal (igur E9) ad skch is magiud ad phas vrsus rqucy o spara graphs. (L A= B= ad l = ad =.) Hi: Exprss his sigal as h sum o wo ucios ad us h liariy propry. x()= Arc ( AB) rc = ( ) X Asic A B sic = X sic sic x() A - - B igur E9 A CT ucio 5-7

72 M. J. Robrs - /7/ X( ) 3 - Phas o X( ) I may commuicaio sysms a dvic calld a mixr is usd. I is simpls orm a mixr is simply a aalog muliplir. Tha is, is rspos sigal, y(), is h produc o is wo xciaio sigals. I h wo xciaio sigals ar ()= ad x ()= cos x sic 5 plo h magiud o h CTT o y(), Y( ), ad compar i o h magiud o h CTT o x (). I simpl rms wha dos a mixr do? ()= () ()= y x x 5sic cos 5 Y= X X= rc δ( )+ δ + [ ] 5 Y( )= rc + rc + X ( ) Y( )

73 M. J. Robrs - /7/ 5. Skch a graph o h covoluio o h wo ucios i ach cas: rc() * rc() (a) rc() rc() rc(- )* rc(+ ) (b) rc rc+ () (c) ri ri ri() ri(-) - ri(-τ) ri(τ-) τ or < -, h o-zro porios o h wo ucios do o ovrlap ad h covoluio is zro. or > 3, h o-zro porios o h wo ucios do o ovrlap ad h covoluio is zro. or - < < : Th o-zro porios ovrlap or < τ < + ad, i ha rag o τ, 5-73

74 M. J. Robrs - /7/ ri( τ)= + τ ad ri( τ )= τ Thror, or - < <, or < < : + + [ ] ri() ri( )= ( + ) d = ( + τττ ) τ τ dτ 3 τ τ ri() ri= ( + ) = = ri(-τ) ri(τ-) -- + τ ri() ri= ri( τ) ri( τ) dτ Th o-zro porios ovrlap or < τ < + ad, i ha rag o τ, hr ar hr cass o cosidr, < τ <, < τ < ad < τ < +. Thror ri() ri= ri( τ) ri( τ) dτ + ri( τ) ri( τ) dτ + ri( τ) ri( τ) dτ Cas : < τ < ri( τ)= + τ ad ri( τ )= τ + Cas : < τ < Cas 3: < τ < + ri( τ)= + τ ad ri( τ )= τ ri( τ)= + τ ad ri( τ )= τ Thror ri() ri= ( + τττ ) d + ( + τττ ) d + ( + τ) ( τ) dτ + 5-7

75 M. J. Robrs - /7/ [ ] + [ + ] + [ + ] ri() ri= τ + τ dτ ( ) τ τ dτ τ ( + ) τ + τ dτ 3 3 τ τ τ τ ri() ri= + + ( + ) + ( + ) τ τ τ τ 3 + ri ri 3 3 () = ( ) + + ( + ) ( + ) ( + ) ( + ) + ( + ) ( + ) ( + ) + ( + ) ri ri () ( )=( ) ri () ri= ( + ) 6 or h rmaiig rgios o, h covoluio simply rpas wih v symmry abou h poi, =. Th aalyical soluios ca b oud by h ollowig succssiv chags o variabl: +,, Ths hr succssiv chags o variabl ca b codsd io o, + Th, or < <, 3 3 ( ) ( ) ri() ri= = ad, or < < 3, ri () + ri= 6 3 = ( + )+ 6 + [ ]

76 M. J. Robrs - /7/ ri() * ri( -) δ() cos() () () (d) 3δ cos 3 () comb() rc() Th cosa,. () 5comb() ri() Th cosa,. 5. I lcroics, o o h irs circuis sudid is h rciir. Thr ar wo orms, h hal-wav rciir ad h ull-wav rciir. Th hal-wav rciir cus o hal o a xciaio siusoid ad lavs h ohr hal iac. Th ull-wav rciir rvrss h polariy o hal o h xciaio siusoid ad lavs h ohr hal iac. L h xciaio siusoid b a ypical houshold volag, Vrms a 6 Hz, ad l boh yps o rciirs alr h gaiv hal o h siusoid whil lavig h posiiv hal uchagd. id ad plo h magiuds o h CTT s o h rsposs o boh yps o rciirs (ihr orm). Hal-Wav Cas: [ ] ()= x cos rc 6comb 6 X= 6 [ δ( 6)+ δ( + 6) ] sic comb 6 X= [ δ( 6)+ δ( + 6) ] sic comb 6 X= k [ ( )+ ( + )] sic k δ 6 δ 6 δ k =

77 M. J. Robrs - /7/ X( )= 3 X( )= 3 k = k = k sic k [ δ ( 6 )+ δ ( + 6 ) ] δ 6 k sic k k δ ( 6 6 )+ δ [ ] ull-wav Cas: [ ] ()= x cos rc 6comb 6 X= 6 [ δ( 6)+ δ( + 6) ] sic comb δ 6 6 X= [ δ( 6)+ δ( + 6) ] sic comb δ 6 k X= 6 δ( 6) δ( + 6)+ sic k k δ ( 6 6 )+ δ k = [ ] X( ) X( ) id h DTT o ach o hs sigals: (a) x[ ]= u [ 3 ] X( )= x[ ] = u [ ] = 3 3 = X( )= m = = m + m ( + ) = 3 3 m = = m

78 M. J. Robrs - /7/ m X( )= = = 3 m = Alra Soluio: x[ ]= u u [ ]= 3 [ ] δ 3 [ ] Usig α u[ ] ad δ[ ] α x u δ 3 3 [ ]= [ ] [ ]= 3 3 x[ ]= 3 = 3 3 = 3 Scod Alra Soluio: x[ ]= u 3 3 X( )= 3 3 [ ] = 3 (b) x[ ]= si u [ ] si = si = cos x[ ]= cos u α cos α cos( ) u[ ] αcos + [ ] α, α < 5-7

79 M. J. Robrs - /7/ 5-79 X cos cos = + = Alra Soluio: x u u [ ]= [ ]= [ ] x u [ ]= [ ] [ ] u [ ] u X = X =

80 M. J. Robrs - /7/ 5- X = X cos = X cos = X si cos = + = (c) x sic sic [ ]= Usig sic comb w w w rc X comb comb = rc rc X comb = rc x sic [ ]=

81 M. J. Robrs - /7/ (d) x[ ]= sic Usig sic wrc w comb w X= comb comb rc rc X= comb comb rc rc X= comb rc rc X= ri comb ri comb = 5. Skch h magiuds ad phass o h DTT s o h ollowig ucios: (a) rc [ ] ( + ) Usig rc [ ] N N + w drcl, N w w, rc [ ] 5drcl, 5 X( ) 5 - Phas o X( )

82 M. J. Robrs - /7/ [ ] ( [ ]) (b) rc 5δ Usig δ[ ] ad x[ ] y[ ] X Y rc [ ] ( 5δ[ ] ) 5drcl (, 5) 5 5drcl, 5 = X( ) 5 - Phas o X( ) - - δ[ ] (c) rc [ ] Usig x[ ] X 6 rc [ ] 3δ [ + 3] 5drcl (, 5) X( ) 5 - Phas o X( ) - (d) rc [ ] ( 5δ[ ] )= rc [ ] ( 5δ[ ] ) - 5-

83 M. J. Robrs - /7/ rc [ ] ( 5δ[ ] ) 5drcl, 5 X( ) 5 - Phas o X( ) - - () rc [ ] comb [ ] Usig comb N [ ] comb N, [ ] [ ] rc comb 5drcl, 5 comb [ ] [ ] rc comb 5drcl (, 5) δ m m = 5 rc comb drcl, 5 m [ ] [ ] δ m = 5 m rc [ ] comb[ ] drcl, 5 m δ m = 5-3

84 M. J. Robrs - /7/ X( ).65 - Phas o X( ) - - () rc [ ] comb 3 [ ] Usig h rsul o () ad x[ ] X, 5 6 m rc comb 3 drcl, 5 m [ ] [ ] δ m = X( ).65 - Phas o X( ) - - [ ] [ ]= [ ] [ ]= [ ] ( ) (g) rc comb rc δ m rc δ m m= m= rc [ ] comb[ ]= rc [ ] δ m rc comb m = [ ]= [ ] [ ] [ ] Thror 5-

85 M. J. Robrs - /7/ [ ] [ ] rc comb 5 drcl, 5 comb [ ] [ ] rc comb 5 drcl (, 5 ) δ m m = 5 rc comb drcl, 5 m [ ] [ ] δ m = 5 m rc [ ] comb[ ] drcl, 5 m δ m = X( ).5 - Phas o X( ) - - (h) rc [ ] comb [ ] 5 [ ] [ ] rc comb5 5drcl, 5 comb 5 [ ] [ ] rc comb5 5drcl (, 5) δ 5 m m = 5 rc comb5 drcl, 5 m [ ] [ ] δ 5 5 m = m rc [ ] comb5[ ] drcl, 5 5 m δ 5 m = Th, usig drcl, m comb m m = [ ] 5-5

86 M. J. Robrs - /7/ rc [ ] comb5[ ] comb5[ m] m δ 5 m = [ ] [ ] rc comb comb 5 [ ] [ ]=. ad, sic comb, ha implis ha rc comb5 X( ) - Phas o X( ) Skch h ivrs DTT s o hs ucios. (a) X comb comb = Usig comb( ) ad x[ ] X combcomb + combcomb ( + ) si combcomb si combcomb 5-6

87 M. J. Robrs - /7/ cos ( + ) + si ( + ) si combcomb cos ( + ) si + si ( + ) si combcomb 3 3 = si si combcomb x[] - (b) X= comb + comb Usig comb( ) ad x[ ] X comb + comb si comb + comb si comb + comb x[]

88 M. J. Robrs - /7/ (c) X= sic sic comb + + Usig T sic comb w cos drcl, T ( )= + w w w w rom Appdix A (bcaus T w is a igr), X= cos drcl, cos drcl, cos d cos d + cos ( ) [ ] d + d cos [ ] cos( ( + 5) )+ si( ( + 5) ) d cos + [ cos( ( 5) )+ si( ( 5) )] d Ths igrals ar zro ulss =±5. Thror ( δ[ + 5]+ δ[ 5] ) cos ( ). ( + ) Th usig rc [ ] N N + w drcl, N w w, Combiig ivrs rasorms, rc [ ] 9drcl, 9 ( δ[ + 5]+ δ[ 5] )+ rc [ ] cos( )+ 9drcl (, 9). 5-

89 M. J. Robrs - /7/ Th, usig x[ ] X ( δ 5 δ 5 9 [ + ]+ [ ])+ [ ] + 9 rc cos drcl, ad ( [ + ]+ [ ])+ [ ] δ 5 δ 5 rc cos 9drcl, 9. Th, ially δ 5 δ 5 9 ( [ + ]+ [ ])+ rc [ ] cos + drcl, 9 + ( δ[ + 5]+ δ[ 5] )+ rc [ ] + cos + 9drcl, Th impulss o h l sid cacl ad w g cos 9drcl, rc 9 [ ] + cos 5 + cos + 9drcl, x[] - -. (d) X= + + comb δ δ 3 δ X= + + comb comb + δ δ δ

90 M. J. Robrs - /7/ comb δ δ δ 6 6 X= comb δ δ δ comb + comb + comb 6 6 X= + comb comb comb Th, usig comb( ) ad x[ ] X w g 3 5 comb + comb + comb comb + comb + comb 6 6 or comb + comb + comb comb + comb + comb or 3 5 comb comb comb cos + cos + cos comb + comb + comb 6 6 x[] Usig h rlaioship bw h CTT o a sigal ad h CTS o a priodic xsio o ha sigal, id h CTS o 5-9

91 M. J. Robrs - /7/ x()= rc comb w T T ad compar i wih h abl ry. X= wsic( w) w w X k X k wsic wk sic T [ ]= = = T k 57. Usig h rlaioship bw h DTT o a sigal ad h DTS o a priodic xsio o ha sigal, id h DTS o ad compar i wih h abl ry. rc N [ ] [ ] comb w N rom h x, Thror, ( + ) X= N + drcl, N X p w [ k]= X( kp). N N w + k X p[ k]= drcl, N w +. N N p w 5-9